1. Consider the following linear programming problem: Maximize Z = 400 x + 100y Subject to 8 x + 10y ≤ 80 2 x + 6y ≤ 36 x≤ 6 x, y ≥ 0
Find the optimal solution using the graphical method (use graph paper). Identify the feasible region and the optimal solution on the graph. How much is the maximum profit? Consider the following linear programming problem: Minimize Z = 3 x + 5 y (cost, $) subject to 10 x + 2 y ≥ 20 6 x + 6 y ≥ 36 y ≥ 2 x, y ≥ 0 Find the optimal solution using the graphical method (use graph paper). Identify the feasible region and the optimal solution on the graph. How much is the minimum cost? 2. The Turner-Laberge Brokerage firm has just been instructed by one of its clients to invest $250 000 for her, money obtained recently through the sale of land holdings in British Columbia. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. She requests that the firm select whatever stocks and bonds they believe are well rated but within the following guidelines: 1. At least 20% of the investment should be in accounts with only Canadian content. 2. At least 40% of the investment should be placed in a combination of U.S. electronics firms, aerospace firms, and pharmaceutical companies. 3. No more than 50% of the invested amount should be in precious metals. 4. Ratio of aerospace to pharmaceutical investment should be at least 2 : 1 . Subject to these restrains, the client’s goal is to maximize projected return on investments. The analysts at Turner-Laberge, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return. Projected Rate of Return (%) Investment 3. Canadian RRSP Thompson Electronics, Inc. (USA) United Aerospace Corp. (USA) Palmer Pharmaceuticals (USA) Alberta Gold Mines (Canada) Formulate this portfolio selection problem using LP. 5.3 6.8 4.9 8.4 11.8
2 4. The manager of a department store in Seattle is attempting to decide on the types and amounts of advertising the store should use. He has invited representatives from the local radio station, television station, and newspaper to make presentations in which they describe their audiences. The television station representative indicates that a TV commercial, which costs $15 000, would reach 25 000 potential customers. The breakdown of the audience is as follows. Male Female Senior 5 000 5 000
Young 5 000 10 000 The newspaper representative claims to be able to provide an audience of 10 000 potential customers at a cost of $4 000 per ad. The breakdown of the audience is as follows Male Female Senior 4 000 3 000 Young 2 000 1 000 The radio station representative says that the audience for one of the station’s commercials, which costs $6 000, is 15 000 customers. The breakdown of the audience is as follows Male Female Senior 1 500 1 500 Young 4 500 7 500 The store has the following advertising policy: Use at least twice as many radio commercials as newspaper ads Reach at least 100 000 customers Reach at least twice as many young people as senior citizens Make sure that at least 30% of the audience is female Available space limits the number of newspaper ads to seven. The store wants to know the optimal number of each type of advertising to purchase to minimize total cost. Formulate a linear programming model for this problem. The Pyrotec Company produces three electrical products – clocks, radios, and toasters. These products have the following resource requirements. Resource Requirements Cost/Unit Labor Hours/Unit Clock Radio Toaster $7 $10 $5 2 3 2
The manufacturer has a daily production budget of $2 000 and a maximum of 660 hours of labor. Maximum daily customer demand is for 200 clocks, 300 radios, and 150 toasters. Clocks sell for $15, radios for $20, and toasters for $12. The company desires to know the optimal product mix that will maximize...