**Question 1 (2 points) **

Q1.1 The following figure is an example of the application of Critical Path Method to the management of a project with 10 activities, but one of the arrows is missing (that is, one precedence relation is not plotted in the figure yet). Please fill in the blanks: The missing arrow is from Activity ____ to Activity ____. (** You do not have to show your calculation.**)

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Note: Here we give an example that shows meanings of different boxes of activity . b represents the earliest start time of activity ; c represents the duration of activity ; d represents the earliest completion time of activity ; e represents the latest start time of activity ; f represents the slack time of activity ; g represents the latest completion time of activity .

Q1.2 This question is based on the complete figure (with the missing arrow added) of the project that you obtain in Q1.1. Now the project manager has $1000 to invest in hiring more manpower for one of the activities to shorten its duration. There are 3 activities the manager can choose to invest in, namely (the duration will be shortened by 5 days with the investment of $1000, i.e., the new duration will be 11 days), (the duration will be shortened by 4 days with the investment of $1000, i.e., the new duration will be 8 days) and (the duration will be shortened by 3 days with the investment of $1000, i.e., the new duration will be 16 days). Note that only one activity can be chosen. Which activity should the project manager choose to invest in to minimize the shortest duration of the project? Why (** Please show your calculation**)?

**Question 2 (2 points)**

In a grocery store, the store owner purchases snacks at the price of 50 cents per unit and sells them at the price of 100 cents per unit. The left-over inventory has the salvage value of X cents per unit. The demands of snacks in past 20 days are listed in the following table,

Demand of snacks (units) in the past 20 days

Day | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Demand | 22 | 18 | 22 | 15 | 15 | 20 | 16 | 18 | 20 | 18 |

Day | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Demand | 18 | 22 | 24 | 20 | 16 | 15 | 15 | 18 | 20 | 14 |

Assuming that the distribution function of the demand is the same as its empirical distribution of the past 20 days. The store owner chooses the order quantity to maximize his/her expected total profits. Which one of the following is ** incorrect**? Why (

- When X=15, the optimal order quantity is 20.
- When X=19, the optimal order quantity is 20.
- When X=25, the optimal order quantity is 20.
- When X=30, the optimal order quantity is 20.
- When X=37, the optimal order quantity is 20.

**Question 3 (2 points)**

A supermarket sells two types of magazines, namely magazine 1 and magazine 2, which are provided by the same publisher. Magazine 1 and magazine 2 have the same selling price of 3 USD. The inventory of magazines at the supermarket is managed by the publisher, who owns all inventory until sold to the consumer or salvaged. The supermarket provides three different magazine shelves to be rented, and their capacities and rents are listed in the following table.

Shelf | Capacity (magazines) | Rent (USD) |

6 | 6 | |

8 | 7 | |

10 | 8 |

For the publisher, most of the costs to produce magazines are fixed costs, including the cost of designing magazine contents and get relevant copyrights, and therefore, the marginal cost of printing a magazine is 0. Meanwhile, the left-over magazines will be destroyed with no extra cost or salvage value. In summary, for both types of magazines, the cost of a magazine is 0, the selling price of a magazine is 3 USD, and the salvage value is 0. The demands of magazine 1 and magazine 2 are independent, and the relative frequencies of demand situations of them are listed in the following table.

Magazine 1 | Magazine 2 | ||

Demand | Relative frequency | Demand | Relative frequency |

0 | 0.1 | 0 | 0 |

1 | 0.15 | 1 | 0 |

2 | 0.15 | 2 | 0.1 |

3 | 0.5 | 3 | 0.2 |

4 | 0.1 | 4 | 0.6 |

5 | 0 | 5 | 0.1 |

Which shelf should the publisher rent and how many units of magazine 1 and how many units of magazine 2 should the publisher store on the rented shelf at the supermarket to maximize the expected total profit of the two magazines? (Note that the expected total profit is the expected total revenue minus the cost of renting the shelf.) (** Please show your calculation.**)

**Question 4 (1 point)**

A hotel has 100 rooms to be sold at the price of $280 for a night’s stay. In the peak season, all rooms are booked in advance (each traveler will book one room). However, some travelers that have booked rooms may fail to show up to receive the service and cancel their bookings. According to historical data, the numbers of travelers who will cancel bookings is a discrete uniformly distributed random variable that can be 0, 1, …, 4. (Note that for simplicity, we have assumed that the number of travelers who will cancel booking is not related to the number of travelers who have booked. And there are abundant travelers who want to book rooms.) If more than 100 travelers that have booked arrive, the hotel must accommodate the extra travelers by booking at a local luxury hotel at the price of $650 per traveler. How many traveler bookings should the hotel accept to maximize the expected total profits? (** Please show your calculation.**)

**Question 5 (2 points) **

A furniture factory consumes 200 tons of paint per year. There are two suppliers that can provide the paint, and the quality at both suppliers is the same. However, the unit prices offered by the two suppliers are different, supplier 1 sells paint at the price of 30 USD per ton and supplier 2 sells paint at the price of 29.5 USD per ton. The ordering cost from supplier 1 is $15 per order, and the ordering cost from supplier 2 is $20 per order. And the annual holding cost of a ton of paint is 50% of the purchasing price. The objective of the factory is to minimize the purchasing cost plus the inventory holding cost and the ordering cost of paint in a year.

Q5.1 Please apply the Economic Order Quantity model and identify: a. the optimal order quantity if the factory purchases paint from supplier 1, b. the optimal order quantity if the factory purchases paint from supplier 2. (** Please show your calculation.**)

Q5.2 The factory can store 9 tons of paint at most at the same time (that is, the maximum allowable inventory level is 9 tons). Please find out which supplier should the factory purchase paint from and how much paint should the factory purchase per order. (** Please show your calculation.**)

**Question 6 (3 points)**

Factory A has received 5 jobs to finish. The processing time, due time, and unit tardiness cost of the 5 jobs are listed in the following table, and these jobs are listed according to their arrival time ( is the first job that arrives and is the last job that arrives).

Job | Processing time (day) | Due time (day) | Unit tardiness cost (USD/day) |

15 | 25 | 2 | |

32 | 50 | 3 | |

26 | 45 | 1.5 | |

18 | 26 | 2 | |

27 | 60 | 2.5 |

The unit tardiness cost means the cost that the factory has to pay for one-day tardiness of a job. For example, if the job tardiness of is 3 days, the factory has to pay 6 () USD for it. At most one job can be processed at a time.

Q6.1 We consider using the three common priority rules, namely FCFS (first come, first served), SPT (shortest processing time), and EDD (earliest due date) to schedule the production of the 5 jobs. Here we give the example of the FCFS case.

Following the FCFS rule, the processing order will be: . The time when each job is finished and tardiness cost of each job are listed in the following table.

Job | Due time (day) | Unit tardiness cost (USD/day) | Finish time (day) | Tardiness cost (USD) |

25 | 2 | 15 | 0 | |

50 | 3 | 47 | 0 | |

45 | 1.5 | 73 | 42 | |

26 | 2 | 91 | 130 | |

60 | 2.5 | 118 | 145 |

The total tardiness cost of the FCFS case equals 0+0+42+130+145=317 USD.

What is the total tardiness cost for the factory if it uses the EDD rule? What is the total tardiness cost for the factory if it uses the SPT rule? (** Please show your calculation.**)

Q6.2 Factory B has received 5 different jobs, too. The processing time, due time, and unit tardiness cost of the 5 jobs for factory B are listed in the following table.

Job | Processing time (day) | Due time (day) | Unit tardiness cost (USD/day) |

25 | 40 | 1.5 | |

R | 35 | 2 | |

23 | K | 1 | |

20 | 50 | 2 | |

N | 65 | M |

Please find out the schedule (i.e., the sequence in which to complete the jobs) that has the lowest total tardiness cost among all possible schedules. You are suggested to use software (e.g. Microsoft Excel) to enumerate all possible schedules (note that the total number of schedules is 5!=120). Note that you should show the optimal schedule you obtained and the lowest total tardiness cost you got. __You do not need to show your calculation. Please answer the question in the following format:__

My values of R, K, N, and M are ____, ____, ____, and ____, respectively.

The optimal schedule (sequence) of the jobs to be completed is _____________.

The total tardiness cost of the above schedule is ______________.

**Question 7 (4 points)**

In a small town, 6 retailers ( to ) buy juice from a local factory (). The factory is responsible for the transportation of the juice, and it owns 5 warehouses ( to ) to store the juice. All juice is first transported from the factory to a warehouse and then delivered to the retailer. The annual demand of each retailer (ton) is listed in the following table.

Retailer | ||||||

Demand | 200 | 345 | 260 | 180 | 220 | 205 |

The factory has to pay a fixed maintenance cost to operate each warehouse. In order to decrease the total cost, the factory decides to shut down some of the warehouses. The annual maintenance cost (USD/year) of each warehouse, the capacity (ton) (i.e., the maximum amount of juice that can be transported from the factory) of each warehouse, the unit transportation cost (USD/ton) from the factory to each warehouse, and the unit transportation cost (USD/ton) from each warehouse to each retailer are listed in the following table.

Warehouse | |||||

Maintenance cost | 5000 | 5500 | 2900 | 3000 | 8000 |

Capacity | 400 | 500 | 350 | 370 | 600 |

Unit transportation cost from | 5 | 3 | 4 | 6 | 5 |

Unit transportation cost to | 2.5 | 3 | 3.5 | 1 | 2 |

Unit transportation cost to | 3 | 4.5 | 3 | 2.5 | 3 |

Unit transportation cost to | 3 | 3 | 2 | 2.5 | 3 |

Unit transportation cost to | 2 | 2 | 3 | 4 | 4.5 |

Unit transportation cost to | 3 | 3 | 2 | 1.5 | 3 |

Unit transportation cost to | 2 | 3 | 2 | 1 | 4 |

And for each retailer, it can get delivery from more than one warehouse. The factory aims to minimize the annual total costs while making sure that all demands from retailers should be fulfilled. Please formulate an LP or IP or MIP model that can be used to determine which warehouse(s) should be shut down and how should the juice be transported after the warehouse(s) is shut down.

Hints (note that the following is just one option of formulating the model; there may be many correct ways to formulate the model):

The decision variables are:

- , binary variable, equal to 1 if is kept and 0 if is shut down;
- the quantity (ton) of juice that receives from ; ;
- the quantity (ton) of juice transported from the factory to , .

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**Question 8 (4 points)**

A logistics company, which transports containers between 8 cities/places ( to ). The demands (number of containers to transport) and unit transportation costs by truck (100 USD/container) are listed in the following tables.

Demand (container) | ||||||||

0 | 15 | 14 | 6 | 8 | 12 | 11 | 18 | |

13 | 0 | 6 | 9 | 13 | 11 | 5 | 10 | |

11 | 14 | 0 | 13 | 15 | 10 | 9 | 12 | |

12 | 12 | 15 | 0 | 20 | 21 | 13 | 10 | |

6 | 9 | 13 | 12 | 0 | 16 | 15 | 11 | |

13 | 13 | 17 | 12 | 15 | 0 | 11 | 8 | |

5 | 8 | 4 | 15 | 13 | 11 | 0 | 10 | |

19 | 22 | 22 | 16 | 14 | 14 | 15 | 0 |

Unit transportation cost by truck (100 USD/container) | ||||||||

0 | 8 | 8 | 2 | 7 | 4 | 10 | 3 | |

8 | 0 | 7 | 7 | 8 | 4 | 8 | 8 | |

8 | 7 | 0 | 1 | 7 | 5 | 6 | 1 | |

2 | 7 | 1 | 0 | 7 | 10 | 5 | 6 | |

7 | 8 | 7 | 7 | 0 | 5 | 6 | 6 | |

4 | 4 | 5 | 10 | 5 | 0 | 2 | 7 | |

10 | 8 | 6 | 5 | 6 | 2 | 0 | 7 | |

3 | 8 | 1 | 6 | 6 | 7 | 7 | 0 |

In the demand table, “13” at the row of and the column of means that 13 containers need to be transported from to. In the unit transportation cost table, “8” at the row of and the column of means that transporting one container from to costs 8 USD. For simplicity, we define as the number of containers to be transported from place to , and as the unit transportation cost (100 USD/container) from to , , . Note that the values of and can be read from the above two tables and you can use them in your model formulation.

Now the logistics company has decided to build a hub-and-spoke (H&S) network to transport containers among these places to simplify the management and reduce the cost. In the H&S network, 3 of the 8 places will be selected to be hubs. Trains, instead of trucks, will be used to transport containers between the three hub places. The unit transportation cost using train is 0.2 times the unit transportation cost by truck. The places that are not selected to be hubs will act as feeders, and ** each feeder place is assigned to exactly one hub place**. Management rules require that

Scenarios | Description |

and are both hubs | The container will be transported by train directly from to , and the resulting cost is |

is a hub, is a feeder, and is assigned to | The container will be transported by truck directly from to , and the resulting cost is |

is a feeder, is a hub, and is assigned to | The container will be transported by truck directly from to , and the resulting cost is |

is a hub, is a feeder, and is assigned to , | The container will be transported by train from to , and then by truck to , and the resulting cost is |

is a feeder, is a hub, and is assigned to , | The container will be transported by truck from to , and then by train to , and the resulting cost is |

is a feeder, is a feeder, and they are both assigned to | The container will be transported by truck from to , and then by truck to , and the resulting cost is |

is a feeder, is a feeder, is assigned to , is assigned to , | The container will be transported by truck from to , then by train to , then by truck to , and the resulting cost is |

Please formulate an LP or IP or MIP model that can be used to determine which places should be selected as hubs and how to assign each of the other places to the hubs so that the total transportation cost can be minimized.

Hints (note that the following is just one option of formulating the model; there may be many correct ways to formulate the model):

Decision variables:

, binary decision variable, equal to 1 when is selected as a hub place, is a feeder place, and is assigned to , .

, binary decision variable, equal to 1 when is selected as a hub port, 0 otherwise, .

, binary decision variable, equal to 1 if and are both hubs and 0 otherwise, .

, binary decision variable, equal to 1 if is a hub, is a feeder, and is assigned to and 0 otherwise, .

, binary decision variable, equal to 1 if is a feeder, is a hub, and is assigned to and 0 otherwise, .

, binary decision variable, equal to 1 if is a hub, is a feeder, and is assigned to , and 0 otherwise, .

, binary decision variable, equal to 1 if is a feeder, is a hub, and is assigned to , and 0 otherwise, .

, binary decision variable, equal to 1 if is a feeder, is a feeder, and they are both assigned to and 0 otherwise, .

, binary decision variable, equal to 1 if is a feeder, is a feeder, is assigned to , is assigned to , and 0 otherwise, .

** **

**Question 9 (3 points)**

A furniture factory has decided to improve the production process of the board that is used in the assembly of wardrobe. The board quality is mainly related to the thickness of the board, so the factory took a sample that consists of 40 boards produced by the board machine. The thicknesses of the 40 boards are measured in millimeters. In the following are the measurements from the 40 boards, namely 40 observations of board thickness.

Observations

1.9 | 2.0 | 1.9 | 1.8 | 2.2 | 1.7 | 2.0 | 1.9 | 1.7 | 1.8 |

1.8 | 2.2 | 2.1 | 2.2 | 1.9 | 1.8 | 2.1 | 1.6 | 1.8 | 1.6 |

2.1 | 2.4 | 2.2 | 2.1 | 2.1 | 2.0 | 1.8 | 1.7 | 1.9 | 1.9 |

2.1 | 2.0 | 2.4 | 1.7 | 2.2 | 2.0 | 1.6 | 2.0 | 2.1 | 2.2 |

Q9.1 Please calculate the mean value of the observations of the sample in the table. (** You do not need to show your calculation.**)

Q9.2 Suppose that the process mean value is equal to the sample mean value you obtained in Q9.1. The standard deviation of the output of the process is 0.209624 millimeter. The process output is approximately normally distributed. In the specification, the upper thickness limit is 2.4 millimeters and the lower thickness limit is 1.4 millimeters. Then, what fraction of the output is expected to be out of tolerance (i.e., non-conforming)? You should use the standard normal distribution table. (** Please show your calculation.**)

Q9.3 Following Q9.2, what is the for the process? (** Please show your calculation.**)

Q9.4 Following Q9.3, we assume that the operators will take samples of 10 boards at a time, and the results are shown in the following table.

Sample | Observation | |||||||||

1 |
1.9 | 2.0 | 1.9 | 1.8 | 2.2 | 1.7 | 2.0 | 1.9 | 1.7 | 1.8 |

2 |
1.8 | 2.2 | 2.1 | 2.2 | 1.9 | 1.8 | 2.1 | 1.6 | 1.8 | 1.6 |

3 |
2.1 | 2.4 | 2.2 | 2.1 | 2.1 | 2.0 | 1.8 | 1.7 | 1.9 | 1.9 |

4 |
2.1 | 2.0 | 2.4 | 1.7 | 2.2 | 2.0 | 1.6 | 2.0 | 2.1 | 2.2 |

Use the 3-sigma rule (i.e. z=3). What are the UCL and LCL in the mean control chart? (** Please show your calculation.**)

Q9.5 Following Q9.4, use the 3-sigma rule (i.e., z=3). What are the UCL and LCL in the range chart? (** Please show your calculation.**)

Q9.6 Based on the charts in Q9.4 and Q9.5, does the current process appear to be in control? (** You do not need to show your calculation.**)

**Question 10-1 (2 points) (solve this question if your value is 1)**

The quality control department at a toy plant has inspected the number of defective toy cars in 20 random samples, and each sample contains 20 observations. The number of defective toy cars found in each sample is listed in the following table.

Q10.2 Does the current process appear to be in control? (** You do not need to show your calculation.**)

**Question 10-2 (2 points) (solve this question if your value is 2)**

The after-sale service department deals with the consumers’ claims for refunds. And the daily number of refund claims are monitored using a c-chart. The following is the historical data of refund claims in the past 20 days.

Day | Number of refund claims | Day | Number of refund claims |

1 | 3 | 11 | 3 |

2 | 2 | 12 | 4 |

3 | 3 | 13 | 2 |

4 | 1 | 14 | 1 |

5 | 3 | 15 | 1 |

6 | 3 | 16 | 1 |

7 | 2 | 17 | 3 |

8 | 1 | 18 | 2 |

9 | 3 | 19 | 2 |

10 | 1 | 20 | 3 |

We aim to construct a 3-sigma control chart (i.e. z=3) with the information provided.

Q10.2 Does the current process appear to be in control? (** You do not need to show your calculation.**)

**Question 11 (1 point)**

Which of the following best describes “bullwhip effect”? (** You do not need to show your calculation.**)

- A network of all the activities involved in delivering a finished product or service to the customer.
- Activities involved in obtaining, producing, and distributing materials and products in the proper place and in proper quantities.
- Increasingly fluctuating demand information in the upstream direction of the supply chain.
- Businesses selling to and buying from other businesses.
- On-line businesses sell to individual consumers.

**Question 12 (2 points)**

A grocery store has decided to sell sandwiches from next year. The store owner is faced with two choices, one is to purchase sandwiches from the local food manufacturer, and the other is to make sandwiches by the owner himself. If the store owner buys sandwiches from the food manufacturer, the delivery will cost him $1000 per year. And the purchasing price paid by the store owner will be $0.4 for each sandwich. No other costs are required. If the store owner makes sandwiches by himself, he will need a kitchen, which will cost him $15,000 per year. Meanwhile, the store owner can make sandwiches at the cost of $0.15 for each. No other costs are required.

Q12.1 If the store owner is sure that he can sell 60,000 sandwiches in a year, should he make sandwiches himself or buy from the food manufacturer? (** Please show your calculation.**)

Q12.2 If the grocery owner is uncertain how many sandwiches will be sold in a year, what is the indifference point between making or buying sandwiches (if the annual demand is equal to the indifference point, the annual total costs of making and buying are the same)? (** Please show your calculation.**)

**Question 13 (2 points)**

A retailer (she) sells souvenir shirts to tourists at the price of F dollars for each. The retailer buys souvenir shirts from the manufacturer (he), who produces the souvenir shirts at the cost of E dollars each. If the souvenir shirts are not sold during the tourist season, then they will be sold to local people at the price of G dollars for each. Note that all shirts that are left over will be sold to the local people because of the low selling price. The tourists’ demand for souvenir shirts is a discrete uniformly distributed random variable that can be 1, 2, …, 15. The retailer decides how many souvenir shirts to order from the manufacturer to maximize her expected total profit. The manufacturer will produce exactly the number of shirts ordered from the retailer.

Given the information, the manufacturer wants to find the optimal selling price at which he sells shirts to the retailer to maximize his expected profits. The manufacturer will choose the optimal selling price from the set , that is, the optimal selling price will be at least one dollar higher than E, at least one dollar lower than F, and an integer number of dollars. You are suggested to use software (e.g. Microsoft Excel) to enumerate all possible selling prices and select the one with the highest manufacturer’s expected total profit. __You do not need to show your calculation. Please answer the question in the following format__:

My values of E, F, and G are ____, _____, and ____, respectively.

The optimal selling price to the retailer is ____ ($/unit). At the optimal selling price, the expected total profit for the manufacturer will be ____ ($), and the expected total profit for the retailer will be ____ ($).

**Question 14 (2 points)**

A convenience store is rearranging the layout of the vending machine area to improve the service level. There is only one vending machine in the store, and each customer will spend a ** constant** 50 seconds to use the machine. It has been estimated that customers will arrive at the vending machine according to a Poisson distribution at an average of one customer every 75 seconds. To arrange the amount of space needed for the line at the vending machine, please calculate the average time a customer spends in the system, the average queue length (in persons), and the average number of persons in the system (both in line and using the machine). (

**Question 15 (2 points)**

The amusement park has set an ice cream stand in the small square. Customers arrive at the ice cream stand according to a Poisson distribution at an average rate of 70 per hour. The ice cream seller needs to make an ice cream manually for the customer and process the payment. And the service time of the seller for each customer can be described by an exponential distribution with a mean of 45 seconds per customer.

Q15.1 What is the average time a customer spends in the system? (** Please show your calculation.**)

Q15.2 In order to improve the service level, an automatic ice cream making machine will be applied in the ice cream stand. And with the machine, the service time of the ice cream seller for each customer can still be described by an exponential distribution, but the average service time becomes 25 seconds per customer. What is the average time a customer spends in the system after the ice cream making machine is applied? (** Please show your calculation.**)

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