The Coe ‰ cients in an Allocation Problem

The allocation or assignment problem is one of the more important problems of management science. It may be stated very briefly: We are given a system with a number of vacant positions and an equal number of available parts. We know how well each part performs in each position; we wish to assign the parts to the positions so that system performance is optimized. Applications range far and wide, from employment to aircraft assignment to naval overhaul programs. The computational aspects of the problem have been solved, under the assumption that a numerical value is associated with each assignment and that the value of the system is given by the sum of the values of the individual assignments [1,2]. The crux of the problem, therefore, becomes the finding of values to use for the individual assignments. Most of the prior literature has either ignored this problem or failed to reach a definitive conclusion. In certain applications it may be possible to determine values in a unique natural way; the transportation problem is an example [1]. But in most cases the problem is much more complex, and there is no natural utility (like dollar cost in the transportation case) available. The employment problem is a case in point. Here we have a number of vacancies and a number of candidates, and we wish to assign the candidates to the vacancies. The problem is of the utmost importance; all organizations but the smallest are plagued by it. It has been suggested that, as a first approximation, we classify each candidate as either ‘‘suitable’’ or ‘‘unsuitable’’ for a job and solve the problem by using the values 1 and 0, respectively. This approach is manifestly unrealistic; it is probably worse than letting the personnel o‰ce struggle along as best it can, using nonnumerical subjective evaluation of the candidates and the jobs. The technique presented in this paper was originally developed for the problem of allocating major electronic equipment to Naval Ships [3,4]. For concreteness, we will refer to this application throughout the sequel, except in the last section, where we will return to the general case. The technique is constituted so as to take advantage of any special internal relationships that might exist between the positions or the parts. Thus it will vary somewhat from application to application; and even for a given application, several variations may be possible. In essence, though, it is completely general; the same basic ideas apply to all allocation