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Lecture 18 - Final Review 2

Lecture 18 - Final Review 2 - 540: OPERATIONSRESEARCH...

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540:311 DETERMINISTIC MODELS IN OPERATIONS RESEARCH Lecture 18: Final Review 2 Prof. W. Art Chaovalitwongse

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Topics Ch. 4: Simplex Method Ch. 5: [email protected] Analysis Graphical [email protected] Analysis LINDO [email protected] Analysis (including 100% rule) Ch. 6: Matrix [email protected] Simplex Method [email protected] Analysis Duality Ch. 8: Network Models Shortest Path Problem Maximum Flow Problem Minimum Spanning Tree Problem Min Cost Network Flow Ch. 9: Integer Programming Modeling Branch and Bound
6.5 – Finding the Dual of an LP max z = c 1 x 1 + c 2 x 2 + + c n x n s.t. a 11 x 1 + a 12 x 2 + + a 1n x n b 1 a 21 x 1 + a 22 x 2 + + a 2n x n b 2 a m1 x 1 + a m2 x 2 + + a mn x n b m x j 0 (j = 1, 2, ,n) min w = b 1 y 1 + b 2 y 2 + + b m y m s.t. a 11 y 1 + a 12 y 2 + + a m1 y m c 1 a 21 y 1 + a 22 y 2 + + a m2 y m c 2 a 1n y 1 + a 2n y 2 + + a mn y m c n y i 0 (i = 1, 2, ,m) Normal max problem It s dual Normal min problem It s dual

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Recall 4.3 – max LPs The Dakota Furniture company manufactures desk, tables, and chairs. The manufacturer of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in the table below. Resource Desk Table Chair Lumber 8 board ft 6 board ft 1 board ft Finishing hours 4 hours 2 hours 1.5 hours Carpentry hours 2 hours 1.5 hours 0.5 hours At present, 48 board feet of lumber, 20 finishing hours, 8 carpentry hours are available. A desk sells for \$60, a table for \$30, and a chair for \$ 20. Dakota believes that demand for desks and chairs is unlimited, but at most 5 tables can be sold. Since the available resources have already been purchased, Dakota wants to maximize total revenue.
Recall 4.3 – max LPs Define: x 1 = number of desks produced x 2 = number of tables produced x 3 = number of chairs produced. The LP is: max z = 60x 1 + 30x 2 + 20x 3 s.t. 8x 1 + 6x 2 + x 3 48 (lumber constraint) 4x 1 + 2x 2 + 1.5x 3 20 (finishing constraint) 2x 1 + 1.5x 2 + 0.5x 3 8 (carpentry constraint) x 2 5 (table demand constraint) x 1 , x 2 , x 3 0

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6.5 – Economic InterpretaLon of the Dual Problem InterpreLng the Dual of a the Dakota (Max) Problem The primal is: max z = 60x 1 + 30x 2 + 20x 3 s.t. 8x 1 + 6x 2 + x 3 48 (Lumber constraint) 4x 1 + 2x 2 + 1.5x 3 20 (Finishing constraint) 2x 1 + 1.5x 2 + 0.5x 3 8 (Carpentry constraint) x 1 , x 2 , x 3 0 The dual is: min w = 48y 1 + 20y 2 + 8y 3 s.t. 8y 1 + 4y 2 + 2y 3 60 (Desk constraint) 6y 1 + 2y 2 + 1.5y 3 30 (Table constraint) y 1 + 1.5y 2 + 0.5y 3 20 (Chair constraint) y 1 , y 2 , y 3 0
6.5 – Economic InterpretaLon of the Dual Problem To determine these prices we define: y 1 = price paid for 1 boards ft of lumber y 2 = price paid for 1 finishing hour y 3 = price paid for 1 carpentry hour The resource prices y 1 , y 2 , and y 3 should be determined by solving the Dakota dual.

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Lecture 18 - Final Review 2 - 540: OPERATIONSRESEARCH...

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