Particle Swarm Optimization Based on Smoothing Approach for Solving a Class of Bi-Level Multiobjective Programming Problem

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1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 17, No 3 Sofia 017 Print ISSN: ; Online ISSN: DOI: /cait Particle Swarm Optimization Base on Smoothing Approach for Solving a Class of Bi-Level Multiobjective Programming Problem Qingping He, Yibing Lv School of Information an Mathematics, Yangtze University, Jingzhou 43403, China s: Heqp_017@163.com lvyibing_001@163.com Abstract: As a metaheuristic, Particle Swarm Optimization (PSO) has been use to solve the Bi-level Multiobjective Programming Problem (BMPP). However, in the existing solving approach base on PSO for the BMPP, the upper level an the lower level problem are solve interactively by PSO. In this paper, we present a ifferent solving approach base on PSO for the BMPP. Firstly, we replace the lower level problem of the BMPP with Kuhn-Tucker optimality conitions an aopt the perturbe Fischer-Burmeister function to smooth the complementary conitions. After that, we aopt PSO approach to solve the smoothe multiobjective programming problem. Numerical results show that our solving approach can obtain the Pareto optimal front of the BMPP efficiently. Keywors: Bi-level multiobjective programming problem, scalarization metho, optimality conitions, smoothing metho, particle swarm optimization. 1. Introuction Bi-level programming problem is a neste optimization problem, which is characterize by the existence of two optimization problems in which the constraint region of the upper level optimization problem (the leaer) contains another optimization problem calle the lower level optimization problem (the follower)[1]. As a powerful tool to escribe the hierarchy existing in the real life problem, bi-level programming has been applie willy in such areas as transportation systems [], inverse optimization value problem [3], transportation network esign [4], etc. Meanwhile, the wie application of the bi-level programming rives more an more researchers to evote to this promising fiel. In fact, in the last two ecaes, many papers have been publishe both from the theoretical an computational points of view (see the monographs of D e m p e [1] an the reviews by D e m p e an 59

2 Z e m k o h o [5], S i n h a, M a l o an D e b [6], an C o l s o n, M a r c o t t e an S a v a r [7]). Bi-level Multiobjective Programming Problem (BMPP), where the leaer, the follower, or the both have multiple conflicting objectives, has successively rawn the researchers attention both from the theoretical an computational points of view. Here, we can roughly classify the feasible algorithms for the BMPP into two types. The first one is the traitional numerical approach an the secon one is the evolutionary approach. Focusing on the traitional numerical approach for the linear BMPP in which both the leaer an the follower have multiple objectives, N i s h i z a k i an S a k a w a [8] give the three optimal solution efinitions. They propose a feasible algorithm base on K-th best algorithm for the linear bi-level single objective programming problem. For the given lower level objective weights, Lv [9] replaces the lower level optimization problem with its optimality conitions an constructs a multiobjective penalize optimization problem. Then, an exact penalize function approach is propose an some numerical results are reporte; subsequently, following the outline in [9] an regaring the lower level objective weights as the upper level ecision variables, L v an W a n [10] propose another exact penalty function approach. For the linear BMPP, where the leaer has single objective an the follower has multiple objectives, A n k h i l i an M a n s o u r i [11] take the margin function of the lower level problem as the penalty term, an construct the corresponing penalize problem. Then, an exact penalty function algorithm is propose for the above BMPP. For the linear BMPP, where the leaer has multiple objectives an the follower has single objective, C a l v e t e an G a l e [1] prove the existence of the weakly efficient solution an the efficient solution uner the conition that the constraint region is not empty, then present the frameworks of some feasible algorithms. However, no numerical results are reporte. In aitions, for a class of BMPP, where the lower level is a convex vector-programming problem, L v an W a n [13] aopt the metho of replacing the lower level problem with its optimality conitions, after that smooth the complementary conitions with some smoothing function. Then, a smoothing approach is propose. It eserves pointing out that the main rawback of the traitional numerical approach for the BMPP is that only someone (weakl efficient solution can be obtaine. A recent stuy by E i c h f e l e r [14] suggests a refinement-base strategy in which the algorithm starts with a uniformly istribute set of points on upper level variable, an some (weakl efficient solutions can be obtaine. Evolutionary approach, which can obtain the (weakl efficient solutions for the multiobjective programming problem efficiently, has been wiely applie to solve BMPP. D e b an S i n h a [15] as well as S i n h a an Deb [16] iscuss BMPP base on evolutionary multiobjective optimization principles. Base on the above stuies, D e b an S i n h a [17] propose a viable an hybri evolutionary-localsearch base algorithm, an present challenging test problems. S i n h a [18] presents a progressively interactive evolutionary multiobjective optimization metho for bilevel multiobjective programming problem. 60

3 It is note that as a metaheuristic, Particle Swarm Optimization (PSO) [19] has prove to be a competitive algorithm for optimization problems compare with other algorithms such as Genetic Algorithm (GA) an Simulating Algorithm (SA). It can converge to the optimal solution rapily. Z h a n g et al. [0] use some improve PSO algorithms to solve BMPP. Subsequently, Z h a n g et al. [1] propose a hybri particle swarm optimization with crossover operator for some high imensional BMPP. Then, some numerical results are presente to illustrate the superiority of the improve PSO. It is note that in the above references on solving bi-level multiobjective programming problem using PSO approach, the moel of the upper an lower level solving their own problems interactively is aopte. In this paper, base on our previous work [13], we will aopt a ifferent tack from the existing PSO approach for the BMPP. Our strategy can be outline as follows. For a class of BMPP, where the lower level is a convex vector optimization problem, we replace the lower level problem with its optimality conitions. After that, we smooth the complementary conitions with some smoothing function an obtain the corresponing smoothe multiobjective programming problem. Then, a particle swarm optimization approach is propose to solve the smoothe multiobjective programming problem an the approximate efficient solutions for the BMPP are obtaine. This paper is organize as follows. In the following Section, we firstly introuce the mathematical moel an some basic efinitions of the efficient solutions of the bi-level multiobjective programs. In aition, the smoothing metho for the BMPP, which we obtain in our previous work [13], is introuce. A particle swarm optimization approach for the smoothe multiobjective programming problem is propose in Section 3. In Section 4, we report some numerical results to illustrate the PSO approach. Finally, we conclue this paper with some remarks.. Bi-level multiobjective programming an smoothing metho The bi-level multiobjective programming problem consiere in this paper can be formulate as: min F( x,, ( xy, ) (1) s.t. y S( x), where S (x) enotes the efficient solution set of the following lower level optimization problem, () ( Px ) min y f ( x,, s.t. g( x, 0, n m nm an x R, y R, an F : R p R, nm q nm f : R R, g : R l R are continuously ifferentiable functions. To facilitate the iscussion, we also introuce the following notations, which is S ( x, : g( x, 0 enote the presente in our previous work [13]. Let 61

4 m m constraint region of problem (1), S x R : y R, g( x, 0 y enote the projection of S onto the leaer s ecision space. In aitions, let I ( x, enote the set of the active constraints, i.e., (3) I( x, : i : gi ( x, 0, i 1,, l. Definition.1. A point ( x, is feasible to problem (1) if ( x, S, an y S(x). Definition.. A point ( x*, y*) is a Pareto optimal solution to problem (1) if it is feasible an there exists no other feasible point ( x,, such that F( x, F( x*, y*) an F( x, F( x*, y*). Through this paper, we make the following assumptions: ( A 1 ) The constraint region S is nonempty an compact. ( A ) For each given upper level variable x, the lower level problem P ) is a convex vector optimization problem, an the partial graient ( x,, i I( x,, is linear inepenent. Assumption ( A 1 ) guarantees that the feasible region of problem (1) is nonempty, an assumption ( A ) can make us aopt some scalarization metho to transform the lower level problem into the corresponing scalar optimization problem [13]. That is, base on assumption ( A ), we can transform problem (1) into the following bi-level multiobjective programming problem, where the lower level is a scalar optimization problem [13], min F( x,, (4) 6 ( xy,, ) q s.t. 1, i i1 0, min, f ( x,, y s.t. g( x, 0. Let ( x, ) enote the optimal solution set of the lower level problem in problem (). On the relationships between the Pareto optimal solutions of problem () an that of the original problem (1), we have the following result. for all Theorem.1 [13]. Let ( xy, ) be a Pareto optimal solution of problem (1). Then _ q q : R, i 1, the point i1 solution of problem (). Conversely, if the point ( x y g i _ ( xy,, ) is a Pareto optimal _ ( xy,, ) is a Pareto optimal solution of problem (), then ( xy, ) is a Pareto optimal solution of problem (1).

5 Going one step further to the problem (), we replace the lower level scalar optimization problem with its Kuhn-Tucker optimality conitions [13], an problem () can be reuce as the following multiobjective programming problem with complementary conitions: min F( x,, (5) ( x, y,, u) s.t. 1,, f ( x, u g ( x, 0, y j y j j1 q i i1 T u g x y l (, ) 0, g( x, 0, 0, u 0, l where the term u R is the Lagrangian multiplier. To facilitate the epiction, we also aopt the following notations [13]. Let H R R, : nmq l l H R R, : nmq l l H3 R R, : nmq l H4 R R, 1 : n m q l m H R R, which is efine as: (6) (7) 5 : n m q l q H ( w) : H ( x, y,, u) :, f u g ( x,, y j y j j1 H ( w) : H ( x, y,, u) : g( x,, H ( w) : H ( x, y,, u) : u, H ( w) : H ( x, y,, u) : 1, H5( w) : H5( x, y,, u) :. Base on the above notations (4), problem (3) can be rewritten as: min Fw ( ), s.t. H ( w) 0, q i1 l H ( w), H ( w) 0, H ( w) 0, H ( w) 0, H ( w) 0, H5( w) 0. Let w* be a feasible point to problem (5), we also efine the following inex set: i 63

6 (8) i 3i i 3i i 3i 5i ( w*) i : H ( w*) 0, H ( w*) 0, i 1,, l, ( w*) i : H ( w*) H ( w*) 0, i 1,, l, ( w*) i : H ( w*) 0, H ( w*) 0, i 1,, l, ( w*) i : H ( w*) 0, i 1,, q. In the following context, the following assumption is satisfie. ( A 3) Let w * is feasible to problem (5), the graient vectors H1 i ( i 1,, m), H ( ) i i, H ( ) 3 i i, H 4, H ( ) 5 i i are linearly inepenent. It is known that problem (5) is a nonsmooth multiobjective problem. In aition, in our previous work [13], to overcome the nonsmooth of problem (5), we aopt the following so-calle perturbe Fischer-Burmeister function : R R R, which has the formulation (9) ( a, b, ) a b a b to smooth problem (5). Moreover, the perturbe Fischer-Burmeister function ( a, b, ) has the property ( a, b, ) 0 a 0, b 0, ab. Base on the perturbe Fischer-Burmeister function (7), we can smooth the complementary conitions H( w), H3( w) 0 in problem (5) as follows: (10) H ( w) H ( w) H ( w) H ( w) 0, i 1,, l. 3i i i 3i Then for every 0 we can obtain the following smooth multiobjective programming problem: min Fw ( ), (11) s.t. H ( w) 0, 3i i i 3i 1 H ( w) H ( w) H ( w) H ( w) 0, i 1,, l, H ( w) 0, 4 H5( w) 0. On the relationships between the Pareto optimal solutions of the smoothe problem (9) an that of problem (5), we have the following result in our previous work [13]. Theorem. [13]. Let w be a Pareto optimal solution of problem (9) an _ suppose that w _ w as 0, an the assumption ( A 3) hols at w. Then w is a KKT point of problem (9) with multiplier ( u, v,, ) ; Moreover, if ' ' ' ' ( u, v,, ) ( u, v,, ) as 0, then ( xy, ) is a Pareto optimal solution to problem (1). 64

7 Base on Theorem., the nonsmooth multiobjective programming problem (5) can be progressively approximate by a sequence of smooth multiobjective programming problem (9). That means we can obtain the approximate Pareto optimal solution of problem (5) by solving problem (9). In fact, in our previous work [13], we aopt the numerical approach, such as -constraint approach, to solve problem (9) an obtain some Pareto optimal solution of the original problem (1). Here, we consier aopting PSO approach to solve problem (9), an obtain the Pareto optimal front of problem (1). 3. Particle swarm optimization for problem (9) 3.1. Overview of the PSO algorithm Particle Swarm Optimization (PSO) is a kin of evolutionary algorithm, which has been wiely stuie in recent years. It is first propose by K e n n e y an E b e r h a r t [19] in 1995 an successfully applie to the optimization problem, an then it has been effectively extene to some other aspects. In PSO, the population is referre as a swarm an the iniviuals are calle particles. Like other evolutionary algorithms, PSO performs searches using a population of iniviuals that are upate from iteration to iteration. To fin the optimal or approximately optimal solution, each particle changes its searching irection accoring to two factors, its own best previous experience an the best experience of the entire swarm in the past. In the process of searching optimal solution, each particle changes its velocity an position accoring to the following formulas: vt 1 wvt c1 r1 ( Pt xt ) c r ( Pg xt ), (1) x x v, t1 t t1 where r 1 an r are ranom numbers in (0, 1); w is the inertia weight; c 1, c are acceleration factors, which are the positive constant parameters; P t is the personal best position for a particle; P g is the global best position in the entire swarm. If the search space is D-imensional, the velocity an position are represente as v v, v, v, v ), x x, x, x, x ), an the same with t t ( t1 t t3 td ( Pt 1, Pt, Pt 3, PtD t ( t1 t t3 td ( Pg 1, Pg, Pg 3, PgD P ), P ). g 3.. The algorithm for solving multiobjective problem (9) As mentione above, we can obtain the approximate Pareto optimal solution of the original problem (1) by solving the smoothe multiobjective programming problem (9). In aition, in the following contents, we use PSO to solve problem (9). 65

8 next last Definition 3.1. Let w an w be the next an the last ajacent particles of the -th particle w, respectively. The crowing istance of the -th iniviual particle (13) w is efinie as p m1 F w F w next last m( ) m( ) max min, Fm Fm next last where p is the number of the objectives in problem (9); Fm( w ), Fm( w ) an F w ) are respectively the fitness values of m-th objective in problem (9); max F m ( min an F m are the maximum an minimum values of the m-th objective, respectively. The crowing istance is introuce as the inex to juge the istance between the particle an the ajacent particle, an it reflects the congestion egree of no ominate solutions. In the population, the larger the crowing istance, the sparser an more uniform. Definition 3.. The infeasibility threshol is efine as 0 (1 5 t / 4 N), t 0.8 N, (14) 0, t 0.8 N, where 0 is an initial value allowe by constraint violation egree, t is the current evolution generation, an N is the maximal evolution generation. From the above formula (11), we can know that the infeasibility threshol ecreases with the increase of the evolution generation. Definition 3.3. The egree of the iniviual particle violating the constraints in problem (9) is efine as l 1 q C( w ) max( H ( w ),0) max( H ( w ),0) 5i 1j i1 j1 max( H ( w ) H ( w ) H ( w ) H ( w ),0) max( H ( w ),0) where is the tolerance of the equation constraints, which reflects the egree of the strictness on the equation constraints. We firstly outline the PSO approach for the multiobjective programming problem. In the feasible solution space, we uniformly an ranomly initialize the particle swarms an select the no ominate solution particles consisting of the elite set. After that by the methos of congestion egree choosing (the congestion egree can make the particles istribution more sparse) an the ynamic infeasibility ominating the constraints, we remove the no ominate particles in the elite set. Then, the objectives can be approximate. The etails of the propose PSO algorithm for problem (9) can be escribe as follows. Propose PSO Algorithm m m 66

9 Step 1. Give the particle population size M (incluing the position x an the velocity v ) an the maximal evolution generation N. Select the initial infeasibility threshol 0, an let t 0. Step. Upate each particle in the particle group: Step.1. Archive the no ominate solutions of the particle swarm in the external elite set an calculate the congestion istance an the egree of the iniviual particle violating the constraints C (w) on each nonominate solution in the external elite set. The istances are mae in escening orer. Then ranomly select one particle as the global optimal position P g from the archive elite set. Step.. Upate the velocity an position of the particle by the Formula (10). If the position of a particle excees the preset bounary, the position of the particle is equal to its bounary value an its velocity is multiplie by 1 to search the particle in the opposite irection. Step 3. Upate the external elitist set: Compare the upate nonominate solutions of the particle swarm with the nonominate solutions in the external elites, an ecie whether the nonominate solutions in the particle swarm shoul be archive in the external elite set. If the solution in the particle swarm satisfies the omination relation, it nees to juge whether the external elitist set is full: if it is not full, the nonominate solution is archive irectly; otherwise, the following steps are aopte: Step 3.1. Archive all nonominate solutions of the external elite set in escening orer accoring to the congestion istance. Step 3.. Ranomly pick a particle in the M particles of the sorte set an replace it with the particle that nees to be archive. Step 4. Upate the local optimal position of the particle: Step 4.1. Upate the global optimum position if the position of the particle upate ominates its historical optimal position. Step 4.. If the upate particle position oes not ominate its historical optimum position, accoring to 50% chance to retain its best position in history. When the egree of all the particles in the nonominate set violating the constraints C (w) is zero, the algorithm terminates an we get the approximate Pareto optimal solutions w * an the values Fw ( *). Otherwise, go to Step 5. Step 5. If t N we get the approximate Pareto optimal solutions w * an the values Fw ( *). Otherwise, set t t 1, an go to Step. 4. Numerical results The Generational Distance (GD) is an inicator to estimate the average istance between the obtaine optimal Pareto solutions an the theoretical optimal solutions. It can be efine as N i1 i GD, N 67

10 where N is the number of the obtaine optimal solutions an i is the Eucliean istance in objective space between each of the foune optimal solutions an the nearest member of the theoretical optimal solution set. It is obvious that GD 0 means the all the foune optimal solutions belong to the theoretical optimal solution set. SPacing (SP) is an inicator to escribe the istribution uniformity of the obtaine Pareto solutions. It can be efine as M E n 1 m m i1 M E m1 m ( i) SP, n i j i i where i min j( F1 ( x, F1 ( x, F ( x, F ( x,,( i, j 1,,, n) ; E is the mean of all i ; m is the Eucliean istance between the extreme solutions in the obtaine Pareto optimal solution set an the theoretical Pareto optimal solution set on the m-th objective; M is the number of the upper level objective function; n is the number of the obtaine solutions. It is obvious that SP 0 means that all the obtaine Pareto solutions istribute evenly in objective space. To verify the feasibility an effectiveness of the above PSO approach for the bilevel multiobjective programming problem, we give some numerical results. The perturbe Fischer-Burmeister function parameter 0.01 an the PSO parameters are set as follows: w 0.7 an c1 c 1.5. We make the programs an use a personal computer (CPU: Intel Pentium.6 GHZ, RAM:1G) to execute the programs. Example 1. This example is taken from [], where x R, y R. The particle population size an the maximal evolution generation for the above example are respectively set as M 100, N 50, (15) T max ( x y, x, x s.t. 0 x 15, T max ( x y, x, y s.t. x y 0, x y4, xy33, y 0. For Example 1, in [] the authors iscuss two ecision mechanisms: semicooperation ecision mechanism an pure-inepenence, two solutions ( x, (15,0), an ( x, (15,8) are foune. Fig. 1 shows the two solutions an the Pareto optimal front by our PSO for Example 1. It can be seem that the solution 68

11 (15, 8) is not in the theoretical optimal solution set an the optimal solutions we foun can integrally escribe the Pareto optimal front. Fig. 1. The obtaine Pareto optimal front of Example 1 Example. This example is taken from [3], where x R, y R, an we set the particle population size an the maximal evolution generation as M 100, N 50. (16) T max ( x 4 y, x, x s.t. 0 x 3, max ( y,, y s.t. x y 3, x y 4, x y1, 3x y 4, y 0. For Example, in [3], the authors use a fuzzy interactive metho an five optimal solutions with ifferent satisfactoriness of the upper level ecision maker are foune. Fig. shows the Pareto optimal front by our PSO in Example 3. It can be seem that the optimal solutions we foun can integrally escribe the Pareto optimal front, an the optimal solutions in [3] are inclue. 69

12 Fig.. The obtaine Pareto optimal front of Example Example 3. This example is taken from [4], where x R, y R, an we set the particle population size an the maximal evolution generation as M 00, N 50, (17) T min ( x y, x ( y 10) ), x s.t. 0 x 15, T min ( y, y( x 30)), y s.t. yx0, 0 y 15. Fig. 3. The obtaine Pareto optimal front of Example 3 70

13 Fig. 3 shows the Pareto optimal front of Example 3. In [0], the authors have also presente a PSO metho for BMPP, in which the BMPP is transforme to solve many multiobjective optimization problems in the upper level an the lower level interactively with a preefine count. Fig. 4. The theoretical Pareto optimal front of Example 3 an the obtaine upper level Pareto optimal * fronts while x varies Fig. 4 shows the theoretical Pareto optimal front of Example 3 an three * obtaine upper level Pareto optimal fronts with ifferent upper ecision variable x. It can be seen that each of the three obtaine upper level Pareto optimal fronts has only one intersection with the theoretical Pareto optimal front of Example 3. This PSO metho has to solve a series of multiobjective optimization problems in the upper level to fin all theoretical optimal solutions. Example 4. This example is taken from [5], where x R, y R an we set the particle population size an the maximal evolution generation as M 00, N 50, (18) T min ( x,( y10) ), x s.t. x y 0, 0 x 15, T min (( x y 30), x 0.5 y ), y s.t. 0 y

14 Fig. 5 shows the obtaine Pareto front by the PSO metho in [0] an our PSO for Example 4, it can be seen that both the two obtaine Pareto optimal fronts are very close to the theoretical Pareto optimal front. To highlight the avantage of our PSO, a series of contrast numerical experiments between our PSO an the PSO metho in [0] are conucte. In orer to eliminate the ifference of each experiment, we execute both the two algorithms for 50 times for each example. Table 1 shows the comparison of results between the two algorithms consiering the two inicators previously escribe an the execution time (ET, in secons). Consiering the generation istance an the spacing, it can be seen that our PSO performs better than the PSO metho in [0] except for example, in which the performance of our PSO is slightly worse. However, our PSO is much more efficient with respect to the execution time. 7 Fig. 5. The obtaine Pareto optimal front of Example 4 Table 1. Results of the General Distance (GD), SPacing (SP) an Execution Time (ET) for the four examples GD SP ET(s) Example PSO in [0] Our PSO PSO in [0] Our PSO PSO in [0] Our PSO Max Example 1 Min Average Max Example Min Average Max Example 3 Min Average Max Example 4 Min Average

15 5. Conclusions In this paper, we propose a ifferent solving approach for the BMPP base on PSO. The main ifference of the solving approach propose here comparing to that in the existing references is that we only nee to solve the corresponing smoothe multiobjective programming problem to obtain the Pareto optimal front of the BMPP. The numerical results prove that our PSO is a high efficiency an accurate metho. It is note that in our solving approach base on PSO for the BMPP, we just aopt the original PSO algorithm. In recent years, the PSO algorithm is applie in many fiels such as social network [6], neural networks optimization [7], reservoir operation [8], etc. The researchers have propose some moifie PSO algorithms [9] an combinatorial PSO algorithms [30] for the optimization problem. Our future work will be to lea these moifie PSO approaches into our solving approach. R e f e r e n c e s 1. D e m p e, S. Founation of Bi-Level Programming. Lonon, Kluwer Acaemic Publishers, 00.. S t o i l o v a, K., T. S t o i l o v, V. I v a n o v. Practical Bi-Level Optimization as a Tool for Implementation of Intelligent Transportation Systems. Cybernetics an Information Technologies, Vol. 17, 017, No, pp L v, Y., Z. C h e n, Z. Wan. A Penalty Function Metho Base on Bi-Level Programming for Solving Inverse Optimal Value Problems. Applie Mathematics Letters, Vol. 3, 010, pp Y a n g, H., M. G. H. B e l l. Transport Bi-Level Programming Problems: Recent Methoological Avances. Transportation Research, Vol. 35, 001, pp D e m p e, S., A. B. Z e m k o h o. The Bi-Level Programming Problem: Reformulations, Constraint Qualifications an Optimality Conitions. Mathematical Programming, Vol. 138, 013, No 1-, pp S i n h a, A., P. M a l o, K. Deb. A Review on Bi-Level Optimization: From Classical to Evolutionary Approaches an Applications. IEEE Transactions on Evolutionary Computation, PP(99), 017, pp C o l s o n, B., P. M a r c o t t e, G. S a v a r. An Overview of Bi-Level Optimization. Annals of Operations Research, Vol. 153, 007, pp N i s h i z a k i, I., M. S a k a w a. Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problem. Journal of Optimization Theory an Applications, Vol. 103,1999, No 1, pp L v, Y. An Exact Penalty Function Approach for Solving the Linear Bi-Level Multiobjective Programming Problem. Filomat, Vol. 9, 015, No 4, pp L v, Y., Z. Wan. Solving Linear Bi-Level Multiobjective Programming Problem via Exact Penalty Function Approach. Journal of Inequalities an Applications, Vol. 015, 015, A n k h i l i, Z., A. M a n s o u r i. An Exact Penalty on Bi-Level Programs with Linear Vector Optimization Lower Level. European Journal of Operational Research, Vol. 197, 009, No 1, pp C a l v e t e, H., C. G a l e. Linear Bi-Level Programs with Multi Objectives at the Upper Level. Journal of Computational an Applie Mathematics, Vol. 34, 010, pp L v, Y., Z. Wan. A Smoothing Metho for Solving Bi-Level Multiobjective Programming Problems. Journal of the Operations Research Society of China, Vol., 014, No 4, pp E i c h f e l e r, G. Multiobjective Bi-Level Optimization. Mathematical Programming, Vol. 14, 010, pp

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