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Additional info for Approaches to Integer Programming
N −1. 88) of the follower from j = 1 to j = N in turn. 88) is the Stackelberg solution. 88) is solved. 90) Ax + By ≤ b ⎪ ⎪ ⎭ y ≥ 0, uT ≥ 0, vT ≥ 0, where u is an m dimensional row vector and v is an n2 dimensional row vector. 91) ⎪ ⎪ Ax + By ≤ b ⎪ ⎪ ⎭ x ≥ 0, y ≥ 0, uT ≥ 0, vT ≥ 0. 91), v is eliminated and the equality constraint u(Ax + By − b) − vy = 0 is transformed into u(b − Ax − By) + (uB + d2 )y = 0. 92) implies that b − Ax − By ≥ 0, uT ≥ 0, (uB+d2 )T ≥ 0, y ≥ 0. Let Ai and Bi be the ith row vector of the matrix A and the matrix B, respectively, and let B j and d2 j be the jth column vector of the matrix B and the jth element of the vector d2 , respectively.
In this section, we consider a case where there are two DMs; one of the DMs ﬁrst makes a decision, and then after acknowledging the decision of the ﬁrst DM, the other DM chooses a decision. Such a situation is formulated as a two-level programming problem. A linear programming problem with two DMs is formulated as follows. For the sake of simplicity, we call the two DMs DM1 and DM2 in this subsection. Let x1 and x2 denote the column vectors of the decision variables of DM1 and DM2, respectively, and let z1 (x1 , x2 ) = c11 x1 + c12 x2 and z2 (x1 , x2 ) = c21 x1 + c22 x2 denote the objective functions of DM1 and DM2, respectively, where ci1 , i = 1, 2 are n1 dimensional coefﬁcient row vector, and ci2 , i = 1, 2 are n2 dimensional coefﬁcient row vector.
For two-level linear programming problems with cooperative DMs, fuzzy programming approaches have been developed (Lai, 1996; Shih, Lai and Lee, 1996; Sakawa, Nishizaki and Uemura, 1998; Sakawa and Nishizaki, 2009). In fuzzy programming for two-level linear programming by Sakawa et al. 79) if z1i < zi (x) ≤ z0i μi (zi (x)) = 1 − z0 ⎪ z ⎪ i i ⎪ ⎩0 if zi (x) > z0i . 80) for DM2 ⎪ subject to Ax ≤ b ⎪ ⎪ ⎭ x ≥ 0. , n = n1 + n2 . Because the two DMs make decisions cooperatively, the decision variable vector is represented simply by x without partition.