Introduction+to+Optimization-2

Introduction+to+Optimization-2 - Introduction to...

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1 Optimization E10 Introduction to Optimization E10, Fall, 2010
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2 Optimization E10 What is Optimization Calculation of how best to operate or configure a system Decision variables ( A for what we can adjust ) Objective function ( B for what’s best ) Constraints on variables ( C for how are we constrained ) Mathematical methods of optimization Calculus: Functional characterization of possible decisions and economic consequences Search: Enumeration and evaluation of alternative decisions Directed search (always move to a better alternative) Proof of optimality without evaluating every alternative
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3 Optimization E10 Introduction to Linear Programming: Example Formulations, and Extensions
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4 Optimization E10 Linear Programming The term “linear program” refers to an optimization problem with an objective that is a linear function of the decision variables, subject to a set of linear constraints on the decision variables The word “program” refers to the plan prescribed by the solution to the problem. (The original problem for which LP was developed was to find the best program of training and assignments for US Air Force officers.)
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5 Optimization E10 Stereo Retailing Retail outlet has 400 sq ft of floor space and a budget for stocking items of $8000 Model X receiver has wholesale cost of $100, sells for $150, and requires 2 sq ft of display space Model Y set of speakers has wholesale cost of $50, sells for $70, and requires 4 sq ft of display space Demand for receivers is expected to be at most 60 units
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6 Optimization E10 Stereo Retailing (cont.) A: Let x = number of receivers to stock y = number of sets of speakers to stock B: Maximize 50x + 20y gross profit C: Subject to 2x + 4y ≤ 400 floor space 100x + 50y ≤ 8000 budget x ≤ 60 sales limit x, y ≥ 0 x, y integer
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7 Optimization E10 Diet (Blending Problem) The hospital wants to concoct a desert shake meeting certain dietary restrictions Ingredients: Egg custard, Ice cream, Butterscotch syrup Egg Ice Butterscotch Requirement Custard Cream Syrup Cholesterol 50 50 90 <= 175 Fat 0 100 50 <= 150 Protein 70 10 0 >= 200 Calories 30 80 200 >= 100 Cost per oz. 0.15 0.25 0.10
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8 Optimization E10 Diet (Blending) Problem (cont.) Let E = ounces of egg custard base in the shake C = ounces of ice cream in the shake S = ounces of butterscotch syrup in the shake Minimize cost Subject to cholesterol fat protein calories S C E 10 . 0 25 . 0 15 . 0 + + 50 150 90 175 E C S + + 100 50 150 C S + 70 10 E C + 200 30 80 200 100 E C S + + E C S , , 0
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9 Optimization E10 Shift-Scheduling Problem City of Berkeley Police Patrol Minimum number of officers required: 12AM – 4AM 6 4AM – 8AM 4 8AM – 12PM 14 12PM – 4PM 8 4PM – 8PM 12 8PM – 12AM 16 Must work 8 hours (two consecutive shifts)
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10 Optimization E10 Shift-Scheduling Problem City of Berkeley Police Patrol Let x i = number of officers reporting at period i for i =1, 2, …, 6 Minimize x 1 + x 6 6 period 1 x 1 + x 2 4 period 2 x 2 + x 3 14 period 3 x 3 + x 4 ≥ 8 period 4 x 4 + x 5 ≥ 12 period 5 x 5 + x 6 16 period 6 x i ≥ 0 and integer, i = 1,2,…,6 x x x x x x 1 2 3 4 5 6 + + + + +
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