Joint optimization of fleet size and maintenance capacity in a fork-join cyclical transportation system R. Pascual∗, A. Mart´ınez† Department of Mining Engineering, Pontificia Universidad Cat´olica de Chile, Santiago, Chile R. Giesen‡ Department of Transport Engineering and Logistics, Pontificia Universidad Cat´olica de Chile, Santiago, Chile July 17, 2012

Abstract This article presents an asset-management oriented multi-criteria methodology for the joint estimation of a mobile equipment fleet size, and the maintenance capacity to be allocated in a productive system. Using a business-centered life-cycle perspective, we propose an integrated analytical model and evaluate it using global cost rate, availability and throughput as performance indicators. The global cost components include: (i) opportunity costs associated to lost production, (ii) vehicle idle time costs, and (iii) maintenance resources idle time costs. This multi-criteria approach allows to build a balanced scorecard that identifies the main tradeoffs in the system. The methodology uses an improved closed network queueing model approach to describe the production and maintenance areas. We test the proposed methodology using an underground mining operation case study. The decision variables are the size of a load-hauldump (LHD) fleet and specialized maintenance crew levels. Our model achieves savings of 20.6% in global cost terms with respect to a benchmark case. We also optimize the system to achieve desired targets of vehicle availability and system throughput (based on system utilization). Results show increments of 7.1% in vehicle availability and 13.5% in system throughput with respect to baseline case. For the case studied, these criteria also have a maximum, which allow for further improvement if desired. Results also show the importance of using balanced performance measures in the decision process. A multi-criteria optimization was also ∗ [email protected] † [email protected] ‡ [email protected]

1

performed, showing the Pareto front of considered indicators. We discuss the trade-offs among different criteria, and the implications in finding balanced solutions. The proposed analytical approach is easy to implement and requires low computational effort. It also allows for an easy reevaluation of resources when the business cycle changes and relevant exogenous factor vary.

1

Introduction

The increasing competition in capital intensive industries has created interesting improvement opportunities for the value chain. For example, there are mining sites where the original equipment manufacturer of the haulage systems is paid for the haulage service, and not by the achievement of a contracted vehicle availability, or the amount of resources allocated to the contract [18]. In this performance based scheme, the haulage machines and the assigned maintenance resources may belong to the service vendor [18]. In this context, a relevant decision for the contract designer is to estimate the number of vehicles and maintenance capacity to attain production targets with satisfactory reliability. The same competitive pressures can be observed in other industries where a fleet of vehicles transports loads in a cyclic system. For example, maritime transportation or air cargo. In these situations, a vehicle is loaded at a production site, and then travels to an unloading site. The cycle is repeated, as long as maintenance actions are not required. Traditional assignment considers the following sequence: (i) setting the production target, (ii) evaluating cycle times using transportation distances, loading and unloading times, road profile, among other parameters, (iii) selecting and sizing the fleet assuming vehicle or machine availability, and (iv) assigning the required resources for the repair shop [19]. As the maintenance capacity affects the availability of the transport units, the process (iii-iv) can be improved by jointly optimizing the involved variables. This can be done using queueing theory and/or discreteevent simulation models that are able to capture non-linear dependences of real systems [20]. The use of these powerful modeling tools includes: (i) designing and implementing an appropriate model structure, (ii) selecting the appropriate performance metrics, (iii) collecting hard data and estimate model parameters, and (iv) optimize/exploit the model to attain optimal design solutions or at least better than known solutions [20]. The simpler (yet representative) the model, the faster and cheaper the design process. This is useful to complement detailed analysis (e.g. based on simulation), allowing to capture essential trade-offs in exploratory and sensitivity analysis. The optimal selection of a fleet size and maintenance capacity has to be aligned with overall business goals. Specifications such as PAS-55 [26] and ISO/IEC 15288 [25] provide a framework to assist in making decisions using a systemic life cycle approach. Within their umbrella, a systematic decision making approach is defined to achieve the strategic corporate goals. This can be translated usually into using expected global costs as key performance in-

2

dicator (KPI). A potential difficulty of using global cost is that it requires the estimation of the economic consequences resulting from vehicle idle times. Such consequences depend on exogenous factors. For example, in the mining industry, the commodity price and the ore grade greatly influence opportunity costs due to lost production. An alternative performance metric is the expected system throughput. It is simpler to obtain, and it can be the contracted KPI. Often, when the commodity price is high, throughput is critical for the margins, while when the commodity price goes down, unit costs are key factors. Another performance metric commonly used is machine or vehicle availability. This indicator is frequently used by maintenance, and it is also a possible contracted KPI. The above list of KPIs (global cost, availability, throughput) sets a business-oriented balanced scorecard that can be used to evaluate different solutions to the joint design problem.[2]. A good balance between involved KPIs is required. An excessive number of vehicles may cause machine idle time due to queues at the production site and at the repair shop. This can lead to excesive global cost rate levels (figure 1a). On the other hand, system throughput increases as the production sites are more utilized (figure 1b). As resources are misused, opportunity costs arise and contracted key performance indicators (KPI) may drop (for example, vehicle availability, represented in figure 1c). The selection of a proper strategy requires an optimization procedure that reveals these existing trade-offs.

3

System throughput

Total costs

(b) System throughput.

Vehicle availability

(a) Global cost rate.

ity

ac

Flee

t siz

e

ai M

c an en

e

p ca

nt

(c) Vehicle availability. Figure 1: Representative diagrams of considered KPIs.

The goal of this work is to improve the performance of a cyclic transportation system by judicious joint resource assignment for fleet and maintenance capacity. We adopt a business centered multi-criteria analysis, considering production and maintenance areas in an integrated optimization methodology. The system is modeled using a closed network of queues with concurrent jobs. Three KPIs are considered under a balanced scorecard framework: global cost rate, vehicle availability and system throughput. We illustrate the proposed methodology through a case study of an underground mine. The relevance of appropriately defining performance measures is shown, comparing our results with the work of [5] as a benchmark. The system’s KPIs are obtained using an improved version of the Mean Value Analysis (IMVA) presented in [13]. We show the need of adopting a multi-criteria approach to account for conflicting trade-offs from KPI selection. Optimal policies obtained through each KPI optimization are studied. Also, a multi-criteria optimization is performed, presenting the

4

Pareto front of optimal solutions. The work is presented as follows. In section 1.1 we present a brief review of relevant works. As the queueing models literature is vast, we kept our review to the main elements used in this work. In section 1.2 we overview an improved version of the mean value analysis methodology, which we use as a part of the system’s global cost evaluation. Section 2 presents the model formulation. In section 3 an application of our proposed model is provided. We present the main conclusions in section 4.

1.1

Literature review

An early application of queueing theory in mining systems is Koenigsberg [1] and Faulkner [3]. The first work considers a cyclic network of loaders in a coal underground mine. Faulkner models the demand process of the loaders arguing that it can be approximated as a Poisson process. Related reviews are provided by Koenigsberg [6], Worthington [10], Newman et al. [7] and Carmichael [4]. An excellent starting point for the model proposed here is the work of Huang and Kumar [5]. They use a queueing theory model to size a fleet of load-hauldump (LHD) machines in a swedish mine. In their model, the LHDs are the customers of mining blocks (loading servers). The arrival rate is determined by the reliability and maintainability parameters of the LHDs, which are modeled using exponential distributions. Their model does not consider that the fleet size affects the mean time to repair, as they do not consider limited resources in the repair shop. Accordingly, they do not consider the resource assignment for the repairs, something that has been extended in this work. Like them, we consider that the blocks are reliable. An approach where the infrastructure components fail is considered in [7]. Regarding the assignment of specialized resources at the repair show, Dietz and Jenkins [13] consider a fork-join queueing network model to estimate system throughput in an aircraft sortie generation process. They consider a cyclic network and extend the mean value analysis (MVA) to more general fork-join networks. Du and Hall [14] study the fleet size and distribution of transport equipment in complex networks. Granger et al. [17] extend these works to consider two-moment distributions. It is also a basis for the integrated model proposed here. Production and repair systems are commonly modelled using closed queueing networks [29]. An important issue is how to estimate performance measures. In section 1.2 we review some techniques, giving some attention to MVA. Most models described in the literature consider constant process cycle time. Mckenzie et al. [15] consider productivity variation, as the distance that the loaders of an open pit have to run increases in time. Pauley and Ormerod [20] describe the application of queueing theory models to analyze bottlenecks in load and haul services in several open-pit sites. Their analysis allowed to reduce existing haul fleets and prioritize efforts in the system bottlenecks, also using the theory of constraints [21]. Beyond that it allowed a fast root-cause analysis for reduced throughput, before developing more complex (expensive) simulation models. Karami and Szymansky [28] use simulation to model a rail 5

haulage system. As a result, they detect bottlenecks and provide recomendations to improve the system, although they do not consider the effect of vehicle availability on expected system throughput. Muduli and Yegulalp [8] use mean value analysis to model a load-haul mining system. Their analysis includes modelling heterogeneous haul fleets with different haulage capacities. Suri et al. [22] propose a numerical scheme to reduce the computational effort of the MVA. Kappas y Yegulalp [9] use MVA to consider general distributions for the different services in the network. They consider that the servers (i.e. the shovels ) do not fail. No fork network is considered for the repair service. Regarding the workforce assignment optimization problem, Dietz and Rosenshine [11] present an analytical method for determining the number of maintenance specialties and the allocation of tasks to them. They study a system of identical vehicles, for which they develop a Markovian model to evaluate its performance. The assignment decisions and workforce structure are optimized based on an utilization criterion using a linear programming algorithm, and taking into account personnel expenditure and qualification constraints. Another example is given by Jardine and Tsang [12]. They present models for crew optimization and vehicle assignment problems using open queueing networks and simulation.

1.2

Performance evaluation of closed queueing networks

Several techniques have been developed to estimate the steady-state performance measures of closed queueing networks. Among the modelling approaches, product-form queueing networks have received much attention[34], as they allow simpler models. As representative examples, we can mention some commonly used techniques [30, 34]. Methods based on MVA [13] evaluate the mean queue lengths and throughputs recursively to obtain steady state indicators. Methods such as Convolution algorithm or LBAC [35] estimate the normalizing constant of network state probabilities to obtain network measures. Other methods, such as the Method of Moments [30] recursively evaluates a set of higher-order moments of queue lengths and then returns mean performance indexes. Other techniques are available, see for example, [17] and [30] and related references. Some of the above mentioned methods are still being extended and new techniques developed. Complex system (e.g multi-class, non product-form and large networks) remain as a modeling challenge, often because of the computational costs involved or because efficient approximate methods need to be developed [30, 36]. A helpful extension for this work are fork-join networks. When multiple concurrent activities are performed in a station, in such a way that a customer leaves when all the activities are finished, the network is no longer of product form. These nodes are known as fork-join nodes. Dietz and Jenkins [13] extend the MVA to approximate the mean performance measures of these networks, to which we refer as IMVA. For the case study of section 3, we apply this extension. Note that our work does not require the use of IMVA, and we select it for its relative convenience (see section 2). For the sake of self-containment we present 6

a brief description of the method, along with some notes about fork-join nodes based on [13].

1.3

IMVA

The mean value analysis is an analytical method that allows an efficient estimation of steady-state expected performance measures of a closed or capacitated queing network [13, 30]. It is suitable for product-form networks, with symmetric service disciplines (first come, first served exponential, processor sharing, infinite server, or last come, first served preemptive) [13]. The main principle of this method is to calculate the mean queue lenghts, and other indicators such as service times, throughput and utilization rates. It does so recursively on the number of customer in the network. More details can be found in references [13, 31]. Let us consider a system that can be modeled as a closed network of M stations, where a fleet of N vehicles goes through requiring different actions. An example is a mining truck fleet in a simplified open pit operation. Each truck operates for example, through load, unload, transport and repair stations. For mid-term planning horizon this system is closed as each truck never leaves the network. For a given network we want to estimate for each station i the mean response (total) time at each station Ri (wating and service time), the vehicle throughput λi , the queue length Qi and the utilization Ui . We first describe the case where all stations consist of a group of identical servers and a queue of wating vehicles. At the end of this section we provide some remarks about concurrent jobs stations (we refer to them as ”fork-join stations”). In this work, no priorities are considered. For a station i, if n vehicles are in it, then the service rate is µi (n) = min(n/si , ri /si ), where ri is the number of attention servers, and si is the mean service time per server. If Pi (n|N ) is the probability of n vehicles are being served in station i, given that the N vehicles are in the network, then for all i it is satisfied the following balance equation: µi (n)Pi (n|N ) = λi (N )Pi (n − 1|N − 1)

(1)

Where λi (N ) is the station i throughput. Then the mean response time Ri is: Ri (N ) =

N X

n Pi (n − 1|N − 1) µ (n) i n=1

(2)

With border conditions on the probabilities Pi (0|N − 1) = 1 and the queue length Qi (N − 1) = 0, if N = 1. The mean cycle time for a vehicle, measured from a reference station (lets name it station 1) is CT1 . CT1 (N ) =

M X vi Ri (N ) i=1

7

v1

(3)

The ratio vi /v1 is the mean number of times a vehicle enters station i per visit to station 1. These ratios are obtained solving Pv = v with P the routing probabilities matrix. The vehicle throughput of station i is then: λi (N ) =

N vi CT1 (N )v1

(4)

The queue length and utilization are then given by Little’s law: Qi (N )

=

Ri (N )λi (N )

(5)

Ui (N )

=

si λi (N )

(6)

Then it is possible to recursively determine the new probability distribution for vehicles in the network. From equation 1: Pi (n|N )

=

λi (n)Pi (n − 1|N − 1) , n>0 µi (n)

Pi (0|N )

=

1−

N X

Pi (n|N )

(7) (8)

n=1

It could be the case that some parts of the system may be better represented using fork-join stations. A fork-join station offers several kinds of jobs (substations) to an arriving vehicle. Each job has its own resources and waiting queue. When a vehicle enters a fork-join station i, it may enter each of the Ki sub-stations with fixed probabilities qik . This defines a set S of required jobs each time a vehicle enters. A vehicle leaves the fork-join station j when it has completed all the required jobs. If the response time at each sub-station is Tik , then the response time of the whole fork-join station i is maxS∈Ω maxk∈S Tik , where Ω is the set of all possible sets of jobs. With some assumptions it is possible to obtain simple approximations to the mean network indicators at each job of any fork-join station [13] in addition to the normal stations. The IMVA algorithm can then be extended by modifying the cycle time and the sub-stations throughput. The cycle time changes to: CT1 (N ) =

X vi Ri (N ) i∈I

v1

+

X vi X πi (S)E[maxk∈S (Tik (N ))] v1 i∈J

(9)

S⊆Ωi

Where I is the set of simple stations, J is the set of fork-join stations, and station 1 is assumed to be a non fork-join station for ease of exposition. The vehicle throughput for each job k in a fork-join station j is: λjk =

2

N vj qjk CT1 (N )v1

(10)

Model formulation

Let us consider a fleet of mobile equipment operating in a cyclic transportation system. The system can be modeled as a closed network with two stations: a 8

production station and a repair station. The production station consists of a set of productive servers in which the vehicles operate. For example, the production station could be the extraction points and transport path of an underground mining operation. A vehicle remains at the production station until maintenance action is required, and it is sent to a repair facility. If available, a spare vehicle is sent to operation. Otherwise, there is an idle position in the system, in the form of idle production capacity. When a vehicle enters the repair shop, it may require one or more types of repair actions. These jobs require the allocation of specialized resources (an example is given in section 3 where each job requires a technicians crew). When all the repair jobs are finished, the equipment becomes available. Under the above conditions, the repair station can be modelled as a fork-join node. Suppose there are K different repair jobs, and each job k has rk resources. A specific job k is required with probability qk each time a vehicle enters the repair facility. These probabilities reflect the steady-state probability of a failure being related to a certain job type. Note that these probabilities need not sum 1, as different repair jobs can be performed at the same time. We assume that these probabilities are constant, and do not depend on each repair station state. Our aim is to estimate the optimal vehicle fleet size, and the number of resources at each job at the repair station. We consider three optimization criteria: global cost rate, vehicle availability and system throughput. We first adress the global cost rate. Huang and Kumar [5] consider a similar problem and proposed the sum of the cost of idle operating servers, and the cost due to idle vehicles that are ready to use. The cost rate of their model can be expressed as: crg = cb Nb Db + cl N Dlb (11) Where cb is the idle production servers cost rate (per server), cl the idle equipment cost rate, Nb is the number of production servers, N is the number of vehicles, Db is the idle time fraction of production blocks, and Dlb is the idle time fraction of available vehicles due to queues at the production servers. The initial cost model of equation 11 (we will refer to it as ”simplified cost model”) has some limitations. First, it only considers as decision variable the fleet size N . This may not be globally optimal, as it is often possible to have some flexibility in the repair resources facility which could lead to potencial fleet size reductions. Second, the simplified cost model does not consider the maintenance function directly in the objective function. This could lead to inefficient strategies, such as having excess repair capacity, or large vehicle unavailability. These shortcomings arise from using a local optimization approach that lacks a business-centered vision. We propose an extended cost model that allows for a system optimization. It allows us to extend the decision problem and jointly estimate fleet size and maintenance capacity. We consider four cost sources: cost for idle servers at the operating system, cost for idle equipment that is available, idle servers at the

9

repair facility, and unavailable equipment. ceg = cb Nb Db + N cl (Dlb + Dlr ) +

K X

qk crk rk Dkr

(12)

k=1

Where Dlr the vehicle unavailability. For a job k, crk is the idle resource cost rate, rk are the resource units, and Dkr the idle resource fraction. Note that equation 12 is not just adding terms to the initial cost function (equation 11), but a different conceptual approach. As the model now considers both production and maintenance areas, it does account for system-wide trade-offs. For example, for a given throughput target, the optimal fleet size is a function of the repair resources. The idle time fractions and vehicle unavailability can be expressed in terms of mean network metrics, as shown in equations 13, 14, 15, and 16. A diagram showing an example of the different idle times involved is shown in Figure 2. These quantities can be obtained, for example, using IMVA presented in section 1.3. In this case, each job at the repair station corresponds to a sub-station in the IMVA framework.

Dlr

Db Dlb

1-Dlb-Dlr

1-Db At repair queue At production queue

Being repaired Operating

Idle, wating for a machine

(a) Vehicles cycle normalized times.

In operation

(b) Production normalized times.

r

1-D

Idle

Busy

(c) Maintenance resources normalized times. Figure 2: Representative diagrams of considered KPIs.

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The expected fraction of idle vehicles ready to operate is: Dlb = (Q − U )/N

(13)

Where Q is the mean queue length at the productive station, U is the mean number of busy productive servers (utilization). The fraction of idle productive servers at the productive station is: Db = 1 −

U Nb

(14)

The vehicle unavailability for the system is: Dlr = 1 −

Q N

(15)

The idle repair resources Dr for job k can be expressed as: Dkr =

rk − Ukr rk

(16)

Where rk is the number of repair resources quantities at job k, and Ukr is the repair resources utilization at job k. We also consider two other optimization criteria: vehicle availability (A) and system throughput. Vehicle availability is defined as the fraction of time a vehicle is able to engage in a productive task. This can be expressed as the steady-state fraction of the fleet that is not at the repair facility. A=

Q N

(17)

The system throughput (τ ) is the mean production flow of the system. If each productive server has a nominal production flow ρ, then the throughput is given by equation 18. τ = Uρ (18) The preceding formulation permits us to evaluate the system KPIs for different fleet sizes and maintenance resource levels. It is possible to optimize the system’s performance, searching for decision variables that meet a decision maker criteria. Different system optimization approaches are possible. We consider the cases of single-KPI optimization, and multi-criteria optimization. Consider the single-KPI problem. Let the decision variables be given by the vector x= (N, r1 , ..., rk ) ∈ V . The set V depends on the particular problem at hand (although it is expected that ceg > 0, A ∈ [0, 1] and τ > 0). The set of KPIs in consideration is O = {ceg (x), −A(x), −τ (x)}. The signs of A and τ were chosen to simplify the notation. The objective function is f (x) ∈ O. The KPIs gi (x) ∈ O \ {f (x)}, i ∈ {1, 2}, may be used as constraints. Let g¯i ∈ R be desired goals for the constraining KPIs. The optimization problem is given by:

11

min f (x) x∈V

subject to gi (x) ≥ g¯i (19) For complex scenarios, it may be the case that a multi-criteria optimization would be wanted instead of a single-KPI optimization. This approach arises naturally when an integrated business-oriented perspective is adopted. In this case, the system objective function is given by m(x) = (ceg (x), −A(x), −τ (x)). A decision x∗ is Pareto optimal if and only if there is no other decision x, such that m(x) ≤ m(x∗ ) (element-wise), and for at least one KPI h(x) ∈ O it is satisfied that h(x) < h(x∗ ) [37]. The set of all Pareto optimal decisions constitutes the Pareto front of the problem. We propose to use the Pareto front as decision tool. This presents the complete set of Pareto optimal trade-offs between the KPIs. The decision maker will select the most suitable, based on his/her particular goals and business environment. Several methods exist to obtain the Pareto front. In many cases, they can be presented as single-objective optimization problems that can be used systematically to find points in the frontier. We do not discuss these methods as it is not the focus of our work. The interested reader is referred to [37, 38].

3

Case study

To illustrate the application of the model, we present a case study adapted from Huang et al. [5]. Let us consider an underground mining operation, with sublevel caving mining method (Figure 3). There are 10 mining blocks producing fragmented ore. A fleet of 12 LHD has been assigned to the operation. At any given time, a LHD can be in any of the following four states: operating, idle at the block queue, idle at the maintenance queue, and being repaired. We assume the time between maintenance and repair actions follows an exponential distribution with mean M T T F [3, 5, 29].

12

Repair shop Task 1 q1 =0.4

Mine

Server 1 Server 2

Task 2

Block 1 q2 =0.3

Server 1 Server 2

Block 2

...

Task 3 Server 1

q3 =0.35

Block 10

Server 2

Task 4 Server 1

q4 =0.15

Server 2

Figure 3: System under study. Each LHD operates in a mining block. At failure, it is sent to a repair shop with four sub-stations.

The repair shop is disaggregated in four service types, as seen in Figure 3. Each job type is performed by a crew of specialized technicians. Without losing generality, we consider that one technician is required to serve a LHD per job type. If there are not enough technicians for a job, a queue of LHDs is made for that particular job type, based on a first come, first served discipline. This implies that the mean time to repair M T T R is not constant. Different service types can be performed simultaneously on a vehicle. A LHD becomes available when all the repair actions are finished. This means that it is appropiate to model the repair shop as a fork-join node. Case study parameters are presented in Table 1. For the repair shop, parameters for different job types are given as a vector of the form (task1 , task2 , task3 , task4 ). Repair shop parameters were chosen to obtain the same M T T R as Huang et al. in the non-optimized (baseline) scenario. The baseline crew assignment is (2,2,2,2). Evaluating the system using IMVA we obtain M T T R = 15.2 hr. We consider this setting as the baseline for our study. We solve the problem of a decision maker facing the optimization of LHD fleet size and number of technicians per repair job. In order to compare to the work of Huang et al we first adress the problem of defining an appropiate cost model. Then, we optimize the decision variables using the expected value of the following KPIs: global cost rate, vehicle availability and system throughput. In the context of a balanced scorecard approach we seek to minimize the global cost rate and meet specified goals for vehicle availability and system throughput. We set a target level for vehicle availability of Atarg =80%. We also set a system 13

utilization target of Utarg = 99%. For a nominal flow of ρ = 200 ton/hr per block, this translates into a system throughput target of τtarg = 1980 ton/hr. Parameter Nb MTTF r q s ρ cb cl cr Atarg Utarg τtarg

Value 10 34.3 [2, 2, 2, 2]T [0.4, 0.3, 0.35, 0.15]T [13.8, 11.1, 12.2, 9.3]T 200 2871 2167 [172, 155, 172, 138]T 80 99 1980

Unit mining blocks hr

hr ton/hr U SD hr·LHD U SD hr·LHD U SD hr·LHD

% % ton/hr

Table 1: Case study parameters.

We consider two optimization cases: variable LHD fleet size with fixed maintenance capacity (Case I), and variable fleet size and number of technicians per job type (Case II). Case I mimics the approach of Huang et al., allowing us to show its shortcomings. For each case we evaluate the optimal policy using two cost models: the simplified cost model (eq. 11) and the proposed extended cost model (eq. 12). A summary of the optimization cases and the baseline scenario notation is given in Table 2. As the problem is relatively small, we used explicit enumeration as optimization procedure. Case Baseline Fixed maintenance capacity (I) Variable maintenance capacity (II)

Simplified cost (a) 0a Ia IIa

Extended cost (b) 0b Ib IIb

Table 2: Optimization cases and notation.

We present the rest of the case as follows: sections 3.1, 3.2 and 3.3 present results for the single-KPI problem, for each considered KPI. Section 3.5 presents the results of a multi-criteria optimization approach. Section 3.6 discusses the main results.

3.1

Global cost rate

To evaluate the network performance, we opt for IMVA as our system is in general small, and simple (e.g. few jobs, single model fleet). Since the number of 14

stations and vehicles is not generally very large for this kind of system, potential computation time gains of other analytical methods are not significant [30]. We also disregard simulation as modeling and running costs are in general higher, particularly for optimization procedures [1, 24]. Results are shown in Figure 4, and were obtained using MATLAB. For the Case II, each point of the curve represents the global cost rate for the optimal technicians quantity at each repair job (i.e. the minimum cost solution for the given LHD fleet size). We evaluated resource levels in the range 1 to 13 for each job type. Based on our tests, beyond 13 technicians the considered KPIs do not differ from the infinite resources case.

15000 10000 0

5000

Global cost (USD/hr)

20000

IIa IIb Ia Ib

10

12

14

16

18

20

LHD fleet size

Figure 4: Global cost as a function of LHD number for the simplified and proposed model.

As expected, from Figure 4 we can see that the simplified model has lower costs (cases Ia and IIa). This is obvious, as it lacks the repair shop idle cost components. In this sense, the model’s objective function is biased and the cost difference is explained by a better description of the system. A more complete global cost rate is given by cases Ib and IIb. For cases Ib and IIb, it can be seen that optimizing the technicians quantity leads to relevant savings for all fleet sizes. This highlights the importance of investing in repair shop flexibility policies. The optimal solutions of each case are summarized in Tables 3 and 15

4. Relative KPI changes with respect to the baseline value are expressed with sign. We define ∆b x as the percentage variation of KPI x with respect to the baseline case. crg (USD·104 /hr) 0.55 0.47 0.33

Case 0a Ia IIa

∆b crg (%) 0.0 -14.6 -40.0

Fleet size 12 14 13

Crew (2,2,2,2) (2,2,2,2) (∞,∞,∞,∞)

Table 3: Case study simplified cost crg results.

Case 0b Ib IIb

ceg (USD·104 /hr) 1.36 1.36 1.08

∆b ceg . (%) 0.0 0.0 -20.6

Fleet size 12 12 13

Crew (2,2,2,2) (2,2,2,2) (4,3,4,3)

Table 4: Case study extended cost ceg results.

Table 3 show the results when using the simplified cost function (cases Ia and IIa). For the case of fixed technicians quantity, the simplified cost model suggests an increase in the LHD fleet size from 12 to 14. When considering a variable number of technicians, the optimal fleet number drops to 13 in exchange of raising the technicians quantity. As maintenance resources are not penalized in the simplified cost function, the algorithm allocates as many technicians as possible, until no further cost reductions are possible. The savings of the simplified cost function with respect to the baseline are only apparent. In this sense, results suggests that it is possible to achieve a maximum of 40 % of simplified cost savings in relation to the baseline (Case 0a) by jointly optimizing the fleet size and maintenance resources. Actual cost savings, however, should be evaluated using extended cost model results. When using the extended cost function (Table 4), Case IIb shows it can achieve maximum savings of 20.6% in relation to the baseline scenario (Case 0b). On the other hand, if Case Ia’s solution is evaluated using the extended cost function, a global cost of 1.43 USD·104 /hr is obtained. That is, Case Ia actually increases the extended cost by 5.2% in relation to the baseline. This is because the maintenance related cost components are neglected. If there was no flexibility to change the technicians crew, and we were only able to keep it at baseline levels, then the best choice is to keep the initial 12 LHD fleet size. Such decision contrasts the simplified cost model recommendation of increasing the fleet in two units. If we optimized the fleet size and technicians number with the simplified cost models, then the fleet size would be optimal, but we would have too many idle technicians compared to the extended cost model solution. Our 16

proposal offers an integrated business perspective which allows for significant cost savings.

3.2

Vehicle availability

70 65 60

Machine availability (%)

75

80

85

So far, our analysis has been focused on the global cost rate. As the decision problem involves different areas of the company (e.g. production and maintenance), it is useful to study several performance metrics. Figure 5 shows vehicle availability as a function of the fleet size. For Case II, each point represents the vehicle availability for an optimal cost-based crew assignment. We focus the study on cases Ib and IIb, as they were obtained using a cost function that is more representative of the system. For completeness we also include the curves of cases Ia and IIa.

50

55

IIb Ia and Ib IIa Goal 80%

10

12

14

16

18

20

LHD fleet size

Figure 5: Vehicle availability.

For cases Ia and Ib, the vehicle availability curve shows a non-monotonic behavior. For small fleet sizes, an additional LHD produces a greater increment in the total fleet size than in the available fleet fraction. This is because the mine utilization is low, and the repair shop technicians number cannot be increased. As the fleet size grows, however, the LHD fleet in repair fraction becomes small 17

10000

compared with the total fleet size. This is consistent with system throughput increasing with fleet size (see Figure 8 in section 3.3). A note on the seemingly strange behavior of availability of Case IIa: this is explained because technicians quantity levels are assigned based on a cost-optimization criterion. After 15 LHDs, the technicians allocated to the repair shop drop drastically, decreasing vehicle availability. The effect can be seen in Figure 6. The simplified global cost and its two components (eq. 11) are plotted as a function of LHD fleet size. After 15 LHDs the main contribution to the simplified global cost changes from the ”Idle LHD at mine queue cost” component to the ”Idle mine blocks cost” component, which explains the maintenance resources reduction.

6000 4000 0

2000

Cost (USD/hr)

8000

Simplified global cost Idle mine blocks cost Idle LHD at mine queue cost

10

12

14

16

18

20

LHD fleet size

Figure 6: Cost components for Case IIa.

In Case IIb vehicle availability increases up to a maximum at 19 LHD. This is explained by the cost-based allocation of maintenance capacity. At a fleet size of 20, the LHD availability decreases. This effect can be seen at Figure 7, which shows unvailability components for Case IIb. LHD availability is given by the ”LHD in repair service” component. As this component increases at a fleet size of 20, the change in technicians number per job type allows for a large decrease in ”Idle repair resources” unavailability component. This change in technicians

18

120

number reduces the global cost rate, but also reduces availability.

80 60 40 0

20

Unavailability components (%)

100

Normalized global cost rate Idle blocks Idle LHD at mining blocks Idle repair resources LHD in repair service

10

12

14

16

18

20

LHD fleet size

Figure 7: Unavailability components.

Based on the previous analysis we can find the fleet size and technicians for each repair job that allow to achieve the desired target of 80%. Note that it is possible to choose among several fleet size and technicians quantities that give a vehicle availability above 80%. We propose to choose the minimum extended cost solution. The optimal LHD fleet size is then 18 with A =80%. Technicians allocation is (4, 3, 3, 2). Considering that baseline scenario (0b) reach a vehicle availability of 74.68%, this means an increment of 7.1%. In this case study, an alternative performance criterion could have been to maximize availability. The maximum vehicle availability is A =81%, with 19 LHDs, and technicians levels of (4, 3, 3, 2). Along with importance of maximizing availability, it is relevant to know that a maximum actually exists. A summary is presented in Table 5.

19

Case 0b IIb, Atarg =80 IIb, max A

A (%) 74.7 80.0 81.0

∆b A (%) 0.0 +7.1 +8.4

Fleet size 12 18 19

Crew (2,2,2,2) (4,3,3,2) (4,3,3,2)

Table 5: Vehicle availability results summary.

3.3

System throughput

The system throughput as a function of LHD fleet size is shown in Figure 8. For Case II each point is the system throughput considering a cost-based optimal assignment. For the fixed maintenance capacity case, the curve increases in the number of LHDs. This is expected as the utilization of the mine increases by having a larger fleet with the same repair shop. Note that this happens even in the decreasing range of availability curve. When technicians quantity can be varied, the curve reach a maximum for 19 LHDs. This occurs because of a technicians cost based allocation. At that point, increasing the throughput does not reduce the global cost. It is interesting to note how the solutions given by the simplified model in Case IIa lead to a notorious decrease of throughput after 15 LHDs. The reasons behind this are the same observed in the availability curve.

20

2100 2000 1900 1800 1700 1600

LHD throughput (Ton/hr)

1400

1500

IIb Ia and Ib IIa Goal=1980 Ton/hr

10

12

14

16

18

20

LHD fleet size

Figure 8: Productivity for fixed and variable technicians allocations.

The optimal solution that meets the target of τtarg = 1980 ton/hr is a fleet of 17 LHDs. The technicians levels are (4, 3, 3, 2). The achieved throughput is 1987 ton/hr, representing a 99.4% of system utilization. As baseline case (0b) throughput is 1752 ton/hr, this corresponds to an increase of 13.5%. As in the availability case, it is also possible to maximize throughput. The maximum throughput achievable is 1997 ton/hr, which is 99.9% of system utilization. It is interesting to observe that the maximizing throughput leads to a global cost rate 84.7% larger than the minimum cost rate. Results are summarized in Table 6. Case 0b IIb, τtarg =1980 IIb, max τ

τ (ton/hr) 1752 1987 1997

∆b τ (%) 0.0 +13.4 +14.0

Fleet size 12 17 19

Crew (2,2,2,2) (4,3,3,2) (4,3,3,2)

Table 6: System throughput results summary.

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3.4

Sensitivity analysis

A sensitivity analysis was conducted with respect to the MTTF. Figure 9 summarizes the impact on the extended global cost rate. As expected, the global cost rate decreases as the MTTF increases. This occurs because the LHDs spend more time at the mining blocks, with respect to the total productionrepair cycle time. The decrease in the number of technicians and fleet size is explained by the reduction in the machines’ non-productive time and resources’ idle time. It can be observed that the optimal solution is quite insensitive to MTTF variations, as it does not vary in more than one LHD and one technician (per service type).

Global cost rate variation (%)

20 15

ΔCost: +19% LHD: 13 TL: (5,4,4,3)

10 ΔCost: +8.3% LHD: 13 TL: (5,4,4,3)

5

0

ΔCost: 0% LHD: 13 TL: (4,3,4,3)

-5

ΔCost: -6.5% LHD: 13 TL: (4,3,4,2)

-10 -20

-10

0

+10

ΔCost: -12% LHD: 12 TL: (4,3,3,2)

+20

LHD's MTTF variation (%)

Figure 9: Impact of the MTTF on the global cost rate. ”TL” stands for ”Technicians levels”.

The impact of the MTTF on the machine availability and system throughput is shown in Figure 10. Both KPIs are shown as a function of the LHD fleet size required to meet the desired targets, for an optimal cost-based crew assignment. The required LHD fleet size decreases with the MTTF for both KPIs. Note that system throughput is less sensitive to MTTF variations than machine availability. As the MTTF increases both KPIs converge to the same optimal fleet size. This result becomes relevant when aligning maintenance and production areas.

22

24

Availability

LHD fleet size

22

Throughput

20 18 16 14 12 10 -20

-10

0

10

20

LHD's MTTF variation (%)

Figure 10: Impact of the MTTF on the machine availability and system throughput.

3.5

Multi-objective optimization

As the problem is small, we exhaustively search for the Pareto front. When evaluating trade-offs among the three considered KPIs, the Pareto front is shown in a three-dimensional space. We refer to it as the P F3 , to distinguish it from Pareto fronts that reflect the trade-offs for only two KPIs (P F2 ). Three significative digits are considered for each KPI.

23

2.00 1.95 1.85

76

ility

Av

78

ail ab

80

(% )

82

1.90

Throughput (kTon/hr)

84

74 10

12

14

16

18

20

22

24

Global cost (kUSD/hr)

Figure 11: Pareto front P F3 .

Figure 11 shows the P F3 Pareto front. Each point in this set is a Pareto optimal trade-off decision. This means it is not possible to improve a KPI without losing performance on another. The decision maker will choose a decision from this set, according to his/her particular business goals. As expected, Pareto optimal decisions show a trade-off between increasing throughput and decreasing global costs. Vehicle availability also shows this behavior, as it is increasing with system throughput. A decision maker could also be interested in the trade-offs of only two KPIs. Although our work aims for an integrated approach using all KPIs, we present the P F2 fronts to complement our work. The P F2 for global cost versus vehicle availability, and global cost versus system throughput are presented in figures 12 and 13. The P F2 for vehicle availability and system throughput consists on one point: A = 82.8% and τ = 2.00 kTon/hr. This means in this study both KPI can be increased without trade-offs.

24

82 80 76

78

Availability (%)

12

14

16

18

20

22

Global cost (kUSD/hr)

Figure 12: P F2 for global cost and vehicle availability.

As expected, the Pareto optimal decisions that increase system throughput or vehicle availability, also increase global cost. It is interesting to note that after numerical precision considerations, the Pareto front for global cost versus system throughput is very simple. This is relevant as these charts may be used to directly support decisions in industrial settings.

25

2.00 1.98 1.96 1.94 1.90

1.92

Throughput (kTon/hr)

11

12

13

14

15

16

17

18

Global cost (kUSD/hr)

Figure 13: P F2 for global cost and system throughput.

3.6

Discussion

Table 7 presents a summary of the optimal solutions when considering one KPI at a time. Results show that, as expected, significant misalignments between business areas exists if KPIs are not carefully selected. When operations seeks a system throughput target of 1980 ton/hr, then using the optimal strategy implies a vehicle availability level of 79.0%. This is close to the maintenance function’s objective, which is to seek a vehicle availability level of 80%. This strategy clash with the asset management function’s goals, which is to minimize the extended global cost. Operations’ goal increases the extended global cost in 16.9% in relation to the baseline, while maintenance’s goal increases it by 31.6% with respect to the baseline. This contrasts with the minimum extended global cost solution, which achieves savings of 20.6% in relation to the baseline. This shows the importance of setting appropiate target levels for vehicle availability and system throughput. A desirable set of goals can be those associated to the minimum extended cost solution. That is, aiming for a vehicle availability of A = 75.3% and system throughput of 1848 ton/hr, for a fleet size 13 LHDs and a crew vector (4,3,4,3). This makes sense since the global cost rate offers a

26

best-for-business performance metric. These trade-offs were expected, and this methodology offers an efficient way to quantify its effects. Table 7 also allows to find compromise strategies. In this case, although maintenance resource allocations do not vary greatly, fleet size does. This could support a method for finding more balanced solutions, where the only decision variable would be the LHD fleet size. We note that this would be an heuristic method only applicable to particular cases with low maintenance capacity sensitivity. KPI ceg A τ

Fleet 13 18 17

Crew (4,3,4,3) (4,3,3,2) (4,3,3,2)

∆b LHD +8.3 +50.0 +41.7

ceg 1.08 1.79 1.59

∆b ceg -20.6 +31.6 +16.9

A 75.3 80.0 79.0

∆b A +0.8 +7.1 +5.8

τ 1848 1994 1987

∆b τ +5.5 +13.8 +13.4

Table 7: Optimal strategies for different KPIs, considering variable technicians number for each repair job. Cost results are in USD·104 /hr, system throughput results are in ton/hr.

The Pareto front P F3 of Figure 11 can be used to select a Pareto optimal balanced scorecard. This multi-criteria approach allows to evaluate the whole set of possible decisions with Pareto optimal trade-offs. A solution like this in many cases may be preferred over the single-KPI approach previously analyzed. However, as noted by [38], obtaining the Pareto front in large problems may be a challenging task. Note that contractual, legal, and other constraints may limit the decision maker’s options.

4

Conclusion

This work presents an integrated decision support methodology that allows to jointly estimate fleet size and maintenance capacity requirements under several criteria. The methodology is based on a closed queueing network model, which can be evaluated using analytical techniques. We have put emphasis on business key performance indicators, yet other performance metrics have also been considered. Given its fast convergence properties, it is well suited for iterative analytical problems, such as the derivation of an optimal resource structure under specified constraints. The proposed analytical approach can be considered as an approximation and may provide valuable insight when used as a complement to more detailed simulations. The methodology is also directly applicable in other related fields, such as different underground mining configurations, open pit mining operations, and more general production-repair systems. We presented a case study that shows the relevance of having a business perspective in the KPI selection. In particular, we showed the importance of a well defined cost function, and the different trade-offs when considering vehicle availability and system throughput. As future work, it is straightforward to develop a more detailed model of the repair processes. For example, LHD repairs can be done in the maintenance line 27

(the local shop) and/or at a second echelon (depot shop). Another direction is to use more advanced approximation techniques, for example, based on extensions to the method of moments [30]. This would allow to study larger and more complex systems using analytical methods.

Acknowledgments The authors wish to acknowledge the partial funding of this work by the Fondo Nacional de Desarrollo Cient´ıfico Y Tecnol´ogico (FONDECYT) of the Chilean government (project 1090079).

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