Turnpike Horizons for Production Planning

A quadratic model for production-inventory planning was made famous by Holt, Modigliani, Muth, and Simon in 1960 in (Holt, C. C., F. Modigliani, J. F. Muth, H. A. Simon. 1960. Planning Production, Inventories, and Work Force. Prentice-Hall, Englewood Cliffs, New Jersey.), especially for its application to a paint factory. A discrete control version of a related quadratic production-inventory model was studied by Kleindorfer, Kriebel, Thompson, and Kleindorfer in (Kleindorfer, P. R., C. H. Kriebel, G. L. Thompson, G. B. Kleindorfer. 1975. Discrete optimal control of production plans. Management Sci. 22 261--273.). In the present paper we solve a continuous version of the model in Kleindorfer, Kriebel, Thompson, and Kleindorfer (Kleindorfer, P. R., C. H. Kriebel, G. L. Thompson, G. B. Kleindorfer. 1975. Discrete optimal control of production plans. Management Sci. 22 261--273.) in complete detail. The reason we are able to obtain a complete solution (which can rarely be done in control models) is that the linear decision rule, which is optimal here as in other quadratic models, permits the elimination of the adjoint function from the state variable equation after one differentiation of the latter. Thus the difficult two-point boundary value problem which usually arises in control problems is converted into an ordinary second order differential equation, which is readily solved. One advantage of having a complete solution to the problem is that it is possible to determine turnpike horizon points. These correspond to zeros of the adjoint function, and have the property that if they are known exactly, then a production-inventory plan which is optimal up to the next horizon point also forms part of the overall optimal plan. In the case of cyclic demand these turnpike horizon points usually occur every half cycle. Similar horizons are likely to exist in real production-inventory problems. A planning procedure for a real problem which extends only as far as a suspected horizon has a good chance of producing an optimal or near optimal solution for that period of time. A second advantage of having the complete solution available is that it is possible to develop a practical production-inventory system which intermingles a prediction procedure (such as the use of a finite Fourier series) with the solution procedure so that a comparison between predicted and actual inventories can be made continuously. Whenever the discrepancy between these two becomes sufficiently large, the model suggests proper corrective actions to be taken.