**IEOR 3402: Final Examination**

**IEOR 3402: Final Examination**

运筹学期末代考 Allowed aid: Textbook, Material posted on the Courseworks, and non-programmable Calculator. No other aid. No collaboration is allowed on the exam.

- Allowed aid: Textbook, Material posted on the Courseworks, and non-programmable Calculator. No other aid.
- No collaboration is allowed on the exam.
- Exam will become available on the Gradescope and the Courseworks on Friday April 23rd 9:00am EST. You will have 2 hours and 10 minutes to finish it till 11:10am EST and an additional 15 minutes to upload it to the Gradescope. Gradescope will stop accepting submission at 11:25am EST. If you encounter unforeseen problems for uploading in the allocated 15 minutes you should email [email protected] a picture of your solutions (just snap it with a phone) by 11:25am EST and then you will be allowed to upload the identical at a later time that will be announced. Submissions later than 11:25am EST either through Email or Gradescope will not be accepted.

**1.**

In other words, the weighted average of the * N *most recent observations is used as the forecast for the next period. Help Chen Consultancy answer the following questions assuming that a weighted moving average of order 4 is used.

(a) What is the expected value of the forecast?

(b) What is variance of the forecast?

(c) What is the average age of the forecast?

(d) Does the average age increase or decrease when the order is increased from 4 to 5?

**2.(Inventory Management: 10 points) 运筹学期末代考**

Demand for flour at a branch of Patel, Medina, Karavaikin (PMK) bakery conglomerate is approximately 10 kg per day. The purchase price is 2 dollars per kg. The interest rate is 5% per year. The cost of delivery is 150 dollars per order.

(a) What is the optimal ordering quantity?

(b) How often should deliveries be made if the bakery can contract with a logistics company to make deliveries of our every 1, 2, 4, 8, 16,… weeks?

(c) Suppose that demand is 100 kg per day, and in addition, each order from the supplier will be delivered approximately continuously at the rate of at most 150 kg per day. Suppose that there is no longer any constraint on the reorder interval. What is the optimal order quantity?

**3.(Inventory Management: 25 points) 运筹学期末代考**

Alferova-Schmitt, a high end 5th avenue retailer, sells a seasonal product for $12 per unit. The cost to the store is $6 per unit. At the end of each season a sale is held for excess units at $2 per unit and they are all sold. Assume that demand for the units is continuously distributed uniformly between 50 to 150.

(a) What is the optimal order quantity?

(b) What is the expected number of units sold if the order quantity in part (a) is used?

(c) What is the expected profit if the order quantity in part (a) is used?

Suppose now that whenever the store places an order of * Q *units, it has the opportunity to buy up to

*2 extra units (through their connection with Mercado-Mohtashami a well known European fashion brand) whenever the demand is greater than*

*Q/**at a higher per unit price of $8 per unit. That is, whenever the demand exceeds the ordered amount, instead of having lost sales, the store can acquire up to*

*Q**2 more units to satisfy that extra demand.*

*Q/*(d) Write the expression for the expected number of units sold under this new scheme if the initial order quantity is * Q*?

(e) Write the expression for the expected profit under this new scheme if the initial order quantity is * Q*?

(f) Does the optimal order quantity increase or decrease under this new scheme? Justify this change in plain words.

**4.(Queueing: 15 points) 运筹学期末代考**

Wang, Wilson, We (WWW) Speculator is a start-upspecializing on giving advice to students on housing. The company has two representatives, Adrian and Jackman, at a service center to handle customer The number of phone calls coming into the center follows a Poisson distribution with an average rate of 10 per hour. The time needed to answer each call follows an exponential distribution with an average of 4 minutes.

The company has three telephone lines dedicated for customers, which can be used either for serving customers or putting customers on hold. Any customer call that is received when all the lines are busy is immediately refused. Let *P*_{n}* *denote the steady state probability that exactly

*lines are used. Using the balance principle (that the rate of entry into a state must equal the rate of entry out of a state in the steady-state), write down the system of equations (balance equations) for solving*

*n*

*{*

*P*

_{n}

*|**= 0*

*n**1*

*,**2*

*,**3*

*,**. Show your work.*

*}*

**5.(Scheduling: 10 points) 运筹学期末代考**

Bilecik-Luo data center queues up batch jobs and processes them on an FCFS basis on a supercomputer. Between 2 and 5 pm, jobs arrive at an average rate 30 per hour and require an average of 1.2 minutes of computer time. Assume the arrival process is Poisson and the processing times are exponentially distributed.

(a) What is the expected number of jobs in the system and in the queue in the steady state?

(b) What are the expected flow time and the time in the queue in the steady state?

(c) What is the probability that the system is empty?

(d) What is the probability that the queue is empty?

(e) What is the probability that the flow time of a job exceeds 10 minutes?

**6.(Scheduling: 18 points) 运筹学期末代考**

Beyonce and Jayz like to read the newspaper on quarantine day mornings. They share the same journal and read the different sections separately. Beyonce reads first and when she finishes a section, she tears that section and gives it to Jayz. There are 4 sections: Main section, Sports, Comics and Classifieds. The time to read each section is assumed to be anexponential random variable with average reading times for both Beyonce and Jayz given in the following chart:

Section | Beyonce | Jayz |

Main | 6 | 4 |

Sports | 1 | 3 |

Comics | 5 | 2 |

Classifieds | 2 | 5 |

(a) In what order should the sections be scheduled in order to minimize the total expected time required for reading?

(b) If all the readings last exactly equal to their expected time, what is the length of the schedule found in part (a) and how many times can Jayz go fill his coffee cup because he has nothing to read?

**7.(Scheduling: 25 points) **

Rosario and Addis of Jaramillo-Boyd Partners have been hired to advise Tang-Osak Aerospace in their scheduling problems.Consider deterministic scheduling of a single machine with * n *jobs

*= 1*

*j**2*

*,**. Let*

*, . . . , n*

*p*

_{j}*be the processing time for job*

*, and let*

*j*

*d*

_{j}*be the due date for job*

*. Let*

*j*

*F**denote the completion time of job*

*j**. Answer the following questions. You should justify your answers, and your justification should be as concise and accurate as possible.*

*j*

**8.(Revenue Management: 15 points) 运筹学期末代考**

Garelli Airlines serves the route between New York and Bermuda with a single-flight-daily 120-seat aircraft. Theone-way fare for discount ticket is $100, and the one-way fare for full-fare tickets is $150. Discount tickets can be booked up until one week in advance, and all discount passengers book before all full-fare passengers. Over a long history of observation, the airline estimates that full-fare demand is normallydistributed, with a mean of 56 passengers and a standard deviation of 23, while discount fare demand is normally distributed, with a mean of 88 passengers and a standard deviation of 44.

(a) A consultant from Domich-Modi tells the airline that they can maximize expected revenue by optimizing the booking limit. What is the optimal protection level for full-fare passengers?

(b) Is the expected revenue convex or concave with respect to the protection level? (no need to prove it)

(c) The airline has been setting a booking limit of 44 on discount demand, to preserve 56 seats for full-fare demand. What is the expected revenue per-flight from low-fare demand?

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