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Solving Examples of Linear Programming

Models

Chapter 4

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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

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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.

An Investment Example

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The following guidelines have been established:1. No more than 20% in municipal bonds2. Investment in CDs should not exceed the other three

alternatives3. At least 30% invested in t-bills and CDs4. More should be invested in CDs and t-bills than in

municipal bonds and growth stocks by a ratio of 1.2 to 15. All $70,000 should be invested.

An Investment ExampleModel Summary

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An Investment ExampleModel

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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

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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.

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Exposure (people/ad or commercial)

Cost

Television Commercial 20,000 $15,000

Radio Commercial 2,000 6,000

Newspaper Ad 9,000 4,000

A Marketing ExampleData 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 followingamount:

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The company must consider the following resource constraints:

1. Budget limit $100,000

2. 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

A Marketing ExampleData and Problem Definition

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A Marketing ExampleModel

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Maximize Z = 20,000x1 + 12,000x2 + 9,000x3

subject to:15,000x1 + 6,000x 2+ 4,000x3 100,000

x1 4x2 10x3 7x1 + x2 + x3 15x1, x2, x3 0

wherex1 = number of television commercialsx2 = number of radio commercialsx3 = number of newspaper ads

A Marketing ExampleModel

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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:

A Multiperiod Scheduling ExampleProblem Definition and Data

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A Multiperiod Scheduling ExampleProblem Definition and Data

Week Computer Orders1 1052 1703 2304 1805 1506 250

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Production Capacity: 160 computers per week50 more computers with overtime

Assembly Costs: $190 per computer regular time; $260 per computer overtime

Inventory Holding Cost: $10/computer per week

A Multiperiod Scheduling ExampleProblem Definition and Data

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.

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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)

A Multi-Period Scheduling ExampleDecision Variables

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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 50 computers in week j (j = 1, 2, 3, 4, 5, 6)r1 + o1 - i1 = 105 week 1r2 + o2 + i1 - i2 = 170 week 2r3 + o3 + i2 - i3 = 230 week 3r4 + o4 + i3 - i4 = 180 week 4r5 + o5 + i4 - i5 = 150 week 5r6 + o6 + i5 = 250 week 6rj, oj, ij 0

A Multi-Period Scheduling ExampleModel Summary

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Component Maximum Barrels Available/day Cost/barrel

1 4,500 $12 2 2,700 10 3 3,500 14

A Blend ExampleProblem 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:

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Grade Component Specifications Selling Price ($/bbl)

Super At least 50% of 1 Not more than 30% of 2 $23

Premium At least 40% of 1 Not more than 25% of 3

20

Extra At least 60% of 1 At least 10% of 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:

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■ 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).

A Blend ExampleProblem Statement and Variables

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Maximize Z = 11x1s + 13x2s + 9x3s + 8x1p + 10x2p + 6x3p + 6x1e+ 8x2e + 4x3e

subject to:x1s + x1p + x1e 4,500 bbl.x2s + x2p + x2e 2,700 bbl.x3s + x3p + x3e 3,500 bbl.

0.50x1s - 0.50x2s - 0.50x3s 00.70x2s - 0.30x1s - 0.30x3s 00.60x1p - 0.40x2p - 0.40x3p 00.75x3p - 0.25x1p - 0.25x2p 00.40x1e- 0.60x2e- - 0.60x3e 00.90x2e - 0.10x1e - 0.10x3e 0

x1s + x2s + x3s 3,000 bbl.x1p+ x2p + x3p 3,000 bbl.x1e+ x2e + x3e 3,000 bbl.

A Blend ExampleModel Summary

all xij 0