In this paper, we discuss a computationally viable algorithm based on a set-partitioning formulation of the Vehicle Routing Problem (VRP). Implementation strategies based on theoretical as well as empirical results are developed. Some computational results are presented. It is shown that a set-partitioning formulation to the VRP, although well known for a long time, deserves considerable research efforts beyond those we present here.

A unifying survey of the literature related to the knapsack problem; that is. maximize wizj W, zj 2 0, and zi integer; where uj, wt, and Iy are known integers, and w,(i= 1, 2, . . ., A') and W are positive. Various uses. including those in group theory and in other integer programming algorithms. a s well a s applications from the literature. are discussed. Dynamic programming. branch and bound, search enumeration, heuristic meth-' ads. and other solution techniques are presented. Computational experience. and extensions of the knapsack problem, such as to the multi-dimensional case, are also considered.uizi. subject to , I

This paper presents an algorithm for the set-covering problem (that is, min c′y: Ey ≧ e, y ≧ 0, yi integer, where E is an m by n matrix of l's and 0's, and e is an m-vector of l's). The special problem structure permits a rather efficient, yet simple, solution procedure that is basically a (0, 1) search of the single-branch type coupled with linear programming and a suboptimization technique. The algorithm has been found to be highly effective for a good number of relatively large problems. Problems from 30 to 905 variables with as many as 200 rows have been solved in less than 16 minutes on an IBM 360 Model 50 computer. The algorithm's effectiveness stems from an efficient suboptimization procedure, which constructs excellent integer solutions from the solutions to linear-programming subproblems.

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