Exercise Solutions :

We define a comprehensive set of decision variables that are utilized in problems 8-1 to 8-3 depending on the problem context.

Decision Variables:

Ht = # of workers hired in month t (t = 1,..,12)

Lt = # of workers laid-off in month t (t = 1,..,12)

Wt = # of workers employed in month t (t = 1,..,12)

Ot = # of hours of overtime in month t (t = 1,..,12)

It = # of units (000s) held in inventory at the end of month t (t = 1,..,12)

Ct = # of units (000s) subcontracted in month t (t = 1,..,12)

Pt = # of units (000s) produced in month t (t = 1,..,12)

Parameters:

Dt = # of units (000s) demanded in time period t (t = 1,…12)

Problem 8-1:

Minimize

Subject to:

Inventory constraints:

Overtime constraints:

Production constraints:

Workforce constraints:

(a) Worksheet 8.1 provides the solution to this problem and the corresponding aggregate plan. The total cost of the plan is $360,400,000.

(b) If the number of overtime hours per employee were increased from 20 to 40 it would result in decreasing the total cost to $356,450,000. So, it is advantageous to do it.

(c) If the number of employees is decreased to 1200 and the overtime hours per employee are held at 20 and 40 then the total costs of the plan are $363,324,000 and $357,422,000, respectively. If the number of employees is increased to 1300 and the overtime hours per employee are held at 20 and 40 then the total costs of the plan are $358,790,000 and $356,270,000, respectively. So, the value of additional overtime increases as workforce size decreases.

(d) We add a new constraint: . The cost will be $363,049,982.

Problem 8-2:

We now include the subcontract option in the model:

Minimize

Subject to:

Inventory constraints:

Overtime constraints:

Production constraints:

Workforce constraints:

Worksheet 8-2 provides the solution to this problem.

(a) Without the subcontract option the total cost is