Problem 1 : Credit Decisions

Vincent Cuomo is the credit manager for the Fine Fabrics Mill. He is currently faced with the question of whether to extend or not to extend $100,000 of credit to a potential new customer (a dress manufacturer). Vincent has three categories for the creditworthiness of a company: poor risk, average risk, and good risk. But he does not know which category fits this potential customer. Experience indicates that 20 percent of companies similar to this dress manufacturer are poor risks, 50 percent are average risks, and 30 percent are good risks. If the credit is extended, the expected profit is for poor risks -$15,000, for average risks $10,000, and for good risks $20,000. If credit is not extended, the dress manufacturer will turn to another mill and Fine Fabrics Mill does not lose or earn anything.

a) Develop a decision analysis formulation of this problem by identifying the decision alternatives, the states of nature, and construct the payoff table.

b) Which alternative should be selected based on Maximax criterion?

c) Which alternative should be selected based on Maximin criterion?

d) Which alternative should be selected based on Maximum likelihood criterion?

e) Which alternative should be selected based on Baye's rule?

f) Vincent is able to consult a credit-rating organization for a fee of $5,000 per company evaluated. This credit-rating organization will tell the category of your customer precisely. Would you use the credit-rating organization? (Hint: calculate the expected value of perfect information).

Problem 2: Modification of 11.12 from the book:

Jerry Jansen, materials handling at the Casper Edison Corporation's new factory, needs to decide whether to purchase a small tractor-trailer train or a heavy-duty forklift for transporting heavy goods between certain producing centers in the factory. Calls for the materials-handling unit to move a load would come essentially at random at a mean rate of four per hour, i.e., they are exponentially distributed with rate four per hour. The total time required to move a load has an exponential distribution, where the expected time would be 12 minutes for the tractor-trailer and 9 minutes for the forklift truck.

The total equivalent uniform hourly cost (capital recovery cost plus operating cost) would be $50 for the tractor-trailer train and $150 for the forklift truck (this is independent of being idle). The estimated cost of idle goods (waiting to be moved or in transit) because of increased process inventory is $20 per load per hour.

Jerry also has established certain criteria that he would like the materials-handling unit to satisfy in order to keep production flowing on schedule as much as possible.

1. He would like to average no more than half an hour for completing the move of a load after receiving the call requesting the move.

2. He also would like the time for completing the move to be no more than one hour at least 80 percent of the time.

3. Finally, he would like to have no more than three load waiting to start their move at least 80 percent of the time.

a) Describe the queuing system above if tractor-trailer train is chosen. To do so, define the customers, arrival rate, mean interarrival time, server (or servers), service rate, mean service time, and explain the service process. Use Kendall's notation to define the queueing system.

b) Obtain the various measures of performance if the tractor-trailer train were to be chosen. Evaluate how well these measures meet the above criteria (criteria 1-3).

c) Calculate the expected total cost per hour (including the cost of idle goods) if the tractor-trailer train were to be chosen.

d) Obtain the various measures of performance if the forklift truck were to be chosen. Evaluate how well these measures meet the above criteria (criteria 1-3).

e)Calculate the expected total cost per hour (including the cost of idle goods) if the forklift truck were to be chosen.

f) Which alternative do you think Jerry should choose? Discuss why.

Problem 3 : 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). Therefore, the total cost of the system includes grinder cost plus the operators being-away cost.

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 additional 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.

a) Calculate different measures of performance (L, LQ, 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, LQ, 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, LQ, 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 4: Establishing a Distribution System

Suppose that you are the executive manager for establishing a distribution system for Home-Office- Delivery (HOD) Demijohn water product your company sells and delivers in Rolla, MO (HOD-Demijohn water products are 10gallons of bottled water delivered to homes/offices upon receiving a call for delivery). Your customer base will consist of 2,000 units (a unit is either a home or an office). Specifically, a unit will give a call to your company's call-center, and then, your company will deliver the bottle to the unit. Assume that each unit will request one bottle at a time. Based on past data from other cities similar to Rolla, where your company operates, you are estimating that each unit in Rolla will consume one HOD- Demijohn bottle in 10 work-days on average, where a work-day is 8-hours.

For deliveries, you are planning to purchase scooters and hire delivery personnel. Each delivery person will have his/her own scooter. That is, the number of scooters is equal to the number of delivery personnel. Again, based on past data from other cities similar to Rolla, you are estimating that a delivery person can make 30 deliveries per work-day on average.

Given the above information, you are considering two-options for establishing the distribution system:

- Option 1: Rent a single (large) store in central Rolla, where you make all deliveries from. In this case, all of the customer calls will be directed to this store, and the deliveries will be made in a first-come first-served fashion.

- Option 2: Rent two (relatively smaller) stores: one in east Rolla and one in west Rolla. In this case, east-Rolla store will serve the 1000 units located in the east of Rolla; and, west-Rolla store will serve the other 1000 units located in the west of Rolla. Here, each store will have its own call- center and the customer calls will be directed to the associated store based on their location. For instance, a customer calling from east-Rolla will speak to the east-Rolla store.

Assume that both the inter-arrival and service time distributions are exponential; the stores have sufficient inventory of HOD-Demijohn water bottles (i.e., there will not any stock-out); and there is no queuing capacity in any case.

a) What is the minimum number of delivery personnel/scooters to keep within the single store to be rented in central Rolla if Option 1 is selected so that your system can be stable?

b) Briefly explain the queuing system or systems if Option 1 is selected as the distribution system Suppose that 10 delivery personnel are hired, which means that 10 scooters are purchased, for the single store. Define the arrival rate, service rate, and use Kendall's notation to define the queueing system or systems if Option 1 is selected.

c) What are the minimum number of delivery personnel/scooters to keep in the east-Rolla and west-Rolla stores to be rented in east of Rolla and west of Rolla, respectively, if Option 2 is selected so that your system can be stable?

d) Briefly explain the queuing system or systems if Option 2 is selected as the distribution system Suppose that 5 delivery personnel are hired, which means that 5 scooters are purchased, for each store. Define the arrival rate, service rate, and use Kendall's notation to define the queueing system or systems if Option 2 is selected.
While establishing your distribution network, you want to satisfy the following standards:

- Standard 1: On average, there should not be more than 2 units waiting for a delivery personnel/scooter to leave a store to deliver to the unit.

- Standard 2: At least 80 percent of the units should have their orders delivered with 30 minutes.

- Standard 3: For at least 70 percent of the units, delivery of their order should start as soon as the order is placed.

e) Suppose that Option 1 is selected and there are 10 delivery personnel with 10 scooters (one for each delivery person). Determine whether the standards 1-3 are satisfied or not. Show your work by showing the inequalities that you are checking for each standard.

f) Suppose that Option 2 is selected and there are 5 delivery personnel with 5 scooters (one for each delivery person) at each store. That is, east-Rolla store has 5 delivery personnel and 5 scooters and west-Rolla store has 5 delivery personnel and 5 scooters. Determine whether the standards 1-3 are satisfied or not. Show your work by showing the inequalities that you are checking for each standard.

g) Now suppose that Option 1 is selected and there are 10 delivery personnel with 10 scooters (one for each delivery person). What is probability of all scooters being parked in front of the single store located in central Rolla?