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mccarl-chapter5u - Chapter 5 LP formulations LP...

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Chapter 5 LP formulations
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LP formulations of four basic problem Resource allocation problem Transportation problem Feed mix problem Joint products problem
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We will examine: Basic Structure Formulation Example application Answer interpretation
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Resource Allocation Problem The classical LP problem involves the allocation of an endowment of scarce resources among a number of competing products so as to maximize profits . Objective: Maximize Profits Competing products index is j; scarce resources index is i Major decision variable Xj is the number of units of the jth product made Non negative production (Xj ≥ 0) Resource usage across all production possibilities is less than or equal to the resource endowment
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Algebraic Set UP cj: profit per unit of the jth product aij: number of units of the ith resource used when producing one unit of the jth product bi: the endowment of the ith resource ij j j j i j j j s.t. a X b for a Ma X 0 for al ll x c i l j X
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Resource Allocation Problem: E-Z Chair Objective : find the number of two types of chairs to produce that will maximize profits. Chair Types : Functional and Fancy Resources : Large & Small Lathe, Chair Bottom Carver, and Labor Profit Contributions: (revenue – material cost - cost increase due to lathe shifts)
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Information for Problem OBJ Functional $82 - $15 = $67 Fancy $105 - $25 = $80 Resource Requirements When Using The Normal Pattern Hours of Use per Chair Type Functional Fancy Small Lathe 0.8 1.2 Large Lathe 0.5 0.7 Chair Bottom Carver 0.4 1.0 Labor 1.0 0.8
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Resource Requirements and Increased Costs for Alternative Methods of Production in Hours of Use per Chair and Dollars Maximum Use of Small Lathe Maximum Use of Large Lathe Functional Fancy Functional Fancy Small Lathe 1.30 1.70 0.20 0.50 Large Lathe 0.20 0.30 1.30 1.50 Chair Bottom Carver 0.40 1.00 0.40 1.00 Labor 1.05 0.82 1.10 0.84 Cost Increase $1.00 $1.50 $0.70 $1.60 Alternative Production Method
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Resource Limits Small lathe: 140 hours Large lathe: 90 hours Chair bottom carver: 120 hours Labor: 125 hours
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Production alternatives and profits Functional, regular method (X1) : $67 (c1) Functional, max small lathe (X2): $66 (c2) Functional, max lg lathe (X3): $66.30 (c3) Fancy , regular method (X4): $80 (c4) Fancy, max small lathe (X5): $78.50 (c5) Fancy, max lg lathe (X6): $78.40 (c6)
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Empirical Set-UP Max 67X 1 + 66X 2 + 66.3X 3 + 80X 4 + 78.5X 5 + 78.4X 6 s.t. 0.8X 1 + 1.3X 2 + 0.2X 3 + 1.2X 4 + 1.7X 5 + 0.5X 6 140 0.5X 1 + 0.2X 2 + 1.3X 3 + 0.7X 4 + 0.3X 5 + 1.5X 6 90 0.4X 1 + 0.4X 2 + 0.4X 3 + X 4 + X 5 + X 6 120 X 1 + 1.05X 2 + 1.1X 3 + 0.8X 4 + 0.82X 5 + 0.84X 6 125 X 1 , X 2 , X 3 , X 4 , X 5 , X 6 0
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X1 X2 X3 X4 X5 X6 RHS Used S.P. Obj 67 66 66.3 80 78.5 78.4 10417.29 Small Lathe 0.8 1.3 0.2 1.2 1.7 0.5 le 140 140.00 33.33 Large lathe 0.5 0.2 1.3 0.7 0.3 1.5 le 90 90.00 25.79 Chair bottom carver 0.4 0.4 0.4 1 1 1 le 120 103.09 0.00 Labor 1 1.05 1.1 0.8 0.82 0.84 le 125 125.00 27.44 answers 62.23 0.00 0.00 73.02 0.00 5.18 reduced cost 0.00 -11.30 -4.08 0.00 -8.40 0.00 Solution from Excel Solver McCarl provides GAMS code and solution from GAMS (same as this solution).
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Interpretation Produce 62 functional chairs using the traditional method, 73 fancy chairs using the traditional method, and 5 fancy chairs
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