Introduction to Management Science 8th Edition by Bernard W. Taylor III 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 Chapter 4 - Linear Programming: Modeling Examples

Problem Definition (1 of 7) A Product Mix Example Problem Definition (1 of 7) 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

A Product Mix Example Data (2 of 7) Chapter 4 - Linear Programming: Modeling Examples

Model Construction (3 of 7) A Product Mix Example Model Construction (3 of 7) Decision Variables: Objective Function: Model Constraints: Chapter 4 - Linear Programming: Modeling Examples

Computer Solution with Excel (4 of 7) A Product Mix Example Computer Solution with Excel (4 of 7) Exhibit 4.1 Chapter 4 - Linear Programming: Modeling Examples

Solution with Excel Solver Window (5 of 7) A Product Mix Example Solution with Excel Solver Window (5 of 7) Exhibit 4.2 Chapter 4 - Linear Programming: Modeling Examples

Solution with QM for Windows (6 of 7) A Product Mix Example Solution with QM for Windows (6 of 7) Exhibit 4.3 Chapter 4 - Linear Programming: Modeling Examples

Solution with QM for Windows (7 of 7) A Product Mix Example Solution with QM for Windows (7 of 7) Exhibit 4.4 Chapter 4 - Linear Programming: Modeling Examples

Data and Problem Definition (1 of 5) 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

Model Construction – Decision Variables (2 of 5) 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

A Diet Example Model Summary (3 of 5) Minimize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

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

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

An Investment Example Model Summary (1 of 5) Kathleen has $70,000 to invest. Municipal bonds (MB): 8.5% Certificates of deposit (CD): 5% Treasury bills (TB): 6.5% Growth stock fund (GSF): 13% No more than 20% in municipal bonds Amount in CD < in other 3 alternatives At 30% in treasury bills and CD Ratio in (TB and CD) to (MB and GSF) > 1.2 to 1 Chapter 4 - Linear Programming: Modeling Examples

An Investment Example Model Summary (2 of 5) Decision Variables: Maximize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

Computer Solution with Excel (2 of 5) An Investment Example Computer Solution with Excel (2 of 5) Exhibit 4.7 Chapter 4 - Linear Programming: Modeling Examples

Solution with Excel Solver Window (3 of 5) An Investment Example Solution with Excel Solver Window (3 of 5) Exhibit 4.8 Chapter 4 - Linear Programming: Modeling Examples

Sensitivity Report (5 of 5) An Investment Example Sensitivity Report (5 of 5) Exhibit 4.9 Chapter 4 - Linear Programming: Modeling Examples

Data and Problem Definition (1 of 6) 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

A Marking Example Model Summary (2 of 6) Decision variables: Maximize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

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

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

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

Integer Solution with Excel (6 of 6) A Marking Example Integer Solution with Excel (6 of 6) Exhibit 4.14 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

A Transportation Example Model Summary (2 of 4) Decision variables: Minimize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

A Transportation Example Solution with Excel (3 of 4) Exhibit 4.15 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

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

Problem Statement and Variables (2 of 6) 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

A Blend Example Model Summary (3 of 6) Maximize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

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

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

Sensitivity Report (6 of 6) A Blend Example Sensitivity Report (6 of 6) 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 Cost: $10/comp. 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

A Multi-Period Scheduling Example Decision Variables (2 of 5) rj = regular production of computers per week j (j = 1 - 6) oj = overtime production of computers per week j ij = extra computers carried over as inventory in week j (j = 1 - 5) Chapter 4 - Linear Programming: Modeling Examples

A Multi-Period Scheduling Example Model Summary (3 of 5) Minimize Z = subject to: Chapter 4 - Linear Programming: Modeling Examples

A Multi-Period Scheduling Example Solution with Excel (4 of 5) 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

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

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 = 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: Recipe requirements: Meow Chow Bow Chow Can Content Constraint 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

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