Chapter 4. Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall 4-1

Transcription

1 Linear Programming: Modeling Examples Chapter 4 4-1

2 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 4-2

3 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? 4-3

4 A Product Mix Example (2 of 8) 4-4

5 A Product Mix Example Data (3 of 8) Processing Time (hr) Per dozen Cost ($) per dozen Profit ($) per dozen Sweatshirt - F 0.10 $36 $90 Sweatshirt B/F T-shirt - F T-shirt - B/F

6 A Product Mix Example Model Construction (4 of 8) 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 x x x 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 dozen sweatshirts x 3 + x dozen T-shirts 4-6

7 A Product Mix Example Computer Solution with Excel (5 of 8) Exhibit

8 A Product Mix Example Solution with Excel Solver Window (6 of 8) Exhibit

9 A Product Mix Example Solution with QM for Windows (7 of 8) Exhibit

10 A Product Mix Example Solution with QM for Windows (8 of 8) Exhibit

11 A Diet Example Data and Problem Definition (1 of 5) Breakfast Food 1. Bran cereal (cup) 2. Dry cereal (cup) 3. Oatmeal (cup) 4. Oat bran (cup) 5. Egg 6. Bacon (slice) 7. Orange 8. Milk-2% (cup) 9. Orange juice (cup) 10. Wheat toast (slice) Cal Fat (g) Cholesterol (mg) Iron (mg) Calcium (mg) Protein (g) Fiber (g) Cost ($) 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. 4-11

12 A Diet Example Model Construction Decision Variables (2 of 5) 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 4-12

13 A Diet Example Model Summary (3 of 5) Minimize Z = 0.18x x x x x x x x x x 10 subject to: 90x x x x x x x x x x calories 2x 2 + 2x 3 + 2x 4 + 5x 5 + 3x 6 + 4x 8 + x g fat 270x 5 + 8x x 8 30 mg cholesterol 6x 1 + 4x 2 + 2x 3 + 3x 4 + x 5 + x 7 + x 10 5 mg iron 20x x x 3 + 8x x x x 8 + 3x x mg of calcium 3x 1 + 4x 2 + 5x 3 + 6x 4 + 7x 5 + 2x 6 + x 7 + 9x 8 + x 9 + 3x g protein 5x 1 + 2x 2 + 3x 3 + 4x 4 + x 7 + 3x x i 0, for all j 4-13

14 A Diet Example Computer Solution with Excel (4 of 5) Exhibit

15 A Diet Example Solution with Excel Solver Window (5 of 5) Exhibit

16 An Investment Example Model Summary (1 of 4) Maximize Z = $0.085x x x x 4 subject to: x 1 $14,000 x 2 - x 1 - x 3 - x 4 0 x 2 + x 3 $21, x 1 + x 2 + x 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 4-16

17 An Investment Example Computer Solution with Excel (2 of 4) Exhibit

18 An Investment Example Solution with Excel Solver Window (3 of 4) Exhibit

19 An Investment Example Sensitivity Report (4 of 4) Exhibit

20 A Marketing Example Data and Problem Definition (1 of 6) Exposure (people/ad or Cost commercial) Television Commercial 20,000 $15,000 Radio Commercial 2,000 6,000 Newspaper Ad 9,000 4,000 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 4-20

21 A Marketing Example Model Summary (2 of 6) Maximize Z = 20,000x ,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 4-21

22 A Marketing Example Solution with Excel (3 of 6) Exhibit

23 A Marketing Example Solution with Excel Solver Window (4 of 6) Exhibit

24 A Marketing Example Integer Solution with Excel (5 of 6) Exhibit 4.12 Exhibit

25 A Marketing Example Integer Solution with Excel (6 of 6) Exhibit

26 A Transportation Example Problem Definition and Data (1 of 3) Warehouse supply of Television Sets: Retail store demand for television sets: 1 - Cincinnati 300 A - New York Atlanta 200 B - Dallas Pittsburgh 200 C - Detroit 200 Total 700 Total 600 Unit Shipping Costs: From Warehouse To Store A B C 1 $16 $18 $

27 A Transportation Example Model Summary (2 of 4) 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

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

29 A Transportation Example Solution with Solver Window (4 of 4) Exhibit

30 A Blend Example Problem Definition and Data (1 of 6) Component Maximum Barrels Available/day Cost/barrel 1 4,500 $12 2 2, , Grade Component Specifications Selling Price ($/bbl) Super Premium Extra At least 50% of 1 Not more than 30% of 2 $23 At least 40% of 1 Not more than 25% of 3 20 At least 60% of 1 At least 10% of

31 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); 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). 4-31

32 A Blend Example Model Summary (3 of 6) 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 x 2s x 3s x 2s x 1s x 3s x 1p x 2p x 3p x 3p x 1p x 2p x 1e x 2e x 3e x 2e x 1e x 3e 0 x 1s + x 2s + x 3s 3,000 bbl. x 1p + x 2p + x 3p 3,000 bbl. all x ij 0 x 1e + x 2e + x 3e 3,000 bbl. 4-32

33 A Blend Example Solution with Excel (4 of 6) Exhibit

34 A Blend Example Solution with Solver Window (5 of 6) Exhibit

35 A Blend Example Sensitivity Report (6 of 6) Exhibit

36 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

37 A Multi-Period Scheduling Example Decision Variables (2 of 5) 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) 4-37

38 A Multi-Period Scheduling Example Model Summary (3 of 5) 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 = 105 week 1 r 2 + o 2 + i 1 - i 2 = 170 week 2 r 3 + o 3 + i 2 - i 3 = 230 week 3 r 4 + o 4 + i 3 - i 4 = 180 week 4 r 5 + o 5 + i 4 - i 5 = 150 week 5 r 6 + o 6 + i 5 = 250 week 6 r j, o j, i j

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

40 A Multi-Period Scheduling Example Solution with Solver Window (5 of 5) Exhibit

41 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 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 4-41

42 A Data Envelopment Analysis (DEA) Example Problem Data Summary (2 of 5) Inputs Outputs School Alton.06 $ Beeks Carey Delancey

43 A Data Envelopment Analysis (DEA) Example Decision Variables and Model Summary (3 of 5) 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 x x 3 subject to:.06 y y y 3 = 1 86x x x 3.06y y y 3 82x x x 3.05y y y 3 81x x x 3.08y y y 3 81x x x 3.06y y y 3 x i, y i

44 A Data Envelopment Analysis (DEA) Example Solution with Excel (4 of 5) Exhibit

45 A Data Envelopment Analysis (DEA) Example Solution with Solver Window (5 of 5) Exhibit

46 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? 4-46

47 Example Problem Solution Model Formulation (2 of 5) 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 ) (x hb + x fb + x cb ) 4-47

48 Example Problem Solution Model Formulation (3 of 5) 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 4-48

49 Example Problem Solution Model Summary (4 of 5) Step 4: Model Summary Maximize Z = $0.05x hm + $0.05x fm + $0.05x cm + $0.06x hb x fb x 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

50 Example Problem Solution Solution with QM for Windows (5 of 5) 4-50

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