Mathematical Programming


Most rapidly developing fields are thought to contain at least two types of persons, those who are “with it” and those who are not. Managerial economics is no exception to this very general rule.
Those who are considered to be “with it” in managerial economics, or operations research as it is often called today, can be identified in part by their age (youth is thought to be a necessary, if not a sufficient, condition for vitality) and in part by their facility with computers and predilection for applied, operational problems. Other characteristics that set the new breed of operations research specialists apart from more conventional managerial economists include an almost evangelistic elan, a belief in the quantifiability of social, economic and technological relationships, relatively strong mathematical backgrounds, and a contempt for the paraphernalia that surrounds, if not for the concepts that underline, classical economic theory. Terms such as “marginal revenue productivity,” “opportunity cost,” and “returns to scale” seldom occur in the professional jargon of operations researchers, although other terms that embody identical concepts are essential elements of their growing professional vocabularly.




Economists have played a prominent role in developing many of the operational techniques that are so familiar to operations research as we know it today—and with good reason. For whatever its application, operations research finds its raison d’etre in the optimal allocation of scarce resources. Whether one’s objective is to maximize a strategic nuclear capability, the profitability of a manufacture’s product mix, or to minimize the number of highway fatalities per year, the cost of transporting goods from factories to warehouses, or of blending a wide variety of materials to create animal feeds or gasolines or sausages, the problems remain the same. Scarce resources committed to one use necessarily are withdrawn from another, and benefits obtained from one output may be enjoyed only at the cost of benefits foregone from others. Management’s function, again, is to choose from among the (perhaps infinite) set of attainable input-output combinations, or marginal tradeoffs, that which is, in some sense, most desirable. Accordingly, an operations research specialist’s function is to employ his analytic skills to assist responsible decision makers in the often-difficult tasks of:


1. Accurately defining and quantifying the set of outputs attainable from an organization’s (necessarily limited) pool of productive resources;
2. Quantifying to the extent possible, the organization’s, or decision maker’s, criteria for choice among these outputs; and
3. Deriving from 1. and 2. the operational implications of the firm’s objectives and output possibilities by solving for an optimal solution to its resource allocation problem.




Operations research, then, shares the classical economist’s emphasis on optimization but adds to this an emphasis on the quantification of underlying structural relationships, resource limitations, managerial objectives (or choice criteria), and solves for their operational implications in very precise and practical terms—so many units of X and so many units of Y should be used to maximize the profitability (or minimize the cost) of producing units of good A and b units of B in such-and-such a fashion. In addition, a creative analyst will attempt to measure the sensitivity of this problem’s solution to the many assumptions that have gone into its specification. How important is it to an organization, for example, to employ exactly X and Y units of resources to produce goods A and B in exactly the specified numbers? How much would be lost by producing a little less of A and correspondingly more of B? Or how much more could be obtained by relaxing one or more of the (physical, institutional, or financial) restrictions built into a particular problem’s solution? Information of this sort, concerning changes in the ground rules under which resource allocation decisions take place, often are of greater use to creative managements than specific solutions for a specific, short range production, inventory transportation problem. Mathematical programming is one of the basic tools that the operations researcher or modern managerial economist uses in meeting these responsibilities, as the greatly simplified but nevertheless typical problem in operations management presented in the next section serves to illustrate.