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# chapter4 - Chapter 4 Duality Given any linear program there...

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Chapter 4 Duality Given any linear program, there is another related linear program called the dual . In this chapter, we will develop an understanding of the dual linear program. This understanding translates to important insights about many optimization problems and algorithms. We begin in the next section by exploring the main concepts of duality through the simple graphical example of building cars and trucks that was introduced in Section 3.1.1. Then, we will develop the theory of duality in greater generality and explore more sophisticated applications. 4.1 A Graphical Example Recall the linear program from Section 3.1.1, which determines the optimal numbers of cars and trucks to build in light of capacity constraints. There are two decision variables: the number of cars x 1 in thousands and the number of trucks x 2 in thousands. The linear program is given by maximize 3 x 1 + 2 . 5 x 2 (profit in thousands of dollars) subject to 4 . 44 x 1 100 (car assembly capacity) 6 . 67 x 2 100 (truck assembly capacity) 4 x 1 + 2 . 86 x 2 100 (metal stamping capacity) 3 x 1 + 6 x 2 100 (engine assembly capacity) x 0 (nonnegative production) . The optimal solution is given approximately by x 1 = 20 . 4 and x 2 = 6 . 5, generating a profit of about \$77 . 3 million. The constraints, feasible region, and optimal solution are illustrated in Figure 4.1. 83

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84 10 20 30 40 10 20 30 40 truck assembly engine assembly metal stamping car assembly feasible solutions trucks produced (thousands) cars produced (thousands) optimal solution Figure 4.1: The constraints, feasible region, and optimal solution of the linear program associated with building cars and trucks. Written in matrix notation, the linear program becomes maximize c T x subject to Ax b x 0 , where c = 3 2 . 5 , A = 4 . 44 0 0 6 . 67 4 2 . 86 3 6 and b = 100 100 100 100 . The optimal solution of our problem is a basic feasible solution. Since there are two decision variables, each basic feasible solution is characterized by a set of two linearly independent binding constraints. At the optimal solution, the two binding constraints are those associated with metal stamp- ing and engine assembly capacity. Hence, the optimal solution is the unique solution to a pair of linear equations: 4 x 1 + 2 . 86 x 2 = 100 (metal stamping capacity is binding) 3 x 1 + 6 x 2 = 100 (engine assembly capacity is binding) . In matrix form, these equations can be written as Ax = b , where A = ( A 3 * ) T ( A 4 * ) T and b = b 3 b 4 .
c Benjamin Van Roy and Kahn Mason 85 Note that the matrix A has full rank. Therefore, it has an inverse A - 1 . Through some calculations, we get (approximately) A - 1 = 0 . 389 - 0 . 185 - 0 . 195 0 . 259 . The optimal solution of the linear program is given by x = A - 1 b , and there- fore, the optimal profit is c T A - 1 b = 77 . 3. 4.1.1 Sensitivity Analysis Suppose we wish to increase profit by expanding manufacturing capacities.

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chapter4 - Chapter 4 Duality Given any linear program there...

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