Business Analytics: review problems. Solutions provided at the end.

Problem 1:

A company produces two products, A and B.  Each unit of A requires 1, 3 and 2 kilograms of wood, plastic and steel respectively and each unit of B requires 3, 4 and 1 kilograms of wood, plastic and steel respectively. A maximum of 240, 360, and 180 kilograms of wood, plastic and steel available. The profit per unit of A and B is $4.00 and $6.00 respectively. The objective is to maximize total profits. Formulate this as a linear programming model.

Problem 2:

A manufacturer produces desks and chairs. Each desk uses 5 units of wood and each chair uses 3 units of wood. Total wood available for a month is 2200 units. Desk production requires 3 hours of labor and a chair needs 1.4 hours.  Total number of hours available per month: 1150. Each desk contributes 40 dollars to profit and each chair contributes $22. Marketing requires that at least 3 chairs be produced for each desk produced. The objective is to maximize total profits. Note: you can ignore the integer requirements. Formulate this as a linear programming model

Problem 3:

A resort hotel is being built in a wooded area. Four locations (“nodes”) are to be connected with paths. All locations must be connected. Building paths is costly, so the objective is to minimize the total distance of building all the paths. Formulate this as a linear program.

Problem 4:

Formulate an integer linear programming problem from the information provided below:

We want to buy two types of machines (M1 and M2); use variables X1 and X2 to denote number of machines of two types.

Objective function:  Marginal daily profitability per unit of M1 and M2 is 100 dollars and 150 dollars and you want to maximize your total daily profitability.

Resource and production constraints:

· M1 costs 15000 dollars per unit and M2 costs 4000 dollars. Your budget: 140,000 dollars.

· Space available: 200 square feet. M1 needs 15 square feet per unit and M2 needs 25 square feet per unit.

· Number of machines of type M1 must be at least twice as many as type M2.

List integer and/or other constraints at the end.

Problem 5:

Table 1 shows the payoff values for 3 alternative investment options and 3 events. Which alternative would you select if you use Mini-max regret approach?

Problem 6:

Table 1 below shows the payoff values with two alternatives A1 and A2. Each alternative has three chance events (E1, E2 and E3) with probabilities shown.

Draw a tree diagram, calculate expected value (EV) for each alternative and select the preferred alternative under risk-neutrality.

Problem 7:

A company has to decide whether to expand or not. The expansion cost is $2.1 million. If the economy improves substantially (20% probability) the (projected) revenues with expansion will be $6.0 million. If the economy turns worse (30% probability) the revenues will be $2.1 million.  With economy remaining roughly the same (50% probability), the revenues will be 4.2 million. Without expansion, the corresponding revenues will be $3.0 million, $1.2 million and 1.9 million.

(a) Draw a decision tree and calculate the expected value of each alternative.

(b) What decision would you recommend to a risk-neutral decision-maker? Do not forget to take into account the cost of expansion.

Problem 8 (from 2018 final exam):

The associate dean at the Karey Business School needs to allocate instructors to the courses offered in the next semester. There are three instructors, each of whom will be teaching exactly one module. The past evaluation scores of these instructors are as follows:

There are 3 modules, each of which has a different number of students, as shown below:

(a) (5 points) The associate dean wants to build a model for assigning the instructors to the modules. Define the decision variables for this model.

(b) (10 points) Formulate a linear program with the objective to maximize total satisfaction (the sum of satisfaction of all students). You can assume that past evaluation scores reflect the satisfaction with each instructor, and that each instructor is capable of teaching each module.

(c)  (10 points) It has been noticed that the instructors received different evaluation scores in the different courses they have taught. The scores are as follows:

Specifically, the dean wants to use the evaluation scores for qualitative courses to predict student satisfaction in module 1, and the evaluation scores for quantitative courses to predict student satisfaction in modules 2 and 3. Formulate a new model using these separate evaluation scores.

Problem 9 (from 2019 final exam)

You have built a fast food business and are now considering selling it. There are several potential acquirers, and you plan to determine the sale price in a sealed bid auction. Under the rules of the auction, the price would be the highest bid.

Due to their limited information the acquirers may underestimate or overestimate what the business is worth.  You have received indication that McDonald’s valuation of your business is uniformly distributed between $100 Million and $120 Million, and that Burger King’s valuation of your business is uniformly distributed between $100 Million and $150 Million. You believe that both companies will bid their valuation during the auction.

In preparation for the auction you want to use simulation to see how much money you can make. You have generated the following random numbers from the [0,1] uniform. distribution (each of the ten numbers has been generated independently):

(a) (2 points) How can you use these numbers to predict the bids and the final sale price? Describe the approach.

(b) (4 points) Given the numbers in the table above, what sale price do you expect, on average?

(c) (2 points) The bidders have an instinct about the Winner’s curse and they each plan to bid only 75% of their valuation (Winner’s curse is a phenomenon where the winner generally overpays during competitive bidding). How would that change your answer to (b)?

(d) (2 points) BARM Analytica, a startup out of Washington DC, offers to identify an additional bidder whose valuation is equally likely to be 100, 120 or 140. How large a fee should you be willing to pay for this service assuming that each bidder bids their true valuation? Use simulation and the following uniform. [0,1] random numbers: 0.11, 0.65, 0.89, 0.23, 0.53 to justify your answer.