Rate of Return and Non-negativity Constraints

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LINEAR PROGRAMMING FORMULATION PROBLEMS AND SOLUTIONS

7-14 The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time maybe used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix situation to find the best combination of air conditioners and fans that yields the highest profit. Use the corner point graphical approach.

Let X1 = the number of air conditioners scheduled to be produced
X2 = the number of fans scheduled to be produced

Maximize| 25X1| +| 15X2| | | (maximize profit)|
Subject to:| 3X1| +| 2X2| ≤| 240| (wiring capacity constraint)| | 2X1| +| X2| ≤| 140| (drilling capacity constraint)| | | | X1, X2| ≥| 0| (non-negativity constraints)|

Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900

7-15 Electrocomp’s management realizes that it forgot to include two critical constraints (see Problem 7-14). In particular, management decides that to ensure an adequate supply of air conditioners for a contract, at least 20 air conditioners should be manufactured. Because Electrocomp incurred an oversupply of fans in the preceding period, management also insists that no more than 80 fans be produced during this production period. Resolve this product mix problem to find the new optimal solution.

Let X1 = the number of air conditioners scheduled to be produced
X2 = the number of fans scheduled to be produced

Maximize| 25X1| +| 15X2| | | (maximize profit)|
Subject to:| 3X1| +| 2X2| ≤| 240| (wiring capacity constraint)| | 2X1| +| X2| ≤| 140| (drilling capacity constraint)| | X1| | | ≥| 20| (a/c contract constraint)|
| | | X2| ≤| 80| (maximum # of fans constraint)|
| | | X1, X2| ≥| 0| (non-negativity constraints)|

Optimal Solution: X1 = 40 X2 = 60 Profit = $1,900

7-16 A candidate for mayor in a small town has allocated $40,000 for last-minute advertising in the days preceding the election. Two types of ads will be used: radio and television. Each radio ad costs $200 and reaches an estimated 3,000 people. Each television ad costs $500 and reaches an estimated 7,000 people. In planning the advertising campaign, the campaign manager would like to reach as many people as possible, but she has stipulated that at least 10 ads of each type must be used. Also, the number of radio ads must be at least as great as the number of television ads. How many ads of each type should be used? How many people will this reach?

Let X1 = the number of radio ads purchased
X2 = the number of television ads purchased

Maximize| 3,000X1| +| 7,000X2| | | (maximize exposure)| Subject to:| 200X1| +| 500X2| ≤| 40,000| (budget constraint)| | X1| | | ≥| 10| (at least 10 radio ads purchased)| | | | X2| ≥| 10| (at least 10 television ads purchased)| | | | X1| ≥| X2| (# of radio ads ≥ # of television ads)| | | | X1, X2| ≥| 0| (non-negativity constraints)|

For solution purposes, the fourth constraint would be rewritten as: | X1| −| X2| ≥| 0|

Optimal Solution: X1 = 175 X2 = 10 Exposure = 595,000 people

7-17 The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in the furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3500 feet of...
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