A Note on Robustness of the Min-Max Solution to Multiobjective Linear Programs

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1 A Note on Robustness of the Min-Max Solution to Multiobjective Linear Programs Erin K. Doolittle, Karyn Muir, and Margaret M. Wiecek Department of Mathematical Sciences Clemson University Clemson, SC January 15, 2016 Abstract: The challenge of the use of scalarizing methods in multiobjective optimization results from the choice of the method, which may not be apparent, and given that a method has been selected, from the choice of the values of the scalarizing parameters. In general these values may be unknown and the decision maker faces a difficult situation of making a choice possibly under a great deal of uncertainty. Due to its effectiveness, the robust optimization approach of Ben-Tal and Nemirovski is applied to resolve the uncertainty carried in scalarized multiobjective linear programs (MOLPs). A robust counterpart is exaed for six different scalarizations of the MOLP yielding a robust (weakly) efficient solution to the original MOLP. The study reveals that the -max optimal solution emerges as a robust (weakly) efficient solution for five out of the six formulations. Keywords: robust optimization, robust counterpart, scalarization, achievement function, conic scalarization, weighted norm, weighted sum, weighted constraint, epsilon constraint, efficient solutions 1 Introduction Multiobjective optimization problems (MOPs) model decision making situations in which several objective functions are optimized under conflict because a feasible solution that is optimal for all functions simultaneously does not exist. Solving MOPs is understood as finding their Pareto-optimal solution set or its representation, or a single Pareto point that is preferred by the decision maker (DM). Solution methods for MOPs can be classified into 1

2 scalarizing approaches that reformulate the original MOP into a single-objective optimization problem (SOP) by means of scalarizing parameters, or nonscalarizing approaches that maintain the vector-valued objective function and use other optimality concepts. Refer to (Ehrgott and Wiecek, 2005) for an extensive survey on this subject. While the interest in the development of nonscalarizing methods has been growing and new approaches are being proposed, the scalarizing methods have mainly been popular among DMs working in multicriteria settings. The challenge in the use of these methods results from the choice of the method, which may not be apparent, and given that a method has been selected, from the choice of the scalarizing parameters values. In general these values may be unknown and the DM faces a difficult situation of making a choice possibly under a great deal of uncertainty. For example, choosing weights as scalarizing parameters has been discussed extensively from a psychological perspective (Eckenrode, 1965) and from an engineering point of view (Marler and Arora, 2010). In any case, a scalarized MOP becomes an uncertain SOP that could benefit from the independently growing field of optimization under uncertainty. Decision making under uncertainty has achieved much attention because of inaccurate or incomplete data that often appear in real-world applications. To address this challenging issue, several types of methodologies such as stochastic optimization (Schneider and Kirkpatrick, 2006), reliability-based optimization (Kuo and Zhu, 2012), fuzzy optimization (Lodwick and Kacprzyk, 2010), uncertain optimization (Liu, 2002), or robust optimization (Ben-Tal and Nemirovski, 1999, 2000), (Ben-Tal et al., 2009) have been developed. The first four types employing the probability and possibility theories require the knowledge of probability distributions or fuzzy membership functions to model and resolve uncertainty. The latter seems to be less demanding since it makes use of deteristic models and methods, and simple definitions of finite or infinite uncertainty sets that are assumed to be known. Due to its effectiveness, robust optimization is applied in this paper to resolve the uncertainty carried in scalarized MOPs. This research direction has already been undertaken for the weighted-sum method in a setting of forest management (Palma and Nelson, 2010) and in a broader theoretical context (Hu and Mehrotra, 2012). The uncertain weights are restricted to a given interval but are allowed to vary throughout the optimization process. We also recognize that robust optimization has been related to multiobjective optimization in other ways than what we present in this paper. Multiobjective optimization has first been used as a tool to solve optimization problems under uncertainty (Kouvelis and Yu, 1997; Köbis and Tammer, 2012; Klamroth et al., 2013; Iancu and Trichakis, 2014) where every realization of uncertainty leads to another objective function. More recent studies address MOPs with uncertain parameters in the objective and/or constraint functions (Kuroiwa and Lee, 2012; Raith and Kuhn, 2013; Ehrgott et al., 2014; Goberna et al., 2014). Some of these works adapt the scheme of (Ben-Tal et al., 2009) in multiobjective settings while others go beyond. Refer to (Goberna et al., 2014) for a recent review on robust multiobjective optimization. The goal of this paper is to review a collection of those SOPs associated with the multiobjective linear program (MOLP) that are well-established in the literature, and solve them within the framework of robust optimization. Because an optimal solution to the SOP is typically a (weakly) efficient solution to the MOLP, this approach reveals scalarization-related 2

3 robust efficient solutions to the MOLP, which provide information of high interest to DMs. The paper is organized as follows. In the next section two types of uncertain SOPs are formulated and preliary definitions and propositions are given. Section 3, as the main part of the paper, contains results on robust counterparts to six specific SOPs. The results are summarized and discussed in Section 4, and the paper is concluded in Section 5. 2 Preliaries Consider the following MOLP with linear objective functions, Cx (1) where C is a p n matrix and X R n is a nonempty feasible polyhedral set. Definition 1. For y 1, y 2 R p, (1) y 1 y 2 if and only if y 1 i y 2 i, i = 1,, p, with y 1 y 2 ; (2) y 1 y 2 if and only if y 1 i y 2 i, i = 1,, p. Suppose, without loss of generality, Cx 0. Let c i x R denote the ith component of Cx R p, where c i is the i-th row of matrix C, i = 1,, p. Definition 2. (i) A solution x X is said to be efficient to MOLP (1) if there does not exist x X such that C x Cx. (ii) A solution x X is said to be weakly efficient to MOLP (1) if there does not exist x X such that C x < Cx. Given MOLP (1), we formulate two types of related SOPs. Let s 1 and s 2 be two scalarizing functions defined as s 1 (Cx, u) : R p R p R 1 and s 2 (Cx) : R p R 1, u U, with U being a set of scalarizing parameters. Let S u X be a subset of the feasible region of (1). Then the scalarized optimization problem (SOP) associated with MOLP (1) can be written in two forms: Scalarizing Model 1 SOP1(u) Scalarizing Model 2 SOP2(u) x s 1 (Cx, u) x X x s 2 (Cx) x S u X (2) One can immediately observe that a feasible solution to SOP (2) is feasible to MOLP (1), however, the converse may not hold true. We therefore make the following assumption. 3

4 Assumption 1. If x is a feasible solution to MOLP (1), then there exists a scalarizing parameter u U such that x is a feasible solution to SOP2(u). Based on (Ehrgott and Wiecek, 2005, and references therein), we state the following general result that applies to a variety of scalarizing functions. Theorem 1. If x is an optimal solution to SOP(u) (2) for some u U, then x is a weakly efficient solution to MOLP (1). Instead of following the traditional approach to selecting specific values for the parameter u, we assume the the values are uncertain and contained in an uncertainty set U. This yields an uncertain SOP (USOP) defined as a collection of SOPs, SOP (u)} u U, one for each realization of the uncertainty parameter u: USOP1 } s 1 (Cx, u) x x X u U USOP2 x } s 2 (Cx) x S u A member of this collection for a fixed u U is referred to as an instance. Given USOP (3), if we wish to apply the robust optimization methodology of (Ben-Tal and Nemirovski, 1999), we reformulate the USOP into a robust counterpart and refer to it as the robust SOP (RSOP): u U (3) RSOP1 RSOP2 s 1 (Cx, u), u U x X x s 2 (Cx) x S u, u U (4) where R is an auxiliary variable. Both robust counterpart problems are not computationally tractable since they have infinitely many constraints parametrized by u in U. It is then of interest to reformulate the counterparts into deteristic problems that are computationally tractable. Before we accomplish this, following (Ben-Tal and Nemirovski, 1999) we adopt the following concepts. Definition 3. (i) A feasible solution to RSOP (4) is said to be a robust feasible solution to USOP (3). (ii) A feasible and optimal solution to RSOP (4) is said to be a robust optimal solution to USOP (3). Note that an optimal solution to RSOP (4) is the feasible solution to this problem that produces the smallest objective value for all realizations u U. This smallest objective value of RSOP (4) is called the robust optimal value of USOP (3). 4

5 In RSOP1, the scalarizing function in the inequality constraint gives rise to p inequality constraints of the form s i (c i x, u i ), u i U i, i = 1,, p where U = U 1 U p. In RSOP2, the constraint is replaced with p 1 constraints of the form x S ui, u i U i, i = 1,, p 1, where U = U 1 U p 1. This structure, which will become obvious in the next section of the paper, is needed to maintain the assumption of constraint-wise uncertainty that has to be satisfied when the robust optimization approach is applied. As discussed in (Ben-Tal and Nemirovski, 1999), the relationship between USOP (3) and RSOP (4) relies on certain properties of these problems and the uncertainty set. The uncertainty set must be constraint-wise and the boundedness assumption must hold. Definition 4. (i) The uncertainty is said to be constraint-wise if the uncertainty set U is the direct product of the partial uncertainty sets U i, U = U 1 U 2 U k, where k = p or p 1. (ii) The boundedness condition holds if there exists a convex compact set in R n which contains the feasible sets of all instances of USOP (3). These definitions lead to the following results. Theorem 2 ((Ben-Tal and Nemirovski, 1999), Proposition 2.1). Let the uncertainty be constraint-wise and the boundedness assumption hold. (i) RSOP (4) is infeasible if and only if there exists an infeasible instance of USOP (3). (ii) If RSOP (4) is feasible and x is an optimal solution to RSOP (4), then x is an optimal solution to at least one instance of USOP (3). In view of problems (3) and (4) and the connections among them, we relate the feasible and optimal solutions to RSOP (4) to the feasible and efficient solutions to MOLP (1). Proposition 1. Let Assumption 1 hold, the uncertainty be constraint-wise and the boundedness assumption hold. (i) A solution x X is feasible to RSOP (4) if and only if it is feasible to MOLP (1). (ii) If x X is a feasible and optimal solution to RSOP (4) then it is a weakly efficient solution to MOLP (1). Proof. (i) A solution x X is feasible to RSOP (4) if and only if, by Theorem 2(i), x is feasible to USOP (3) for all u U. This is equivalent to x being feasible to SOP(u) (2) for all u U and, because of Assumption 1, feasible to MOLP (1). (ii) Let x X be feasible and optimal to RSOP (4). Then by Theorem 2(ii), x is optimal to USOP (3) for a particular realization ū U. Thus, x is optimal to SOP(ū) (2) and, by Proposition 1, weakly efficient to MOLP (1). In the next section we present four scalarization methods using Model 1 and two scalarization methods using Model 2. In all cases we assume a scalarizing parameter is uncertain and follow the scheme presented above, that is, we formulate an uncertain problem and derive a deteristic and computationally tractable robust counterpart. 5

6 3 Robust counterparts to scalarized MOLPs In the two subsections we study two types of scalarization models treating them as optimization problems under uncertainty. In the first model the uncertain parameter is in the objective function, while in the second model it is confined to the constraints. For each scalarization we first review the relationship between the optimal solution of the scalarized problem and the (weakly) efficient solutions to MOLP (1). For each case, we then derive a computationally tractable, deteristic robust counterpart which reveals a relationship between the optimal solutions to the uncertain problem and the optimal solutions to the counterpart. 3.1 Model 1 In all of the following SOPs, the uncertain scalarization parameter is the vector of weights, w R p. Since weights are typically nonnegative and normalized to sum to 1, we may wish to to use the uncertainty set w R p : e T w = 1, w 0}, where e R p is a vector of ones. However, the normalization violates the constraint-wise uncertainty required by Proposition 2, and we instead let U = w R p : 0 w 1}, (5) where U = U 1 U p and U i = w i R : 0 w i 1} for i = 1,, p. In some of the presented SOPs formulations the weights are required to be nonnegative while in the others they must be strictly positive in order to guarantee that optimal solutions to the SOPs are (weakly) efficient to MOLP (1). On the other hand, in all formulations of the USOPs it is assumed that the uncertain weights are elements of the set U given in (5) and so may have some components equal to zero. We discuss this situation later in this subsection. The case when w = 0 is considered trivial because it eliates the objective function of the SOP Methods using achievement functions We first consider achievement functions to scalarize MOLP (1). Given a real-valued achievement function s R : R p R, the scalarized problem is given by s R(Cx). (6) The achievement functions must be increasing to lead to efficient solutions. Definition 5. An achievement function s R : R p R is said to be (1) strictly increasing if for y 1, y 2 R p, y 1 < y 2 then s R (y 1 ) < s R (y 2 ); (2) strongly increasing if for y 1, y 2 R p, y 1 y 2 then s R (y 1 ) < s R (y 2 ). We consider the strictly increasing function s R (y) = max wk (y k yk R ) } k=1,,p 6

7 and the strongly increasing function s R (y) = max wk (y k yk R ) } + ρ k=1,,p w k (y k yk R ), where w R p > is a vector of positive weights, y R R p is a reference point and ρ 1 > 0 and sufficiently small. In the notation of (2) we have S u = X and u = w. Theorem 3. (Wierzbicki, 1986a,b) (i) Let an achievement function s R be strictly increasing. If ˆx X is an optimal solution to problem (6), then ˆx is weakly efficient to MOLP (1). (ii) Let an achievement function s R be strongly increasing. If ˆx X is an optimal solution to problem (6), then ˆx is efficient to MOLP (1). Note the difference in the specification of the weights, which has been mentioned earlier. In Theorem 3, w R p >, while in (5) and Proposition 2, w R p. If w Rp is used to construct the achievement functions, then the objective functions corresponding to the zero components of the weight vector are eliated and Theorem 3 is applied to a reduced MOLP. In this case, an optimal solution to problem (6) is efficient to the reduced MOLP. However, we can now utilize another result stating that a weakly efficient solution to an MOP with, say, m < p objective functions is weakly efficient to an MOP with p objective functions, to which p m objective functions have been added (Fliege, 2007; Engau and Wiecek, 2008). We can therefore conclude that using a nonnegative weight in Theorem 3 yields a weakly efficient solution to MOLP (1). We now formulate the uncertain achievement function problem (7 ) with the strongly increasing function and derive its computationally tractable, deteristic robust counterpart (8). Proposition 2. A solution x X is robust optimal to the uncertain achievement function problem ( wj (c j x yj R ) } ) } + ρ w i (c i x yi R ) (7) max j=1,,p if and only if it is an optimal solution to the -max problem k=1 w U max 0, j=1,,p cj x yj R }. (8) Proof. Using an auxilliary variable, problem (11) can be rewritten equivalently as max wj (c j x yj R ) } + ρ w i (c i x yi R ), j=1,,p x X 7 w U.

8 Since the inequality constraint must hold for the maximum value, it must hold for all j = 1,, p, meaning we write the robust counterpart as w j (c j x y R j ) + ρ w i (c i x yi R ) In the worst case scenario, this problem becomes ( max w j (c j x yj R ) + ρ w w i 1 i = 1,, p w i 0 i = 1,, p w j U j, w i U i, j = 1,, p, i = 1,, p ) w i (c i x yi R ) j = 1,, p Now, making use of linear programg duality theory, we take the dual of the inner maximization problem with dual variables v ij and obtain v ij x X, v ij v ij ρ(c i x y R i ) + 1 (j=i) (c j x y R j ) i = 1,, p v ij 0 i = 1,, p where 1 is the indicator function, meaning 1 (j=i) (c j x yj R c i x yi R if j = i ) = 0 otherwise. j = 1,, p Since the optimal value of each of the p subproblems must be less than or equal to, we imize the maximum of their objective function values max j=1,,p v ij v ij v ij ρ(c i x y R i ) + 1 (j=i) (c j x y R j ) i = 1,, p, j = 1,, p v ij 0 i = 1,, p, j = 1,, p 8

9 and combine the constraints to obtain max j=1,,p v ij v ij v ij max0, ρ(c i x y R i ) + 1 (j=i) (c j x y R j )}, i = 1,, p, j = 1,, p. We can now eliate the v ij variables by moving the p 2 inequality constraints to the objective function. Thus, the above problem reduces to } max max0, ρ(c i x yi R ) + 1 (j=i) (c j x yj R )}. j=1,,p As we maximize over j = 1,, p, the sum only differs by c j x yj R function. Therefore, this problem is equivalent to due to the indicator as desired. max 0, j=1,,p cj x yj R } Since the strictly increasing function is a special case of the strongly increasing function, we immediately obtain the following analogous result. Proposition 3. A solution x X is robust optimal to the uncertain achievement function problem ( max wi (c i x yi R ) }) } (9),,p w U if and only if it is an optimal solution to the -max problem max 0, c i x yi R,,p }. (10) We proceed in a similar fashion with other scalarization methods Conic method We exae the conic scalarization problem (Kasimbeyli, 2013). Given a weight parameter w R p > and α such that 0 α w i, i = 1,..., p}, the problem is given by wt (Cx y R ) + α Cx y R 1 (11) where 1 denotes the l 1 norm and y R R p is a reference point. Theorem 4. (Kasimbeyli, 2013) If ˆx X is an optimal solution to problem (11) then ˆx is efficient to MOLP (1). 9

10 In the notation of (2) we have S u = X and u = w. Similar to the achievement function method, the conic scalarization requires the weights to be strictly positive. If a weight is nonnegative and has some zero components then problem (11) is modified by dropping the corresponding components in both terms of the objective function and, by the same arguments as those given for the scalarization with achievement functions, an optimal solution to the reduced SOP (11) is weakly efficient to MOLP (1). Choosing α to be uncertain will always result in α = w i, i = 1,, p} since that is the largest and therefore the most conservative value in its range, so we omit that case. Proposition 4. A solution x X is robust optimal to the uncertain conic problem } wt (Cx y R ) + α (c i x yi R ) w U (12) if and only if it is an optimal solution to the -max problem ( ) p max,,p ci x yi R } + α (c i x yi R ). (13) Proof. We can write (12) equivalently as ( wi c i x + αc i x 2yi R since c i x > 0. The robust counterpart of (14) is ) } w U (14) ( wi c i x + αc i x 2yi R In the worst case this reduces to ) wi U i (15) max w This can be written equivalently as ( wi c i x + αc i x 2yi R w i 1 i = 1,, p w i 0 i = 1,, p ) (16) 10

11 (αc i x 2yi R ) + max w i c i x w w i 1 i = 1,, p w i 0 i = 1,, p Taking the dual of the above problem gives (αc i x 2yi R ) + v i v v i c i x i = 1,, p v i 0 i = 1,, p Now we can remove the inner imization to obtain (17) (18) which simplifies to x,v (v i + αc i x 2yi R ) v i c i x i = 1,, p x X (19) or ( ( max,,p ci x} + αc i x 2y R i ) (20) p max,,p ci x y R i } + α ) (c i x yi R ). (21) Weighted-norm method We now attempt to investigate the traditional weighted-norm formulation given a parameter w R p, w T (Cx r) p (22) 11

12 where p indicates the l p norm with p = 1 or, and r R p is a utopia point defined as r i = ci x ɛ i, i = 1,, p for some ɛ i 0, i = 1,, p. In the notation of (2), S u = X and u = w. Theorem 5. (i) (Ehrgott, 2005) Let w R p. If ˆx X is a unique optimal solution to the l 1 -norm problem (22) then ˆx is efficient to MOLP (1). (ii) (Choo and Atkins, 1983) Let w R p >. If ˆx X is an optimal solution to the l -norm problem (22) then ˆx is weakly efficient to MOLP (1). Proposition 5. A solution x X is robust optimal to the uncertain weighted l 1 -norm problem } w i (c i x r i ) (23) if and only if it is an optimal solution to the -max problem ( ) p max,,p ci x r i }. (24) Proof. The robust counterpart of problem (23) w U ( ) w i c i x r i wi U i This, under the worst-case scenario, reduces to max w w i (c i x r i ) w i 1 i = 1,, p w i 0 i = 1,, p Applying duality results for SOPs on the inner maximization yields v x X v i v i c i x r i i = 1,, p v i 0 i = 1,, p 12 (25)

13 where v R p is the dual variable. Problem (25) is equivalent to the following,v v i v i c i x r i i = 1,, p v i 0 i = 1,, p Removing the unnecessary variable and recognizing that c i x r i 0 yields x,v v i v i c i x r i i = 1,, p x X or equivalently problem (24). Proposition 6. A solution x X is robust optimal to the uncertain weighted l -norm problem max wi (c i x r i ) } } (26),,p w U if and only if it is an optimal solution to the -max problem max } c i x r i. (27),,p Proof. Using an auxilliary variable, problem (26) can assume the following equivalent form ( )} max wi c i x r i,,,p x X w U. This family of MOPs has the robust counterpart of the form w i ( c i x r i ) wi U i, i = 1,, p 13

14 This, under the worst-case scenario, reduces to ( ) max w i c i x r i w i w i 1 w i 0 i = 1,, p Applying duality results to the inner maximization yields v i x X v i v i c i x r i i = 1,, p v i 0 (28) where v R p is the dual variable. Problem (28) is equivalent to the following,v v i i = 1,, p v i c i x r i, i = 1,, p v i 0, i = 1,, p Removing the unnecessary variable and recognizing that c i x r i 0 yields x,v max v i},,p v i c i x r i, i = 1,, p x X, or equivalently problem (27) Weighted-sum method Given a parameter w R p, the weighted-sum problem is, Again, in the notation of (2), S u = X and u = w. wt Cx. (29) 14

15 Theorem 6. (Geoffrion, 1968) If ˆx X is a unique optimal solution to the weighted-sum problem (29) then ˆx is efficient to MOLP (1). Proposition 7. A solution x X is robust optimal to the uncertain weighted-sum problem wt Cx } w U (30) if and only if it is an optimal solution to the -max problem p max,,p ci x}. (31) Proof. The proof is omitted becaue the weighted-sum method is a special case of the weighted l 1 norm method with r = Model 2 We now present the scalarizations that use parameters u R p 1 and have additional constraints originating from p 1 objective functions. The first one makes use of weights while the other employs right-hand side coefficients Weighted-constraint method The weighted-constraint scalarization problem is given by w jc j x w i c i x w j c j x i = 1,, p, i j. In this approach, weights w R p > are assigned to each of the p objective functions, and one weighted objective function is selected to be imized. Then p 1 constraints are added to the problem requiring the remaining weighted objective functions to be less than or equal to the weighted objective function being imized. (32) Theorem 7. (Burachik et al., 2014) A feasible solution ˆx X is a weakly efficient solution to MOLP (1) if and only if there exists w R p > such that ˆx is an optimal solution to problem (32) for all j = 1,, p. Here, the weights w i, i = 1,, p, i j are uncertain, where w j is the weight associated with the objective function being imized. We do not allow w j to be uncertain because the constraint-wise uncertainty would be violated. We therefore define U = w j R p 1 > : w j = (w 1,, w j 1, w j+1,, w p )}, (33) where U = U 1 U j 1 U j+1 U p and U i = R > for i = 1,, p, i j. In the notation of (2) we have S u = x X : w i c i x w j c j x for i = 1,, p, i j}. 15

16 Proposition 8. A solution x X is robust optimal to the uncertain weighted-constraint problem w jc j } x (34) w i c i x w j c j x i = 1,, p, i j if and only if it is an optimal solution to the -max problem w i U i,i j,,,p max j=1,,p, i j ci x}. (35) Proof. By the construction of the scalarization, the uncertainty is already restricted to the constraints, so no reorganization of the problem is needed. We therefore begin with w jc j } x (36) w i c i x w j c j x i = 1,, p, i j and note that in the worst case scenario this can be written as w i U i,i j w jc j x max w i w i c i x w i 1 w jc j x i = 1,, p, i j. w i 0 Applying duality results to the inner maximization problem yields (37) w jc j x v i v i v i c i x w jc j x i = 1,, p, i j v i 0 where v i, i = 1,, p, i j is the dual variable. Now we remove the inner imization to obtain,v w j c j x v i w j c j x i = 1,, p, i j v i c i x v i 0 i = 1,, p. i = 1,, p, i j The inequalities v i w j c j x and v i c i x imply that c i x w j c j x so the dual variables can be removed and the problem can be written equivalently as (38) (39) w jc j x c i x w j c j x i = 1,, p, i j (40) 16

17 which can be rewritten as the desired -max problem max,,p, i j ci x}. (41) ε-constraint method Consider the ε-constraint method in which only one objective is imized, while the others are converted into new constraints. For each j = 1,, p, the ε-constraint method generates an SOP as follows: cj x c i x ε i i = 1,, p, i j, where parameter ε R p 1 is chosen so problem (42) is feasible. Theorem 8. (Chankong and Haimes, 1983) If ˆx is a unique optimal solution to (42) for some j 1,..., p} and some ε such that (42) is feasible, then ˆx is efficient to MOLP (1). We assume that the epsilon is uncertain, that is, we define (42) U = ε R p 1 : ε L ε ε U }, (43) where U = U 1 U j 1 U j+1 U p and U i = ε i R : ε L i ε i ε U i } for i = 1,, p, i j. In the notation of (2), S u = x X : c i x ε i i = 1,, p, i j}. Proposition 9. A solution x X is robust optimal to the uncertain ε-constraint problem } cj x (44) c i x ε i i = 1,, p, i j ε i U i,i j,,,p if and only if it is an optimal solution to a related ε-constraint problem of the form cj x c i x ε L i i = 1,, p, i j. Proof. The family of MOPs (42) has the robust counterpart of the form cj x which under the worst-case scenario reduces to cj x max ε i ε i c i x ε i U i, i = 1,, p, i j ε i ε i ε U i ε i ε L i ci x i = 1,, p, i j. ε 0 17 (45)

18 We observe that the solution to the inner maximization problem is ε i = ε L i. Thus, this reduces to which is equivalent to cj x ε L i c i x i = 1,, p, i j as desired. cj x c i x ε L i i = 1,, p, i j (46) In the next section we summarize the obtained results. 4 Discussion Based on Propositions 2-9 given in Section 3, we make two general observations that apply to all presented scalarizations of MOLP (1). First, a solution x X is robust optimal to the uncertain scalarized MOLPs (of the form USOP (3)) if and only if it is optimal to the corresponding deteristic robust counterparts that are derived in these propositions. Second, all these robust counterpart problems assume a certain form of the scalarized problem SOP(ū) where ū is a particular realization of the uncertain parameter u. We therefore recognize this realization in the following definition. Definition 6. The uncertainty realization ū of the uncertainty parameter u that yields the robust optimal solution to USOP (3) is called the robust realization. For the four scalarization methods in Subsection 3.1, u R p and all deteristic robust counterparts assume a similar final form. In particular, for y R = r = 0 and α = 0, all counterparts reduce to max c i x }, (47),,p which is the well-known -max problem. Recall that an optimal solution to the -max problem associated with MOLP (1) is weakly efficient to the MOLP (Ehrgott, 2005). The corresponding common robust realization of the uncertain parameter ū = [0,, 0, 1, 0,, 0] T has all components equal to 0 with 1 in the component corresponding to the largest objective value c i x among all objectives i = 1,, p for all feasible solutions of MOLP (1). For the two scalarization methods in Subsection 3.2, u R p 1. The deteristic robust counterpart for the uncertain weighted-constraint problem is a -max problem on p 1 objective functions, and, based on the discussion in Subsection 3.1.1, its optimal solution is weakly efficient to MOLP (1). The accompanying robust realization of the uncertain parameter is analogous to that provided by the first four methods. The deteristic robust counterpart for the uncertain ε-constraint problem assumes the form of the ε-constraint 18

19 problem with the robust realization of the uncertain parameter equal to the lower bound on the uncertainty set, ū = ε L. As a result, all robust counterparts we have presented assume a scalarized form of MOLP (1), SOP(ū), that is associated with ū, the robust realization of the uncertain parameter. They all act in accordance with Proposition 1, that is, their optimal solutions are weakly efficient to MOLP (1). While this proposition was known in Section 2 before the scalarizations were studied in detail in Section 3, our work in Section 3 revealed the computationally tractable robust counterparts and the actual values of the robust realizations ū. An optimal solution to SOP(ū) with the robust realization of uncertainty deserves its own notion. Definition 7. An optimal solution to SOP(ū) obtained for the robust realization of uncertainty is called a robust weakly efficient solution to MOLP (1). This discussion is closed with the following corollary. Corollary 1. An optimal solution to the -max problem (47) associated with MOLP (1) is a robust weakly efficient solution to MOLP (1). 5 Conclusion Using the framework of single-objective robust optimization of (Ben-Tal et al., 2009), we have studied six scalarizations of the MOLP making the scalarizing parameters uncertain. For the first five scalarizations, an optimal solution to the -max problem associated with the MOLP emerges as a robust weakly efficient solution to the MOLP. This leads to two conclusions. First, from the robustness point of view, these scalarizations are equivalent to each other and their choice should be justified by considerations such as numerical convenience rather than decision making implications. Second, the -max optimal solution to the MOLP exhibits unusual strength and significance for decision making in the presence of multiple criteria. Recall that the -max optimal solution is also an equitable solution to the MOLP (Singh, 2007) which makes this solution even more special. There is a clear direction on how to continue the work presented in this paper. Uncertain scalarized MOLPs could be exaed in the context of other notions of robustness. Additionally, they can also be studied more generally, making use of other approaches to uncertainty. All such studies should provide more insight into decision making with multiple criteria. References A. Ben-Tal and A. Nemirovski. Robust solutions of uncertain linear programs. Operations Research Letters, 25(1):1 13, A. Ben-Tal and A. Nemirovski. Robust solutions of linear programg problems contaated with uncertain data. Mathematical Programg, 88(3): ,

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