Chapter 11, Problem 1P

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

1. a. How many units of capacity should be built to minimize the total cost of providing capacity plus lost sales?
2. b. State a general rule regarding the amount of capacity to build.
3. c. What principle does this problem illustrate?

Summary Introduction

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.

$\begin{array}{c}\text{Build 0 units of capacity}=\left(7\times 0\right)+20\left[\begin{array}{l}\left(0\times 0.05\right)+\left(1\times 0.10\right)+\left(2\times 0.15\right)+\\ \left(3\times 0.20\right)+\left(4\times 0.20\right)+\left(5\times 0.15\right)+\\ \left(6\times 0.10\right)+\left(7\times 0.05\right)\end{array}\right]\\ =0+20\left(0+0.10+0.30+0.60+0.80+0.75+0.60+0.35\right)\\ =20\times 3.5\\ =70\end{array}$

Build 1 unit of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 1 unit of capacity}=\left(7\times 1\right)+20\left[\begin{array}{l}\left(0\times \left(0.05+0.10\right)\right)+\left(1\times 0.15\right)+\left(2\times 0.2\right)+\\ \left(3\times 0.20\right)+\left(4\times 0.15\right)+\left(5\times 0.10\right)+\\ \left(6\times 0.05\right)\end{array}\right]\\ =7+20\left(0+0.15+0.40+0.60+0.60+0.50+0.30\right)\\ =7+\left(20\times 2.55\right)\\ =58\end{array}$

Build 2 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 2 units of capacity}=\left(7\times 2\right)+20\left[\begin{array}{l}\left(0\times \left(0.05+0.10+0.15\right)\right)+\left(1\times 0.20\right)+\\ \left(2\times 0.2\right)+\left(3\times 0.15\right)+\left(4\times 0.10\right)+\left(5\times 0.05\right)\end{array}\right]\\ =14+20\left(0+0.20+0.40+0.45+0.40+0.25\right)\\ =14+\left(20\times 1.7\right)\\ =48\end{array}$

Build 3 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 3 units of capacity}=\left(7\times 3\right)+20\left[\begin{array}{l}\left(0\times \left(0.05+0.10+0.15+0.20\right)\right)+\left(1\times 0.20\right)+\\ \left(2\times 0.15\right)+\left(3\times 0.10\right)+\left(4\times 0.05\right)\end{array}\right]\\ =21+20\left(0+0.20+0.30+0.30+0.20\right)\\ =21+\left(20\times 1\right)\\ =41\end{array}$

Build 4 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 4 units of capacity}=\left(7\times 4\right)+20\left[\begin{array}{l}\left(0\times \left(0.05+0.10+0.15+0.20+0.20\right)\right)+\\ \left(1\times 0.15\right)+\left(2\times 0.10\right)+\left(3\times 0.05\right)\end{array}\right]\\ =28+20\left(0+0.15+0.20+0.15\right)\\ =28+\left(20\times 0.50\right)\\ =38\end{array}$

Build 5 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 5 units of capacity}=\left(7\times 5\right)+20\left[\begin{array}{l}\left(0\times \left(0.05+0.10+0.15+0.20+0.20+0.15\right)\right)+\\ \left(1\times 0.10\right)+\left(2\times 0.05\right)\end{array}\right]\\ =35+20\left(0+0.10+0.10\right)\\ =35+\left(20\times 0.20\right)\\ =39\end{array}$

Build 6 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 6 units of capacity}=\left(7\times 6\right)+20\left[\begin{array}{l}\left(0\times \left(\begin{array}{l}0.05+0.10+0.15+0.20+\\ 0.20+0.15+0.10\end{array}\right)\right)+\\ \left(1\times 0.05\right)\end{array}\right]\\ =42+20\left(0+0.05\right)\\ =42+\left(20\times 0.05\right)\\ =43\end{array}$

Build 7 units of capacity:

It is calculated by adding capacity cost and penalty cost.

$\begin{array}{c}\text{Build 7 units of capacity}=\left(7\times 7\right)+20\left[\left(0\times \left(\begin{array}{l}0.05+0.10+0.15+0.20+\\ 0.20+0.15+0.10+0.05\end{array}\right)\right)\right]\\ =49+20\left(0\right)\\ =49\end{array}$

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.

Summary Introduction

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.

Summary Introduction

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.