Introduction to Management Science 9th Edition by Bernard W. Taylor III Chapter 4 Linear Programming: Modeling Examples © 2007 Pearson Education Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples Chapter Topics A Product Mix Example A Diet Example An Investment Example A Marketing Example A Transportation Example A Blend Example A Multi-Period Scheduling Example A Data Envelopment Analysis Example Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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? Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example (2 of 8) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example Data (3 of 8) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 sweatshirts x3 + x4  500 dozen T-shirts Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example Computer Solution with Excel (5 of 8) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.1

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example Solution with Excel Solver Window (6 of 8) Exhibit 4.2 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example Solution with QM for Windows (7 of 8) Exhibit 4.3 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Product Mix Example Solution with QM for Windows (8 of 8) Exhibit 4.4 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Diet Example Data and Problem Definition (1 of 5) 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. Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Diet Example Model Construction – Decision Variables (2 of 5) x1 = cups of bran cereal x2 = cups of dry cereal x3 = cups of oatmeal x4 = cups of oat bran x5 = eggs x6 = slices of bacon x7 = oranges x8 = cups of milk x9 = cups of orange juice x10 = slices of wheat toast Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 + 100x3 + 90x4 + 75x5 + 35x6 + 65x7 + 100x8 + 120x9 + 65x10  420 2x2 + 2x3 + 2x4 + 5x5 + 3x6 + 4x8 + x10  20 270x5 + 8x6 + 12x8  30 6x1 + 4x2 + 2x3 + 3x4+ x5 + x7 + x10  5 20x1 + 48x2 + 12x3 + 8x4+ 30x5 + 52x7 + 250x8 + 3x9 + 26x10  400 3x1 + 4x2 + 5x3 + 6x4 + 7x5 + 2x6 + x7+ 9x8+ x9 + 3x10  20 5x1 + 2x2 + 3x3 + 4x4+ x7 + 3x10  12 xi  0, for all j Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Diet Example Computer Solution with Excel (4 of 5) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.5

Chapter 4 - Linear Programming: Modeling Examples A Diet Example Solution with Excel Solver Window (5 of 5) Exhibit 4.6 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples An Investment Example Model Summary (1 of 4) Maximize Z = $0.085x1 + 0.05x2 + 0.065 x3+ 0.130x4 subject to: x1  14,000 x2 - x1 - x3- x4  0 x2 + x3  21,000 -1.2x1 + x2 + x3 - 1.2 x4  0 x1 + x2 + x3 + x4 = 70,000 x1, x2, x3, x4  0 where x1 = amount invested in municipal bonds ($) x2 = amount invested in certificates of deposit ($) x3 = amount invested in treasury bills ($) x4 = amount invested in growth stock fund($) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples An Investment Example Computer Solution with Excel (2 of 4) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.7

Chapter 4 - Linear Programming: Modeling Examples An Investment Example Solution with Excel Solver Window (3 of 4) Exhibit 4.8 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples An Investment Example Sensitivity Report (4 of 4) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.9

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Data and Problem Definition (1 of 6) 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 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Model Summary (2 of 6) Maximize Z = 20,000x1 + 12,000x2 + 9,000x3 subject to: 15,000x1 + 6,000x 2+ 4,000x3  100,000 x1  4 x2  10 x3  7 x1 + x2 + x3  15 x1, x2, x3  0 where x1 = Exposure from Television Commercial (people) x2 = Exposure from Radio Commercial (people) x3 = Exposure from Newspaper Ad (people) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Solution with Excel (3 of 6) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.10

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Solution with Excel Solver Window (4 of 6) Exhibit 4.11 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Integer Solution with Excel (5 of 6) Exhibit 4.12 Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.13

Chapter 4 - Linear Programming: Modeling Examples A Marketing Example Integer Solution with Excel (6 of 6) Exhibit 4.14 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Transportation Example Problem Definition and Data (1 of 3) Warehouse supply of Retail store demand Television Sets: for television sets: 1 - Cincinnati 300 A - New York 150 2 - Atlanta 200 B - Dallas 250 3 - Pittsburgh 200 C - Detroit 200 Total 700 Total 600 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Transportation Example Model Summary (2 of 4) Minimize Z = $16x1A + 18x1B + 11x1C + 14x2A + 12x2B + 13x2C + 13x3A + 15x3B + 17x3C subject to: x1A + x1B+ x1C  300 x2A+ x2B + x2C  200 x3A+ x3B + x3C  200 x1A + x2A + x3A = 150 x1B + x2B + x3B = 250 x1C + x2C + x3C = 200 All xij  0 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Transportation Example Solution with Excel (3 of 4) Exhibit 4.15 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Transportation Example Solution with Solver Window (4 of 4) Exhibit 4.16 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Problem Definition and Data (1 of 6) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Problem Statement and Variables (2 of 6) 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); xij = 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). Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Model Summary (3 of 6) Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e + 8x2e + 4x3e subject to: x1s + x1p + x1e  4,500 x2s + x2p + x2e  2,700 x3s + x3p + x3e  3,500 0.50x1s - 0.50x2s - 0.50x3s  0 0.70x2s - 0.30x1s - 0.30x3s  0 0.60x1p - 0.40x2p - 0.40x3p  0 0.75x3p - 0.25x1p - 0.25x2p  0 0.40x1e- 0.60x2e- - 0.60x3e  0 0.90x2e - 0.10x1e - 0.10x3e  0 x1s + x2s + x3s  3,000 x1p+ x2p + x3p  3,000 x1e+ x2e + x3e  3,000 xij  0 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Solution with Excel (4 of 6) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.17

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Solution with Solver Window (5 of 6) Exhibit 4.18 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Blend Example Sensitivity Report (6 of 6) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.19

Chapter 4 - Linear Programming: Modeling Examples A Multi-Period Scheduling Example Problem Definition and Data (1 of 5) 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: Week Computer Orders 1 105 2 170 3 230 4 180 5 150 6 250 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 ij = extra computers carried over as inventory in week j (j = 1, 2, …, 5) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 (j = 1, 2, 3, 4, 5, 6) oj  150 (j = 1, 2, 3, 4, 5, 6) r1 + o1 - i1 = 105 r2 + o2 + i1 - i2 = 170 r3 + o3 + i2 - i3 = 230 r4 + o4 + i3 - i4 = 180 r5 + o5 + i4 - i5 = 150 r6 + o6 + i5 = 250 rj, oj, ij  0 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Multi-Period Scheduling Example Solution with Excel (4 of 5) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.20

Chapter 4 - Linear Programming: Modeling Examples A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Exhibit 4.21 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 1 = teacher to student ratio Output 1 = average reading SOL score Input 2 = supplementary $/student Output 2 = average math SOL score Input 3 = parent education level Output 3 = average history SOL score Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Data Envelopment Analysis (DEA) Example Decision Variables and Model Summary (3 of 5) Decision Variables: xi = a price per unit of each output where i = 1, 2, 3 yi = a price per unit of each input where i = 1, 2, 3 Model Summary: Maximize Z = 81x1 + 73x2 + 69x3 subject to: .06 y1 + 460y2 + 13.1y3 = 1 86x1 + 75x2 + 71x3 .06y1 + 260y2 + 11.3y3 82x1 + 72x2 + 67x3  .05y1 + 320y2 + 10.5y3 81x1 + 79x2 + 80x3  .08y1 + 340y2 + 12.0y3 81x1 + 73x2 + 69x3  .06y1 + 460y2 + 13.1y3 xi, yi  0 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Chapter 4 - Linear Programming: Modeling Examples Exhibit 4.22

Chapter 4 - Linear Programming: Modeling Examples A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Exhibit 4.23 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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? Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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) Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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 + xfm- xcm  0 Bow Chow xhb/(xhb + xfb + xcb)  1/2 or xhb- xfb - xcb  0 Can Content Constraint xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples 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  0 xhb- xfb - xcb  0 xhm + xfm + xcm + xhb + xfb+ xcb  36,000 ounces xij  0 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples Example Problem Solution Solution with QM for Windows (5 of 5) Exhibit 4.24 Chapter 4 - Linear Programming: Modeling Examples

Chapter 4 - Linear Programming: Modeling Examples End of chapter Chapter 4 - Linear Programming: Modeling Examples