A bi-level multi-choice programming problem

Transcription

1 Int. J. Mathematics in Operational Research, Vol. 7, No. 1, A bi-level multi-choice programming problem Avik Pradhan and M.P. Biswal* Department of Mathematics, Indian Institute of Technology, Kharagpur , India avik02iitkgp@gmail.com mpbiswal@maths.iitkgp.ernet.in *Corresponding author Abstract: A Bi-level linear programming problem is treated as a multi-objective optimisation problem where the decision is taken by two different decision makers who are at two different levels. In this paper we consider a bi-level linear programming problem where some of the cost coefficient of the objectives, and some of the right hand side parameters of the constraints are multi-choice parameters. The aim of this paper is to establish a suitable solution procedure to solve the stated bi-level programming problem. To tackle the multi-choice parameters of the bi-level programming problem, we use some interpolating polynomials. Multi-choice parameters are replaced with interpolating polynomials. Then we use fuzzy programming method to solve the transformed bi-level programming problem. We present a numerical example to illustrate the solution procedure of the bi-level linear programming problem involving some multi-choice parameters. Keywords: bi-level programming; interpolating polynomial; fuzzy programming; multi-choice programming; MSP. Reference to this paper should be made as follows: Pradhan, A. and Biswal, M.P. (2015) A bi-level multi-choice programming problem, Int. J. Mathematics in Operational Research, Vol. 7, No. 1, pp Biographical notes: Avik Pradhan received his MSc from Indian Institute of Technology, Madras, India in Currently, he is working as a Research Scholar at the Department of Mathematics, Indian Institute of Technology, Kharagpur, India, under the supervision of Professor M.P. Biswal. His area of research is operations research. M.P. Biswal received his MSc in Mathematics from Utkal University, Odisha, India. He received his PhD in Mathematical Programming from Indian Institute of Technology, Kharagpur, West Bengal, India. Currently, he is a Professor of Mathematics, at Department of Mathematics, Indian Institute of Technology, Kharagpur, India. His research interest is multi-objective programming, multi-choice programming, stochastic-optimisation, numeric optimisation and computational statistics. Copyright 2015 Inderscience Enterprises Ltd.

2 2 A. Pradhan and M.P. Biswal 1 Introduction In real life, there are some decision making problems where two or more decision makers (DMs) are there at different levels to take decision, e.g., in a two player non-zero sum game there are two player as DM and they are at two different level, i.e., second player is at lower level and first player is at upper level. This type of hierarchical decision making problem, where two levels are there, is called a bi-level decision making problem. The formal mathematical formulation of the bi-level programming problem was studied by Fortuni-Amat and McCarl (1981) and Candler and Townsley (1982). Bi-level programming problem has some following properties (Shih et al., 1996): 1 the DMs take decision interactively within a predominantly hierarchical structure 2 the decision are made sequentially from upper level to lower level 3 each level DM independently imise its owns net benefits, but their action are affected by the action of the other DM 4 these affect can be reflected in both the objective and the feasible space. In bi-level programming the decision are made in the following way, first upper-level DM or the leader set his/her goal or decision and then ask the lower level DM or the follower to give his/her optimal solution which has to be calculated in isolation; after getting the optimal solution the follower submit it to the leader; then leader will modify the solution under the overall benefit of the organisation; this process will continue until DMs get a satisfactory solution (Bard, 1983; Burton, 1977). Most of the real life case studies of bi-level programming can be found in export-import analysis of business, agriculture, government policy, economic systems, finance, warfare, transportation, network designs, and is especially suitable for conflict resolution. It has been observed that, the parameters of a mathematical programming problem which forms the parameter space of the problem may be of multi-choice type, i.e., there may exist a set of choices for a parameter, out of which only one is to be selected to optimise the objective function. Such type of mathematical programming problem is known as multi-choice programming (MCP) (Biswal and Acharya, 2011). MCP problem is originated by Healey (1964). Multi-choiceness can be occur for any parameter in a mathematical programming problem. In a MCP we have to find a combination of values of the parameters such that it optimise the objective function. MCP is used in real life decision making problem, e.g., selecting a new car, selecting a new security personnel, implementing a new policy for a community, etc. In this paper, we consider a bi-level programming problem whose parametric space contains some multi-choice parameters. After presenting the introduction in Section 1, literature review, mathematical model of the problem, formulation of the interpolating polynomial for multi-choice parameters, fuzzy programming approach to solve bi-level programming problem, a numerical example and conclusions are presented in the subsequent sections of the paper.

3 A bi-level multi-choice programming problem 3 2 Literature review In the last three decades several developments have been done in the field of hierarchical optimisation. After the initial formation of bi-level programming problem by Fortuni-Amat and McCarl (1981) and Candler and Townsley (1982), further research has been done in several directions. Candler and Townsley (1982) has proposed an implicit search algorithm which generates enumerating bases from lower-level activities, but no progress has been made for a large system. Then Bialas and Karwan (1982, 1984) has proposed two methods to solve a bi-level programming problem, are known as vertex enumeration method and k-th best method. Anandalingam (1988) discussed multi-level programming problem (MLPP) as well as bi-level decentralised programming problem based on Stackelberg solution procedure. It was Lai (1996) who first used the concept of fuzzy set theory to multi-level programming problem. After Lai (1996) fuzzy set theory concept for multi-level programming, Shih et al. (1996) and Shih and Lee (2000) extended his concept by introducing non-compensatory -min aggregation operator and the compensatory fuzzy operator respectively for MLPP. After the development of the fuzzy programming approach for hierarchical optimisation problem fuzzy goal programming approach has been developed by Pal and Moitra (2003) and Pramanik and Roy (2007). Then it has been used to solve quadratic bi-level multi-objective programming problem (Pramanik et al., 2011), they also linearise the fuzzy membership function using Taylor series expansion such that the problem remains linear. The fuzzy goal programming approach also has been used to solve bi-level fractional programming problem (Abo-Sinna and Baky, 2010; Dey and Pramanik, 2011). In case of MCP problem, it was Healey (1964) who originate the problem.the problem belongs to a class of combinatorial optimisation problems with a requirement to choose a value from a number of choices, and to find a combination which optimises an objective function subject to a set of constraints. In practice, MCP can be extended as an application of generalised assignment problems, multiple choice knapsack problems, sales resource allocation, multi-item scheduling, timetabling, etc. Chang (2007, 2008) has proposed the formulation of multi-choice goal programming (MCGP). In these problems the DMs have to set multi-choice aspiration levels (MCAL) for each goal. Paksoy and Chang (2010) have applied the revised multi-choice goal programming approach of Chang (2008) to deal with the multi-choice parameters to solve a supply chain network design problem. Liao (2009) follows the method of Chang (2007) to solve the multi-segment goal programming problem. Then Biswal and Acharya (2009) has extended Chang s method to transform MCP problem where right hand side parameters are multi-choice type, to a standard mathematical programming problem. Further Biswal and Acharya (2011) have used interpolating polynomials to tackle the multi-choice parameters of the constraints, and formulated a mixed integer non-linear programming problem. Chang et al. (2012) have studied multi-coefficient goal programming and used Chang (2007) transformation technique to deal with multi-choice parameter. In this paper we consider a bi-level programming problem in which some parameters are multi-choice type. We use interpolating polynomial approach of Biswal and Acharya (2011) to tackle the multi-choice parameters and then use fuzzy programming approach to solve the transformed problem.

4 4 A. Pradhan and M.P. Biswal 3 Mathematical model In a bi-level programming problem there are two DM in two different levels. They are called first level DM or upper-level DM (leader) and second level DM or lower level DM (follower). The decision vector X R n is partitioned between two different DM as R n1 and X 2 R n2, where n 1 + n 2 = n. The decision vectors and X 2 are controlled by the upper level and lower level DM respectively. A bi-level linear programming problem can be presented as: Find X = (x 1, x 2,..., x n ) T so as to subject to where : Z 1 = X 2 : Z 2 = c j x j (1) d j x j (2) a ij x j b i, i = 1, 2, 3,..., m (3) x j 0, j = 1, 2, 3,..., n (4) X 2 = X; = (x 1, x 2,..., x n1 ) T ; X 2 = (x n1 +1, x n1 +2,..., x n ) T (5) In this paper we consider a bi-level linear programming problem where some parameters are multi-choice type. We formulate the mathematical models for four different cases. They are as follows: Case 1: In this case we consider the problem where some or all of the cost coefficient c j and d j (j = 1, 2,..., n) are multi-choice type. The model is given by: subject to : Z 11 = X 2 : Z 12 = {c (1) j, c (2) j,..., c (k j) j }x j (6) {d (1) j, d (2) j,..., d (l j) j }x j (7) a ij x j b i, i = 1, 2, 3,..., m (8)

5 A bi-level multi-choice programming problem 5 x j 0, j = 1, 2, 3,..., n. (9) where k j and l j are number of alternative choices for the j th parameter c j (j = 1, 2,..., n) and d j (j = 1, 2,..., n) respectively. Case 2: In this case, we consider a bi-level programming problem where some of the technological coefficients a ij (i = 1, 2,..., m; j = 1, 2,..., n) are multi-choice type. The model is given by: subject to : Z 21 = X 2 : Z 22 = c j x j (10) d j x j (11) {a (1) ij, a(2) ij,..., a(pij) ij }x j b i, i = 1, 2, 3,..., m (12) x j 0, j = 1, 2, 3,..., n. (13) where number of alternatives for the technological coefficient a ij is p ij i, j. Case 3: In this case, we consider a bi-level programming problem in which the resource level of the constraints b i (i = 1, 2,..., m) are considered as multi-choice parameter. The model is given by: subject to : Z 31 = X 2 : Z 32 = c j x j (14) d j x j (15) a ij x j {b (1) i, b (2) i,..., b (ri) i }, i = 1, 2, 3,..., m (16) x j 0, j = 1, 2, 3,..., n. (17) where r i is the number of alternative choices available for i th resource level b i (i = 1, 2,..., m).

6 6 A. Pradhan and M.P. Biswal Case 4: Lastly, we consider a bi-level programming problem whose some of the parameters c j, d j (j = 1, 2,..., n), a ij (i = 1, 2,..., m; j = 1, 2,..., n), b i (i = 1, 2,..., m) are multi-choice type. The mathematical model is given by: subject to : Z 41 = X 2 : Z 42 = {c (1) j, c (2) j,..., c (k j) j }x j (18) {d (1) j, d (2) j,..., d (lj) j }x j (19) {a (1) ij, a(2) ij,..., a(p ij) ij }x j {b (1) i, b (2) i,..., b (r i) i }, i = 1, 2, 3,..., m (20) x j 0, j = 1, 2, 3,..., n. (21) In all these four cases, some of the parameters of the problem are multi-choice type. For each multi-choice parameter the feasible region will be different, e.g., if we consider a bi-level multi-choice linear programming problem in which two parameters are multi-choice type, for first parameter suppose there are p alternative choices and for second parameter q alternatives are there, then we have to solve pq different bi-level programming problem to obtain the optimal solution. Due to the presence of the multi-choice parameter, the problem cannot be solved directly. Further the bi-level programming problem has to be solved for two different DM. In the rest of the paper we discuss about the method to solve these type of problem. We use Lagrange s interpolating polynomial for the multi-choice parameters. 4 Interpolating polynomial for the multi-choice parameters To solve the bi-level multi-choice linear programming problem at first we take care of the multi-choice parameter. Chang (2007, 2008), Liao (2009) and Biswal and Acharya (2009, 2011) has introduced methods to transform the problem containing multi-choice parameter to a mixed integer programming problem. We apply the method of interpolating polynomial approach Biswal and Acharya (2011) to tackle the multi-choice parameters. Now we formulate the interpolating polynomial for each of the four cases of the bi-level multi-choice linear programming problem. 4.1 Case 1 To tackle the j th multi-choice parameters c j (j = 1, 2,..., n) in the first model, we introduce an integer variable u j which takes k j number of values (u j = 0, 1,..., k j 1). We formulate a Lagrange interpolating polynomial f cj (u j ) which passes through all the k j number of points (see Table 1).

7 A bi-level multi-choice programming problem 7 Table 1 Data table for multi-choice coefficient c j u j k j 1 f cj (u j ) c (1) j c (2) j c (3) j c (k j ) j Following Lagrange s formula (Atkinson, 2009), we obtain the interpolating polynomial for j th multi-choice parameter c j as: f cj (u j ) = (u j 1)(u j 2) (u j k j + 1) c (1) ( 1) (kj 1) j (k j 1)! + u j(u j 2) (u j k j + 1) c (2) ( 1) (kj 2) j (k j 1)! + u j(u j 2)(u j 3) (u j k j + 1) c (3) ( 1) (kj 3) j + (22) 2!(k j 3)! + u j(u j 1)(u j 2) (u j k j + 2) c (k j) j, j = 1, 2,..., n. (k j 1)! Similarly, for the j th multi-choice parameters d j (j = 1, 2,..., n), we introduce an integer variable v j which takes l j number of values (v j = 0, 1,..., l j 1). We formulate a Lagrange interpolating polynomial f dj (v j ) which passes through all the l j number of points given by Table 2. The polynomial is given by: f dj (v j ) = (v j 1)(v j 2) (v j l j + 1) d (1) ( 1) (lj 1) j (l j 1)! + v j(v j 2) (v j l j + 1) d (2) ( 1) (lj 2) j (l j 1)! + v j(v j 2)(v j 3) (v j l j + 1) d (3) ( 1) (lj 3) j + (23) 2!(l j 3)! + v j(v j 1)(v j 2) (v j l j + 2) d (l j) j, j = 1, 2,..., n. (l j 1)! Table 2 Data table for multi-choice coefficient d j v j l j 1 f dj (v j ) d (1) j d (2) j d (3) j d (l j ) j After transforming the multi-choice parameters c j and d j (j = 1, 2,..., n), we obtain a mixed integer bi-level programming problem. Hence the transformed mixed integer bi-level programming problem for (6) to (9) is given by: : Z 11 = X 2 : Z 12 = f cj (u j )x j (24) f dj (v j )x j (25)

8 8 A. Pradhan and M.P. Biswal subject to a ij x j b i, i = 1, 2, 3,..., m (26) x j 0; 0 u j k j 1; 0 v j l j 1; u j, v j N 0, j = 1, 2, 3,..., n. (27) where N 0 is the set of natural numbers including zero. 4.2 Case 2 In the second case, to tackle the multi-choice parameter a ij we introduce an integer variable w ij which takes p ij number of different values(w ij = 0, 1,..., p ij 1). An interpolating polynomial f aij (w ij ) which passes through all the p ij number of points (see Table 3). Table 3 Data table for multi-choice coefficient a ij w ij p ij 1 f aij (w ij ) a (1) ij a (2) ij a (3) ij a (p ij ) ij The interpolating polynomial can be given by: f aij (w ij ) = (w ij 1)(w ij 2) (w ij p ij + 1) a (1) ( 1) (pij 1) ij (p ij 1)! + w ij(w ij 2) (w ij p ij + 1) a (2) ( 1) (pij 2) ij (p ij 1)! + w ij(w ij 2)(w ij 3) (w ij p ij + 1) a (3) ( 1) (pij 3) ij + (28) 2!(p ij 3)! + w ij(w ij 1)(w ij 2) (w ij p ij + 2) a (pij) ij, (p ij 1)! i = 1, 2,..., m; j = 1, 2,..., n. Hence, the transformed mixed integer bi-level programming problem for (10) to (13) is given by: Find X = (x 1, x 2,..., x n ) so as to : Z 21 = X 2 : Z 22 = c j x j (29) d j x j (30)

9 subject to A bi-level multi-choice programming problem 9 f aij (w ij )x j b i, i = 1, 2, 3,..., m (31) x j 0; 0 w ij p ij 1; w ij N 0 ; i = 1, 2,..., m; j = 1, 2, 3,..., n. (32) 4.3 Case 3 In the third model we consider b i (i = 1, 2,..., m) as a multi-choice parameter. We introduce a new integer variable q i for the multi-choice parameter b i and formulate an interpolating polynomial f bi (q i ) which passes through all the r i number of points (see Table 4). Table 4 Data table for multi-choice coefficient b i q i k j 1 f bi (q i) b (1) i b (2) i b (3) i b (r i) i The interpolating polynomial can be formulated as: f bi (q i ) = (q i 1)(q i 2) (q i r i + 1) b (1) ( 1) (ri 1) i (r i 1)! + q i(q i 2) (q i r i + 1) b (2) ( 1) (ri 2) i (r i 1)! + q i(q i 2)(q i 3) (q i r i + 1) b (3) ( 1) (ri 3) i + (33) 2!(r i 3)! + q i(q i 1)(q i 2) (q i r i + 2) b (r i) i, i = 1, 2,..., m. (r i 1)! Therefore, the transformed mixed integer bi-level programming problem is given by: subject to : Z 31 = X 2 : Z 32 = c j x j (34) d j x j (35) a ij x j f bi (q i ), i = 1, 2, 3,..., m (36) x j 0; 0 q i r i 1; q i N 0 i = 1, 2, 3,..., m. (37)

10 10 A. Pradhan and M.P. Biswal 4.4 Case 4 The last model (Case 4) is a combination of the last three cases, hence we transform the model (18) to (21) using the derived interpolating polynomials. Hence the transformed problem is given by: subject to : Z 41 = X 2 : Z 42 = f cj (u j )x j (38) f dj (v j )x j (39) f aij (w ij )x j f bi (q i ), i = 1, 2, 3,..., m (40) x j 0; (41) 0 u j k j 1; 0 v j l j 1; 0 w ij p ij 1; 0 q i r i 1; (42) u j, v j, w ij, q i N 0 ; i = 1, 2, 3,..., m; j = 1, 2,..., n. (43) The interpolating polynomials f cj (u j ), f dj (v j ), f aij (w ij ) and f bi (q i ) are defined in (22), (23), (28) and (33) respectively. 5 Fuzzy programming approach to bi-level programming In the previous section we have derived mixed integer bi-level programming problem for bi-level MCP problem. Then we solve these bi-level programming problems. Bi-level programming problems have two different DMs. The objective function for both the DM are different and conflicting in nature. So, we find a solution which will satisfy both the DM. Hence we have to obtain a compromise solution. To obtain the compromise solution, we use fuzzy programming approach, introduced by Shih et al. (1996). To formulate the fuzzy programming model for the mixed integer bi-level programming problem we define the membership function for the upper and lower level DM. 5.1 Construction of the fuzzy membership function At first we consider the problem (24) to (27). We formulate the fuzzy programming model. In order to construct the membership function for the DM s objective function, we solve both the DM s problems with their own objective function individually over the same feasible region. The first level DM s problem is to imise the objective function given by (24) over the feasible region given by the constraints (26) and (27). Similarly the follower s problem is to imise the objective function (25) subject

11 A bi-level multi-choice programming problem 11 to the constraints (26) and (27). After solving both the problem we find the optimal solution for the leader and the follower as (Z11; U X1 U, X2 U ; u ) and (Z12; L X1 L, X2 L ; v ) respectively where u = (u 1, u 2,...u n ), v = (v 1, v 2,..., v n ). Now we set the minimum tolerance value for the membership function of the objective function of follower as Z12 U = Z 12 (X1 U, X2 U, v ). For minimum tolerance value of the leader s objective, the leader gives a minimum value for his objective. The worst choice for minimum tolerance is given by Z11 L = Z 11 (X1 L, X2 L, u ), the leader can also change this value. Here we consider the given value Z11 L as the minimum tolerance for the leader s objective. Since the objective function are conflicting in nature, Z11 U Z11 L and Z12 U Z12 L also (X1 U, X2 U, u ) and (X1 L, X2 L, v ) are two different points. The leader control the decision variable, to get the compromise solution for both the DM leader have to give a range for. Let the positive and negative deviation for X1 U be P and N respectively, where P, N R n 1, and they need not be same. Hence we construct the fuzzy membership function for Z 11,Z 12 and as: µ (1) Z 11 (X, u) = µ (1) Z 12 (X, v) = 1, Z 11 Z11 U Z 11 Z L 11, Z Z11 U ZL 11 L < Z 11 < Z11 U 11 0, Z 11 Z11 L 1, Z 12 Z12 L Z 12 Z U 12, Z Z12 L ZU 12 U < Z 12 < Z12 L 12 0, Z 12 Z12 U (44) (45) µ (1) ( ) = { (X U 1 +P ) P, X U 1 X U 1 + P (X U 1 N) N, X U 1 N X U 1 (46) These membership functions are described in Figures 1 to 3 respectively. Figure 1 Linear membership function for first level DM s objective µ (1) Z11 1 O Z L 11 Z U 11 Z 11

12 12 A. Pradhan and M.P. Biswal Figure 2 Linear membership function for second level DM s objective µ (1) Z12 1 O Z U 12 Z L 12 Z 12 Figure 3 Triangular membership function for first level decision variable µ (1) X1 1 O X U 1 N X U 1 X U 1 +P X U Fuzzy programming model We formulate all the required membership functions to construct the fuzzy programming model of the bi-level multi-choice linear programming problem. Let α and β be the minimum acceptable degree of satisfaction for the objective Z 11 and Z 12 respectively. Then we have µ (1) Z 11 α and µ (1) Z 12 β. Let γ be the minimum acceptable degree of satisfaction of the decision variable, then we have µ (1) γ. Let us set λ 1 = min{α, β, γ}. Now we apply -min operator introduced by Bellman and Zadeh (1970) to construct the fuzzy programming model for the problem. The fuzzy programming model is given by, subject to, : λ 1 (47) f cj (u j )x j (Z11 U Z11)λ L 1 Z11 L (48)

13 A bi-level multi-choice programming problem 13 f dj (v j )x j (Z12 L Z12)λ U 1 Z12 U (49) µ (1) λ 1 I (50) a ij x j b i, i = 1, 2, 3,..., m (51) x j 0, 0 u j k j 1; 0 v j l j 1; u j, v j N 0 j = 1, 2, 3,..., n (52) where I R n1 and all of its elements are 1. This is treated as an mixed integer non-linear programming problem. Using any non-linear programming solver, we solve the problem. In the similar manner we formulate the fuzzy programming model for rest of the three cases. Fuzzy programming model of Case 2 [equations (29) (32)] can be formulated as: subject to, : λ 2 (53) c j x j (Z21 U Z21)λ L 2 Z21 L (54) d j x j (Z22 L Z22)λ U 2 Z22 U (55) (X U 1 + P ) λ 2 P (56) λ 2 N (X1 U N) (57) f aij (w ij )x j b i, i = 1, 2, 3,..., m (58) x j 0; 0 w ij p ij 1; w ij N 0 ; j = 1, 2, 3,..., n. (59) Fuzzy programming model of Case 3 [equations (34) (37)] can be formulated as: subject to, : λ 3 (60) c j x j (Z31 U Z31)λ L 3 Z31 L (61) d j x j (Z32 L Z32)λ U 3 Z32 U (62) (X U 1 + P ) λ 3 P (63) λ 3 N (X1 U N) (64) a ij x j f bi (q i ), i = 1, 2, 3,..., m (65) x j 0; 0 q i r i 1; q i N 0 i = 1, 2, 3,..., m. (66)

14 14 A. Pradhan and M.P. Biswal Finally, fuzzy programming model for Case 4 [equations (38) (42)] is given by: : λ 4 (67) subject to, f cj (u j )x j (Z41 U Z41)λ L 4 Z41 L (68) f dj (v j )x j (Z42 L Z42)λ U 4 Z42 U (69) (X1 U + P ) λ 4 P (70) ( λ 4 N) (X1 U N) (71) f aij (w ij )x j f bi (q i ), i = 1, 2, 3,..., m (72) x j 0; 0 u j k j 1; 0 v j l j 1; (73) 0 w ij p ij 1; 0 q i r i 1; u j, v j, w ij, q i N 0 i = 1, 2, 3,..., m; j = 1, 2,..., n. (74) 5.3 Solution procedure bi-level multi-choice linear programming problem Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Establish the interpolating polynomials corresponding to all the multi-choice parameters present in the problem. Formulate the mixed integer bi-level programming problem with the help of the interpolating polynomials, as discussed in Section 4. Find the individual optimal solution for both the DM over the given feasible region, and form the pay-off matrix of the objective functions. Set the minimum and imum tolerance limit for the objective function from the pay-off matrix, and revise the lower tolerance limit for leader objective. Set the positive and negative deviation for the leader s control variables. Build the membership function for both the objective function and leader s control variable as (44),(45) and (46). Construct a suitable fuzzy programming model of the problem. Solve the model to get a satisfactory solution. 6 A numerical example In this Section we present a numerical example to illustrate the method. We consider an example where only cost coefficient are multi-choice type.

15 A bi-level multi-choice programming problem 15 An export company exports three of its products (say A, B, C) to the foreign market. The government of the company s country claims some taxes from the company for these exports. For the first product, i.e., for product A there are demand from three different country s market. But the company wants to sell product A in one of the foreign market. For three different foreign markets, company has to pay different amount of taxes to government for product A, e.g., let name of the three market for product A be M 1, M 2, M 3. If the company want to sell product A in M 1 then company have to pay 2 units tax for a product A exported, for M 2 it is 3 units, and for M 3 it is 1 unit. In each of these three markets the selling price of product A are different, so profit after paying the tax will be different. If the company sells the one unit of product A in these three markets then the profit for the company are 23 units, 22 units, 25 units respectively. Similarly for the products B, company can sell it in four different markets and for product C number of available markets are two. For product B (one unit) the tax and profit for the company for four different markets are given by (3, 22), (4, 20), (2, 25) and (5, 21) respectively. For product C (one unit) tax and profit are given by (1, 24) and (2, 23) respectively for two different foreign markets. The government has the control over the number of unit of product B exported, and remaining are controlled by the company itself. To produce these three products company needs two types of materials and man hour. Also company needs some space for storage. The requirement of materials, space and man hour to produce one unit of each products and their availability are given in the Table 5. Table 5 Requirements Product-A Product-B Product-C Availability Material Material Storage Man hour Solution: Let x 1, x 2, x 3 be the number of unit of product A, product B and product C exported to foreign market respectively. Then the model can be formulated as a bi-level MCP problem. The model is given by: : f 1 = {2, 3, 1}x 1 + {3, 4, 2, 5}x 2 + {1, 2}x 3 x 2 (75) : f 2 = {23, 22, 25}x 1 + {22, 20, 25, 21}x 2 + {24, 23}x 3 x 1,x 3 (76) subject to, x 1 + 2x 2 + x 3 20 (77) 3x 1 + 2x 2 + 4x 3 50 (78) x 1 + x 2 + 2x 3 27 (79) 3x 1 + x 2 + x 3 25 (80) x j 0, j = 1, 2, 3 (81)

16 16 A. Pradhan and M.P. Biswal x 2 is controlled by leader and x 1,x 3 are controlled by the follower. Now introducing interpolating polynomial for multi-choice type cost coefficient we present the model as: : f 1 = (2 + 5z 11 x 2 2 3z )x 1 + (3 + 31z (1 + z 13 )x 3 11z : f 2 = (23 3z z 2 x 1,x 3 21)x 1 + (22 65z z (24 z 23 )x 3 subject to, + 4z )x 2 (82) 8z )x 2 (83) x 1 + 2x 2 + x 3 20 (84) 3x 1 + 2x 2 + 4x 3 50 (85) x 1 + x 2 + 2x 3 27 (86) 3x 1 + x 2 + x 3 25 (87) x j 0, j = 1, 2, 3 (88) 0 z i1 2; 0 z i2 3; 0 z i3 1; i = 1, 2. (89) z ij I; i = 1, 2; j = 1, 2, 3. (90) This problem is treated as a mixed integer non-linear programming problem. We use Lingo 11.0 (Schrage, 2008) to obtain the solution for both level DM individually. The solution for leader is given by f1 U = 53 at (x, z 1 ) = (6, 7, 0, 1, 3, 1). The follower s solution is given by f2 L = at (x, z 2 ) = (4.615, 4.231, 6.923, 2, 2, 0). After constructing the pay-off matrix we set f1 L = 20, f2 U = 279. The decision variable x 2 is controlled by the leader, so the leader set the positive and negative deviation for the variable x 2 as 1 and 3 respectively. By the help of (44), (45) and (46) we construct the membership function for the problem. Then the crisp model for the problem becomes, subject to, : λ (91) (2 + 5z z )x 1 + (3 + 31z (1 + z 13 )x 3 20λ (23 3z z21)x (22 65z (24 z 23 )x λ z z z )x 2 (92) 8z )x 2 (93) x 2 + λ 8 (94) x 2 3λ 4 (95) x 1 + 2x 2 + x 3 20 (96) 3x 1 + 2x 2 + 4x 3 50 (97) x 1 + x 2 + 2x 3 27 (98)

17 A bi-level multi-choice programming problem 17 3x 1 + x 2 + x 3 25 (99) x j 0, j = 1, 2, 3 (100) 0 λ 1 (101) 0 z i1 2; 0 z i2 3; 0 z i3 1; i = 1, 2. (102) z ij I; i = 1, 2; j = 1, 2, 3. (103) Solving this problem using Lingo 11.0, we obtain the solution as λ =.6456, (f 1, f 2 ) = ( , ), (x 1, x 2, x 3 ) = (5.468, 5.937, 2.658). But the leader choose the second market for product A whereas the follower choose the third market to sell the product A, same case arises for product B, but one product can be sold in one market. So, leader and follower have to choose the same market for one product. To do this we set z 1j = z 2j, j = 1, 2, 3 and reformulate the problem. Then using Lingo 11.0 Schrage (2008) we obtain the solution as λ = 0.509, (f 1, f 2 ) = (36.8, ), (x 1, x 2, x 3 ) = (5.3, 5.6, 3.5). For product A third market, for product B fourth market and for product C first market is chosen by both the DM. 7 Conclusions In this paper we consider a bi-level linear programming problem, where some of the parameters are multi-choice type. We replace the multi choice parameters by interpolating polynomials. Instead of using these interpolating polynomials, one can use binary variables for the transformation, but the size of the transformed problem becomes larger due to the presence of more number of binary variables. Acknowledgements The authors thank the referees for their valuable comments which have improved the presentation of the paper. References Abo-Sinna, M.A. and Baky, I.A. (2010) Fuzzy goal programming procedure to bilevel multiobjective linear fraction programming problems, International Journal of Mathematics and Mathematical Sciences, 01 15, ID , doi: /2010/ Anandalingam, G. (1988) A mathematical programming model of decentralized multi-level systems, Journal of the Operational Research Society, Vol. 39, No. 11, pp Atkinson, K.E. (2009) An Introduction to Numerical Analysis, 2nd ed., John Wiley & Sons, Inc. Bard, J.F. (1983) Coordination of a multidivisional organization through two levels of management, Omega, Vol. 11, No. 5, pp Bard, J.F. (1984) Optimality conditions for the bilevel programming problem, Nav. Res. Logist. Q., Vol. 31, No. 1, pp Bellman, R. and Zadeh, L.A. (1970) Decision-making in a fuzzy environment, Mgmt Sci., Vol. 17, No. 4, pp.b Bialas, W.F. and Karwan, M.H. (1982) On two-level optimization, IEEE Trans. Autom. Control, Vol. AC-27, No. 1, pp

18 18 A. Pradhan and M.P. Biswal Bialas, W.F. and Karwan, M.H. (1984) Two-level linear programming, Mgmt Sei., Vol. 30, No. 8, pp Biswal, M.P. and Acharya, S. (2009) Transformation of multi-choice linear programming problem, Applied Mathematics and Computation, Vol. 210, No. 1, pp Biswal, M.P. and Acharya, S. (2011) Solving multi-choice linear programming problems by interpolating polynomials, Mathematical and Computer Modelling, Vol. 54, Nos. 5 6, pp Burton, R.M. (1977) The multilevel approach to organizational issues of the firm a critical review, Omega, Vol. 5, No. 4, pp Candler, W. and Townsley, R. (1982) A linear bilevel programming problem, Computers & Operations Research, Vol. 9, pp Chang, C-T., Chen, H-M. and Zhuang, Z-Y. (2012) Multi-coefficients goal programming, Computers & Industrial Engineering, Vol. 62, No. 2, pp Chang, C-T. (2007) Multi-choice goal programming, Omega The International Journal of Management Science, Vol. 35, No. 4, pp Chang, C-T. (2008) Revised multi-choice goal programming, Applied Mathematical Modelling, Vol. 32, No. 12, pp Dey, P.P. and Pramanik, S. (2011) Goal programming approach to linear fractional bilevel programming problem based on Taylor series approximation, International Journal of Pure and Applied Sciences and Technology, Vol. 6, No. 2, pp Fortuni-Amat, J. and McCarl, B. (1981) A representation and economic interpretation of a two-level programming problem, Journal of Operational Research Society, Vol. 32, pp Healey, W.C. (1964) Multiple choice programming, Operations Research, Vol. 12, pp Lai, Y.J. (1996) Hierarchical optimization: a satisfactory solution, Fuzzy Sets and Systems, Vol. 77, No. 3, pp Liao, C-N. (2009) Formulating the multi-segment goal programming, Computers & Industrial Engineering, Vol. 56, No. 1, pp Paksoy, T. and Chang, C.T. (2010) Revised multi-choice goal programming for multi-period, multi-stage inventory controlled supply chain model with popup stores in Guerrilla marketing, Applied Mathematical Modelling, Vol. 34, No. 11, pp Pal, B.B. and Moitra, B.N. (2003) A fuzzy goal programming procedure for solving quadratic bilevel programming problems, International Journal of Intelligence Systems, Vol. 18, No. 5, pp Pramanik, S. and Roy, T.K. (2007) Fuzzy goal programming approach to multi-level programming problem, European Journal of Operational Research, Vol. 176, No. 2, pp Pramanik, S., Dey, P.P. and Giri, B.C. (2011) Fuzzy goal programming approach to quadratic bi-level multi-objective programming problem, International Journal of Computer Applications, Vol. 29, No. 6, pp Schrage, L. (2008) LINGO Release 11.0, LINDO System, Inc. Shih, H.S. and Lee, E.S. (2000) Compensatory fuzzy multiple level decision making, Fuzzy Sets and Systems, Vol. 114, No. 1, pp Shih, H-S., Lai, Y-J. and Lee, E.S. (1996) Fuzzy approach for multi-level programming problems, Comput. Oper. Res., Vol. 23, No. 1, pp