Sanjay Dominik Jena
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A Mixed Integer Programming approach for sugar cane cultivation and harvest planning
MSc Thesis
Thesis presented to the Graduate Program in Informatics of the Department of Informatics, PUC–Rio, as partial fulfillment of the requirements for the degree of Master in Informatics Adviser: Prof. Marcus V. S. Poggi de Arag˜ao
Rio de Janeiro March 2009
Sanjay Dominik Jena
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A Mixed Integer Programming approach for sugar cane cultivation and harvest planning
Thesis presented to the Graduate Program in Informatics of the Department of Informatics, PUC–Rio, as partial fulfillment of the requirements for the degree of Master in Informatics. Approved by the following commission:
Prof. Marcus V. S. Poggi de Arag˜ ao Adviser Department of Informatics — PUC–Rio
Prof. Eduardo Uchoa Barboza Departamento de Produ¸c˜ao — UFF
Prof. Alexandre Street de Aguiar Departmento de Engenharia El´etrica — PUC–Rio
Prof. Jos´ e Eugˆ enio Leal Coordinator of the Centro T´ecnico Cient´ıfico — PUC–Rio
Rio de Janeiro — March 18, 2009
All rights reserved.
Sanjay Dominik Jena
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Sanjay Dominik Jena joined an apprenticeship in Mathematics and Information Technology at AXA Germany. Afterwards he graduated from the Fachhochschule K¨oln (Cologne, Germany) in General Computer Science, while working at AXA Germany as a software developer for Intranet applications. He then obtained a Master degree at the PUC–Rio in computer science focused on combinatorial optimization and actively participated on the department’s work for Gapso.
Bibliographic data
Jena, Sanjay Dominik A Mixed Integer Programming approach for sugar cane cultivation and harvest planning / Sanjay Dominik Jena; adviser: Marcus V. S. Poggi de Arag˜ao. — Rio de Janeiro : PUC–Rio, Department of Informatics, 2009. v., 153 f: il. ; 29,7 cm 1. MSc Thesis - Pontif´ıcia Universidade Cat´olica do Rio de Janeiro, Department of Informatics. Bibliography included. 1. Informatics – Thesis. 2. Operations Research. 3. Combinatorial Optimization. 4. Mixed Integer Programming. 5. Scheduling. 6. Network Flows. 7. Sugar cane harvesting. 8. Harvest Planning. I. Arag˜ao, Marcus V. S. Poggi de. II. Pontif´ıcia Universidade Cat´olica do Rio de Janeiro. Department of Informatics. III. Title.
CDD: 510
Acknowledgements
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First, I wish to acknowledge my supervisor, Prof. Dr. Marcus V. S. Poggi de Arag˜ao, who introduced me to the field of optimization, and to express my deep gratitude for his dedication and continuous advice, guidance and encouragement. His research and motivation inspired and influenced me during my master studies. I also gratefully acknowledge the whole of the Gapso company, in particular Pedro Cunha and Haroldo Gambini Santos, for their extensive support and numerous discussions that were crucial to the success of my research. My appreciation also goes to Ricardo Hermes from the Grupo Virgolino de Oliveira and Eduardo Sans from the GaTech for their continuous support during the specification and development of the model. Secondly, I would like to thank the DAAD (Deutscher Akademischer Austauschdienst) for their financial support. Their strong aspiration to strengthen the academic interchange between Brazil and Germany has always been a great motivation to me. Furthermore, I appreciate the numerous opportunities to contribute to this ongoing cooperation. My deepest gratitude goes to my family, that has fully accepted and supported my choice to study abroad. I am aware of all moments that I was not able to share with them. I would like to thank Mrs. Beatriz Barbieri for proofreading the draft of this thesis, and notably my friends and my girlfriend Angelita for always finding a special way to encourage and motivate me during troubled times and for always having faith in me. Finally, I would like to thank the Brazilian people for embracing me during my entire stay and integrating me to their culture. Every minute spent in their country has turned into an unforgetable experience to me.
Abstract
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Jena, Sanjay Dominik; Arag˜ao, Marcus V. S. Poggi de. A Mixed Integer Programming approach for sugar cane cultivation and harvest planning. Rio de Janeiro, 2009. 153p. MSc Thesis — Department of Informatics, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro. Mathematical Programming techniques such as Mixed Integer Programming (MIP), one of today’s principal tools in Combinatorial Optimization, allows fairly close representation of industrial problems. In fact, contemporary MIP solvers are able to solve large and difficult problems from practice. Whereas the use of MIP in industrial sectors such as logistics, packing or chip design is widely common, its practice in agriculture is still relatively young. However, planning of agricultural cultivation and harvesting is a complex task. Sugar cane is one of the most important agricultural commodities of Brazil, the worldwide largest producer of this crop that is used to produce sugar and alcohol. Most of the planning methods in use, manual or computer aided, still result in high waste of resources, on field and in commercialization. The purpose of this work is to provide decision support, based on Optimization techniques, for sugar cane cultivation and harvesting. A decision support system is implemented. It divides the planning into a tactical and an operational planning. The problem is proved to be NP-hard and determines the best moment to harvest the fields, maximizing the total profit given by the sugar content within the cane. It considers resources such as cutting and transport crews, processing capacities in sugar cane mills, the use of maturation products and the application of vinasse on harvested fields. The MIP model extends the classical Packing formulation, incorporating a network flow for the harvest scheduling. Several pre-processing techniques are used to reduce the problem size. Heuristically obtained initial solutions are passed to the solver in order to facilitate the solution. A problem segregation strategy based on semantic information is presented leading to very competitive results. As the linear relaxation optimum solution turned out to be highly fractional, this work also invests in valid inequalities in order to strengthen the MIP formulation. Several further solution approaches such as Local Branching and heuristics based on the optimum solution of the linear relaxation were explored. One of Brazil’s large sugar cane producers was involved in the entire development process in order to guarantee a realistic presentation of the pro-
cesses. All experiments were performed with instances from practice provided by this producer.
Keywords
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Operations Research. Combinatorial Optimization. Mixed Integer Programming. Scheduling. Network Flows. Sugar cane harvesting. Harvest Planning.
Resumo
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Jena, Sanjay Dominik; Arag˜ao, Marcus V. S. Poggi de. Uma abordagem de Programa¸ c˜ ao Mista Inteira para o planejamento de cultivo e colheita de cana-de-a¸ cu ´ car. Rio de Janeiro, 2009. 153p. MSc Thesis — Department of Informatics, Pontif´ıcia Universidade Cat´olica do Rio de Janeiro. T´ecnicas de Programa¸c˜ao Matem´atica como Programa¸c˜ao Mista Inteira (PMI), atualmente uma das principais ferramentas da Otimiza¸c˜ao Combinat´oria, permitem obter representa¸c˜oes de problemas industriais bem pr´oximas `a realidade. De fato, resolvedores PMI contemporˆaneos resolvem instˆancias reais grandes e dif´ıceis. Enquanto o uso de PMI em setores industriais como log´ıstica, empacotamento e projeto de circuitos ´e bastante comum, a sua pr´atica na agricultura ainda ´e relativamente nova. Entretanto, o planejamento de cultivo e colheita agr´ıcola ´e uma tarefa complexa. Cana-de-a¸cu ´car ´e um dos mais importantes recursos agr´ıcolos do Brasil que ´e mundialmente o maior produtor de cana usada para produ¸c˜ao de a¸cu ´car e ´alcool. A maioria dos m´etodos usados para planejamento, tanto manuais quanto assistidos por computador, ainda resultam em grande desperd´ıcio de recursos em campo e em comercializa¸c˜ao. O objetivo deste trabalho ´e promover, baseado em t´ecnicas de otimiza¸c˜ao, suporte `a decis˜ao para cultivo e colheita de cana-de-a¸cu ´car. Um sistema de suporte `a decis˜ao ´e implementado, dividindo o planejamento em um planejamento t´atico e operacional. O problema ´e provado ser NP-dif´ıcil e incorpora a determina¸c˜ao do melhor momento para a colheita dos talh˜oes, maximizando o lucro total dado pelo conte´ udo de a¸cu ´car na cana. O planejamento considera recursos como frentes de corte e transporte, processamento em usinas, uso de maturadores e aplica¸c˜ao de vinha¸ca em talh˜oes. O modelo PMI se basea na formula¸c˜ao cl´assica do problema da mochila, incluindo fluxos em rede para o escalonamento de colheita. V´arias t´ecnicas de pr´e-processamento s˜ao usadas para reduzir o tamanho do problema e solu¸c˜oes iniciais obtidas por heur´ısticas s˜ao passadas ao resolvedor para facilitar a resolu¸c˜ao. Uma estrat´egia de segrega¸c˜ao do problema baseada em informa¸c˜oes semˆanticas ´e apresentada, resultando em um desempenho muito competitivo. Como a solu¸c˜ao ´otima da relaxa¸c˜ao linear ´e fortemente fracion´aria, este trabalho tamb´em investe em desigualdades v´alidas para fortalecer a formula¸c˜ao PMI. Outras abordagens de resolu¸c˜ao como Local Branching e heur´ısticas baseadas na solu¸c˜ao ´otima da relaxa¸c˜ao linear foram exploradas.
Um dos grandes produtores brasileiros de cana-de-a¸cu ´car foi inclu´ıdo no processo completo de desenvolvimento de forma a garantir uma representa¸cao real dos processos. Todos os experimentos foram efetuados com instˆancias reais fornecidas por este produtor.
Palavras–chave
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Pesquisa Operacional. Otimiza¸c˜ao Combinat´oria. Mixed Integer Programming. Scheduling. Fluxo em rede. Colheita de cana-de-a¸cu ´car. Planejamento de colheita.
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Contents
1 Introduction 1.1 Context and Motivation 1.1.1 Sugar cane harvest process 1.1.2 Optimization techniques to solve harvest problems 1.2 Objectives of this thesis 1.3 Previous work review 1.3.1 Decision support tools in agriculture 1.3.2 Decision support tools in sugar cane industry 1.3.3 Decision support tools for other harvest types 1.4 Outline
13 13 13 15 16 17 17 18 19 20
2 Problem Description 2.1 The sugar cane harvest problem 2.1.1 Current practice 2.2 A DSS for sugarcane cultivation and harvesting 2.2.1 Tactical module 2.2.2 Operational Module 2.3 Related problems 2.4 Complexity
21 21 23 24 24 27 29 31
3 Optimization Methods 3.1 Definitions 3.2 Exact methods 3.2.1 Exact methods for linear programming 3.2.2 Exact methods for mixed integer programming 3.3 Heuristics and Metaheuristics 3.4 MIP Solver 3.4.1 ILOG CPLEX
37 37 40 40 41 42 43 44
4 Problem Formulation 4.1 Preliminary discussions 4.1.1 Modeling issues 4.1.2 Simplifications 4.1.3 Modeling the SCHP as a GAP extension 4.2 Formulation for the tactical planning 4.2.1 Mathematical model 4.2.2 Enabling repeated field harvesting 4.3 Formulation for the operational module 4.3.1 Modeling alternatives 4.3.2 Mathematical model
45 45 45 46 47 49 50 61 63 63 64
5 Solution Strategies 5.1 Instances for the computational experiments 5.1.1 Indicators for the level of difficulty
79 81 82
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5.1.2 5.1.3 5.2 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.5 5.5.1
Instances for the SCHP-TP Instances for the SCHP-OP Experiments on the SCHP-TP Preprocessing for the SCHP-OP Distance Filtering Variable pruning and Reduction tests Field grouping Exact solution approaches for the SCHP-OP Analysis of the optimization process Initial solutions through constructive heuristics Linear relaxation analysis Valid Inequalities Alternative solution strategies for the SCHP-OP Segregation and Aggregation of the cutting crews’ planning
83 86 88 91 92 98 100 101 101 106 115 116 125 127
6 Conclusions 6.1 Future work
135 137
Bibliography
139
A
Glossary
146
B
Instance properties for the SCHP-OP and solutions for the segregation strategy 148
List of Figures
1.1
Example routes for cutting transportation crews
14
2.1 2.2
Data flow from the tactical module into the operational module Modeling the harvest problem as a knapsack problem
25 30
4.1
Network flow for the cane quantity along the weeks, enabling the model to harvest a field more than once along the planning period Network flow starting at mill pf Starting node of the network flow at the first instant at field f Initialc Flow conservation at instant node i at field f of cutting crew c Example network flow for a mechanical cutting crew: nodes of not available instants are skipped Example network flow for a manual cutting crew: the crew goes home at the end of the day and moves to any field before the next day
4.2 4.3 4.4 4.5
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4.6
5.1 5.2 5.3 5.4 5.5 5.6
5.7 5.8
5.9 5.10
62 74 75 76 77
78
Pol distribution per week for an example instance 89 Quantity of harvested sugar cane per week in the optimal solution for an example instance 89 Processed sugar cane at the mills suggested by an optimal solution 91 for instance GVO10 3 Example of a reduction of non-terminal nodes with degree 2 100 Routes of cutting crews for solutions in the initial stage of the optimization process 104 Routes of cutting crews for solutions in an advanced stage of the optimization process. The lower figure illustrates the routes for a close to optimum solution 105 Objective function value throughout branching and polishing process106 Example route for a cutting crew: first, the crew greedily chooses the fields and waits until the end of the planning (a); in a replanning step (b), the cuts are balanced along the planning period by insertion of the waiting variables after each cut 112 Route of a mechanical cutting crew in the optimal solution for the linear relaxation of instance GVO102 2 117 Fractional example route for a cutting crew after insertion of valid inequalities in order to force traveling. The first three travels between field 3 and 4 are performed to compensate the cutting flow used at field 1. 119
List of Tables
5.1 5.2 5.3
5.14 5.15
Instances for the SCHP-TP 85 GVO instance sets 100 and 102 for the SCHP-OP 90 Outgoing edges per field for different distance filtering approaches 94 (example of 25 fields of Cutting Crew 204 at instance GVO102 2. Impact comparison of distance filtering approaches: No filtering, distance limitation to 5km, 10km and 50km and node balancing with up to 50k, 100k and 250k travel variables 96 Impact comparison of distance filtering approaches: MST only, MST with node balancing with up to 50k, 100k and 250k travel variables 97 Influence of minimum processing demands to the difficulty of an instance 102 Influence of the cutting crews’ occupation rate to the difficulty of an instance 102 Solution properties during the optimization process 103 Quality of the solutions obtained by the heuristics 113 Influence of initial solutions to the quality of the final solutions 115 Influence in the polishing phase of starting solutions passed to CPLEX116 Influence of the cuts in the upper bounds and the optimization 124 Influence of increasing and decreasing cutting crew order in segregation strategy to the cut cane quantity 130 Comparison of solution quality of different solving strategies 131 Results of the LP-and-Fix heuristic 134
B.1 B.2 B.3 B.4 B.5
GVO instance sets 100 and 102 for the SCHP-OP 149 GVO instance sets 103 and 106 for the SCHP-OP 150 Sets of artificial instances 151 Results for all artificial instances for the segregation solution strategy152 Results for all GVO instances for the segregation solution strategy 153
5.4
5.5
5.6
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5.7 5.8 5.9 5.10 5.11 5.12 5.13
1 Introduction
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1.1 Context and Motivation Sugar cane is a sub-tropical and tropical genus of tall growing crop, counting 37 species plus a number of hybrid species. According to the Food and Agriculture Organization of the United Nations (FAO) [Sta08], sugar cane is one of the most important commodities in the world. With more than 420 billion tons of harvested sugar cane in the year 2005, Brazil is by far the largest producer of this grass worldwide, followed by India, China and Thailand. Among all agricultural commodities produced in Brazil, sugar cane is its most produced measured in biomass and its fourth most lucrative. Internationally, sugar cane is a highly competetive market. Recent international studies [HTAJ07, GLMS08, BFGN02] showed great opportunities to improve the value chain and reduce costs in the operational planning in order to remain competetive. 1.1.1 Sugar cane harvest process The sugar cane harvest typically begins in May, sometimes April and prolongates to November, the time of the year when the sugar cane plants normally reach their maturation peaks. The maturation of sugar cane is measured in percentage of sucrose in the sugar cane, denoted to Pol and reduced sugar, denoted to AR. The maturation periods vary widely around the world from six to 24 months. Manual and mechanical cutting crews cut the plants on the fields, chopping down the stems but leaving the roots to re-grow in time for the following harvest. The harvest is then immediately transported to the industrial sector, i.e. sugar cane mills, by trucks, rail wagons or by manual carriage (cart pulled by a bullock or a donkey). Figure 1.1 illustrates example routes for cutting and tranporting. The cutting crews travel from one field to another harvesting the cane. The transportation crews commute between the fields and the mills.
14
Chapter 1. Introduction
Sugar cane mill Plantation field Cutting crew route Transportation crew route
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Figure 1.1: Example routes for cutting transportation crews In the mills, the sugar cane is crushed and the cane juice is extracted. The bagasse leftover, also referred to fiber, is burned in boilers. The induced steam drives the turbines that generate the power for the mills. The sugar cane is further processed either to sugar or to ethanol. The sugar is also referred to as the total recoverable sugar (ATR1 ). For the sugar production, the sugar cane juice undergoes further processes such as heating, filtering and evaporation. The result is a syrup which is centrifuged to separate the sugar crystals from the molasses. To produce ethanol, the juice also undergoes processes as heating, filtering and evaporation. Afterwards, the juice is fermented in large vats, centrifuged and distillated to separate the ethanol. A side effect of the alcohol distillation process is a residual industrial liquid called vinasse. Vinasse is a corrosive contaminant that contains high levels of organic matter, potassium, calcium and moderate amounts of nitrogen and phosphorus [GR00]. However, vinasse is an efficient fertilizer, thus its application to harvested plantation fields has become common practice. The use of maturation products is a common approach in agriculture to influence the natural maturation curve of plants. In the context of sugar cane, often used products are growth regulators which decrease the growth of the cane and therefore lead to an increase of the relative quantity of sugar within the plant. Growth regulators are commonly used to prepone a field’s yield in order to provide raw material, i.e. sugar cane, for the mills. Thus, the use of such maturation products is directly related to the moment of sugar cane processing in the mills. The cultivated areas can contain hundreds of lots with different varieties, each with distinct growth and maturation properties. One of the most difficult, but most important decisions is the determination of the ideal moment to cut each field and apply growth regulators in order to benefit best from the 1
The abbreviation ATR is originated in the portuguese term A¸cu ´car Total Recuper´ avel, used in Brazil.
Chapter 1. Introduction
15
maturation peaks. The cutting crew’s capacities and logistic factors are directly involved in such decisions, as they may constraint the harvest and therefore must be taken into account. These factors turn the planning of sugar cane cultivation and harvest into a very complex and difficult task. Most of the planning methods in use, manual or computer aided, still result in high waste of resources on field, during transportation and in further commercialization.
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1.1.2 Optimization techniques to solve harvest problems Advances in Information Technology shifted the focus from manual to computer aided planning in many industrial areas. In the past, manual planning was based on the experience of specialists [FT07]. Today, the agriculture industry benefit from computer based planning in order to reduce costs and risks and increase their total gain. Operations Research (OR)2 is dedicated to the search for optimal or nearoptimal solutions to complex problems, as they arise in practice in industry. It is an interdisciplinary branch of formal sciences and applied mathematics and benefits from Simulation, Optimization, Probability and Statistics. Operations Research helps management achieve its goals using scientific methods. One of the primary tools of OR is Combinatorial Optimization (CO), which deals with optimization problems where the set of feasible solution is discrete or can be reduced to a discrete one. In addition to OR, Combinatorial Optimization is related to other fields such as algorithm theory and computational complexity. In the last decades, many algorithms to classic problems of CO appeared [AHLS97]. Such algorithms range from exact methods to approximation algorithms. Most of such methods were successfully applied to solve problems from practice. Several classical CO problems, even instances that are accepted as very complicated, can be solved today. Exact methods solve them to optimality in reasonable time and heuristics and metaheuristics find solutions close to optimality in extremely short processing time. In contrast to classical CO problems, which are usually well-studied, problems that arise in practice have not been yet thoroughly explored. Such problems can usually be seen as extensions or combinations of classical problems. These differences make it difficult to solve problems arising in practice by using methods that were originally developed for the original classical problem. 2 Operations Research is the official term in North America, South Africa and Australia. In Europe, this research area is known as Operational Research.
Chapter 1. Introduction
16
Among many approaches such as heuristics and metaheuristics to handle complex practice problems, mathematical programming has proved to be a powerful tool to solve such problems. Breakthrough advances in information technologies have provided new possibilities to the use of Combinatorial Optimization (CO). Such great developments in computers’ capabilities make it possible to model large problems from practice as mixed integer programs and solve them effectively using mathematical solvers. Solvers such as CPLEX and XPRESS-MP underlie a continuous development process and had their efficiency improved by the years. For this reason they are now capable of solving even larger problems. These great advances and the ability of mathematical programming to handle complex and individual problems suggest to apply such techniques to the complex task of planning sugar cane cultivation and harvest.
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1.2 Objectives of this thesis This work focuses on support for crucial decisions that must constantly be made in sugar cane cultivation and harvest. It aims to give suggestions to such decisions in order to support the planning in sugar cane industry. A decision support system (DSS) will be implemented, based on optimization techniques and representing the principal problems as mathematical models. The system must support the planning for a total horizon of up to one harvest season, i.e. approximately twelve months. The objective of the planning is the maximization of the total benefit returned by the cultivation and harvest process, respecting a set of industrial constraints. Throughout the whole planning, the system should maximize the total profit generated by the sale of factory produced sugar. The DSS should determine the exact moments to harvest the plantation fields, based on the maturation level of the sugar cane and the periods in which to apply maturation products. The system must determine cutting crews to cut the fields and transportation crews to carry the cut sugar cane to the factories. Cutting and transportation crews may be limited in capacity and time available. The planning system shall be separated into two modules: a tactical module for the long-term planning of a whole harvest season and an operational module for the detailed planning for up to four weeks. The tactical module divides the whole planning horizon into weeks. It should expose the best periods to harvest each plantation field and assign cutting and transportation crews to the fields in order to guarantee capacities to cut and transport the
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Chapter 1. Introduction
17
sugar cane of the chosen fields. An additional function of this module enables it to select maturation products and to determine the best moment to apply them to the fields. The operational module is responsible for the detailed planning of up to four weeks, dividing the whole planning horizons into days. The input data of the operational module consists of the plantation fields selected by the tactical module and any other fields added to the data. It must suggest cutting routes for the cutting crews on a daily basis. Both modules need to guarantee a certain number of hectares to be cut at each day/weak in order to allow the use of vinasse to the fields. The sugar cane mills may demand a minimum and maximum quantity of sugar cane to be processed at each day. The objective of this thesis is the study of adequate optimization techniques to solve the problem mentioned above. The two suggested modules should be implemented, having a strong focus on the operational module, and the resulting system should be able to solve problems of dimensions as they appear in practice. This work focuses on the optimization within sugar cane harvesting and does not aim at simulation aspects (for example for the evaluation of different scenarios). It also does not include attempts of robust optimization. Thus, it does not address sensitivity analysis and uncertainty. 1.3 Previous work review 1.3.1 Decision support tools in agriculture Methods of Operations Research have been applied to the algricultural sector since more than five decades. Heady’s work from 1954 [Hea54] is frequently cited as one of the first applications of linear programming to agricultural planning. His model assigned farm land to various crops subject to operational constraints using a profit maximizing objective function. In 1978, Audsley et al. [ADB78] compared different cultivation techniques for four types of crop: winter wheat, spring barley, sugar beets and potatoes. The linear programming model includes the allocation of resources such as land, labor and machinery. Although some variables in reality should be integers, the authors solve the problem as a linear problem and round the solution values. In the same year, McCarl presented a successful application of linear programming to grain crop production planning [MCDR78]. Such crops include corn, soybeans, wheat and silage. The resulting program allowed for the recommendation of production of one of the crop types. Its input data was provided by a 500
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Chapter 1. Introduction
18
questions document answered by the farmers. The program has been used by the farmers for over 35 years in annual workshops. OR was institutionally coupled to agricultural engineering already decades ago [Aud85] and has become common practice in the agricultural sector. Today, there is a considerable amount of research by the OR society, applied to almost all types of fruits, vegetables, crops, corns and so on. Masini [Mas03] applied mixed integer programming to the supply chain planning optimization in fruit industry. Bixby et al. [BDS06] provide a a complex system with several optimization models to support planning of operations and deliveries in meat industry. An example for the problems’ diversity in agricultural planning is given by the problem of wine harvesting. Ferrer et al. [FMMTV08] present a model that both minimizes the operational costs and maximizes the wine quality. The model uses a quality loss function to relate these two goals one to each other. A first review of work applying decision support tools to agriculture was given by Glen [Gle87] in 1987. Since his comprehensive survey, at least three recent surveys appeared. Lowe and Preckel [LP04] highlight some of the important works based on linear programming, stochastic programming, risk programming, dynamic programming and simulation. Their publication also includes a call for future research, which was responded by the OR society. The review of Lucas and Chhajed [LC04] focuses on location analysis applied to agriculture. Finally, Ahumada and Villalobos [AV09] aim to complete the previous compilations. They list more than 40 works, where most of them have their scope on the tactical planning. In addition to the surveys, France and Thornley [FT07] provide a rich introduction into mathematical models applied to agriculture. 1.3.2 Decision support tools in sugar cane industry The efforts of the OR society to support sugar cane industry mainly developed in the last ten years. They can be divided roughly into value chain optimization, harvest and crew scheduling and the prediction of sugar cane performance indicators. Efforts were mainly recognized from countries where sugar cane typically is planted, for example South America, South Africa [GLMS08], Australia and the United States of America. Higgins strongly contributed to recent literature about optimization approaches in sugar cane industry, mostly based on experiences in the australian sugar cane industry. His works include various planning aspects in sugar cane industry. In one of his works [HL06b], Higgins aims better integration and
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Chapter 1. Introduction
19
optimization of the cane harvesting and transport sectors of the value chain. The formulation is based on the P-Median problem and spatial clustering. Further works to improve the value chain [HASDPA04, HTAJ07] include investigations of the capacities of cutting and transport crews as well as the mill capacities [HTAJ07]. His works on harvest scheduling include an extension of the generalized assignment problem with over 500.000 integer variables, subject to constraints of transport and mill crushing capacities. The model is solved heuristically. Later exact approaches include a LP model [Zha05] and an integrated statistical and optimization approach [Zha05] that maximizes the sugar content in the crops for a harvest season. Higgins also separately tackled scheduling of cane transport [Hig06a], the simulation of transport capacityplanning [HD05] and the optimization of harvester rosters [Hig02]. De Alencar et al. use a genetic algorithm (GA) to maximize the produced sugar [RCB06]. Caliaria et al. use linear programming, based on the XA Callable library, to maximize the sugar production [CSS04]. Pacheco and Neto [PB08] seem to be the first ones to use the term Sugar Cane Harvest problem (SHP) for the problem. They incorporated it as a 0/1-Knapsack problem and show experimental results with multi-objective evolutionary algorithms (MOEA). The model considers mill and crew capacities. The paper also proposes the incorporation of logistic data such as distances between sugar cane lots, though such logistic issues were not implemented into their model. Pacheco et al. also presented many works using multi-objective Artificial Intelligence (AI) approaches [POF08, PPF07]. All these works incorporate the value of the sugar content (PCC, Apparent Percentage of sucrose in the sugar cane), the biomass per square meter TCH (Tons of sugar cane per hectare) and the total fiber in their objective functions. Pacheco et al. [PRBN05] and Oliveira et al. [OPF06] present works that aim to predict the performance indicators PCC, TCH and fiber using Artificial Neural Networks (ANN) and GAs. Finally, D´ıaz and P´erez [DP00] focus on sugar cane transportation. 1.3.3 Decision support tools for other harvest types The forest sector has also been an intensive user of Operations Research tools. The tactical and operational planning include planting, harvesting and transporting. Cutting crews possess cutting capacities and transportation crews are limited by its transportation capacities. Clients usually demand a certain timber quantity during several time periods. Research work in the forest sector has been very versatile. Mitchel [Mit04] and Karlsson et al. [Kar03] focus on short-term operational planning in forest
Chapter 1. Introduction
20
harvest optimization, including crew scheduling. As transport is a crucial issue in forest harvesting, many researchers such as Karlsson et al. [Kar04, HKR06] also focus on road planning to meet the demands for the timber transport. Further optimization in silviculture include [HMZW04] and several works of Weintraub et al. such as [WN76, MEW98, EGMW97].
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1.4 Outline The details discussed within each chapter are described below. Chapter 2 explains the sugar cane harvest problem by a detailled description of the tactical and the operational module. The provided background gives an understanding of sugar cane industry terminology and techniques used in this thesis. Furthermore, the relation of the problem to other classical problems as well as the problem’s complexity are discussed. Chapter 3 defines the basic terminology used throughout this thesis and introduces briefly into different optimization methods. It also gives an overview of contemporary MIP solvers. Chapter 4 discusses modeling issues and shows the abstraction of the Generalized Assignment Problem to the given problem. Afterwards, the mathematical formulations for both modules are given. Chapter 5 focuses on the solution of the problem and the computational experiments performed. First, the instances used for experimental evaluation are explained. Experiments performed on the SCHP-TP are presented. For the SCHP-OP, the content is then divided into pre-processing, exact methods and alternative solution strategies. Pre-processing includes all techniques that aim at reducing the problem size in order to facilitate its solution. The section about exact methods explains the basic techniques applied to the problem. Initial solutions provided as starting solutions due to the problem’s difficulty are then discussed. Finally, the characteristics of the linear relaxation are shown and valid inequalities to strengthen the MIP formulation are given. The last section of this chapter includes approaches usually applied to largescale problems as they can be found in practice. Solution strategies based on semantic information of the problem are discussed. Furthermore, a heuristic based on the linear relaxation is presented. Computational results are presented for all methods. Chapter 6 concludes this work by resuming its contribution and stating possible extensions for the future.
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2 Problem Description
This chapter explains the given problem in detail. Throughout this work, this problem will be referred to as the Sugar Cane Cultivation and Harvest Problem (SCHP) as it incorporates both cultivation and harvest issues appearing in sugar cane industry. After the problem’s components are presented, the current practice at one of Brazil’s sugar producers is explained in order to demonstrate the current planning process. The SCHP is then divided into the problems of the tactical and the operational planning. Both of them are described in detail. Finally, related problems and the asymptotic time complexity are discussed. 2.1 The sugar cane harvest problem One of the most important decisions in this problem is the determination of the optimal moment to harvest the plantation fields. Clearly, it is desirable to harvest each field at the peak of its sugar content, as the sugar indicated by the Pol and AR values vary as the cane grows. In the beginning of the planning, each field got a certain initial age. A field can only be cut within a given interval of its age defined in the input data. In the following, the complete problem is described with an introduction to the decisions that must be made and to the problem’s constraints that need to be satisfied. Sugar cane mills. After harvesting a field, its sugar cane is immediately transported to one of the sugar cane mills to be crushed and further processed to sugar. The mills operation is one of the most important constraints as it must not interrupt sugar cane processing. Each mill contains minimum and maximum process capacities which must be respected by the planning solution. Plantation fields that have been selected for harvesting must be assigned to one of the available sugar cane mills. Furthermore, sugar cane fiber is used to generate electricity in order to operate the sugarcane mills. Thus, the processed sugar cane must contain a certain minimum quantity of fiber.
Chapter 2. Problem Description
22
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Maturation products. Some sugar cane varieties allow the use of maturation products to anticipate its harvest. In general, such products slow down the growing process of the cane mass, whereas the growth of the sucrose within the cane is not affected. Thus, the percentage of sugar in the cane increases. The eligible maturation products are given for each variety within the input data. Maturation products can only be applied when the cane reaches minimum age. Once applied, they directly influence the originally given interval in which the field can be harvested. For each combination of maturation products and sugar cane varieties, an interval is defined that determines the feasible period to harvest the field after the product was applied. The planning solution shall consider the use of maturation products, i.e. determine the type and exact moment of its application, in order to find the most lucrative harvesting schedule. Cutting crews. The sugar cane is harvested by cutting crews. A cutting crew can either be manual, that is a group of human workers, or mechanical. Each cutting crew may be eligible to cut only a certain subset of the fields. Cutting crews may not work every day and work a limited time at each day. Each cutting crew has its own properties. It’s minimum and maximum cutting capacities, travel speed as well as cutting and travel costs are given in the input data and must be taken into account. After finishing their work at the end of each day, mechanical cutting crews remain at the current field and start working in the beginning of the next day. Manual cutting crews return to a place where they are accommodated. Transportation. Transportation crews carry the cut sugar cane from the fields to the sugar cane mills. Each crew possesses individual properties such as a certain transport capacity, speed and costs. The solution must suggest a transportation schedule that assigns exactly one transportation crew to each field selected to be harvested. Transportation crew capacities, speed and costs must be considered in the solution of the problem. Vinasse application. After crushing the sugar cane within the mills, the waste dump vinasse remains. A common practice to remove this byproduct is its application at already harvested area. In order to allow its frequent application, a sufficiently large field area must be harvested in certain periods. Not all fields are eligible for vinasse application. The vinasse demands, i.e. the periods and the necessary field size, and the fields that are eligible for vinasse application are informed by the input data. A planning solution must
Chapter 2. Problem Description
23
guarantee that sufficient fields are harvested in order to satisfy the vinasse application demand in all periods.
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2.1.1 Current practice The Pol and AR values for each cane variation are one of the most important performance indicators to select the best moment for a field’s harvest. However, these values cannot be foreseen very well for a long period such as twelve months. A detailed planning for such a broad time period would result in a huge problem size that could hardly be solved in adequate time. In many industrial planning problems, the division of the entire problem into smaller pieces is a natural approach. Many decisions have to be made only for a rough time period such as months or weeks, without determination of the exact detailed planning for shorter periods such as days or hours. Planning of such a broad time horizon is referred to as tactical planning. It is generally followed by an operational planning, where the problem is subdivided into smaller pieces and the resulting sub problems are solved in detail. The current practice at the Grupo Virgolino de Oliveira (GVO), a Brazilian sugar cane company, follows the above explained planning process. Planning is performed either manually by experienced experts from the agricultural sector or by computer based decision support tools1 . The decision support tool in use does not consider any distances between the plantation fields. Thus, the logistic planning does not fairly represent the reality. It allows partial field harvest as it is based on linear programming. Although it is a good tool to choose the plantation fields to harvest, the planning may not always be feasible in practice as it may not respect all industrial constraints. In addition, there are always stochastic factors that cannot be foreseen, such as delays of the cutting and transportation crews or industrial deficiencies. As a result, it may not always be possible to harvest all fields as it was planned. In the GVO’s current practice, if a field was not harvested in the period to which it was assigned by an operational planning, it is added to the operational planning of the following month. In some cases, also the fields of the successive month are added in order to consider updated Pol and AR values. The field set within the input data for an operational planning for period p may hence contain: – all fields that were indicated by the tactical planning to be harvested within period p − 1, but in fact were not harvested. 1 The GVO currently uses an optimization tool based on linear programming. The LP is solved by the XA Callable Library of Sunset Software
Chapter 2. Problem Description
24
– all fields indicated by the tactical planning to be harvested within period p. – all fields indicated by the tactical planning to be harvested within period p + 1. The operational planning should then identify the best fields to be harvested within the current period. All remaining fields are shifted into next planning again.
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2.2 A DSS for sugarcane cultivation and harvesting The proposed decision support tool for sugar cane cultivation and harvest planning follows the previously discussed approach and divides the whole planning into a tactical planning and an operational planning. The decisions of the tactical planning directly influence the input data for the operational module. First, the tactical module performs the planning for the whole planning horizon, i.e. up to a whole harvest season. Afterwards, the total planning time is divided into smaller periods of up to 30 days. The operational planning is then performed for each of these sub periods. The input data for each operational planning is based on the decisions of the tactical module in the according time period. That is, the set of fields selected by the tactical module for a certain time period forms the input data of an operational planning. In this operational planning, the exact days to harvest each of these fields are determined. In addition, the user may modify the input data for the operational module. Figure 2.1 illustrates the data flow of the system. As the operational planning is frequently performed, it is desirable that the execution time of the optimization does not exceed 30 minutes. The tactical planning, as it is performed only every few months, does not possess an explicit time limit. For reasons of the user’s convenience, one can assume a time limit of 60 minutes. In the following, both modules are explained. 2.2.1 Tactical module The tactical module of the SCHP, denoted by the SCHP Tactical Planning Problem (SCHP-TP), supports the planning for a total planning horizon of up to twelve months, i.e. one harvest season. It may be applied to shorter periods such as a few months, usually in the replanning during a running season.
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Chapter 2. Problem Description
Input
Data generation by the user
Tactical module Output Input
Manual modification by the user
Operational module
Output Figure 2.1: Data flow from the tactical module into the operational module
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The tactical module determines for each week of the whole planning horizon: – the plantation fields that should be harvested and the cutting crews to cut these fields. – the sugar cane mills in which the cane should be processed. Each field is associated to exactly one mill. – the transportation crews that carry the cut sugar cane to the mills. Each field is assigned to exactly one transportation crew. – the maturation products that shall be applied and the fields at which the products shall be applied. – the types of vinasse that shall be applied on the fields after harvesting them and the according fields. The module maximizes the total profit given by the processed sugar cane in the mills and the current value of the ATR, representing the quantity of theoretic total usable sugar. The ATR is computed against the quantity of reduced sugar (AR) and Pol, using coefficients for both values to normalize the relation between them: AT R = coef P ol · tonsP ol + coef AR · tonsAR On the cost side, there are the costs for cutting the fields, transporting the sugar cane to the mills, the processing of the cane within the mills and the application of maturation products and vinasse.
Chapter 2. Problem Description
26
During the maximization of the above mentioned indicators, the system must satisfy the following constraints: – All plantation fields must be harvested. Weights can be attributed to the fields for each week to indicate a priority of harvesting in certain weeks. – All cutting crews have a minimum and maximum cutting capacity of sugar cane per week. – All transportation crews have a maximum capacity of sugar cane that can be carried.
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– The sugar cane cut at a day shall be transported to and processed in the sugar cane mill within the same day. The sugar cane mills have a minimum and maximum capacity of sugar cane that they can process in each week. Consequently, the sugar cane quantity cut by the cutting crews shall satisfy the mill’s demands. – In all weeks, the mills must produce a certain minimum quantity of fiber, given by a percentage within the processed sugar cane. – The varieties of the different sugar cane species have a minimum and maximum age to be cut. The sugar cane can be cut only during these given intervals. These intervals may vary by week and field. – Maturation products can be applied only during a certain interval of the sugar cane’s age. After applying a maturation products, the sugar cane must be harvest within a given number of weeks. – In all weeks, a minimum quantity of vinasse must be applied on the recently harvested plantation fields. In order to allow these applications, the number of hectares freed by field harvesting in each week must be sufficiently high. The complete harvest of any plantation field may not exceed one week, i.e. only plantation fields that can be harvested within one week are considered. In addition to the sugar cane cut by the cutting crews, it is possible to acquire sugar cane from third party suppliers. Each supplier provides sugar cane of certain types of cane varieties up to a certain capacity. The sugar cane possesses its own properties (Pol, AR, Fiber, etc.) and can be processed at any mill.
Chapter 2. Problem Description
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2.2.2 Operational Module The operational module of the SCHP, also denoted by the SCHP Operational Planning Problem (SCHP-OP), realizes a detailed planning for a time horizon of up to 30 days. Its input data is based on the planning of the tactical module for the chosen time period. The operational module may redefine assignments between cutting crews, fields and mills. Valid assignments are informed in the input data. The application of maturation products is not determined by this module. Instead, such decision are already covered by the tactical module and shall be informed in the input data for the operational module. While the tactical module works with estimated maturation curves of the sugar cane, the operational module is intended to work with updated recent values of the sugar cane’s maturity, i.e. Pol, reduced sugar AR and fiber. These values result from the pre-analysis, where cane examples of a certain area are analyzed before the sugar cane is cut. The cutting crews may be located either at a mill or at a plantation field. Latter case occurs when the cutting crew did not finish the harvest of the field in the previous planning. The cutting crew must then initiate the cut of the field at which it is located in the beginning of the planning. The manual cutting crews are usually hosted at a place close to the fields. At the end of a working day, they spend the night at such place and return to work in the next morning. It is assumed that the time spent and the costs generated to get to their accommodation and to return to a field in the next morning are already considered by the input data through average values, i.e. the spent time and costs must not be considered in the planning. Thus, manual cutting crews may travel to any field in the next morning without any travel costs. A field can be partially cut, if it is cut at the end of the planning and the (partial) cut ends at the last available day of the crew. In this case, the rest of the field will be cut in the next operational planning. Moreover, harvesting a field can take more than one day and must only be interrupted by the time that a crew is not available (usually during the night). Cuts must not be interrupted by days at which the field cannot be cut, the crew does not work or the mill is not available. Thus, a cutting crew can only start harvesting a field after having finished the harvest of the previous field. Finally, assignments of the transportation crews will not be considered in this module. In practice, transportation crews are sufficiently available and can be hired on demand when necessary. Hence, the operational planning does not involve their planning. It is assumed that a successful planning of the tactical
Chapter 2. Problem Description
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module already guarantees sufficient availability of transportation crews. Given the input data for the cutting crews, fields and mills, the operational module should determine the harvest sequence of the fields for each cutting crew so that it maximizes the total profit given by the sugar production in the mills minus costs such as for cutting, transportation and processing the sugar cane, the movement of the cutting crews and vinasse application. For each day of the planning, the system should suggest: – the fields to be cut and the cutting crews to cut these fields. The cutting sequence during the planning should consider the displacement costs and time from one field to another. – the mills at which the sugar cane of each field shall be processed. – the vinasse types that shall be applied at which fields after the harvest.
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The system must consider the following constraints: – All cutting crews have a minimum and maximum cutting capacity of sugar cane in tons per day. – The sugar cane cut at a day shall be transported to and processed in the sugar cane mill at the same day. The sugar cane mills have an inferior and superior limit of sugar cane to processed at each day. Consequently, the sugar cane quantity cut by the cutting crews shall satisfy the mill’s demands. – In all weeks, the mills must produce a certain minimum quantity of fiber, given by a percentage within the processed sugar cane. – In all days, a minimum quantity of vinasse must be applied on the recently harvested plantation fields. At least this number of hectares must be harvested in each week to permit the vinasse application. Sugar cane be acquired from third party suppliers, just as explained in the description of the tactical module. Note that the operational module does not demand all fields to be harvested. This is justified by the current planning practice in industry, explained in Section 2.1.1. Another feature of this module is the possibility to rank the importance of the constraints. In some instances, it may not be possible to satisfy all constraints: such a ranking then indicates which constraints shall be violated less than others. The following table resumes the main differences between both modules:
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Chapter 2. Problem Description
Tactical module Operational module Planning horizon Objective Decisions Must cut all fields Cutting crews Cutting crew harvest sequence Transportation crews Maturation products application Vinasse application
One season maximize profit weekly yes yes no yes yes yes
7 to 30 days maximize profit daily no yes yes no no yes
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2.3 Related problems The essential structures for the tactical and operational module are very similar. Both modules try to maximize the profit by choosing the best moment to harvest the plantation fields, considering the minimum and maximum processing demands of the sugar cane mills and the limited capacities of the cutting crews. This effort can be seen as 0/1 Multiple Knapsack problem. It is a generalization of the of the 0-1 Knapsack problem and identical to the Generalized Assignment Problem (GAP), one of the most studied combinatorial problems. Martello and Toth [MT90] elaborately introduce into these problems and show that they are NP-hard. In the 0/1 Multiple Knapsack problem, a number of objects must be packed into a given number of bins. Each object has a weight and a profit for each bin. Bins are limited in their capacity of weight. The objective of the problem consists in finding the configuration that maximizes the total profit, respecting the weight capacities of the bins. Pacheco and de Lima Neto [PB08] already modeled harvest scheduling as a 0/1 multiple knapsack problem. In the case of the tactical module, the sugar cane mills can be modeled as bins, one for each week. All sugar cane mills own a maximum processing capacity, referring to the bins’ capacities. The weights and profits to process a field’s sugar cane at different weeks and mills may vary. The profit is obviously linked to the sugar content of the field at the processing week. Figure 2.2 (a) examples such an assignment. At each week, the sugar cane mill processes a different subset of fields. In the same way, other resources such as the cutting and transportation crews can be modeled. Figure 2.2 (b) examples this abstraction to the cutting crews: A cutting crew is assigned to harvest several fields within its cutting capacities in each week. A detailed extension of the mathematical model for the GAP is given below in subsection 4.1.3.
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Mill 1
(b) Cutting
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Crew 1
Week 1
Week 2
Field 2
Field 3
Field 13
Field 4
Field 15
Field 11
Week 1
Week 2
Week n
...
Field 6
Week n
Field 3
cutting capacity
(a)
processing capacity
Chapter 2. Problem Description
Field 2 Field 5 Field 8
Field 4
...
Field 6
Field 9
Figure 2.2: Modeling the harvest problem as a knapsack problem The operational planning is much more complex than the tactical, because it involves the cutting sequences for the cutting crews. A part of the operational planning can evidently be modeled in the same way. The sugar cane mills’ and cutting crews’ capacities can be modeled based on a 0/1 Multiple Knapsack formulation, using a bin for each day. The main difference is the addition of the cutting sequences for the cutting crews which incorporate knowledge of the distances between one field to another. As such a sequencing is often referred to as a harvest scheduling, the problem of determining the field harvest sequence naturally can be seen as a general type of scheduling as well. The harvest sequence of the cutting crews also resembles the Vehicle Routing Problem (VRP), a class of NP-hard problems. In the VRP, a fleet of vehicles supplies customers, starting from and returning to a depot where the vehicles are reloaded. Each vehicle has a certain capacity and each customer has a certain demand. The objective of these problems is to find optimal vehicle routes that satisfy the customer demands and either minimize the total distance or the number of necessary vehicles. Rich [Ric99] broadly introduces into the VRP. For the SCHP, the vehicles can be represented by the cutting crews and the clients by the fields that have to be harvested. The VRP
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Chapter 2. Problem Description
31
versions that partially resemble the SCHP-OP are the Capacitated Vehicle Routing Problem (CVRP), the Vehicle Routing Problem with Time Windows (VRPTW) and the Periodic Vehicle Routing Problem (PVRP). The CVRP limits the capacities for each vehicle. In the VRPTW, the customers have to be supplied within a certain time window. This equals the eligible days to harvest a plantation field. Finally, the PVRP allows that the deliveries may take more than one day. Though these problems are partially very similar to the SCHP-OP, they do not allow the incorporation of the most important aspects: the varying profit, i.e. the sugar content within the cane, as the sugar cane grows. As the cutting crews’ temporal resources may be a crucial factor in the harvest planning, it is desirable that they spend most of their time in cutting sugar cane and as less time as possible in moving from one field to another. This effort is directly linked to the distances and the length of the paths between the fields, leading to the attempt to minimize the total covered distance. From this point of view, the SCHP-OP also intersects with problems such as the Traveling Salesman Problem (TSP) and the problem of finding the shortest Hamiltonian Path. Latter one aims to find a shortest path between two given vertices that visits each vertex exactly once. In the problem of the Hamiltonian Cycle, the starting vertex is the same as the terminal vertex. The TSP is one of the most studied combinatorial problems. It consists in finding a path that covers all vertices so that the total distance is minimal. The Multiple Traveling Salesman Problem (M-TSP) is a generalization of the TSP, where more than one salesman can be used within the solution. The M-TSP can also be extended to a variety of vehicle routing problems. It strongly relates to the problem of finding the cutting sequences for the cutting crews. All these problems are known to be NP-hard. Particularly the operational planning of the SCHP is a very versatile problem that intersects with many classical NP-hard problems known in Combinatorial Optimization. Thus, one may presume that the problems of the SCHP also are NP-hard. This assumption is proved in the following section. 2.4 Complexity It is now shown that the problems of the tactical and operational planning of the SCHP are NP-hard. In order to prove the NP-hardness of the problems, although it is not necessary, first it is shown their decision versions are NPcomplete. The decision versions of the SCHP-TP and SCHP-OP consist in determining whether there exist feasible solutions that represent a profit
Chapter 2. Problem Description
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greater than ∆. From the NP-completeness of the decision versions of the SCHP-TP and SCHP-OP follows that their optimization versions are NP-hard. Polynomial certifier for the tactical module. In order to prove that decision version of the problem belongs to the class NP, one must find an efficient certifier for the solution of the problem, i.e. a polynomial time verification for the correctness of any given solution for the tactical planning. This is easily possible by verifying each of the problem’s constraints. In addition, the certifier must verify whether the profit represented by the solution is greater than ∆. Let W be the set of weeks of the whole planning, let F be the set of plantation fields that can be harvested, let CF be the set of all cutting crews, let T F be the set of all transportation crews, let P be the set of all mills, let R be the set of all third party sugar cane suppliers and let M be the set of all maturation products. A solution of the problem is given by a set of decisions for each week within W , containing the fields to be cut within the week, the cutting crew assigned to cut this field, the transportation crew assigned to transport the cane from this field and the sugar cane mill assigned to process the cane from this field. Furthermore, the solution identifies the fields to receive a maturation product or vinasse (including the applied vinasse quantity). 1. The total profit represented by the solution has to be greater than ∆. It can be calculated in polynomial time. It must consider the profit given by the production of sugar cane harvested in fields and acquired from third party suppliers. In addition, it must subtract the costs generated through cutting, transporting, purchasing and processing of sugar cane. Further costs are given by the traveling costs of the cutting and transportation crews, the use of maturation products and the application of vinasse. 2. A sugar cane mill may not be able to process all types (varieties) of sugar cane. Thus, it must be verified whether the assignments between the fields and the mills are valid. Such verification can be done in O(|F | · |P |). 3. For each field, it must be verified whether the week in which the field is cut lays within the eligible interval to harvest this field (considering the anticipation of the interval by use of a maturation product). This verification can be performed in constant time. 4. The verification whether the lower and upper processing capacities of a mill within a certain week are satisfied can be done in O(|F | + |R|). As there are |P | mills and |W | weeks, the complete verification is performed in O(|P | · |W | · (|T | + |R|)).
Chapter 2. Problem Description
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5. The average fiber percentage for one mill in a week is verified in O(|F | + |R|) time. Thus, all verifications at |W | weeks for |P | mills are done in O(|P | · |W | · (|F | + |R|)) time. 6. The obligation to cut all fields is verified in O(|F | · |CF | · |W |) steps, verifying each combination of fields, cutting crews and weeks. 7. The set of fields that a cutting crew can cut may not include all exisiting fields. The verification whether assignments between cutting crews and fields are valid can be performed in O(|F | · |CF | · |W |). 8. Checking whether the cutting capacities of a cutting crew is satisfied in a certain week costs O(|F | · |W |) steps, resulting in a total cost of O(|F | · |CF | · |W |) for all cutting crews in all weeks.
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9. The time availability of a cutting crew at a day must not be exceeded. This verification needs at most O(|F | · |CF | · |W |) time for all cutting crews for all weeks. 10. The quantity of transported sugar cane is limited by the capacities of the cutting crews. These constraints can be verified in O(|F | · |T F | · |W |) for all transportation crews for all weeks. 11. All applications of maturation products at fields must be verified, i.e. the accuracy of the application moment as well as whether the product is allowed to be applied at the field at all. 12. The total sum of cut sugar cane in hectares must be verified at each week in order to guarantee sufficient harvested area for vinasse application. This verification, for all weeks, needs O(|F | · |P | · |W |) time.
Polynomial certifier for the operational module. Let D be the set of days of the whole planning. Let F be the set of plantation fields that can be harvested, let CF be the set of all cutting crews, let P be the set of all mills and let R be the set of all third party sugar cane suppliers. A solution of the problem is given by a set of decisions for each day d ∈ D, containing the fields to be cut at day d, the cutting crew assigned to cut this field, the transportation crew assigned to transport the cane from this field and the sugar cane mill assigned to process the cane from this field. Furthermore, the solution informs at which fields how much vinasse shall be applied. A certifier for a solution of the operational planning must verify the following constraints:
Chapter 2. Problem Description
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1. The total profit represented by the solution has to be greater than ∆. It can be calculated in polynomial time. It must consider the profit given by the production of sugar cane harvested in fields and acquired from third party suppliers. In addition, it must subtract the costs generated through cutting, transporting, purchasing and processing of sugar cane. Further costs are given by the traveling costs of the cutting and transportation crews and the application of vinasse. 2. A sugar cane mill may not be able to process all types (varieties) of sugar cane. Thus, it must be verified whether the assignments between the fields and the mills are valid. Such verification can be done in O(|F | · |P |).
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3. For each field, it must be verified whether the days at which the field is cut lays within the eligible interval of days to harvest this field. This verification can be performed in O(|D|) time. 4. The verification whether the lower and upper processing capacities of a mill at a certain day are satisfied can be done in O(|F | + |R|). As there are |P | mills and |D| days, the complete verification is finished in O(|P | · |D| · (|F | + |R|)). 5. The average fiber percentage for one mill at a day is verified in O(|F | + |R|) time. Thus, all verifications at |D| days for |P | mills are done in O(|P | · |D| · (|F | + |R|)) time. 6. The set of fields that a cutting crew can cut may not include all existing fields. The verification whether assignments between cutting crews and fields are valid can be performed in O(|F | · |CF | · |D|). 7. Checking whether the cutting capacities of the cutting crews are satisfied at a certain day costs O(|F |) steps. This leads to a total cost of O(|F | · |CF | · |D|) for all cutting crews at all days. 8. The time availability of a cutting crew at a day must not be exceeded. This includes the time spent in cutting the field as well as time needed to travel from one field to another. Considering all possible combinations between two fields for the travels and all cutting crews, the verification for one day can be done in O(|F |2 · |CF |) steps. The total time for all days is thus O(|F |2 · |CF | · |D|). 9. The total sum of hectares cut at plantation fields must be verified at each day in order to permit vinasse application. This verification, for all days, needs O(|F | · |P | · |D|) time.
Chapter 2. Problem Description
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As the feasibility of a solution to the SCHP can be verified within polynomial time, the decision versions of the SCHP-TP and SCHP-OP belong to the class NP. Reduction of a NP-complete problem. As shown above, solutions for both the tactical and the operational planning of the SCHP (and their decision versions) can be certified in polynomial time. In order to show that the decision versions of the problems are NP-complete, one must reduce the decision version of another NP-hard problem to them. The problem selected to be reduced is the GAP, which is known to be NP-hard [MT90]. Section 4.1.3 shows that both the SCHP-TP and SCHP-OP are generalizations of the GAP. Consequently, the decision versions of the SCHP-TP and SCHP-OP are generalizations of the decision version of the GAP. Consider a GAP instance with m bins b1 , · · · , bm and n objects x1 , · · · , xn . Let wM axi be the capacity of bin bi . Let weighti,j be the weight and prof iti,j the profit to put object xi into bin bj . Let T be set of temporal units, i.e. weeks for the SCHP-TP and days for the SCHP-OP. The following configuration represents a feasible reduction of this GAP instance to both the SCHP-TP and the SCHP-OP: – Representation of the bins. There is exactly one sugar cane mill p. There are exactly m temporal units (weeks or days) within the whole planning horizon. Each of the bins within the GAP b1 , · · · , bm is represented as this mill p1 · · · pm at each temporal unit 1, · · · , m. The maximum weight capacity wM axi of each bin i transforms into the maximum processing capacity of this mill for the temporal unit i. The minimum processing of the mill is always zero. – Representation of the objects’ weights. Each object within the GAP that has to be assigned to a bin is represented by a plantation field. Its productivity at the temporal unit i is set to the weight of the object when it is put into the bin bi . – Representation of the objects’ profits. As an object’s weight and profit usually differ within a GAP instance, it is not valid to represent the profit only by the field’s sugar cane productivity. However, it can be represented by the sugar content percentage within the sugar cane of the field at a certain temporal unit. More exactly, the Pol value for a field j prof iti,j at temporal unit i is set to P roductivity . The ATR costs are set to one j,i for all temporal units. The AR values are set to zero for all fields at all temporal units.
Chapter 2. Problem Description
36
– Further fixings. There is one transportation crew and one cutting crew for each plantation field. All transport costs of the transportation crews are set to zero. Also, the transport costs are set to zero. The transportation crews’ speed and carriage capacities are set to infinity. All cutting costs for the cutting crews are set to zero. There are no maturation products and no vinasse demands.
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Given the total profit composed of the profits of the selected objects, it can easily be verified whether exists a solution with profit greater than ∆. This reduction of the GAP to the SCHP-TP and the SCHP-OP concludes the prove of the NP-completeness of the decision versions of tactical and operational planning problems. Therefore, the SCHP-TP and SCHP-OP are NP-hard.
3 Optimization Methods
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This chapter gives a brief overview over different optimization methods and introduces into its essential concepts. It is intended to provide basic knowledge in order to facilitate the understanding of the strategies used to solve the problem (see Section 5). The chapter starts with basic definitions. Afterwards, it reviews methods to solve problems exactly as well as heuristically. Finally, current MIP solver are presented. 3.1 Definitions This section defines important terms used in this thesis. Further introduction to linear and integer programming can be found in the works of Wolsey [Wol98] and Wolsey and Nemhauser [WN99]. Well-founded literature into combinatorial optimization include the works of Gr¨otschel et al. [GLS88], Schrivjer [Sch94], Papadimitriou [PS98] and Wolsey and Nemhauser [WN99]. Mixed Integer Program Let m, n and p be non negative integers. Consider a matrix A ∈
M AX subject to
c·x+h·y
(3-1)
A·x+G·y ≤b
(3-2)
x ∈ Z+n
(3-3)
y ∈
(3-4)
is called a Mixed Integer Program (MIP), also known as a Mixed Integer Linear Program (MILP). x = (x1 , · · · , xn ) and y = (x1 , · · · , xp ) are called the variables or unknowns.
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Chapter 3. Optimization Methods
If all variables are integers, then the problem reduces to a pure Integer Program (IP):
M AX subject to
c·x
(3-5)
A·x≤b
(3-6)
x ∈ Z+n
(3-7)
When no integer variables are present, the problem is referred to as a Linear Program (LP):
M AX subject to
h·y
(3-8)
G·y ≤b
(3-9)
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y ∈
(3-10)
The expression in (3-1) is called the objective function of the MIP, which is either to be minimized or to be maximized. The inequalities in (32), composed by the right-hand-side vector b and the products between the coefficient matrices and the variable vectors, represent the set of constraints which have to be satisfied. Finally, the variables’ domain is defined in (3-3) and (3-4). A problem is said to be in standard form if the MIP is a minimization problem, all constraints are given in form of equalities and all variables are non-negative. Duality in Linear Programming Consider the previously introduced linear program given by (3-8) to (310), also called the primal problem. Associated with this primal problem, there is a corresponding problem referred to as its dual problem given by:
M IN subject to
u·b
(3-11)
u·G≤h
(3-12)
u ∈
(3-13)
Every linear program can be transformed into its dual problem. The optimum solution of the dual problem is equal to the optimum solution of its primal problem. Detailed explanations of the dual concept can be found in Bradley et al. [BHM77] and Wolsey [Wol98].
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Chapter 3. Optimization Methods
Linear relaxation The linear relaxation of a MIP involves the relaxation of the integrality constraints for the set of integer variables. Consider the MIP given by the expressions (3-1) to (3-4). The linear relaxation of the problem includes the expressions (3-1), (3-2) and (3-4). Instead of (3-3), the linear relaxation of the problem additionally includes: x ∈
(3-14)
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Integrality Gap Since the linear relaxation relaxes the integrality constraints for the set of integer variables, an optimal solution for the linear relaxation has commonly a higher (in a maximization problem) objective function value than an optimal solution for the original mixed integer problem. The relative difference between the solution values of the optimal solutions for a MIP and its linear relaxation is referred to as the integrality gap of the MIP problem. Valid Inequalities and Cuts Given a mixed integer program, a valid inequality is an inequality that is satisfied by all feasible integer solutions of the MIP. A valid inequality that is not part of the current problem formulation is called a cut. Cuts are commonly used to approximate the convex hull of a MIP’s linear relaxation to the convex hull of the original MIP. In other words, one wants to get the optimal solution of a MIP’s linear relaxation closer to the optimal solution of this problem. This is typically the case if the cut forbids the formerly optimal solution to the linear relaxation problem. A MIP formulation that leads to small gap between its optimal solution and the optimal solution of its linear relaxation is said to be tight. Cuts can be added to the formulation a priori, i.e. they are added to the MIP formulation as constraints. However, this may result in a very large number of constraints, complicating the solution of the problem. In this case, the linear relaxation may be solved and only the most violated cuts are added to the problem’s model, such as in Cutting Plane algorithms. A Cutting Plane method repeatedly solves the linear relaxation and adds cuts until no more cuts are found. For further introduction to valid inequalities and Cutting Plane methods, the reader is referred to [Wol98].
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40
Solution quality In order to measure the quality of a solution to a MIP, the deviation of the solution’s objective function value from the best known solution is used. The calculation used in this work to compute this deviation for a solution is the same as used by CPLEX. For a maximization problem, it is: dev =
valuebestKnown − valuesolution valuesolution
A widely spread approach to estimate the deviation from optimality for a LP solution is to solve its dual problem as well. If both objective functions meet, the deviation is zero.
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3.2 Exact methods 3.2.1 Exact methods for linear programming The probably most known method to solve linear programs is the Simplex algorithm. Created by the mathematician George Dantzig in 1947, it is listed as one of the most important algorithms of the century. The method uses the concept of a simplex, i.e. a polytope of n + 1 vertices in n dimensions. Each inequality specifies a half-space in n-dimensional Euclidean space. Their intersection is the set of all feasible solutions. It is proven that one optimum solution lays on one of the extreme points of the polytope. The simplex method starts at one of these extreme points and then moves along the edges trying to improve the solution. A detailed introduction into this method can be found in Cormen et al. [CSRL01]. Though it has exponential complexity in worst case, the simplex method is remarkably efficient in practice as it showed polynomial average-time complexity [ST04]. Another important classes of methods to solve linear programs are interior- and exterior-point methods. Former were developed by Narendra Karmarkar in 1984 [Kar84]. These classes of methods are also referred to as Barrier methods. In contrast to the simplex method, barrier methods traverse the interior of the feasible region in order to find the optimum solution. Barrier methods are known to be among the most successful methods to solve linear programming problems. An extensive introduction into Linear Programming and its solution methods is given by Vanderbrei [Van01].
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41
3.2.2 Exact methods for mixed integer programming 3.2.2.1 Dynamic Programming Dynamic programming is a method of solving problems with overlapping subproblems and optimal substructures, i.e. optimal solutions of subproblems are used to find the optimal solution of the original problem. The subproblems are usually solved recursively. Dynamic programming can be used to solve linear and non-linear problems and has proven to be very useful applied to problems with integer features. The term dynamic programming was originally used in the 1940s by Bellman, as reported by Dreyfus [Dre02]. An introduction to this technique can be found in the original work of Bellman [Bel57, Bel66].
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3.2.2.2 Branch and Bound Branch-and-Bound (BB) is a general algorithm to find optimal solutions for (mixed) integer programming problems. It is based on systematic exploration of the enumeration tree of feasible solution. The algorithm skips large parts of the tree at nodes that cannot yield better feasible solutions than already found. At each stage of the algorithm, the feasible region is divided into subsets based on a decision. In the case of binary variables this decision may be the value of a variable, yielding one sub region where the variable is fixed to zero and another sub region where the variable is fixed to one. This type of branching is referred to as variable branching. At some nodes within the branching tree, the bounding step is performed to test whether the sub tree at this node can be pruned. A method is used to find a bound for the best solution that may be obtained considering the decisions made up to the current node. In a maximization (resp. minimization) problem, an upper (lower) bound is computed. If this upper (lower) bound is lower (higher) than the best solution already known, the sub tree at this node can be pruned as it will not yield any better solutions. Clearly, the quality of the bound strongly affects the efficiency of the Branch-and-Bound algorithm. BB is one of the basic exact optimization approaches and is widely spread in implementations of MIP solvers. Generic MIP solver usually use the optimal solution of a problem’s linear relaxation (see Duality in Linear Programming in Section 3.1) to estimate the bound for a problem. Certainly, the tighter the MIP formulation, the better the bound.
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42
3.2.2.3 Branch and Cut Branch-and-Cut is a hybrid of Branch-and-Bound and Cutting Plane methods. After repeatedly addition of cuts and solution of the problem’s linear relaxation, a branching is performed. The algorithm proceeds by alternating cutting steps and branching.
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3.2.2.4 Column Generation algorithms The idea of Column Generation methods is inspired by the fact that many linear programs may contain an extensively large number of variables, too large to consider all of them. Since most of the variables of an optimal solution are non-basic (non-active variables within a solution), only a small subset of variables has to be generated. The main idea behind Column Generation is to only generate variables that have potential to improve the objective function. Column Generation methods are not implemented within this thesis and are listed here for reasons of completeness. The interested reader is referred to [Wol98]. Column Generation used within a BB framework is also known as Branch-and-Price. A BB framework with integrated Cutting Plane method and Column Generation is widely called a Branch-and-Cut-and-Price algorithm. 3.3 Heuristics and Metaheuristics Problems are often too large or too difficult to solve them by exact methods in reasonable time. However, in some cases one might not need the optimal solution, but a good quality solution in short processing time. A Heuristic is a technique designed to find solutions to a problem. In general, heuristic algorithms do not guarantee any quality of their solutions. However, they usually produce good quality solutions in fair processing time. Heuristics with provable solution quality (for instance within 5% of the optimal solution) and provable run time bounds are known as approximation algorithms. A metaheuristic is an algorithmic framework to find solutions to a general class of optimization problems. It guides other heuristics in order to find feasible solutions. While the framework of a metaheuristic is the same for each problem, its specific implementation for each problem usually differs. The most common metaheuristics include Local Search, Tabu Search, Simulated Annealing, Genetic Algorithms and Ant Colonies. An introduction
Chapter 3. Optimization Methods
43
and comparison of several metaheuristics can be found in Blum and Roli [BR03].
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3.4 MIP Solver Due to the advances in Information Technology and algorithm design and understanding, many MIP solvers for mathematical programming are available in order to tackle classical and practice problems. ILOG’s commercial CPLEX solver [Ilo] offers a rich scope for solving linear problems and a very effective BC framework to solve mixed integer problems. The BC framework is able to generate a multitude of different generic cuts. In order to find and improve feasible solutions it also incorporates many heuristics and genetic algorithms. Other widely used commercial products include the XPRESS solver from Dash Optimization [Das] and the XA solver from Sunset Software [Sun], both offering software to solve linear and mixed integer problems. Beside commercial products, a strong interest in non-commercial software to solve optimization problems can be noticed. Some of them are introduced in the following. The Computational Infrastructure for Operations Research (COIN-OR) project [Coi] is an Open Source project providing software for the OR community. It includes a list of independent projects providing representation and algorithm implementations for areas such as graphs, metaheuristics, modeling systems, convex optimization, discrete (non-) linear optimization and stochastic optimization. The BCP project offers a parallel Branch-and-Cut-and-Price framework to solve MIPs; the CLP (COIN-OR LP) provides a simplex solver. The COIN-OR also developed the Open Solver Interface (OSI) that specifies the interface for linear relaxation solver. SCIP (Solving Constraint Integer Programs) [Sci] is a framework for Constraint Integer Programming and also allows to be used as a pure MIP solver or as a framework for Branch-and-Cutand-Price. SCIP does not include a proper LP solver, but accepts the interface of popular LP solvers such the ones from CPLEX, XPRESS or COIN-OR. The GNU Linear Programming Kit (GLPK) [Glp] is a software package, developed under GNU General Public License, for solving large-scale LP and MIPs. It uses the revised simplex method and the primal-dual interior point method for LPs and a BB with gomory cut for MIPs. ABACUS (A Branch-And-CUt System) [Aba] is a framework for the implementation of BB algorithms. Cutting planes and columns are dynamically generated within the BP and BCP framework by implementation of the problem specific separation for cutting
Chapter 3. Optimization Methods
44
planes and column generation. ABACUS does not incorporate an implementation to solve linear relaxations, but accepts the OSI. Finally, MINTO (Mixed Integer Optimizer) [Min] is a BB based MIP solver. As some of the previously introduced non-commercial MIP solver, MINTO does not have a LP solver of its own, but uses popular LP solvers such as the ones defined by the OSI interface. All experiments within this work were carried out using ILOG’s CPLEX which is now explained in further detail.
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3.4.1 ILOG CPLEX ILOG CPLEX, named by the Simplex method and the C programming language, is a high performance MIP solver. CPLEX offers several algorithms to solve the node and initial linear relaxation. Such algorithms include the interior-point methods as well as the barrier method. The BC framework automatically generates cuts such as Clique cuts, Cover cuts, Disjunctive cuts, Flow cover and path cuts, Gomory fractional cuts, Implied bound cuts, Mixed integer rounding cuts and Zero half cuts. The BC phase can be followed by a polishing phase, where CPLEX aims to improve existing solutions making use of genetic algorithms [Rot07]. This polishing phase has proved to be very effective when used with a previous branching phase to provide a pool of feasible solutions. The CPLEX configuration used within the experiments in this work can be found in Section 5.
4 Problem Formulation
In this chapter, the SCHP is stated as a mixed integer program. The first section discusses modeling aspects for the tactical and the operational module. Afterwards, their MIP formulations are presented. 4.1 Preliminary discussions
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4.1.1 Modeling issues Harvest schedule planning has been addressed by several works (see Section 1.3). Some of the problems treated in those works showed strong similarities to planning aspects found in the sugar cane harvesting process. Silvicultural planning, for example, involves the allocation of a workforce crew to geographical units that have to be harvested. Mitchel [Mit04] presents two different modeling alternatives to represent crew allocation: – Model I. Each crew, harvest unit and period combination is modeled as a single binary integer variable. In the integer solution, one variable is positive for each crew in each period. – Model II. The decisions for the harvest units and periods are modeled jointly for each crew. The harvest sequence throughout the planning horizon is a composite decision. The integer solution contains one positive variable for each crew. Mitchel gives detailed formulations for both alternatives and lists additional three approaches for crew allocation based on composite decisions. The number of variables for the first model is clearly linear in size of the input data and therefore it is reasonable to use BB / BC algorithms. The number of variables for the second approach is very large, because there is an exponential number of combinations between the harvest units and periods. The solution of such large problems typically requires methods such as Column Generation. The formulation given in this work is based on the first model, extending a classical problem from Combinatorial Optimization (as later on explained).
Chapter 4. Problem Formulation
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4.1.2 Simplifications
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In practice, industrial problems involve numerous details. Mathematical models that aim at providing solutions to such problems must focus on decisive information in order to guarantee a reasonable model size. The mathematical models presented in this section assume the following simplifications: – Temporal representation. The representation of the temporal aspect is a fundamental decision within the problem formulation. In general, one may distinguish between approaches based on continuous time models and models with time discretization. In the tactical model, an assignment for each harvest activity within a certain week is necessary in order to consider the weekly varying properties of the fields and cutting crews. These properties such as the varying Pol and AR values are typically available on weekly basis within the input data. Consequently, in this case, time discretization is adequate suitable. For the tactical planning, this means that a field harvest cannot be divided into two continuous weeks. Such a behavior could be modeled by using continuous variables for the harvested quantities of the last cut of a week and the first cut of the following week. However, since it is assumed that all fields can be harvested within a single week, the here presented model for the SCHPTP does not consider splitting a cut into two weeks. The operational planning additionally necessitates a harvest sequence for each cutting crew in order to consider proper distances between the fields selected for harvesting. This modeling aspect is discussed in the Modeling alternatives (see Section 4.3.1) for the operational planning. – Cutting and transportation of cane. After the sugar cane is cut, transportation crews carry it to mills where it is processed. In practice, cut sugar cane must not be left at the field for too much time before being processed. In order to simplify the model, it is assumed that the sugar cane is transported to the mill at the same instant in which it is cut. – Vinasse application. Vinasse can only be applied to fields that have recently been harvested. The time limit allotted between the cutting of the field and the application of vinasse may vary. For sake of simplification, it is assumed that the vinasse is applied at the same instant in which the field is completely cut.
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Chapter 4. Problem Formulation
4.1.3 Modeling the SCHP as a GAP extension It is now demonstrated how the Generalized Assignment Problem can be abstracted to model partial problems of the tactical and operational planning. First, the classic GAP formulation is presented. Then, the GAP is used to assign fields to sugar cane mills in order to maximize the total sugar production profit. Such assignment can be applied to cutting crews and fields as well as transportation crews and fields in the same manner. Classic GAP
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Consider m bins and n items. Let wM axi be the capacity of bin bi . Let wij be the weight and pij the profit to put item j into bin i. The classic GAP can formulated as:
MAX
m X n X
pij .xij
(4-1)
i=1 j=1
subject to: n X
wij .xij ≤ wM axi , i = 1, . . . , m;
(4-2)
j=1
m X
xij ≤ 1, j = 1, . . . , n;
(4-3)
i=1
xij ∈ {0, 1}, i = 1, ..., m, j = 1, . . . , n;
(4-4)
Problem abstraction to an extended GAP The bins represent the different mills mill1 . . . millnM ills over the time where the sugar cane can be processed. The items represent the plantation fields f ield1 . . . f ieldnF ields where the sugar cane is cut. The productivity of the plantation fields differ each week, as do the mills’ processing capacities. In order to decide which plantation fields to cut in which of the nW eeks weeks so that the total profit is maximized, it is necessary to distinguish the variables within each week:
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Chapter 4. Problem Formulation
– quantityf,p,w represents the total quantity of sugar cane that will be processed at mill p when field f is cut within week w. – gainf,p,w represents the total profit of associating plantation field f to mill p within week w. This value involves factors such as the sugar content within the cane and the ATR value at that day. – wM axp,w is the upper limit of sugar cane that can be processed in mill p within week w. Observing the problem as a graph, there is one node for each field f ieldf , one node for each mill in each week millp,w and one edge (f, (p, w)) from each field f ieldf to each mill in each week millp,w . Additionally, the last inequality (4-3) is transformed into an equality to assure that all plantation fields are harvested exactly once. This extension results in the following problem:
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MAX
nM ills nF ields nW eeks X X X p=1
f =1
gainf,p,w · xf,(p,w)
(4-5)
w=1
subject to: nF ields X
quantityf,p,w · xf,(p,w) ≤ wM axp,w ,
f =1
p = 1, . . . , nM ills; w = 1, . . . , nW eeks;
nM ills nW eeks X X p=1
xf,(p,w) = 1, f = 1, . . . , nF ields;
(4-6)
(4-7)
w=1
xf,(p,w) ∈ {0, 1}, p = 1, . . . , nM ills, f = 1, . . . , nF ields, w = 1, . . . , nW eeks;
(4-8)
Consider wM inp,q as the lower limit of sugar cane that should be processed in mill p within week w: nF ields X
wp,f,w · xf,(p,w) ≥ wM inp,w ,
f =1
p = 1, . . . , nM ills; w = 1, . . . , nW eeks;
(4-9)
Chapter 4. Problem Formulation
49
The input data for the linear program can then be computed as follows: – quantityf,p,w = productivityf,w , f = 1, . . . , nF ields, p = 1, . . . , nM ills, w = 1, . . . , nW eeks – gainf,p,w = (valueAT Rp,w · (Coef P ol.P olf,w + Coef AR.ARf,w ) − transportCostf,p,w ) · productivityf,w , f = 1, . . . , nF ields, p = 1, . . . , nM ills, w = 1, . . . , nW eeks
where the used data is defined as follows: – valueAT Rp,w - value of sugar ATR produced by mill p within week w (unit: $/ton) – P olf,w - percentage of Pol in the sugar cane cut within week w from plantation field f (unit: %)
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– ARf,w - percentage of AR within the sugar cane cut within week w from plantation field f (unit: %) – Coef P ol - coefficient to normalize the Pol – Coef AR - coefficient to normalize the AR – processCostp - process cost of mill p (unit: $/ton) – productivityf,w - quantity of sugar cane that field f produces if it is cut within week w (unit: tons) – transportCostf,p,w - cost to transport a ton of sugar cane from field f to mill p within week w (unit: $/ton) 4.2 Formulation for the tactical planning This section covers the mathematical formulation of the tactical module as a MIP. The model presented here assumes that each plantation field can be cut exactly once throughout the whole planning horizon. However, different crop types or longer planning periods may make it desirable to permit a field to be harvested more than once. Section 4.2.2 shows how the model can be extended in order to allow for multiple harvesting for a field.
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50
4.2.1 Mathematical model Input data For the mathematical formulation, consider the following input data separated into the following topics: sets, mills, plantation fields and maturation data, maturation products, cutting crews, transportation crews and third party suppliers. Sets – W - Set of all weeks considered during the planning. – F - Set of existing plantation fields. – P - Set of existing mills.
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– CF - Set of all cutting crews. – T F - Set of all transportation crews. – R - Set of third party sugar cane suppliers. – M - Set of maturation products. – T V inw - Set of plantation fields f ∈ F that are eligible for the application of maturation products within week w ∈ W .
Mills and sugar production process – M inP rocw,p - Minimum quantity of sugar cane (in tons) that has to be processed within week w ∈ W by mill p ∈ P . – M axP rocw,p - Upper limit of sugar cane (in tons) that can be processed by mill p ∈ P within week w ∈ W . – P F iberM inw,p - Minimum percentage of fiber that has to be processed by mill p ∈ P within week w ∈ W . – P rocCostw,p - Processing Cost (per ton of sugar cane) in mill p ∈ P within week w ∈ W . – V alueAT Rw,p - Value of one ton of ATR within week w ∈ W for mill p ∈ P . This is the price at which the sugar is presumably sold. – Coef AR - Coefficient of AR for the ATR calculation. – Coef P ol - Coefficient of Pol for the ATR calculation.
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51
Plantation fields and maturation data – P olw,f,m - Percentage of extracted sucrose from sugar cane in plantation field f ∈ F , that used the maturation product m ∈ M , within week w ∈ W. – ARw,f - Percentage of reduced sugar within the sugar cane from plantation field f ∈ F within week w ∈ W . – ARw,r - Percentage of reduced sugar within the sugar cane from third party supplier r ∈ R within week w ∈ W . – P F iberw,f - Percentage of fiber within the sugar cane variety in plantation field f ∈ F within week w ∈ W . – N umHAf - Size of plantation field f ∈ F (in hectares).
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– Qw,f - Productivity (of sugar cane in tons) of plantation field f ∈ F within week w ∈ W . – DistF ieldsf1 ,f2 - Distance (in km) between two plantation fields f1 ∈ F and f2 ∈ F . – Distf,p - Distance (in km) from plantation field f ∈ F to mill p ∈ P . – InitialAgef - Initial age (in weeks) of the sugar cane in plantation field f ∈ F at the beginning of the planning. – M inCutAgef,m - Minimum age (in weeks) of the sugar cane to be harvested at plantation field f ∈ F , treated by maturation product m ∈ M. – M axCutAgef,m - Maximum age (in weeks) up to which the sugar cane can be harvested at plantation field f ∈ F , treated by the maturation product m ∈ M .
Maturation products – Coef Reducm - Reduction factor of sugar cane productivity by applying maturation product m ∈ M on a field. – M inAgeApplf,m - Minimum age (in weeks) of the sugar cane at plantation field f ∈ F at which a maturation product m ∈ M can be applied. – CostApplf,m - Cost to apply maturation product m ∈ M on plantation field f ∈ F .
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52
Cutting crews – M axCutw,c - Upper limit of the cutting capacity (in tons) of cutting crew c ∈ CF within week w ∈ W . – M inCutw,c - Minimum quantity of sugar cane (in tons) that has to be cut by cutting crew c ∈ CF within week w ∈ W . – CutCostw,c - Cost to cut one ton of sugar cane by cutting crew c ∈ CF within week w ∈ W . – CF Availw,c - Availability (in hours) of cutting crew c ∈ CF within week w ∈ W . This data is computed as M axCutw,c · T imeCutw,c . – T imeCutw,c - Time (in hours) spent by cutting crew c ∈ CF to cut one ton of sugar cane within week w ∈ W .
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– DisplSpeedc - Displacement speed (in Km/h) of cutting crew c ∈ CF to travel from one field to another.
Transportation crews – T imeT ravelw,f,t - Time (in hours) spent by transportation crew t ∈ T F to move from any mill (average value) to plantation field f ∈ F within week w ∈ W . – CostT ransportw,f,t - Transportation cost per km for one ton of sugar cane from plantation field f ∈ F by transportation crew t ∈ T F within week w ∈ W . – N umEquipsw,t - Number of vehicles in transportation crew t ∈ T F within week w ∈ W . – CostEquipw,t - Cost to use one piece of equipment (usually one vehicle) of the transportation crew t ∈ T F within week w ∈ W .
Third party suppliers – P F iberr - Percentage of fiber within the sugar cane provided by third party supplier r ∈ R. – P olw,r - Average percentage of extracted sucrose from sugar cane provided by third party supplier r ∈ R within week w ∈ W . – CostSCr - Cost of one ton of sugar cane provided by third party supplier r ∈ R.
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53
Vinasse – M inV inw - Minimum quantity of vinasse (in hectares) that should be applied to the plantation fields within week w ∈ W , i.e. the minimum quantity (in hectares) that have to be harvested within this week in order to provide sufficient area for vinasse application. Others – P riorityAge - Priority of all constraints related to the cutting age of the sugar cane. – P riorityCrews - Priority of all constraints related to the cutting crews. – P riorityIndustrial - Priority of all constraints of the industrial category.
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– P riorityM anipulation - Priority of all constraints of the manipulation category. – P riorityM illw,f - Priority of the plantation field f ∈ F to be cut within week w ∈ W . – P riorityV inasse - Priority of all constraints related to the application of vinasse. The priorities are commonly valued between 0 and 100. Variables The variables used in this model will now be listed. In contrast to the model of the operational planning, this model extensively uses continuous variables to facilitate the representation of harvested, transported and processed sugar cane quantities. This type of formulation makes the influence of the use of maturation products on sugar cane quantity (see constraint (4.2.1)) easy to model, as well as facilitating the extension of the model to repeated field harvesting (see Section 4.2.2). – aw f,m ∈ <+ - quantity of cut sugar cane (in tons) within week w ∈ W in plantation field f ∈ F , using the maturation product m ∈ M . This quantity does not consider reduction of sugar cane caused by the application of the maturation product. – aw f ∈ <+ - quantity of total cut sugar cane (in tons) within week w ∈ W in plantation field f ∈ F . This quantity does not consider reduction of sugar cane caused by the application of the maturation product. w ∈ {0, 1} - variable that indicates whether the maturation product – kf,m m ∈ M is used in plantation field f ∈ F within week w ∈ W .
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– sw f,m,p ∈ <+ - quantity of sugar cane (in tons) from plantation field f ∈ F using the maturation product m ∈ M processed by mill p ∈ P within week w ∈ W . w – qr,p ∈ <+ - variable that indicates the quantity of sugar cane (in tons) acquired by the third party supplier r ∈ R to be processed in mill p ∈ P within week w ∈ W (bounded by the capacity of the third party).
– hw f,t,m,p ∈ <+ - quantity of sugar cane (in tons) produced by plantation field f ∈ F using the maturation product m ∈ M transported within week w ∈ W by the transportation crew t ∈ T F to mill p ∈ P .
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– xw f,t,m ∈ {0, 1} - variable that indicates whether a plantation field f ∈ F , using the maturation product m ∈ M , is associated to the transportation crew t ∈ T within week w ∈ W . A value of m = 0 indicates that no maturation product is applied. w – yf,c,m ∈ {0, 1} - variable that indicates whether plantation field f ∈ F , using the maturation product m ∈ M , is associated to the cutting crew c ∈ CF within week w ∈ W . A value of m = 0 indicates that no maturation product is applied. The variables of this type are bounded in the maximum value they can have in order to satisfy the minimum and maximum cutting age of the fields. w – zf,c,m ∈ <+ - quantity of sugar cane (in tons) from plantation field f ∈ F , using the maturation product m ∈ M , cut by the cutting crew c ∈ CF within week w ∈ W . This variable already considers the reduction of the sugar cane quantity caused by the application of the maturation product. w – slCutN onApplf,m ∈ <+ - slack variable for the constraint that forbids cutting a field beyond the valid time interval altered by the application of a maturation product.
– slCutObligf ∈ {0, 1} - slack variable that indicates whether the plantation field f ∈ F was not cut at any time within the planning. – slF ibw p ∈ <+ - slack variable for the constraint of the minimum fiber percentage within week w ∈ W in mill p ∈ P . – slM inCutw c ∈ <+ - slack variable for the constraint of minimum cutting capacity by the cutting crew f ∈ F C within week w ∈ W . – slM axCutw c - slack variable for the constraint of maximum cutting capacity by the cutting crew c ∈ CF within week w ∈ W . – slM inP rocw p ∈ <+ - slack variable for the constraint of minimum processing capacity in mill p ∈ P within week w ∈ W .
Chapter 4. Problem Formulation
55
– slM axP rocwp ∈ <+ - slack variable for the constraint of maximum processing capacity in mill p ∈ P within week w ∈ W .
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– slV inApplicw ∈ <+ - slack variable that indicates the quantity of hectares not harvested, but required for vinasse application within week w ∈ W.
56
Chapter 4. Problem Formulation
Objective Function The objective function for the tactical planning maximizes the total profit. The whole function is given by:
MAX
P P P P
(V alueAT Rw,p · (Coef P ol · P olw,f,m + Coef AR · ARw,f )
w∈W f ∈F m∈M p∈P anipulation) · max(1,P riorityM 100
P riorityM ill
w,f · − P rocCostw,p ) · sw f,m,p 100 P P P (V alueAT Rw,p · (Coef P ol · P olw,r + Coef AR · ARw,r ) +
p∈P w∈W r∈R
w −P rocCostw,p − CostSCr ) · qr,p P P P w CostApplf,m · kf,m − w∈W f ∈F m∈M P P P P P CostT ransportw,f,t · hw − t,f,m,p · Distf,p w∈W t∈T F f ∈F m∈M p∈P P P P P CostEquipw,t · N umEquipsw,t · xw − f,t,m w∈W t∈T F f ∈F m∈M P P P P w CutCostw,c · zf,c,m −
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w∈W c∈CF f ∈F m∈M
w −γ1 · slCutN onApplf,m · P riorityAge −γ2 · slCutObligf · P riorityAge −γ3 · slF ibw p · P riorityIndustrial −γ4 · slM axCutw c · P riorityCrews −γ5 · slM axP rocw p · P riorityIndustrial −γ6 · slvM inCutw c · P riorityCrews −γ7 · slM inP rocwp · P riorityIndustrial −γ8 · slV inApplicw · P riorityV inasse
The profit is composed of the sugar production, just as shown in the abstraction of the problem to a GAP in Section 4.1.3. The sugar is produced by processing sugar cane, shown within the first two lines, respectively, through harvesting sugar cane and through acquiring sugar cane from third parties. Both lines already subtract the processing costs. The following lines represent the cost of applying maturation products, transport costs (considering the distance between the field and the mill), the cost to hire the transportation crews and the costs to harvest the fields. Finally, the slack variables are listed as costs in the objective function in order to penalize constraint violations. γ1 , . . . , γ8 are weights that can adjust this penalization and are usually selected between 10 and 500.
57
Chapter 4. Problem Formulation
Constraints Field cutting obligations A plantation field f ∈ F must be cut once within the complete planning period. This constraint matches the assignment obligation constraint (4-7) in the abstraction of the problem to a GAP shown in Section 4.1.3.
X X X
w + slCutObligf = 1; ∀f ∈ F yf,c,m
(4-10)
c∈CF m∈M w∈W
Cane transportation within the same week
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All sugar cane cut at a plantation field f ∈ F within week w ∈ W must be transported within the same week. This constraint passes the previous cutting obligation to the binary processing variables.
X X
t∈T F m∈M
xw f,t,m =
X X
w ; ∀w ∈ W, ∀f ∈ F yf,c,m
(4-11)
c∈CF m∈M
Cutting capacities of the cutting crews A cutting crew c ∈ CF must cut at least M inCutw,c tons within week w ∈ W . These constraints match the capacity constraints for cutting crews within the GAP shown in Section 4.1.3.
XX
w + slM inCutw zf,c,m c ≥ M inCutc,w ; ∀w ∈ W, ∀c ∈ CF (4-12)
f ∈F m∈M
At the same time, the cutting crew can cut at most M axCutc,w tons within week w ∈ W .
XX
w − slM axCutw zf,c,m c ≤ M axCutc,w ; ∀w ∈ W, ∀c ∈ CF (4-13)
f ∈F m∈M
Minimum and maximum processing limits of a mill A mill p ∈ P must process at least M inP rocw,p tons within week w ∈ W . The sugar cane may be harvested in fields as well as be acquired from third
58
Chapter 4. Problem Formulation
party suppliers. These constraints match the capacity constraints for mills within the GAP shown in Section 4.1.3.
XX
sw,f,m,p +
f ∈F m∈M
X
w + slM inP rocw qr,p p ≥ M inP rocw,p ;
r∈R
∀w ∈ W, ∀p ∈ P
(4-14)
At the same time, the mill is limited to a maximum of M axP rocw,p processed tons of sugar cane within week w ∈ W .
XX
f ∈F m∈M
sw f,m,p +
X
w − slM axP rocw qr,p p ≤ M axP rocw,p ;
r∈R
∀w ∈ W, ∀p ∈ P
(4-15)
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Sugar cane reduction caused by maturation products This constraint performs the reduction of the quantity of sugar cane, caused by the application of a maturation product.
X
w = aw zf,c,m f,m · Coef Reducm ; ∀w ∈ W, ∀f ∈ F, ∀m ∈ M (4-16)
c∈CF
Maximum amount of sugar cane cut from a field The quantity of sugar cane cut by a cutting crew c ∈ CF from plantation field f ∈ F within week w ∈ W must be less than Qw,f , if the crew c is associated with field f within week w. Otherwise, this limit will be zero.
w w ; ∀w ∈ W, ∀c ∈ CF, ∀f ∈ F, ∀m ∈ M ≤ Qw,f · yf,c,m zf,c,m
(4-17)
Quantity of sugar cane transported from a field The quantity of sugar cane from field f ∈ F transported by crew t ∈ T F within week w ∈ W , using the maturation product m ∈ M , must equal the cane quantity at plantation field t within week w, using the maturation product m.
59
Chapter 4. Problem Formulation
XX
w hw f,t,m,p = af,m · Coef Reducm ;
t∈T F p∈P
∀w ∈ W, ∀f ∈ F, ∀m ∈ M
(4-18)
Maximum amount of sugar cane transported from a field The quantity of sugar cane from field f ∈ F transported by crew t ∈ T F , within week w ∈ W , must be less than Qw,f , if the crew f is associated with the plantation field t within week w. Otherwise, this limit will be zero.
X
w hw f,t,m,p ≤ Qw,f · xf,t,m ; ∀w ∈ W, ∀t ∈ T F, ∀f ∈ F, ∀m ∈ M (4-19)
p∈P
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Minimum age for application of a maturation product to a field A maturation product m ∈ M can only be applied to a plantation field f ∈ F within week w ∈ W if this field already possesses the minimum age required for the application of the maturation product.
w+M inAgeApplf,m0
X
m0 ∈M
0
X
w kf,m 0 ≤ 1 −
w0 =w+1
X X
yfw00 ,c,m00 ;
c00 ∈CF m00 ∈M
∀f ∈ F, ∀w ∈ W
(4-20)
Quantity of sugar cane cut from a field through the use of a maturation product The quantity cut sugar cane of from a plantation field f ∈ F , which used the maturation product m ∈ M , within week w ∈ W , must be non-zero only if field f , using the maturation product m, was harvested within week w.
aw,f,m = Qw,f ·
X
w ; ∀w ∈ W, ∀f ∈ F, ∀m ∈ M yf,c,m
(4-21)
c∈CF
Association between quantities of cut sugar cane The quantity of sugar cane cut at a plantation field f ∈ F within week w ∈ W must equal the total sugar cane using all maturation products.
60
Chapter 4. Problem Formulation
X
aw,f,m = aw,f ; ∀w ∈ W, ∀f ∈ F
(4-22)
m∈M
Quantity of sugar cane processed per week The quantity of sugar cane processed within week w ∈ W has to be equal to the quantity transported in that week.
sw f,m,p =
X
hw f,t,m,p ; ∀w ∈ W, ∀f ∈ F, ∀m ∈ M, ∀p ∈ P
(4-23)
t∈T F
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Minimum percentage of fiber The average fiber percentage within the sugar cane processed at mill p ∈ P must meet a certain minimum percentage demand within all weeks w ∈ W. P P F iberr · qrw + slF ibw P F iberw,f · sw p f,m,p + r∈R f ∈F m∈M P P w P w ≥ P F iberM inw,p ; sf,m,p + qr P P
f ∈F m∈M
r∈R
∀w ∈ W ∀p ∈ P
(4-24)
Relationship between maturation product application and field cutting If a maturation product m ∈ M was applied to a plantation field f ∈ F , this plantation field must be cut in one of the weeks within the planning period.
X X
∀w∈W ∀c∈CF
w yf,c,m =
X
w kf,m ; ∀f ∈ F, ∀m ∈ M
(4-25)
∀w∈W
Prohibition of cutting during non-valid weeks when using maturation products If a maturation product m ∈ M was applied to a plantation field f ∈ F within week w ∈ W , then this plantation field cannot be cut during the weeks outside the valid cutting interval [w+cuttingTimeMin, w+cuttingTimeMax] defined for each maturation product.
61
Chapter 4. Problem Formulation
w+cuttingT imeM in−1
X
X
c∈CF w0 =w+cuttingT imeM ax+1 w ≤ 1 − kf,m ; ∀w ∈ W, ∀f ∈
0
w w yf,c,m + slCutN onApplf,m
F, ∀m ∈ M
(4-26)
Vinasse application For each week w ∈ W , there may be a minimum number of hectares that have to be harvested from a set of eligible fields T V inw in order to permit the vinasse application.
X X
X
w + slV inApplicw ≥ M inV inw ; N umHAf · yf,c,m
c∈CF m∈M t∈T V in(s)
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∀w ∈ W
(4-27)
4.2.2 Enabling repeated field harvesting For long planning periods such as one year or more, it may be necessary to harvest a field more than once throughout the complete planning period. This section demonstrates how the model can be extended to permit repeated field harvesting. First, the cutting obligation constraint (4-10) must be modified in order to permit harvesting the same field more than once. At the same time, it must guarantee that each field is harvested at least once. This can be accomplished by replacing the equal sign by a greater or equal sign:
X X X
w + slCutObligf ≥ 1; ∀f ∈ F yf,c,m
(4-28)
c∈CF m∈M w∈W
Limit to a maximum of one weekly cut per field The number of selected cutting variables for a field must now be limited to one in each week, assuming that a field will be harvested no more than once a week.
X X
c∈CF m∈M
w ≤ 1; ∀f ∈ F ∀w ∈ W yf,c,m
(4-29)
62
Chapter 4. Problem Formulation
Network flow for the field’s sugar cane quantity and growth One of the crucial factors in a model that permits multiple harvesting of the same field is the quantity of sugar cane in a field. After each cut, the field begins with a quantity of zero sugar cane. A simple and elegant way to model this behavior is through the use of a network flow for the quantity of sugar cane in a field during each week. Let uw,f be the quantity of sugar cane (in tons) of field f within week w. The current proposed model links the quantity of sugar cane in field u to the quantity of sugar cane cw,f cut within a week. In the beginning of the planning, the flow must be inserted into one of these two variables, either to cut the field within the first week or to pass it to the next week:
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cw0 ,f + uw0 ,f = Qf,0 ; ∀f ∈ F
(4-30)
Once the sugar cane quantity flow for each field is inserted, it must pass to the next week, growing by a determined degree. For the sake of simplicity, it is assumed that the quantity of sugar cane in field f linearly grows by a constant degree within each week, denoted by growthRatef . The accumulated flow is then passed to the following week, where it is either conserved or used to cut the field:
growthRatef + uw−1,f = cw,f · uw,f ; ∀w ∈ W \{w0 ,w|W |−1 } , ∀f ∈ F
(4-31)
Note that this constraint is neither valid for the first week w0 nor for the last week of the planning w|W |−1 . The flow leaves the model in the last week, as there is no flow conservation constraint. Figure 4.1 illustrates the network flow for the quantity of sugar cane in field f . growthRate f
Q 0, f
u 0, f
growthRate f
u 1, f
growthRate f
u |W|-3, f
u 2, f
growthRate f
u |W|-2, f
... c 0, f
c 1, f
c 2, f
c |W|-2, f
c
|W|-1, f
Figure 4.1: Network flow for the cane quantity along the weeks, enabling the model to harvest a field more than once along the planning period
Chapter 4. Problem Formulation
63
4.3 Formulation for the operational module Next is presented, the mathematical formulation for the operational module as a MIP. First, modeling alternatives for this module are discussed and later the final model is presented. 4.3.1 Modeling alternatives
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Time discretization. The harvest sequence for the cutting crews makes the operational planning much more complex than the tactical planning. There are essentially two possibilities for a model of such a sequence: 1. Continuous time. The time is continuous and the moments at which the cutting crews’ harvest activities take place do not have to be exactly determined. However, the order of the fields when compared in time is important. Clearly, the representation of such an order results in a discrete set of decisions. 2. Time discretization. Each day is discretized into a certain number of time instants. As these instants follow a natural order within time, the harvest sequence is implicitly given. The constraints for a continuous approach, as outlined above, may significantly complicate the solution by a MIP solver since it requires several constraints to be realized. The model for the operational planning presented in this work is based on the discretization of time during each day. This approach incorporates the following disadvantage: the time to cut a field uncommonly equals exactly the time represented by a number of instants. For example, if the day is discretized into instants of two hours and cutting a field takes 100 minutes, the flow variable for this cut will occupy two whole instants. This leads to a practical waste of 20 minutes. This problem is diminished by discretization of a day into more instants. However, the greater the discretization, the larger the total problem. On the other hand, the problem size is easily configurable by modifying the discretization factor. Since the operational model does not influence the maturation product application planning, the quantity of sugar cane harvested from a certain field on a certain day remains constant and independent of the decisions made. Hence, the model for the operational module uses less continuous variables than the model for the tactical planning. As an example, consider the representation of a quantity of cut sugar cane by a cutting crew c from a field f in a day d. Let
Chapter 4. Problem Formulation
64
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Qd,f be the productivity of sugar cane (in tons) of field f within day d. The operational module models this quantity as Qd,f · yc,f,d , where y is a binary variable. The model for the tactical planning uses an additional continuous variable z to represent the quantity of cut sugar cane, limited by a constraint zc,f,d ≤ Qd,f · yc,f,d . A further consideration for the SCHP-OP model regards the possibly large number of travel variables. One possible way to reduce the number of variables is to join the cutting and travel variables into one and force traveling right after each cut. The disadvantage of this modeling alternative is that fields can only be entered if they are subject to being cut. In order to be able to represent all permutations of field sequences, a complete distance graph for the fields is necessary. However, a complete distance graph may result in a large number of variables, which may make the problem difficult to solve. Hence, the model presented in this work strictly separates cutting and traveling variables and uses pre-processing techniques to reduce the number of travel variables (see Section 5.3). 4.3.2 Mathematical model Input data For the mathematical formulation, the following input data was separated into the following topics: sets, mills, plantation fields and maturation data, maturation products, cutting crews, transportation crews and third party suppliers. Sets – D - Set of days considered in this planning horizon. – Df - Set of days in which a plantation field can be cut. These sets of days are important for the use of the variables x and y and are only used in this mathematical model. In the implementation, these sets of days will be realized by the creation of the variables x and y only for the set of days in which each plantation field can be cut. – CF - Set of existing cutting crews. This set is divided into a set CFman of manual cutting crews and a set CFmec of mechanical cutting crews. – I - Set of instants. All days are split into a total of |I| instants, numP artitions instants within each day. The operators + and − at the indices of the time instant are assumed to operate within the set
Chapter 4. Problem Formulation
65
of available instants of the cutting crew, i.e. i − 1 is the first available instant before i. – Id - Set of instants within day d ∈ D. – P - Set of existing mills. – R - Set of third party sugar cane suppliers – F - Set of existing plantation fields. – Fd - Set of plantation fields whose cut can be started within this day. These sets of days are important for the use of the variables x and y and are only used in this mathematical model. In the implementation, these sets of fields will be realized by the creation of the variables x and y only for the set of days in which each plantation field can be cut.
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– Vd - Set of all plantation fields on which vinasse can be applied within day d.
Mills and sugar production process – Coef AR - Coefficient of AR for the ATR calculation. – Coef P ol - Coefficient of Pol for the ATR calculation. – P rocCostp - Processing Cost (per ton) of mill p ∈ P (equal at all days). – V alueAT Rd,p - Value of one ton of ATR within day d ∈ D for mill p ∈ P . – M inP rocd,p - Minimum quantity of sugar cane (in tons) that has to be processed by mill p ∈ P within day d ∈ D. – M axP rocd,p - Upper limit of sugar cane (in tons) that can be processed by mill p ∈ P within day d ∈ D. – P F iberM ind,p - Minimum percentage of fiber that has to be processed by mill p ∈ P within day d ∈ D.
Plantation fields and maturation data – Qd,f - Productivity (of sugar cane in tons) of plantation field f ∈ F within day d ∈ D. – T Initialc - The plantation field at which the cutting crew c ∈ CF is located at the beginning of the planning. – DateP reAnalysisf - The date at which the pre-analysis for the field was performed. – P olP ref - Percentage of Pol in the pre-analysis.
Chapter 4. Problem Formulation
66
– P olP recM onthf - Percentage of Pol in the preceding month. – P olSuccM onthf - Percentage of Pol in the successive month. – ARP ref - Percentage of AR in the pre-analysis. – ARP recM onthf - Percentage of AR in the preceding month. – ARSuccM onthf - Percentage of AR in the successive month. – P old,f - Percentage of sucrose extracted from sugar cane from plantation field f ∈ F within day d ∈ D. – ARd,f - Percentage of reduced sugar in the sugar cane from plantation field f ∈ F within d ∈ D. – P F iberd,f - Percentage of fiber within the sugar cane variety from plantation field f ∈ F within d ∈ D. – N umHAf - Size of plantation field f ∈ F (in hectares).
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– Distf,p - Distance (in km) from plantation field f ∈ F to mill p ∈ P . – Distf1 ,f2 - Distance (in km) between two plantation fields f1 ∈ F and f2 ∈ F . The maturation curves for the sugar cane P old,f and ARd,f are based on a linear function. The percentage of Pol in the sugar cane within each day is computed by the following formula: P old,f = P olP ref + P olP recM onthf −P olSuccM onthf (DateP reAnalysisf − InitialDate) · 30 The AR percentage for each day is computed through an analogous manner: ARd,f = ARP ref + ARP recM onthf −ARSuccM onthf (DateARAnalysisf − InitialDate) · 30 Maturation products – M atGainf - The factor by which the Pol value in the sugar cane (given in percentage) from plantation field f ∈ F is multiplied (and thereby increased or decreased) if a maturation product has been applied. – M atReducF actorf - The reduction factor of the productivity of plantation field f ∈ F if a maturation product has been applied.
Chapter 4. Problem Formulation
67
Cutting crews – CuttingCostd,c - Cost to cut one ton of sugar cane by cutting crew c ∈ CF within day d ∈ D. – CuttingT imed,c - Time (in hours) spent by cutting crew c ∈ CF to cut one ton of sugar cane within day d ∈ D. – T ravelCostCFd,f - Displacement cost (per Km) of cutting crew c ∈ CF within day d ∈ D. – Speedd,c - Displacement speed (in Km/h) of cutting crew c ∈ CF within day d ∈ D. – T imeDispCFd,f - Availability (in hours) of cutting crew c ∈ CF within day d ∈ D. Currently calculated as M axCutd,f ∗ CuttingT imed,c .
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– M inCutd,c - Minimum quantity of sugar cane (in tons) that has to be cut by cutting crew c ∈ CF within day d ∈ D. – M axCutd,c - Upper limit of the capacity (in tons) that can be cut by cutting crew c ∈ CF within day d ∈ D.
Third party suppliers – T hirdP artyCapacr - Capacity of sugar cane (in tons) that the third party supplier r can provide per day (Unit: tons / day). – T hirdP artyCostr - Cost of one ton of sugar cane provided by third party supplier r ∈ R. – P old,r - Average percentage of sucrose extracted from sugar cane provided by third party supplier r ∈ R within day d ∈ D. – ARd,r - Percentage of reduced sugar in the sugar cane of third party supplier r ∈ R within d ∈ D. – P F iberr - Percentage of fiber within the sugar cane variety provided by third party supplier r ∈ R.
Vinasse – M inV ind - Minimum quantity of vinasse (in hectares) that should be applied to the plantation fields within day d ∈ D.
Chapter 4. Problem Formulation
68
Others – InitialDate - The initial date of the planning. – T ransportCost - Medium cost to transport one Kg of sugar cane one km from the plantation fields to the mills (Unit: $ / ton / km). – N umP artitions - Number of partitions in which each day d ∈ D will be divided to form several instants i ∈ I. The following priorities are commonly valued between 0 and 100. – P riorityAge - Priority of all restrictions related to the cutting age of the sugar cane. – P riorityCrews - Priority of all restrictions related to the cutting crews. – P riorityIndustrial - Priority of all restrictions of the industrial category.
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– P riorityV inasse - Priority of all restrictions related to the application of vinasse. The following sets are not directly provided by the input data. The data is computed in a pre-processing step in order to simplify the formulation and implementation. i – δc,f - Number of instants that cutting crew c ∈ CF will need in order to cut the plantation field f ∈ F , considering that it starts cutting at instant i ∈ I. Operations + and − at the indices of the time instant are considered to operate within the set of available instants of the cutting crew. In order to determine the number of instants necessary to cut a field, the minimum between the cutting crew’s cutting rate CuttingT ime and it’s maximum cutting capacity M axCut is considered.
– λi,d c,f - Quantity (in tons) of sugar cane cut within day d by cutting crew c ∈ CF at plantation field f ∈ F , considering that the cutting crew began cutting at instant i ∈ I. Operations + and − at the indices of the time instant are considered to operate within the set of the available instants of the cutting crew. i - Number of instants that cutting crew c ∈ CF will need to travel – θc,f 1 ,f2 from plantation field f1 to f2 ∈ F , starting the journey at instant i ∈ I. Operations + and − at the indices of the time instant are considered to operate within the set of the available instants of the cutting crew.
Chapter 4. Problem Formulation
69
Variables The following lists the variables used within this model. – f hdc,f ∈ {0, 1} - Binary variable that indicates whether cutting crew c ∈ CF leaves home at day d ∈ D to start its work at plantation field f ∈ F at the first available instant of the day. – f pc,f,p ∈ {0, 1} - Binary variable that indicates whether cutting crew c ∈ CF leaves mill p ∈ P and begins its activities at field f ∈ F .
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i1 ,i2 – oc,f ∈ {0, 1} - Variable that indicates whether cutting crew c ∈ CF 1 ,f2 will leave plantation field f1 ∈ F at instant i1 ∈ I to arrive at plantation field f2 ∈ F at instant i2 ∈ I. This notation is equal to oic,f1 ,f2 , where the instant at which the cutting crew arrives at plantation field f2 is i i + θc,f . 1 ,f2 d – qr,p ∈ <+ - Variable that indicates the quantity of sugar cane acquired by third party supplier r ∈ R to be processed in mill p ∈ P within day d ∈ D. The variable is limited by the value T hirdP artyCapacr .
– nic,f ∈ {0, 1} - Variable that indicates whether cutting crew f ∈ F waits one instant at plantation field f ∈ F (from instant i to instant i + 1). – xf,p ∈ {0, 1} - Binary variable that indicates whether the sugar cane from plantation field f ∈ F will be processed in mill p ∈ P . i1 ,i2 – yc,f ∈ {0, 1} - Binary variable that indicates whether cutting crew c ∈ CF starts cutting the field f ∈ F in time instant i1 ∈ I and is i available again at instant i2 . This notation is equal to yc,f , where the i cutting crew will be available again at instant i + δc,f . d – zf,p ∈ <+ - Variable that indicates the quantity of sugar cane from plantation field f ∈ F that will be processed in mill p ∈ P within day d ∈ D.
– slCF Dispdc ∈ <+ - Slack variable which indicates that the availability of cutting crew c ∈ CF was violated. – slM inCutdc ∈ <+ - slack variable for the constraint of minimum cutting limit by cutting crew c ∈ CF within day d ∈ D. – slM axCutdc ∈ <+ - slack variable for the constraint of maximum cutting limit by cutting crew c ∈ CF within day d ∈ D. – slM axP rocdp ∈ <+ - slack variable for the constraint of maximum processing at mill p ∈ P within day d ∈ D. – slM inP rocdp ∈ <+ - slack variable for the constraint of minimum processing at mill p ∈ P within day d ∈ D.
Chapter 4. Problem Formulation
70
– slCutObligf ∈ <+ - slack variable for the constraint which indicates that plantation field f ∈ F was not cut at any time during the planning period. – slV inApplicd ∈ <+ - slack variable which indicates that the vinasse constraint was not satisfied within day d ∈ D. Objective Function The objective function for the formulation aims to maximize the total profit:
MAX
P P P
f ∈F d∈Df p∈P
d zf,p · M atReducF actorf ·
V alueAT Rd,p · (Coef P ol · (P old,f + M atGainf ) + Coef AR · ARd,f ) P P P d zf,p · M atReducF actorf · − PUC-Rio - Certificação Digital Nº 0711327/CA
f ∈F d∈Df p∈P
(P rocCostp + T ransportCostd,f · Distf,p ) P P P d (V alueAT Rd,p · (Coef P ol · P old,r ) · qr,p + p∈P d∈D r∈R P P P d (P rocCostp + T hirdP artyCostr ) · qr,p − p∈P d∈D r∈R P P P i CuttingCostd,c · Qt,d · M atReducF actorf · yc,f − i∈I c∈F C f ∈F P P P P i T ravelCostCFb − i ,f · Distf1 ,f2 · oc,f1 ,f2 N umP artitions c t ∈T t ∈T i∈I f ∈CF 1
2
−γ1 · slCF Dispdc · P riorityCrew −γ2 · slM axP rocdp · P riorityIndustrial −γ3 · slM inP rocdp · P riorityIndustrial −γ4 · slCutObligf · P riorityAge −γ5 · slM axCutdc · P riorityCrews −γ6 · slM inCutdc · P riorityCrews −γ7 · slV inApplicd · P riorityV inasse As in the model for the tactical planning, the profit is presented by the produced sugar, just as shown in the problem abstraction to a GAP in Section 4.1.3. The sugar is produced by processing the sugar cane harvested in fields (line one) and acquired from third parties (line three). The calculation for the processing of harvested sugar cane involves the reduction factor for the total cane mass when a maturation product is applied. It also adds in the increase of the Pol value caused by the product. Line two covers the processing and transportation costs for harvested sugar cane. Line four to six represent acquiring and processing costs for third party sugar cane, harvesting costs of the cutting crews and travel costs for the cutting crews. Finally, the
Chapter 4. Problem Formulation
71
slack variables are listed as costs in the objective function in order to penalize constraint violation. γ1 , . . . , γ7 are weights to adjust this penalization, usually chosen between 10 and 500. Constraints The following presents the model’s constraints. Plantation field cutting obligation A plantation field f ∈ F must be cut at least once during the planning period. This constraint is not essential to the model for the operational model, because it may not be possible to harvest all fields within the available time. In this case, the penalization costs for this constraint have to be low.
XX
i + slCutObligf = 1; ∀f ∈ F yc,f
(4-32)
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c∈CF i∈I
Minimum and maximum processing limits of a mill The quantity of sugar cane processed within day d ∈ D by mill p ∈ P must be at least M inP rocd,p . These constraints match the capacity constraints for mills within the GAP shown in Section 4.1.3.
X
d zf,p +
f ∈Fd
X
d qr,p + slM inP rocdp ≥ M inP rocd,p ;
r∈R
∀d ∈ D, ∀p ∈ P
(4-33)
Just as the processing minimum, there is a processing upper limit. The quantity of sugar cane processed during day d ∈ D by mill p ∈ P must be no more than M axP rocd,p .
X
f ∈Fd
d + zf,p
X
d − slM axP rocdp ≤ M axP rocd,p ; qr,p
r∈R
∀d ∈ D, ∀p ∈ P
(4-34)
Minimum percentage of fiber Each mill p ∈ P demands a certain average percentage of fiber from the processed sugar cane within each day d ∈ D. The sugar cane cut on these
72
Chapter 4. Problem Formulation
days and the sugar cane acquired by third party suppliers must meet these minimum values.
P d d P F iberr · qr,p + slF ibdp zf,p · P F iberd,f + r∈R f ∈Fd P d P d ≥ P F iberM ind,p ; qr,p zf,p + P
f ∈Fd
r∈R
∀d ∈ D, ∀p ∈ P
(4-35)
Cutting capacities of the cutting crews
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The quantity of sugar cane cut by a cutting crew c ∈ CF within a day d ∈ D must be at least M inCutd,c . These constraints match the capacity constraints for cutting crews within the GAP shown in Section 4.1.3.
XX
d i · λi,d yc,f c,f + slM inCutc ≥ M inCutd,c ;
f ∈Fd i∈I
∀d ∈ D, ∀c ∈ CF
(4-36)
At the same time, the quantity of sugar cane cut by cutting crew c ∈ CF within a day d ∈ D is limited to no more than M axCutd,c .
XX
i d yc,f · λi,d c,f − slM axCutc ≤ M axCutd,c ;
f ∈Fd i∈I
∀d ∈ D, ∀c ∈ CF
(4-37)
Time availability of the cutting crew The time a cutting crew c ∈ CF spends (partially or entirely) cutting field f within day d and the time spent by that crew traveling to other fields must not exceed its daily availability T imeDispd,c . If a cutting crew starts traveling from one field to another within a day d, the entire traveling time will be subtracted from the crew’s availability on that day d.
XX
i · λi,d yc,f c,f · CuttingT imed,c +
f ∈Fd i∈I
X
X X
oic,f1 ,f2 · Distf1 ,f2 · Speedd,c
i∈I at day d f1 ∈Fd f2 ∈Fd
−slCF Dispdc ≤ T imeDispd,c ; ∀d ∈ D, ∀c ∈ CF
(4-38)
73
Chapter 4. Problem Formulation
Vinasse Application In order to guarantee a sufficient area wherein to apply the vinasse, minimum quantities of cut fields (in hectares) may be defined for each day d ∈ D. The field cuts must finish by day d and the fields f ∈ F must be eligible for vinasse application, i.e. f ∈ Vd .
X XX
i,id N umHAf · yc,f + slV inApplicd ≥ M inV ind ; ∀d ∈ D (4-39)
f ∈Vd c∈CF i∈I
Association between field cutting and processing
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If sugar cane from plantation field f is processed in a mill, then the cutting variable must be non-zero for this field for any cutting crew.
XX
i = yc,f
c∈CF i∈I
X
xf,p ; ∀f ∈ F
(4-40)
p∈P
Quantity of cut sugar cane This constraint sums up the quantity of sugar cane cut by all cutting crews within each day and field.
X p∈P
d zf,p ≤
XX
i λi,d c,f · yc,f ; , ∀d ∈ D ∀f ∈ F
(4-41)
c∈CF i∈I
Let maxP roductionf be the maximum productivity of field f throughout all planning days. The following constraint allows the quantity of sugar cane cut from field f and processed at mill p to be non-zero only if the sugar cane is processed at this mill.
X
d zf,p = maxP roductionf · xf,p ; , ∀f ∈ F ∀p ∈ P
(4-42)
d∈D
Each cutting crew possesses its own network that is independent of the networks of other cutting crews. Each network flows through time (i.e. all
74
Chapter 4. Problem Formulation
instants) and the fields that the cutting crew may cut. The network flows are handled by the following constraints: Initial cutting crew position at a mill In the beginning of the planning, each cutting crew c ∈ CF is located either at a plantation field or at a sugar cane mill. If the cutting crew is located at a mill, it must travel to one of its valid fields before the first instant of the planning. Let CFp be the set of cutting crews that are located at a mill. The mill at which c is located is denoted as pc . The flow must be inserted into one of the variables f p for the mill pc .
X
f pc,f,pc = 1 ∀c ∈ CFp
(4-43)
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f ∈F
Figure 4.2 illustrates the initial network flow at a mill. p1 . . . pn ∈ P represent all existing mills. iF irstc,f is the node at the first available instant of cutting crew c for field f . The flow starts at mill pf , from where it is passed to one of the eligible fields fβ1 . . . fβk . p1 ... 1
pf
fpc, fß 1, pf
iFirst c , f ß 1
fpc, fß 2, pf fpc, fß 3 , pf
...
iFirst c , f ß 2
iFirst c , f ß 3
fpc, fß k-1, pf ...
pn-1
fpc, fß , pf k
pn
iFirst c , f ß k-1
iFirst c , f ß k
Figure 4.2: Network flow starting at mill pf
Initial cutting crew position at a field and passed flow from a mill The flow for each cutting crew c ∈ CF must pass to the node of the first available instant iF irstc of the cutting crew. The flow can then be used to cut
75
Chapter 4. Problem Formulation
a field, wait at the field or move to another field. Let f Initialc be the location of c at the beginning of the planning. If the cutting crew is located at a plantation field, the flow (i.e. a flow of one unit) start at the first node of the first available instant for this field. For the nodes of the first available instants at other fields, the starting flow is zero. If the cutting crew is initially located at mill pc , then the flow from the corresponding variables from mills f p passes into the first available node of each field:
iF irstc irstc yc,f + niF + c,f
X
irstc oiF c,f,f2
f2 ∈F
1, if t = tInitialc and f Initial is a planting f ield = f pc,f,pc , if f Initialc is mill uf 0, otherwise.
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∀c ∈ CF
∀f ∈ F
(4-44)
Figure 4.3 illustrates the initial network flow for cutting crews that are initially located at a field. y iFirst c , f 1
0
n
... y iFirst c , fInitial c
1
n
o c, fInitialc , fn
... y 0
iFirst c , f n
n
n
Figure 4.3: Starting node of the network flow at the first instant at field f Initialc
General flow conservation - cutting, waiting and traveling Once the flow entered the network, it must pass along time. The flow that enters a node at instant i must also leave it. Figure 4.4 illustrates the general case of this flow conservation. Flow can enter from cutting variables for the field1 , started at different earlier instants iβ1 . . . iβk . Flow also can enter from a waiting variable at the previous instant or from one of the other fields 1
If the cutting rate of the crews varies with the days, it may be possible that more than one cutting variable for a field terminates in the same node
76
Chapter 4. Problem Formulation
fµ1 . . . fµm by traveling. If flow enters the node, it leave it again by cutting the field (the flow will then enter into the node at instant ie , which represents the first available instant after the field cutting is finished), waiting one instant or moving to another eligible field fa1 . . . faj .
y ci , f, i ... y ci , f, i ß1
y ci ,, if
e
ßk
,i n ci-1 ,f
i c, f
o ci ,, if,Ωf a 1 ... o ci ,, if,Ω f
o c,iπ f,µi , f ... oc,iπ f,µi , f 1
1
1
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n ci,,i+1 f
g
h
aj
m
Figure 4.4: Flow conservation at instant node i at field f of cutting crew c In the flow conservation constraints, one must distinguish between mechanical and manual cutting crews. Mechanical cutting crews remain on the field during the night, whereas manual crews return to a place where they spend the night. The general flow conservation explained above is valid for all available instants of the mechanical cutting crews and all instants except the first instant of each day of the manual cutting crews. Let IFcM an be a set with all instants except the first available instant of each day, for each manual cutting crew cM an. Let IcM ec be a set with all available instants of all days, available for each mechanical cutting crew. The constraint that represents the general flow conservation is given by equation (4-45). Note that the operations ’+’ and ’-’ within the instants operate only within the set of available instants of a cutting crew, i.e. unavailable instants (instants at which the crew does not work) are skipped. Figure 4.5 exemplifies a network flow for a mechanical cutting crew. The nodes in the grey area are not available for work. Hence, all flow variables skip the nodes of unavailable instants.
i−δ i
yc,f c,f + ni−1 c,f +
X
f1 ∈F
i−θi
i 1 ,f ) + nic,f + oc,f1(c,f = yc,f ,f
X
f2 ∈F
oic,f,f2
77
Chapter 4. Problem Formulation
∀i ∈ IFcM an , IcM ec \{i0 }, ∀f ∈ F , ∀c ∈ CF
(4-45)
y n
n
n
n
n
Day 1
Day 2
Available instants
Not available instants
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Figure 4.5: Example network flow for a mechanical cutting crew: nodes of not available instants are skipped Manual cutting crews are usually hosted at a place where they spend the night at the end of the day, subsequently referred to as the crew’s home. The next day, they will be able to work at any field (eligible for these crews) independent of the distance. The flow of all activities that end at the last instant of a day, i.e. those of which the next available instant is the first instant of the next day, enters into a variable that indicates that the crew goes home. This variable must then enter into the first available instant of the next day of all fields. Figure 4.6 shows an example of a network for a manual cutting crew. Cuttings and waitings that would enter into the first available instant of a day, now enter home node. From home, the flow passes to any field that the cutting crew can cut. As stated before, equation (4-45) is valid for the nodes of manual cutting crews as well, excepted the ones valid for the first available instants of each day. The flow that would have entered in these first nodes of each day, now enter into the node of the crew’s home. This is guaranteed by equation (4-46), where Ic represents a set with the first available instants of all days of the cutting crew c. Note that this constraint is not generated for the node of the very first available instant of the entire planning.
X
i−δ i−1 +1
(yc,f c,f
+ ni−1 c,f +
f ∈F
∀i ∈ Ic , ∀c ∈ CF
X
f1 ∈F
i−θi−1
+1
1 ,f ) oc,f1(c,f )= ,f
X
f hdc,fi
f ∈F
(4-46)
78
Chapter 4. Problem Formulation
fh fh
y
y
n
n
Field 1
n
n
n
n
y
y
n
n
Field 2
n
n
n
Day 1
Day 2
Available instants PUC-Rio - Certificação Digital Nº 0711327/CA
n
Not available instants
Figure 4.6: Example network flow for a manual cutting crew: the crew goes home at the end of the day and moves to any field before the next day Finally, equation 4-47 distributes the flow from the home node to all plantation fields for each cutting crew.
i f hdc,fi = yc,f + nic,f +
X
oic,f,f2
f2 ∈F
∀i ∈ Ic \{i0 }, ∀f ∈ F , ∀c ∈ CF
(4-47)
If a cutting crew is located at a plantation field, then this crew is forced to start harvesting the field in the first available instant before proceeding to other fields. In the implementation of the model, the field harvesting is forced by constraint (4-44), not generating waiting or travel variables for the first available instant.
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5 Solution Strategies This chapter deals with the solution methods applied to the tactical and operational modules. First of all, preliminaries related to the presented solution methods and performed experiments are given. Then, the instances used to evaluate all presented methods are explained. The following section introduces into the extensive pre-processing procedures performed in order to reduce the size of the problem. Afterwards, the solution approaches for the tactical and operational module are discussed. The solution methods for the operational module are divided into exact and alternative solution approaches. The first part focuses on the solution of the original problem, including tools to improve the solutions such as initial starting solutions and valid inequalities. The second part regards solution methods based on segregation as well as heuristic attempts. Preliminary notes Throughout this section, the reader shall consider some important facts related to the presented solution methods and the experimental results: 1. Focus on the operational planning. As the operational module turned out to be more difficult than the tactical module, the research on solution strategies is focused on the operational module. Unless otherwise noted, all experiments and results presented in this section refer to the operational module. 2. Computational resources and Compiler version. All implementations were compiled with Visual Studio 2008, using Microsoft Windows Vista 32bit. All experiments were carried out on a Personal Computer with a Intel(R) Core(TM)2 Duo 2.33 GHz CPU and 2 GByte memory. 3. Upper bounds considered in calculation of deviation from optimum. Most of the results for computational experiments use the final solutions’ deviation from the best known solution to compare the solutions’ quality of each approach. However, the execution of an optimization process for different approaches may result in different upper bounds for the same
Chapter 5. Solution Strategies
80
instance. In order to guarantee an equitable comparison, the calculation of the final deviation value is based on the same upper bound1 for all approaches. Solver Configuration Contemporary MIP solver such as ILOG CPLEX offer numerous possibilities to configure the algorithms involved within the optimization process. Throughout the experiments of this work, ILOG CPLEX version 11.2 was used with the following configuration: – The Barrier method with Crossover (parameter CPX ALG BARRIER) is used to solve the linear relaxation of the MIP problem. The primal interior-point method (parameter CPX ALG PRIMAL) was tested, but did not perform well on the instances.
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– The MIP emphasis is set to CPX MIPEMPHASIS HIDDENFEAS, i.e. the solver strongly invests in finding high quality feasible solutions that are otherwise very difficult to find. This option performed best among the ones tested. – The upper limit on the number of cutting plane passes (parameter CPX PARAM CUTPASS) for the solution at the root node is set to three. Experiments showed that, without such upper limit, CPLEX performed up to ten iterations, but only slightly improved the upper bounds in the last iterations. However, the most effective iterations seemed to be the first ones. For large instances, a number of iterations greater than three was not feasible, because these further iterations spent too much execution time. – The branching phase is terminated when a relative deviation (from the best known solution) lower than 0.01% is found (parameter CPX PARAM EPGAP). However, this limit is not valid for the polishing phase. – The number of MIP solutions to be found before aborting the optimization process (parameter CPX PARAM INTSOLLIM) was set to positive infinity. – Unless otherwise noted, all experiments carried out in this work include a 15 minutes branching phase followed by a 15 minutes polishing phase. 1
In general, this upper bound is the lowest bound found throughout all experiments
Chapter 5. Solution Strategies
81
Problem configuration for the SCHP-OP in computational experiments
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There are several configurations that can be made in order to influence the input data and the computational experiments. Unless otherwise noted, the following configurations are valid for all experiments for the operational planning presented in this chapter: – Discretization. Each day within the operational planning is discretized into a number of time instants. The larger the number of instants per day, the closer the representation to continuous time, but also the bigger the problem. In contrast, a small number of instants per day may keep the problem small and easy to solve. However, in this case, the model may not be able to represent good quality solutions that are feasible in practice, i.e. solution that would work in practice cannot be represented by the model due to the time discretization. Experiments showed that a reasonable tradeoff includes twelve instants per day, i.e. a discretization into two hours per time instant. This configuration is assumed for all presented experiments. – Distance filtering. There are many ways to decrease the number of distances considered in the model in order to guarantee a reasonable problem size. These techniques are evaluated in Section 5.3.1. All experiments presented after the distance filtering techniques assume the filtering technique that obtained one of the best results: the use of the minimum spanning tree of the distance graph plus node balancing (to be introduced in the next section) with up to 100,000 travel variables. – Initial solutions. Initial solutions are generally passed to the solver before starting the optimization of the cutting crews’ routes. The strategy HeurRandSeq (see Section 5.4.2) turned out to be the most effective when passing its solutions to the solver. By default, this strategy is used in all further experiments. 5.1 Instances for the computational experiments An important issue in the solution of industrial problems is the evaluation of the solution methods based on adequate test data. The algorithms for both modules were tested with instances from practice. As the operational module turned out to be very hard to solve, there are many more instances available for this module, instances from practice as well as artificial instances. The instances from practice were provided by the Grupo Virgolino de Oliveira (GVO), a large Brazilian sugarcane producer. The GVO owns four
Chapter 5. Solution Strategies
82
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industrial units in the state of S˜ao Paulo. Monthly, about 1600 plantation fields are harvested, representing 1.3 million tons of sugar cane cut by almost 40 cutting crews. The mills possess a total processing capacity of 60,000 tons per day, equivalent to about 10 million tons per year. The manual cutting crews cut about 1.600 tons of sugar cane daily, whereas mechanical cutting crews cut 2.300 tons per day. Approximately 50% of the processed sugar cane comes from proper harvesting and therefore has potential to be optimized. The instances provided by the GVO have already been pre-processed in relation to the grouping of fields (see Section 5.3.3), i.e. each field within the input data represents a group of several fields. Indicators were elaborated that characterize crucial instance properties in order show the size and difficulty of an instance. At first, these indicators are introduced. Afterwards, the instances for the tactical and the operational module are discussed. 5.1.1 Indicators for the level of difficulty Three indicators are used to measure the difficulty of the instances. They are mainly based on the capacities of the cutting crews, the number of fields that the cutting crews can cut and the processing demands of the sugar cane mills: – Cutting crew field intersection. Each cutting crew owns its individual set of fields that it potentially can cut. These sets of fields may be disjoint, may intersect or even be equal. The larger the sets, the larger the problem size. Furthermore, the smaller the intersection between the sets, the easier the assignment of the fields to the cutting crews. Hence, one may conclude that the problem gets more difficult with higher intersection. Let f ieldSetc be the set of fields that cutting crew c can cut. The indicator for the cutting crews’ field intersection IndF I is defined as: |F | IndF I = (1.0 − P ) · 100 ∀c∈CF |f ieldSetc | The indicator is a value between 0 and 100. The higher the value, the larger the intersection and the larger the problem size. Clearly, the number of cutting crews influences the maximum value for this indicator. For two cutting crews with complete field intersection, the indicator has the value 50. Three cutting crews with complete field intersection lead to a value of 33.33, four cutting crews with complete field intersection result in a value of 25 and so on. The minimum value for this indicator is
83
Chapter 5. Solution Strategies
always 0 when the field sets for all cutting crews (no matter how many cutting crews there are) are completely disjoint. – Cutting crew occupation rate. There may be more plantation fields than can be cut by the cutting crews. The relation between the total number of tons of all fields and the total cutting capacity of the cutting crews reveal information about the difficulty of the problem. The indicator IndCO is defined as the relation between the sum of the daily average productivity of all fields and the daily average cutting capacity of the cutting crews: P
∀f ∈F
IndCO = P
∀c∈CF
P
Qd,f · 100 ∀d∈D M axCutd,c ∀d∈D
P
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If the indicator is greater than 100, then it may not be possible to cut all fields. However, a value lower than 100 does not guarantee that all fields can be cut in the best planning possible. – MinProc satisfiability. The minimum processing demands of the sugar cane mills are one of the most important constraints in the practice of sugar cane harvesting. There are strong efforts to avoid the violation of such constraints through the use of high constraint penalization costs. However, the quantity of sugar cane that can be cut is limited by the fields’ total productivity and the cutting crews’ capacities. The relation between the mills’ processing demands and the fields’ total productivity or cutting crews’ cutting capacities estimates the feasibility of the mills’ processing demands. This indicator, denoted as IndM P , is defined as follows: P P ∀p∈P ∀d∈D M inP rocd,p IndM P = · 100 maxP rod where maxP rod = min(
X X
∀f ∈F ∀d∈D
Qd,f ,
X X
M axCutd,c )
∀c∈CF ∀d∈D
If the indicator is greater than 100, then it may not be possible to satisfy the processing minimum demands of the mills. Experiments showed that the problem gets more difficult as the indicator increases. 5.1.2 Instances for the SCHP-TP The GVO provided 14 instances for the tactical module. The number of fields reaches from 69 to 1155. As the GVO performs the planning in
Chapter 5. Solution Strategies
84
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monthly instead of weekly temporal units, the harvest decisions are made for each month. The provided instances contain a planning horizon of between four and ten months. The provided instances neither contain tight minimum processing demands nor tight occupation rates of the cutting crews. The principal objective for these instances is the determination of the optimal period to harvest the field. Based on these 14 instances, further nine instances were generated by increasing the processing demands on the sugar cane mills or decreasing the cutting crew cutting capacities in order to increase their occupation rate. Table 5.1 presents all instances and its properties. The instance sets GVO1, GVO2, GVO3 and GVO10 contain more than one instance, each with a different minimum processing demand index ind M P . Instance set GVO7 contains two instances with different cutting crew capacities. Some of the instances also contain cutting crew minimum cutting demands and make use of maturation products.
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GVO1 1 GVO1 2 GVO1 3 GVO2 1 GVO2 2 GVO2 3 GVO3 1 GVO3 2 GVO3 3 GVO4 GVO5 GVO6 GVO7 1 GVO7 2 GVO8 GVO9 GVO10 1 GVO10 2 GVO10 3 GVO11 GVO12 GVO13 GVO14
# # Months Mills 4 4 4 4 4 4 4 4 4 4 10 9 9 9 6 9 8 8 8 8 8 8 8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1
# # # Fields CFs TFs 69 69 69 261 261 261 261 261 261 261 402 419 456 456 511 667 1155 1155 1155 1155 853 466 463
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 16 16 16 16 16 5 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1
# # ind CO ind M P # CFs with # # Matur. Distances minCut Var Const Products demand 0 4342 31.57 0 3 3375 2988 0 4342 31.57 46.24 3 3375 2988 0 4342 31.57 92.49 3 3375 2988 0 25407 31.57 0 3 11047 10725 0 25407 31.57 46.24 0 11047 10725 0 25407 31.57 92.49 0 11047 10725 0 42323 31.57 0 3 10523 10143 0 42323 31.57 98.66 3 10523 10143 0 42323 31.57 104.82 0 10523 10143 0 25407 126.28 0 3 10471 10092 0 86302 29.49 106.39 5 31890 28246 0 64497 37.28 97.23 5 26550 28473 0 77269 37.89 95.78 5 26299 27676 0 77269 54.84 95.78 5 26299 27676 0 123076 30.33 0 3 24979 27183 0 181402 16.24 0 0 47678 52780 1 181402 11.38 0 0 103365 96392 1 181402 11.38 76.53 0 103365 96392 1 181402 11.38 91.3 0 103365 96392 1 534514 11.38 0 0 80865 73135 0 181397 1.11 99.9 0 32245 30760 0 66497 8.16 96.71 4 22022 22936 0 54130 6.2 99.31 0 20813 21336
85
Table 5.1: Instances for the SCHP-TP
Chapter 5. Solution Strategies
Instance
Chapter 5. Solution Strategies
86
5.1.3 Instances for the SCHP-OP The algorithms presented for the operational module were tested with instances from practice as well as with artificial instances. These instance sets are now described. Detailed properties for each instance can be found in Appendix B.
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5.1.3.1 Instances from practice The GVO provided four instance sets with a total of 25 instances. The first instance within each set is referred to as the instance seed. All other instances within the same set are variations of this seed and include modifications of certain properties. The sets correspond to different moments in the harvest season and usually contain the fields of three months of the tactical planning’s output (Section 2.1.1 explains the planning practiced at the GVO). The first set contains instances with a large number of fields, the second and third sets tend to contain a medium number of fields and the last set tends to contain a low number of fields. Instance set GVO100. The instances from this set contain two sugar cane mills, Mon¸c˜oes and Bonifacio, for a total planning of 16 days. The first instance contains 233 fields, the remaining ones contain 334 fields. All instances contain 16 manual and five mechanical cutting crews. The field intersection index IndF I is fairly low, i.e. the field sets for the cutting crews are almost disjoint. The cutting crew occupation rate IndCO is high, 74% in the first instance, 550% in the remaining instances. Clearly, even in the optimal solution not all fields can be cut. The indicator for the processing demand IndM P is very tight for the first instance and zero for the other ones. Minimum and maximum capacities for the cutting crews are tight for all instances in this set. Instance set GVO102. This instance set refers to the mill Itapira. All instances contain 15 days, 108 plantation fields and five cutting crews (two manual and three mechanical ones). The instances possess a medium field intersection of the cutting crews’ field sets and a cutting crew occupation rate of about 120%. The processing demand index IndM P lays between 87% and 99% for all instances in this set.
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Instance set GVO103. The instances from this set contain one sugar cane mill, Catanduva. The instances cover a short-term planning for seven days, containing between 58 and 176 fields. Two manual and three mechanical cutting crews with very high occupation rates (170% - 540%) can be used to harvest a selection of the fields. The crews’ field sets are almost disjoint, but the processing minimum demands of the sugar cane mill are fairly tight, i.e. above 75%.
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Instance set GVO106. The last set provided by the GVO, also based on fields for the mill Catanduva, is the one with the most heterogeneous instances. All instances contain one sugar cane mill. The first instance contains 15 days, the remaining ones contain seven days. The number of fields varies from 19 to 123. All instances provide two manual and three mechanical cutting crews. The field intersection index IndF I varies from low to medium. The cutting crew’s occupation as well as the mill’s processing demands vary from low to very high. 5.1.3.2 Artificial instances The artificially created instances were designed to provide a broad variety of characteristics in order to analyze the influence of such properties in the difficulty of the problem. 15 artificial instances with different properties were selected. The instances were divided into groups. Each group contains a different number of plantation fields, i.e. 20, 50 or 100 fields. Each instance contains a planning horizon of either 15 or 30 days. Each configuration is available with different values for the mill’s minimum processing demands. In total, there are 10 instances with 10 fields (Art10), four instances with 20 fields (Art20), five instances with 50 fields (Art50) and six instances with 100 fields (Art100). All instances provide a complete graph for the distances between the plantation fields. Instance set Art20. This set contains four instances with 20 fields each. The planning is performed for 15 days and provides one manual and one mechanical cutting crew. There are two mills. The field intersection is high and the cutting crews’ total occupation rate is low. Additionally, there are five days with a demand for vinasse application. The instances in this set contain different processing minimum demands at one sugar cane mill. This demand varies from zero to 93% throughout the instances.
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Instance set Art50. This set contains five instances, all of them with 50 fields. Three of the instances are very similar: Two mills, one manual and one mechanical cutting crew. Each crew can cut all fields, i.e. the field intersection in the crews’ field sets is the highest possible. The crews’ occupation is at almost 100% and the processing demands at one of the mills vary from zero to 61%. In addition to these three instances, the set contains two more variations based on the instance without processing minimum demands. One of the two additional instances performs the planning for 30 days, i.e. the cutting crews are occupied only in half of the time. The other instance adds one mechanical crew, alleviating the crews’ total occupation. Instance set Art100. The last of the artificial instance sets increases the number of fields to 100. Two instances contain one manual and one mechanical cutting crew, two mills and a planning horizon of 15 days. One of the instances does not have a minimum processing demand, the other ones’ demand IndM P is about 65%. In both instances, the crews’ total occupation is very high. Also, both crews are able to cut all fields. In two further instances, one manual and one mechanical crew are added, holding the index IndM P at a value greater than 50%. A final variation contains two more instances with two manual and two mechanical crews and a planning horizon of 30 days. The field sets were limited to between 25 and 35 fields for each crew. 5.2 Experiments on the SCHP-TP In the overall planning of the SCHP, the tactical planning has a crucial influence to the period in which a field will be harvested. This decision is essentially guided by the maturity of the cane. A simple example demonstrates the strong influence of these values. The example instance considers ten fields with a constant productivity of 150 tons (i.e. the productivity is the same in all weeks). The planning horizon contains eight weeks. The Pol values for all fields vary according to the values illustrated in Figure 5.1, whereas the AR values are constant. The maximum processing capacities at the sugar cane mill are limited to 10,000 tons per week, i.e. not all fields’ sugar cane can be processed at one single day. It is supposed that the optimal solution cuts as many fields as possible within the week with the highest Pol value. Figure 5.2 confirms this assumption: the fields were cut within the weeks with the best Pol values. Table 5.2 shows the results for all instances for the SCHP-TP. The experiments include a 10 minutes branching phase and a 5 minutes polishing phase. As can be seen in the final deviation values, this time limit turned out to
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Figure 5.1: Pol distribution per week for an example instance
Figure 5.2: Quantity of harvested sugar cane per week in the optimal solution for an example instance be sufficiently high to solve the instances. All instances were solved to at least 0.09% close to proved optimality. An extra column within the table shows the time required to solve the problem within 0.5% of proved optimality. For all instances, this was reached within the branching phase, i.e. within the first ten minutes. Among the instance sets with varying minimum processing demands, relatively high differences in the solution time can be observed. It seems a tendency that, the tighter the processing demands, the harder the problem. Though this does not seem to be a rule as the low solution time for instance GVO10 2 shows. A further interesting observation can be made at instance set GVO7. The instance with a higher cutting crew occupation rate is solved much faster than the one with a low cutting crew occupation rate. Figure 5.3 presents the interval of the minimum and maximum processing capacities of the mills and the processed sugar cane tons (represented by the horizontal bar) at both mills suggested by an optimal solution for instance GVO10 2. The minimum processing demands are respected in all months. An interesting observation is that as much sugar cane as possible was processed
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at the second mill. This in founded in the fact that the processing costs of the second mill are lower than at the first mill. As mentioned, all instances for the SCHP-TP provided by the GVO contain a planning horizon of at most ten months, i.e. ten temporal units. In these ten units, harvesting of all fields for up to a whole harvest season is planned. Other sugar cane companies may use the tactical module for a whole year on weekly basis, i.e. 52 weeks, but still planning the harvest for the same number of fields. As the solver did not show any difficulties to plan the harvest of 1200 fields for ten weeks in short execution time, it is assumed that there will be no difficulties to plan the harvest for such 1200 fields for 52 weeks, when eventually providing more execution time.
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Instance
GVO1 1 GVO1 2 GVO1 3 GVO2 1 GVO2 2 GVO2 3 GVO3 1 GVO3 2 GVO3 3 GVO4 GVO5 GVO6 GVO7 1 GVO7 2 GVO8 GVO9 GVO10 1 GVO10 2 GVO10 3 GVO11 GVO12 GVO13 GVO14
# Var
# Const
3375 3375 3375 11047 11047 11047 10523 10523 10523 10471 31890 26550 26299 26299 24979 47678 103365 103365 103365 80865 32245 22022 20813
2988 2988 2988 10725 10725 10725 10143 10143 10143 10092 28246 28473 27676 27676 27183 52780 96392 96392 96392 73135 30760 22936 21336
Reported sec to solve dev opt% ≤ 0.5 % 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.09 0.00 0.02 0.08 0.02 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.01 0.00
4.26 3.12 4.96 <1 <1 5.04 <1 1.06 4.24 37.47 7.10 4.07 179.00 8.67 <1 1.05 342.28 44.54 504.21 168.53 <1 37 <1
Table 5.2: GVO instance sets 100 and 102 for the SCHP-OP
Economic Analysis An analysis of the contribution of the different factors to the objective function, based on the GVO instances, demonstrated that the profit made by
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Figure 5.3: Processed sugar cane at the mills suggested by an optimal solution for instance GVO10 3 the sugar production is clearly the dominating variable within the objective function. The profit made by sugar production is always much greater than the costs to harvest the field, transport the sugar cane and process it at a mill. Even with resource costs varying along the time, the solver still prefers to harvest the field close to its maturation peak as the profit increase is still much larger than the possible additional costs. 5.3 Preprocessing for the SCHP-OP Large-scale instances for classical problems as well as instances for problems as they appear in practice often cannot be solved by MIP solvers in reasonable time. In the latter case of instances as they come from industry, often redundant information or data that can be disregarded from the problem are provided. Pre-processing has proven to be very useful in order to decrease the problem size while the optimal solutions of the problems are only marginally affected or not all. This section presents the pre-processing techniques applied to the SCHP-OP in order to tackle large instances.
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5.3.1 Distance Filtering The distances between the fields contained within the input data are essential for the movement of the cutting crews from one field to another. The more distances are available, the more options the cutting crews have to travel from one field to another and the better the optimal solution to the problem. However, the more distances are included, the larger the problem. Each distance results in up to |I| travel variables for each cutting crew that may use this edge to travel. A complete graph is often not feasible due to the large size of the model. A limitation of the distances may worsen the optimal solution for the resulting problem. However, in time limited executions of the optimization, such a limitation of the distances may result in better solutions, because the problem is smaller and therefore easier to solve. Distance filtering is an important task. On the one hand, one cannot select too many distances, because the problem grows quickly. On the other hand, sufficient distances must be available in order to allow the cutting crews to travel to all fields. In this work, several approaches to the problem of distance filtering were experimented: Maximum distance. Statistics of the input data from practice showed that the distances between the fields are well distributed between one and 150 km. A simple approach to filter the distances from the input data is the acceptance of distances only below a certain distance value maxDist. This approach has two obvious disadvantages: First, there is no guarantee that the field graph is connected or the distances are well distributed over the fields. Second, this approach does not allow any control over the number of selected distances. Node balancing. A graph with a balanced number of outgoing edges for each node tends to be more connected than a graph with a highly varying number of outgoing edges for the nodes. This approach guarantees a certain number of outgoing edges (distances) for each node (field) and therefore tends to result in a connected graph. First, the distances are sorted in increasing order. Then, for each of the fields, one distance from the sorted list of distances is selected. This process is repeated until a determined number of selected distances is reached. Minimum spanning tree. The minimum spanning tree (MST) over a set of fields and distances can be used to select the distances from the input data.
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For a set of n fields, the MST contains exactly n − 1 edges. This approach is very suitable to guarantee that a crew can reach all fields from any position, because the MST over the field graph for each cutting crew guarantees complete connectivity. The implementation in this work used the Prim algorithm to compute a MST for each cutting crew’s network.
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Shortest path edges. This distance filtering is based on the shortest paths from each field to all other fields. The set of accepted distances then contains all distances of these shortest paths. In practice, this approach turned out to be not very effective, because the shortest path between two fields usually is the straight distance between the two fields. Thus, in the experiments performed, almost all of the distances from input data were selected. Variable pruning. Another approach to reduce the problem size is limiting the number of instants for which the travel variables are being generated. In addition to the common pruning mechanisms (see Section 5.3.2), one can still limit the instants for which certain travels are allowed. This can be performed either by randomization or by other strategies. The main disadvantage of this approach is the fact that the resulting model may not be able to represent some of the good quality solutions. Thus, this approach was not further explored in this work. Table 5.3 compares the number of outgoing edges per field for different distance filtering approaches. The compared approaches include the limitation by a maximum distance of 10km (Max10) and 50km (Max50), the use of the Minimum Spanning Tree (MST), node equilibration with up to 50,000 travel variables (50k) and node equilibration with up to 100,000 travel variables (100k). The node equilibration approaches were then combined with a MST (MST 50k and MST 100k). For each approach, the average value and the standard deviation is presented. An approach with a low standard deviation tends to provide a good node equilibration. The results show that the approaches Max 10 and Max 50 do not balance the number of outgoing edges very well. Considering all approaches, it seems that the combination of the MST with node equilibration results in the most balanced distribution of outgoing edges. The distance filtering approaches are now compared in terms of their impact on the optimization process: the original problem without distance filtering (Orig), limitation by a maximum distance of 5km (Max 5), 10km (Max 10) and 50km (Max 50), node balancing with up to 50,000 (Bal 50k), 100,000 (Bal 100k) and 250,000 travel variables (Bal 250k). Furthermore, the
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Field Max 10 1 17 2 18 3 19 4 18 5 17 6 17 7 13 8 13 9 20 10 17 11 12 12 17 13 17 14 17 15 17 16 19 17 19 18 12 19 12 20 11 21 9 22 9 23 0 24 8 25 6 Avg 14.16 Std dev % 4.83
Max 50 35 35 35 31 35 35 24 39 31 35 41 27 27 26 26 35 35 37 24 43 23 23 24 20 15 30.44 6.92
MST 1 2 2 2 3 3 2 3 2 2 2 2 2 2 2 1 2 2 2 1 1 2 2 2 1 1.92 0.56
50k MST 50k 6 6 5 5 2 2 5 5 4 5 4 4 4 4 3 4 4 4 5 5 3 4 5 5 5 5 4 4 8 8 5 5 3 3 4 4 4 4 5 5 4 4 4 4 4 4 4 5 4 4 4.32 4.48 1.06
100k 8 5 6 8 8 10 8 6 8 10 7 9 9 9 9 7 6 7 7 7 9 7 9 10 7 7.84 1.35
MST 100k 8 5 6 8 8 10 8 6 8 10 8 9 9 9 9 7 6 7 7 7 9 7 9 11 7 7.92 1.41
Table 5.3: Outgoing edges per field for different distance filtering approaches (example of 25 fields of Cutting Crew 204 at instance GVO102 2. use of the minimum spanning tree (MST) and the minimum spanning tree plus node balancing with up to 50,000 (MST Bal 50k), 100,000 (MST Bal 100k) and 250,000 travel variables (MST Bal 250k) are evaluated. For all approaches, all instances were executed using CPLEX with a 15 minutes branching phase followed by a 15 minutes polishing phase. Table 5.4 reports the results for the first seven of the eleven filtering approaches mentioned above, Table 5.5 reports the results for the other four approaches. The results are given for each instance set. The number within parentheses presented after the instance’s class name denotes the number of instances contained by the set. For all instance sets, the table also presents the number of instances where the branching phase was not able to find a solution (# no sol) within the available time. For all instances that led to an integer solution, the average deviatin (from the best known solution) and the
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deviation’s variance is given. The results for all artificial and GVO instances are summarized separately. Finally, the results of all instances are summarized, also reporting the average number of travel variables (Avg # o vars). Distance filtering clearly affects the optimal solutions to the problem. However, in time limited experiments, the final results improved as showed above. In the original problem (without distance filtering) with more than 500,000 travel variables, the solver was not able to find any feasible solution within the available time for 16 of the 40 instances. The results confirm that the problem size, indicated by the number of travel variables, is directly linked to the hardness of the problem. In the experiments for all three presented techniques, the number of instances where no feasible solution was found increased with greater numbers of travel variables. In fact, this observation seems to be valid throughout all techniques: The greater the number of travel variables, the less instances result in a feasible integer solution when the execution time is limited. In order to identify an appropriated filtering technique for the problem, one must also consider the average deviation (from the best known solution) and its variance. The lower the variance, the more confidential the average deviation. Filtering by maximum distance limitation resulted in a large number of instances without feasible solution as well as in a high average deviation. Thus, this approach was disregarded in further experiments. The filtering based on node balancing improved the average deviation and decreased the number of instances where no solution was found. Especially for the GVO instances, the results are quite satisfactory. However, this technique did not perform well on the sets of artificial instances. The best results were obtained through the combination of the MST and node balancing. For the GVO instances, the results remain very similar to the ones of the node balancing approach. For the artificial instances, the results significantly improved. The results for the distance filtering by a shortest path network are similar to the ones of the original problem, as very few distances were excluded. Hence, its results are not listed here. In conclusion, the experiments’ results suggest that, for the available time limit, the distance filtering approach based on the MST plus node balancing is the most indicated one.
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Instance set Orig Artificial Art20 (4) # no sol 0 Avg dev % 6.24 Variance 91 Art50 (5) # no sol 4 Avg dev % 54.50 Variance 0 Art100 (6) # no sol 6 Avg dev % Variance Total # no sol 10 Avg dev % 15.89 Variance 445 GVO GVO100 (6) # no sol 6 Avg gap Variance GVO102 (5) # no sol 0 Avg dev % 41.47 Variance 201 GVO103 (7) # no sol 0 Avg dev % 10.68 Variance 170 GVO106 (7) # no sol 0 Avg dev % 12.97 Variance 261 Total # no sol 6 Avg dev % 19.63 Variance 383 All instances Avg # o vars 558600 # no sol 16 Avg dev % 18.85 Variance 398
Max 5
Max 10
Max 50
Bal 50k
Bal 100k
Bal 250k
0 75.99 3044
0 75.59 3153
0 10.87 346
0 2.90 24
0 6.24 91
0 6.24 91
1 24.09 399
2 83.63 2944
2 430.45 50967
0 28.02 138
0 116.69 35842
2 349.28 89644
6 -
6 -
6 -
2 68.33 2016
4 29.61 115
6 -
7 50.04 2395
8 79.03 3079
8 190.69 65156
2 32.70 1353
4 60.69 19025
8 153.26 67290
1 29.67 1698
2 18.76 97
6 -
1 13.80 13
1 3.27 3
1 2.70 1
0 93.23 5095
0 58.94 70
0 42.06 81
0 67.20 84
0 39.31 233
0 36.17 141
0 45.69 726
0 34.92 326
0 8.51 75
0 9.72 79
0 5.28 9
0 4.43 7
0 122.64 28969
0 62.28 8198
0 12.32 267
0 16.11 232
0 13.39 254
0 12.97 262
1 74.70 11486
2 45.66 2910
6 18.75 344
1 24.41 599
1 14.31 306
1 13.17 263
158727 8 68.54 9328
282953 10 53.45 3149
544234 14 65.04 23610
40204 3 31.32 1139
70299 5 28.93 6299
150.192 9 44.81 18829
Table 5.4: Impact comparison of distance filtering approaches: No filtering, distance limitation to 5km, 10km and 50km and node balancing with up to 50k, 100k and 250k travel variables
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Instance set Artificial Art20 (4) # no sol Avg dev % Variance Art50 (5) # no sol Avg dev % Variance Art100 (6) # no sol Avg dev % Variance Total # no sol Avg dev % Variance GVO GVO100 (6) # no sol Avg dev % Variance GVO102 (5) # no sol Avg dev % Variance GVO103 (7) # no sol Avg dev % Variance GVO106 (7) # no sol Avg dev % Variance Total # no sol Avg dev % Variance All instances Avg # o vars # no sol Avg dev % Variance
MST
MST Bal 50k
MST Bal 100k
MST Bal 250k
0 0.16 0
0 2.96 24
0 6.24 91
0 6.24 91
0 12.97 113
0 15.90 182
2 109.20 5.008
2 325.60 115.774
1 177.13 87.064
3 60.17 2.890
4 34.77 509
6 -
1 67.94 37.784
3 22.65 1.307
6 46.90 3.884
8 143.11 74.646
1 3.11 0
1 6.01 64
0 12.61 332
1 4.74 25
0 40.81 117
0 37.38 253
0 39.60 254
0 34.78 153
0 12.81 73
0 8.01 34
0 7.87 32
0 5.08 7
0 48.45 3.436
0 15.25 245
0 13.55 252
0 13.54 252
1 27.02 1.399
1 15.83 282
0 16.94 343
1 13.66 244
27.659 2 57.80 25.425
55.954 4 21.34 933
83.552 6 25.42 1.408
160.896 9 42.89 19.973
Table 5.5: Impact comparison of distance filtering approaches: MST only, MST with node balancing with up to 50k, 100k and 250k travel variables
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5.3.2 Variable pruning and Reduction tests
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5.3.2.1 Variable pruning The variables used in the model presented for the SCHP-OP represent certain decisions. These decisions are available for combinations of resources of the problem, for example the decision whether a certain cutting crew harvests a certain plantation field. However, there are many of such combinations that are not valid. For example, a cutting crew may not work at all days. Variable pruning, i.e. not generating the variables for certain decision combinations, is a widely common approach to prohibit certain combinations and allows for the implementation of several functionalities of the model. In the implementation of this work, variables were not generated if they involve a cutting crew and an instant at which the cutting crew is not available. In the following, the pruning rules for the most important variable groups are explained. Note that these types of variable pruning are valid, i.e. they do not affect any feasible solution of the problem. i1 ,i2 Cutting variables. A variable yc,f , i1 , i2 ∈ I, c ∈ CF, f ∈ F is not generated if one of the following conditions is true:
– the plantation field cannot be cut at one of the days between i1 and i2 . – the cutting crew does not work at one of the days between i1 and i2 . – the cutting crew cannot cut the field f at all – there is no mill available at one of the days between i1 and i2 , i.e. there is no x variable for one of these days. – cutting crew c is located at f at the beginning of the planning and i1 is not the first available instant of c. – cutting crew c is located at a field f2 ∈ F \{f } at the beginning of the if irst planning and i1 ≤ δc,f , where if irst is the first available instant of c. – there is a cutting crew c2 ∈ CF \{c} that is located at f at the beginning of the planning. Mill assignments. A variable xf,p , f ∈ F, p ∈ P is pruned in the following cases: – the plantation field cannot be cut at any day. – mill p cannot process the sugar cane from field f .
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– mill p is not active at any day. – the input data does not contain a distance between field f and mill p.
Third party sugar cane. A variable qd,r,p , d ∈ D, r ∈ R, p ∈ P is pruned in the following cases: – the supplier r cannot provide sugar cane at day d. 1 ,i2 Travel variables. A variable oic,f , i1 , i2 ∈ I, c ∈ CF, f1 , f2 ∈ F is not 1 ,f2 generated in one of the following cases:
– the cutting crew c does not work at any of the days between i1 and i2 . – there is no direct distance between the fields f1 and f2 .
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– the cutting crew c cannot cut one or even both of the fields. – the cutting crew is a manual crew and the travel is performed during the night, i.e. i1 and i2 do not belong to the same day. This is founded in the fact that manual crews spend the night at their accommodation and return to any field in the following morning. Thus, the travel variable is not necessary. – the cutting crew c is located at any field f3 ∈ F at the beginning of the if irst planning and i1 ≤ δc,f , where if irst is the first available instant of c. 3 Waiting variables. A variable nic,f , i ∈ I, c ∈ CF, f ∈ F is not generated in one of the following cases: – the cutting crew c cannot cut the plantation field f . – the cutting crew c is located at any field f3 ∈ F at the beginning of the if irst planning and i1 ≤ δc,f , where if irst is the first available instant of c. 3 Travels from home. A variable f hdc,f , d ∈ D, c ∈ CF, f ∈ F is pruned if: – the cutting crew c cannot cut the plantation field f . – the cutting crew c does not work at day d.
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5.3.2.2 Reduction tests Reduction tests aim at reducing the original problem to a smaller, equivalent problem by identifying redundant or trivial decisions. In the case of the network flow of the problem presented in this work, the aim is to identify and eliminate redundant nodes.
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Non-terminal node with degree 2. Nodes with degree two may be deleted and its two edges may be replaced by a single edge. This type of reduction test is generally referred to as the reduction of non-terminal node with degree 2 [PU03]. Figure 5.4 illustrates an example of such a reduction. The nodes n2 , n3 and n7 may be pruned as their in and out degree is two. In the implementation, most of the redundant nodes had already been eliminated due to the previously explained variable pruning. Thus, this reduction test had a relatively small impact.
n1
n2
n3
n4
n1
n5
n6
n7
n8
n5
n4
n6
n8
Figure 5.4: Example of a reduction of non-terminal nodes with degree 2
5.3.3 Field grouping Some problem instances from practice contained more than 1000 plantation fields. Instances with such a large number of fields are difficult to solve. In order to reduce the problem size, several fields with the same characteristics are grouped to one field block. A grouping is possible if the fields agree in the following properties: – All pairs of distances between the fields are less than one kilometer. – All fields have the same Pol, AR and fiber values. – All fields are eligible to be cut at the same days. – All fields can be cut by the same cutting crews. A field group sums up the productivity of all contained fields for each day. Experiments performed by the GVO showed that it is possible to group four fields to one group in average.
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5.4 Exact solution approaches for the SCHP-OP The previous section presented pre-processing techniques in order to reduce the problem to a reasonable size. This section presents the solution methods applied to solve the SCHP-OP. First, the optimization process of previous experiments is analyzed, identifying combinatorial characteristics of the problem and inspecting the obtained solutions. Afterwards, it is demonstrated how initial solutions can facilitate the solution of the problem. Finally, the linear relaxation is analyzed in order to find valid inequalities. 5.4.1 Analysis of the optimization process
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Combinatorial characteristics The computational experiments showed that the problem’s difficulty is very sensible to variations within the values of the input data. That is, not only the size of the input data but also their values. The following sensibilities were observed: – Minimum and maximum processing demands of sugar cane mills. The minimum and maximum processing demands of the mills strongly influence the difficulty of the problem. Table 5.6 shows the final deviations (from the best known solutions) after an optimization process for instances with different minimum processing demands. All instances within each set contain the same properties, except the varying minimum processing demands indicated by Ind M P . The results confirm the assumption that the problem gets harder to solve as Ind M P increases. Further experiments showed that instance Art20 1 4 is hard to solve even with more optimization time effort, because it is hard to find a cutting crew schedule that only marginally violates the processing minimum demands at all days. – Minimum and maximum cutting capacities of cutting crews. Similarly, it can be assumed that the minimum and maximum cutting capacities of the cutting crews complicate the solution of the problem. – Cutting crews occupation rate. The higher a cutting crew’s occupation rate, the more difficult is finding a good arrangement of the harvest sequence, because the cutting crew has less spare time. Table 5.7 shows such an effect. Both instances are identical, except the number of planning days. Though instance 50 2 1 possess twice the number of days
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than 50 2 2 (and thus results in a larger problem size), its solution is much easier for the MIP solver, because the cutting crews’ occupation rate was bisected. Instance 20 1 1 20 1 2 20 1 3 20 1 4
Ind M P Dev % 0 0.00 40.91 0.00 81.82 2.28 92.73 22.67
Instance Ind M P Dev % 50 3 1 0 22.00 50 3 2 43.64 110.27 50 3 3 61.09 195.33
Table 5.6: Influence of minimum processing demands to the difficulty of an instance
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Instance Ind CO Dev % 50 2 1 49.62 0.33 50 3 1 99.64 18.47 Table 5.7: Influence of the cutting crews’ occupation rate to the difficulty of an instance
Solution analysis The optimization process is now analyzed. Table 5.8 shows five solutions obtained during the optimization process for instance Art50 3 3. Solution #1 is the first feasible solution found during the optimization, solution #2 is the second feasible solution found, solution #3 is the fifth feasible solution found and solution #4 is the best solution found in the experiments made in the previous section (optimization with distance filtering MST + node equilibration with up to 100,000 travel variables). Finally, solution #5 represents a close to optimality solution obtained in further experiments (to be exact, it is the solution obtained by using initial heuristic solutions and the cut type 1, see Section 5.3.1 and Section 5.4.4.1, respectively). The table shows the solution value of the objective function, the violation of the minimum processing demand constraints (MinProc), the number of fields not harvested (CutOblig) and the total quantity of cut sugar cane (Tons cut). Solution #1 begins highly negative and is then improved during the solution process. The close to optimal solution #5 satisfies the complete minimum processing demands, but does not cut all fields. An interesting observation is the change of the solution’s properties from solution #4 to #5: though the objective function and the total quantity of harvested sugar cane is much larger in solution #5, the number of harvested fields only increased by three. This demonstrates that after a certain point during the optimization
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Solution Violation Violation Tons cut OF MinProc CutOblig -61170449 39200 49 2800 -41180742 31600 42 13000 -34847338 29300 38 16600 15964827 7700 23 38850 41712980 0 20 56600
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Table 5.8: Solution properties during the optimization process process, further increase in the objective function is provoked by changing small fields by large fields. The total productivity of all 50 fields within this instance sums up about 68,750 tons. Consequently, the 20 fields that were not harvested represent a total cane quantity of about only 12,150 tons, whereas the 30 harvested fields represent 56,600 tons. Once again, this proves that a good quality solution tends to select a few large fields instead of many small fields. Obviously, this behavior can be explained by the travel time required to move from one field to another. Figure 5.5 and Figure 5.6 illustrate the routes for both cutting crews. The values at the y-axis represent the different fields. The value −10 at the y-axis represents the accommodation to which manual cutting crews return at the end of a day. The x-axis denotes the planning horizon divided into time instants. The instance contains 15 days. The configuration in this example contains twelve instants per day. Figure 5.5 represents solution #1 and #3. Figure 5.6 represents solution #4 and #5. The route for the first solution (compare upper route in Figure 5.5) contains almost no field cuts. The cutting crews travel from field to field. The mechanical cutting crew CF1 commutes between the same two fields. The manual cutting crew CF2 starts the planning by harvesting a field. Afterwards, its time is spent by traveling to fields and returning to its accommodation. An interesting observation is that both the cutting crews do not wait at any field though waiting is cheaper than traveling. The second solution’s route (compare lower route in Figure 5.5) improves the objective function by adding more field cuts for both cutting crews CF1 and CF2. The more advanced solution #4 (compare upper route in Figure 5.6) contains much more field cuts and focuses on large fields. However, it still performs many travels which seem to be unnecessary. Finally, the close to optimal solution #5 (compare lower route in Figure 5.6) almost completely avoids unnecessary travelling and cuts even more large fields than solution #4.
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Figure 5.5: Routes of cutting crews for solutions in the initial stage of the optimization process
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Figure 5.6: Routes of cutting crews for solutions in an advanced stage of the optimization process. The lower figure illustrates the routes for a close to optimum solution
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Figure 5.7 shows the objective function and the upper bound during the optimization process for instance Art50 3 3. The first solutions, beginning highly negative, quickly improve. After 900 seconds, the polishing phase is initiated and seems to improve the solution almost linearly.
Figure 5.7: Objective function value throughout branching and polishing process
Economic Analysis The analysis of the contribution of the different factors to the objective function (again based on the GVO instances) demonstrated a similar proportion as within the solutions for the tactical planning. The profit generated by sugar production is the dominating factor, whereas the harvesting, transportation and processing costs have only marginal influence. The travel costs for the cutting crews marginally influence as well. Traveling over long distances is avoided only because it may cost more than one time instant to travel, and it is not possible to harvest while traveling. As the solutions for the GVO instances for the SCHP-OP often presented a high violation of the minimum processing demands at the mills, the penalty cost of not harvesting becomes very high. For many instances, these penalty costs have a significant impact in the objective function. 5.4.2 Initial solutions through constructive heuristics Modern MIP solvers usually invest heuristic effort to find feasible starting solutions. For difficult or large problems, the solver may take a long time to find an initial feasible solution or, even worse, not find any solution at all in the time available for the solver’s execution.
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Operational plannings are usually very restricted in its available execution time. Thus, in a time limited optimization, it is not guaranteed that the solver finds a solution of reasonable quality. However, the user may strongly depend from the obtained solutions and may use even bad quality solutions in his planning. In some cases, he may not even be aware of the bad quality of the solution. Therefore, it is reasonable that the solutions obtained by the optimization should be at least as good as an obvious planning manually performed. A common approach to guarantee a certain quality level is the use of heuristic methods in order to provide starting solutions. Several constructive heuristics were implemented and tested for the SCHP-OP. The first ones use very limited knowledge of the problem’s characteristics. Later heuristics profit from more knowledge. Algorithm 1 shows the pseudo code of the skeleton for all implemented heuristics. The method chooseNextField is implemented individually by each heuristic and returns the next plantation field that shall be cut by the cutting crew as well as the path to get to this field. Since manual cutting crews can travel for zero cost at the end of each day, the chooseNextField method never suggests a manual crew to travel during the night in order to start harvesting a field at the next day. In addition, the method never returns fields that have already been cut or are subject to be cut afterwards as they are initial starting locations of other cutting crews (and will therefore be cut by these crews).
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Algorithm 1: Heuristic skeleton Input : An ordered sequence of cutting crews c1 , . . . , cn Output: Solution for each cutting crew c = c1 , . . . , cn do currInstant ←− c.firstAvailableInstant(); if c.type is manual and c.initialLocation is a plantation field f then currF ield ←− f ; cut field f ; currInst ←− CutcurrInst .f inishingInstant() ; c,f currInst ←− c.getN extAvailableInstant(currInstant); else currF ield ←− c.initialM ill; end while currInst ≤ c.lastAvailableInstant() do nextF ield ←− chooseN extF ield() ; if nextField = NULL then wait one instant at current field ; currInst ←− c.getN extAvailableInstant(currInst) ; else currInstant currInst ←− travelcurrentF ield,nextF ield .f inishingInstant ; currInst ←− c.getN extAvailableInstant(currInst) ; cut plantation field nextField; currInst ←− CutcurrInst .f inishingInstant ; c,f currF ield ←− nextF ield; end end end
Mill selection to process cut sugar cane. After the time instants for the field harvesting are determined (in fact, the field harvesting starts at these time instants, but does not necessarily terminate at the same instants), the heuristic must select a sugar cane mill for each of the selected fields in order to process their sugar cane. The heuristic aims at minimizing the total violation of the minimum processing demands at the sugar cane mills. The assignment of a sugar cane mill to a field in order to process the sugar cane of that field may decrease the total violation of these demands. To each field that was selected for harvesting, the heuristic greedily assigns the mill that results in the lowest total violation. This assignment of sugar cane mills to plantation fields is performed for
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all heuristic variations. The pseudo code for such assignment is presented in algorithm 2.
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Algorithm 2: Mill assignment Input : Set of fields FS selected for harvesting Output: Assignment of fields to mills compute total violation of the minProc constraints ; for each plantation field f ∈ F S do minV iolation ←− +∞ ; minV iolationM ill ←− N U LL ; for each mill p do actViolation ←− total violation (at all mills) assuming that f will be processed at mill p ; if actViolation < minViolation then minV iolation ←− actV iolation ; minV iolationM ill ←− p ; end end assign field f to mill minV iolationM ill ; update total violations of the minProc constraints considering the new field assignment ; end
Heuristic 1 - Initial cuts only This heuristic performs only the initial cuts for the cutting crews that are located at a plantation field at the beginning of the planning. After the initial cut, the cutting crew waits at the same field until the end of the planning. Cutting crews that are initially located at a mill will move to an aleatory field and wait until the end of the planning. The heuristic can be implemented using the above skeleton with an implementation of the chooseNextField method that always returns NULL. Heuristic 2 - Any field and 1-edge moves This heuristic aims at cutting fields by use of an aleatory selection. The chooseNextField method selects any field that – can be reached using travels of one edge and – can be started to cut immediately after arriving at that field.
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Heuristic 3 - Any field and n-edge moves This heuristic acts just as the previous one, but allows traveling by more than one edge. The heuristic uses the shortest path between the two fields to determine the edges of the path.
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Heuristic 4 - Field with best tons/cutting time relation This variation is intended to be more effective than the previous heuristics especially for instants in which not all fields can be cut. In the previous heuristic, the distance to the next field has not been considered. This may result in an unnecessary waste of time to travel. This improved heuristic greedily selects the field f which has the best relation between the quantity of sugar cane cut at the field and the time spent to travel to the field as well as to cut it. Let tf be the quantity of sugar cane that can be cut at field f , tDurcurrF ield,f the duration (in number of instants) required to travel from the current field currF ield to field f and cDurf the number of instants required to cut field f . Then, the chooseNextField method selects the field with the highest relation tf . tDurcurrF ield,f +cDurf Heuristic 5 - Field with best tons/cutting time relation with randomization This heuristic differs from the previously presented heuristic only in the selection of the next field. Instead of always selecting the field with the best tons/cutting time relation, it randomly selects one of the three best fields. Heuristic 6 - Field with best tons/cutting time relation and cut equilibration The previous heuristics aim at cutting and traveling as early as possible. In some cases, the cutting crews may not be occupied during the whole planning performing cuts and travels, i.e. their occupation rate may be low. In such cases, the solutions of the previously presented heuristics result in many field cuts at the beginning of the planning, whereas the crews have spare time by the end of the planning. When considering the sugar cane harvested by all cutting crews, this leads to an accumulation of harvested sugar cane at the first days of the planning horizon and a tendency to a decreasing sugar cane quantity as time passes. For instances with tight minimum processing demands at the sugar cane mills or minimum cutting demands of the cutting crews, this may result in unncessary demand violation.
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It is possible to diminish such an effect by balancing the cut sugar cane along the planning. In order to do so, the field cuts of cutting crews with a low occupation rate must be distributed throughout the planning. For each cutting crew, this heuristic first plans the field cuts according to heuristic 4. Afterwards, it distributes the field cuts along the total planning period. Let nW be the number of instants at which the cutting crew is not occupied within the given solution, i.e. the number of instants at which the cutting crew is waiting. Let also nC be the number of fields that the cutting crew is supposed to harvest according to the solution. In the following, the set of all time instants at which the crew is not occupied is distributed along the planning by inserting a certain amount of them after each field cut. Let k be the number of waiting instants that will be inserted after each cut, computed . as follows: k = nW nC If k < 2 or nC < 2, then the first planning for this crew is accepted and the heuristic proceeds to the planning for the next cutting crew. Otherwise, the heuristic re-calculates the instants at which the fields are harvested through inserting exactly k waiting instants after each field cut of the first planning solution. Figure 5.8 illustrates such a cut balancing. The first planning suggests to cut three fields. After these cuts, the crew has no more fields left to cut (see 5.8 (a)). After the last cut until the end of the planning, the cutting crew waits for seven instants. Hence, 73 of the waiting instants are inserted after each cut. Heuristic 7 - Field with best tons/cutting time relation with randomization and cut equilibration This heuristic differs from heuristic 6 only in the selection of the next field. Just as heuristic 5, this heuristic randomly chooses one of the best three fields. Computational Experiments Comparison of the heuristically produced solutions. The heuristics presented in this section were compared by means of the following experiment. The cutting crews were sorted by the increasing number of plantation fields that they can cut, i.e. the size of their field sets. The heuristics 2, 3, 4 and 6 were executed once, whereas each of the randomized heuristics 5 and 7 were executed five times.
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Figure 5.8: Example route for a cutting crew: first, the crew greedily chooses the fields and waits until the end of the planning (a); in a re-planning step (b), the cuts are balanced along the planning period by insertion of the waiting variables after each cut Table 5.9 lists the results of these experiments. The heuristics with aleatory field selection (heuristic 2 and 3) demonstrated solution with deviation from optimum. The prioritization of the field with the best relation between spent time and quantity of cut sugar cane (heuristic 4) significantly improved the quality of the solutions. The randomization in heuristic 5, executing this heuristic several times, obtains even better results. In both of the latter cases, the extension of the heuristics in order to equilibrate the field cuts along the planning time slightly improves the quality of the obtained solutions. Influence of the cutting crew sequence. The previous experiments were carried out for a fixed sequence of the cutting crews. Clearly, the sequence of the cutting crews influences the decisions of the heuristics. This suggests experimenting variations of this sequence. Consider a set of 48 initial solutions, in the following referred to as HeurRandSeq, created through the following strategy: six different cutting crew sequences were considered. In one sequence, the cutting crews are sorted in increasing order of the number of the fields that they can cut. The other five cutting crew sequences are randomly selected. For each of these sequences heuristic 4 and 6 were applied once and each of the randomized heuristics 5 and 7 were applied three times. Column HeurRandSeq in Table 5.9 represents
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Instance Art20 1 1 Art20 1 2 Art20 1 3 Art20 1 4 Art50 2 1 Art50 3 1 Art50 3 2 Art50 3 3 Art50 4 1 Art100 1 1 Art100 1 2 Art100 2 1 Art100 2 2 Art100 3 1 Art100 3 2 GVO100 1 GVO100 2 GVO100 3 GVO100 4 GVO100 5 GVO100 6 GVO102 2 GVO102 3 GVO102 4 GVO102 5 GVO102 6 GVO103 1 GVO103 2 GVO103 3 GVO103 4 GVO103 5 GVO103 6 GVO103 7 GVO106 1 GVO106 2 GVO106 3 GVO106 4 GVO106 5 GVO106 6 GVO106 7 Average
Heur2 4.54 4.53 149.62 125.70 47.55 51.22 58.30 83.14 141.03 263.47 107.27 211.47 108.24 63.96 169.87 41.73 17.98 32.18 29.43 28.27 25.77 70.48 54.62 31.29 36.23 67.79 538.34 338.15 79.73 13.90 265.76 243.35 872.58 9.29 0.50 27.64 40.78 261.05 759.88 117.12 139.84
Heur3 7.82 7.82 170.41 134.38 11.03 47.67 42.80 58.33 14.34 85.08 196.08 49.39 121.60 2.78 1915.56 29.40 25.48 25.87 25.48 25.40 25.65 46.19 59.53 33.69 39.56 43.53 538.34 518.61 79.73 13.90 301.47 219.10 478.73 9.29 0.50 27.64 41.47 261.05 759.88 140.23 165.87
Heur4 4.48 4.47 140.64 123.54 11.09 28.27 24.04 23.67 10.71 14.14 14.30 19.93 19.90 3.26 201.24 40.65 27.72 27.49 27.78 27.45 27.76 38.71 54.10 28.35 51.64 38.09 83.85 47.91 46.21 7.74 91.96 293.66 51.30 35.61 0.45 3.78 15.88 1131.95 959.42 189.59 99.82
Heur5 6.14 6.14 144.93 125.42 11.08 29.22 24.96 24.21 10.38 13.55 13.71 19.98 19.96 3.24 261.94 19.26 21.71 22.37 21.82 29.41 32.18 35.88 50.39 26.89 32.16 34.80 55.75 30.32 35.81 5.21 49.03 53.56 52.57 7.95 0.41 14.67 13.73 31.51 386.11 95.77 46.85
Heur6 0.88 0.87 182.13 134.43 10.88 28.27 24.04 23.67 10.71 14.14 14.30 19.93 19.90 3.10 115.05 39.92 27.72 27.49 27.78 27.45 27.76 37.21 53.94 28.35 51.58 37.99 46.83 47.91 46.18 7.73 91.96 293.66 51.30 35.13 0.36 3.78 15.88 1131.95 959.42 189.59 97.78
Heur7 9.16 9.16 207.11 147.68 10.95 29.22 24.96 24.21 10.38 13.55 13.71 19.98 19.96 2.96 118.31 20.68 35.51 16.47 18.34 16.43 16.42 36.88 49.00 28.06 31.22 34.26 66.73 30.32 35.80 5.21 49.03 53.56 52.57 9.19 0.28 12.66 14.55 31.51 386.11 95.77 45.20
Table 5.9: Quality of the solutions obtained by the heuristics
Min HeurRandSeq 0.88 0.88 0.87 0.87 140.64 194.05 123.54 144.36 10.88 10.88 28.27 22.91 24.04 18.85 23.67 16.62 10.38 10.36 13.55 11.89 13.71 12.05 19.93 18.66 19.90 18.63 2.78 2.77 115.05 2.67 19.26 18.82 17.98 21.71 16.47 15.25 18.34 13.19 16.43 13.07 16.42 14.69 35.88 35.99 49.00 45.34 26.89 25.64 31.22 30.53 34.26 32.29 46.83 23.16 30.32 25.33 35.80 31.78 5.21 4.17 49.03 24.62 53.56 43.52 51.30 19.16 7.95 7.72 0.28 0.27 3.78 3.78 13.73 11.90 31.51 31.51 386.11 87.06 95.77 85.26 41.04 28.81
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the deviation for each best solution of these sets of initial solutions. The randomization within the cutting crew sequence obtained significantly better results than all previous approches. Its average results even outperform the solutions given by the minimum values of all previous heuristics’ solutions for each instance, listed in column MIN. In addition to these experiences, experiments with first- and bestimprovement Local Search methods on the cutting crew sequence were performed, using a 2-OPT neighborhood2 . These approaches turned out to be not competitive with the previously presented strategy due to two facts. First, the evaluation of the neighborhood is relatively costly and, second, the solutions improved about two iterations only. Analysis of the heuristically obtained solutions
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The selection of a field with a good relation between spent time and quantity of cut sugar cane often tends to prefer large fields instead of several smaller ones. This is motivated by two facts: – The discretization of the days into smaller time instants leads to a waste of time. The smaller a field, the higher the probability that its cut involves a high relative waste. Hence, larger fields tend to have the better relation between spent time and quantity of cut sugar cane. – The transport from one field to another spends time and influences the relation between spent time and cut sugar cane. Often, one large field contains a better relation than the sum of several small fields including their travels. This aspect is not limited to the presented heuristics, but also motivates MIP solvers to tend to larger fields. Influence of initial solutions Passing initial solutions to the MIP solver often results in better solutions as Table 5.10 shows. The table compares a standard optimization over all instances with and without initial solutions. For both experiments, the number of instances in which no feasible solution was found and the final deviation from the best known solution values are reported. Without initial solutions, CPLEX strongly invests heuristic effort to find feasible solutions. For some instances, no feasible solution was found at all in 15 minutes. In either case, the heuristic efforts often take a lot of time. Thus, there is few time left for the subsequent optimization. 2
A 2-OPT neighborhood defines a neighborhood as the set of all solutions obtained by swapping the positions of two cutting crews within the sequence.
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Instance set
Without initial solutions With initial solutions # no sols Dev % Dev % Art20 (4) 2.84 10.72 Art50 (5) 36.06 2.68 Art100 (6) 130.57 3.06 GVO100 (6) 3 1.83 1.75 GVO102 (5) 1 23.29 9.75 GVO103 (7) 2.22 4.58 GVO106 (7) 0.84 0.99 Avg Artificial 48.61 4.97 Avg GVO 6.13 4.07 Total 20.29 4.41 Table 5.10: Influence of initial solutions to the quality of the final solutions
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Influence of the solution pool The polishing phase of the MIP solver CPLEX is based on genetic algorithms [Rot07]. Genetic algorithms typically profit from a pool of feasible solutions which is subsequently used to generate new solutions by operations such as mutation or cross-over. Computational experiments performed in this work showed that the number of solutions of a certain quality tend to influence the effectiveness of the polishing phase. A solution pool was created, based on the strategy of solution creation HeurRandSeq. Afterwards, the experiment was performed for three versions. In the first one, only the solution with the highest objective function value was passed to the CPLEX MIP solver. In the second version, the best ten of the 48 solutions were passed to the solver. The third version involves the best 20 solutions. All versions include a 15 minutes polishing phase (without polishing). Table 5.11 summarizes the results for each set of instances. The total average deviation indicates a slight improvement as the number of solutions passed to the solver was increased. These results suggest passing more than one initial solution to the solver. 5.4.3 Linear relaxation analysis The linear relaxation of this problem turned out to be strongly fractional. Figure 5.9 illustrates the route of a mechanical cutting crew in the optimum solution of the linear relaxation for instance GVO102 2. The total flow from the initial location at the mill is divided into nine fields. Each field is then repeatedly cut until the end of the planning. In most of the instances, the last
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Instance set Art20 Art50 Art100 GVO100 GVO102 GVO103 GVO106 Total
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Dev % Dev % Dev % Dev % Initial solution One solution 10 solutions 20 solutions 85.04 0.10 0.09 7.35 15.36 10.19 10.12 9.98 14.91 14.05 14.00 14.09 313.26 16.12 16.12 4.48 34.23 30.97 28.39 29.25 24.53 5.19 5.41 5.63 32.50 10.18 8.91 9.29 82.26 11.62 11.14 9.88
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Table 5.11: Influence in the polishing phase of starting solutions passed to CPLEX cut is only partially performed and no instants are spent in traveling or waiting in order to benefit best from the available time. A similar behavior can be observed for manual cutting crews. In order to pass the flow from the last instant of a day to the first instant of the following day, manual crews have to travel to their accommodation, because there are no waiting variables that connect two subsequent days. Leaving from their accommodation, manual cutting crews are able to travel to any of their eligible fields by zero time and cost. However, it seems that they always return to the same field. This can be justified by the same motivation found in the behavior of mechanical crews: the current field already perfectly contributes to the satisfaction of the constraints and the maximization of the total profit. Such trend to strongly fractional solutions is based on the daily processing minimum demands of the mills and the daily minimum cutting of the cutting crews. The solution equilibrates the cuts along the entire planning horizon, at least over days with minimum demand constraints. The selected fields are typically the ones with the best relation between spent time and return of Pol and AR quantity. In this manner, the solution minimizes the violation of the minimum constraints and benefits from the highest profit possible. 5.4.4 Valid Inequalities Linear relaxation solutions that are strongly fractional often result in large integrality gaps, i.e. large gaps between the linear relaxation’s and the original integer program’s optimum solutions. However, one should strive for small gaps between these optimum solutions, as a tight MIP formulation holds several advantages when using a Branch-and-Bound algorithm: – From the beginning of the branching, the deviation calculated for the found solutions are closer to the real deviation from optimum. ‘Loose
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Figure 5.9: Route of a mechanical cutting crew in the optimal solution for the linear relaxation of instance GVO102 2
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MIP formulations’ tend to be able to prove optimality only as they significantly advance in the branching tree. – Tight MIP formulations result in good quality bounds. Due to such bounds, more subtrees can be pruned within the branching tree. In an analysis of the optimal solution of the linear relaxation, as performed in section 5.4.3, inconsistencies may be found beside the fractionation of integer variables. Often, such inconsistencies are represented by solution behavior that would not make any sense in an integer solution of the problem and can be identified if knowing well the semantic structure of the problem. This section invests in cuts to strengthen the MIP formulation, i.e. to decrease the gap between the MIP’s and its linear relaxation’s optimum solutions. The following presents three cut types identified during the analysis of the linear relaxation’s optimal solutions. 5.4.4.1 Cut 1: Forcing travels throughout whole planning The analysis of the optimal solutions of the linear relaxations exposes two conspicuous behaviors: First, a cutting crew’s flow is split into several fields during the same time instant. Second, a cutting crew repeatedly cuts the same field in order to perfectly distribute the harvested sugar cane quantity along the planning time. It is observed that the linear relaxation’s solution prefers not investing in traveling, as this is consuming in time and costs. However, in an integer solution, a mechanical cutting crew is required to travel from one field to another in order to cut both of them. This requirement also holds for manual cutting crews if they do not travel during the night (i.e. making use of their accommodation zero cost travel). Consider a mechanical cutting
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crew that cuts n fields throughout the entire planning period. This crew must perform at least n − 1 travels in order to visit all fields. Hence, the following valid inequality can be stated:
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A manual cutting crew, contrary to a mechanical one, is able to move from one field to another spending neither time nor money: the crew may return to its accommodation at the end of a day and travel to any other eligible field in the morning of the following day using the f h variables. The previously stated inequality for mechanical crews can be extended for manual cutting crews by additionally considering the f h variables:
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5.4.4.2 Cut 2: Forcing travels at each field The previously introduced cuts guarantee, for each cutting crew, that the overall number of travels is at least as high as the overall number of performed field cuts. However, in the linear relaxation, some fields may have such a good relation between returned profit and spent harvest time that the cutting crews will try to cut these fields as often as possible. In this case, a cutting crew repeatedly harvests such a field f ∗ with a partial flow throughout the entire planning period. The previously inserted cut 1 then forces the solver to perform travelings with an equal flow value in order to compensate these cuts. In the linear relaxation’s optimal solution, these extra travels are performed without interfering the seamless harvesting in the desired field f ∗ . Figure 5.10 exemplifies this situation. The cutting crew prefers to repeatedly harvest field 2 (in this case three times). At the same time, a number of travels with an equal flow value are performed in order to compensate the repeated cuts. In flow for mechanical crews. Such a behavior can be avoided by adding constraints that force traveling at each field. The flow entering in a field and the flow leaving from a field are considered separately. The constraints that control the field entering flow are founded in the following observation: a cutting crew
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Figure 5.10: Fractional example route for a cutting crew after insertion of valid inequalities in order to force traveling. The first three travels between field 3 and 4 are performed to compensate the cutting flow used at field 1. can only invest a certain flow quantity in harvesting a field, if this flow quantity (or more) has been inserted into that field by traveling before. This results in the following inequality for mechanical cutting crews, containing a constant Φc,f which is explained afterwards:
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p∈P
XX
oic,f2 ,f1 + Φc,f1 ≥
f2 ∈F i∈I
X
i yc,f ; 1
i∈I
∀c ∈ CFmec , ∀f1 ∈ F
(5-3)
If the cutting crew is initially located at a plantation field, none of the f p variables will be active. Furthermore, if the current field is the field at which the crew is located at the beginning of the planning, then this must be considered as an entering flow as well. Thus, if c.initialLocation = f , then Φc,f = 1, otherwise Φc,f = 0. In flow for manual crews. In addition to traditional traveling, manual cutting crews also have the possibility to travel by making use of the f h variables. The inequalities for manual cutting crews are:
X p∈P
f pc,f1 ,p +
XX
oic,f2 ,f1 +
f2 ∈F i∈I
∀c ∈ CFman , ∀f1 ∈ F
X d∈D
f hdc,f1 + Φc,f1 ≥
X
i ; yc,f 1
i∈I
(5-4)
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Out flow for mechanical crews. Next, the inequalities to limit the outgoing flow are considered. Clearly, all flow that is invested in field cuts must also leave these field by making use of traveling variables. At the end of the planning, the flow may leave the network without use of any traveling; this flow unit is subtracted on the right hand side of the inequality. For mechanical cutting crews, the following inequalities hold at each field:
XX
oic,f1 ,f2 ≥
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f2 ∈F i∈I
X
i − 1; ∀c ∈ CFmec , ∀f1 ∈ F yc,f 1
(5-5)
i∈I
Out flow for manual crews. The f h variables inform at which field the flow enters, but not from which field the flow comes. Hence, outgoing flow inequalities for the manual cutting crews must consider any variables that may allow for traveling to other fields, i.e. all variables whose flow enters into the accommodation flow of the following night. This is the case for waiting and cutting variables whose flow would have entered into the first instant of a day, but were bypassed to the accommodation flow instead (see constraint (4-46) in Section 4.3.2). The inequality for the outgoing flow for manual cutting crews is stated as:
XX f2 ∈F i∈I
≥
X
oic,f1 ,f2 +
X
(d+1).f irstInstant
nc,f1
+
d∈D
XX
i,(d+1).f irstInstant
yc,f1
d∈D i∈I
i yc,f − 1; ∀c ∈ CFmec , ∀f1 ∈ F 1
(5-6)
i∈I
Since many cutting variables appear on both sides of the inequality, this constraint is likely to be not very tight. 5.4.4.3 Cut 3: Forcing travels at each field at each day In flow for mechanical crews. In order to strengthen the constraints for the in and out flow at each field, these inequalities can be stated for each single day. The input flow at each day for mechanical cutting crews is considered first. At the first day, the input flow may come from a mill or from the cutting crew if it is initially located at that field. Again, if c.initialLocation = f , then Φc,f = 1, otherwise Φc,f = 0.
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X
f pc,f1 ,p +
p∈P
XX X
i1 ,i2 oc,f + Φc,f1 ≥ 2 ,f1
f2 ∈F i1 ∈I i2 ∈Id0
X
i yc,f ; 1
i∈Id
∀c ∈ CFmec , ∀f1 ∈ F
(5-7)
During all days after the first one, flow will not come from a mill. Instead, it may come from a waiting variable n that leaves from the last instant of the previous day and enters at the first instant of the current day:
X X
(d−1).lastInstant
1 oic,f + nc,f1 2 ,f1
f2 ∈F i1 ∈Id
≥
X
i yc,f ; 1
i∈Id
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∀d ∈ D, ∀c ∈ CFmec , ∀f1 ∈ F
(5-8)
In flow for manual crews. In order to model the in flow inequalities for manual cutting crews, the in flow inequality for mechanical crews can be extended considering the use of the accommodation flow variables f h to travel. This results in the following inequality for the first day of the planning for manual cutting crews:
X
f pc,f1 ,p +
p∈P
XX X
i1 ,i2 oc,f + Φc,f1 ≥ 2 ,f1
f2 ∈F i1 ∈I i2 ∈Id0
X
i
d0 yc,f ; 1
i∈I
∀c ∈ CFman , ∀f1 ∈ F
(5-9)
For all further days, the following inequalities hold:
X X
(d−1).lastInstant
1 oic,f + nc,f1 2 ,f1
f2 ∈F i1 ∈Id
∀d ∈ D, ∀c ∈ CFman , ∀f1 ∈ F
+ f hdc,f ≥
X
i yc,f ; 1
i∈Id
(5-10)
Out flow for mechanical crews. The out flow inequalities for mechanical cutting crews have to consider all outgoing traveling variables at a certain day. In addition, they have to involve the waiting variables that leave from the last available instant of that day in order to pass the flow to the following day. The total flow accumulated by these variables must be greater or equal than all flow invested in field cuts terminating at the current day. This inequality can be formulated as:
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X X
1 oic,f + nd.lastInstant ≥ c,f1 1 ,f2
f2 ∈F i1 ∈Id
XX
i1 ,i2 yc,f ; 1
i1 ∈I i2 ∈Id
∀d ∈ D, ∀c ∈ CFmec , ∀f1 ∈ F
(5-11)
Out flow for manual crews. Just as in the out flow inequalities for manual cutting crews stated for each field (compare inequality (5-6)), the field cutting variables that initiate the field cut at the current day and terminate it at the last instant of the same day (i.e. the flow would be available again in the first instant of the following day) must be considered, because their flow passes into the accommodation variables f h.
X X f2 ∈F i1 ∈Id
1 oic,f + nd.lastInstant + c,f1 1 ,f2
X i∈Id
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∀d ∈ D, ∀c ∈ CFman , ∀f1 ∈ F
i,(d+1).f irstInstant
yc,f1
≥
XX
i1 ,i2 yc,f ; 1
i1 ∈I i2 ∈Id
(5-12)
Since many cutting variables appear on both sides of the inequality, this constraint is likely to be not very tight. 5.4.4.4 Computational evaluation of the cuts. Computational experiments were carried out in order to compare the impact of the cuts to the obtained upper bounds and the optimization process. The experiments involve a 15 minutes branching phase followed by a 15 minutes polishing phase. Each of the three cut types was considered: inequalities for the whole planning (labeled as Cut 1 ), inequalities for each field (labeled as Cut 2 ) and inequalities for each field at each day (labeled as Cut 3 ). The inequalities for the outgoing flow for the manual cutting crew in the latter two types are not likely to provide a tight cut, but involve a large number of constraints. Thus, the experiments for these inequality types were also performed without the outgoing flow inequalities for the manual crews (labeled as Cut 2 mo and Cut 3 mo). In addition, the combination of the cut types was tested: Cut 1 + 2 + 3 refers to the combination of all three cut types; Cut 1 + 2 mo + 3 mo represents the combination of all three cut types without the manual crews’ outgoing flow inequalities for cut 2 and 3. Table 5.12 resumes the experiments’ results. The results for the integer program without cuts are labeled as Without cuts. UBI represents the improvement of the first upper bound found (i.e. the value of the optimal so-
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lution to the linear relaxation of the original problem) compared with the first upper bound found within the execution Without cuts. The Dev % of each experiment represents the average deviation from optimum reported by ILOG CPLEX at the end of the optimization. If the linear relaxation was not solved within the given time, i.e. no upper bound was found, this instance was not considered in the average deviation value. Dev* % demonstrates the average deviation after the optimization, considering the best upper bound known for the instance. For each experiment type, the average time to solve the linear relaxation is reported (given in the line LP solve). The following number in parentheses represents the number of instances for which no optimal solution to the linear relaxation was found within the 15 available minutes. In this case, the average value assumes a solution time of 15 minutes for the instance.
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UBI 0.00 5.42 6.01 1.57 7.94 8.25 88.55 4.08 29.07 20.10
UBI 0.01 5.58 8.09 3.24 13.36 9.43 89.00 4.35 31.01 22.36
Cuts 2 Dev % Dev* % 10.80 10.80 6.02 6.01 4.76 4.41 3.70 3.69 12.87 11.77 1.62 1.65 0.89 0.91 6.93 6.63 4.16 3.79 5.16 4.85 105.66 sec (1) Cuts 1 + 2 + 3 Dev % Dev* % 41.08 41.08 6.90 6.90 2.28 4.49 1.73 3.20 17.22 15.80 0.94 0.94 0.94 0.94 18.49 15.05 4.50 4.46 8.89 8.43 208.39 sec (4)
UBI 0.00 5.55 7.77 3.59 19.03 9.69 88.71 4.76 32.22 22.36
UBI 0.01 5.59 9.93 2.94 19.12 10.54 89.60 4.35 33.80 24.55
Cuts 2 mo Dev % Dev* % 6.57 6.57 6.09 6.08 2.72 3.28 3.47 3.48 13.32 12.20 1.34 1.37 0.82 0.83 5.41 5.08 4.10 3.72 4.53 4.23 116.23 sec (2) Cuts 1 + 2 mo + Dev % Dev* % 9.32 9.32 7.87 7.86 1.95 3.58 2.56 2.48 20.14 17.56 1.20 1.17 1.11 1.13 7.32 6.54 5.29 4.75 5.91 5.42 221,96 sec (4)
Table 5.12: Influence of the cuts in the upper bounds and the optimization
UBI 0.00 5.55 7.33 3.59 19.03 9.69 88.71 4.39 32.22 22.70 3 mo UBI 0.01 5.58 9.96 3.92 19.12 10.54 89.60 4.35 32.80 24.11
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Without cuts Cuts 1 Dev % Dev* % UBI Dev % Dev* % Art20 10.72 10.72 2.68 2.68 1.99 1.98 Art50 8.15 2.72 Art100 10.87 4.84 6.58 4.50 5.18 4.25 GVO100 4.93 3.43 GVO102 25.64 18.17 16.20 9.62 3.07 1.10 GVO103 6.13 4.58 GVO106 7.43 0.99 0.68 0.58 Avg Artificial 9.86 4.97 3.83 3.16 Avg GVO 10.11 3.36 5.54 3.25 4.92 3.21 Avg All 10.02 3.96 LP solve 67.29 sec (1) 125.56 sec (1) Instance set Cuts 3 Cuts 3 mo Dev % Dev* % UBI Dev % Dev* % Art20 2.90 2.90 0.01 68.76 68.76 Art50 7.28 7.32 5.58 4.05 4.08 Art100 5.26 4.51 8.85 3.92 3.15 GVO100 1.85 1.70 3.34 1.73 3.10 23.42 16.04 GVO102 22.49 14.99 14.13 GVO103 0.95 1.19 9.80 1.64 1.45 0.96 1.00 GVO106 1.01 1.09 89.53 Avg Artificial 5.31 5.00 4.54 25.59 20.94 Avg GVO 5.49 3.87 31.44 6.00 4.47 12.53 10.64 Avg All 5.43 4.29 22.72 LP solve 218,80 sec (3) 125,39 sec (4)
Chapter 5. Solution Strategies
Instance set
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In all experiments, the addition of the inequalities complicated the resolution of the linear relaxation. Using any of these cuts, the original average resolution time of 67 seconds increased to at least 100 seconds. However, the use of the inequalities turned out to be very effective in respect to the quality of the final solutions, in particular for the GVO instances. The first upper bounds of the MIP, i.e. the optimal solutions to its linear relaxation, were lowered by at least 20% for all cut types. Cut types 2 and 3 improved these bounds slightly better than cut type 1. The joint use of all cut types led to a bound improvement of almost 25% in the total average and almost 33% for the GVO instances. The bound improvement for the artificial instances has been relatively small. Almost all cut types resulted in an improved average solution quality as well as in a better average of the proved deviation from optimum. The poor performance at instance set Art20 for the cut combinations Cuts 3 mo and Cuts 1 + 2 + 3 is caused by a bad quality final solution for instance Art20 1 4. The remarkable improvement at instance set GVO106 resulted almost in proved optimality. However, the solutions themselves within this instance set marginally improved. In almost all instance sets and cut types, the reported final deviation as well as the final solution improved. The reported final deviation from optimum reduced from 10.02% to 4.53%. As suspected, the outgoing flow inequalities for manual cutting crews of cut type 2 and 3 did not show much effect in the upper bound improvement. Instead, they led to a higher average time to solve the linear relaxation. However, there is no clear dominance of one of the cut types with or without outgoing flow inequalities for manual cutting crews. The combinations of all three cut types resulted in the best bound improvement, but increased the average solution time of the linear relaxation as well as the average deviations from optimum. In conclusion, it is observed that the use of Cut 1 results in a very good average solution quality and may therefore be recommended for the use within the problem’s mathematical model for time limited executions. At this point it is worth to mention that, applying the valid inequalities, the linear relaxation’s optimal solution is less fractional than before, but still does not result in variables set to one. 5.5 Alternative solution strategies for the SCHP-OP Frameworks based on Branch-and-Bound for mathematical programming aim at solving a generic problem. However, there are several possibilities to
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involve semantic information of a problem to solve it. Inspired by problems from practice, there are many approaches and strategies to manage such huge problems [Pog08]. Typical strategies that may include such semantic information are: – Temporal variation. A variable period size is used: larger as it goes into the future – Temporal decomposition. The problems first solved for period smaller than the total planning horizon. The solution values are fixed and the next period solved. This is repeated until the entire problem is solved.
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– Split and assembling. The problem is divided into subproblems. The problem knowledge may indicate the best sequence to solve the sub problems. – Segregation and aggregation. The problem’s planning is divided into a number of subproblems. Each subproblem, with a considerable smaller problem size, is now solved. Finally, the solutions of the subproblems are composed to one problem, which is then solved. – Hierarchic decomposition. The problem is solve for the most important group of variables first. Afterwards, the other variables are solved. – Objective function or constraint sequencing. The problem is solved for a simpler objective functions and subsets of constraints. In the context of harvest scheduling in sugar cane industry, given by the SCHP-OP, there are several opportunities to apply such semantic information to solve the problem: – The planning horizon can be split into smaller time periods, for example into weeks. Having information about the indicated week for each plantation field, the fields can be separated into each week so that the final problem is much smaller. – The objective function can be simplified through disregarding certain variables. For example, travel or harvest costs only marginally influence the objective function may therefore be disregarded from the objective function. – The problem can be segregated into a separated planning for each cutting crew. After solving all subproblems, the solutions to these subproblems are composed to one solution feasible for the entire problem.
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5.5.1 Segregation and Aggregation of the cutting crews’ planning This section investigates a solution strategy based on segregation of the original problem. The strategy benefits from the fact that the networks of the cutting crews are independent one from each other. Considering each cutting crew’s planning separately, the problem can be divided into one subproblem for each cutting crew. In this case, the plantation fields and the mills are the only resources used in common by the cutting crews. Particularly for instances that contain cutting crews with mostly disjoint field sets, this solution strategy is a very interesting and promising approach. The cutting crews’ planning is performed one after another. After the solution of each problem, the set of fields that have not been selected and the processing demands of the sugar cane mills are updated. Algorithm 3 outlines the pseudo code of the implemented method. Algorithm 3: Problem Segregation by cutting crews Input : A set of n cutting crews c A set of fields F available for harvesting Daily min/max demands of the mills M illCapac Output: Solution determine cutting crew order c1 , . . . , cn ; residualF ields ←− F ; for each cutting crew ci , i = 1, . . . , n in order do create MIP for the subproblem for ci ; solve subproblem considering residualF ields and M illCapac ; save solution for this subproblem ; update residualF ields ; update M illCapac ; end create MIP for entire problem ; aggregate the solutions of the subproblems to one solution for the overall problem ; pass this solution to the solver ; solve entire problem ; The sequence in which the planning of the cutting crews are performed clearly influences the final solution. In addition to the sequence given within the input data of the SCHP-OP, an ordering based on the number of plantation fields within the cutting crew’s field set, i.e. the number of fields that a crew may possibly cut, is considered. There are different motivations to consider either an increasing order or a decreasing order:
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1. Consider that there is a reasonably high intersection in the field sets of the cutting crews (for example, at least half of the fields of a crew can also be cut by other crews). Consider a cutting crew cs with a small set of fields and a cutting crew cb with a large set of fields. If the planning for cb is performed before the planning for cs , then the first planning may select fields that can also be cut by cs . Consequently, cs has even less options to cut fields than it had before. This may lead to a solution where cs does not make optimal use of its available time, because there are no more fields that it can cut. In fact, cb could have selected other fields than the ones shared with cs . An increasing order by the number of fields for each crew tries to avoid this problem. 2. Consider that the problem possesses tight processing demands of the sugar cane mills (i.e. the possible quantity of total cut sugar cane quantity is close to these processing demands) and/or tight vinasse demands (i.e. the possible number of total harvested field area is close to these vinasse demands). A not optimal solutions may violate the corresponding constraints. Consider a cutting crew cs with a field set sufficiently small so that, even cutting all its fields, the crew still has spare time. If the planning for cs is performed as one of the last plannings, it may have a high potential to distribute its cuttings. Hence, the solutions tend to violate less the mills and/or vinasse demands. This motivation leads to a decreasing order by the number of fields within the cutting crews’ field sets. Experiments were performed with these different cutting crew orderings. All generated problems and subproblems used the standard configuration for distance filtering, i.e. the MST plus node balancing with up to 100,000 travel variables (see Section 5.3.1). Furthermore, the ten best solutions obtained by the heuristic initial solution approach HeurRandSeq (see Section 5.4.2) were passed to the solver. All executions were limited to 30 minutes. The normal optimization used 15 minutes of branching and 15 minutes of polishing. The segregation approaches divided either the first 20 minutes or the first 25 minutes into smaller optimization periods, one for each cutting crew. The remaining time (i.e. either 10 or 5 minutes) was used to improve the overall problem. The subproblems for each cutting crew spent 23 of the time in branching and the remaining 31 of the time in polishing. For all branching phases. Table 5.14 shows the average deviations (from the best known solutions) for all instance sets for different solution strategies. The deviations for each
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instance can be found in Appendix B in Table B.4 and Table B.5. The original problem (i.e. the problem without the use of valid inequalities and without crew segregation) is labeled as Normal. Segr20 represents a problem segregation that divides 20 minutes into the subproblems for the cutting crews’ plannings and uses ten minutes to improve the overall solution. Segr25 invests the time by dividing 25 minutes distributed to the planning of each crew and five minutes to improve the overall solution. No cuts denotes the original model without any of the valid inequalities presented in Section 5.4.4. Cut 1 represents the use of cut type 1 etc. The label Incr indicates an increasing order of the cutting crews by its number of fields, whereas Decr indicates a decreasing order. It can be observed that some of the segregation configurations led to a remarkable improvement of the average deviations. For many configurations, instance Art20 1 4 turned out to be an extreme outlier again, worsening the average deviations for the instance set Art20. The results for the experiments with 20 and 25 minutes for the separated crew planning are very similar in their total average deviations. The 25 minutes configuration performed better on the GVO instances, whereas the 20 minutes configuration performed better on the artificial instances. An interesting observation is that the original order of the cutting crews determined within the input data led to good results in almost all instance sets. The increasing ordering of the cutting crews tends to perform better on the GVO instances. Disregarding instance Art20 1 4, the decreasing order demonstrated better results on the artifical instances. A very strong solution quality difference is perceived for instance set GVO102. Table 5.13 focuses on these instances and compares configurations with increasing and decreasing ordering sequences with and without cuts. For each configuration, the deviation of the final solution value from the best known solution, the number of fields that were not harvested (CutOblig) and the total number of harvested sugar cane in tons (Cut Sum) is reported. All instances contain two cutting crews with an occupation rate (see Section 5.1.1 for a formal definition of the occupation rate) lower than 60% and three cutting crews with an occupation rate higher than 150%. Although the average number of not harvested fields is almost the same in all approaches, the total sum of cut sugar cane is much higher within the solutions of the configuration with the increasing crew ordering. This behavior, observed throughout all instances of this set, is likely to confirm the previously stated assumption made for a decreasing ordering. In a decreasing ordering, the planning for the two crews with few fields are performed as the last ones. As the field intersection (IndF I ) is very high, it is probable that the firstly
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planned cutting crews (the ones with large field sets) selected fields that can also be cut by the two crews with small field sets. Hence, the two crews with small field sets had less options to harvest fields and resulted in more spare time. The increasing ordering preferred the planning of the two crews with small field sets. In this configuration, these crews were able to select all fields within their field sets. Hence, the following crews selected other fields (as they had more options). As a result, the crews with small field sets were able to cut more fields and the other crews harvested the same cane quantity as in the decreasing order approach. This behavior suggests that an increasing order is selected whenever one or more cutting crews possess a field intersection index lower than 100. In summary, the segregation solution strategy remarkably improved the results of the traditional solution approach (branching and polishing on the entire problem). Some configurations reached an average deviation of almost 3% for the artificial instances and 1.38% for the GVO instances.
GVO102 GVO102 GVO102 GVO102 GVO102 Avg
2 3 4 5 6
GVO102 GVO102 GVO102 GVO102 GVO102 Avg
2 3 4 5 6
Segr25. Dev % 5.68 8.33 4.09 2.68 2.29 4.61 Segr25. Dev % 4.48 9.40 3.17 2.98 3.06 4.62
No Cuts. Incr CutOblig Cut Sum 64 78700 58 83091 58 80274 59 80887 62 81151 60.20 80820.69 Cuts 1. Incr CutOblig Cut Sum 63.00 78702 58.00 82950 59.00 80483 59.00 80861 62.00 81001 60.20 80799.318
Segr25. Dev % 34.44 13.32 21.44 22.51 37.92 25.93 Segr25. Dev % 6.32 40.73 31.09 18.39 29.79 25.26
No Cuts. Decr CutOblig Cut Sum 63 70844 58 82028 62 74839 56 74803 59 70344 59.60 74571.41 Cuts 1. Decr CutOblig Cut Sum 64.00 78295 61.00 75143 65.00 70526 56.00 76218 57.00 73611 60.60 74758.416
Table 5.13: Influence of increasing and decreasing cutting crew order in segregation strategy to the cut cane quantity
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Normal
Art20 Art50 Art100 GVO100 GVO102 GVO103 GVO106 Artificial GVO All
10.72 2.68 3.06 2.35 9.75 2.25 0.76 4.97 3.36 3.96
Segr20 Segr20 Segr25 Segr25 Segr25 Segr25 Segr25 Segr25 No cuts No cuts No cuts No cuts Cuts 1 Cuts 1 Cuts 12mo3mo Cuts 123 Incr Orig Incr Decr Incr Decr Incr Incr 37.02 3.02 16.83 2.88 2.90 14.40 24.87 98.29 6.24 6.26 6.05 3.90 6.07 3.77 7.52 8.25 3.49 5.34 3.43 2.56 3.66 2.70 4.12 4.09 1.54 1.42 1.56 1.48 1.74 1.68 1.34 1.55 4.50 4.13 4.61 25.93 4.62 25.26 3.76 3.54 1.00 1.23 0.95 1.07 0.67 0.70 0.61 0.75 0.45 0.59 0.52 0.51 0.54 0.43 0.47 0.41 13.35 5.03 7.88 3.09 4.26 6.17 10.79 30.60 1.67 1.68 1.71 5.98 1.68 5.77 1.38 1.40 6.05 2.94 4.02 4.90 2.65 5.92 4.90 12.35
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Instance
Table 5.14: Comparison of solution quality of different solving strategies
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LP-and-Fix Heuristic In some cases, the MIP formulation of a problem may result in only slight fractionation of the integer variables within the linear relaxation’s optimum solution. Many values for such variables may even be integral. The decisions made by an optimal solution for the linear relaxation may not be very different from the decisions made in an optimal integer solution. Many solution approaches are based on the optimal solution of the linear relaxation. They fix certain variables within the integer solution, based on the decisions made by the linear relaxation’s solution. After the variable fixings, the MIP problem is usually much smaller and therefore easier to solve. Such approaches are part of a heuristic branch usually referred to as constructional MIP heuristics, as they aim at heuristically construction of feasible solutions. Typical examples for constructional MIP heuristics are LP-and-Fix, Cut-andFix and Relax-and-Fix. Pochet and Wolsey [PW06] provide an introduction into this topic. Large instances for the SCHP-OP, whose entire MIP model was hard to solve by the solver, motivated the efforts in this work to implement the LP-andFix heuristic. The approach is based on the assumption that the harvesting of a field is not important for the integer solution if it was not considered relevant in the optimal solution of its linear relaxation. At the same time, if a cutting variable of a field has value one, then it is assumed that harvesting this field is also very important to the integer solution. The implemented method first solves the linear relaxation of the problem. Afterwards, it analyzes, for each plantation field, all cutting variables within the linear relaxation’s optimal solution x. If a field was not cut throughout the entire planning period, all cutting, travel and processing variables for that field are fixed to zero. All variables within x with a value of one are fixed to one. The experiment performed includes the solution of the linear relaxation with a time limit of 15 minutes. After variable fixing, the time remaining from the available 15 minutes were spent into the branching phase of the partially fixed problem. Finally, these solutions were improved in a 15 minutes polishing phase. Table 5.15 presents the results for this experiment. It presents the percentage of fields that were not harvested by the linear relaxation’s optimal solution, the time spent to solve the linear relaxation (sec LP ), the reported deviation from optimum for the partially fixed problem (dev % ) and the real deviation of the solutions found (dev ∗ %) based on the best upper bounds
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throughout the experiments performed within this work. Normal denotes the results for the solution of the original model (without variable fixing), assuming the standard configuration for the problem (see Section 5). The average deviation for the LP-and-Fix heuristic (dev ∗ %) is based on the deviation values of all instances for which a solution was found. In order to provide a fair comparison of the performance of both solution approaches, the average deviation for the normal configuration is based on the solutions of the same instances as the average deviation for the LP-and-Fix heuristic. An analysis of the linear relaxation’s optimal solution showed that cutting variables were rarely fixed to one. However, as the results show, a high number of fields’ variables (more than 40% of the fields in average) were fixed to zero. This indeed demonstrates that the linear relaxation’s optimal solutions consider a large part of the fields as nonrelevant. An important observation within the results for the LP-and-Fix heuristic is that, for many instances, the solver was not able to find feasible solutions. The final deviations from optimum tend to be significantly worse than the results of the optimization of the original model. This is likely to be related to the following two facts: 1. Solution time of the linear relaxation. When solving a mixed integer program with ILOG CPLEX, the solver performs several pre-processing steps in order to reduce the problem size and strengthen the linear relaxation. In the experiments performed, the original MIP was relaxed to a linear problem and solved afterwards. At that moment, the MIP solver did not consider the problem as a MIP, but as a pure LP. Consequently, it did not apply all pre-processing steps that it would apply to a MIP. This usually leads to a longer solution time of the linear relaxation which, in turn, results in less time for finding feasible solutions and the optimization. 2. Limited solution space. The set of harvested fields in good quality solutions for the linear relaxation may strongly differ from the set of harvested fields in good quality solutions for the original MIP. The exclusion of several fields from the planning may hence exclude good quality solutions from the solution space. In addition to the previous presented solution approaches, the improvement heuristic Local Branching, introduced by Fischetti and Lodi [FL03], was tested for this problem. Throughout the experiments performed, this approach did not lead to good results.
Chapter 5. Solution Strategies
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Instance Art20 1 1 Art20 1 2 Art20 1 3 Art20 1 4 Art50 2 1 Art50 3 1 Art50 3 2 Art50 3 3 Art50 4 1 Art100 1 1 Art100 1 2 Art100 2 1 Art100 2 2 Art100 3 1 Art100 3 2 GVO100 1 GVO100 2 GVO100 3 GVO100 4 GVO100 5 GVO100 6 GVO102 2 GVO102 3 GVO102 4 GVO102 5 GVO102 6 GVO103 1 GVO103 2 GVO103 3 GVO103 4 GVO103 5 GVO103 6 GVO103 7 GVO106 1 GVO106 2 GVO106 3 GVO106 4 GVO106 5 GVO106 6 GVO106 7 Avg
LP-and-Fix Normal Fixed fields % sec LP dev % dev* % dev* % 0.00 7 0.00 0.00 0.00 0.00 7 0.00 0.00 0.00 0.00 9 0.36 0.36 0.11 0.00 8 99.21 99.21 11.23 0.00 156 21.36 8.00 60 32.99 22.65 17.16 8.00 70 70.81 8.00 73 209.23 182.96 67.87 0.00 0 3.07 70.00 100 - 163.43 70.00 104 - 118.56 9.00 141 - 109.71 9.00 214 1.00 74 1.00 96 45.06 364 82.34 295 10.72 9.70 1.15 82.34 433 4.24 3.35 2.61 82.34 309 2.94 1.73 1.22 82.04 286 2.85 1.68 2.51 82.34 392 4.06 2.90 1.64 53.70 63 25.25 50.00 62 42.72 51.85 49 21.32 50.93 59 11.62 51.85 93 15.56 53.45 14 2.96 0.40 0.00 83.23 52 7.78 6.01 1.57 52.46 24 15.44 7.84 0.05 52.46 19 0.64 0.00 0.00 81.03 339 17.26 12.36 1.84 83.91 352 78.70 74.66 7.60 85.23 48 12.58 10.69 4.46 6.25 148 20.87 16.07 0.52 0.00 1 0.00 0.00 0.00 60.87 <1 0.00 0.00 0.00 73.39 37 3.09 3.41 0.82 0.00 2 41.81 0.00 0.00 0.00 2 59.92 1.89 0.00 72.36 63 11.62 6.80 4.56 41.11 25.57 18.59 5.08
Table 5.15: Results of the LP-and-Fix heuristic
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6 Conclusions
This thesis investigated mixed integer formulations and algorithms for the planning of sugar cane cultivation and harvesting, referred to as the Sugar Cane Cultivation and Harvesting problem (SCHP). The objective in such planning is the maximization of the total profit given by the quantity of sugar production. The quantity of sugar depends on the sucrose (Pol) and reduced sugar (AR) content within the cane as well as on the total quantity of cane. As such values vary throughout the cane maturation, it is desirable to harvest a field at the maturation peak of the cane. In addition to the determination of the optimal moment to harvest the fields, the planning includes assignments of the cutting and transportation crews. It also suggests the application of vinasse in harvested field area and the application of maturation products in order to increase the sucrose percentage within the cane. The entire planning is subject to several industrial and resource constraints. These constraints include the availability of cutting and transportation crews as well as their cutting and transportation capacities, respectively. Sugar cane mills contain certain processing minimum demands and maximum processing capacities. The implemented decision support system was divided into a tactical (SCHP-TP) and an operational module (SCHP-OP). Both problems were presented in detail. The former performs the planning for up to a whole harvest season, i.e. approximately twelve months. Its decisions are made for each week. The latter performs a daily planning for up to 30 days and involves the determination of the harvest sequence for each cutting crew. The GVO, a Brazilian sugar cane producer, was involved in the whole development process in order to guarantee a realistic presentation of the processes. Instances from practice, based on the current harvest planning of the GVO, were provided in order to evaluate the modules. The MIP model of the problem extends the classical Packing formulation, incorporating a network flow for the harvest scheduling. Both problems, the SCHP-TP as well as the SCHP-OP, were proved to be NP-complete. The MIP solver ILOG CPLEX version 11.2 was used to solve the problems. The tactical module turned out to be easily solved to optimality as experiments
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Chapter 6. Conclusions
136
with instances from practice demonstrated. In contrast to the tactical planning problem, experiments with artificial instances and instances from practice demonstrated that the basic use of CPLEX for the operational planning model was not sufficient to solve the problem in a reasonable time of up to 30 minutes. Several pre-processing techniques were used to reduce the size of the problem. Such techniques include distance filtering, reduction tests for network flows, variable pruning and the grouping of similar fields. Distance filtering turned out to be a very important task as it effectively reduced the problem size. Though the limitation of distances used within the model results in a worse optimal solutions for the problem, the final solutions obtained in time limited experiments improved significantly. Initial solutions were constructed by several greedy heuristics. As the MIP solver showed strong difficulties in finding feasible initial solutions, a set of the constructed initial solution were passed to the solver, leading to a fast improvement of the final solutions’ quality. The linear relaxation’s optimal solution proved to be strongly fractional. Further analysis of these solutions led to valid inequalities to strengthen the MIP formulation. These inequalities improved the upper bounds by more than 20% and led to an improvement of the final solutions. As an additional effect, the final solutions were proved to be close to optimality. Furthermore, a problem segregation strategy based on semantic information was presented. This resolution strategy divides the problem in one subproblem per cutting crew. The subproblems are then solved one after another. Different cutting crew ordering sequences were experimented in order to identify the best order for each type of instance. These sequences consider the number of fields that each cutting crew is able to be assigned to. The experiments showed that the sequence of cutting crews sorted by increasing number of field sets leads to better results than if sorted by decreasing or arbitrary order. The results obtained by this solution strategy outperforms the solution strategy that involves the optimization of the entire problem. During the development process, alternative algorithms were experimented. These include Local Branching and heuristics based on the optimal solution of the linear relaxation such as LP-and-Fix. The performed experiments based on these algorithms did not result in competitive solutions. The experiences made within this work showed that the presented algorithms are very effective to solve even larger problems as they appear in practice in reasonable time. Some of the solution strategies that were suggested solved the instance benchmark only a few percents from optimality.
Chapter 6. Conclusions
137
The presented model considers numerous industrial and resource constraints that are crucial to the sugar cane cultivation and harvest planning. Due to this close representation of the real processes, it is likely that the obtained solutions are feasible in practice. Members of the GVO performed extensive evaluations of the generated solutions and confirmed the feasibility as well as the high quality of the obtained solutions. These results suggest that the use of the decision support system presented in this work is likely to make better use of the resources and the sugar cane itself and thus comprises a strong potential to increase the total profit to sugar cane companies and, consequently, reduce prices to the society.
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6.1 Future work This work aimed at providing a decision support tool for sugar cane harvest planning in practice. Experiences that will be made in the future through the use of this tool may show that some of the model’s functionalities or constraints still do not perfectly meet the requirements in practice. A vinasse application closer to the practice is one of such aspects and will be implemented soon. In the models presented in this thesis, the demands for vinasse application have to be satisfied by fields whose harvesting terminates at the same week/day of the vinasse demand. For tight vinasse demands, i.e. demands where the required area is close to the total harvested area, the demands may be difficult to meet. However, in practice, vinasse may be applied on a field up to two months after it is harvested. The use of such constraints instead of the ones involved in the current models will facilitate meeting tight vinasse demands. From the view point of the problem’s mathematical model and applied solution strategies, it is surely worth to invest in the search of more valid inequalities that may help to break the strong fractional characteristics of the linear relaxation’s optimum solution. Since the problem incorporates several assignments as they can be found in the classical GAP, the use of classic cuts that appeared for the GAP may be promising. The distance filtering pre-processing may also be worth further research. Good quality solutions demonstrated that only a small amount of the distances are used in the cutting crews’ routes. In addition, not all fields may be harvested in instances with a large number of fields. A deeper evaluation of the characteristics of the harvested fields may help identifying fields that are more important than other ones and, consequently, are more likely to be harvested in a good quality solution. Such knowledge may help distributing the distances
Chapter 6. Conclusions
138
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according to the importance of a field. Finally, the models presented in this work hold potential to be used to simulate the competitiveness and feasibility of scenarios with different resource configurations. For example, the effect of using more mechanical cutting fronts or more mills can be tested. Such a simulation optimization is realized by introducing new variables that dimension the use of certain resources and may identify great potential to further reduce costs.
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A Glossary
ABACUS A Branch-And-CUt System, a non-commercial MIP solver ANN Artificial Neural Network AR Percentage of reduced sugar within the cane; sometimes also referred to as reducing sugar. ATR Total recoverable sugar (as a result of cane processing)
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BB Branch and Bound, an algorithm to solve mixed integer programs. BC Branch and Cut, an algorithm to solve mixed integer programs. COIN-OR Computational Infrastructure for Operations Research, an Open Source project for the Operations Research community CVRP Capacitated Vehicle Routing Problem, a variation of the VRP DSS Decision Support System GA Genetic Algorithm GAP Generalized Assignment Problem GLPK GNU Linear Programming Kit, a non-commercial MIP solver GVO Grupo Virgolina de Oliveira, a Brazilian sugar company IP Integer Program LP Linear Program MIP Mixed Integer Program MILP Mixed Integer Linear Program, synonym for MIP MINTO Mixed Integer Optimizer, a non-commercial MIP solver M-TSP Multiple Traveling Salesman Problem, generalization of the TSP
Appendix A. Glossary
147
NP Non-deterministic polynomial time OR Operations Research OSI Open Solver Interface, an interface for linear relaxation solver PCC Apparent sucrose percentage in the sugar cane, also refered to the sugar content within the cane Pol see PCC PVRP Periodic Vehicle Routing Problem, a variation of the VRP SCHP Sugar Cane Cultivation and Harvest Problem SCHP-TP Tactical Planning Problem of the SCHP SCHP-OP Operational Planning Problem of the SCHP
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SCIP Solving Constraint Integer Programs, a non-commercial MIP solver TCH Tons of sugar cane per hectare TSP Traveling Salesman Problem, classical path finding problem from Combinatorial Optimization VRP Vehicle Routing Problem, classical routing problem from Combinatorial Optimization VRPTW Vehicle Routing Problem with Time Windows, a variation of the VRP
B Instance properties for the SCHP-OP and solutions for the segregation strategy
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This appendix lists all used instances and their most important properties. Table B.1 refers to the instances of the sets GVO100 and GVO102. Table B.2 shows the instance properties for the sets GVO103 and GVO106. The properties for the artificial instants are shown in Table B.3. Finally, Table B.4 and Table B.5 list the gaps separately for all artificial and GVO instances for the experiments of the problem segregation strategy (see Section 5.5.1).
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# Plants
# Fields
# # CFs CFs Man. Mech.
# IndF I days Vin
GVO100 100 1
16
2
233
16
5
0
100 2
16
2
334
16
5
100 3
16
2
334
16
100 4
16
2
334
100 5
16
2
100 6
16
GVO102 102 2 102 3 102 4 102 5 102 6
15 15 15 15 15
IndCO
IndM P
2.10
73.85
96.51
0
2.34
550.68
0.00
5
0
2.34
550.68
0.00
16
5
0
2.34
550.68
0.00
334
16
5
0
2.34
550.68
0.00
2
334
16
5
0
2.34
550.68
0.00
1 1 1 1 1
108 108 108 108 108
2 2 2 2 2
3 3 3 3 3
0 0 0 0 0
12.20 12.20 12.20 12.20 12.20
117.86 120.36 117.03 115.68 121.05
87.88 93.59 99.73 96.12 96.71
IndCO for each CF
352; 41; 139; 0; 64; 35; 16; 50; 5; 81; 0; 0; 260; 46; 199; 22; 33; 71; 11; 13; 0 2715; 173; 545; 0; 473; 249; 80; 242; 19; 394; 0; 0; 1177; 716; 1757; 113; 240; 950; 1284; 1313; 0 2715; 173; 545; 0; 473; 249; 80; 242; 19; 394; 0; 0; 1177; 716; 1757; 113; 240; 950; 1284; 1313; 0 2715; 173; 545; 0; 473; 249; 80; 242; 19; 394; 0; 0; 1177; 716; 1757; 113; 240; 950; 1284; 1313; 0 2715; 173; 545; 0; 473; 249; 80; 242; 19; 394; 0; 0; 1177; 716; 1757; 113; 240; 950; 1284; 1313; 0 2715; 173; 545; 0; 473; 249; 80; 242; 19; 394; 0; 0; 1177; 716; 1757; 113; 240; 950; 1284; 1313; 0 31; 36; 32; 31; 34;
50; 58; 51; 50; 55;
151; 173; 154; 151; 165;
186; 145; 171; 174; 163;
Table B.1: GVO instance sets 100 and 102 for the SCHP-OP
521 405 480 486 456
# Var
# Const
197855
46895
233283
67086
232943
67086
233797
67086
232267
67086
231953
67086
117351 123021 117591 117621 118911
17772 19437 17772 17772 18267
149
# Days
Appendix B. Instance properties for the SCHP-OP and solutions for the segregation strategy
Instance Name
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GVO103 103 1 103 2 103 3 103 4 103 5 103 6 103 7 GVO106 106 1 106 2 106 3 106 4 106 5 106 6 106 7
# Days
# Plants
# Fields
# CFs Man.
# CFs Mech.
# IndF I days Vin
7 7 7 7 7 7 7
1 1 1 1 1 1 1
58 167 61 61 174 174 176
2 2 2 2 2 2 2
3 3 3 3 3 3 3
0 0 0 0 0 0 0
15 7 7 7 7 7 7
1 1 1 1 1 1 1
48 19 23 109 23 23 123
2 2 2 2 2 2 2
3 3 3 3 3 3 3
0 0 0 0 0 0 0
# Var
# Const
59; 257; 8; 327 609; 682; 256; 696 69; 270; 14; 338 66; 2368; 14; 323 690; 716; 272; 727 688; 698; 267; 712 624; 714; 265; 709
56433 113263 51731 57261 109173 110965 114883
5154 15099 4694 5121 13944 14700 15882
89; 21; 118; 16; 90 54; 53; 0; 37; 78 613; 357; 151; 151; 129 528; 499; 345; 308; 456 85; 93; 0; 143; 89 79; 85; 0; 132; 82 860; 872; 397; 785; 585
81861 10806 6901 100459 11133 11133 95393
9102 1795 2067 9787 1821 1821 9045
IndCO
IndM P
IndCO for each CF
0.00 3.47 0.00 0.00 3.33 3.33 3.30
176.07 506.05 190.94 214.40 542.88 533.95 522.37
76.28 76.28 88.65 110.26 80.91 94.16 76.28
171; 394; 201; 192; 449; 448; 398;
2.04 5.00 0.00 2.68 8.00 8.00 4.65
69.49 36.26 207.43 396.31 69.01 65.12 600.78
129.45 215.24 0.00 78.04 136.86 140.88 94.44
Table B.2: GVO instance sets 103 and 106 for the SCHP-OP
Appendix B. Instance properties for the SCHP-OP and solutions for the segregation strategy
Instance Name
150
PUC-Rio - Certificação Digital Nº 0711327/CA
Art20 InputTest20 1 1 InputTest20 1 2 InputTest20 1 3 InputTest20 1 4 Art50 InputTest50 2 1 InputTest50 3 1 InputTest50 3 2 InputTest50 3 3 InputTest50 4 1 Art100 InputTest100 1 1 InputTest100 1 2 InputTest100 2 1 InputTest100 2 2 InputTest100 3 1 InputTest100 3 2
# Days
# # Plants Fields
# CFs Man.
# CFs Mech.
# IndF I days Vin
IndCO
IndM P
IndCO for each CF
# Const
44741 44756 44756 44756
3747 3762 3762 3762
15 15 15 15
2 2 2 2
20 20 20 20
1 1 1 1
1 1 1 1
5 5 5 5
33.33 33.33 33.33 33.33
39.86 39.86 39.86 39.86
0.00 40.91 81.82 92.73
80; 80; 80; 80;
30 15 15 15 15
2 2 2 2 2
50 50 50 50 50
1 1 1 1 1
1 1 1 1 2
0 0 0 0 0
50.00 50 50.00 50.00 66.67
49.62 99.64 99.64 99.64 60.31
0.00 0.00 43.64 61.09 0.00
142; 286; 286; 286; 286;
179480 135756 135771 135771 167525
22632 11292 11307 11307 18808
15 15 15 15 30 30
2 2 2 2 2 2
100 100 100 100 100 100
1 1 2 2 2 2
1 1 2 2 2 2
0 0 0 0 0 0
50.00 50.00 75 75 16.67 16.67
199.28 199.28 99.64 99.64 49.82 49.82
0.00 65.22 0.00 54.55 0.00 32.73
572; 305 177731 572; 305 177746 572; 572; 305; 305 255413 572; 572; 305; 305 255428 69; 74; 52; 54 174309 69; 74; 52; 54 174339
22474 22489 43205 43220 30515 30545
Table B.3: Sets of artificial instances
48 48 48 48
# Var
76 152 152 152 152; 152
Appendix B. Instance properties for the SCHP-OP and solutions for the segregation strategy
Instance Name
151
PUC-Rio - Certificação Digital Nº 0711327/CA
Art20 1 1.xml Art20 1 2.xml Art20 1 3.xml Art20 1 4.xml Avg Art50 2 1.xml Art50 3 1.xml Art50 3 2.xml Art50 3 3.xml Art50 4 1.xml Avg Art100 1 1.xml Art100 1 2.xml Art100 2 1.xml Art100 2 2.xml Art100 3 1.xml Art100 3 2.xml Avg Artificial Avg
Normal
0.00 0.00 0.33 42.54 10.72 0.25 2.68 4.52 5.78 0.14 2.68 1.98 2.18 4.75 6.03 0.73 2.67 3.06 4.97
Segr20 Segr20 Segr25 Segr25 Segr25 Segr25 Segr25 No cuts No cuts No cuts No cuts Cuts 1 Cuts 1 Cuts 12mo3mo Incr Orig Incr Decr Incr Decr Incr 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.31 0.39 0.37 0.22 0.39 0.36 3.23 147.76 11.70 66.94 11.28 11.23 57.22 96.24 37.02 3.02 16.83 2.88 2.90 14.40 24.87 4.00 1.21 1.46 10.16 1.47 10.22 1.51 4.59 4.88 4.84 1.86 4.89 1.52 4.93 10.93 12.59 12.11 2.49 12.56 2.72 14.16 11.04 11.23 11.23 3.94 10.99 4.15 11.88 0.62 1.42 0.61 1.07 0.42 0.24 5.13 6.24 6.26 6.05 3.90 6.07 3.77 7.52 3.06 3.54 3.67 1.09 3.59 1.09 3.59 3.04 5.50 3.25 1.09 3.02 1.09 3.67 6.91 11.80 6.11 5.71 6.40 6.11 7.65 5.32 8.72 5.08 5.56 6.14 5.93 5.97 0.75 0.62 0.96 0.97 0.97 1.07 1.43 1.87 1.88 1.51 0.94 1.82 0.89 2.40 3.49 5.34 3.43 2.56 3.66 2.70 4.12 13.35 5.03 7.88 3.09 4.26 6.17 10.79
Segr25 Cuts 123 Incr 0.00 0.05 0.45 392.66 98.29 1.51 11.12 11.34 12.18 5.12 8.25 3.67 3.67 7.14 6.01 1.71 2.33 4.09 30.60
Appendix B. Instance properties for the SCHP-OP and solutions for the segregation strategy
Instance
Table B.4: Results for all artificial instances for the segregation solution strategy 152
PUC-Rio - Certificação Digital Nº 0711327/CA
5.37 1.75 1.22 1.54 2.23 2.00 2.35 9.40 18.72 3.84 7.81 8.97 9.75 0.00 2.49 0.00 0.00 4.01 4.85 4.38 2.25 0.91 0.00 0.00 1.04 0.00 0.00 3.38 0.76 3.36
Segr20 Segr20 Segr25 Segr25 Segr25 Segr25 Segr25 No cuts No cuts No cuts No cuts Cuts 1 Cuts 1 Cuts 12mo3mo Incr Orig Incr Decr Incr Decr Incr 6.52 5.08 6.55 5.46 6.70 5.48 5.47 0.44 0.66 0.44 0.66 0.64 0.57 0.44 0.64 0.79 0.57 1.04 1.01 0.75 0.57 0.54 0.95 0.54 0.47 0.81 0.79 0.47 0.49 0.49 0.65 0.69 0.65 0.90 0.49 0.62 0.58 0.57 0.57 0.62 1.59 0.57 1.54 1.42 1.56 1.48 1.74 1.68 1.34 4.05 3.24 5.68 34.44 4.48 6.32 5.31 7.97 9.45 8.33 13.32 9.40 40.73 6.09 3.25 2.33 4.09 21.44 3.17 31.09 3.19 3.10 2.21 2.68 22.51 2.98 18.39 2.98 4.13 3.43 2.29 37.92 3.06 29.79 1.23 4.50 4.13 4.61 25.93 4.62 25.26 3.76 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.44 2.39 2.23 2.61 1.37 1.64 1.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.22 3.20 1.57 2.16 1.40 0.85 0.98 2.26 1.78 1.78 1.25 0.92 0.96 0.98 1.07 1.24 1.07 1.45 1.00 1.42 1.00 1.00 1.23 0.95 1.07 0.67 0.70 0.61 0.51 0.37 0.71 0.58 0.37 0.37 0.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.85 0.97 0.90 1.06 1.01 0.90 0.55 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.76 2.81 2.01 1.90 2.37 1.76 2.37 0.45 0.59 0.52 0.51 0.54 0.43 0.47 1.67 1.68 1.71 5.98 1.68 5.77 1.38
Table B.5: Results for all GVO instances for the segregation solution strategy
Segr25 Cuts 123 Incr 6.54 0.44 0.57 0.47 0.69 0.57 1.55 4.49 5.81 3.19 2.98 1.23 3.54 0.00 1.94 0.00 0.00 1.26 1.07 1.00 0.75 0.37 0.00 0.00 0.74 0.00 0.00 1.75 0.41 1.40
153
GVO100 1.xml GVO100 2.xml GVO100 3.xml GVO100 4.xml GVO100 5.xml GVO100 6.xml Avg GVO102 2.xml GVO102 3.xml GVO102 4.xml GVO102 5.xml GVO102 6.xml Avg GVO103 1.xml GVO103 2.xml GVO103 3.xml GVO103 4.xml GVO103 5.xml GVO103 6.xml GVO103 7.xml Avg GVO106 1.xml GVO106 2.xml GVO106 3.xml GVO106 4.xml GVO106 5.xml GVO106 6.xml GVO106 7.xml Avg GVO Avg
Normal
Appendix B. Instance properties for the SCHP-OP and solutions for the segregation strategy
Instance
