Question 1:
The New York NY plant of Computer Products produces disk drives for personal and small business computers. Ben Dover, 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 2,700 2,610 2,520
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. 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 and either using overtime or outsourcing as explained next for any disk units over this quantity that are required in a quarter. If additional capacity is needed during quarters one and two, the company will use overtime to meet the additional demand during each of these periods. During quarters three and four, the company will use outsourcing if needed to meet the demand during each of these periods. The company will only produce what is required during a quarter and underutilization will occur if all of the production capacity is not required. 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 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 underutilization. No overtime is worked during any quarter. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.
Question 2:
The Kentucky 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 3:
The information technology department of Microsoft 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 4:
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 5:
5-You are to plan production for a four-month period: February through May. For February and March, you should produce to the exact demand forecast. During February and March, you will follow a matching strategy and hire and layoff the works as required. For April and May, you should use overtime and inventory with a stable workforce. Stable means that the number of workers needed for March will be held constant through May. However, government constraints put a maximum of 5,000 hours of overtime labor per month in April and May. There is no overtime allowed in February and March. If demand exceeds supply, then backorders occur. If there is excess capacity in May, the workers are paid but there is no work performed hence there are no extra units produced in May. There are 100 workers on January 31. You are given the following demand forecast
February - 80,000
March - 64,000
April - 100,000
May - 40,000
Productivity is four units per worker hour. Each worker works eight hours per day, 20 days per month. Assume zero inventory on February 1. There is to be no inventory at the end of May. Hiring costs are $50 per worker, while layoff costs are $70 per worker. The inventory holding cost is $10 per unit per month based on the number of units in inventory at the end of the month. Straight time labor is $10 per hour and overtime is 150% of the straight time labor rate. Backorder costs are $20 per unit per month.
Find the total cost of this plan.