
doi: 10.15807/jorsj.34.58
Summary: \textit{G. Bitran} [Manage. Sci. 26, 694-706 (1980; Zbl 0445.90076)] introduced a notion of efficiency to multiobjective linear programming problem with interval objective functions \(\hbox{Max}\{\Phi x/ x\in\mathbb{R}^ n, Ax=b, x\geq 0\}\), where \(A\) is an \(m\) by \(n\) matrix, \(b\) and \(x\) are respectively \(m\) and \(n\) vectors, \(\Phi\) is a set of \(p\) by \(n\) matrices with components \(c_{ij}\) in the interval \([\ell_{ij},u_{ij}]\), \(i=1,2,\dots,p\) and \(j=1,2,\dots,n\). Bitran pointed out that two kinds of efficient solutions can be considered. The solution concepts for the interval multiobjective linear program are proposed in a manner different to Bitran's approach. First, given a reflective domination relation in the objective space, a strong (nonreflective) domination relation is defined. The domination relations in the decision space are composed from the domination relations in the objective space, and using them the nonreflective domination relations are defined. The properties of nonreflective domination relations are investigated. Next, four concepts of nondominated solutions for the interval multiobjective linear programming problem, i.e. possibly nondominated solutions, necessarily nondominated solutions, strong possibly nondominated solutions and weak necessarily nondominated solutions, are defined using four nonreflective domination relations. The properties of these nondominated solutions are discussed and it is shown that an arbitrary element of each nondominated solution set can be expressed by a convex combination of the corresponding nondominated extreme points. Moreover, the relationships with Bitran's two efficient solutions are clarified. Two of four nondominated solutions are equivalent to Bitran's two efficient solutions and the others are intermediated ones between them.
Linear programming, nondominated solutions, Multi-objective and goal programming, interval multiobjective linear program, domination relations
Linear programming, nondominated solutions, Multi-objective and goal programming, interval multiobjective linear program, domination relations
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