Solving Examples of Linear Programming Models

Transcription

1 Solving Examples of Linear Programming Models Chapter 4 Copyright 2013 Pearson Education 4-1

2 Chapter Topics 1. A Product Mix Example 2. A Diet Example 3. An Investment Example 4. A Marketing Example 5. A Transportation Example 6. A Blend Example 7. A Multiperiod Scheduling Example 8. A Data Envelopment Analysis Example Copyright 2013 Pearson Education 4-2

3 An Investment Example Kathleen Allen, an individual investor, has $70,000 to divide among several investments. The alternative investments are municipal bonds with an 8.5% annual return, certificates of deposit with a 5% return, treasury bills with a 6.5% return, and a growth stock fund with a 13% annual return. The investments are all evaluated after 1 year. However, each investment alternative has a different perceived risk to the investor; thus, it is advisable to diversify. Kathleen wants to know how much to invest in each alternative in order to maximize the return. 4-3

4 An Investment Example Model Summary 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. 4-4

5 An Investment Example Model 4-5

6 No more than 20% in municipal bonds Investment in CDs should not exceed the other three alternatives At least 30% invested in t-bills and CDs Copyright 2013 Pearson Education 4-6

7 More should be invested in CDs and t-bills than in municipal bonds and growth stocks by a ratio of 1.2 to 1 All $70,000 should be invested. 4-7

8 A Marketing Example Data and Problem Definition The Biggs Department Store chain has hired an advertising firm to determine the types and amount of advertising it should invest in for its stores. The three types of advertising available are television and radio commercials and newspaper ads. The retail chain desires to know the number of each type of advertisement it should purchase in order to maximize exposure. It is estimated that each ad or commercial will reach the following potential audience and cost the following amount: Exposure (people/ad or Cost commercial) Television Commercial 20,000 $15,000 Radio Commercial 2,000 6,000 Newspaper Ad 9,000 4,

9 A Marketing Example Data and Problem Definition The company must consider the following resource constraints: 1. Budget limit $100, Television time for four commercials 3. Radio time for 10 commercials 4. Newspaper space for 7 ads 5. Resources for no more than 15 commercials and/or ads 4-9

10 A Marketing Example Model 4-10

11 A Marketing Example Model 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 4-11

12 A Multiperiod Scheduling Example Problem Definition and Data PM Computers assembles its own brand of personal computers from component parts it purchases overseas and domestically. PM sells most of its computers locally to different departments at State University as well as to individuals and businesses in the immediate geographic region. PM has enough regular production capacity to produce 160 computers per week. It can produce an additional 50 computers with overtime. The cost of assembling, inspecting, and packaging a computer during regular time is $190. Overtime production of a computer costs $260. Furthermore, it costs $10 per computer per week to hold a computer in inventory for future delivery. PM wants to meet all customer orders, with no shortages, to provide quality service. PM s order schedule for the next 6 weeks is as follows: 4-12

13 A Multiperiod Scheduling Example Problem Definition and Data Week Computer Orders

14 A Multiperiod Scheduling Example Problem Definition and Data 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 PM Computers wants to determine a schedule that will indicate how much regular and overtime production it will need each week to meet its orders at the minimum cost. The company wants no inventory left over at the end of the 6-week period. 4-14

15 A Multi-Period Scheduling Example Decision Variables 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) 4-15

16 A Multi-Period Scheduling Example Model Summary 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 50 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

17 A Blend Example Problem Definition and Data A petroleum company produces three grades of motor oil super, premium, and extra from three components. The company wants to determine the optimal mix of the three components in each grade of motor oil that will maximize profit. The maximum quantities available of each component and their cost per barrel are as follows: Component Maximum Barrels Available/day Cost/barrel 1 4,500 $12 2 2, ,

18 To ensure the appropriate blend, each grade has certain general specifications. Each grade must have a minimum amount of component 1 plus a combination of other components, as follows: 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

19 A Blend Example Problem Statement and Variables 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). 4-19

20 Copyright 2013 Pearson Education 4-20

21 Copyright 2013 Pearson Education 4-21

22 Copyright 2013 Pearson Education 4-22

23 A Blend Example Model Summary 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 -0.50x 3s x 2s x 1s -0.30x 3s x 1p x 2p x 3p x 3p -0.25x 1p x 2p x 1e x 2e x 3e x 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. all x ij 0 x 1e + x 2e + x 3e 3,000 bbl. 4-23