Use of Linear Programming for Analysis of Integrated Residuals Management at Industrial Plants
We mentioned in the body of this chapter that linear programming offers a particularly powerful analytical tool for the analysis of REQM problems at the individual activity level. There are two reasons for this: (1) it provides an explicit and efficient optimum-seeking procedure; and (2) once formulated, results from changing variables of interest can be calculated very rapidly. This appendix sketches a linear programming model used in the petroleum refining and steel industry studies discussed in the chapter proper. Familiarity with basic linear programming is assumed.
A first problem, of course, is that the functions involved are inherently nonlinear. Thus if one wishes to capture them fully, the resulting model is complex and difficult to solve even for simple situations. To avoid this problem, the approach used in constructing working models was to attempt to identify only some relatively small set of discrete production alternatives and to structure these in the form of unit activity vectors, that is, vectors giving inputs and costs required for the processing of one unit of an input or production of one unit of output of interest. The objective of the firm may be taken to be profit maximization, cost minimization for given output, or any other convenient variant. The constraint set may include limits on input availability; product mix; quality requirements to be met by products; limits on discharges of one or more residuals; and, most important, continuity conditions (or mass and energy balance equations) requiring, for example, that the full amount of each residual generated be accounted for explicitly, either by intake to a modification or transport process or by discharge.
To say that once a quantity of residual is generated in the model, it must be modified, transported, or discharged is not to say that all residuals must necessarily be included in the model. For many purposes, for example, the carbon dioxide and water residuals from combustion processes will not be of interest and may be ignored in the construction of a response model. Which residuals are of interest will depend to a large extent on the spatial and time dimensions of the study, on the particular REQM problem being investigated, or both.
AUTHORS' NOTE: This appendix is adapted from Clifford S. Russell, Residuals Management in Industry: A Case Study of Petroleum Refining ( Baltimore, Johns Hopkins University Press for Resources for the Future, 1973).