Linear Programming: Modeling Examples Chapter Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-1 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 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-2 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? Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-3 A Product Mix Example Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall (2 of 8) 4-4 A Product Mix Example Data (3 of 8) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-5 A Product Mix Example Model Construction (4 of 8) Decision Variables: x1 = sweatshirts, front printing x2 = sweatshirts, back and front printing x3 = T-shirts, front printing x4 = T-shirts, back and front printing Objective Function: Maximize Z = $90x1 + $125x2 + $45x3 + $65x4 Model Constraints: 0.10x1 + 0.25x2+ 0.08x3 + 0.21x4 ≤ 72 hr 3x1 + 3x2 + x3 + x4 ≤ 1,200 boxes $36x1 + $48x2 + $25x3 + $35x4 ≤ $25,000 x1 + x2 ≤ 500 dozen Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-6 sweatshirts A Product Mix Example Computer Solution with Excel (5 of 8) Exhibit 4.1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-7 A Product Mix Example Solution with Excel Solver Window (6 of 8) Exhibit 4.2 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-8 A Product Mix Example Solution with QM for Windows (7 of 8) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 4.3 4-9 A Product Mix Example Solution with QM for Windows (8 of 8) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 4.4 4-10 A Multi-Period Scheduling Example Decision Variables (2 of 5) Decision Variables: rj = regular production of computers in week j (j = 1, 2, …, 6) oj = overtime production of computers in week j (j = 1, 2, …, 6) ij = extra computers carried over as inventory in week j (j = 1, 2, …, 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-37 A Multi-Period Scheduling Example Model Summary (3 of 5) Model summary: Minimize Z = $190(r1 + r2 + r3 + r4 + r5 + r6) + $260(o1+o2 +o3 +o4+o5+o6) + 10(i1 + i2 + i3 + i4 + i5) subject to: rj ≤ 160 computers in week j (j = 1, 2, 3, 4, 5, 6) oj ≤ 150 computers in week j (j = 1, 2, 3, 4, 5, 6) r1 + o1 - i1 = 105 r2 + o2 + i1 - i2 = 170 r Pearson + oEducation, - iPublishing + i2 Inc = 230 Copyright © 2010 as Prentice Hall week week week 4-38 A Multi-Period Scheduling Example Solution with Excel (4 of 5) Exhibit 4.20 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-39 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 4-40 A Data Envelopment Analysis (DEA) Example Problem Definition (1 of 5) 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 = teacher to student ratio Input = supplementary funds/student Input = average educational level of parents Output = average reading SOL score Output = average math SOL score Output = average history SOL score Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-41 A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-42 A Data Envelopment Analysis (DEA) Example Decision Variables and Model Decision Variables: Summary (3 of 5) xi = a price per unit of each output where i = 1, 2, yi = a price per unit of each input where i = 1, 2, Model Summary: Maximize Z = 81x1 + 73x2 + 69x3 subject to: 06 y1 + 460y2 + 13.1y3 = 86x1 + 75x2 + 71x3 ≤ 06y1 + 260y2 + 11.3y3 82x1 + 72x2 + 67x3 ≤ 05y1 + 320y2 + Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10.5y 4-43 A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 4-44 A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Exhibit 4-45 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 Copyright © 2010 Pearson Education, Inc Publishing as profit? Prenticemaximize Hall 4-46 Example Problem Solution Model Formulation (2 of 5) Step 1: Define the Decision Variables xij = 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(xhm + xfm + xcm) + 0.06(xhb + xfb + xcb) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-47 Example Problem Solution Model Formulation (3 of 5) Step 3: Formulate the Model Constraints Amount of each ingredient available each week: xhm + xhb ≤ 9,600 ounces of horse meat xfm + xfb ≤ 12,800 ounces of fish xcm + xcb ≤ 16,000 ounces of cereal additive Recipe requirements: Meow Chow: xfm/(xhm + xfm + xcm) ≥ 1/2 or - xhm + xfmxcm ≥ Bow Chow: ≥ xhb/(xhb + xfb + xcb) ≥ 1/2 or xhb- xfb - xcb Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-48 Example Problem Solution Model Summary (4 of 5) Step 4: Model Summary Maximize Z = $0.05xhm + $0.05xfm + $0.05xcm + $0.06xhb + 0.06xfb + 0.06xcb subject to: xhm + xhb ≤ 9,600 ounces of horse meat xfm + xfb ≤ 12,800 ounces of fish xcm + xcb ≤ 16,000 ounces of cereal additive - xhm + xfm- xcm ≥ xhb- xfb - xcb ≥ xhm + xfm + xcm + xhb + xfb+ xcb ≤ 36,000 ounces xij ≥ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-49 Example Problem Solution Solution with QM for Windows (5 of 5) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-50 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 4-51 ... of Television Sets: Retail store demand for television sets: - Cincinnati 300 A - New York 150 - Atlanta 200 B - Dallas 250 - Pittsburgh 200 C - Detroit 200 Total 700 Total 600 Copyright © 2010... 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... slices of wheat toast 4-12 A Diet Example Model Summary (3 of 5) Minimize Z = 0.18x1 + 0.22x2 + 0.10x3 + 0.12x4 + 0.10x5 + 0.09x6 + 0.40x7 + 0.16x8 + 0.50x9 + 0.07x10 subject to: 90x1 + 110x2