Bixby_1 - 15 September 2003 INFORMS Practice 2002 1 Solving...

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Unformatted text preview: 15 September 2003 INFORMS Practice 2002 1 Solving Linear and Integer Programs Robert E. Bixby ILOG, Inc. and Rice University Ed Rothberg ILOG, Inc. 2 Outline ¡ Linear Programming: Bob Bixby ¡ Example and introduction to basic LP, including duality ¡ Primal and dual simplex algorithms ¡ Computational progress in linear programming ¡ Implementing the dual simplex algorithm ¡ Mixed-Integer Programming: Ed Rothberg 15 September 2003 INFORMS Practice 2002 2 3 An Example 4 Diet Problem * Bob wants to plan a nutritious diet, but he is on a limited budget, so he wants to spend as little money as possible. His nutritional requirements are as follows: 1. 2000 kcal 2. 55 g protein 3. 800 mg calcium * From Linear Programming, by Va ŝ ek Chvátal 15 September 2003 INFORMS Practice 2002 3 5 Nutritional values Diet Problem Bob is considering the following foods: Food Serving Size Energy (kcal) Protein (g) Calcium (mg) Price per serving Oatmeal 28 g 110 4 2 $0.30 Chicken 100 g 205 32 12 $2.40 Eggs 2 large 160 13 54 $1.30 Whole milk 237 cc 160 8 285 $0.90 Cherry pie 170 g 420 4 22 $0.20 Pork and beans 260 g 260 14 80 $1.90 6 Variables Diet Problem We can represent the number of servings of each type of food in the diet by the variables: x 1 servings of oatmeal x 2 servings of chicken x 3 servings of eggs x 4 servings of milk x 5 servings of cherry pie x 6 servings of pork and beans 15 September 2003 INFORMS Practice 2002 4 7 Nutritional values Diet Problem Bob is considering the following foods: Food Serving Size Energy (kcal) Protein (g) Calcium (mg) Price per serving Oatmeal 28 g 110 4 2 $0.30 Chicken 100 g 205 32 12 $2.40 Eggs 2 large 160 13 54 $1.30 Whole milk 237 cc 160 8 285 $0.90 Cherry pie 170 g 420 4 22 $2.00 Pork and beans 260 g 260 14 80 $1.90 x 1 x 2 x 3 x 4 x 5 x 6 KCAL constraint: 110 x 1 + 205 x 2 + 160 x 3 + 160 x 4 + 420 x 5 + 260 x 6 ≥ 2000 (110 x 1 = kcals in oatmeal) 8 LP formulation Diet Problem 110 x 1 + 205 x 2 + 160 x 3 + 160 x 4 + 420 x 5 + 260 x 6 ≥ 2000 4 x 1 + 32 x 2 + 13 x 3 + 8 x 4 + 4 x 5 + 14 x 6 ≥ 55 2 x 1 + 12 x 2 + 54 x 3 + 285 x 4 + 22 x 5 + 80 x 6 ≥ 800 Minimize subject to: , , , , , 6 5 4 3 2 1 ≥ x x x x x x Cost Nutritional requirements Bounds 0.3 x 1 + 2.40 x 2 + 1.30 x 3 + 0.90 x 4 + 2.0 x 5 + 1.9 x 6 15 September 2003 INFORMS Practice 2002 5 9 Solution Diet Problem When we solve the preceding LP (using CPLEX, of course) we get a solution value of $6.71 , which is achieved with the following menu: 14.24 servings of oatmeal servings of chicken servings of eggs 2.71 servings of milk servings of cherry pie servings of pork and beans 10 Some Basic Theory 15 September 2003 INFORMS Practice 2002 6 11 Where c ∈ R n , b ∈ R m , A ∈ R m×n , and x is a vector of n variables . c T x is known as the objective function , Ax=b as the constraints , and x ≥ as the nonnegativity conditions. b is called the right-hand side ....
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Bixby_1 - 15 September 2003 INFORMS Practice 2002 1 Solving...

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