In the situations described below in problems 1 and 2, identify the objective function, whether it is a maximization or minimization problem. Also identify the free variables, the explicit constraints and the implicit constraints.
16.1. A company manufactures two products, widgets and greebles, on two machines, I and II. It has been determined that the company will realize a profit of $6 on each widget and a profit of $8 on each greeble. To manufacture a widget requires 9 minutes on machine I and 7 minutes on machine II. To manufacture a greeble requires 13 minutes on machine I and 5 minutes on machine II. There are 5 hours of machine time available on machine I and 3 hours of machine time available on machine II in each work shift. How many units of each product should be produced?
16.2. Harbor Tours operates a fleet of ocean vessels for touring the coast around the Alaska. The fleet has two types of vessels: Cruisers have 60 deluxe cabins and 160 standard cabins, while Corvettes have 80 deluxe and 120 standard cabins. Under a chartered agreement with Glacier Travel Agency, Harbor Tours is to provide Glacier Travel with a minimum of 410 deluxe and 720 standard cabins for their 15-day cruise in July. If costs $65,000 to operate a cruiser and $82,000 to operate a corvette for that period of time. How many of each type vessel should be used?
16.3. A financier plans to invest up to $5 million in three projects. She estimates that project A will yield a return of 12% on the investment, project B will yield a return of 16% on the investment and project C will yield a return of 23% on her investment. Because of the risks associated with the investments, she decides to put not more than 20% of the total investment into project C. She also decided that her investments in projects B and C should not exceed 60% of the total investment. Finally she decided that her investment in project A should be at least 60% of her investment in projects B and C.
In this problem, we are obviously trying to maximize the financier’s return on her investments (the objective function) by altering the free variables (the total amount of money invested in each of the three projects). We have four explicit constraints: (1) Investment in B should be less than or equal to 20% of the total investment. (2) Investment in B and C should be less than or equal to 60% of the total investment. (3) Investment in A should be greater than or equal to 60% of the total investment in B and C. (4) The total of all the investments must not exceed $5 million. We also have some implicit constraints: (1) All three investments must be positive or zero. (2) All three investments must be a monetary value with at most two decimal places, when measured in dollars.
16.4. Ivory Keys manufactures upright and console pianos in two plants, Boise and Canton. The output of Boise is at most 300/month, whereas the output of Canton is at most 250/month. These pianos are shipped to three warehouses that serve as distribution centers for the company. To fill current and projected future orders, the warehouse in Seattle requires a minimum of 200 pianos/month, the warehouse in Dallas requires at least 150 pianos/month, and the warehouse in Pittsburgh requires at least 200 pianos/month. The shipping costs from Boise to Seattle, Dallas and Pittsburgh are $60, $60, and $80, respectively. The shipping costs from Canton to Seattle, Dallas and Pittsburgh are $80, $70, and $50, respectively.
Clearly, in this problem we want to minimize the shipping costs (objective function) by altering the free variables (the number of each type of piano shipped from Boise to Seattle, Dallas and Pittsburgh and the number of each type of piano shipped from Canton to Seattle, Dallas and Pittsburgh; that’s six variables!). We have five explicit constraints: (1) The number of pianos shipped from Boise cannot exceed 300. (2) The number of pianos shipped from Canton cannot exceed 250/month. (3) Seattle needs at least 200 pianos/month. (4) Dallas needs at least 150 pianos/month. (5) Pittsburgh needs at least 200 pianos/month. Finally, the only implicit constraints are that each shipping value must be positive and an integer.
16.5. Return to the furniture production example explored in this chapter. Modify the spreadsheet to include the additional constraint that you must produce at least four times as many chairs as tables. Solve this problem with the additional constraint.