Reference no: EM13550778

PROBLEM 1: Green Cross-docking

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 interarrival time between incoming trucks is exponentially distributed.

a) Define the above queueing system by describing the customers, arrival rate, expected interarrival time, server(s), service rate, and expected service time. Use Kendall's notation to define the queueing 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. 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, L_q, W, W_q, P_r(W>t), P_r(W_q>t), and P_n; 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 80 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 CO₂ emissions. 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 CO₂ emissions. 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 80 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 80 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 2 : Loan Services at a Bank

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.

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 interarrival 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.

You want to see if the current loan service system is satisfying these standards.

a) 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).

Suppose that the current loan service is not satisfying all or some of the standards; hence, you are thinking about taking possible actions that might improve other standards.

b) Without solving anything, state whether the following ideas are correct or wrong and briefly explain your reasoning. That is, do not assume any values and solve for the standards and state that the idea is correct or wrong based on the solution.

a. You want to reduce the average number of people waiting for the loan expert, either in the seats or in the standing area. To do so, you tell the loan expert to work more efficiently so that he can complete a customer's loan request in a shorter amount of time. Will this really reduce the average number of people waiting for the loan expert?

b. You are planning to give a promotion to customers who come to your bank for loan services. This promotion is expected to result in an increase in the customer arrival rate. Will this promotion decrease the average number of customers a loan expert can help with loan requests?

c. You are thinking to hire another loan expert with the same service rate that will also help customers in the same loan service area, i.e., the customers will wait for the next available loan expert. Will hiring the second loan expert decrease the average total time a customer spends in the bank for completing his/her loan request?

For some reason, you thought that the ideas you had are not directly helping for you to achieve your standards. Therefore, you decided to seek help from other managers in the bank and you got two suggestions:

- Facilities Manager suggested adding another waiting seat, right next to the second waiting seat, to the loan area.

- Human Resources Manager suggested replacing the current loan expert with a new one. The new loan expert can complete a customer's loan request within 12 minutes on average; and, the service time is still exponentially distributed.

You want to implement one or both of these suggestions.

c) If you decide to implement the Facilities Manager's suggestion, will the loan service satisfy standards 1, 2, and 3? Show your calculations if needed.

d) If you decide to implement the Human Resources Manager's suggestion, will the loan service satisfy standards 1, 2, and 3? Show your calculations if needed.

PROBLEM 3: Teaching Assignment Office Hours

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 interarrival 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 interarrival 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 deparment (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 for TA and the time to ask questions to the TA) 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?

PROBLEM 4: Cookie Inventory

Suppose that you are controlling the inventory of daily cookies in your coffee shop. Since each customer buys two cookies at the same time, you observed that the daily demand for cookies is either 10 or 12 or 14 or 16 or 18. Based on your observations since you started your coffee shop, you have built the table below, which gives the probabilities of each daily demand value (in number of cookies).

You buy cookies every morning before opening your coffee shop. You buy cookies in multiples of 5. You have the following information about your cookie sales:

- Each pack of cookie you purchase contains 5 cookies
- Each pack of cookie costs you $15
- Each cookie is sold for $5
- If you have unsold cookie at the end of the day, you can sell each unsold cookie for $1 to farmers as animal food

Answer the following questions based on the above information.

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, how many packs of cookies you should buy in the morning? Show how you reach to your solution.

c) Based on the maximin criterion, how many packs of cookies you should buy in the morning? Show how you reach to your solution.

d) Based on the maximum likelihood criterion, how many packs of cookies you should buy in the morning? Show how you reach to your solution.

e) Based on the Baye's decision rule, how many packs of cookies you should buy in the morning? Show how you reach to your solution.

f) Suppose that you meet a medium who can tell you how many cookies that you will sell before you buy cookies. Would you pay her $2 every morning to do a reading for your cookie sales? Why or why not? Explain your solution and Show how you reach to your solution.