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Tight Mixed Integer Linear Programming Formulations for the Unit Commitment Problem James Ostrowski, Member, IEEE, Miguel F. Anjos, Member, IEEE, and Anthony Vannelli, Member, IEEE

Abstract—This paper examines the polytope of feasible power generation schedules in the unit commitment (UC) problem. We provide computational results comparing formulations for the UC problem commonly found in the literature. We introduce a new class of inequalities, giving a tighter description of feasible operating schedules for generators. Computational results show that these inequalities can significantly reduce overall solution times.

Maximum ramp-down rate of generator (MW/h). Maximum ramp-up rate of generator (MW/h). Maximum shutdown rate of generator (MW).

Index Terms—Integer-programming, optimization, unit commitment.

Maximum startup rate of generator (MW). Set of time periods.

NOTATION

Nmber of hours generator is required to be on at the start of the planning period (h).

Parameters

Minimum number of time periods required for generator to be on before it can be turned off (h).

Feasible region for production schedules for generator at time . Coefficients for the startup cost function for generator .

Variables Startup cost of unit in period ($).

Number of hours generator is required to be off at the start of the planning period (h).

Power produced at time of generator (MW).

Demand at time period (MW). Minimum number of time periods required for generator to be off before it can be turned on (h).

Maximum available power at time from generator (MW).

unless exceeds the planning horizon, is . in which case

Startup status at time of generator .

On/off status at time of generator .

Shutdown status at time of generator .

Set of generators. Cost of turning on generator after it has been inactive for time periods ($). unless exceeds the planning horizon, is . in which case Maximum power output of generator (MW). Minimum power output of generator (MW). Spinning reserve at time period (MW). Manuscript received March 18, 2010; revised July 15, 2010, October 08, 2010, December 28, 2010, and April 01, 2011; accepted July 07, 2011. Date of publication August 15, 2011; date of current version January 20, 2012. This work was supported in part by MITACS, NSERC, and the Humboldt Foundation. Paper no. TPWRS-00218-2010. J. Ostrowski and M. F. Anjos are with the Department of Management Sciences, University of Waterloo, Waterloo, ON N2L 3G1, Canada. A. Vannelli is with the School of Engineering, University of Guelph, Guelph, ON N1G 2W1, Canada. Digital Object Identifier 10.1109/TPWRS.2011.2162008

I. INTRODUCTION HE unit commitment (UC) problem is a very important problem in the power industry. The UC problem minimizes system-wide operational costs of power generators by providing an optimal schedule of power production for each generator so that the demand for electricity is met. Generators must operate within certain technical limits. Operational constraints such as ramping constraints and minimum uptime/ downtime constraints, in addition to the scale of the problem, make large UC problems challenging to solve. The UC problem is generally modeled as a large-scale non-convex problem. A popular method for solving UC problems in the past has been Lagrangian relaxation. However, the availability of modern software has made mixed integer linear programming (MILP) an attractive option. The first MILP UC formulation was described in [1] and has been used extensively ever since. This formulation uses three sets of binary variables to model the on/off status of generators. As computational power increased so did the complexity of

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the MILP formulations, adding features such as ramping constraints and minimum up and downtimes. Some formulations such as [2] and [3] kept the 3-binary variable format while others sought more compact formulations [4] by reducing the number of binary variables. Other efforts at improving the UC formulation focus on specific aspects of the UC problem. In [5], Frangioni et al. give an improved formulation for the quadratic objective function, while [6] strengthens the UC formulation by examining the minimum up and downtime polytope of the 3-binary variables model. The UC formulation provides a rough approximation of the costs to generate electricity. Ideally, one would like to more accurately incorporate the physical constraints on the system. Unfortunately, in the real world, the time required to solve the UC models is a hard practical limitation, restricting the size and scope of UC formulations. Strengthening the basic UC formulation will have a positive effect in solving more advanced models. Even though the UC model was developed in an era of monopolistic producers, it remains important today, even after the deregulation of the power industry. The UC problem can easily be extended to generate production schedules in a competitive market environment [3], [7], [8]. Given sufficient time, MILP methods are able to produce provably optimal solutions to UC. From a fairness perspective, this is important in competitive markets as two near-optimal solutions can produce considerably different payments to generator owners [9]. An improved formulation for the UC problem is especially valuable for advanced and computationally demanding problems such as stochastic formulations [10] or transmission switching [11]. The valid inequalities discussed in this paper may also be applied to MILP formulations of the self-scheduling problem. However, it should be noted that although a pseudo-polynomial formulation of the whole convex hull of the feasible points for the self-scheduling problem is known [12], [13], it has not been found useful in the context of pure MILP or mixed-integer quadratic programming approaches. The basic UC formulation we consider is described in Section II. In Section III, we provide computational results that indicate that the 3-binary variable formulation based on [3] is more computationally effective than [4], even though the former has considerably more binary variables. In Section IV, we present a class of inequalities that can be used to strengthen the UC formulation. Computational results are given in Section V. Conclusions will be given in Section VI. Problem data used for computational test are given in Appendix A, while Appendix B contains proofs showing the strength of the proposed inequalities. II. 3-BINARY VARIABLE FORMULATION

(3) (4) represents the set of feasible production quantities where gives the for generator in time period . The function gives the spinning reserve requiredemand at time while gives the cost of generator proment. The function units of electricity at time . It is generally assumed ducing is modeled as a to be a quadratic function. Typically, piecewise linear function. Strong inequalities for the linear approximation, called perspective cuts, are given in [14] and [5]. In the case of uncertainty, the parameters can incorporate expected costs and risk aversion. The focus of this paper lies with the set of constraints in (4). are described as The operational constraints defining follows. Generation Limits: If a generator is turned on, the power output from that generator must be within certain operational limits: (5) is a binary variable that is equal to one if unit Recall that is on during time period . If unit is off during time , constraint to be zero. (5) forces Ramping Constraints: The power output of a generator are constrained cannot fluctuate too rapidly. The variables by ramp-up (ramp-down) rates as well as startup (shutdown) rates [3]: (6) Constraint (6) limits the increase in power output allowed between time intervals. If the generator was on in the previous , then the increase in power output time period cannot be larger than the ramp-up rate. If the generator is turned , then the generator can on in the current time period units of energy in time period . A produce at a maximum similar argument holds for ramp-down constraints: (7) Uptime and Downtime Constraints: The minimum uptime constraints are commonly expressed by the set of linear equations [3]

(8)

The unit commitment problem can be formulated as (1)

(9)

(2) (10)

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TABLE I COMPUTATIONAL RESULTS

(11) (12)

(13) where and . At the start of the planning period, there may be some generators that have recently been turned on or turned off. Equations (12) and (13) force these generators to remain on for the appropriate amount of time to satisfy the minimum up and downtime constraints. Note that these constraints only serve to fix the appropriate variables. Equations (8) and (10) enforce the minimum uptime constraint while (9) and (11) enforce the minimum downtime constraints. An alternative to the minimum up/downtime constraints mentioned above is based on a series of papers [15] and [6]. In [6], Rajan and Takriti claim that the constraints (14) and (15), along with the following logical constraints and variable-bound constraints, define the convex hull of all feasible solutions in the minimum up and downtime polytope:

TABLE II GENERATOR DATA

TABLE III GENERATOR COST

(14) and

(15) While it is not the scope of this paper to verify this claim, computational results in Section III indicate that adding the new set of minimum up/downtime inequalities can have a significant impact on the computational time needed to solve UC problems. Startup Costs: A startup cost is incurred when a generator is put into operation. The cost is dependent on how long the unit has been inactive. While the startup cost function is nonlinear, it can be discretized into hourly periods, giving a stepwise function. The MILP formulation for the startup cost is [16] (16) and (17) Logical Constraints: In addition to the above constraints, constraint (18) is needed to ensure that and take the appropriate values when a generator is either turned on or off: (18)

III. CHOOSING THE FORMULATION Note that if all the variables are known, then the and variables are easily determined. Because of this, the formulation can be rewritten to omit all and terms. This is the approach taken by [4]. The advantage of this method is that the number of binary variables needed in the formulation is reduced by a factor of three. However, there are some drawbacks to this type of formulation. Mainly, using binary variables to indicate start up or shut down can make it easier to generate strong valid inequalities. Table I compares the formulations using data provided by [4] adapted to include ramping data. Tables II and III give information regarding generator parameters and costs, while the types of generators as well as demand can be found in Appendix A. The results shown in Table I indicate that the original formulation seems to be more effective. Problems are solved to within 0.5% of optimality using a 2-h time limit. All tests in this paper were carried out using CPLEX 12.1 on an Intel Xeon 2.4 GHz ( 2) processor with 4 GB of memory. The formulation proposed by [4], that requiring fewer binary variables, is referred to as the efficient formulation. The original formulation is the formulation outlined in Section II with minimum up/downtime constraints (8)–(13), while Up/Downtime is

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the original formulation with the minimum up/downtime constraints given in [6]. It may seem counterintuitive that larger formulations may be easier to solve. The results begin to make sense when considering the high degree of correlation between the , , and variables. While fixing the variable to one, one also fixes to zero for all . Similarly, many variables can be fixed to zero if were fixed to zero. Indeed, because the and variables are determined by the variables, we could relax the integrality constraints on both and . That being said, we do not recommend relaxing the integrality constraints for the following reasons. The assumed benefit of relaxing the integrality constraints is that fewer variables will need to be branched on, leading to a smaller enumeration tree. There is a cost, however, of relaxing these constraints. MILP solvers contain tools that attempt to fix integer variables during the enumeration process. Also, cutting plane methods are able to exploit the integrality of variables. By relaxing the integrality constraint, the solver does not know that the variable must be integral, and is not able to look for opportunities to exploit that fact. A better way to avoid branching on or variables would be to adjust their branching priority so that they are chosen only after all variables have been fixed (effectively prohibiting them to be candidates for branching). Furthermore, the underlying assumption that branching on and variables is bad may not be true. There may be certain instances for which branching on a variable may provide a better branch than any of the variables. By allowing for branching on all variables, the solver has more choices to pick the best branching variable. In theory, if a solver is reasonably adept at picking branching variables, a larger pool of branching candidates should produce a better enumeration tree. That being said, choosing the right branching disjunction is very difficult, and even advanced solvers such as CPLEX cannot be expected to consistently produce good branches without requiring a large amount of time to do so. A possible topic for future research would be to develop branching rules specifically for the UC problem. Note that these results seem to contradict the results in [4]. One possible reason for the discrepancy is the difference in the solvers used. The authors of [4] used CPLEX version 9.0 while the results in this paper were obtained using CPLEX 12.1. A major advancement in CPLEX between these versions has been the implementation of the feasibility pump [17], a heuristic used to find feasible solutions. Lastly, we investigate the impact of the minimum up/down time constraints provided by [6]. The computational results show that these inequalities are better than those used in [4]. IV. TIGHTER DESCRIPTIONS OF THE POWER GENERATION POLYTOPE The goal of this paper is to study the set of feasible production schedules for individual power generators. The more closely the feasible region of a problem instance resembles its convex hull, the more easily the instance will likely be solved. By studying the convex hull of power generation schedules, we present groups of inequalities that significantly strengthen the linear programming (LP) relaxation of UC problem instances

as well as decreasing overall computation time. As only constraints defining the polytope of feasible schedules for a single generator are considered, generator subscripts will be omitted in this section. Upper Bound Constraints:

(19) where . These constraints serve to make the upper bound on power output a function of the and terms. If a given , then the generator cannot generator is turned off at time in time . Similarly, if the generator is produce more than , then it cannot produce more than turned off at time in time (because it needs to be able to ramp down to or ). For large [when ], below in time does not knowing the generator will be turned off at time affect the upper production limits at time because even if the generator were producing at maximum capacity at time , there will be sufficient time for the generator to ramp down to a level . that will allow it to shut off at time , If the generator is not turned off in the interval then the upper bound for is if the generator is on at time and 0 if it is off. If the generator is on at and is not turned off in the interval, then all variables on the right-hand side (RHS) term, making the RHS . If the of (19) are zero except the generator is off at time , then either all variables on the RHS of (19) are zero, or one of the terms and the term are equal to one. These terms cancel each other out, making the RHS 0. is bounded above by the minimum up time to The term ensure that only one term takes the value one in the interval . and Ramp-Down Constraints: If

(20) is a valid inequality for . The generator must always obey the ramping constraints at all time intervals the generator is on. Under certain circumstances, however, the ramping constraints , can be strengthened. If a generator was turned on at time the difference between the startup rate and the minimum power limits may provide a tighter bound than the ramp-down must be below and must be rate [because and equal 1, greater than ]. In this case, both making the RHS in (20) a tighter constraint than if the ]. The ramping actual ramping rate was used [if constraints are further strengthened by forcing to be negative if the generator is turned on at time . In this case, the constraint enforces that . Compare for a moment the constraints in (20) with those in , the (7). As a result of the assumption that will always term be negative. As a result, the set of constraints (20) will always be

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stronger than constraints (7), i.e., any fractional solution in the LP relaxation that satisfies the constraints in (20) will also satisfy constraints (7). In this way, the set of constraints (20) can be said to dominate ramp-down constraints (7). If constraints (20) are included in the formulation, constraints (7) can be omitted with no loss in performance. If , , and

(21) Constraints in (21) are similar to those in (20) except that they take into account information from time . The last strengthened ramping constraint comes when bounding ramping over two time periods:

(22) Ramp-Up Constraints: If

and

(23) and the Constraint (23) is similar to (20). If , then the upper bound generator will be turned off in time provided by the is stronger than of the upper bound provided by the RU rate. Also, the addition of term forces the RHS to be negative if the generator is the ]. Similar to turned off at time [implying that constraints (20), it can be shown that constraints (23) dominate constraints (6). Ramp-up constraints given in (24) are analogous to (22). , , and , then If

(24) are valid inequalities. Under minimal assumptions, the Ramp-up and Ramp-down inequalities described in (20)–(24) are facets for certain projections of . Consider the inequality (20):

Inequality (20) is a facet for the subspace formed by projecting onto the variables with time indices and . Similarly, for the remaining constraints that include variables with time , , and , the constraint is a facet for the proindices , jection of onto the set of all variables with time indices , and . The proofs are given in Appendix B. V. COMPUTATIONAL RESULTS To test the impact of the tightened formulation on the UC, we used a sample problem based on [4]. The test problems in [4]

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include ten generators (three of which with similar characteristics). The eight unique generators are described in Tables II and III. The quadratic cost functions were approximated by a piecewise linear function. The marginal cost of power produced in the , bottom half of the possible production quantities is while the marginal cost of producing power above the midpoint (given in Table III). Also, we assume that the is hours of its previous cost of starting up a generator within . For any generator that has shutdown is done with a cost of hours, the startup cost is . Spinbeen off longer than ning reserves were set to be 3% of demand. To generate problem instances, we used replicated data from [4] to create larger instances. By replicating generators and scaling the hourly demand appropriately, we generated instances of varying sizes, each primarily containing generators of type one and two. Creating test instances by replicating generators introduces symmetry that can result in significantly harder than usual problems. We solve the smaller problems to 0.5% optimality and the larger problems to 1.0% optimality. The small problems are identical to those solved in Section III. Obviously the time required to solve a problem is the most important measure of the quality of the formulation. Unfortunately, however, looking only at time required to solve problems does not provide an indication of why one formulation is better than another. In addition to time, we also use the integrality gap to measure the strength of a formulation. , The integrality gap is defined to be is the optimal value of the LP relaxation and where the optimal value of the MILP. As the LP relaxation provides a lower bound on the cost, better formulations will provide better (larger) lower bounds. The integrality gap is a major factor in the time expected to solve the MILP. A node in an enumeration tree can only be pruned when there is a guarantee that the node does not contain an optimal solution. This is done when the LP relaxation at the node, i.e., the lower bound on the objective value of the node, is larger than the best known solution. Tighter formulations for the MILPs produce better (larger) lower bounds, making it more likely that the LP solution at a node is worse than the best known integer solution. The UC formulation described in [6] was compared with and without the additional valid inequalities described in Section IV. Columns 3 and 4 of Table IV show the integrality gaps at the root node for both the formulations exactly as in [6], referred to as “UD”, and that based on Section IV, referred to as “Tight”. The integrality gaps are labeled “Gap UD” and “Gap Tight”, respectively. Column 5 of Table IV indicates the percent improvement in the gap at the root node from including the additional inequalities. The last four columns of Table IV indicate solution times and number of nodes in the branch-and-bound tree for solving the problems to the pre-specified optimality gap using each of the formulations. The results in Table IV indicate that the strengthened formulation can considerably improve the solution time for the UC problem. The LP bound is much better with the additional constraints, allowing for nodes to be pruned earlier in the tree. For most of the larger instances, CPLEX is able to exceed the 1.0% optimality tolerance without needing to branch.

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TABLE IV COMPUTATIONAL RESULTS

TABLE V NUMBER OF GENERATORS

TABLE VI DEMAND (% OF TOTAL CAPACITY)

One note of caution regarding these inequalities is that the constraints may make the LP relaxation addition of all harder to solve. It is possible that, while the corresponding enumeration tree may be smaller using the inequalities, the additional time solving the LPs may outweigh the benefits. Fortunately, it is not necessary to include all the inequalities to the problem formulation. For instance, the ramp-up constraints need not be included for time periods where overall demand is decreasing. Also, some generators may be so cheap (expensive) that it is clear that they will (not) be in operation. Rather than explicitly adding all the constraints to the problem formulation, the computational tests were done by declaring them as user cuts. This designation tells CPLEX that the inequalities are not necessary to ensure that the resulting solution is feasible, but that they only exist to potentially strengthen the formulation. CPLEX then examines the set of inequalities and adds the inequalities it thinks will be beneficial, so, while it can poten, it will likely add just a small fraction of tially add all the inequalities. Searching through these constraints does come with a computational cost. However, given the nature of the UC problem, most of this cost can in principle be avoided. The UC problem is solved on a daily basis, and changes to the generators are rare. It is reasonable to assume that the collection of cuts that improve the solving of today’s UC problem will also be effective at solving tomorrow’s problem. VI. CONCLUSION In this paper, we presented an improved formulation for the operating region of an electric generator. The additional constraints are easily generated and can have a significant effect on the quality of the UC formulation and the computational time required to solve it. While we study these formulations only in the context of the UC problem, it is straightforward to apply them to other scheduling problems.

TABLE VII AFFINELY INDEPENDENT SOLUTIONS TO CONSTRAINT (20)

APPENDIX A PROBLEM DATA Problems used for computational testing were generated randomly by choosing how many of each generator were in the problem. The breakdown of generators in each problem is given in Table V. Hourly demand was determined using Table VI. APPENDIX B PROOF OF THE FACETS We assume that for every generator , the UC problem remains feasible if generator is turned off throughout the entire time horizon. Note that if this is not the case, then there is at least one time period, , where generator must be turned on in can order to satisfy the demand. In this case, the variable be fixed to one. is the polytope describing feasible production Recall that schedules for generator . Let be any set of variables de-

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TABLE VIII AFFINELY INDEPENDENT SOLUTIONS TO CONSTRAINT (21)

TABLE IX AFFINELY INDEPENDENT SOLUTIONS TO CONSTRAINT (22)

TABLE X AFFINELY INDEPENDENT SOLUTIONS TO CONSTRAINT (23)

TABLE XI AFFINELY INDEPENDENT SOLUTIONS TO CONSTRAINT (24)

scribing generator . The projection of onto the set , is the subspace of defined only by the variables in . For any solution in the projection, there is at least one solution in that is consistent with on variables . A valid inequality is a facet for a polyhedron with dimension if and only if there are affinely independent solutions in that satisfy the inequality with equality [18]. We use this prop-

erty to show that the ramping constraints presented in Section IV are facets for some projections of . Consider a constraint of type (20). Let . The dionto is 7 [there are 8 variables and one mension of equality constraint from (18)]. Table VII shows 7 affinely independent solutions that satisfy (20) with equality.

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Similarly, constraints of type (21) are facets of the projeconto tion of . Note that the dimension of is 10 [12 variables and two equality constraints of type (18)]. Ten affinely independent solutions are given in Table VIII. The appropriate number of affinely independent solutions to prove that onto is given (22) is a facet on the projection of in Table IX, and the affinely independent solutions proving that onto (23) and (24) are facets on the projection of are given in Tables X and XI, respectively.

[14] A. Frangioni and C. Gentile, “Perspective cuts for a class of convex 0–1 mixed integer programs,” Math Program., Ser. A, vol. 106, no. 2, pp. 225–236, Apr. 2006. [15] J. Lee, J. Leung, and F. Margot, “Min-up/min-down polytopes,” Discrete Optimiz., vol. 1, no. 1, pp. 77–85, Jun. 2004. [16] M. P. Nowak and W. Romisch, “Stochastic Lagrangian relaxation applied to power scheduling in a hydro-thermal system under uncertainty,” Ann. Oper. Res., vol. 100, pp. 251–272, Nov. 2000. [17] M. Fischetti, F. Glover, and A. Lodi, “The feasibility pump,” Math Program., Ser. A, vol. 104, no. 1, pp. 91–104, Sep. 2005. [18] G. Nemhauser and L. Wolsey, Integer and Combinatorial Optimization. New York: Wiley-Interscience, 1999.

ACKNOWLEDGMENT The authors would like to thank D. Fuller for his helpful feedback and contributions to the presentation. The authors also would like to thank the referees for their invaluable comments. REFERENCES [1] L. L. Garver, “Power generation scheduling by integer programming—Development and theory,” Trans. Amer. Inst. Elect. Eng. Part III: Power App. Syst., vol. 81, no. 3, pp. 730–734, Apr. 1962. [2] G. W. Chang, Y. D. Tsai, C. Y. Lai, and J. S. Chung, “A practical mixed integer linear programming based approach of unit commitment,” in Proc. IEEE PES General Meeting, Jun. 2004, vol. 1, pp. 221–225. [3] J. M. Arroyo and A. J. Conejo, “Optimal response of a thermal unit to an electricity spot market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp. 1098–1104, Aug. 2000. [4] M. Carrion and J. M. Arroyo, “A computationally efficient mixed integer linear formulation for the thermal unit commitment problem,” IEEE Trans. Power Syst., vol. 21, no. 3, pp. 1371–1378, Aug. 2006. [5] A. Frangioni, C. Gentile, and F. Lacalandra, “Tighter approximated MILP formulations for unit commitment problems,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 105–113, Feb. 2009. [6] D. Rajan and S. Takriti, Minimum Up/Down Polytopes of the Unit Commitment Problem With Start-Up Costs, Jun. 2005, IBM Research Report. [7] A. Borghetti, A. Frangioni, F. Lacalandra, C. Nucci, and P. Pelacchi, “Using of a cost-based unit commitment algorithm to assist bidding strategy decisions,” in Proc. IEEE Powertech Bologna Conf., A. Frangioni, C. Nucci, and M. Palone, Eds., Jun. 2003, vol. 2, 8 pp. [8] B. Hobbs, M. Rothkopf, R. O’Neill, and H. Chao, The Next Generation of Unit Commitment Models. Norwell, MA: Kluwer, 2001. [9] R. Sioshansi, R. O’Neill, and S. Oren, “Economic consequences of alternative solution methods for centralized unit commitment in dayahead electricity markets,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 344–352, May 2008. [10] S. Takriti, J. R. Birge, and E. Long, “A stochastic model for the unit commitment problem,” IEEE Trans. Power Syst., vol. 11, no. 3, pp. 1497–1508, Aug. 1996. [11] E. Bartholomew, R. O’Neill, and M. Ferris, “Optimal transmission switching,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1346–1355, Aug. 2008. [12] W. Fan, X. Guan, and Q. Zhai, “A new method for unit commitment with ramping constraints,” Elect. Power Syst. Res., vol. 62, no. 3, pp. 215–224, Jul. 2002. [13] A. Frangioni and C. Gentile, “Solving nonlinear single-unit commitment problems with ramping constraints,” Oper. Res., vol. 54, no. 4, pp. 767–775, Jul. 2006.

James Ostrowski (M’10) received the Ph.D. degree in the Industrial and Systems Engineering Department at Lehigh University, Bethlehem, PA. While at Lehigh, he was awarded second place in the George Nicholson Student Paper competition for his work dealing with symmetry in integer programming. Since Lehigh, he has been working as a Postdoctoral Fellow in the Department of Management Sciences at the University of Waterloo, Waterloo, ON, Canada, under the supervision of M. F. Anjos and A. Vannelli. His research interests include integer and stochastic programming.

Miguel F. Anjos (M’06) received the Ph.D. degree from the University of Waterloo, Waterloo, ON, Canada, in 2001. He has been with the Department of Management Sciences at the University of Waterloo since 2004, became an Associate Professor in 2007, and was Associate Chair for Graduate Studies & Research from 2007 to 2009. He is cross-appointed since 2004 to the Department of Electrical and Computer Engineering. His research interests are in optimization, and he is particularly interested in its application to engineering problems. He has worked on industrial research projects in revenue management (sponsored by British Midland Airways), network planning (sponsored by Bell Canada), and ambulance deployment (sponsored by the Region of Waterloo Emergency Medical Services). He has been awarded a Humboldt Research Fellowship for Experienced Researchers for 2009–2010. Dr. Anjos is an Associate Editor of Discrete Applied Mathematics and a member of the editorial board of Optimization and Engineering. He is also a member of the Research Review Committee of MITACS.

Anthony Vannelli (M’06) received the Ph.D. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 1983. From 1983 to 1984, he was a Research Fellow with the Mathematical Sciences Department, Thomas J. Watson Research Center, Yorktown Heights, NY, where he worked on circuit layout problems in a group led by R. Brayton. He was the recipient of a University Research Fellowship from the Natural Sciences and Engineering Research Council (NSERC) of Canada, which he held from 1984 to 1993. He joined the Department of Industrial Engineering, University of Toronto, Toronto, ON, Canada, in 1984 and returned to the Department of Electrical and Computer Engineering, University of Waterloo, in 1987. In 1993–1994, he was a Visiting Research Scientist with Shell Research, Amsterdam, The Netherlands. He was the Chair of Electrical and Computer Engineering at the University of Waterloo from 1998 to 2004 and Associate Dean for the Faculty of Engineering, University of Waterloo, from 2004–2006. Since January 2007, he has been the Dean of the College of Physical and Engineering Science at the University of Guelph, Guelph, ON, Canada. His main research focuses on the development of efficient linear, nonlinear, and discrete optimization techniques to solve large-scale circuit layout, power, and design problems.