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Unformatted text preview: IE521 Advanced Optimization Lecture 9 Dr. Zeliha Akc ¸a December 2011 1 / 31 Sensitivity Analysis I Sensitivity analysis investigates how the optimal solution is sensitive to the parameters of the linear program. I Parameters are objective function coefficients of variables, right hand side values of constraints, coefficients of variables in constraints. I Sensitivity Analysis is important since in real life, we may not be able to estimate the right parameters in our optimization problem. I Also, we may need to know what happens if we change the parameters (buy more resources, increase the price of products, etc.) I We can understand how the optimal solution to an LP changes if the parameters change using sensitivity analysis. 2 / 31 Example I Company WW sells 4 types of products. I In order to produce these four products, raw material and labor hour are used. I Sales price of each product, amount of each resource used for each product and maximum available resources are given below: P1 P2 Pr3 P4 Max Avail Raw material 2 3 4 7 4600 Labor Hour 3 4 5 6 5000 Sales Price 4 6 7 8 I To meet demand, the company needs to produce exactly 950 units of product and at least 400 units of product 4. I Formulate an LP that can be used to maximize company’s revenue. 3 / 31 Example: LP Formulation We can formulate the problem as follows: max 4 x 1 + 6 x 2 + 7 x 3 + 8 x 4 s.t. x 1 + x 2 + x 3 + x 4 = 950 x 4 ≥ 400 2 x 1 + 3 x 2 + 4 x 3 + 7 x 4 ≤ 4600 3 x 1 + 4 x 2 + 5 x 3 + 6 x 4 ≤ 5000 x 1 , x 2 , x 3 , x 4 ≥ 4 / 31 Solving LPs Using Excel Solver I The Solver can be used for both equationsolving (often called goalseeking or backsolving) and constrained optimization (using linear programming, nonlinear programming, and integer programming methods). I MS Excel has free Solver. I Use Excel help to learn how to include Solver to your Excel (if you had not done before), I In order to use Excel Solver to solve your LP model: Allocate cells: ⇒ One blank cell for each decision variable (nor write any formula) ⇒ One cell for objective function value (need to write formula to define your objective function in terms of the decision variable cells and objective function coefficients of decision variables) ⇒ One cell for each constraint‘s right hand side value (constant values) ⇒ One cell for each constraint‘s left hand side function (need to write the formula to define the constraint function in terms of decision variable cells) 5 / 31 Using Excel Solver, Cont. I 4 blank cells (yellow cells, B2:E2) are for decision variables x 1 , x 2 , x 3 , x 4 . I Red cell is for objective function (F2) ⇒ contains the formula B1*B2+C1*C2+D1*D2+E1*E2 I 4 cells for constraint right hand sides dark blue cells ⇒ contains constant values I 4 cells for constraint functions, light blue cells ⇒ contains the formulas, for example, F6 contains B6*B2+C6*C2+D6*D2+E6*E2....
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 Fall '11
 ZelihaAkça
 Linear Programming, Optimization, objective function, LP, new optimal solution

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