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.
In the first part of this paper, the concept of logical reduction is presented Minimal preferred variable inequalities are introduced, and algorithms are given for their calculation. A simple illustrative example is carried along from the start, further examples are provided later. The second part of the paper describes certain properties of the generated logical inequalities. It then explains some of the decreases of computational effort which may be achieved by the use of minimal preferred inequalities and outlines a number of concrete applications with some numerical results. Finally, a number of more recent concepts and results are discussed, among them the notion of “probing” and a related zero-one enumeration code for large scale problems under the extended control language of MPSX/370.
The algorithms of this paper belong to the direct-search or implicit-enumeration type. They compare to the recently proposed algorithm of Efroymson and Ray, as does the mixed integer algorithm proposed by C. E. Lemke and the author to that of Land and Doig. The general plan of procedure is expected to be equally valid for the capacitated plant-location problem and for transshipment problems with fixed charges; with some of the proposed devices more important for these difficult problems than for the simple plant location problem. Considerable computational experience has been accumulated and is discussed at some length. It suggests that additional work on the construction of “adaptive” programs, matching algorithm to data structure, is desirable.
A method of solution for the mixed integer-programming problem is proposed. It is based on an exhaustive search of the integer variables, coupled with an efficient use of the product form of the basis matrix inverse, for the linear programming calculations. In this connection a detailed algorithm for the (0 − 1) problem is developed and experimental results are discussed.
We describe a production allocation problem which was worked on at Frito-Lay and an integer programming model formulated for its solution. In connection with model formulation, we discuss estimation and collection of the required cost coefficients. Finally, we report on the use of two different linear programs, representing the same integer problem, which gave dramatically different running times to solve the integer programs. Some reasons for the improved running times are given.integer programming, distribution, production allocation
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