# Statistics-The Wax bookstore purchases sweatshirts with the school

| February 25, 2017

A

The Wax bookstore purchases sweatshirts with the school name and logo from a vendor. The vendor sells the sweatshirts to the store for \$38 per shirt. The cost to the bookstore for placing an order is \$120 and the annual carrying cost is 25 percent of the cost of a sweatshirt. The bookstore manager estimates that 1700 sweatshirts will be sold during the year. The vendor has offered the bookstore the following volume discount schedule:

Order Size

1-200

201-599

600-799

800+

Discount

0%

3%

3.5%

5%

1) What is the bookstore’s optimal order quantity given this quantity discount information? Why?

2) What will be its annual inventory cost (including the holding cost, ordering cost and cost of the sweatshirts) for this optimal order quantity?

3) What is the lowest total annual inventory cost (as defined in b) for each of the other price (equivalently discount) options?

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B.

The Houston, Texas plant of Computer Products produces disk units for personal and small business computers. Gerald Knox, the plant’s production planning director, is looking over next year’s sales forecasts for these products and will be developing an aggregate capacity plan for the plant. The quarterly sales forecasts for the disk units are as follows:

1st Quarter

2nd Quarter

3rd Quarter

4th Quarter

2,340

2,610

2,700

2,520

Ample machine capacity exists to produce the forecast. Each disk unit takes an average of 20 labor-hours. In addition, you have collected the following information:

a. Inventory holding cost is \$100 per disk unit per quarter. The holding cost is based on the inventory at the end of the quarter.

b. The plant works the same number of days in each quarter, 12 five-day weeks, 6 hours per day.

c. Beginning inventory is zero disk units.

d. In a backlog situation, the customer will wait for his order to be filled but will expect a price reduction each quarter he waits. The backlog costs are \$300 per disk for the first quarter the customer waits, \$700 for the second quarter the customer waits, and \$900 for the third quarter the customer waits. In any quarter, if there is a backlog, this backlog will be filled before the demand for that period is filled.

e. The cost of hiring a worker is \$800 while the cost of laying off a worker is \$950.

f. The straight time labor rate is \$20 per hour for the first quarter and increases to \$22 per hour beginning in the third quarter.

g. Overtime work is paid at time and a half (150%) of the straight time work.

h. Outsourcing (contract work) is paid at the rate of \$480 per disk unit for the labor and you provide the material.

i. Demand is projected to increase this year. Demand during the fourth quarter of the prior year was 2,340 units. The demand for the first quarter of the next year (year following the year you are analyzing) is projected to be at the 2,700 unit level.

1) You want to maintain a work force capable of producing 2,520 in a quarter outsourcing any disk units over this quantity. Excess units produced in a quarter would be carried over to meet demand in a subsequent quarter. Any additional demand is met through outsourcing. All workers will be fully utilized each quarter. In other words, there is no under utilization. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.

2) The company will maintain a work force capable of producing 2,340 units in a quarter. It will allow backlogs to occur until the fourth quarter when it will outsource all demand that cannot be met with its own workforce. All workers will be fully utilized each quarter. In other words, there is no under utilization. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.

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C.

Johnson Trucking is planning its truck purchases for the coming year. It allocated \$600,000 for the purchase of additional trucks, of which three sizes are available. A large truck costs \$150,000 and will return the equivalent of \$15,000 per year to profit. A medium-sized truck costs \$90,000 and will return the equivalent of \$12,000 per year. A small truck costs \$50,000 and will return the equivalent of \$9,000 per year. WeHaul has maintenance capacity to service either four large trucks, five medium-sized trucks, or eight small trucks, or some equivalent combination. WeHaul believes that it will be able to hire a maximum of seven new drivers for these added trucks. The company cannot spend more than one/half of the total funds it actually spends to purchase medium-sized trucks. (Hint: this is not necessarily one half of the total funds it has allocated for the purchase of additional trucks).

Provide the linear programming formulations and show the linear programming software solution.

1) Formulate a linear programming model to be used for determining how many of each size of truck to purchase if the company wants to maximize its profit. Ignore the time value of money. Provide the linear programming variables, the objective function, and the constraints for the problem.

2) At optimality, how much profit will result and what is the optimal combination of trucks? Provide the linear programming formulations and show the linear programming software solution.

If the answer is in fractional units of trucks that is acceptable – do not round to whole number of trucks.

3) Using your sensitivity analysis output, provide two sensitivity analysis interpretations. One must be for the objective function and one must be for one of the constraints. Povide the source of your answers from the sensitivity analysis output.

4) Now suppose that there is a requirement that must purchase at least one of each size truck. Write the constraint(s) for this requirement. However, you do not need to resolve the problem.

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D.

Development of a new deluxe version of a particular toy is being considered by T-O-Y Inc. The activities necessary for the completion of this project along with the relevant information for each activity are listed in the following table. The Total Crash Time represents the time for that activity after it has been crashed the maximum crash time allowed for that activity. For example, we can reduce the time for Activity A by one week to 3 weeks after crashing it.

Activity

Time to complete using Normal Time (Weeks)

Time activity is completed using Crash Time (Weeks)

Total Normal Cost

Crash Cost per period

Immediate Predecessors

A

4

3

\$2,000

\$600

B

2

1

2,200

\$800

C

3

3

500

\$0

D

8

4

2,300

\$750

A

E

6

3

900

\$100

B

F

3

2

3,000

\$1,200

C

G

4

2

1,400

\$300

D,E

1) Using the normal time, what is the critical path, the project completion time, the total time required to complete all of the activities, and the slack time for each path and each activity? (The slack time for each path should be relative to the time to complete the critical path.)

2) If you wish to reduce the time required to complete this project by one week, which activity should be crashed, and how much will this increase the total cost?

3) What is the maximum time that can be crashed? How much would the total cost increase?

4) Let’s return to the results found in a). Suppose the company believes that reducing the project completion time from the time reported in part a), will help the company realize an additional \$700 per week in revenue for the first two weeks the project time is reduced, \$650 per week in revenue for the next two weeks the project is reduced and \$500 per week in revenue after that for every week the project is reduced. How many weeks should the company reduce the project completion time and what is the resulting cost of the project considering the additional revenue as an offset to the cost associated with crashing the project? The company is interested in selecting the project duration and reduced activity times that minimize the cost of the project.
E.

There are seven lab tests (shown below) that need to be run in the lab at Cheryl’s hospital. The technician who does the testing can only do one at a time. Each test has a testing duration and due date (promised time). In all seven cases, the test results are coordinated with the results of other tests done in separate labs. Missing the deadline could delay a diagnosis. The tests are numbered as they come in the lab, but not all tests go through this department (hence the missing sequence numbers).

Test Time (Hours)

Promised Time From Now (End of Hour)

Test #104

3

9

Test #108

2

7

Test #112

5

6

Test #132

6

14

Test #141

8

22

Test #143

1

12

Test #150

4

19

It is now the end of the first hour and you have not started any of these tests. You are asked to schedule and evaluate several sequencing rules. You are ready to schedule these seven tests. They are all ready to be run.

1) Determine the sequence of jobs using the following rules:

· Longest processing time (LPT)

· Critical ratio rule (CR)

2) Prioritize the rules used in a) based on average flow time measured from now.

3) Prioritize the rules used in a) based on average lateness with no credit for completing any test early. If a job is late, its late time is zero (“0”).

5) Prioritize the rules used in a) based on average early time. If a job is late, its early time is zero (“0”).

6) Which rule would you recommend using? Why?

F.

A food processor uses approximately 28,000 glass jars a month for its fruit juice product. Because of storage limitations, a lot size (order quantity) of 4,000 jars has been used. The monthly holding cost is 18 percent of the jar’s cost of \$1.00. The ordering cost is \$60 per order. The company operates an average of 20 days a month, twelve months a year. The lead time is 6 working days. The product is used uniformly throughout the 20 days.

1) What is the reorder point and what is the time between orders?

2) The manager would prefer ordering every 4 days (5 times per month) but would have to justify any change in the order size from its current 4,000 order quantity. One possibility is to simplify and improve the order processing activity to reduce the ordering cost. From a cost perspective, assuming everything else remains the same, what ordering cost would enable the manager to justify ordering every fourth day compared to its current 4,000 order quantity?

3) If the manager were to use the EOQ strategy, what would the order quantity be? What is the inventory position immediately after placing this order?
G.

Rick Wing is looking at improving the temperature control on one of his machines. This system currently has four integrated controllers in series:

.0/msohtmlclip1/01/clip_image001.gif”>.0/msohtmlclip1/01/clip_image001.gif”>.0/msohtmlclip1/01/clip_image001.gif”>A B C D

The reliability of these four controllers are 0.90 (A), 0.92 (B), 0.94 (C), and 0.96 (D).

Rick is looking at installing a back-up for each of the controllers. The back-up controllers all have a reliability of 0.90

1) What is the reliability of the new system which provides a back-up for each of the controllers and what is the percent improvement compared to the current design?

2) Suppose that Rick is considering another back-up design. Here he would create a back-up system in parallel to the current system. This is shown below. The reliability of all four back-ups, A1, B1, C1, and D1, are each 0.90.

.0/msohtmlclip1/01/clip_image002.gif”>.0/msohtmlclip1/01/clip_image001.gif”>.0/msohtmlclip1/01/clip_image001.gif”>.0/msohtmlclip1/01/clip_image003.gif”>.0/msohtmlclip1/01/clip_image004.gif”>A B C D

.0/msohtmlclip1/01/clip_image005.gif”>
.0/msohtmlclip1/01/clip_image006.gif”>

.0/msohtmlclip1/01/clip_image007.gif”>.0/msohtmlclip1/01/clip_image008.gif”>.0/msohtmlclip1/01/clip_image007.gif”>A1 B1 C1 D1

What is the reliability of this back-up design?
H.

A South American country has 17 planes in its air force. They are short on money and do not want to spend any more than necessary to keep the planes in running condition. The cost to perform preventive maintenance on all 17 planes is \$16,000 in local dollars. If a plane has a malfunction between preventive maintenance, it costs \$5,000 (local dollars) to while the plane is out of service for repairs and other related costs (referred to as the breakdown cost). The malfunction history on the planes is as follows.

Months Between PM

Average Number of Breakdowns Between PM Cycles

1

2.52

2

5.64

3

8.20

4

12.24

1) How often should preventive maintenance be performed based on a cost analysis?

2) What is the breakdown cost that would equate the monthly total maintenance strategy for a two-month PM cycle with a three-month PM cycle? For this analysis, assume that except for the cost of a breakdown all other data (facts and figures) are the same as given in the original problem statement.

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