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Suppose we are considering the question of how much capacity to build in the face of uncertain demand. Assume that the cost is $20 per unit of lost sales due to insufficient capacity, Also assume that there is a cost of $7 for each unit of capacity built. The probability of various demand levels is as follows:
| Deman d- X Units | Probability of X |
| 0 | .05 |
| 1 | .10 |
| 2 | .15 |
| 3 | .20 |
| 4 | .20 |
| 5 | .15 |
| 6 | .10 |
| 7 | .05 |
- a. How many units of capacity should be built to minimize the total cost of providing capacity plus lost sales?
- b. State a general rule regarding the amount of capacity to build.
- c. What principle does this problem illustrate?
a)
To determine: The number of capacity units required to minimize the total cost of providing capacity plus lost sales.
Introduction:
Capacity planning is the process of planning the required production output based on the requirement or the demand that is predicted.
Explanation of Solution
Given information:
It is given that the cost of the lost sales is $20 per unit, due to insufficient capacity. Assume that the cost is $7 for each unit of capacity built. In addition to this, the following information is given:
| Demand - X units | Probability of X |
| 0 | 0.05 |
| 1 | 0.10 |
| 2 | 0.15 |
| 3 | 0.20 |
| 4 | 0.20 |
| 5 | 0.15 |
| 6 | 0.10 |
| 7 | 0.05 |
Build 0 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 1 unit of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 2 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 3 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 4 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 5 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 6 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Build 7 units of capacity:
It is calculated by adding capacity cost and penalty cost.
Hence, the number of capacity units required (that has the minimum total cost of $38) to minimize the total cost of providing capacity plus lost sales is 4 units.
b)
To determine: The general rule regarding the amount of capacity to build.
Introduction:
Capacity planning is the process of planning the required production output based on the requirement or the demand that is predicted.
Explanation of Solution
Given information:
It is given that the cost of the lost sales is $20 per unit due to insufficient capacity. Assume that the cost is $7 for each unit of capacity built. In addition to this, the following information is given:
| Demand - X units | Probability of X |
| 0 | 0.05 |
| 1 | 0.10 |
| 2 | 0.15 |
| 3 | 0.20 |
| 4 | 0.20 |
| 5 | 0.15 |
| 6 | 0.10 |
| 7 | 0.05 |
General instruction regarding the amount of capacity to build:
The general instruction regarding the amount of capacity to build is to minimize the total cost of capacity plus lost sales.
c)
To determine: The principle that is illustrated in the problem.
Introduction:
Capacity planning is the process of planning the required production output based on the requirement or the demand that is predicted.
Explanation of Solution
Given information:
It is given that the cost of the lost sales is $20 per unit due to insufficient capacity. Assume that the cost is $7 for each unit of capacity built. In addition to this, the following information is given:
| Demand - X units | Probability of X |
| 0 | 0.05 |
| 1 | 0.10 |
| 2 | 0.15 |
| 3 | 0.20 |
| 4 | 0.20 |
| 5 | 0.15 |
| 6 | 0.10 |
| 7 | 0.05 |
Determine the principle that is illustrated in the problem:
In the given problem, the trade-off between excess capacity and lost sales is illustrated.
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Chapter 11 Solutions
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