Pessimistic Referential-Uncooperative Linear Bilevel Multi-follower Decision Making with An Application to Water Resources Optimal Allocation

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1 Pessimistic Referential-Uncooperative Linear Bilevel Multi-follower Decision Making with An Application to Water Resources Optimal Allocation Yue Zheng, Yuxin Fan, Xiangzhi Zhuo, and Jiawei Chen Nov. 27, 2016 Abstract: In this paper, we study a pessimistic linear bilevel multi-follower programg problem in a referential-uncooperative situation (for short, PLBMF-RU) where the followers are uncooperative while having cross reference to decision making information of the other followers. We first give a model and the solution definitions of PLBMF-RU, and then use a simple example to illustrate it. Furthermore, we develop an algorithm based on the penalty function method with finite teration to solve PLBMF-RU. Finally, a case study of water resources optimal allocation illustrates the feasibility of the proposed algorithm. Key words: Bilevel multi-follower programg; Pessimistic formulation; Multiple followers; Water resources optimal allocation MR(2000): 90C26; 90C30 1 Introduction Bilevel programg plays an exceedingly important role in different application fields, such as transportation, economics, ecology, engineering and others [11]. It has been developed and researched by many authors, e.g., see the recent monographs [5, 10, 22]. When the set of solutions of the lower level problem is not a singleton, it is difficult for the leader to optimize his choice unless he knows the follower s reaction to his choice. In this situation, at least two approaches have been suggested: optimistic (or strong) formulation and pessimistic (or weak) formulation [17]. The pessimistic bilevel programg problem is very difficult to solve. As a result, most research on bilevel programg relates to the optimistic formulation. Interested readers can refer to the surveys [8, 11, 20, 23, 24, 27] and the references therein. However, the pessimistic formulation is very popular in practice. There are at least three reasons for this. First, for the actual bilevel programg problems, a non-cooperative behavior of the followers can frequently be observed in applications. Second, in fact, the leader in general has incomplete information about the followers, i.e., he may not know the values of all parameters in the lower level problem. The leader would like to create a safety margin to bound the damage resulting from undesirable selections of the followers. Finally, if the leader can obtain a pessimistic solution, this result will provide him with a policy-making reference that prompts him to make a rational decision to effectively control risk. Note that, several studies has been contributed to pessimistic (weak) bilevel programg problems from different subjects: existing results of solutions[1, 2, 3, 13, 15, 16, 30], optimality conditions [9, 12], approximation algorithm [25, 7], penalty method [28] and so on. The above stated pessimistic bilevel programg problems are limited to a specific situation with oneleader-and-one-follower. However, for the actual bilevel programg problems, the lower level problem often Corresponding author. School of Managenment, Huaibei Normal University, Huaibei , China. zhengyuestone@126.com Huazhong University of Science and Technology, Wuhan, , China School of Managenment, Huaibei Normal University, Huaibei , China. School of Mathematics and Statistics, Southwest University, Chongqing China. J.W.Chen713@163.com 1

2 involves multiple decision makers. Moreover, the most basic situation is that these followers are uncooperative and they do cross reference information by considering other followers decision results in each of their own decision objectives and constraints. It is worthwhile noting that, Lu et al. [18] call this case as a referentialuncooperative situation, and this paper will study the pessimistic linear bilevel multi-follower programg problem on this situation. Note that several papers have been devoted to bilevel multi-follower decision making problems dealing with different situations, we cite, for example, Refs. [19, 29]. The reader is also referred to the book on bilevel multi-follower decision making [26]. The remainder of the paper is organized as follows. The pessimistic linear bilevel multi-follower programg problem is presented in a referential-uncooperative situation (PLBMF-RU) in Section 2. Under some assumptions, we then propose a penalized problem of PLBMF-RU, and develop an algorithm based on an exact penalty method with finite teration in Section 3. A case study is given to illustrate the PLBMF-RU model and the feasibility of the proposed algorithm in Section 4. Finally, we give the conclusions. 2 Pessimistic referential-uncooperative linear bilevel multi-follower decision making model Consider the following pessimistic linear bilevel programg problem where k followers are involved in a referential-uncooperative situation: [c T x+ x,y 1,,y k sup y i Ψ i(x,y i) d T i y i] s.t. x X, (2.1) y i Ψ i (x,y i ),i = 1,2,,k, where y i = (y 1,,y i 1,y i+1,,y k ) and Ψ i (x,y i ) is the set of solutions of the ith follower s problem y i 0 ut i y i s.t. A i x+ B ij y j b i. (2.2) Here, x,c R n, y i,d i,u i R mi, A i R qi n, B ij R qi mj, b i R qi, i = 1,2,,k, and X is a closed subset of R n. Problem (2.1) can be equivalently transformed into the following problem P: [c T x+ x,y 1,,y k where y i is a solution of the following problem P(x,y i ), j=1 d i y i ] s.t. x X, (2.3) y i Ψ i (x,y i ),i = 1,2,,k, y dt i y i. (2.4) i Ψ i(x,y i) 2

3 In order to assure that the problem is well posed, it will now be assumed that a solution to problem P, and hence, problem (2.1) exists. Next, we give the following definitions. Definition 2.1. (a) Constraint region of problem (2.1): S = {(x,y 1,,y k ) : x X,A i x+ (b) Projection of S onto the leader s decision space: (c) Feasible set for the ith follower: (d) The ith follower s rational reaction set: B ij y j b i,y i 0,i = 1,2,,k}. j=1 S(X) = {x X : (x,y 1,,y k ) S}. S i (x,y i ) = {y i : A i x+ (e) Inducible region or feasible region of the leader: B ij y j b i,y i 0}. j=1 Ψ i (x,x,y i ) = {y i : y i Arg[u T i y i : y i S i (x,y i )]}. IR = {(x,y 1,,y k ) : (x,y 1,,y k ) S,y i Ψ i (x,y i ),i = 1,2,,k}. Note that, (x,y1,,yk ) IR means that, i = 1,2,,k, u T i y i u T i y i for any y i S i (x,y i ) where y i = (y 1,,y i 1,y i+1,,y k ). Then, a popular solution concept is the socalled Nash equilibrium defined as (y 1,,y k ) with respect to x. The interested reader can refer to Refs. [4, 14]. Definition 2.2. A point (x,y 1,,y k ) IR is called a pessimistic solution to (2.1), if (x,y 1,,y k ) IR, where f i (x,y i ) = sup y i Ψ i(x,y i) f i (x,y i ) = c T x + d T i y i. sup d T y i Ψ i(x,y i ) i y i, i = 1,2,,k, f i (x,y i) c T x+ f i (x,y i ), To illustrate these concepts, assume that k = 2 followers exist, and that for each i = 1,2, the follower i has control over the vector y i = (y 1 i,y2 i ) R2. Consider the following example which is a pessimistic linear bilevel programg problem where two followers are involved in a referential-uncooperative situation. Example x+ x,y 1,y 2 s.t. 0 x 1, sup y 1 Ψ 1(x,y 1), y 2 Ψ 2(x,y 2) y i Ψ i (x,y i ),i = 1,2 (y y2 1 +y1 2 +2y2 2 ) 3

4 where Ψ 1 (x,y 1 ) and Ψ 2 (x,y 2 ) are the sets of solutions of the followers problem respectively, Here, the constraint region is given by y 1 0 y1 1 y2 1 s.t. y 1 1 +y2 1 x, y 1 1 +y 2 1 x+y 1 2 +y , y 2 0 y1 2 y2 2 s.t. y 1 2 +y2 2 x, y 1 2 +y 2 2 x+y 1 1 +y S = {(x,y 1,y 2 ) : 0 x 1,y 1 1 +y2 1 x,y1 1 +y2 1 x+y1 2 +y ,y1 2 +y2 2 x, y 1 2 +y 2 2 x+y 1 1 +y ,y 1,y 2 0}. The vector (0.6,0,0,0,0) S, but it is not a feasible point since, given x = 0.6 and y 2 = (0,0), y 1 = (0.2,0) (rather than y 1 = (0,0)) imizes the first follower s problem. The vector (0.5,0.1,0,0,0) is feasible, since (0.1,0,0,0) is a Nash equilibrium point with respect to x = 0.5. It is not difficult to show that the inducible region IR for this example is given by IR = {(x,y 1,y 2 ) : y1 1 +y2 1 = x, y2 1 +y ; x+y2 1 +y , y2 1 +y2 2 < 0.4; y2 1 +y2 2 = x y1 1 +y ; x+y1 1 +y y1 1 +y1 2 < 0.6; (x,y 1,y 2 ) S}. Given IR, it is easy to check that an extreme point (0.5,0,0.1,0,0) of S is a solution for this example and that the pessimistic value is Penalty problems and algorithm with finite teration In this section, a penalized problem of PLBMF-RU is proposed under some suitable assumptions, and then, we develop an algorithm based on an exact penalty method, which has finite teration property. In order to establish theoretical results, we need the following blanket assumptions throughout the paper. Assumption A: (A1) For any x X, S i (x,y i ), {(y 1,y 2,,y k ) : y i Ψ i (x),i = 1,2,,k}, and there exists compact subsets W i such that S i (x,y i ) W i for all x X, i = 1,2,,k. (A2) The set X is a bounded non-empty polyhedron. Then we have the following lemma which provides the existence of solutions to problem (2.3). 4

5 Lemma 3.1. If assumptions (A1) and (A2) are satisfied, then problem (2.3) has at least one solution. Proof. Under assumptions (A1) and (A2), for each i = 1,2,,k, it follows from [10, Theorem 4.3] that f i (x,y i ) is continuous. Using the results of [21, Lemma 1], we have IR is a non-empty compact set. Therefore, the proof follows from the Weierstrass s Theorem. The dual problem of problem (2.2) can be written as: (b i A i x z i 0 s.t. B T ii z i u i. B ij y j ) T z i Let Z i := {z i : Bii Tz i u i,z i 0}, and denote the ith follower s duality gap by π i (x,y 1,,y k,z i ) := u T i y i +(b i A i x k B ij y j ) T z i. It follows from the dual theory that problem P(x,y i ) is written as follows: y i,z i d T i y i s.t. π i (x,y 1,,y k,z i ) = 0, y i S i (x,y i ), z i Z i. For ρ i > 0(i = 1,2,,k), we consider the following penalized problem P ρi (x,y i ): [d T y i y i ρ i π i (x,y 1,,y k,z i )] i,z i s.t. y i S i (x,y i ), z i Z i. Let V( ) represent the set of vertices of the set to be concerned. Lemma 3.2. Under assumption (A1), for ρ i > 0, problem P ρi (x,y i ) has at least one solution in V(S i (x,y i ) V(Z i ). Proof. For ρ i > 0, we have T sup [di y i ρ i π i (x,y 1,,y k,z i )] y i S i(x,y i ) z i Z i sup y i S i(x,y i ) d T i y i = y dt i y i. i S i(x,y i ) Then, it follows from (A1) that the objective function of the linear programgproblem P ρi (x,y i ) is bounded from above. Hence, the latter problem has at least one solution in V(S i (x,y i ) V(Z i ). The dual of problem P ρi (x,y i ) is written as: [(b i A i x t i,v i 0 s.t. B T iit i d i +ρ i u i, B ii v i ρ i (b i A i x B ij y j ) T t i +u T i v i] B ij y j ). 5

6 Denote the optimal value of problem P ρi (x,y i ) by f ρi (x,y i ). For ρ k+i > 0(i = 1,2,,k), we now consider the following problem ˆP: and its penalized problem P: For simplicity, let x,y {ct x+ i,z i,t i,v i,,2,,k [(b i A i x s.t. B T ii t i d i +ρ i u i, B ii v i ρ i (b i A i x x,y {ct x+ i,z i,t i,v i,,2,,k + (x,y 1,,y k ) S, z i Z i, π i (x,y 1,,y k,z i ) = 0, t i,v i 0,i = 1,2,,k, [(b i A i x ρ k+i π i (x,y 1,,y k,z i )} s.t. B T ii t i d i +ρ i u i, B ii v i ρ i (b i A i x (x,y 1,,y k ) S, z i Z i, t i,v i 0,i = 1,2,,k. B ij y j ), B ij y j ), B ij y j ) T t i +u T i v i] B ij y j ) T t i +u T i v i ] Z k+1 (ρ 1,,ρ k ) := {(x,y 1,,y k,v 1,,v k ) : (x,y 1,,y k ) S, B ii v i ρ i (b i A i x B ij y j ),v i 0,i = 1,2,,k}, Z k+2 (ρ 1,,ρ k ) := {(z 1,,z k,t 1,,t k ) : z i Z i, B T ii t i d i +ρ i u i, t i 0,i = 1,,k}. and denote the objective function of problem P by F ρ3,ρ 4 (x,y 1,,y k,v 1,,v k,z 1,,z k,t 1,,t k ). Then we have the following lemma which provides the existence of solutions to problem P. Theorem 3.1. Under Assumption A, for fixed values of ρ i > 0(i = 1,2,,2k), problem P has at least one solution in V(Z k+1 (ρ 1,,ρ k )) V(Z k+2 (ρ 1,,ρ k )). Proof. Note that problem P is a disjoint bilinear programg problem for fixed values of ρ i > 0(i = 1,2,,2k). Using the result of [1, Theorem 3.2], we have that problem P has at least one solution in V(Z k+1 (ρ 1,,ρ k )) V(Z k+2 (ρ 1,,ρ k )). For each η > 0, let W η = {β Bβ η(b Aα)} and W = {ξ Bξ b Aα}. 6

7 Lemma 3.3. If (α η,β η) V(W η ), then there exists (α,ξ ) V(W) such that α η = α. Proof. By the definition of W, we have (α η, β η η ) W. Let (α1,ξ 1 ),,(α r,ξ r ) be the distinct vertices of W. Since any point in W can be written as a convex combination of these vertices, let (α η, β η η ) = ˆr κ i (x i,y i ), where ˆr κ i = 1,κ i > 0,i = 1,,ˆr, and ˆr r. Then, it follows that (α η,β η) = ˆr κ i (α i,ηξ i ). Note that (α i,ηξ i ) Z k. Hence, the last equality implies that ˆr = 1. Because (α η,β η ) is a vertex of W η, a contradiction results unless ˆr = 1. Thus, there exists (α,ξ ) V(W) such that α = α η and β η = ηξ. Let Z k+3 := {(x,y 1,,y k,v 1,,v k ) : (x,y 1,,y k ) S,v i 0, B ii v i b i A i x B ij y j, i = 1,2,,k}. The following result is true by Theorem 3.1 and Lemma 3.3. Corollary 3.1. Under Assumption A, for fixed values of ρ i > 0(i = 1,2,,2k), problem P has at least one solution in V(Z k+3 V(Z k+2 (ρ 1,,ρ k )). We now characterize the relations between problems P and ˆP. Theorem 3.2. Let Assumption A, ρ i > 0(i = 1,2,,2k) and (x,y 1,,y k,z 1,,z k,t 1,,t k,v 1,,v k ) depending on ρ i be a solution of problem P. Then there exist finite values ρ k+i > 0(i = 1,2,,k) such that (x,y 1,,y k ) is a solution of problem ˆP for all ρ k+i ρ k+i. Proof. The result follows immediately from [6, Theorem 3]. For the given penalty parameters ρ i > 0(i = 1,2,,2k), if a solution to problem P is a feasible point to problem ˆP, then it is also a solution of problem ˆP. Furthermore, if a solution to problem ˆP is a feasible point to problem P, then it will be a solution of problem P. In this case, we try to force feasibility by increasing the penalty parameters, and then present an algorithm based on penalty method as follows. Algorithm 1 Step 0. Let γ > 1, ρ i > 0(i = 1,2,,2k) and set m = 0. Step 1. Solve problem P, and denote the solution by (x,y1,,yk,z 1,,zk,t 1,,t k,v1,,vk). Step 2. If set π i (x,y 1,,y k,z i ) 0, then ρ k+i = γρ k+i for all i = 1,2,,k, and go to Step 1. Otherwise, (x m,y m 1,,ym k,zm 1,,zm k,tm 1,,tm k,vm 1,,vm k ) = (x,y 1,,y k,z 1,,z k,t 1,,t k,v 1,,v k), 7

8 m = m+1, and go to Step 3. Step 3. For all j = 1,2,,k, let H m := {yj m yj m is not a solution to P(x m,yj )} m where yj m = (ym 1,,ym j 1,ym j+1,,ym k ). Step 4. If H m =, then stop, and (x m,y1 m,,yk m) is a solution of problem P. Otherwise, set ρ i = γρ i for all i = 1,2,,k, and go to Step 1. We next shows that Algorithm 1 terates within a finite number of iterations. Theorem 3.3. Assume that {(x m,y1 m,,yk m,zm 1,, zk m,tm 1,,t m k,vm 1,,vk m )} is generated by Algorithm 1. Then, under Assumption A, there exist finite values ρ i (i = 1,2,,2k) such that for all ρ i > ρ i, (x m,y1 m,,yk m ) is a solution of problem P. Proof. Note that, problem ˆP is equivalent to the following problem: [c T x+ x,y 1,,y k where (y i,z i ) is the solution of the following problem P (x,y i ) d i y i ] s.t. (x,y 1,y 2,,y k ) IR, (3.1) [d T y i y i ρ i π i (x,y 1,,y k,z i )] i,z i s.t. y i S i (x,y i ), z i Z i. For each ρ i > 0 (i = 1,2,,k), assume that (x,y1,,yk,z 1,,z k,t 1,,t k,v 1,,vk ) is a solution of problem ˆP. Then, (x,y 1,y 2,,y k ) is asolution ofproblem (5). Moreover,assumethat (y i,z i ) solvesproblem P (x,y i ) (i = 1,2,,k). For i = 1,2,,k, let ŷi solves problem P(x,y i ), and define the following sets D i := {(y i,z i ) V(S i (x,y i ) Z i) : π i (x,y 1,,y i 1,y i,y i+1,,y k,z i) > 0}. Now, we consider the following two cases: Case I: D i =. It follows that any solution to problem P (x,y i ) also solves problem P(x,y i ). Case II: D i. (a) If d T i y i d T i ŷ i for any (y i,z i ) D i, then it follows immediately that π i (x,y 1,,y i 1,y i,y i+1,,y k,z i) = 0, and thus, any solution to problem P (x,y i ) also solves problem P(x,y i ). (b) If (y i,z i) D i d T i y i > d T i ŷ i, then let ρ i be selected such that ρ i > ρ i = (y i,z i) D i d T i y i d T i ŷ i π i (x,y 1,,y i 1,y i,y i+1,,y k,z i). 8

9 Hence, we deduce that d T i y i ρ i π i (x,y 1,,y i 1,y i,y i+1,,y k,z i) < d T i y i ρ i π i(x,y 1,,y i 1,y i,y i+1,,y k,z i ) d T i ŷ i d T i y i ρ i π i (x,y 1,,y i 1,y i,y i+1,,y k,z i), whichimplies that, anysolutiontoproblemp (x,y i )alsosolvesproblemp(x,y i )wheneverρ i > ρ i. Inview ofthe casesiorii(a), set ρ i = 0. Now, let ρ i = (ρ i,ρ i ), andwhen ρ i > ρ i, it followsthat (x,y1,y 2,,y k ) is a feasible point of problem P, and then it is also a solution to problem P. From Theorem 3.2, we know that there exists finite values ρ k+i, for all ρ k+i ρ k+i (i = 1,2,,k), such that if (x m,y1 m,,ym k,zm 1,,zm k,tm 1,,tm k,vm 1,,vm k ) is a solution ofproblem P, then it is alsoasolution of problem ˆP. Consequently, (x m,y1 m,,ym k ) is a solution of problem P for all ρ i > ρ i (i = 1,2,,2k). 4 PLBMF-RU for water resources optimal allocation In this section, a water resources optimal allocation problem is given to show how to construct a pessimistic referential-uncooperative linear bilevel multi-follower decision making model and to illustrate the feasibility of Algorithm 1. It is well-known that fresh water is very crucial and precious resources of our living and production. With the accelerating promotion of economic globalization, one of the biggest problems is that fresh water is in increasingly high demand and is in a growingly short supply. The conflict between increasing demand and the relative shortage of fresh water becomes current difficulties faced by China and the world. Particularly, in China, the limited available water is unable to meet the growing demand for such water. The contradiction between supply and demand of water resources has become an important factor restricting China s boog economy. Therefore, the higher requirements to the water resources optimal allocation are put forward. As the water resources are public welfare. In addition to meeting the water demand of individual user, the water management department also need to allocate a certain public water rights to ensure the public interest (Ecological protection, environmental purification and other social welfare). Without loss of generality, suppose that the total amount of water allocated to the water management department is denoted by Q. The public water right is w, and the water resources rate is κ. The initial water rights for the ith water user is q i (i = 1,2,,n). Clearly, the total amount of public water rights and all water users is not more than Q. That is to say, w+ n q i Q. Furthermore, suppose that the revenue function of ith water user is f i (q i ) where f i (0) = 0 and f i (q i) > 0, and the ith user has a imum water demand δ i respectively. Because an individual water user is rational, the purpose of user is to imize his own interests. Then, imizing the ith water user s revenue function 9

10 is deemed to be the ith follower problem of pessimistic bilevel problem, i.e., g i(q i ) = f i (q i ) κq i q i 0 n s.t. w+ q i Q, q i δ i. On the other hand, the total social benefits contains the gains from the public water consumption and the interest from all water users. Note that, the water management department regulates the water market by distributing the initial water rights, and then imize the total social benefits. The mathematical formulation of the upper level model is given by where h(0) = 0 and h (w) > 0. h(w)+ n w 0 g i (q i ) Stated thus, based on problems (2.1) and (2.2), we can construct a pessimistic referential-uncooperative bilevel multi-follower water resources optimal allocation as follows: [h(w)+ w,q 1,,q n where q i is a solution of the ith follower s problem, n q i g i (q i )] s.t. w 0, (4.1) g i(q i ) = f i (q i ) κq i q i 0 n s.t. w+ q i Q, q i δ i. InordertoeasilyshowtheapplicationfortheproposedPLBMF-RUmodel,wecanassignavaluetoavariable respectively. The experimental data employed for the model (4.1) is provided as follows: n = 2, h(w) = 1.5w, κ = 0.2, Q = 1, f 1 (q 1 ) = 0.6q 1, f 2 (q 2 ) = 0.4q 2, δ 1 = 0.25, and δ 2 = Therefore, a water resources optimal allocation problem is established by simplifying it into the following linear PLBMF-RU decision making model: [1.5w+ w,q 1,q 2 s.t. w 0, 2 q i g i (q i )] q 1 0 g 1(q 1 ) = 0.4q 1 s.t. w+q 1 +q 2 1, (4.2) q , q 2 0 g 2(q 2 ) = 0.2q 2 s.t. w+q 1 +q 2 1, q

11 Table 1: Computational results of problem (4.3) via Algorithm 1 ρ 1 ρ 2 ρ 3 ρ 4 x y 1 y 2 f π 1 π To simply use Algorithm 1 for solving problem (4.2), we transform it into the following problem: [ 1.5x+ x,y 1,y 2 s.t. x 0, 2 y i g i (y i )] y 1 0 g 1(y 1 ) = 0.4y 1 s.t. x+q y +y 2 1, (4.3) y , y 2 0 g 2(y 2 ) = 0.2y 2 s.t. x+y 1 +y 2 1, y By using Algorithm 1, the numerical results of problem (4.3) are listed as in Table 1 where f = 1.5x+ 2 g i (y i ). It follows from Table 1 that an optimal solution of problem (4.2) occurs at the point (w,q 1,q 2 ) = (0.6, 0.25, 0.15). The result shows that an optimal solution for the water management department, who is riskaverse, is to take the public water right variable as 0.6 through anticipating all possible responses of the water users from the worst-case point of view. Each user is assumed to execute simultaneously his individually optimal choice after decisions of the water management department. That is, the two users will take values of their decision making variables 0.25 and 0.15 respectively as their response for the water management department. 5 Conclusions A pessimistic linear bilevel multi-follower programg problem in a referential-uncooperative situation(plbmf- RU) occurs commonly in management and planning of many organizations. This paper gives the model and the solution definitions of PLBMF-RU decision problem. Moreover, for solving such a problem, this paper present an algorithm which extends the penalty function method from dealing with optimistic one-leader-andone-follower situation to complex pessimistic referential-uncooperative multiple followers situation. This paper further illustrates the details of the proposed algorithm by an application example of water resources optimal allocation. Some meta-heuristic search methods to solve the large-scale pessimistic referential-uncooperative bilevel multi-follower problems will be considered as our future research task. Acknowledgments This work was partially supported by the Key Project of Anhui Province University Excellent Youth Support Plan (No. gxyqzd ), the National Natural Science Foundation of China (Nos , , 11

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