DMS2030

Individual Assignment 2

Due Date: 11:59 pm, March 9th, 2025

1. Customers walk into a bank and depart after receiving service at an average rate of 16 per hour.  There is one clerk in the bank serving the customers.  The arrival process is random but not Poisson.  Service time has an exponential distribution with a mean of 3 minutes.  On average there are 4 customers waiting in queue to be served.  On average how long does a customer wait in queue? (5 points)

2. City bank has a staff planning policy based on a maximum utilization level for each operation. For example, the allowable utilization for operation A is 70%. If the workload for this operation is expected to increase  in the  next  planning  period,  clerks  will  be  hired  to  ensure  the  utilization  to  be  about  70%. Conversely, if the workload is expected to decline, clerks will be assigned elsewhere as long as the 70% utilization rule is not violated. The manager believes that such a policy will allow the bank to maintain a specified customer waiting time, which  is  harder to  monitor.  An  internal study shows that for  most operations, there is significant variability in the job arrival pattern as well as the processing times for the jobs. Does this policy work correctly to achieve the specified customer waiting time? Please briefly justify your answer. No calculation is needed. (5 points)

3. An orange juice vending machine can make customers fresh orange juice upon their arrival. The time of making a cup of orange juice is fixed to be 10 min. On average, customers arrive every 15 min. However, the inter-arrival time is actually random, and follows exponential distribution. On average, how many customers are waiting to be served? On average, how many people are standing in front of the vending machine? (5 points)

4. The unloading dock of the warehouse of P&S Supermarket has a single crew of two workers. The whole crew is needed to unload a truck. Truck arrivals are Poisson with a mean of 3 trucks per hour while the unloading time per truck is exponential with a mean of 15 minutes. If a truck arrives and the crew is unloading another truck, the arriving truck joins the line of trucks waiting for service. Assume there is enough space to accommodate essentially any number of trucks waiting in the line.

(a) What is the average number of trucks in the waiting line? What is the average time a truck must wait in the line before receiving service? (5 points)

(b) The management is considering adding a second crew of two workers. There is sufficient space on the loading dock to allow two trucks to unload at the same time. Each unloading dock worker is paid $100 per hour while the hourly cost of a truck waiting is $500. Complete the following input table for using the Queue.xls. (5 points)

(c) The following is the output of the Queue.xls Excel program. Each worker is paid $100 per hour while the hourly cost of a truck waiting is $500. Is there any economic justification for adding the second crew? (5 points)

5. Expando, Inc. is considering the possibility of building an additional factory that would produce a new addition to its product line. The company is currently considering two options. The first is a small facility that it could build at a cost of $6 million. If demand for new products is low, the company expects to receive $10 million in discounted revenues (present value of future revenues) with the small facility. On the other hand, if demand is high, it expects $12 million in discounted revenues using the small facility. The second option is to build a large factory at a cost of $7 million. Were demand to be low, the company would expect $10 million in discounted revenues with the large plant. If demand is high, the company estimates that the discounted revenues would be $14 million. In either case, the probability of demand being high is 0.4, and the probability of it being low is 0.6. Not constructing a new factor would result in no additional revenue being generated because the current factories cannot produce these new products.

(a) Construct a decision tree for Expando. (10 points)

(b) What is Expando’s best decision? (5 points)

6. Peter has just finished a whole day’s work and wants to drive home as soon as possible. He can first drive on the Washington Avenue to get to the city hall, and then choose from one of the two roads to get home: the May Road or the Lin Road. He can also drive on the Ben Avenue to get to the museum first, and then choose from one of the two roads to get home: the Kings Road or the Queens Road. The following is a list of how much time he will NORMALLY spend on each route:

Washington Avenue: 10 minutes; May Road: 20 minutes; Lin Road: 15 minutes; Ben Avenue: 12 minutes; Kings Road: 12 minutes; Queens Road: 18 minutes.

There might be traffic jams on the four Roads. If a traffic jam happens, Peter has to DOUBLE THE NORMAL TIME on the road. A traffic jam never happens on the Avenues. The probability of a traffic jam on each road is: the May Road: 0.3; the Lin Road: 0.4; the Kings Road: 0.4; the Queens Road: 0.3.

Before arriving at the city hall (the museum), Peter cannot observe whether there are traffic jams on the May Road and the Lin Road (the Kings Road and the Queens Road) or not. He only knows the probabilities. A traffic jam is observable (if it happens) when Peter arrives at the city hall (the museum).

(a) Construct a decision tree for Peter. (10 points)

(b) Should Peter choose the Washington Avenue or the Ben Avenue at the beginning? (5 points)

(Hint: You will need to use the Rule of Multiplication of probabilities. For example, the probability that both the May Road and the Lin Road are jammed is 0.3 × 0.4 = 0.12. )

7. A post office requires different numbers of full-time employees on different days of the week.  The minimum number of full-time employees needed on each day is as follows:

Labor law states that each full-time employee must work five consecutive days and then receive two days off.  For example, an employee who works Monday to Friday must be off Saturday and Sunday.  The post office wants to meet its daily requirements using only full-time employees. Suppose that the daily salary for each worker is $10. The post office  is  considering how to minimize the weekly pay for full-time employees.

(a) What are the decisions variables in this problem? (5 points)

(b) Formulate the problem as a constraint optimization problem. (10 points)

(c) Now, suppose that employees must be paid double salary on weekends. Formulate the problem as a constraint optimization problem. (5 points)

8. O’Connel Airlines is considering air service from its hub of operations in Cicely, to Rome and Seattle. O’Connel has one gate at the Cicely Airport, which operates 12 hours per day. Each flight requires 1 hour of gate time. Each flight to Rome consumes 15 hours of pilot crew time and is expected to produce a profit of $2500. Serving Seattle uses 10 hours of pilot crew time per flight and will result in a profit of $2000 per flight. Pilot crew labor is limited to 150 hours per day. The market for service to Rome is limited to nine flights per day. O’Connel wants to obtain the highest possible profit.

(a) Formulate the problem as a constraint optimization problem. (5 points)

(b) Draw a graph to illustrate the constraints. (10 points)

(c) What is the best strategy for O’Connel? (5 points)

9. Any more suggestions/comments for the class? (Optional)