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346Chapter15HWSolutions

# 346Chapter15HWSolutions - Chapter 15 Multicriteria Decision...

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Chapter 15 Multicriteria Decision Problems Learning Objectives 1. Understand the concept of multicriteria decision making and how it differs from situations and procedures involving a single criterion. 2. Be able to develop a goal programming model of a multiple criteria problem. 3. Know how to use the goal programming graphical solution procedure to solve goal programming problems involving two decision variables. 4. Understand how the relative importance of the goals can be reflected by altering the weights or coefficients for the decision variables in the objective function. 5. Know how to develop a solution to a goal programming model by solving a sequence of linear programming models using a general purpose linear programming package. 6. Know what a scoring model is and how to use it to solve a multicriteria decision problem. 7. Understand how a scoring model uses weights to identify the relative importance of each criterion. 8. Know how to apply the analytic hierarchy process (AHP) to solve a problem involving multiple criteria. 9. Understand how AHP utilizes pairwise comparisons to establish priority measures for both the criteria and decision alternatives. 10. Understand the following terms: multicriteria decision problem analytic hierarchy process (AHP) goal programming hierarchy deviation variables pairwise comparison matrix priority levels synthesization goal equation consistency preemptive priorities consistency ratio scoring model 15 - 1

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Chapter 15 Solutions: 1. a. Raw Material 1 2 3 Amount Needed to Achieve Both P 1 Goals 2 / 5 (30) + 1 / 2 (15) 1 / 5 (15) 3 / 5 (30) + 3 / 10 (15) = = = 12 + 7.5 3 18 + 4.5 = 19.5 = 22.5 Since there are only 21 tons of Material 3 available, it is not possible to achieve both goals. b. Let x 1 = the number of tons of fuel additive produced x 2 = the number of tons of solvent base produced d 1 + = the amount by which the number of tons of fuel additive produced exceeds the target value of 30 tons d 1 - = the amount by which the number of tons of fuel additive produced is less than the target of 30 tons d 2 + = the amount by which the number of tons of solvent base produced exceeds the target value of 15 tons d 2 - = the amount by which the number of tons of solvent base is less than the target value of 15 tons Min d 1 - + d 2 - s.t. 2 / 5 x 1 + 1 / 2 x 2 20 Material 1 1 / 5 x 2 5 Material 2 3 5 x 1 + 3 / 10 x 2 21 Material 3 x 1 - d 1 + + d 1 - = 30 Goal 1 x 2 - d 2 + + d 2 - = 15 Goal 2 x 1 , x 2 , d 1 + , d 1 - , d 2 + ,   d 2 - 0 c. In the graphical solution, point A minimizes the sum of the deviations from the goals and thus provides the optimal product mix. 15 - 2
Multicriteria Decision Problems 0 10 20 30 40 50 10 20 30 40 50 60 70 Material 2 Material 3 Material 1 Goal 2 Goal 1 A (30, 10) B (27.5, 15) Tons of Solvent Base Tons of Fuel Additive x 2 x 1 d. In the graphical solution shown above, point B minimizes 1 2 2 d d - - + and thus provides the optimal product mix.

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346Chapter15HWSolutions - Chapter 15 Multicriteria Decision...

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