# 4-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4.

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4-1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Linear Programming: Modeling Examples Chapter 4

4-2 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Chapter Topics A Product Mix Example A Diet Example An Investment Example A Marketing Example A Transportation Example A Blend Example A Multiperiod Scheduling Example A Data Envelopment Analysis Example

4-3 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Product Mix Example Problem Definition (1 of 8) Four-product T-shirt/sweatshirt manufacturing company. ■ Must complete production within 72 hours ■ Truck capacity = 1,200 standard sized boxes. ■ Standard size box holds12 T-shirts. ■ One-dozen sweatshirts box is three times size of standard box. ■ \$25,000 available for a production run. ■ 500 dozen blank T-shirts and sweatshirts in stock. ■ How many dozens (boxes) of each type of shirt to produce?

4-4 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Product Mix Example (2 of 8)

4-5 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Product Mix Example Data (3 of 8) Resource requirements for the product mix example.

4-6 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Variables: x 1 = sweatshirts, front printing x 2 = sweatshirts, back and front printing x 3 = T-shirts, front printing x 4 = T-shirts, back and front printing Objective Function: Maximize Z = \$90x 1 + \$125x 2 + \$45x 3 + \$65x 4 Model Constraints: 0.10x 1 + 0.25x 2 + 0.08x 3 + 0.21x 4  72 hr 3x 1 + 3x 2 + x 3 + x 4  1,200 boxes \$36x 1 + \$48x 2 + \$25x 3 + \$35x 4  \$25,000 x 1 + x 2  500 dozen sweatshirts x 3 + x 4  500 dozen T-shirts A Product Mix Example Model Construction (4 of 8)

4-7 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Product Mix Example Computer Solution with Excel (5 of 8) Exhibit 4.1 Objective function Click on “Data” tab to access Solver =D7*B14+E7*B15+F7*B16+G7*B17 =J7-H7 These cells have no effect; added for “cosmetic” purposes. Model formulation included on all Excel files on Companion Web site =F11*B16+G11*B17

4-8 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.2 A Product Mix Example Solution with Excel Solver Window (6 of 8) Includes all five constraints.

4-9 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.3 A Product Mix Example Solution with QM for Windows (7 of 8) Model solution is: x1=175.56 boxes of front-only sweatshirts x2=57.78 boxes of front and back sweatshirts x3 = 500 boxes of front-only t-shirts Z=\$45,522.22 profit

4-10 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.4 A Product Mix Example Solution with QM for Windows (8 of 8)

4-11 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, 12 grams of fiber, and must have no more than 20 grams of fat and 30 milligrams of cholesterol. A Diet Example Data and Problem Definition (1 of 5)

4-12 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall x 1 = cups of bran cereal x 2 = cups of dry cereal x 3 = cups of oatmeal x 4 = cups of oat bran x 5 = eggs x 6 = slices of bacon x 7 = oranges x 8 = cups of milk x 9 = cups of orange juice x 10 = slices of wheat toast A Diet Example Model Construction – Decision Variables (2 of 5)

4-13 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall MinimizeZ = 0.18x 1 + 0.22x 2 + 0.10x 3 + 0.12x 4 + 0.10x 5 + 0.09x 6 + 0.40x 7 + 0.16x 8 + 0.50x 9 + 0.07x 10 subject to: 90x 1 + 110x 2 + 100x 3 + 90x 4 + 75x 5 + 35x 6 + 65x 7 + 100x 8 + 120x 9 + 65x 10  420 calories 2x 2 + 2x 3 + 2x 4 + 5x 5 + 3x 6 + 4x 8 + x 10  20 g fat 270x 5 + 8x 6 + 12x 8  30 mg cholesterol 6x 1 + 4x 2 + 2x 3 + 3x 4 + x 5 + x 7 + x 10  5 mg iron 20x 1 + 48x 2 + 12x 3 + 8x 4 + 30x 5 + 52x 7 + 250x 8 + 3x 9 + 26x 10  400 mg of calcium 3x 1 + 4x 2 + 5x 3 + 6x 4 + 7x 5 + 2x 6 + x 7 + 9x 8 + x 9 + 3x 10  20 g protein 5x 1 + 2x 2 + 3x 3 + 4x 4 + x 7 + 3x 10  12 x i  0, for all j A Diet Example Model Summary (3 of 5)

4-14 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.5 A Diet Example Computer Solution with Excel (4 of 5) Decision variable, C5:C14 =SUMPRODUCT(C5:C14,F5:F14) or =C5*F5+C6*F6+C7*F7+C8*F8+ C9*F9+C10*F10+C11*F11+C12* F12+C13*F13+C14*F14 Constraint value, 420, typed in cell F17 =SUMPRODUCT(C5:C14,E5:E14) or =C5*E5+C6*E6+C7*E7+C8*E8+ C9*E9+C10*E10+C11*E11+C12* E12+C13*E13+C14*E14

4-15 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.6 A Diet Example Solution with Excel Solver Window (5 of 5) Decision variables; “servings” in column C Constraint for “calories” in column F; SUMPRODUCT (C5:C14,F5:F14)<420

4-16 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall An investor has \$70,000 to divide among several instruments. Municipal bonds have an 8.5% return, CD’s a 5% return, t-bills a 6.5% return, and growth stock 13%. The following guidelines have been established: 1.No more than 20% in municipal bonds 2.Investment in CDs should not exceed the other three alternatives 3.At least 30% invested in t-bills and CDs 4.More should be invested in CDs and t-bills than in municipal bonds and growth stocks by a ratio of 1.2 to 1 5.All \$70,000 should be invested. An Investment Example Model Summary (1 of 5)

4-17 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Maximize Z = \$0.085x 1 + 0.05x 2 + 0.065 x 3 + 0.130x 4 subject to: x 1  \$14,000 x 2 - x 1 - x 3 - x 4  0 x 2 + x 3  \$21,000 -1.2x 1 + x 2 + x 3 - 1.2 x 4  0 x 1 + x 2 + x 3 + x 4 = \$70,000 x 1, x 2, x 3, x 4  0 where x 1 = amount (\$) invested in municipal bonds x 2 = amount (\$) invested in certificates of deposit x 3 = amount (\$) invested in treasury bills x 4 = amount (\$) invested in growth stock fund An Investment Example Model Summary (2 of 5)

4-18 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall An Investment Example Computer Solution with Excel (3 of 5) Exhibit 4.7 Total investment requirement, =D10*B13+E10*B14+F10*B15+G10*B16 First guideline, =D6*B13 Objective function, Z, for total return

4-19 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.8 An Investment Example Solution with Excel Solver Window (3 of 4) Guideline constraints

4-20 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall An Investment Example Sensitivity Report (4 of 4) Exhibit 4.9 Shadow price for the amount available to invest

4-21 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall  Budget limit \$100,000  Television time for four commercials  Radio time for 10 commercials  Newspaper space for 7 ads  Resources for no more than 15 commercials and/or ads A Marketing Example Data and Problem Definition (1 of 7)

4-22 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Maximize Z = 20,000x 1 + 12,000x 2 + 9,000x 3 subject to: 15,000x 1 + 6,000x 2 + 4,000x 3  100,000 x 1  4 x 2  10 x 3  7 x 1 + x 2 + x 3  15 x 1, x 2, x 3  0 where x 1 = number of television commercials x 2 = number of radio commercials x 3 = number of newspaper ads A Marketing Example Model Summary (2 of 7)

4-23 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.10 A Marketing Example Solution with Excel (3 of 7) Objective function =F6*D6+F7*D7+F8*D8 or =SUMPRODUCT(D6:D8,F6:F8)

4-24 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.11 A Marketing Example Solution with Excel Solver Window (4 of 7) Includes all five constraints

4-25 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Marketing Example Integer Solution with Excel (5 of 7) Exhibit 4.12 Decision variables Click on “int” for integer.

4-26 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.13 A Marketing Example Integer Solution with Excel (6 of 7) Integer restriction

4-27 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.14 A Marketing Example Integer Solution with Excel (7 of 7) Integer solution Better solution—17,000 more total exposures—than rounded-down solution

4-28 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Warehouse supply ofRetail store demand Television Sets:for television sets: 1 - Cincinnati 300A - New York 150 2 - Atlanta 200B - Dallas250 3 - Pittsburgh 200C - Detroit200 Total 700Total600 A Transportation Example Problem Definition and Data (1 of 3)

4-29 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Minimize Z = \$16x 1A + 18x 1B + 11x 1C + 14x 2A + 12x 2B + 13x 2C + 13x 3A + 15x 3B + 17x 3C subject to: x 1A + x 1B + x 1C  300 x 2A + x 2B + x 2C  200 x 3A + x 3B + x 3C  200 x 1A + x 2A + x 3A = 150 x 1B + x 2B + x 3B = 250 x 1C + x 2C + x 3C = 200 All x ij  0 A Transportation Example Model Summary (2 of 4)

4-30 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.15 A Transportation Example Solution with Excel (3 of 4) =C5+C6+C7 =C5+D5+E5

4-31 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.16 A Transportation Example Solution with Solver Window (4 of 4) Decision variables Supply constraints Demand constraints

4-32 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Blend Example Problem Definition and Data (1 of 6)

4-33 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall ■ Determine the optimal mix of the three components in each grade of motor oil that will maximize profit. Company wants to produce at least 3,000 barrels of each grade of motor oil. ■ Decision variables: The quantity of each of the three components used in each grade of gasoline (9 decision variables); x ij = barrels of component i used in motor oil grade j per day, where i = 1, 2, 3 and j = s (super), p (premium), and e (extra). A Blend Example Problem Statement and Variables (2 of 6)

4-34 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Maximize Z = 11x 1s + 13x 2s + 9x 3s + 8x 1p + 10x 2p + 6x 3p + 6x 1e + 8x 2e + 4x 3e subject to: x 1s + x 1p + x 1e  4,500 bbl. x 2s + x 2p + x 2e  2,700 bbl. x 3s + x 3p + x 3e  3,500 bbl. 0.50x 1s - 0.50x 2s - 0.50x 3s  0 0.70x 2s - 0.30x 1s - 0.30x 3s  0 0.60x 1p - 0.40x 2p - 0.40x 3p  0 0.75x 3p - 0.25x 1p - 0.25x 2p  0 0.40x 1e - 0.60x 2e- - 0.60x 3e  0 0.90x 2e - 0.10x 1e - 0.10x 3e  0 x 1s + x 2s + x 3s  3,000 bbl. x 1p + x 2p + x 3p  3,000 bbl. x 1e + x 2e + x 3e  3,000 bbl. A Blend Example Model Summary (3 of 6) all x ij  0

4-35 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.17 A Blend Example Solution with Excel (4 of 6) =B7+B10+B13 Decision variables—B7:B15 =B7+B8+B9 =0.5*B7-0.5*B8-0.5*B9

4-36 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.18 A Blend Example Solution with Solver Window (5 of 6)

4-37 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Blend Example Sensitivity Report (6 of 6) Exhibit 4.19 The shadow price for component 1 is \$20. The upper limit for the sensitivity range for component 1 is 4500+1700=6200.

4-38 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Production Capacity: 160 computers per week 50 more computers with overtime Assembly Costs: \$190 per computer regular time; \$260 per computer overtime Inventory Holding Cost: \$10/computer per week Order schedule: A Multiperiod Scheduling Example Problem Definition and Data (1 of 5)

4-39 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Variables: r j = regular production of computers in week j (j = 1, 2, …, 6) o j = overtime production of computers in week j (j = 1, 2, …, 6) i j = extra computers carried over as inventory in week j (j = 1, 2, …, 5) A Multi-Period Scheduling Example Decision Variables (2 of 5)

4-40 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Model summary: Minimize Z = \$190(r 1 + r 2 + r 3 + r 4 + r 5 + r 6 ) + \$260(o 1 +o 2 +o 3 +o 4 +o 5 +o 6 ) + 10(i 1 + i 2 + i 3 + i 4 + i 5 ) subject to: r j  160 computers in week j (j = 1, 2, 3, 4, 5, 6) o j  150 computers in week j (j = 1, 2, 3, 4, 5, 6) r 1 + o 1 - i 1 = 105week 1 r 2 + o 2 + i 1 - i 2 = 170week 2 r 3 + o 3 + i 2 - i 3 = 230 week 3 r 4 + o 4 + i 3 - i 4 = 180week 4 r 5 + o 5 + i 4 - i 5 = 150week 5 r 6 + o 6 + i 5 = 250week 6 r j, o j, i j  0 A Multi-Period Scheduling Example Model Summary (3 of 5)

4-41 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Multi-Period Scheduling Example Solution with Excel (4 of 5) Exhibit 4.20 Decision variables for regular production – B6:B11 Decision variables for overtime production – D6:D11 B7+D7+I6; regular production + overtime production + inventory from previous week G7-H7

4-42 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.21 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5)

4-43 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall DEA compares a number of service units of the same type based on their inputs (resources) and outputs. The result indicates if a particular unit is less productive, or efficient, than other units. Elementary school comparison: Input 1 = teacher to student ratio Input 2 = supplementary funds/student Input 3 = average educational level of parents Output 1 = average reading SOL score Output 2 = average math SOL score Output 3 = average history SOL score A Data Envelopment Analysis (DEA) Example Problem Definition (1 of 5)

4-44 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5)

4-45 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Decision Variables: x i = a price per unit of each output where i = 1, 2, 3 y i = a price per unit of each input where i = 1, 2, 3 Model Summary: Maximize Z = 81x 1 + 73x 2 + 69x 3 subject to:.06 y 1 + 460y 2 + 13.1y 3 = 1 86x 1 + 75x 2 + 71x 3 .06y 1 + 260y 2 + 11.3y 3 82x 1 + 72x 2 + 67x 3 .05y 1 + 320y 2 + 10.5y 3 81x 1 + 79x 2 + 80x 3 .08y 1 + 340y 2 + 12.0y 3 81x 1 + 73x 2 + 69x 3 .06y 1 + 460y 2 + 13.1y 3 x i, y i  0 A Data Envelopment Analysis (DEA) Example Decision Variables and Model Summary (3 of 5)

4-46 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.22 A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Value of outputs, also in cell H8 =B5*B12+C5*B13+D5*B14 =E8*D12+F8*D13+G8*D14

4-47 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 4.23 A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Scaling constraint Constraint for outputs < inputs

4-48 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example Problem Solution Problem Statement and Data (1 of 5) Canned cat food, Meow Chow; dog food, Bow Chow. ■ Ingredients/week: 600 lb horse meat; 800 lb fish; 1000 lb cereal. ■ Recipe requirement: Meow Chow at least half fish Bow Chow at least half horse meat. ■ 2,250 sixteen-ounce cans available each week. ■ Profit /can: Meow Chow \$0.80 Bow Chow \$0.96. How many cans of Bow Chow and Meow Chow should be produced each week in order to maximize profit?

4-49 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 1: Define the Decision Variables x ij = ounces of ingredient i in pet food j per week, where i = h (horse meat), f (fish) and c (cereal), and j = m (Meow chow) and b (Bow Chow). Step 2: Formulate the Objective Function Maximize Z = \$0.05(x hm + x fm + x cm ) + 0.06(x hb + x fb + x cb ) Example Problem Solution Model Formulation (2 of 5)

4-50 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 3: Formulate the Model Constraints Amount of each ingredient available each week: x hm + x hb  9,600 ounces of horse meat x fm + x fb  12,800 ounces of fish x cm + x cb  16,000 ounces of cereal additive Recipe requirements: Meow Chow: x fm /(x hm + x fm + x cm )  1/2 or - x hm + x fm - x cm  0 Bow Chow: x hb /(x hb + x fb + x cb )  1/2 or x hb - x fb - x cb  0 Can Content: x hm + x fm + x cm + x hb + x fb + x cb  36,000 ounces Example Problem Solution Model Formulation (3 of 5)

4-51 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Step 4: Model Summary Maximize Z = \$0.05x hm + \$0.05x fm + \$0.05x cm + \$0.06x hb + 0.06x fb + 0.06x cb subject to: x hm + x hb  9,600 ounces of horse meat x fm + x fb  12,800 ounces of fish x cm + x cb  16,000 ounces of cereal additive - x hm + x fm - x cm  0 x hb - x fb - x cb  0 x hm + x fm + x cm + x hb + x fb + x cb  36,000 ounces x ij  0 Example Problem Solution Model Summary (4 of 5)

4-52 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example Problem Solution Solution with QM for Windows (5 of 5) Solution to the Bark’s Pet Food Company problem using QM for Windows To determine the number of cans of each flavor, we must sum the ingredient amounts for each and divide by 16 ounces (the size of a can). x hm +x fm +x cm =0+8,400+8,400=16,800 oz of Meow Chow 16,800 / 16 = 1,050 cans of Meow Chow

4-53 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.