Some Notes on "Dynamic" Linear Programming

Abstract

The most common application of linear programming in agricultural situations has been to the problem of resource allocation between competing farm activities. Given relevant input-output information for a specific farm, together with real or assumed price and cost patterns, the technique of linear programming enables calculation of the combination of enterprises which maximizes net profit, within the limitations imposed by the availability of farm resources. It is necessary in some linear programming analyses to make explicit allowance for the peculiar influence of time on the structure of the system under study. Of the many ways in which this may be achieved, this article considers four, which have been, or are likely to be, of relevance in an agricultural context: (i) Parametric programming, which allows consideration of resource or price variation between time periods; (ii) extension of the time-span of an activity to cover a series of sequential processes, for example the treatment of rotational sequences as single activities; (iii) the referencing of some resources and/or activities to specific time periods; a common example is the fragmentation of labour supply into months; and (iv) the so-called multi-stage or linear programming where a single matrix is used to describe, in an orderly fashion, a system's structure over a time-span of several periods. It is the latter with which we are primarily concerned here. In its simplest form a dynamic linear programming problem may be set up as a large matrix composed of a series of smaller matrices lying down the diagonal. In its more advanced form allowance can be made for interactions between resources and activities in different periods. In general, dynamic linear programming problems are characterized by large sparse matrices (i.e., matrices in which many coefficients are zero) and usually a block diagonal or block triangular pattern is evident. The size of such matrices is frequently forbidding; however, computational algorithms are available which allow overall solutions to be obtained by solving a series of smaller problems. With the aid of a little ingenuity a great variety of time-dependent restrictions, resources, activities and opportunities can be accounted for in a dynamic linear programming analysis. From an agricultural economist's viewpoint it would not seem extravagant to claim that dynamic linear programming can be used to provide a more adequate analytical description of whole-farm situations over time than most other tools at present available in his kit.