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A Noniterative Algorithm for the Linear-Quadratic Profit-Maximization Model for Smoothing Multiproduct Production

In multiproduct-production and workforce-smoothing problems, the objective is to determine the levels of employment and production for each product in each period that will maximize the total profit over a planning horizon of N periods given quadratic revenue and cost functions for M products. The planned levels of production, in turn, determine the planned levels of inventory or backorders for each product in each period. A real-world problem may involve 20,000 products and 12 periods, leading to over 480,000 variables. This requires solving simultaneous equations with as many variables. We exploited the staircase structure of optimality conditions to develop an algorithm that requires solving, without iterations, simultaneous equations with only three variables. The computation time, regardless of the number of variables in the model, is of the order of one second on a Vax 11/780. This algorithm also facilitates sensitivity analysis and generation of alternate plans for production, the workforce, and sales.