In publishing “Interindustry Economics,” Hollis Chenery and Paul Clark have furnished the profession with a uniquely useful work — both a readable textbook for the beginner, and at the same time a systematic treatise for the input-output practitioner. It is unfortunate, however, that the authors have leaned over backwards to find merits in Leontief’s original model as compared with the activity analysis approach to interindustry economics. Activity analysis provides a convenient framework for handling certain kinds of “conceptual” difficulties — import substitution, processing substitution, labor-capital substitution, and the output of by-products — difficulties that have arisen in any of the empirical applications proposed by Chenery and Clark. On balance, their work provides impressive evidence against the presumption that the collection of data for a square matrix is cheaper than for a rectangular one.
This is a proposal for the application of discrete linear programming to the typical job shop scheduling problem — one that involves both sequencing restrictions and also non-interference constraints for individual pieces of equipment. Thus far, no attempt has been made to establish the computational feasibility of the approach in the case of large-scale realistic problems. This formulation seems, however, to involve considerably fewer variables than two other recent proposals and on these grounds may be worth some computer experimentation.
This paper is designed to show how a typical sequential probabilistic model may be formulated in linear programming terms. In contrast with Dantzig and Radner, the time horizon here is an infinite one. For another very closely related study, the reader is referred to a paper by R. Howard.
The essential idea underlying this linear programming formulation is that the “state” variable i and the “decision” variable j are introduced as subscripts to the unknowns xij. These unknowns xij represent the joint probabilities with which the state variable takes on the value of i and the decision variable the value of j. Although the particular application described is a rather specialized one, there seem to be quite a number of other cases where the technique should be an efficient alternative to the functional equation approach of Bellman.
This paper studies the planning problem faced by a machine shop required to produce many different items so as to meet a rigid delivery schedule, remain within capacity limitations, and at the same time minimize the use of premium-cost overtime labor. It different from alternative approaches to this well-known problem by allowing for setup cost indivisibilities.
As an approximation, the following linear programming model is suggested: Let an activity be defined as a sequence of the inputs required to satisfy the delivery requirements for a single item over time. The capacity input coefficients for each such activity may then be constructed so as to allow for all setup costs incurred when the activity is operated at the level of unity or at zero. It is then shown that in any solution to this problem, all activity levels will turn out to be either unity or zero, except for those related to a group of items which, in number, must be equal to or less than the original number of capacity constraints. This result means that the linear programming solution should provide a good approximation whenever the number of items being manufactured is large in comparison with the number of capacity constraints.