Duality - Duality in Linear Programming 4 In the preceding...

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Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow prices give us directly the marginal worth of an additional unit of any of the resources. Second, when an activity is ‘‘priced out’’ using these shadow prices, the opportunity cost of allocating resources to that activity relative to other activities is determined. Duality in linear programming is essentially a unifying theory that develops the relationships between a given linear program and another related linear program stated in terms of variables with this shadow-price interpretation. Theimportanceofdualityistwofold. First, fullyunderstandingtheshadow-priceinterpretation of the optimal simplex multipliers can prove very useful in understanding the implications of a particular linear-programming model. Second, it is often possible to solve the related linear program with the shadow prices as the variables in place of, or in conjunction with, the original linear program, thereby taking advantage of some computational efficiencies. The importance of duality for computational procedures will become more apparent in later chapters on network-flow problems and large-scale systems. 4.1 A PREVIEW OF DUALITY We can motivate our discussion of duality in linear programming by considering again the simple example given in Chapter 2 involving the firm producing three types of automobile trailers. Recall that the decision variables are: x 1 = number of flat-bed trailers produced per month, x 2 = number of economy trailers produced per month, x 3 = number of luxury trailers produced per month. The constraining resources of the production operation are the metalworking and woodworking capacities measured in days per month. The linear program to maximize contribution to the firm’s overhead (in hundreds of dollars) is: Maximize z = 6 x 1 + 14 x 2 + 13 x 3 , subject to: 1 2 x 1 + 2 x 2 + x 3 24 , x 1 + 2 x 2 + 4 x 3 60 , (1) x 1 0 , x 2 0 , x 3 0 . After adding slack variables, the initial tableau is stated in canonical form in Tableau 1. In Chapter 2, the example was solved in detail by the simplex method, resulting in the final tableau, repeated here as Tableau 2. 130

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4.1 A Preview of Duality 131 Tableau 1 Basic Current variables values x 1 x 2 x 3 x 4 x 5 x 4 24 1 2 2 1 1 x 5 60 1 2 4 1 ( - z ) 0 6 14 13 Tableau 2 Basic Current variables values x 1 x 2 x 3 x 4 x 5 x 1 36 1 6 4 - 1 x 3 6 - 1 1 - 1 1 2 ( - z ) - 294 - 9 - 11 - 1 2 As we saw in Chapter 3, the shadow prices, y 1 for metalworking capacity and y 2 for woodworking capacity, can be determined from the final tableau as the negative of the reduced costs associated with the slack variables x 4 and x 5 . Thus these shadow prices are y 1 = 11 and y 2 = 1 2 , respectively.
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