Question 1:
A grocery store orders paper grocery bags from a distributor. The store uses 2,300 cases of bags per year, and its ordering cost is $65 per order. The store's carrying cost rate is 35% of the price per case of bags. The distributor has the following pricing structure for cases of bags:
|
Order Quantity (Number of cases)
|
Price per case
|
|
1-49
|
$129.45
|
|
50-249
|
$127.95
|
|
250-999
|
$126.45
|
|
1000 and over
|
$125.95
|
a) How many cases of bags should the store order each time if we want to select the lowest total annual cost strategy (ordering cost plus holding cost plus material acquisition cost)?
b) What is the resulting total cost per year (ordering cost plus holding cost plus material acquisition cost) for this strategy and the next lowest cost strategy?
c) If the store has only enough storage space for 100 cases of bags, how many should it order each time? Why? If a calculated order quantity is a fraction of a case, leave it as a fractional amount.
Question 2:
The Austin, Texas plant of Computer Products produces disk drives 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 drives are as follows:
|
1st Quarter
|
2nd Quarter
|
3rd Quarter
|
4th Quarter
|
| 2,340 |
2,520
|
2,700
|
2,610
|
Ample machine capacity exists to produce the forecast. Each disk drive takes an average of 20 labor-hours. In addition, you have collected the following information:
a. Inventory holding or carrying cost is $100 per disk drive per quarter. The holding cost is based on the inventory at the end of each 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 drives.
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 floppy 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 in the fourth 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 $475 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 and this was met with a workforce that was fully employed with no under utilization and no overtime. 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.
a) You want to maintain a work force capable of producing 2,520 in a quarter. This work force is fully employed and there is no under utilization. If more units are produced in a quarter than are needed, they will be used to help meet demand in future periods. When demand in a quarter cannot be met from the units produced in that quarter or from units produced in previous quarters, the company will use overtime to meet the unfilled demand. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.
b) The company will use a modified form of the matching demand strategy. Not exceeding a workforce capable of producing 2,520 units in a quarter, the company will match the workforce in each quarter to the demand for that quarter. To meet the demand in any quarter when the workforce at full utilization cannot meet the demand, the company will use overtime to meet the unfilled demand. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.
Question 3
Assume that Product Z is made up of one unit of A and five units of B. A is made of three units of C and four units of D. D is made of two units of E. The demand for Z is 100 units in Week 8 and 200 units in Week 10. For Z, A, C, and D, there are no units on hand, no safety stock, and no units allocated. Item E has 900 units on hand, with a safety stock of 200 units and no units allocated. Item B has 200 units on hand, 100 units of safety stock and 100 units already allocated. There are 100 units of Item C currently scheduled to arrive in Week 1 and 200 units of item D are scheduled to arrive in Week 1
|
Week
|
6
|
7
|
8
|
9
|
10 |
|
Demand
|
--
|
--
|
100
|
-
|
200 |
Additional information is the following:
|
Item
|
Z
|
A
|
B
|
C
|
D
|
E
|
|
On hand
|
--
|
--
|
200
|
_-
|
--
|
900
|
|
Safety stock
|
--
|
--
|
100
|
--
|
--
|
200
|
|
Allocated
|
--
|
--
|
100
|
--
|
--
|
--
|
|
Scheduled Receipts
|
--
|
--
1
|
100 in Week
1
|
200 in
Week 1
|
--
|
Provide the additional number of all items, including Z, that are required to meet the total demand shown for Product Z. Ignore the timing of these requirements.
Question 4
The Bluegrass Distillery produces custom-blended whiskey. A particular blend consists of rye and bourbon whiskey. The company has received an order for a minimum of 400 gallons of the custom blend. The customer specified that the order must contain at least 40 percent rye and not more than 250 gallons of rye. The blend cannot contain more than 45% bourbon. The distillery can produce no more than 500 gallons per week, regardless of the blend. The production manager wants to complete the order in one week. The blend is sold for $12 per gallon. The distillery company's cost per gallon is $4 for rye and $2 for bourbon. The company wants to determine the blend mix that will meet customer requirements and maximize profits.
If a calculated quantity is a fractional amount, leave it as a fractional amount.
a) Formulate the linear programming model for this problem providing the objective function and the constraints and the definition of the variables.
b) Using linear programming, solve the problem for the optimal answer and provide the value of the objective function and the variables at optimality. Provide your linear programming solution.
c) Using the sensitivity analysis output from your solution in b), provide any two sensitivity analysis interpretations. One of the interpretations must relate to the objective function and the second one must relate to one of the constraints.
d) Now suppose that the company wants to minimize the cost of production. It also wants to ensure that the profit is at least $3,500. Provide the changes to the formulation in part a) that would need to be made for these modifications. You do not need to solve for the optimal answer but only need to changes to the formulation you provided in a).
You must submit your linear programming formulation and show the linear programming software solution to this problem to receive credit.
Question 5
The marketing department at Bellevue University is working on a project with the following information. The Activity Time is also referred to as the Normal Activity Time and is the time it takes to complete the activity based on the normal times. The Crash Time is the time it takes to complete the activity if this activity is crashed.
|
Activity
|
Immediate
Predecessors
|
Activity
Time
(Days)
|
Activity
Cost (Total)
|
Total Time
After
Crashing
(Days)
|
Activity
Crash Cost
(Total)
|
|
A
|
--
|
6
|
$10,000
|
4
|
$12,000
|
|
B
|
-
|
4
|
$5,500
_
|
2
|
$7,300
|
|
C
|
A, B
|
3
|
$6,000
|
2
|
$7,500
|
|
D
|
C
|
6
|
$4,000
|
4
|
$6,400
|
|
E
|
C
|
4
|
$3,500
|
2
|
$5,100
|
a) What is the critical path for this project, what is the project completion time, how much total work time (in days) is required for this project, and what is the cost for completing this project using the Activity Time? Also identify the slack time for each path. The slack time for each path should be based on the time to complete the critical path.
b) Suppose that the school wants to complete the project in 13 days by crashing the most economical activity. Which activity(ies) should it crash and how much will the project cost increase?
c) Suppose that the school wants to complete the project in 12 days by crashing the most economical activity(ies). Which activity(ies) should it crash and how much will the project cost increase from your answer in a)?
d) Start with the original Activity Times in answering this part. In addition to the cost information provided above, indirect project costs total $600 per day. The company also incurs $300 in penalty costs for each day the project lasts beyond eleven days. What is the minimum cost schedule for this project? What is the cost of this schedule and what is the project completion time?
Question 6.
Today is the start of Day 4 and John Smith is ready to start scheduling the following jobs. Jobs are assigned a letter of the alphabet, starting with the letter A, based on their arrival.
|
Job
|
Processing Time (in days)
|
Due Date (End of Day)
|
|
A
|
3
|
10
|
|
B
|
10
|
12
|
|
C
|
2
|
25
|
|
D
|
4
|
8
|
|
E
|
5
|
15
|
|
F
|
8
|
18
|
|
G
|
7
|
20
|
a) Sequence these jobs based on the following scheduling rules: Slack and Critical Ratio (CR) (time to due date/processing time). Use the definitions of these scheduling rules as we defined them on the homework assignments.
b) Based on average flow time measured from today (start of Day 4), which of the sequencing rules is preferred?
c) Based on the average lateness with no credit given for jobs completed early, which of the sequencing rules is preferred?
d) Based on the average early time with zero days early if the job is completed late, which of the sequencing rules is preferred?
Question 7.
The information technology department of Bellevue University buys paper for its copier machine frequently. The office manager would like to determine the best quantity to order each time an order is placed. She has estimated that the ordering cost is $12 each time an order is placed. The monthly demand for paper is 135 reams (500 sheets to a ream). The cost of paper is $6.50 per ream, and the carrying cost is 25 percent of the paper cost per month. Base your answers to the following questions on the economics that are provided and using the traditional inventory models we have studied.
a) How many reams should be ordered at a time?
b) Suppose the information technology department of the university only has space to hold 35 reams of paper at any time. How many reams should be ordered at a time? Why?
c) There are 350 working days per year and the lead time is 3 days. What are the reorder point and the inventory position immediately after placing the order?
d) If the university were to order at least 60 reams of paper every time it places an order, the paper company will lower the price of the paper by $0.33 for all reams of paper. The university can now acquire all of the storage space that it needs. However, it will cost the university an additional $1.00 per ream per month for storage. What is the difference in the annual inventory cost between this policy and the policy found in a)? Consider all relevant costs.
Question 8.
It currently costs a total of $1,000 for an automobile engine repair following a breakdown. This is based on performing oil changes every 60,000 miles. However, for every 10,000 miles an oil change is delayed, the engine repair costs following a breakdown increases by $100 (for example if the oil change is 70,000 miles, the repair cost is $1,100). Following is breakdown information for a fleet of five automobiles. The average number of breakdowns between oil changes for this fleet is as follows:
|
Thousands of Miles between Oil
Changes
|
Average Number of Breakdowns between Oil Changes
|
|
60
|
0.2500
|
|
70
|
0.7265
|
|
80
|
1.8131
|
|
90
|
3.6469
|
|
100
|
5.5822
|
A custom oil change with filter, long-wearing oil, and careful adjustments costs $200 at each oil change for each automobile assuming that the oil is changed every 60,000 miles. For every 10,000 miles the oil change is delayed, the cost of an oil change increases by $25. For example, if the oil change is every 70,000 miles, the oil change is $225.
Which interval between oil changes would you select for a fleet of five automobiles? Base this recommendation on the economics as presented.
Question 9.
The estimated times and immediate predecessors for the activities in a project at John Black's company are given in the following table. Assume that the activity times are independent. All times are in weeks.
|
Activity
|
Predecessors
|
Optimistic Time
(a)
|
Most Likely Time
(m)
|
Pessimistic Time (b)
|
|
A
|
--
|
2
|
4
|
6
|
|
B
|
--
|
1
_
|
2
|
9
|
|
C
|
A
|
2
|
4
|
6
|
|
D
|
B
|
5
|
8
|
17
|
|
E
|
C
|
3
|
4
|
11
|
a) What is the critical path? You may use any method to determine this.
b) What is the expected completion time of the project?
c) What is the probability of completing the project in 16 weeks or more?
Question 10.
Below is the computer solution to a linear programming problem linear programming:
|
Maximize
|
X1 X2
40
|
30
|
RHS
|
Shadow
Price
|
|
Constraint 1
|
0.4
|
0.5 <=
|
20
|
33.3333
|
|
Constraint 2
|
0.0
|
0.2 <=
|
5
|
0
|
|
Constraint 3
|
0.6
|
0.3 <=
|
21
|
44.4444
|
|
Solution
|
25
|
20
|
|
|
Sensitivity Analysis
Variable Value Reduced Original Lower Upper
Cost Value Bound Bound
|
X*I
|
25
|
0
|
40
|
24
|
60
|
|
X2
|
20
|
0
|
30
|
20
|
50
|
Constraint Shadow Slack/ Original Lower Upper
Price Surplus Value Bound Bound
|
Constraint 1
|
33.3333
|
0
|
20
|
14.00
|
21.50
|
|
Constraint 2
|
0
|
1
|
5
|
4.00
|
Infinity
|
|
Constraint 3
|
44.4444
|
0
|
21
|
18.75
|
30.00
|
Note: The Reduced Cost is often referred to as the Coefficient Sensitivity.
For the above information, answer the following questions. Provide your answers based on the above information and explain your answers in terms of this information.
a) Write the objective function in equation form and write the constraints as equalities or inequalities.
b) What are the values of the variables at optimality and what is the value of the objective function at optimality?
c) If there was an opportunity to purchase additional units of each resource (as expressed by the constraints), based on the sensitivity analysis which resource(s) would you consider for purchase? Why? Identify all of the resources that you would consider. If you did not consider a resource for purchase, why did you exclude it? Remember that Constraint 1 is equivalent to Resource 1, Constraint 2 is Resource 2, and Constraint 3 is Resource 3.
d) Suppose the contribution of variable X2 in the objective function increases by 10. What are the values of the variables at optimality now and what is the value of the objective function at optimality with this change? Use the above sensitivity analysis in answering this question and provide the sensitivity analysis information you used.