CONSTRUCTING THE SET OF EFFICIENT OBJECTIVE VALUES IN LINEAR MULTIPLE-OBJECTIVE TRANSPORTATION PROBLEMS

A forward dynamic programming approach is utilized to find an algebraic representation for a polyhedron in objective space associated with a multiple objective transportation problem having k linear objectives. This polyhedron has the same efficient structure as the set of all feasible objective values, and moreover all of its vertices are efficient. The algebraic representation of this polyhedron is of the form {y is-an-element-of R(k): Hy greater-than-or-equal-to Ua + Vd}, where the matrices H, U and V are independent of the vector a of availabilities and vector d of demands. The procedure is illustrated by the numerical example of Isermann [H. Isermann, ''The enumeration of all efficient solutions for a linear multiple-objective transportation problem'', Naval Research Logistics Quarterly 26 (1979) 123-139].