A Support-Based Algorithm for the Bi-Objective Pareto Constraint

Transcription

1 A Support-Based Algorithm for the Bi-Ojetive Pareto Constraint Renaud Hartert and Pierre Shaus UCLouvain, ICTEAM, Plae Sainte Bare 2, 1348 Louvain-la-Neuve, Belgium {renaud.hartert, Astrat Bi-Ojetive Cominatorial Optimization prolems are uiquitous in real-world appliations and designing approahes to solve them effiiently is an important researh area of Artifiial Intelligene. In Constraint Programming, the reently introdued i-ojetive Pareto onstraint allows one to solve i-ojetive ominatorial optimization prolems exatly. Using this onstraint, every non-dominated solution is olleted in a single tree-searh while pruning su-trees that annot lead to a non-dominated solution. This paper introdues a simpler and more effiient filtering algorithm for the i-ojetive Pareto onstraint. The effiieny of this algorithm is experimentally onfirmed on lassial i-ojetive enhmarks. Bi-Ojetive Cominatorial Optimization (BOCO) aims at optimizing two ojetive funtions simultaneously. Sine these ojetive funtions are often onfliting, there is usually no perfet solution that is optimal for oth ojetives at the same time. In this ontext, deision makers are looking for all the est ompromises etween the ojetives to hoose a posteriori the solution that est fits their needs. Hene, the notion of optimal solution is replaed y the notion of effiieny and we are searhing for the set of all the effiient solutions (usually alled effiient set or Pareto frontier) instead of one single solution (Ehrgott 2005). During the past years, many approahes were developed to takle BOCO prolems exatly. However, many of them were developed in the ontext of Mathematial Programming (Mavrotas 2007; Ralphs, Saltzman, and Wieek 2006) and only a few an e applied effiiently in Constraint Programming. Among these approahes, the ɛ-onstraint is proaly the most widely used (Haimes, Lasdon, and Wismer 1971; Le Pape et al. 1994; Van Wassenhove and Gelders 1980). The idea is to deompose the original prolem into a sequene of suprolems to optimize with regard to the first ojetive funtion. At eah iteration, a new suprolem is generated y onstraining the seond ojetive to take a etter value than its value in the optimal solution of the previously solved suprolem. Notie that the numer of single- This work was (partially) supported y the ARC grant 13/ from Communauté française de Belgique. Copyright 2014, Assoiation for the Advanement of Artifiial Intelligene ( All rights reserved. ojetive prolems to solve is linear in the effiient set s ardinality (Haimes, Lasdon, and Wismer 1971). The i-ojetive Pareto onstraint is an alternative (and more effiient (Gavanelli 2002)) approah to ompute the effiient set exatly. The idea ehind this onstraint is to uild an approximation of the effiient set inrementally during the searh and to use this approximation to detet and to prune su-trees that an only lead to solutions that are less effiient than the ones already ontained in the approximation. Eventually, the approximation eomes the effiient set and its optimality is proven when the searh is ompleted. The algorithm of the Pareto onstraint relies on two operations. The first operation is used to update the approximation (y inserting new solutions in it) while the seond onsists to use this approximation to redue the searh spae of the prolem. In this work, we show how to use speifi BOCO properties to improve the effiieny of the iojetive Pareto onstraint. Preisely, we show that oth operations (update and filtering) an enefit from eah other in an iterative way to uild a simpler and more effiient algorithm for the onstraint. This doument is strutured as follows. First, we riefly introdue onstraint programming and its main onepts. Then, we formalize multi-ojetive ominatorial optimization in the ontext of onstraint programming and present some important definitions. The third setion is dediated to the Pareto onstraint in its general multi-ojetive form. Our support-ased algorithm is presented in the fourth setion. Setion five diretly follows with our experiments and results on two lassial enhmarks i.e. the i-ojetive knapsak prolem and the i-ojetive travelling salesman prolem. Finally, the last setion offers some onlusions and perspetives. Constraint Programming Bakground Constraint Programming is a powerful paradigm to solve onstraint satisfation prolems and ominatorial optimization prolems. A onstraint programming prolem is usually defined y a set of variales with their respetive domain (i.e. the set of values that an e assigned to a variale), and a set of onstraints on these variales. The ojetive is to find an assignment of the variales that respets all the onstraints of the prolem. The onstraint programming proess interleaves a tree-searh exploration (ommon in ar-

2 tifiial intelligene) with an inferene proedure (also alled propagation or filtering) to remove values that annot appear in a solution. The inferene part prunes the ranhes of the searh-tree and thus redues the searh spae to explore. In an optimization ontext, an ojetive funtion an also e onsidered. Usually, this is formulated y the addition of an ojetive variale having as domain the set of possile values that an e taken y the ojetive funtion. Eah time a solution is found, a onstraint is added to the prolem dynamially to fore the ojetive variale to e assigned to a etter value. 1 The optimality is proven when the searh is exhausted, the last solution eing the optimal one. Multi-Ojetive Cominatorial Optimization A Multi-Ojetive Cominatorial Optimization (MOCO) prolem is a quadruple P = X, D, C, F where X = {x 1,..., x n } is a set of variales, D = {D 1,..., D n } is a set of domains, C is a set of onstraints on the variales in X, and F = {f 1,..., f m } is a set of ojetive funtions to minimize simultaneously. Eah ojetive funtion f i (X) assoiates a disrete ost to the assignment of the variales in X. In the following, we assume that m ojetive variales oj 1,..., oj m have een added to the set of variales X and onstrained to e equal to the value taken y their orresponding ojetive funtion i.e. oj i = f i (X). In partiular, the minimal and maximal values of the domain of an ojetive variale oj i are the ounds of this ojetive with regard to a given partial assignment of the variales in X. A solution of a MOCO prolem P is a omplete assignment of the variales in X that satisfies all the onstraints in C. In this work, we represent a solution sol y its assoiated ojetive vetor (sol 1,..., sol m ) where sol i is the value assigned to the orresponding ojetive variales oj i. As it is not likely that a solution is simultaneously optimal for all ojetives at the same time, we are interested in a way to define a partial order on the ojetive vetor of the solutions. Among these orderings, the weak Pareto dominane is a popular hoie (see Figure 1). It is defined as follows. Definition 1 (Weak Pareto dominane). Consider sol = (sol 1,..., sol m ) and sol = (sol 1,..., sol m), two solutions of a MOCO prolem P. We say that sol dominates sol, denoted sol sol, if and only if: i {1,..., m} : sol i sol i. (1) Let sols(p) denote the set of all the solutions of a MOCO prolem P. A solution sol of P is effiient if and only if no solution sol in sols(p) dominates sol: sol sols(p) : sol sol sol sol. (2) In other words, a solution is effiient if it is impossile to improve the value of one ojetive without degrading the value of at least one other ojetive (i.e. an effiient solution is an optimal ompromise etween the ojetives). Solving a MOCO prolem usually means finding the set of all the effiient solutions that is generally alled the effiient set or Pareto front (the effiient set of an aritrary solution spae is depited in Figure 2). Unfortunately, the size 1 Notie that the searh does not stop ut ontinues after the addition of the onstraint to the prolem. of the effiient set often grows exponentially with the size of the prolem to solve (Ehrgott 2005). Hene, in pratie, only an approximation of the effiient set an e found in a reasonale amount of time and of memory. We all suh an approximation of the effiient set an arhive. Definition 2 (Arhive). An arhive A is a domination-free set of solutions i.e. a set of solutions suh that no solution in A dominates another solution in A: sol A, sol A : sol sol sol sol. (3) Clearly, the effiient set is also an arhive. a f e Figure 1: Solution e dominates the solutions,, and d while eing dominated y solution f. Solutions a and g do not dominate solution e and are not dominated y solution e. g d The Pareto Constraint Figure 2: An aritrary iojetive solution spae. Effiient solutions are olored in lak. The effiient set is the set of all the effiient solutions. The Pareto onstraint (Shaus and Hartert 2013) is a gloal onstraint defined over the m ojetive variales of a MOCO prolem P and an arhive A: Pareto(oj 1,..., oj m, A). (4) The aim of the Pareto onstraint is to use the arhive A to prune solutions that are dominated y at least one solution ontained in the arhive A. In other words, the Pareto onstraint ensures that a newly disovered solution is not dominated y any solution in the arhive. This definition is formalized as follows: m oj i < sol i (5) Filtering sol A i=1 The filtering rule of the Pareto onstraint was originally proposed in (Gavanelli 2002). The idea is to use artifiial ojetive vetors, alled dominated points, to adjust the upper ounds of the ojetive variales in order to prevent the disovery of dominated solutions (5). Definition 3 (Dominated point). Let oji min and oji max denote the lower and upper ounds of the ojetive variale oj i, the dominated point DP i is defined as follows:

3 where DP i = (DP i 1,..., DP i m) { oj DPj i max = j ojj min if j = i otherwise In other words, the dominated point DP i an e seen as an artifiial solution for whih eah ojetive variale is assigned to its est possile value exept for oj i whih is assigned to its worst possile value. Proposition 1. Let sol e a solution in the arhive A suh that sol dominates DP i. Then, the value of the ojetive variale oj i has to e smaller than the value sol i : (6) sol A, sol DP i oj i < sol i. (7) Proof. Consider a solution sol in the arhive A suh that sol DP i. Aording to (6), we know that sol j ojj min ( j {1,..., m}, j i). Hene, eah newly disovered solution sol new (i.e. a solution ontained in the Cartesian produt of the ojetive variales) suh that sol i soli new is dominated y sol and an thus e safely removed. Figure 3 illustrates the domain of the ojetive variales efore (left part) and after (right part) applying the filtering rule of Proposition 1. DP 1 DP 1 Figure 3: Filtering of the Pareto onstraint. Blak solutions orrespond to the solutions ontained in the arhive A. Grey areas represent the parts of the ojetive spae that are dominated y at least one solution in A. The dominated points DP 1 and are represented y the white solutions. The hashed area orresponds to the redution of the ojetive variales domains after applying the filtering rule of Proposition 1. If a dominated point DP i is dominated y several solutions at the same time, the filtering rule of Proposition 1 has to e applied until no solution in the arhive dominates DP i. Clearly, the order in whih the dominating solutions are seleted affets the numer of alls of the filtering rule. This situation is illustrated in Figure 4 where the worst possile seletion order is a,, and. From this oservation, it appears that the Pareto onstraint an reah its fixed point in one step if it is ale to aess diretly the solution that dominates DP i with the lowest value in oj i. Suh a solution is alled the tightest solution of oj i. Definition 4 (Tightest solution). The tightest solution of oj i is the solution in the arhive A that dominates DP i and that has the lowest value for oj i : argmin sol A {sol i sol DP i }. (8) If DP i is not dominated y at least one solution in A, then, oj i has no tightest solution. Tightest solutions have een used in (Shaus and Hartert 2013) to propose the following idempotent filtering rule: sol A, sol DP i oj i < T i i (9) where T i is the tightest solution of oj i. a Figure 4: Worst possile seletion order. Solution is the tightest solution of oj 2. Multi-Ojetive Branh-and-Bound The Pareto onstraint an e used to find the exat effiient set of any MOCO prolem. The idea is to improve the quality of the arhive A used y the onstraint dynamially during the searh. Initially, the arhive is the empty set. Then, eah time a new solution is disovered, it is inserted into the arhive and the searh ontinues. Aording to (5), every newly disovered solution is guaranteed to e dominated y none of the solutions in the arhive. Hene, a newly disovered solution inserted in the arhive improves its quality and onsequently strengthens the filtering. Two situations are possile when inserting a new solution in the arhive: 1. The new solution does not dominate any solution in the arhive and nothing has to e done after the insertion (see Figure 5 left); 2. The new solution dominates one or several solutions in the arhive. In this ase, these dominated solutions have to e removed from the arhive to ensure that the arhive remains domination-free (see Figure 5 right). In oth ases, adding the new solution in the arhive stritly inreases the size of the suspae of the ojetive spae that is dominated y the arhive. 2 Eventually, the arhive eomes the effiient set and its optimality is proven when the searh is exhausted. The use of the Pareto onstraint with a dynamially improving arhive an e seen as a multiojetive ranh-and-ound that generalizes the lassial ranh-and-ound algorithms used in onstraint programming (Gavanelli 2002). 3 2 The size of the suspae that is dominated y an arhive is a ommon indiator, known as the Hypervolume, used to measure the quality of an arhive (Zitzler et al. 2003). 3 In single ojetive optimization, the arhive is either the empty set or a singleton ontaining the est-so-far solution. The only dominated point DP 1 orresponds to the upper ound of the ojetive variale to minimize.

4 B C A sol new new sol F D G A E B C D E F G Figure 5: Insertion of a new solution in an arhive. Hashed areas orrespond to the additional parts of the ojetive spae that are dominated after the insertion of the new solution in the arhive. Implementation and data struture The multi-ojetive ranh-and-ound algorithm ased on the Pareto onstraint relies on two distint operations: Aess the tightest solution of eah ojetive to adjust the upper ound of the ojetive variales (9); Insert a newly disovered solution in the arhive and remove potentially dominated solutions from the arhive. Clearly, the effiieny of these operations is impated y the underlying data struture used to implement the inner mehanisms of the onstraint and to store its arhive. In (Gavanelli 2002), the author suggests the use of point quad trees (simply quad-trees in the sequel) to implement the arhive. A quad-tree (Finkel and Bentley 1974; Haeniht 1983; Samet 2006) is a data struture that generalizes inary searh trees to store m-dimensional vetors. Any node of a quad-tree divides its suspae into 2 m disjoint suspaes. More preisely, the root of the tree divides the spae into 2 m suspaes originating at the root. The 2 m hildren of the root (exatly one for eah suspae of the root) divides their suspaes into 2 m suspaes and so on (see Figure 6). As for inary searh trees, quad-trees are very sensitive to the order in whih solutions are inserted. In the est ase, when the tree is well-alaned, a quad-tree ensures a logarithmi omplexity for the aess operation. In the worst ase however, the quad-tree is strutured as a linear list and the aess operation needs to traverse the entire data struture to find the tightest solutions. The removal operations are also a weakness of this data struture. Indeed, removing a solution in a quad-tree leads to the destrution of the sutree rooted at this solution and often leads to an expensive omputational ost to repair it. Unfortunately, the insert operation may require several expensive removals to maintain the arhive domination-free (i.e. to remove the solutions that are dominated y the inserted solution). Support-Based Algorithms Bi-Ojetive Cominatorial Optimization (BOCO) prolems are MOCO prolems with only two ojetives (m = 2). Usually, BOCO prolems are easier to reason aout and have properties that annot e generalized to MOCO prolems with more than two ojetives. One of these properties Figure 6: Illustration of the spae partition (left) of a idimensional quad-tree (right). The node A is the root of the quad-tree. Nodes B, C, D, and E are the sons of A. Nodes F and G are the sons of D. is the i-ojetive ordering property that is defined as follows. Property 1 (i-ojetive ordering). Let A e an aritrary i-ojetive arhive. We denote A >i (resp. A <i ) the arhive A ordered y dereasing (resp. inreasing) value of oj i. Then, sorting the solutions of A y dereasing order of their value in a first ojetive (i.e. A >1 ) amounts to sorting these solutions y inreasing order of their value in the seond ojetive (i.e. A <2 ) and vie-versa. Proof. Let sol and sol e two solutions in A. Sine A is domination-free, sol does not dominate sol and sol does not dominate sol. Hene, if sol 1 < sol 1 we know that sol 2 > sol 2. Symmetrially, if sol 1 > sol 1 then sol 2 < sol 2. In the remainder of this setion, we introdue an inremental algorithm to implement the i-ojetive Pareto onstraint. The idea ehind this algorithm is to use the iojetive ordering property to store the arhive A in a iordered linked-list i.e. a linked-list suh that eah solution has a pointer to its diret suessor in A >1 and in A >2 (see Figure 7). Preisely, the algorithm relies on speial solutions, alled supports, to maintain the tightest solutions of eah ojetive (Definition 4) inrementally during the exploration of the searh tree. Definition 5 (Support). Let sol e a solution in a iojetive arhive A. We say that sol is the support of oj 1 if and only if: sol = argmin sol A {sol 1 sol 2 < oj min 2 }. (10) The support of oj 2 is defined symmetrially. Figure 8 illustrates the supports of oth ojetive variales oj 1 and oj 2. Proposition 2. Supports are never inluded in the Cartesian produt of the domain of the ojetive variales i.e. supports annot e dominated y newly disovered solutions. Proof. Let sol sup e the support of oj 1 (resp. oj 2 ). By Definition 5, we know that sol sup 2 < oj2 min (resp. sol sup 1 < oj1 min ). Hene, sol sup annot e dominated y a newly disovered solution that is at est (oj1 min, oj2 min ).

5 a Figure 7: Illustration of an aritrary i-ojetive arhive A stored in a i-ordered linked-list. Eah solution has a pointer to its diret suessor in A >1 and in A >2. d e a Figure 8: Solutions and e are respetively the supports of oj 2 and oj 1. Both supports are not inluded in the Cartesian produt of the ojetive variales. Proposition 3. If it exists, the tightest solution of oj i is the support of oj i or its diret suessor in A >i. Proof. Let DP 1 e dominated y a solution sol suh that sol is the tightest solution of oj 1. If sol 2 < oj2 min, y Definition 5 we know that sol is the support of oj 1. Else, sol 2 = oj2 min and sol is the diret suessor of the support of oj 1. The link etween supports and tightest solutions is illustrated in Figure 8 where d is the tightest solution of oj 1 and is the tightest solution of oj 2. Proposition 4. Let sol new = (oj1 min, oj2 min ) e a newly disovered solution. If the arhive is ordered aording to the value of one ojetive (i.e. A >1 or A >2 ), then, sol new dominates all the solutions ontained etween oth supports. Proof. Let S i denote the set of all the suessors of the support of oj i in the ordered arhive A >i. By Definition 5, we know that S i = {sol A sol i oji min }. The intersetion of S 1 and S 2 is thus the set of solutions that is dominated y (oj1 min, oj2 min ): S 1 S 2 = {sol A sol 1 oj min 1 sol 2 oj min 2 }. Sine sol new = (oj1 min, oj2 min ), the solutions ontained in the intersetion of the suessors of oth supports are dominated y sol new. The insertion of a new solution is illustrated in Figure 9. This operation is performed in onstant time y updating the pointers of oth supports. We use Propositions 2, 3, and 4 to design an inremental algorithm for the i-ojetive Pareto onstraint. As mentioned aove, the algorithm maintains the tightest solutions of eah ojetive y adjusting the supports inrementally during the exploration of the searh tree. More preisely, we desrie the algorithm to adjust the upper ound of oj 1 as follows: 4 4 The upper ound of oj 2 is adjusted symmetrially. d e 1. Eah time the lower ound of oj 2 is adjusted, we know that we have to reonsider the support of oj 1. To do so, we iterate on the diret suessors of the old support in A >1 until we reah the new support. Let denote this numer of iterations. Clearly, the sum of the annot exeed the size of A along a ranh of the searh tree. 2. When the new support is found, we aess and use the tightest solution of oj 1 to adjust the upper ound of oj 1 (see Proposition 3 and (9)). 3. When a new solution sol new is disovered, we insert it in A y simply updating the pointers of its supports (see Proposition 4). Assuming a trail-ased CP solver, reversile pointers an e used to maintain the supports. 5 Hene, eah time a aktrak ours, the algorithm is ale to restore its previous supports in amortized onstant time. Oserve that, aording to Proposition 2, we know that these supports are not dominated y a previously disovered solution. s 2 sol new s 1 s 2 sol new Figure 9: Insertion of a new solution in onstant time. Solutions s 1 and s 2 are respetively the support of oj 1 and oj 2. Experiments and Results This setion presents the experimental evaluation of our support-ased algorithm against the quad-tree algorithm ommonly used to implement the i-ojetive Pareto onstraint. Our experiments used lassial instanes of the i-ojetive inary knapsak prolem (Xavier Gandileux 2013) and instanes of the i-ojetive traveling salesman prolem (Paquete and Stützle 2009). All algorithms were implemented in the open-soure OsaR solver (OsaR Team 2012) that runs on the Java Virtual Mahine using a omputer running Ma Os X 10.9 on an Intel i7 2.6 Ghz proessor. First, we ompare the numer of searh nodes explored using oth algorithms on instanes of the i-ojetive inary knapsak prolem within a time limit of 30 seonds. In this experiment, the i-ojetive Pareto onstraint starts with an empty arhive and explores the searh-spae with a random heuristi. The usage of a random heuristi should favor the quad-tree, highly sensitive to the order in whih solutions are inserted. Indeed, using a heuristi dediated to one of the ojetives would lead to an unalaned quad-tree negatively 5 This funtionality an easily e adapted for opy-ased CP Solvers. s 1

6 impating its effiieny. Tale 1 gives the mean and the standard deviation of the numer of nodes explored over 10 runs for eah algorithm. We oserve that our support-ased algorithm is always the fastest despite the fair heuristi that we have hosen. It explores on average 20% more nodes. Tale 1: Numer of searh nodes explored within a time limit of 30 seonds on instanes of the i-ojetive inary knapsak prolem with 500 items. Quad-tree Support-ased instane mean st. dev. mean st. dev. 500A B C D Our seond experiment ompares oth algorithms on instanes of the i-ojetive traveling salesman prolem. In this experiments, the Pareto onstraint starts with an initial arhive that is a good approximation of the exat effiient set. In fat, the effiient set of these instane is urrently unknown and the given set is the union of the approximation of many state-of-the-art algorithm to solve this prolem (Lust and Teghem 2010). Sine the internal struture of the Pareto onstraint is initialized in advane, we onsider oth ases of an unalaned and well-alaned quad-tree. As for the previous experiments, the mean and standard deviation over 10 runs are presented in the Tale 2. Tale 2: Numer of searh nodes explored within a time limit of 30 seonds on instanes of the i-ojetive travelling salesman prolem with 100, 150, 200, 300, 400, and 500 ities. Quad-tree Quad-tree (al.) Support-ased instane mean st. dev. mean st. dev. mean st. dev. KroAB KroAB KroAB KroAB KroAB KroAB Again, the support-ased is the fastest approah. As mentioned aove, we oserve that the quad-tree algorithm is very sensitive to the order in whih solutions are added in its struture. On one hand, a well-alaned quad-tree is ompetitive with the support-ased algorithm (ut still always dominated). On the other hand, a poorly alaned quad-tree sustantially deteriorates the numer of searh nodes explored. Interestingly, the alaned quad-tree and the support-ased algorithms were ale to add 3 solutions in the approximation of the KroAB300 effiient set and 4 solutions in the approximation of the KroAB500 effiient set. This experiment illustrates the flexiility of the Pareto onstraint and one of its possile uses to improve an existing approximation of the effiient set or to prove its optimality. Conlusion This paper introdued a support-ased algorithm to implement the i-ojetive Pareto onstraint. This inremental filtering algorithm relies on the i-ojetive ordering property of BOCO prolems. Experiments demonstrate that the support-ased algorithm is more effiient than the lassial algorithm used to implement the Pareto onstraint. Also, this algorithm is simpler to implement than the quad-tree ased propagator sine it only relies on a linked list and two reversile pointers. Referenes Ehrgott, M Multiriteria optimization, volume 2. Springer Berlin. Finkel, R. A., and Bentley, J. L Quad trees a data struture for retrieval on omposite keys. Ata informatia 4(1):1 9. Gavanelli, M An algorithm for multi-riteria optimization in CSPs. ECAI Haeniht, W Quad trees, a datastruture for disrete vetor optimization prolems. In Essays and Surveys on Multiple Criteria Deision Making. Springer Haimes, Y. Y.; Lasdon, L. S.; and Wismer, D. A On a iriterion formulation of the prolems of integrated system identifiation and system optimization. IEEE Transations on Systems, Man, and Cyernetis 1(3): Le Pape, C.; Couronné, P.; Vergamini, D.; and Gosselin, V Time-versus-apaity ompromises in projet sheduling. Lust, T., and Teghem, J Two-phase Pareto loal searh for the iojetive traveling salesman prolem. Journal of Heuristis 16(3): Mavrotas, G Generation of effiient solutions in multiojetive mathematial programming prolems using GAMS. Effetive implementation of the ε-onstraint method. OsaR Team OsaR: Sala in OR. Availale from Paquete, L., and Stützle, T Design and analysis of stohasti loal searh for the multiojetive traveling salesman prolem. Computers & operations researh 36(9): Ralphs, T. K.; Saltzman, M. J.; and Wieek, M. M An improved algorithm for solving iojetive integer programs. Annals of Operations Researh 147(1): Samet, H Foundations of multidimensional and metri data strutures. Morgan Kaufmann. Shaus, P., and Hartert, R Multi-Ojetive Large Neighorhood Searh. In 19th International Conferene on Priniples and Pratie of Constraint Programming. Van Wassenhove, L. N., and Gelders, L. F Solving a iriterion sheduling prolem. European Journal of Operations Researh 4: Xavier Gandileux A olletion of test instanes for multiojetive ominato-

7 rial optimization prolems. Availale from Zitzler, E.; Thiele, L.; Laumanns, M.; Fonsea, C. M.; and da Fonsea, V. G Performane assessment of multiojetive optimizers: An analysis and review. Evolutionary Computation, IEEE Transations on 7(2):