Problem 1- Multi-Supplier Decision Making

Suppose that you are planning to purchase 100 units of a product. To do so, you can purchase the product from one of the two suppliers available. That is, if you choose a supplier, you will buy the whole 100 units from the same supplier. When you purchase the product, some portion of the products can be damaged during delivery. You can return the damaged products to the supplier and they will partially refund you for the damaged product. However, different suppliers have different refund policies and they only accept limited number of damaged products. The specifications of the suppliers are as follows:

• Supplier 1: It sells the product for $15 per unit. It refunds you $7 per each returned damaged product. It can refund you for at most 15 units of damaged products.

• Supplier 2: It sells the product for $16 per unit. It refunds you $12 per each returned damaged product. It can refund you for at most 25 units of damaged products.

Upon previous observations on delivery, you have estimated that 10% of the products you purchased will be damaged with probability 0.45, 20% of the products you purchased will be damaged with probability 0.30, and 30% of the products you purchased will be damaged with probability 0.25. You will sell any undamaged product for $30 per unit in your store. The delivery cost and return cost back to the supplier are negligible. You want to decide which supplier to purchase 100 units of the product to maximize profit. Your profit will be equal to revenues gained from selling undamaged products minus the purchase cost paid to the supplier plus the money you get refunded for damaged products.

a) Formulate a decision analysis problem by constructing the payoff table with the information given above. That is, describe your alternatives, states of natures, and calculate the payoff (profit) for each alternative and state of nature pair, and note the prior probabilities for each state of nature.

b) Based on the maximax criterion, which supplier you would purchase 100 units from? Show how you reach to your solution.

c) Based on the maximin criterion, which supplier you would purchase 100 units from? Show how you reach to your solution.

d) Based on the maximum likelihood criterion, which supplier you would purchase 100 units from? Show how you reach to your solution.

e) Based on the Baye's decision rule, which supplier you would purchase 100 units from? Show how you reach to your solution.

f) What is the expected value of perfect information for you?

Problem 2- Dr. Konur's Nightmare at Taco-Bell Drive-Through Lines

While in Memphis, Dr. Konur used to eat from Taco-Bell a lot. Generally, he has seen long lines of cars in the drive-through. Therefore, he started to work on the Taco-Bell's drive-through policy. Dr. Konur had the following information about the current system:

• There is a single drive-through lane and there are two teller windows, each taking the orders from the next car in line

• The cars arrive randomly at the drive-through entrance and inter-arrival times are exponentially distributed with expected inter-arrival time equal to 90 seconds

• On average, each teller can complete the orders from 30 cars per hour, i.e., a single teller's service rate is 30 per hour

Answer the following questions based on the information above.

a) Describe the above system as a queueing system by defining

I. Customers, expected inter-arrival time, and arrival rate

II. Server or servers, expected service time, and service rate for each server

III. Express the queueing model using Kendall's notation

Dr. Konur has the following restrictions about getting food from fast-food restaurants' drive-throughs:

• Restriction 1: The time between the moment he arrives at the drive-through until the moment he gets his order and leaves the system should not be more than 3 minutes on average

• Restriction 2: The probability of him seeing 3 or more cars in the line waiting for tellers should be less than 0.15

• Restriction 3: The probability of him waiting less than 5 minutes until he reaches a teller should be more than 0.10.

b) Determine whether each of the above restrictions is being satisfied with the current taco-bell's drive-through.

I. Determine whether restriction 1 is satisfied or not. Show your calculations.

II. Determine whether restriction 2 is satisfied or not. Show your calculations.

Suppose that Dr. Konur is not satisfied with the current system and he goes and talks to the manager and suggests the manager to have two separate drive-through lanes; one for each teller. In this case, manager estimates that half of the cars will go through one lane and the other half will go through the other lane. Therefore, manager estimates that cars will randomly arrive at each drive-through lane every 3 minutes on average and the inter-arrival times will be still exponential.

Since the manager decided to consider Dr. Konur's suggestion, Dr. Konur lowered his restrictions. Now, he only wants to have the following restriction to be satisfied:

• Dr. Konur's New Restriction: The probability of duration from the moment he arrives at a drive-through lane until he leaves the drive-through lane being more than 5 minutes should be less than 0.50
However, manager also has a restriction:

• Manager's Restriction: On average, there should be less than 3 cars waiting for a teller in the drive-through lanes in total

c) If the manager implements Dr. Konur's suggestion for having two separate lanes, one for each teller, determine whether Dr. Konur's new and the mananger's restrictions are satisfied.

I. Determine whether Dr. Konur's new restriction is satisfied or not. Show your calculations.
Suppose that Manager is not satisfied with Dr. Konur's suggestion; therefore, he comes up with his own plan. Manager now considers to keep the single drive-through lane but he will hire another teller. That is, there will be 3 tellers and one lane and a teller will start helping the car in the front of the lane.

d) If the manager implements his own plan, i.e., hire another teller and keep the single drive-trhough lane, determine whether Dr. Konur's new and the mananger's restrictions are satisfied.

I. Determine whether Dr. Konur's new restriction is satisfied or not. Show your calculations.

II. Determine whether Manager's restriction is satisfied or not. Show your calculations.

Problem 3- Just-in-Time Manufacturing

Suppose that you have a machine that completes a specific process for manufacturing a product. This machine can complete the process of a sub-product within 3 minutes on average and the process completion time is exponentially distributed. Sub-products arrive randomly with a rate of 15 per hour at the station in front of the machine. The inter-arrival times of the sub-products is exponentially distributed.

There should always be an operator next to the machine and an operator's cost is $15 per hour. The machine has a cost of $20 per each hour it is actively processing a sub-product; when the machine is inactive, it has no cost. The machine cost when it is active does not depend on how fast it works. Furthermore, the sub-products have a penalty for the time they spend from the moment they arrive at the station until their processes have been completed. Specifically, due to just-in-time requirements, a sub-product's process should be completed within 10 minutes.

• If a sub-product's process completion is tardy between 0 and 5 minutes, the penalty is $2

• If a sub-product's process completion is tardy more than 5 minutes, the penalty is $4

Please answer the following questions about the above system.

a) Calculate the expected hourly cost of the queueing system. Expected hourly cost of the system includes the labor cost per hour plus the expected hourly cost of the machine plus the expected penalties charged per hour. Show your calculations.

b) Now suppose that the machine can be set to complete the process of a sub-product faster or slower. Assuming that the machine cost, labor cost, and the penalty costs will be the same for each setting of the machine, explain whether the following statements are true or false and explain your reasoning briefly without assigning values and solving with the assigned values.

I. If the machine is set to work slower, the average machine cost per hour will increase

II. If the machine is set to work faster, the average number of sub-products in the queueing system will increase

III. If the machine is set to work slower, the percentage of sub-products being tardy will increase

IV. If the machine is set to work faster, the expected penalties per hour will increase

V. If the machine is set to work slower, the average number of sub-products waiting to be processed will decrease

Problem 4- Cashier Management

Suppose that you are the manager of a department store and you need to determine how many cashiers needed for helping customers check out. There is a single check-out line and the next available cashier helps the customer in the front of the line. Upon observations, you estimate that the customers get in the checkout line every 5 seconds on average. Furthermore, you estimate that a cashier can check out a customer in 20 seconds on average.

a) What is the minimum number of cashiers needed so that the check-out line can reach a steady state?

b) Suppose that you do not know what type of a queueing system you have and you hired 8 cashiers. Furthermore, you are observed that, with 8 cashiers, there are 4 customers waiting for a cashier for check out on average. What is the expected time a customer will spend from the moment he/she enters the line until his/her checkout is complete?

c) Now, suppose that you decided to hire 6 cashiers and your system is now M/M/6. What is the probability of at least one cashier being idle?

Problem 5- Loan Services at a bank.

Calculate the expected number of people waiting in the standing area of the loan area. (Hint: Leave this bonus part to the end, first solve the other problems.)