# Trsplxm - 1 A Standard Maximum Problem is a linear program...

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1 A Standard Maximum Problem is a linear program in which we wish to maximize the objective function F = c 1 x 1 + c 2 x 2 + ... + c n x n subject to constraints of the form a 11 x 1 + a 12 x 2 + + a 1 n x n b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n b 2 ....... + ....... + .... + ....... ... a m 1 x 1 + a m 2 x 2 + + a mn x n b m where x i 0, b j 0 for i = 1,2,. .., n ; j = 1 ,2 ,..., m . Equivalently Maximize : F = c 1 x 1 + c 2 x 2 + + c n x n subject to constraints with slack variables a 11 x 1 + a 12 x 2 + + a 1 n x n + s 1 = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n + s 2 = b 2 a k 1 x 1 + .... ....... ...... ..... + s k = b k a m 1 x 1 + a m 2 x 2 + + a mn x n + s m = b m where x i 0, b j 0, s k 0, for i = .., n , j = 1 m , k = .., m . Example TB19 . A company manufactures three types of patio furniture : chairs , rockers , and chaise lounges . Each requires wood , plastic , and aluminum as shown in the following table . Wood Plastic Aluminum Price Chair 1 unit 1 unit 2 unit \$7 Rocker 1 unit 1 unit 3 unit \$8 Chaise Loung 1 unit 2 unit 5 unit \$12 Available 400 units 500 units 1450 units The company has available 400 units of wood , 500 units of plastic , and 1450 units of aluminum . Each chair , rocker , and chaise lounge sells at \$7, \$8, and \$12, respectively . Assuming that all furniture can be sold , determine a production order so that total revenue will be maximum . What is the maximum revenue ? Solution . Let x 1 =# chairs produced , let x 2 rockers produced , and let x 3 chaise lounges produced . Objective Function : Maximize R = 7 x 1 + 8 x 2 + 12 x 3 Constraints x 1 + x 2 + x 3 400 x 1 + x 2 + 2 x 3 500 2 x 1 + 3 x 2 + 5 x 3 1450 x 1 x 2 0, x 3 0

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2 x 1 x 2 x 3 s 1 s 2 s 3 R | s 1 111 1000 | 400 s 2 11 2 0100 | 500 s 3 235 0010 | 1450 ———— ———— | R 7 8 1 2 0001 | 0 Rto 400 250 290 Pivot = m 23 = 2. FirstofallRO : 1 2 R 2 R 2 , Then apply : R 2 + R 1 R 1 , 5 R 2 + R 3 R 3 ,12 R 2 + R 4 R 4 x 1 x 2 x 3 s 1 s 2 s 3 R | s 1 0.5 0.5 0 1 0.5 0 0 | 150 x 3 0.5 0.5 1 0 0.5 0 0 | 250 s 3 0.5 0.5 0 0 2.5 1 0 | 200 —— — ——— —— | R 1 2 006 01 | 3000 Rto 300 500 400 Pivot = m 12 = 1 2 . FirstlyApplyRO :2 R 1 R 1 . Next RO .: 1 2 R 1 + R 2 R 2 , 1 2 R 1 + R 3 R 3 ,2 R 1 + R 4 R 4 x 1 x 2 x 3 s 1 s 2 s 3 R | x 2 1102 10 0 | 300 x 3 001 11 0 0 | 100 s 3 1 21 0 | 50 ————————- | -— R 1004401 | 3600 The production of 0 chairs , 300 rockers , and 100 chaise lounges gives the maximum revenue of \$3600. === End of 7.4 Simplex Method Trans. Lecture === Example B312Z3 . Agriculture . Afarmerownsa 100 acre farm and plans to plant at most three crops . The seed for crops A , B , and C costs \$40, \$20, and \$30 per acre , respectively . A maximum of \$3200 can be spent on seed . Crops A , B , and C require 1, 2, and 1 work days per acre , respectively , and there are a maximum of 160 workdays available . If the farmer can make a profit of \$100 per acre on crop A , \$300 per acre on crop B , and \$200 per acre on crop C , how many acres of each crop should be planted to maximize profit ?
3 Crop Crop A Crop B Crop C Maximum # Acres x 1 x 2 x 3 100 Acres Cost per Acre \$ 40 \$ 20 \$ 30 \$ 3200 Work Days / Acre 1 2 1 160 Days Profit / Acre \$ 100 \$ 300 \$ 200 Obj .( P ) Then we have the following linear programming problem : Objective Function Maximize P = 100 x 1 + 300 x 2 + 200 x 3 subject to the Constraints : x 1 + x 2 + x 3 100 40 x 1 + 20 x 2 + 30 x 3 3200 x 1 + 2 x 2 + x 3 160 x

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## This note was uploaded on 12/17/2009 for the course MATH MATH131 taught by Professor Dr.rajalatif during the Spring '09 term at King Fahd University of Petroleum & Minerals.

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Trsplxm - 1 A Standard Maximum Problem is a linear program...

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