Develop a linear-programming model for the least-cost production plan


Assignment #3

Q1. Harold Grey owns a small farm that grows apricots in the Salinas Valley. The apricots are dried on the premises and then sold to a number of large supermarket chains. Based on past experience and committed contracts, he estimates that the sales over the next five years in thousands of packages will be as follows:

Year Forecasted Demand (thousands of packages)

1 300 2 120 3 180 4 100 5 250

Grey currently has three workers on the payroll. Assume that each worker stays on the job for at least one year. He estimates that he will have 10,000 packages on hand at the end of current year. Assume that, on the average, each worker is paid $25,000 per year and is responsible for producing 30,000 packages. Inventory costs have been estimated to be 4 cents per package per year, and shortages are not allowed.

Based on the effort of interviewing and training new workers, Grey estimates that it costs $500 for each worker hired. Severance pay amounts to $1,000 per worker.

(a) Assuming that shortages are not allowed, determine the minimum constant workforce that he will need over the next five years.

(b) Evaluate the cost of the plan found in (a).

(c) Develop a linear-programming model for the least-cost production plan.

Q2. The ABC Toy Company makes a few types of toy cars on one of its production line. Based on orders received and forecasts of future demand, it is estimated that the demand (in units) for the next four seasons is: Fall 10,000; Winter 8000; Spring 7,000; Summer 12,000. Inventory at the beginning of Fall is 660 units. At the beginning of Fall, ABC has 30 workers.

Relevant costs: hiring: $100/worker hired layoff: $200/worker laid off inventory holding: $5/unit/season backorder: $10/unit regular time cost: $5/hour

Assume that the productivity is 0.5 units per worker hour, with eight hours per day and 60 days per season. Inventory cost is estimated based on the ending inventory for each period. The relevant costs are shown in the accompanying table.

(a) Find the total cost of the level plan.

(b) Find the total cost of the chase plan.

(c) Suppose that ABC has negotiated with the union on the option of using the regular workforce on overtime ($8/hour) during Winter and Spring, if necessary. Overtime is not available for the other seasons. The new company policy restricts hiring (or firing) to take place only once a year. Consequently, it is decided that hiring takes places only in the be- ginning of Summer while firing can be done at the end of Summer. Formulate the problem as a linear program. Be sure to define all the variables.

Q3. Leather-All produces a line of handmade leather products. At the present time, the company is producing only belts, handbags, and attache cases. The predicted demand for these three types of items over a six-month planning horizon is as follows:

Month No. of Belts Handbags Attache Cases Working days

1 22 2500 1250 240 2 20 2800 680 380 3 19 2000 1625 110 4 24 3400 745 75 5 21 3000 835 126 6 17 1600 375 45

The belts require an average of two hours to produce, the handbags three hours, and the attache cases six hours; all the workers have the skill to work on any item. Leather-All has 46 employees who each have a share in the firm and cannot be fired. There are an additional 30 locals that are available and can be hired for short periods at a higher cost. Regular employees earn $8.50 per hour on regular time, and $14.00 per hour on overtime. Regular time comprises a seven-hour workday and the regular employees will work as much overtime as is available. The additional workers are hired for $11.00 per hour and are kept on the payroll for at least one full month. Costs of hiring and firing are negligible. Because of the competitive nature of the industry, Leather-All does not want to incur any demand backorders.

(a) Using worker hours as an aggregate measure of production, convert the forecasted de- mand to demand in terms of aggregate units.

(b) What would be the size of the work force needed to satisfy the demand for the coming six months on regular time only? Would it be to the company’s advantage to bring the permanent work force up to this level? Why or why not?

(c) Determine a production plan that meets the demand using only regular time employees and the total cost of that plan.

(d) Determine a production plan that utilizes only additional employees to absorb excess demand and the cost of that plan.

(e) Formulate the problem of optimizing Leather-All’s hiring schedule as a linear program. Define all variables and include the necessary constraints.

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