A manager is applying the transportation model of linear programming to solve an aggregate planning problem. Demand in period 1 is 100 units and in period 2 demand is 150 units. The manager has 125 hours of regular employment available for $10/hour each period. In addition, 50 hours of overtime are available for $15/hour each period. If holding costs are $2 per unit each period, how many hours of regular employment should be used in period 1 (assume demand must be met in both periods 1 and 2 for the lowest possible cost and that production is 1 unit per hour)?
A) 100
B) 125
C) 150
D) 50
E) none of the above

Question

Quizplus · Accepted Answer

We can set up the transportation model as follows:
- Let x be the number of units produced in period 1
- Let y be the number of units produced in period 2
- Let z be the number of units held over from period 1 to period 2
- Transportation costs: 
    - Regular employment: $10/hour
    - Overtime: $15/hour
- Number of hours available:
    - Regular employment: 125 hours each period
    - Overtime: 50 hours each period
- Holding costs: $2 per unit each period

The objective function is to minimize the total cost:
Total cost = (transportation cost for x units from period 1) + (transportation cost for y units from period 2) + (holding cost for z units from period 1 to period 2)

Minimize 10x + 10y + 15(max(0, x+y-100)-z)*0 + 2z + 15(max(0, x+y-150))*0

Subject to:
- x + z = 100 (demand in period 1)
- y + max(0, x+z-100) = 150 (demand in period 2)
- x, y, z >= 0
- x <= 125 (regular employment available in period 1)
- y <= 125 (regular employment available in period 2)
- x+y <= 175 (regular employment + overtime available in period 1 and 2)

Solving the model, we get x = 100, y = 125, and z = 0. Therefore, the manager should use 125 hours of regular employment in period 1 to produce 100 units to meet the demand, and use the remaining 25 hours to produce 25 units to be held over for period 2. This is the lowest possible cost, taking into account transportation costs and holding costs.

A Manager Is Applying the Transportation Model of Linear Programming