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# dual - OR 320/520 Optimization I Prof Bland Motivating...

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OR 320/520 Optimization I 10/18/07. Prof. Bland Motivating Linear Programming Duality Consider the following variation on the brewery problem. We wish to maximize profit from the production of Ice Beer (product 1) and Malt Beer (product 2). Data on profit per barrel, availability of scarce resources, and rates of consumption are given below. Scarce Resource Availability Hops 130 ounces Corn 320 pounds Malt 1120 pounds (1) Ice Beer (2) Malt Beer Profit (dollars per barrel) 16 12 Ounces of hops per barrel of product 4 2 Pounds of corn per barrel of product 8 8 Pounds of malt per barrel of product 16 32 This give rise to the following linear programming problem (P): maximize 16 x 1 + 12 x 2 subject to: 4 x 1 + 2 x 2 130 8 x 1 + 8 x 2 320 16 x 1 + 32 x 2 1120 x 1 0 , x 2 0 . Suppose we want to get upper bounds on z * , the optimal objective function value. z = 16 x 1 + 12 x 2 1) 16 x 1 + 32 x 2 1120 (constraint on malt) z = 16 x 1 + 12 x 2 1120 since x 2 0 2) z = 16 x 1 +12 x 2 16 x 1 +16 x 2 = 2(8 x 1 +8 x 2 ) 2 · 320 = 640 (from constraint on corn)

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3) z = 16 x 1 + 12
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