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Unformatted text preview: ESI 6417 Duality 1 Ref.: Chap 6 in BJS Dual Problem • Associated with each linear program is another linear program called the “ dual .” • Mathematically, Primal problem : min c T x s.t. Ax > b , (Note the direction of the inequality) x > 0. Dual problem : max b T w s.t. A T w < c w > 0.  Terminology: In the literature, the first and second problems are referred to as the minimization and maximization problems in “canonical” form.  The dual of the dual problem Primal problem : max b T w s.t. A T w < c w > 0. Dual problem : min c T x s.t. Ax > b , x > 0. • Example: Primal Dual . , 7 2 5 4 3 . . 8 6 min 2 1 2 1 2 1 2 1 ≥ ≥ + ≥ + + x x x x x x t s x x . , 8 2 6 5 3 . . 7 4 max 2 1 2 1 2 1 2 1 ≥ ≤ + ≤ + + w w w w w w t s w w ESI 6417 Duality 2 Ref.: Chap 6 in BJS • Interpretation of the dual problems: Example The Hion Hog Farm Company may purchase and mix one or more of three types of grain, each containing different amount of four nutritional vitamins as shown in the table below. Corn Oats Soybean Minimum Req. (2 wks) Vitamin A 2.0 3.0 7.0 1250 g. Vitamin B 1.0 1.0 0.0 250 g. Vitamin C 5.0 3.0 0.0 900 g. Vitamin D 0.6 0.25 1.0 232.2 g. Cost/kg $41 $35 $96 The production manager specifies that any feed mix for his livestock must meet at least the minimal nutritional requirements and he seeks the least costly among all such mixes. Suppose he wishes to purchase enough to fill his needs for two weeks. Decision variables : x c , x o , x s = amounts of corn, oats, and soybean to purchase, respectively. Formulation : ݉݅݊ 41ݔ 35ݔ 96ݔ ௦ ݏ. ݐ. 2ݔ 3ݔ 7ݔ ௦ 1250 ݔ ݔ 250 5ݔ 3ݔ 900 0.6ݔ 0.25ݔ ݔ ௦ 232.5 ݔ , ݔ , ݔ ௦ 0 The different perspective: The Hog Heaven pharmaceutical company wishes to sell vitamin pills to the Hion Hog Farm Company. To do so, Hog Heaven must set prices for their vitamin pills so that they would be an attractive alternative to grains. However, Hog Heaven still wants to maximize profit. What is the problem that Hog Heaven needs to solve? Decision variables : w a , w b , w c , w d = prices per gram for vitamins A, B, C, and D, respectively. Formulation : ESI 6417 Duality 3 Ref.: Chap 6 in BJS • Observe that the number of dual variables ( w ) is the same as the number of constraints in the primal problem, i.e., there is a dual variable associated with each constraint. (The nonnegativity constraints are not included in this discussion.) • The dual of any linear program can be found by converting the problem to the appropriate canonical form....
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 Spring '07
 SIRIPHONGLAWPHONGPANICH
 Linear Programming, Optimization, Vitamin, Dual problem, BJS, Duality Ref.

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