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Lecture 2

# Lecture 2 - Modeling and Sensitivity Analysis for Linear...

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1 Modeling and Sensitivity Analysis for Linear Programming Excel Solver Blending problem Car rental problem

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2 Allocating Machines to Production Producing 5 types of product using 4 machines Unit processing time (hour) Availability Product 1 Product 2 Product 3 Product 4 Product 5 (hour) Machine 1 1.2 1.3 0.7 0 0.5 160 Machine 2 0.7 2.2 1.6 0.5 1.0 200 Machine 3 0.9 0.7 1.3 1.0 0.8 120 Machine 4 1.4 2.8 0.5 1.2 0.6 280 Unit Profit 18 25 10 12 15
3 5 ,..., 1 for , 0 280 6 . 0 2 . 1 5 . 0 8 . 2 4 . 1 120 8 . 0 0 . 1 3 . 1 7 . 0 9 . 0 200 0 . 1 5 . 0 6 . 1 2 . 2 7 . 0 160 5 . 0 7 . 0 3 . 1 2 . 1 to subjec 15 12 10 25 18 Maximize 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 5 3 2 1 5 4 3 2 1 = + + + + + + + + + + + + + + + + + + + = j P P P P P P P P P P P P P P P P P P P P P P P P P Z j LP Model

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4 Using Excel Solver to Solve LP Microsoft Excel Solver Solver is an Add-in in Microsoft Excel Solver can be used to solve small-size optimization problems, including linear programming Where is the Solver? Under menu “Tools”, see if you can find “Solver” If not, click “Add-ins”, then check “Solver Add- in”
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6 Optimal Solution & Sensitivity Analysis Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease Product 1 58.96 0.00 18.00 6.81 4.74 Product 2 62.63 0.00 25.00 16.82 1.16 Product 3 0.00 -13.53 10.00 13.53 INF Product 4 10.58 0.00 12.00 5.92 0.71 Product 5 15.64 0.00 15.00 0.36 5.09 In a feasible solution, a variable not reaching its bound (either lower or upper) is referred to as a Basic Variable .
7 Reduced Cost Final Reduced Name Value Cost Product 1 58.96 0.00 Product 2 62.63 0.00 Product 3 0.00 -13.53 Product 4 10.58 0.00 Product 5 15.64 0.00 A non-basic variable has a Reduced Cost , indicating the marginal cost of forcing a non-basic variable to leave its bound and become basic. the total profit will decrease at the rate of \$13.53/unit within a certain range; or the unit profit of Product 3 has to be increased by \$13.53 to make it breakeven. If we have to produce some units of product 3, then

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8 Allowable Changing Range of Objective Function Coefficients Final Reduced Objective Allowable Allowable Name Value Cost Coefficient Increase Decrease Product 1 58.96 0.00 18.00 6.81 4.74 Product 2 62.63 0.00 25.00 16.82 1.16 Product 3 0.00 -13.53 10.00 13.53 INF Product 4 10.58 0.00 12.00 5.92 0.71 Product 5 15.64 0.00 15.00 0.36 5.09 Allowable changing range of an objective function coefficient defines an interval for each coefficient of the objective function. When the coefficient changes within the interval, the optimal solution will NOT change.
9 Constraint Analysis & Shadow Price Final Shadow R.H. Allowable Allowable Name Value Price Side Increase Decrease M1 Used 160.00 4.82 160.00 13.00 60.65 M2 Used 200.00 5.20 200.00 30.36 15.31 M3 Used 120.00 8.96 120.00 117.50 18.87 M4 Used 280.00 0.36 280.00 15.64 20.68 Shadow Price indicates the marginal objective function value change when the right-hard side of each constraint changes.

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