Volume 5, Number 2 OPERATIONS RESEARCH LETI'ERS July 1~6 A COMPARLqON OF SEVERAL DYNAMIC PROC-~,MMING ~$ODELS OF THE MADE-TO-ORDER P~ODLTCTION SITUATION * Norman Keith WOMER Department of Economics. Clemson Onwersily. Clemson. SC 29631. USA Thomas K. G', ]LLEDGE, Jr. and M. Murat TARIMCILAR Department of Quantita, c~ Business Analysis. Louisiana State University. Baton Rouge. LA 70803. USA Received January 1985 Revised April 1986 This paper examines the relationbhips among severa| dynamic V:ogramming models of the made-to-order production situation. The models consider the impacts of both learning an.d production ~rate changes on Qptimal program costs. The comparisons among the models suggest that a type of ¢¢oao~Jc d~,~tit~ links the solutions. The results of this research suggest a method of transformation which may provide a useful means for finding tractabi¢ soimions to similar dynamic programming pmnlems. 1. Introduction Consider the situation where a finn has con- tracted to produce V units in T time periods. In the operations research literature, this is known as the made-to-order production situation. Large weapon systems are examples of made-to-order production, but applications are not restricted Lo governmental projects. This problem has been of interest to eeo,:~,~t~g for a number of years [1,2,6,11,12], but in general this ear|y research was not applications oriented. Most of this research was concerned with testing hypotheses about the economic behavior of the maqe-to-order firm. In recent years a group of applications oriented decision models have been developed for analyz- ing made-to-order production [3,4,5,15,16,171. These models follows directly from the work of Rosen [13] and Washburn [14], but the justifica- tiori for the modeling framework has its origins in the recent work of Muth [8,9]. This research com- pares these new models, and as a result of these comparisons, a more unified modeling framework * Presented at the ORSA/TIMS Joint National Meeting, November 26-28, 1984, Dallas, Texas. for the made-to-order pro.~tuction situation is pro- posed. 2. The models In this section we present two models of the made-to-order firm. In one model production rate is measured at the end of the period, and in the other, rate ~ an average over the time interval. The models are similar, but both are necessary since these models define the notation for the analysis that follows; t_~a~ is, these models ~efine the relationshins among the various parameters that are con=par~ :! throughout the paper. In a recent paper [5], a dynamic programming production model was presented. Thi~ model de- scribed a situation where required resources de- pend on both production rate and cumulative output. The production function for each unit was ,d,--az~,/~(i) s, (1) and the corresponding resource requirement func- tion was --~-~ z, = A-~w;(~) . (2) 016%6377/86/$3.50 © 1986, Elsevier Science Publishers B.V. (No~lb-Holland) 87