Duality in Linear Programming 4

Duality in Linear Programming4In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimalsimplex multipliers is a very useful concept. First, these shadow prices give us directly the marginal worthof an additional unit of any of the resources. Second, when an activity is priced out using these shadowprices, the opportunity cost of allocating resources to that activity relative to other activities is in Linear Programming is essentially a unifying theory that develops the relationships between agiven Linear program and another related Linear program stated in terms of variables with this shadow-priceinterpretation. The importance of Duality is twofold. First, fully understanding the shadow-price interpretationof the optimal simplex multipliers can prove very useful in understanding the implications of a particularlinear- Programming model. Second, it is often possible to solve the related Linear program with the shadowprices as the variables in place of, or in conjunction with, the original Linear program, thereby taking advantageof some computational efficiencies.

However, these shadow prices for the constraints of (2) are exactly the optimal values of the decision variables of the firm’s allocation problem. Hence, in solving the dual (2) by the simplex method, we apparently have solved the primal (1) as well. As we will see later, this will always be the case since ‘‘the

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  Allocation, Optimal

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