### A Primal Method for the Assignment and Transportation Problems

M. L. Balinski, R. E. Gomory
1964 Management science
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more » ... tor.org. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science. This paper describes a simple calculation for the assignment and transportation problems which is "dual to" the well-known Hungarian Method. While the Hungarian is a dual method, this method is primal and so gives a feasible assignment at each stage of the calculation. Bounds on the number of steps required for the assignment and transportation problems are given. They are the same as the best bounds known for the Hungarian Method. Introduction Perhaps the best known, most widely used, and most written about method for solving the assignment problem is the "Hungarian Method". Originally suggested by Kuhn [8] in 1955, it has appeared in many variants (e.g., [4], [5], [9], [10], [11]). It provided essential ideas for the early methods used in solving network flow problems [5], it has been extended to solve the transportation problem [51, [11], and it has even been "generalized" to solve the linear programming problem [3]. It is a dual method with a feasible assignment being obtained only at the last computational step. This paper presents a primal method for the assignment and transportation problems which is a method "dual to" the Hungarian Method. Where the Hungarian Method provides at each intermediate computational step a dual feasible vector (U, V) and a corresponding (infeasible) primal vector X orthogonal to (U, V), the present method provides at each step a feasible X (a complete assignment or transportation solution) and a corresponding orthogonal (U, V). In addition to the advantage of being primal, and so providing a constantly improving solution, the method seems to be extremely simple to describe, explain, and-it would seem-program. With this method, we are able to bound the number of steps required to solve the assignment and transportation problems. Different bounds can be obtained from different variants of the procedure. The best bounds are n(n + 1)/2 labeling passes for the n X n assignment problem and (Zjcj) min (m, n) passes for the m source n sink transportation problem with total demand of Ej Cj . These bounds are the same as the best known bounds for the Hungarian Method. Remarkably enough, the variant that gives the best bounds requires a special rule of choice very similar to the simplex method's "most negative column" rule.