Investment Goals and Optimal Locations - Problem Solving

• For general instructions about assignments, please see Section 3 of the Course Out-line, especially Section 3.3 on file formats.

• This document has been created only for internal use at Memorial University. This pdf file is not to be forwarded to anyone not currently registered in Business 2400. Any transmission of the solutions to this assignment to a third party constitutes academic dishonesty.

1. (20 marks) Cindy has $9000 to invest in a portfolio. Her investment alternatives and their expected returns are:

She will put at least $1500 into each investment, and cannot invest more than $9000 in total. In addition, she has three goals which may be violated if need be. Goal 1 is for the expected annual return to be at least $550. Goal 2 is to invest no more than $4000 in the two retirement plans combined. Goal 3 is to invest in investment 3 at least twice as much as in investment 1.

Goal 1 is five times as important as Goal 2, and Goal 2 is twice as important as Goal 3.

• Write the algebraic model for this problem.

• Solve the model using LINGO or the Excel Solver.

• State the solution in words.

2. (25 marks) A licence to shoot a moose costs $115 for residents and $920 for non-residents. The government must decide how many licences to issue in both cate-gories. There is a demand for up to 30,000 resident licences, and up to 12,000 non-resident licences; these are system constraints. The government has several goal priorities which in descending order of importance are (i) earn at least $12,006,000 in revenue (ii) issue at least 80% of licences to residents, and (iii) limit the total number of licences to 40,000.

• Formulate this goal programming model.

• Give the algebraic model for the first sub-problem, and solve this using LINGO or the Excel Solver.

• Embedding the solution from (b), give the algebraic model for the second sub-problem, and solve this using LINGO or the Excel Solver.

• Embedding the solution from (c), give the algebraic model for the third sub-problem, solve this using LINGO or the Excel Solver, and state the overall solution in words.

3. (20 marks) A university library has received an endowment which generates an annual income of $10,000. The library has decided that it will use all or some of this money to pay for new annual subscriptions to electronic versions of journals. There are five such journals under consideration. To help decide what to choose, for each journal they have three pieces of information: (i) for each journal they know the annual subscription cost in dollars, (ii) they know the number of times each journal is cited by articles in other journals, and (iii) they know the average faculty member’s evaluations of journals based on a scale of 0 to 10.

The budget can be taken as a system constraint. There are three goals:

• The number of unsubscribed journals should be as low as possible.

• The average number of citations of the subscribed journals should meet or exceed the average number of citations of all five journals.

• The average faculty evaluations of the subscribed journals should be at least 8.

The first goal is five times as important as the third goal, and the second goal is twice as important as the third goal.

• Define the set of decision variables and the three deviational variables. write the three sets of deviational constraints, and finally putting all this into an algebraic model with an objective function and the budget constraint. (Hint: For two of the deviational constraints, you will need to cross-multiply before inserting the deviational variable.)

• Solve the model in LINGO or the Excel Solver, and state the recommended solution.

4. (15 marks) A company wishes to open a cinema complex, and needs to determine its location. There are five neighbourhoods nearby from which patrons might come. The location of each neighbourhood is described by distances in metres east and north of a standard point. The potential business from a neighbourhood varies lin-early with its population, but is inversely proportional to the Euclidean distance between the neighbourhood and the cinema complex. The cinema complex must be at least 500 metres away from each neighbourhood. The total potential business is the sum of the potential from each of the five neighbourhoods. The population and location of the neighbourhoods are as follows:

Solve for the optimal location of the cinema complex in LINGO or the Excel Solver.

5. (20 marks) Before doing this problem, look at the Steiner tree Wikipedia article, found here. See the picture of the four point solution.

A mining company has four mines located only a few kilometres apart. At 500 metres below the surface of the earth, they wish to build a tunnel system to connect these four mines. The most convenient points for these connections are at reference points (100, 2300), (300, 400), (3700, 1800), and (4100, 600), where (a; b) is a metres east and b metres north of a common reference point. The company wishes to determine where the connecting tunnels should be made.

• Give the algebraic model of this problem.

• Solve this model using LINGO or the Excel Solver.

• State the solution in words.