Applied Mathematics and Computation 188 (2007) 786–800 www.elsevier.com/locate/amc A probabilistic bi-level linear multi-objective programming problem to supply chain planning E. Roghanian *, S.J. Sadjadi, M.B. Aryanezhad Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran P.C. 16844, Iran Abstract Bi-level programming, a tool for modeling decentralized decisions, consists of the objective(s) of the leader at its first level and that is of the follower at the second level. Three level programming results when second level is itself a bi-level programming. By extending this idea it is possible to define multi-level programs with any number of levels. In most of the real life problems in mathematical programming, the parameters are considered as random variables. The branch of mathematical programming which deals with the theory and methods for the solution of conditional extremum problems under incomplete information about the random parameters is called ‘‘stochastic programming’’. Supply chain planning problems are concerned with synchronizing and optimizing multiple activities involved in the enterprise, from the start of the process, such as procurement of the raw materials, through a series of process operations, to the end, such as distribution of the final product to customers. Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where for example, one level may correspond to a local plant control/scheduling/planning problem and another level to a corresponding plant-wide planning/network problem. Such a multi-level decision network structure can be mathematically represented by using ‘‘multi-level programming’’ principles. In this paper, we consider a ‘‘probabilistic bi-level linear multi-objective programming problem’’ and its application in enterprise-wide supply chain planning problem where (1) market demand, (2) production capacity of each plant and (3) resource available to all plants for each product are random variables and the constraints may consist of joint probability distributions or not. This probabilistic model is first converted into an equivalent deterministic model in each level, to which fuzzy programming technique is applied to solve the multi-objective nonlinear programming problem to obtain a compromise solution.  2006 Published by Elsevier Inc. Keywords: Bi-level programming; Multi-objective decision-making; Multi-level multi-objective decision-making; Fuzzy decisionapproach; Stochastic programming; Supply chain management * Corresponding author. E-mail addresses: E-roghanian@iustarak.ac.ir (E. Roghanian), Sjsadjadi@iust.ac.ir (S.J. Sadjadi), Mirarya@iust.ac.ir (M.B. Aryanezhad). 0096-3003/$ - see front matter  2006 Published by Elsevier Inc. doi:10.1016/j.amc.2006.10.032 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 787 1. Introduction and literature review 1.1. Bi-level programming A bi-level programming problem is formulated for a problem in which two decision-makers make decisions successively. For example, in a decentralized firm, top management makes a decision such as budget of the firm, and then each division determines a production plane in the full knowledge of the budget [1]. Research on multi-level mathematical programming to solve organizational planning and decision-making problems has been conducted widely. The research and application have concentrated mainly on bi-level programming [1]. In the BLP problem, each decision maker tries to optimize its own objective function(s) without considering the objective(s) of the other party, but the decision of each party affects the objective value(s) of the other party as well as the decision space. The general formulation of a bi-level programming problem (BLPP) is [2]: min F ðx; yÞ s:t: Gðx; yÞ 6 0; min f ðx; yÞ s:t: gðx; yÞ 6 0; x y ð1Þ where x 2 Rn1 and y 2 Rn2.The variables of problem are divided into two classes, namely the upper-level variables x 2 Rn1 and the lower-level variables y 2 Rn2. Similarly, the functions F : Rn1 · Rn2 ! R and f : Rn1 · Rn2 ! R are the upper-level and lower-level objective functions respectively, while the vector-valued functions G : Rn1 · Rn2 ! Rm1 and g : Rn1 · Rn2 ! Rm2 are called the upper-level and lower-level constraints respectively. All of the constraints and objective functions may be linear, quadratic, nonlinear, fractional, etc. 1.2. Stochastic programming In most of the real life problems in mathematical programming, the parameters are considered as random variables. The branch of mathematical programming which deals with the theory and methods for the solution of conditional extremum problems under incomplete information about the random parameters is called ‘‘stochastic programming’’. Most of the problems in applied mathematics may be considered as belonging to any one of the following classes [7]: 1. Descriptive Problems, in which, with the help of mathematical methods, information is processed about the investigated event, some laws of the event being induced by others. 2. Optimization Problems in which from a set of feasible solutions, an optimal solution is chosen. Besides the above division of applied mathematics problems, they may be further classified as deterministic and stochastic problems. In the process of the solution of the stochastic problem, several mathematical methods have been developed. However, probabilistic methods were for a long time applied exclusively to the solution of the descriptive type of problems. Research on the theoretical development of stochastic programming is going on for the last four decades. To the several real life problems in management science, it has been applied successfully [13]. The chance constrained programming was first developed by Charnes and Cooper [4]. Subsequently, some researchers like Sengupta [12], Contini [5], Sullivan and Fitzsimmons [14], Leclercq [9], Teghem et al. [15] and many others have established some theoretical results in the field of stochastic programming. Stancu-Minasian and Wets [13] have presented a review paper on stochastic programming with a single objective function. The fuzziness occurs in many of the real life decision making problems. Decision making in a fuzzy environment was first developed by Bellman and Zadeh [3]. Zimmermann [16] presented an application of fuzzy linear programming to the linear vector-maximum problem and showed that the solution obtained by fuzzy 788 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 linear programming is always efficient. Hanan [6], Narasimhan [10], Leberling [8] and many others have made contributions in fuzzy goal programming and fuzzy multi-objective programming. 1.3. Supply chain and supply chain management In recent years there has been a great interest in enterprise-wide supply chain planning problems because of their impact on substantially improving the overall competitiveness of economic potential of individual organizations. Supply chain planning problems are concerned with synchronizing and optimizing multiple activities involved in the enterprise, from the start of the process, such as procurement of the raw materials, through a series of process operations, to the end, such as distribution of the final product to customers [17]. Supply chain planning has thus obtained a great deal of attention in the open literature. A number of outstanding issues deserve some further attention [17]: First, supply chain planning problems naturally incorporate multiple decision modeling steps, which are connected in a ‘‘hierarchical’’ way. Since individual activities are often governed by separate supply chain components which have their own, often mutually conflicting, objectives, the operation and control of the entire networks is based on multi-perspectives (see [18]). Most of the planning models are however, based on the assumption that ‘‘all’’ activities of supply chain networks are governed by a ‘‘global organizer’’ neglecting such multiple perspectives (for example [19–22]). Second, different participant components in the supply chain network may not operate with the same level of information. Some may possess more information, while other may possess less information; this may lead to information distortion as discussed in [23]. Recent works where this important issue is being discussed include Gjerdrum et al. [24] and Zhou et al. [25]. Finally, the presence of uncertainty in supply chain planning models further amplifies the complexity of the problem. Uncertainty typically exists in supply chain parameters, such as processing times, performance coefficients, utility coefficients, delivery and inventory costs, supply of raw materials, etc. Here, approaches which have started to appear in the open literature include scenario-based multi-period formulations [5], stochastic programming formulations [20,21] and supply chain dynamics and control formulations [26– 28]. In this paper, we consider a ‘‘probabilistic bi-level linear multi-objective programming problem’’ and its application in enterprise-wide supply chain planning problem, where market demand and warehouse capacity are random variables and the constraints follow a joint probability distributions or not. This probabilistic model is first converted into an equivalent deterministic model in each level, to which fuzzy programming technique is applied to solve the multi-objective nonlinear programming problem to obtain a compromise solution. The paper is further organized as follows: in the next section we will present an example for motivating reader, in Section 3 multi-objective chance constrained programming will be described, in Section 4 we will explain a multi-level nonlinear multi-objective decision-making under fuzziness, in Section 5 model formulation will be present, in Section 6 a solution method will be presented, in Section 7 an illustrative numerical example is given to demonstrate the obtained results and finally in Section 8, conclusions are drawn regarding the model. 2. A motivating example [17] Consider a manufacturing enterprise involving two plants P1, P2 and one warehouse or distribution centre (DC), with two products, A and B. The enterprise aims to minimize the overall cost which corresponds to a production cost and a distribution cost. Based on the notation in Table 1, the supply chain model may be mathematically formulated into the following constraints (see Fig. 1). The production part is subject to the following constraints: (a) Common resources are shared by both plants Y 1A þ Y 1B þ Y 2A þ Y 2B 6 500: ð2Þ E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 789 Table 1 Notation for motivating example [17] Description Variable Y1A Y1B Y2A Y2B XA XB Production amount A in plant P1 (ton) Production amount B in plant P1 (ton) Production amount A in plant P2 (ton) Production amount B in plant P2 (ton) Inventory holding of A in DC (ton) Inventory holding of B in DC (ton) Parameter hA hB Demand amount A (ton) Demand amount B (ton) P1 DC PRODUCT A, B CUSTOMER P2 Fig. 1. Process configuration of an enterprise in motivating example [17]. (b) Some resources may be controlled by individual plant conditions: Y 1A þ Y 1B 6 200; Y 2A þ Y 2B 6 250: ð3Þ (c) Production levels achieved by both plants should not be lower than levels required by the inventory Y 1A þ Y 2A P X A ; Y 1B þ Y 2B P X B : ð4Þ The objective of the production part is to minimize a production cost which can be typically formulated as follows: min Z PC ¼ 1:5X A þ 2X B þ 7Y 1A þ 3Y 1B þ 10Y 2A þ 6Y 2B : ð5Þ On the other hand, distribution centers are subject to the following constraints: (a) Inventory levels are limited by their overall capacity 3X A þ 2X B 6 500: ð6Þ (b) Inventory levels should meet demands X A P hA ; X B P hB : ð7Þ The objective of the distribution centre is to minimize a distribution cost which may be formulated as follows: min Z DC ¼ 15X A þ 13X B þ 3Y 1A þ 2Y 1B þ 3:5Y 2A þ 2:5Y 2B : ð8Þ A way to pose this problem is formulating the problem as a bi-level optimization model which is as follows: min X A ;X B s:t: Z DC ¼ 15X A þ 13X B þ 3Y 1A þ 2Y 1B þ 3:5Y 2A þ 2:5Y 2B 3X A þ 2X B 6 500; X A P hA ; 790 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 X B P hB ; Y 1A þ Y 2A P X A ; min Y 1A ;Y 1B ;Y 2A ;Y 2B s:t: Y 1B þ Y 2B P X B ; Z PC ¼ 1:5X A þ 2X B þ 7Y 1A þ 3Y 1B þ 10Y 2A þ 6Y 2B Y 1A þ Y 1B þ Y 2A þ Y 2B 6 500; Y 1A þ Y 1B 6 200; Y 2A þ Y 2B 6 250; Y 1A þ Y 2A P X A ; Y 1B þ Y 2B P X B ; X A ; X B ; Y 1A ; Y 1B ; Y 2A ; Y 2B P 0: ð9Þ The formulation in (9) consists of two sub-problems: the higher level decision problem and the lower level decision problem. The objective function for the higher level problem is ZDC and for the lower level is ZPC.1 The higher level problem is also known as the leader’s problem or the outer problem, the lower level problem is sometimes called a follower’s problem or an inner problem. The two problems are connected in a way that the leader’s problem sets parameters influencing the follower’s problem and the leader’s problem, in turn, is affected by the outcome of the follower’s problem. To compare the two problems in terms of scope of information, a follower makes decisions using only its local information while a leader does using the complete information including the follower’s possible reaction to the leader’s decision. This is an important feature which can contribute to address the complicated industrial situations which may be difficult to model using other modeling methodologies. Therefore, BLPPs have been applied in extensive and diverse areas. Nevertheless (9) deals with following weaknesses: (I) Each level consists of ‘‘one’’ objective function while in real situations it is possible to have ‘‘more than one’’. (II) Input data of parameters are often stochastic or fuzzy in most real-world situations. (III) Objective functions and constraints are often nonlinear (quadratic, geometric, etc.) rather than linear in real-world situations. 3. Multi-objective chance constrained programming problem with a joint constraint [30] A multi-objective chance constrained programming problem with a joint probability constraint can be stated as n X ðkÞ min zðkÞ ðxÞ ¼ cj xj ; k ¼ 1; 2; :::; K j¼1 s:t: pr " n X j¼1 xj P 0 1 a1j xj P b1 ; n X a2j xj P b2 ; . . . ; j¼1 j ¼ 1; 2; :::; n; n X j¼1 # amj xj P bm P 1  a; ð10Þ The issue of dominance between multiple elements of supply chains, such as production and distribution, needs to be addressed. The illustrated mathematical model takes the distribution problem as the outer problem which is ultimate optimization problem and the production operation is regarded as a subordinate optimization problem. An importance is placed on distribution since actual profits are thought to be obtained through distribution inventory manipulation. However, there may be other types of bi-level formulation model encapsulating various types of supply chain operations. For instance, some special industries involving difficult processes may pose their outer problem as maintaining process operability and inner problem as distribution problem due to the critical importance of keeping their process condition safe. It may be also possible that transportation problem can be an outer problem for industries where delivery needs special care and corresponding high costs [17]. E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 791 where bi’s are independent normal random variables with known means and variances. Constraint is a joint probabilistic constraint and 0 < a < 1 is a specified probability. We assume that the decision variables xj’s are deterministic. Let the mean and standard deviation of the normal independent random variable bi be given by E(bi) and r(bi) respectively. Sinha et al. [30] have shown equivalent deterministic model of probabilistic problem (10) can be presented as n X ðkÞ cj xj ; k ¼ 1; 2; :::; K min zðkÞ ðxÞ ¼ j¼1 s:t: rffiffiffi  2 3bi bi p exp  ð2y þ 1Þ; P 2 i 2 3  b2i m Y y i P 1  a; ð11Þ i¼1 n X aij xj  bi rðbi Þ ¼ Eðbi Þ; j¼1 0 6 y i 6 1; i ¼ 1; 2; :::; m; xj P 0; j ¼ 1; 2; :::; n: When bi’s are independent uniform random variables with known minimum and maximum, problem (10) should be presented as n X ðkÞ cj xj ; k ¼ 1; 2; :::; K min zðkÞ ðxÞ ¼ j¼1 s:t: m  Y i¼1 Pn  mini maxi  mini j¼1 aij xj  ð12Þ P 1  a; xj P 0; j ¼ 1; 2; :::; n: 4. A multi-level nonlinear multi-objective decision-making under fuzziness [29] Let xi 2 Rni (i = 1, 2) be a vector variables indicating the first decision level’s choice and the second decision level’s choice ni P 1 (i = 1, 2). Let F1 : Rn1 · Rn2 ! RN1 be the first level objective functions and F2 : Rn1 · Rn2 ! RN2 be the second level objective functions. Let the FLDM and SLDM have N1 and N2 objective functions, respectively. Let G be the set of feasible choices {(x1, x2)}. So the BLN-MODM problem may be formulated as follows: ½1st level Max F 1 ðx1 ; x2 Þ ¼ Max ðf11 ðx1 ; x2 Þ; . . . ; f1N 1 ðx1 ; x2 ÞÞ; where x2 solves; x1 x1 ½2nd level ð13Þ Max F 2 ðx1 ; x2 Þ ¼ Max ðf21 ðx1 ; x2 Þ; . . . ; f2N 2 ðx1 ; x2 ÞÞ x2 x2 s:t: G ¼ fðx1 ; x2 Þjgi ðx1 ; x2 Þ 6 0; i ¼ 1; 2; . . . ; mg; where G is the bi-level convex constraint set. F1, F2 are nonlinear and concave functions. The decision mechanism of BLN-MODM is that the FLDM and SLDM adopt the two-planer Stackelberg game. Osman et al. [29] propose a solution method for solving above problem. For more information refer to [29]. 5. Model formulation Production and distribution models of an enterprise can be mathematically formulated as follows (see Table 2 for the notation used). 792 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 Table 2 Notation for supply chain planning model Indices i l w r Product (1, . . . , I) Plant (1, . . . , L) Warehouse (1, . . . , W) Market (1, . . . , R) Constant parameters ali bli cwi ali blwi hwri trwri Mri Pl Q 0 6 c, k, p 6 1 Capacity coefficient of product i at plant l Resource coefficient of product i at plant l Resource coefficient of product i at warehouse w Production cost coefficient for of product i at plant l Transportation cost coefficient for product i from plant l to warehouse w Inventory holding cost coefficient for product i at warehouse w for market r Transportation cost coefficient for product i from warehouse w to market r Demand of product i at market r (random variable) Production capacity of plant l (random variable) Resources available to all the plants for product i (random variable) Probability Variables ZPC ZDC Z 0DC Ylwi Xwri Objective function of a production part (cost) Objective function of a distribution part (cost) Objective function of a distribution part (capacity) Production amount of i at plant l for warehouse w Inventory of product i at warehouse w for market r 5.1. A production model A production part of supply chains is typically subject to the following constraints: (a) Production amounts from the plants should meet the levels required at the warehouses L X Y lwi P R X X wri 8w; i: ð14Þ r¼1 l¼1 (b) Production levels at the plants are limited by individual plant capacities ! W I X X P ali Y lwi 6 P l ; ð8lÞ P 1  kl ð8lÞ2 : ð15Þ i¼1 w¼1 (c) Common used resources may be shared by all the plants ! W L X X bli Y lwi 6 Qi ; ð8iÞ P 1  ci ð8iÞ: P ð16Þ l¼1 w¼1 An operating objective of production parts is to minimize their costs, which typically consists of its manufacturing cost and distribution cost between plants and warehouses min Z PC ¼ Y lwi 2 P l¼1 w¼1 i¼1 P P I W i¼1 L X W X I X w¼1 ali Y lwi ali Y lwi þ L X W X I X blwi Y lwi : ð17Þ l¼1 w¼1 i¼1 P P   I W 6 P l ; 8l P 1  kl refer to joint distribution of P’s in other hand P i¼1 w¼1 ali Y lwi 6 P l P 1  kl ; 8l [11] refer to disjoint distribution of them, these two states are one at time. E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 793 Operation of the production part can thus be formulated as the following mathematical programming problem: min Y lwi Z PC ¼ ali Y lwi þ l¼1 w¼1 i¼1 L X s:t: L X W X I X Y lwi P X wri 8w; i; W I X X L X W X ! P 1  kl ð8lÞ; ! P 1  ci ð8iÞ; ali Y lwi 6 P l ; ð8lÞ i¼1 w¼1 P blwi Y lwi l¼1 w¼1 i¼1 r¼1 l¼1 P R X L X W X I X bli Y lwi 6 Qi ; ð8iÞ l¼1 w¼1 X wri P 0 8w; r; i; Y lwi P 0 8w; l; i: ð18Þ 5.2. A distribution model A distribution part is typically subject to the following constraints: (a) Sums of individual warehouses’ holding should meet demands in markets ! W X X wri P M ri ; ð8r; iÞ P 1  p ð8r; iÞ: P ð19Þ w¼1 (b) The first objective function indicates capacity of all warehouses min X wri Z 0DC ¼ W X R X I X w¼1 r¼1 ð20Þ cwi X wri : i¼1 (c) The following indicates an objective function for the distribution part of the supply chain: min X wri Z DC ¼ I R X W X X w¼1 r¼1 hwri X wri þ I R X W X X w¼1 r¼1 i¼1 trwri X wri ; ð21Þ i¼1 where the first term denotes inventory holding cost including material handling cost at warehouses and the second indicates transportation cost from warehouses to markets: Operation of the inventory part can thus be formulated as the following mathematical programming problem: min X wri min X wri s:t: Z DC ¼ W X R X I X w¼1 r¼1 Z 0DC ¼ P W X W X R X I X w¼1 r¼1 i¼1 W X R X I X w¼1 r¼1 hwri X wri þ trwri X wri i¼1 cwi X wri i¼1 X wri P M ri ; ð8r; iÞ w¼1 X wri P 0 8w; r; i; Y lwi P 0 8w; l; i: ! ð22Þ P1p ð8r; iÞ; 794 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 Note that the decisions of the distribution part are based on those of the production part: for example, inventory policies are made using the outcome of production decisions. Similarly, decisions on the production part are affected by parameters which are decided by the distribution part: for example, production levels are decided from given information regarding the inventory conditions. Therefore the overall supply chain planning model can be posed as the following bi-level optimization problem: min X wri min X wri s:t: Z DC ¼ I R X W X X w¼1 r¼1 Z 0DC ¼ P Y lwi s:t: Z PC ¼ cwi X wri X wri P M ri ; ð8r; iÞ L X W X I X ! ali Y lwi þ l¼1 w¼1 i¼1 L X Y lwi P X wri P1p L X W X I X blwi Y lwi l¼1 w¼1 i¼1 ð23Þ 8w; i; I X W X W L X X ! P 1  kl ð8lÞ; ! P 1  ci ð8iÞ; ali Y lwi 6 P l ; ð8lÞ i¼1 w¼1 P ð8r; iÞ; r¼1 l¼1 P R X trwri X wri i¼1 i¼1 w¼1 min I R X W X X w¼1 r¼1 i¼1 I R X W X X w¼1 r¼1 W X hwri X wri þ bli Y lwi 6 Qi ; ð8iÞ l¼1 w¼1 X wri P 0 8w; r; i; Y lwi P 0 8w; l; i; where the inner problem corresponds to the production optimization problem and the outer problem to the distribution optimization problem. 6. Solution method In this section we will present a solution method in flowchart frame work for solving model (23) (see Fig. 2). Transform model (23) to a deterministic program according to sec. 3. Solve deterministic program according to sec. 4 Adjust tolerances of objective functions and decision variables No If solution is acceptable for all DM’s yes End Fig. 2. Solution method. E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 795 7. Numerical examples Example 1. We have changed Eq. (9) of Section 2 such that min Z DC ¼ 15X A þ 13X B þ 3Y 1A þ 2Y 1B þ 3:5Y 2A þ 2:5Y 2B min Z 0DC ¼ 3X A þ 2X B X A ;X B X A ;X B s:t: min P ðX A P hA Þ P 1  p; P ðX B P hB Þ P 1  p; Z PC ¼ 1:5X A þ 2X B þ 7Y 1A þ 3Y 1B þ 10Y 2A þ 6Y 2B s:t: P ðY 1A þ Y 1B þ Y 2A þ Y 2B 6 QÞ P 1  k; Y 1A ;Y 1B ;Y 2A ;Y 2B ð24Þ P ðY 1A þ Y 1B 6 P 1 ; Y 2A þ Y 2B 6 P 2 Þ P 1  c; Y 1A þ Y 2A P X A ; Y 1B þ Y 2B P X B ; X A ; X B ; Y 1A ; Y 1B ; Y 2A ; Y 2B P 0; where  p ¼ k ¼ c ¼ 0:05; hA  N ð400; 50Þ; Z 0:05  2; hB  N ð500; 60Þ; Q  U ð4000; 6000Þ; P 1  U ð1000; 1500ÞhA  U ð1500; 2000Þ: According to the procedure in Section 4, individual best solutions and individual worst solutions for above models are This data can then be formulated as the following membership functions of fuzzy set theory [31]: lZ DC ½Z DC  ¼ lZ 0 ½Z 0DC  ¼ DC lZ PC ½Z PC  ¼ 8 > < 1; 43202:7Z DC > 25092:7 : 0; 8 1; > < 6112:5Z 0DC > : 3372:5 0; 8 > < 1; 26906:2Z PC > 19696:2 : 0; ; Z DC 6 18 110; 18 110 6 Z DC 6 43202:7; 43202:7 6 Z DC ; Z 0DC 6 2740; ; 2740 6 Z 0DC 6 6112:5; 6112:5 6 Z 0DC ; Z PC 6 7210; ; 7210 6 Z PC 6 26906:2; 26906:2 6 Z PC : ð25Þ 796 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 According to [29] and by using Lingo-8, solutions of FLDM and SLDM are FLDM problem solution Objective value: Variable LANDADC XA XB Y1A Y1B Y2A Y2B 1.000000 Value 1.000000 500.0000 620.0000 0.000000 500.0000 0.000000 620.0000 SLDM problem solution Objective value: Variable LANDAPC XA XB Y1A Y1B Y2A Y2B 1.000000 Value 1.000000 500.0000 620.0000 0.000000 500.0000 0.000000 620.0000 Z DC ¼ 18 110; Z 0DC ¼ 2740; Z PC ¼ 7210: Solution is complete and it is not necessary to continue. Example 2. Consider problem (24) with following changes: min Z DC ¼ 15X A þ 13X B þ 3Y 1A þ 2Y 1B þ 3:5Y 2A þ 2:5Y 2B min Z 0DC ¼ 3X A þ 2X B X A ;X B X A ;X B s:t: min Y 1A ;Y 1B ;Y 2A ;Y 2B s:t: P ðX A P hA Þ P 1  p; P ðX B P hB Þ P 1  p; Z PC ¼ 1:5X A þ 2X B þ 7Y 1A þ 3Y 1B þ 10Y 2A þ 6Y 2B P ðY 1A þ Y 1B þ Y 2A þ Y 2B 6 QÞ P 1  k; ð26Þ P ð5Y 1A þ 3Y 1B 6 P 1 ; 2Y 2A þ 7Y 2B 6 P 2 Þ P 1  c; Y 1A þ Y 2A P X A ; Y 1B þ Y 2B P X B ; X A ; X B ; Y 1A ; Y 1B ; Y 2A ; Y 2B P 0: According to the procedure in Section 4, individual best solutions and individual worst solutions for above models are 797 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 This data also can then be formulated as the following membership functions of fuzzy set theory [31]: 8 Z DC 6 18 160; < 1; DC lZ DC ½Z DC  ¼ 72 715Z ; 18 160 6 Z DC 6 72 715; : 54 555 0; 72 715 6 Z DC ; 8 1; Z 0DC 6 2740; > < lZ DC ½Z 0DC  ¼ 11 680Z 0DC 8940 2740 6 Z 0DC 6 11 680; 11 680 6 Z 0DC ; Z PC 6 7311:5; 450Z PC lZ PC ½Z PC  ¼ 4740138:5 ; 7311:5 6 Z PC 6 47 450; : 0; 47 450 6 Z PC : ð27Þ ; > : 0; 8 1; < According to [29] and by using Lingo-8, solutions of FLDM and SLDM are FLDM problem solution Objective value: Variable LANDADC XA XB Y1A Y1B Y2A Y2B 0.9306607 Value 0.9306582 500.0000 620.0000 1094.782 0.000000 0.000000 1239.437 SLDM problem solution Objective value: Variable LANDAPC XA XB Y1A Y1B Y2A Y2B 0.814852 Value 0.8148886 500.0000 620.0000 0.000000 2089.142 0.000000 1080.695 798 E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 Z DC ¼ 21 943; Z 0DC ¼ 2740; Z PC ¼ 14741:6: As we noted before, two solutions are usually different because of nature between two levels objective functions. We assume the FLDM’s control decisions XA, XB with the tolerance 10,20, respectively. With FLDM solution With SLDM solution ZDC Z 0DC ZPC 21 943 22 440 2740 2740 17090.1 14741.6 According to [29] following problem should be solved: Max d X A  ð500  10Þ P d; 10 ð500 þ 10Þ  X A P d; 10 X B  ð620  20Þ P d; 20 ð620 þ 20Þ  X B P d; 20 22 440  ð15X A þ 13X B þ 3Y 1A þ 2Y 1B þ 3:5Y 2A þ 2:5Y 2B Þ P d; 22 440  21 943 17090:1  ð1:5X A þ 2X B þ 7Y 1A þ 3Y 1B þ 10Y 2A þ 6Y 2B Þ P d; 17090:1  14741:6 P ðX A P hA Þ P 1  p; P ðX B P hB Þ P 1  p; P ðY 1A þ Y 1B þ Y 2A þ Y 2B 6 QÞ P 1  k; P ðY 1A þ Y 1B 6 P 1 ; Y 2A þ Y 2B 6 P 2 Þ P 1  c; Y 1A þ Y 2A P X A ; Y 1B þ Y 2B P X B ; X A ; X B ; Y 1A ; Y 1B ; Y 2A ; Y 2B P 0: According to [29] and by using Lingo-8, solution is Objective value Variable DELTA XA XB Y1A Y1B Y2A Y2B Z DC ¼ 20856:4; 1.000000 Value 1.000000 500.0000 620.0000 104.1667 395.8333 483.4933 1000.000 Z 0DC ¼ 2740; Z PC ¼ 14741:6: Obtained solution is better than or equal to prior solution and we assume it is a compromise solution. E. Roghanian et al. / Applied Mathematics and Computation 188 (2007) 786–800 799 8. Conclusions In this paper, we considered a ‘‘probabilistic bi-level linear multi-objective programming problem’’ and its application in enterprise-wide supply chain planning problem where (1) market demand, (2) production capacity of each plant and (3) resource available to all plants for each product were random variables and the constraints included joint and disjoint probability distributions. This probabilistic model was first converted into an equivalent deterministic model in each level, to which fuzzy programming technique was applied to solve the multi-objective nonlinear programming problem to obtain a compromise solution. Two simplified example was used to illustrate the process of interaction. Method can be applied to explicit situations by changing certain assumptions to solve the specific problem properly. 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