Application of waiting line theory to job shop servicing problems



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The problem of assigning the appropriate number of automatic machines to a single worker, sometimes called the machine interference problem, has been known as one of the earliest applications of waiting line theory to manufacturing problems. The purpose of this study is to describe how the waiting line approach may be used to reduce manufacturing costs through improved operator assignment procedures which involve a team of collaborating operators being assigned to a group of automatic machines or independent work stations requiring an operator's servicing work from time to time. The basic waiting line values such as expected machine waiting and running times, and degree of operator utilization for exponentially distributed service times and random calls for service from the machines were calculated on a digital computer, and inclusive tables are presented for wide ranges of collaborating operator assignments. From these waiting line values, the optimum operator assignment which will yield the minimum total cost per unit of output for any given condition was also computed for various servicing situations in the system, and the results were plotted in a form so that the optimum assignment is readily determined for varying size of machine groups and varying ratios of the cost of a machine waiting for service and the cost of an operator. The discussions and figures were presented describing the economy of a larger machine grouo, and the cost of alternative operator assignments relative to the cost of optimum assignment. Since the optimality of the economic decision rule depends on the accuracy of estimating the costs involved in the operator assignment and the parameters showing the average rate of calls for sendee and the servicing rate in the system, the method of determining these variables in practice as well as the effects of their substantial errors in the estimation upon the optimal decision were examined in detail. To demonstrate the applicability of the waiting line model to similar kinds of problems, in particular, the problem of grinding machine requirements In the job shop production center was outlined. A model was developed to determine the optimum number of machines to provide, taking into account the cost of the machinist waiting and the cost associated with excess machine capacity.