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Business Analytics – Fall II 2023

Sample Exam Questions

Maximum 40 points: Points for each question are given below.

General Instructions

· This is a 3-hour open-book, open-note exam.

· There are three problems for a total of 40 points.

· The difficulty levels of the questions in each problem are gradually increasing. Use your time well.

· For the simulation problem, please use the provided continuous uniform numbers as instructed in the problem description. The uniform numbers are in a file named “Continuous Uniform 0-1.xlsx,” available for download on Canvas under Assignments/Final Exam.

· Upload a Word/PDF document with your write-ups and an Excel file with all necessary supporting results on Canvas under Assignments/Final Exam. Points will be deducted for answers without justification.

· If you get correct answers with justification, points will not be deducted for not having GAPs; however, if you do not get the correct answers but provide some correct elements of GAPs, partial credits will be given.

· Any uploads received up to 30 minutes late will receive a 5-point deduction. Any uploads received beyond 30 minutes late will receive a 15-point deduction.

· You must keep your camera on during the 3 hours, and no virtual backgrounds are allowed. Any violation will receive a 15-point deduction.

Problem I: Optimization (13 points)

JH Logistics is trying to fulfill five orders (denoted as O) from two of its warehouses (denoted as W) to maximize its profit. The information about order quantities to be fulfilled, the warehouse capacities, and the profit margins ($/unit) for fulfilling each unit from each warehouse is summarized as follows:

Profit margin	O1	O2	O3	O4	O5	Capacity
W1	2.8	4.2	3.5	2.2	3.1	1600
W2	2.5	3.6	2.2	4	3.2	2500
Order quantity	1000	800	1200	630	750

a) How many units should JH Logistics send from each warehouse to fulfill each order? You can assume that orders can be fulfilled in fractional numbers in this part. (4 points)

Hint: It might be helpful to check the total capacity and the total order quantity before formulating the problem. Can all the orders be fulfilled?

b) Suppose there is a technology that can increase the capacity of warehouse 1 to 1800. How much is JH Logistics willing to pay for this technology at most? Justify your answer without resolving the problem. (3 points)

c) It is required that at least 25% of order 4 fulfilled by the two warehouses must come from warehouse

1. Besides, no orders can be fulfilled in fractional numbers. Determine the profit-maximizing fulfillment plan by modifying your formulation in part (a) with these requirements. (3 points)

d) Assume that the requirements in part (c) remain. In addition, order 2 cannot be fulfilled partially or separately from two warehouses. That is, the 800 units of order 2 should be fulfilled either all from warehouse 1 or all from warehouse 2. Otherwise, 0 units can be fulfilled by either warehouse. Determine the profit-maximizing fulfillment plan by modifying your formulation in part (c) with this additional requirement. (3 points)

Problem II: Risk Modeling (13 Points)

Mike T is planning to invest in a recently started practice in Landscape Design, which he will operate for five years. Over the five years, demand could take three different levels, which he estimates as the following:

Poor market (30% chance): demand will be low (10,000 units) for each of the five years. Fair market (40% chance): demand will be medium (15,000 units) for each of the five years. Good market (30% chance): demand will be high (20,000 units) for each of the five years.

To run his landscaping business, Mike must first purchase an automated drafting system. There are three systems, each with different drafting capacities and associated purchase costs:

1) The Small System can handle up to 10,000 units each year and costs $5,000.

2) The Medium System can handle up to 15,000 units each year and costs $10,000.

3) The Large System can handle up to 20,000 units each year and costs $15,000.

Mike is planning to charge his customers $0.50 per unit sold. The only type of cost he is considering is the fixed cost of purchasing the drafting system. For example, if Mike chooses the Medium System and the market is poor, then the total revenue is 5 * 10,000 * $0.5 = $25,000, and the total cost is the purchase cost of the Medium System ($10,000). (Hint: You might want to think carefully about the total revenue in a good market with the Medium System.)

a) What is the best decision under the minimax regret criterion? Please provide a regret table to justify your answer. (4 points)

b) What is the best decision under the expected profit criterion? You can answer this question with a payoff table or a decision tree. (4 points)

c) Suppose there is a marketing research company called Clairvoyant that can tell Mike which of the three demand scenarios will occur. What is the maximum amount Mike should be willing to pay for such information if his objective is to maximize the expected profit? (3 points)

d) It turns out that Clairvoyant is a hoax: no one can perfectly predict the future. The most reliable and experienced marketing research company in the city where Mike lives is called Semi- Clairvoyant, which provides investment suggestions based on comprehensive market analysis. According to historical data, the probabilities that Semi-Clairvoyant recommends investment given the market condition is poor, fair, or good are 0.1, 0.75, and 0.9, respectively. Given that Semi- Clairvoyant recommends investment, what is the probability that the market condition will be good? What is the probability of a good market if Semi-Clairvoyant does not recommend investment? (2 points)

Problem III: Simulation (14 Points)

PartFoods Market, a supermarket chain, is offering a perishable product called “auyester.” According to the market estimates, the demands for auyesters at the current price during different periods of the day are given by the following discrete distributions:

Demand	150	160	170	180	190	200
7 am – 1 pm	0.02	0.08	0.14	0.20	0.26	0.30
1 pm – 7 pm	0.11	0.19	0.22	0.20	0.16	0.12

For example, the probability that the demand for auyester is 160 units is 0.08 between 7 am and 1 pm.

The price for each auyester is $2, and the cost of ordering each auyester is $0.5. Because auyester is perishable, any left-over auyesters at the end of each day must be disposed of at a unit disposal cost of $1.2. The manager of PartFoods Market must decide how many auyesters to order (in batches of 10 units) at the beginning of each day.

a) Use the provided uniform random numbers in columns U1 and U2 to generate 500 instances from the demand distributions in the two time periods, respectively. What is the average daily demand? (4 points)

b) To maximize the expected profit, how many auyesters should PartFoods order each day and what is the corresponding profit? You may want to use your answer in part (a) as a starting point and try a few different order quantities with step size 10. (4 points)

c) Suppose the unit ordering cost follows a continuous uniform distribution over [0.3, 0.7], and disposal cost follows the Gamma distribution with shape 2 and scale 1.2 (See Figure 1). Everything else stays the same as in the base setting. How many auyesters should PartFoods order each day and what is the corresponding profit? Use the provided uniform random numbers in columns U3 and U4 to generate random numbers from the continuous uniform distribution and the Gamma distribution, respectively. (3 points)

d) Assume the same setting in part (c) holds. In addition, because of perishability, a random fraction of the auyester inventory can become stale and unsellable. For example, if PartFoods orders 300 auyesters at the beginning of the day, and 5% of them go stale, then 15 (=300*5%) auyesters are unsellable. Customers can pick auyesters as they want and will not purchase the stale auyesters. The stale oysters will be disposed of at the end of the day, together with the leftovers. Assume the perished fraction follows the Beta distribution with parameters 0.5 and 25 (see Figure 2). How many auyesters should PartFoods order and what is the corresponding profit? Use the provided uniform random numbers in column U5 with beta.inv(U, parameter 1, parameter 2) to determine the perished fractions. Round the perished quantities to the nearest integers. (3 points)