PART I [25 MARKS] A corporation has $30 million available for the coming year to allocate to its three subsidiaries. Because of commitments to stability of personnel employment and for other reasons, the corporation has established a minimal level of funding for each subsidiary. These funding levels are $3 million, $5 million, and $8 million respectively. Owing to the nature of its operation, subsidiary 2 cannot utilize more than $17 million without major new capital expansion. The corporation is unwilling to undertake such expansion at this time. Each subsidiary has the opportunity to conduct various projects with the funds it receives. A rate of return (as a percent of investment) has been established for each project. In addition, certain of the projects permit only limited investment. The data of each project are given.

An analyst has formulated an LP model for this problem as follows:

Let \[{{x}_{i}}=\text{ amount (in }\!\!\\(\!\!\text{million)toinvestinproject}i\text{,}i=1,2,...,8.\]

Use any available application software (e.g., The Management Scientist , Excel Solver, Lingo, Lindo, STORM) and print out the Optimal Solution/Results and Sensitivity Analysis outputs. Based only on the software output you printed out , answer questions 2-11. Explain/ support your answers, citing specific parts of the output used to answer the question and, where appropriate, showing how the information was applied.

1. Write out the LP model in standard form, with all technical constraints written out as equality constraints.
2. State the optimal solution in the language of the given problem – i.e., in English sentences which do NOT involve variable notations/symbols. What is the optimal total return on investment?
3. At the reported optimal solution, which constraints are binding and which are not? Interpret the values of slack/surplus variables in terms of (full, partial or non-) utilization of available funds or meeting or exceeding investment requirements.
4. If the rate of return on Project 1 changes to 9%, will the current optimal solution remain optimal? Why or why not?
5. If the rate of return on Project 3 changes to 9%, will the current optimal solution remain optimal? Why or why not?
6. If the total amount available to invest in the three subsidiaries were reduced to $28.5 million, then the optimal OFV would change to what amount? Explain your answer.
7. If the maximum amount of total investments in subsidiary 2 projects were reduced to $14 million, then the optimal OFV would change to what amount? Explain your answer.
8. If the minimum total investment in subsidiary 3 projects were reduced to $6.5 million, then the optimal OFV would change to what amount? Explain your answer.
9. If the upper limit of investment in Project 5 were reduced to $5 million, then the optimal OFV would change to what amount? Explain your answer.
10. If the upper limit of investment in Project 6 were increased to $5 million, then the optimal OFV would change to what amount? Explain your answer.
11. If the upper limit of investment in Project 7 were increased to $9 million, then the optimal OFV would change to what amount? Explain your answer.

PART II [25 MARKS]

2. [15 marks]

The Westfall Company has a contract to produce 10,000 garden hoses for a large discount chain. Westfall has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.

1. This problem requires two different kinds of decision variables. Clearly define each kind.
2. The company wants to minimize total cost. Give the objective function.
3. Give the constraints for the problem.
   d. Write a constraint to ensure that if machine 4 is used, machine 1 cannot be.
   3. [10 marks]
   RVW (Restored Volkswagens) buys 15 used VW's at each of two car auctions each week
   held at different locations. It then transports the cars to repair shops it contracts with.
   When they are restored to RVW's specifications, RVW sells 10 each to three different
   used car lots. There are various costs associated with the average purchase and
   transportation prices from each auction to each repair shop. Also there are
   transportation costs from the repair shops to the used car lots. RVW is concerned with minimizing its total cost given the costs in the table below.
   1. Given the costs below, draw a network representation for this problem.

b. Formulate this problem as a transshipment linear programming model.

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Solution: The downloadable solution consists of 12 pages, 1018 words and 4 charts.
Deliverable: Word Document