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a)
To determine: The optimal plan using the transportation method.
Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.
a)
Answer to Problem 17P
The optimal plan using the transportation method has been developed.
Explanation of Solution
Given information:
The following information has been given:
| Quarter |
| Regular time | Overtime | Subcontract |
| 1 | 500 | 400 | 80 | 100 |
| 2 | 750 | 400 | 80 | 100 |
| 3 | 900 | 800 | 160 | 100 |
| 4 | 450 | 400 | 80 | 100 |
Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.
Develop optimal plan using transportation model:
Develop cost matrix:
| Cost matrix | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Supply |
| Beginning inventory | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 250 |
| Regular time 1 | 1 | 1.2 | 1.4 | 1.6 | 1.8 | 400 |
| Over time 1 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 80 |
| Subcontract 1 | 2 | 2.2 | 2.4 | 2.6 | 2.8 | 100 |
| Regular time 1 | 1.5 | 1 | 1.2 | 1.4 | 1.6 | 400 |
| Over time 1 | 2 | 1.5 | 1.7 | 1.9 | 2.1 | 80 |
| Subcontract 1 | 2.5 | 2 | 2.2 | 2.4 | 2.6 | 100 |
| Regular time 1 | 2 | 1.5 | 1 | 1.2 | 1.4 | 400 |
| Over time 1 | 2.5 | 2 | 1.5 | 1.7 | 1.9 | 80 |
| Subcontract 1 | 3 | 2.5 | 2 | 2.2 | 2.4 | 100 |
| Regular time 1 | 2.5 | 2 | 1.5 | 1 | 1.2 | 400 |
| Over time 1 | 3 | 2.5 | 2 | 1.5 | 1.7 | 80 |
| Subcontract 1 | 3.5 | 3 | 2.5 | 2 | 2.2 | 100 |
| Demand | 500 | 750 | 900 | 450 | 2570 | |
| 2600 |
Excel worksheet to generate the above table:
Develop optimal plan:
| Optimal plan | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Dummy |
| Beginning inventory | 100 | 150 | ||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 800 | |||||
| Over time 1 | 40 | 100 | 20 | |||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 50 | 30 | ||||
| Subcontract 1 | 100 | |||||
| Demand | 500 | 750 | 900 | 450 |
The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.
b)
To determine: The total cost of the optimal plan.
Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.
b)
Answer to Problem 17P
The optimal cost of the plan is $2,641.
Explanation of Solution
Given information:
The following information has been given:
| Quarter | Forecast (units) | Regular time | Overtime | Subcontract |
| 1 | 500 | 400 | 80 | 100 |
| 2 | 750 | 400 | 80 | 100 |
| 3 | 900 | 800 | 160 | 100 |
| 4 | 450 | 400 | 80 | 100 |
Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.
Develop optimal plan using transportation model:
Develop cost matrix:
| Cost matrix | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Supply |
| Beginning inventory | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 250 |
| Regular time 1 | 1 | 1.2 | 1.4 | 1.6 | 1.8 | 400 |
| Over time 1 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 80 |
| Subcontract 1 | 2 | 2.2 | 2.4 | 2.6 | 2.8 | 100 |
| Regular time 1 | 1.5 | 1 | 1.2 | 1.4 | 1.6 | 400 |
| Over time 1 | 2 | 1.5 | 1.7 | 1.9 | 2.1 | 80 |
| Subcontract 1 | 2.5 | 2 | 2.2 | 2.4 | 2.6 | 100 |
| Regular time 1 | 2 | 1.5 | 1 | 1.2 | 1.4 | 400 |
| Over time 1 | 2.5 | 2 | 1.5 | 1.7 | 1.9 | 80 |
| Subcontract 1 | 3 | 2.5 | 2 | 2.2 | 2.4 | 100 |
| Regular time 1 | 2.5 | 2 | 1.5 | 1 | 1.2 | 400 |
| Over time 1 | 3 | 2.5 | 2 | 1.5 | 1.7 | 80 |
| Subcontract 1 | 3.5 | 3 | 2.5 | 2 | 2.2 | 100 |
| Demand | 500 | 750 | 900 | 450 | 2570 | |
| 2600 |
Excel worksheet to generate the above table:
Develop optimal plan:
| Optimal plan | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Dummy |
| Beginning inventory | 100 | 150 | ||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 800 | |||||
| Over time 1 | 40 | 100 | 20 | |||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 50 | 30 | ||||
| Subcontract 1 | 100 | |||||
| Demand | 500 | 750 | 900 | 450 |
The given demand and supply should be splitted and the remaining supply and demand should be used as a dummy value.
Calculate the total optimal cost:
It is calculated by adding the multiple of values in the optimal plan table and the value in the cost matrix to the respective value.
Hence, the total optimal cost is $2,641.
c)
To determine: The number of units remained unused in regular time capacity.
Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.
c)
Answer to Problem 17P
No, the regular time capacity remains unused.
Explanation of Solution
Given information:
The following information has been given:
| Quarter | Forecast (units) | Regular time | Overtime | Subcontract |
| 1 | 500 | 400 | 80 | 100 |
| 2 | 750 | 400 | 80 | 100 |
| 3 | 900 | 800 | 160 | 100 |
| 4 | 450 | 400 | 80 | 100 |
Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.
Develop optimal plan:
| Optimal plan | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Dummy |
| Beginning inventory | 100 | 150 | ||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 800 | |||||
| Over time 1 | 40 | 100 | 20 | |||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 50 | 30 | ||||
| Subcontract 1 | 100 | |||||
| Demand | 500 | 750 | 900 | 450 |
The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.
Determine whether the regular time capacity remain unused:
From the above table, it is clear that no regular time capacity remains unused. All the regular time capacity has been used in respective quarters.
d)
To determine: The extent of backordering in units and dollars.
Introduction: Aggregate planning using transportation method helps to attain minimum cost using the optimal plan. The major advantage of transportation method is to achieve the optimal solution using optimal plans.
d)
Answer to Problem 17P
The total unit of the backordered is 40 units and the total cost of producing the backorders are $20.
Explanation of Solution
Given information:
The following information has been given:
| Quarter | Forecast (units) | Regular time | Overtime | Subcontract |
| 1 | 500 | 400 | 80 | 100 |
| 2 | 750 | 400 | 80 | 100 |
| 3 | 900 | 800 | 160 | 100 |
| 4 | 450 | 400 | 80 | 100 |
Initial inventory is given as 250 units, regular time cost is $1 per unit, overtime cost is $1.50 per unit, and subcontract cost is $2 per unit. Carrying cost is given as $0.5 per unit per quarter and backorder cost is $0.5 per unit per quarter. Initial inventory would incur $0.2 per unit.
Develop optimal plan:
| Optimal plan | Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | Ending inventory | Dummy |
| Beginning inventory | 100 | 150 | ||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 80 | |||||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 800 | |||||
| Over time 1 | 40 | 100 | 20 | |||
| Subcontract 1 | 100 | |||||
| Regular time 1 | 400 | |||||
| Over time 1 | 50 | 30 | ||||
| Subcontract 1 | 100 | |||||
| Demand | 500 | 750 | 900 | 450 |
The given demand and supply should be separated and the remaining supply and demand should be used as a dummy value.
Determine the extent backordering in units and dollars:
The colored cell is the only cell in the optimal plan, which is used for backordering. Hence, the backordering in units is 40 units. It is given that backorder cost is $0.50 per unit per quarter.
Hence, the total unit backordered is 40 units and the total costs of producing the backorders are $20.
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Chapter 13 Solutions
EBK PRINCIPLES OF OPERATIONS MANAGEMENT
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