PROBLEM 1: Modification of Problem 9.16
Vincent Cuomo is the credit manager for the Fine Fabrics Mill. He is currently faced with the question of whether 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 credit is extended, the expected profit for poor risks is -$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.
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: Problem 11.9
Explain why the utilization factor p for the server in a single-server queueing system must equal to 1-Po, where Po is the probability of having 0 customers in the system.
PROBLEM 3: Modification of Problem 11.13
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. The estimated cost of idle goods (waiting to be moved or in transit) because of increased in-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. 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. He also would like the time for completing the move to be no more than one hour 80 percent of the time. Finally, he would like to have no more than three load waiting to start their move at least 80 percent of the time.
a) Obtain the various measures of performance if the tractor-trailer train were to be chosen. Evaluate how well these measures meet the above three criteria.
b) Repeat part a if the forklift truck were to be chosen.
c) Compare the two alternatives in terms of their expected total cost per hour (including the cost of idle goods). Which alternative do you think Jerry should choose? Discuss why.
PROBLEM 4: Green Cross-docking from Previous Final
Suppose that you are an Operations Manager at a cross-docking facility, where trucks from different origins come for unloading, and loaded trucks leave for different destinations. In particular, you are responsible for managing the Inbound Operations, that is, unloading of the incoming trucks to the facility. Currently, there is one dock assigned for unloading the incoming trucks. At this dock, there is a forklift that completes the unloading operations. Since different incoming trucks have different sizes and different loads, you observed that it takes different times to unload the incoming trucks. From the past data on truck unloading operations at the dock, you estimate that the forklift at the dock can unload 5 incoming trucks per hour on average and the unloading time is exponentially distributed. The incoming trucks form a single line in front of the dock and the forklift starts unloading the truck at the beginning of the line. Using the past arrival data of the incoming trucks, you estimate that a new incoming truck arrives in the inbound operations area every 15 minutes on average and the inter arrival time between incoming trucks is exponentially distributed.
a) Define the above queuing system by describing the customers, arrival rate, expected inter arrival time, server(s), service rate, and expected service time. Use Kendall's notation to define the queuing system.
Currently, your company wishes to reduce the carbon emissions generated at the cross-docking facility. Carbon emissions are generated because the incoming trucks waiting for unloading are not turning off their engines as they need to move towards the dock frequently. You know that, on average, an incoming truck generates 20 grams of CO2 per hour when its engine is on. The trucks turn off their engines when they are being unloaded. On the other hand, since the forklift has smaller engine, its engine is turned off for the times the forklift is not unloading the incoming truck; hence, it does not generate CO2 when it is inactive. On the other hand, while the forklift is unloading incoming trucks, it generates 5 grams of CO2 per hour.
b) Calculate the expected amount of CO2 generated per hour due to inbound operations? You can use templates for the values of L, Lq, W, Wq, Pr(W>t), Pr(Wq>t), and Pn; however, for other equations you use, please show your calculations and how you reach to your final answer.
The company now targets that the average CO2 emissions from inbound operations should not exceed 54 grams per hour and you observed that you are exceeding this limit. Therefore, you are considering some changes in the inbound operations to reduce the CO2 emissions generated from inbound operations.
Specifically, since you do not have control over the incoming trucks, you want to make changes in the forklift operations. In particular, you have two other alternative forklifts as detailed below:
- Alternative forklift 1: It can unload an incoming truck in 11 minutes on average and unloading time is exponentially distributed. Since it is faster, it generates more CO2emissions. Particularly, it generates 20 grams of CO2 per hour when it is active for unloading. When it is inactive, the engine can be turned off.
- Alternative forklift 2: It can unload an incoming truck in 10 minutes on average and unloading time is exponentially distributed. Since it is faster, it generates more CO2emissions. Particularly, it generates 30 grams of CO2 per hour when it is active for unloading. When it is inactive, the engine cannot be turned off (as frequent engine turn offs can create mechanic issues) and it generates 20 grams of CO2 per hour when it is inactive.
c) Does forklift alternative 1 satisfy the company's new target for not exceeding 54 grams of CO2 per hour on average? Show you calculations on how you determine the expected amount of CO2 generated per hour due to inbound operations when you start using forklift alternative 1?
d) Does forklift alternative 2 satisfy the company's new target for not exceeding 54 grams of CO2 per hour on average? Show you calculations on how you determine the expected amount of CO2 generated per hour due to inbound operations when you start using forklift alternative 1?
e) Based on the expected carbon emissions per hour, which alternative you would select if you want to minimize carbon emissions from inbound operations?
PROBLEM 5: Loan Services at a Bank from Previous Final
Suppose that you are the general manager of a local bank that provides loan services to its customers. Currently, you have a dedicated area in the bank for loan services. In this loan area, there is one loan expert, who helps the customers with their loan requests. In the loan area, there are 2 waiting seats available for the customers waiting for a loan request. If the loan expert is busy and both of the waiting seats are occupied, a customer waits in a general standing area within the loan area. You are practically assuming that the general standing area can hold as many customers as possible. The figure below illustrates the loan area in the bank.
LOAN AREA
Note that a customer being helped by the loan expert sits on a separate chair by the loan expert's desk, i.e., the customer being helped does not sit on the waiting seats. Based on the past observations, you are estimating that the customers come to the bank with loan requests every 30 minutes on average, i.e., every 30 minutes, a customer walks into the loan area. Again, based on the past data, you observed that the loan expert can help a customer with his/her loan request within 15 minutes on average. Both the customer inter arrival times and the service times are exponentially distributed. As the general manager, you have standards about the customer service for any type of service you provide within your bank. Specifically, you have the following three standards for the loan service:
- Standard-1: On average, there should not be more than 1 person waiting for the loan expert.
- Standard-2: On average, at most 5% of the customers should wait standing in the general standing area for waiting.
- Standard-3: On average, at least half of the customers should complete their requests with the loan expert within 20 minutes after arriving at the loan area.
Determine whether or not the standards 1, 2, and 3 are being satisfied currently. Show your calculations or how you reach to your answer (i.e., if a standard implies L<=3, you do not need to show your calculations because the excel template already calculates L; however if a standard implies some probability calculations, you need to determine that probability using the information in excel template if possible).
PROBLEM 6: Teaching Assignment Office Hours from Previous Final
Dr. Konur needs to figure out how to arrange the office hours for EMGT 365 for the next semester. He will have two teaching assistants (TAs). He is considering two options: assign different offices for each TA or assign one office for both TAs. He observed the following from previous semesters.
- Option 1: In this case, each TA will have its own office. Students arrive at the office of each TA every 10 minutes on average and the inter arrival time is exponentially distributed.
- Option 2: In this case, both TAs will help students in the same office. Each TA will help one student at a time. Students arrive at the office every 5 minutes on average and the inter arrival time is exponentially distributed.
The time it takes to help a student by a TA is exponentially distributed with mean 5 minutes. Answer the following questions. You can use excel templates provided. Please present the excel template results for each option.
a) If Dr. Konur prefers to have less number of students in the department (i.e., the ones waiting for the TA plus the ones TAs are helping) on average, which option he should select?
b) If Dr. Konur wants to have at least 75% of the students complete their operation (i.e., the time waiting and the time to ask questions) within 10 minutes, which option he should select?
c) If Dr. Konur wants have lower number of students waiting for a TA on average, which option he should select?
d) Suppose that each TA is paid $10 per hour and each student's time (including the time to wait and the time to ask questions) costs $8 per hour, which option Dr. Konur should select if he wants to have lower hourly cost?
8. Modification of11.12 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. The estimated cost of idle goods (waiting to be moved or in transit) because of increase 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. 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. He also would like the time for completing the move to be no more than one hour 80 percent of the time. Finally, he would like to have no more than three load waiting to start their move at least 80 percent of the time.
a) Obtain the various measures of performance if the tractor-trailer train were to be chosen. Evaluate how well these measures meet the above criteria.
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B I C I D
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E
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3
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Data
4
5
1
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I
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Results
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4
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=
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(mean arrival rate)
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4
3.2
1
0.8
0.8
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5
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(mean service rate)
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-.: =
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6
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(It servers)
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7
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8
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Pr(W > t) = 0.36787944A
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9
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when t =
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1
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10
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11
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Pruh(Wa> 9 =
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0.29430355
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12
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when t =
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. 1
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7 .
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Cumulative
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13
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0.2 0.16 0.128
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02
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14
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I
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0.36
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15
16
17
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2
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0.488
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3
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0.1024
i t
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0.5904
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4
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0.08192
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0.67232
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The train does not meet any of the criteria. The average time is more than half-anhour (W= 1 hour), it is no more than an hour less than 80% of the time (Pr(W> 1) = 36.8%), and there are three loads or fewer less than 80% of the time (Pol-P1+P2+P3+P4= 67.2%).
b) Repeat part a if the forklift truck were to be chosen
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B
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C I D
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E
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G
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H
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3
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Data l
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Results
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= 4 (mean arrival rate)
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L =
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1.5
0.9
0.3750C
0.2250C
0.6
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4
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5
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g = 6.66666667 (mean service rate)
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Lq=
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6
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s= 1 (# servers)
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7
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W= Wq =
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8
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Pr(VV> t)=
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0.069483451
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9
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when t
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= 1
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10
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p .
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11
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Prob(Wq> t) =
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0 04169007
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12
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when t = 1
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n
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Pn
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Cumulative
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13
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0
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0.4
0.24
0.144
0.0864
0.05184
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0.4
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14
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1
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0.64
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15
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2
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0.784
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16
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3
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0.8704
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17
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4
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0.92224
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The forklift truck meets all the criteria. The average time is less than half-an-hour (W = 0.375 hours), it is no more than an hour more than 80% of the time (Pr(W > 1) = 6.9%), and there are three loads or fewer more than 80% of the time (Po+P1+P2+P3+P4 = 92.2%).
c) Compare the two alternatives in terms of their expected total cost per hour (including the cost of idle goods).
Tractor-trailer train: L($20)+$50 = (4)($20) + $50 = $130/hour
Forklift truck: L($20)+$150 = (1.5)($20) + $150 = $180/hour
d) Which alternative do you think Jerry should choose? Discuss why.
While the forklift truck has higher overall costs, it does a better job of meeting the additional criteria.