Question

1. A company producing toy trucks is working to create the production schedule for the next few months. Its current inventory level is 5,000 toy trucks and, at its current workforce level of 100 employees, is capable of producing 50,000 trucks. The company pays its employees $2,500 per month.

The company expects demand during the next three months will be:

Month	1	2	3
Demand	48,000	44,000	52,000

Management has decided to maintain a minimum inventory level (measured at the end of the month) of 2,000 trucks to be able to handle variations in demand. Inventory storage costs are $0.75/unit. The company can currently store up to 8,000 units per month.

The company would like to minimize the changes to its capacity in order not to cause undue labour problems. Therefore any increases in capacity will be done through short-term contracts. The company has worked out that increasing capacity this way will cost $7.00/unit. Because of an existing labour agreement, capacity can only be increased this way to a maximum of 5,000 units. Each short-term employee is expected to produce 500 units per month, the same as normal employees.

Any decrease in capacity will be handled by having workers perform maintenance or other non-productive activities, and so the company will not see any increase in costs. However, the company would like to minimize the maximum number of unnecessary employees to 10, or an equivalent capacity of 5,000 units.

The company wants to create a production requirements model to help it find the best production schedule.

1.a: Create a linear program to solve the production schedule for the next three months. Do not solve.

1.b: The company estimates its delivery costs to be $0.40 per toy truck, on average. How would this information affect the production requirements model?

Answer 1

Answer:-

Given that:-

A company producing toy trucks is working to create the production schedule for the next few months. Its current inventory level is 5,000 toy trucks and, at its current workforce level of 100 employees, is capable of producing 50,000 trucks. The company pays its employees $2,500 per month.

1.a: Create a linear program to solve the production schedule for the next three months. Do not solve.

Linear Program is as follows:

Decision variables:

Let

Wi be the number of employees used for production in month i

Oi be the number of units produced through through short-term contracts in month i

Fi be the total number of unnecessary employees in month i

Vi be the ending inventory in month i

Cost per employee

Objective function:

Min 2500W1+2500W2+2500W3+7O1+7O2+7O3+0.75V1+0.75V2+0.75V3

s.t.

Constraints:

500W1+O1-V1 = 43000 (Net production in month 1 = Demand - beginning inventory = 48000-5000 = 43000)

500W2+O2-V2+V1 = 44000

500W3+O3-V3+V2 = 52000

O1, O2, O3 5000

V1, V2, V3 2000

V1, V2, V3 8000

W1, W2, W3 50

W1, W2, W3 40 (maximum unnecessary employees is 10, so minimum number of employees in any month = 50-10 = 40)

Wi, Oi, Vi 0

1.b: The company estimates its delivery costs to be $0.40 per toy truck, on average. How would this information affect the production requirements model?

elivery cost will be added in the objective function

Productivity per employee = 500 units per month

So, total delivery cost on productivity of each employee = 0.4*500 = $ 200

So, effective cost for production per employee = 2500+200 = $ 2700

Unit cost per unit through short term contract = 7+0.4 = $ 7.4

Revised Objective function is following:

Min 2700W1+2700W2+2700W3+7.4O1+7.4O2+7.4O3+0.75V1+0.75V2+0.75V3

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