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 three 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.
- Supplier 1: It sells the product for $17 per unit. It refunds you $15 per each returned damaged product. It can refund you for at most 30 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. You 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: Machine Shop Grinders
Jake's Machine Shop contains a grinder for sharpening the machine cutting tools. A decision must now be made on buying 1 or 2 more additional grinders. The grinding time required by a machine operator to sharpen cutting tool has an exponential distribution with mean grinding time equal to 1 minute. The machine operators arrive randomly to sharpen their tools at a mean rate of one every two minutes and the inter-arrival time is exponentially distributed.
The maintenance cost for a single grinder is estimated to be $1 per minute. The estimated cost of an operator being away from his/her machine for grinding his tools is $0.80 per minute (once an operator leaves for grinding, he/she does not come back until he/she grinds the tools and the time to go to the grinder is negligible).
Suppose that you are in the control of making decisions for buying additional grinders. You want to decide to buy 1 or 2 more additional grinders, or not to buy any more grinders. If you have more than one grinders, the machine operators will form a single line and the one in the front of the line will go to the next available grinder.
You can use the excel templates for calculating the measures of performance.
a) Calculate different measures of performance (L, Lc, W, WQ), when you do not buy any additional grinders. That is, when you have a single grinder. Also, calculate the cost per minute of the system with a single grinder.
b) Calculate different measures of performance (L, Lo, W, WQ), when you buy one additional grinder. That is, when you have two grinders. Also, calculate the cost per minute of the system with two grinders.
c) Calculate different measures of performance (L, Lc, W, WQ), when you buy two additional grinders. That is, when you have three grinders. Also, calculate the cost per minute of the system with three grinders.
d) Comparing costs per minute in parts a, b, and c, what is the decision you should make to minimize the cost per minute?
PROBLEM 3: Airport Management
Dino-ville International Airport has two runways, one used exclusively for takeoffs and the other exclusively for landings. Airplanes arrive randomly in Dino-ville air space to request landing instructions at a mean rate of 10 per hour. The interarrival times are exponentially distributed. The time required for an airplane to land after receiving clearance to land has an exponential distribution with a mean of three minutes, and this process must be completed before giving clearance to land to another airplane. Airplanes awaiting clearance must circle the airport.
The Federal Aviation Administration has a number of criteria regarding the safe level of congestion of airplanes waiting to land. These criteria depend on a number of factors regarding the airport involved, such as the number of runways available for landing. For Dino-ville, the criteria are as follows:
1. The average number of airplanes waiting to receive clearance to land should not exceed 1
2. 95% of the time, the actual number of airplanes waiting to receive clearance to land should not exceed 4
3. For 99% of the airplanes, the amount of time spend circling the airport before receiving clearance to land exceed 30 minutes
In this problem, you are asked to analyze the queueing system in Dino-ville International Airport.
a) Explain the queueing system at Dino-ville Airport by defining the customers, arrival rate, server, service rate and use Kendall's notation to define the queueing system.
b) Evaluate if criteria 1-3 are currently satisfied.
c) A major airline is considering adding this airport as one of its hubs. This would increase the mean arrival rate to 15 airplanes per hour. Evaluate if criteria 1-3 would be satisfied if this happens.
d) Airport management is now considering a second runway for landings. It is estimated that this eventually would increase the mean arrival rate to 25 airplanes per hour as more airlines will become active at the airport. Evaluate if criteria 1-3 would be satisfied if this happens.
You can use excel templates for this problem.
PROBLEM 4: 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 active, i.e., processing a sub-product. 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; the penalties for sub-products are given by the following schedule:
- If the sub-product takes less than 10 minutes to be processed, the penalty is $1
- If the sub-product takes between 10 and 15 minutes to be processed, the penalty is $2
- If the sub-product takes more than 15 minutes to be processed, the penalty is $3
Please answer the following questions about the above system.
You can use the excel templates in your calculations. However, you need to show how you are using the information given by the excel template in your calculations as excel templates gives your only information for L, Lq, W, Wq, Pn, P(W>t), and P(Wq>t).
a) Describe the above system as a queueing system by defining
- Customers, expected inter-arrival time, and arrival rate
- Server(s), expected service time, and service rate
- Express the queueing model using Kendall's notation
b) Explain if the above queueing system satisfies the following criteria by showing your calculations for each criterion.
- On average, there should not be more than 2 sub-products waiting to start their processes at the station in front of the machine.
- On average, there should be less than or equal to 3 sub-products waiting to start their processes at the station in front of the machine for at least 75% of the time.
- On average, less than 50% of the sub-products should wait more than 5 minutes to start their processes at the station in front of the machine.
- As the machine is a sensitive machine, the time it is active should not be 4 times more than the time it is inactive
c) 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.
PROBLEM 5: Queueing Analysis
Please answer the following questions and explain your reasoning briefly.
a) Consider an M/M/1 queueing system. If the arrival rate increases, then the expected waiting time in the queue increases. Yes or No? Explain.
b) Which of the following actions will always increase the expected number of people in an M/M/1 queueing system?
a. Increasing the number of servers. Explain why.
b. Decreasing the service rate. Explain why.
c) Suppose that the current criterion of an M/M/1 queueing system is as follows: the expected waiting time in the queue should be less than one hour at least 90% of the time. You know that this criterion is not satisfied currently. Which of the following actions will help satisfy this criterion?
a. Adding another server with its own queue. Explain why.
b. Decreasing the expected inter-arrival time of customers. Explain why.