Chapter 5, Problem 11P

a)

Summary Introduction

To determine: The number of machines of each type required to handle the demand and its capacity and the machine that would have lower total cost.

Introduction: Capacity planning is the process of planning the required production output based on the requirement or the demand that is predicted.

Answer to Problem 11P

Two machines are need in type A and type B and one machine is needed in type C. Buying two B machines would have lower total cost.

Explanation of Solution

Given information:

A company should decide whether to buy machine of type A, type B, or type C. The cost of type A machine is $40,000, cost of type B machine is $30,000, and the cost of type C machine is $80,000.

Annual demand and processing unit is given as follows for each product:

Product	Annual demand	Processing time per unit
		A	B	C
1	16,000	3	4	2
2	12,000	4	4	3
3	6,000	5	6	4
4	30,000	2	2	1

Determine the number of machines of each type required and its capacity:

It is given that the machines will operate 60 minutes per hour, 10 hours per day and 250 days per year.

Determine the minutes available for all types of machine:

It is calculated by multiplying number of days per year, hours per day, and minutes per hour. Hence, the available minutes per year for machine type A, type B, and type C is 150,000 minutes per year.

$\begin{array}{c}\text{Available minutes}=\text{Number of days per year}\times \text{Hours per day}\times \text{Minutes per hour}\\ =250\times 10\times 60\\ =150,000\text{ minutes per year}\end{array}$

Determine the processing requirement of product 1 using machine type A:

It is calculated by multiplying annual demand of product 1 and the processing time per unit on type A. Hence, the processing requirements of product 1 using machine type A is 48,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 1}=\text{Annual demand}\times \text{Processing time}\\ =16,000\times 3\text{ minutes}\\ =48,000\text{ minutes}\end{array}$

Determine the processing requirement of product 2 using machine type A:

It is calculated by multiplying annual demand of product 2 and the processing time per unit on type A. Hence, the processing requirements of product 2 using machine type A is 48,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 2}=\text{Annual demand}\times \text{Processing time}\\ =12,000\times 4\text{ minutes}\\ =48,000\text{ minutes}\end{array}$

Determine the processing requirement of product 3 using machine type A:

It is calculated by multiplying annual demand of product 3 and the processing time per unit on type A. Hence, the processing requirements of product 3 using machine type A is 30,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 3}=\text{Annual demand}\times \text{Processing time}\\ =6,000\times 5\text{ minutes}\\ =30,000\text{ minutes}\end{array}$

Determine the processing requirement of product 4 using machine type A:

It is calculated by multiplying annual demand of product 4 and the processing time per unit on type A. Hence, the processing requirements of product 4 using machine type A is 60,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 4}=\text{Annual demand}\times \text{Processing time}\\ =30,000\times 2\text{ minutes}\\ =60,000\text{ minutes}\end{array}$

Determine the total processing requirement using machine type A:

It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type A is 186,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 1}+\\ \text{Processing requirement of product 2}+\\ \text{Processing requirement of product 3}+\\ \text{Processing requirement of product 4}\end{array}\right)\\ =48,000+48,000+30,000+60,000\\ =186,000\text{ minutes}\\ \simeq \text{3,100 hours}\end{array}$

Determine the needed number of machine type A:

It is calculated by dividing the total processing requirement and available minutes for machine type A. Hence, the needed number of machine type A is 2 machines.

$\begin{array}{c}\text{Number of machine type A needed}=\frac{\text{Processing time needed}}{\text{Available minutes for machine type A}}\\ =\frac{186,000}{150,000}\\ =1.24\\ \simeq 2\text{ machines}\end{array}$

Determine the processing requirement of product 1 using machine type B:

It is calculated by multiplying annual demand of product 1 and the processing time per unit on type B. Hence, the processing requirements of product 1 using machine type B is 64,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 1}=\text{Annual demand}\times \text{Processing time}\\ =16,000\times 4\text{ minutes}\\ =64,000\text{ minutes}\end{array}$

Determine the processing requirement of product 2 using machine type B:

It is calculated by multiplying annual demand of product 2 and the processing time per unit on type B. Hence, the processing requirements of product 2 using machine type B is 48,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 2}=\text{Annual demand}\times \text{Processing time}\\ =12,000\times 4\text{ minutes}\\ =48,000\text{ minutes}\end{array}$

Determine the processing requirement of product 3 using machine type B:

It is calculated by multiplying annual demand of product 3 and the processing time per unit on type B. Hence, the processing requirements of product 3 using machine type B is 36,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 3}=\text{Annual demand}\times \text{Processing time}\\ =6,000\times 6\text{ minutes}\\ =36,000\text{ minutes}\end{array}$

Determine the processing requirement of product 4 using machine type B:

It is calculated by multiplying annual demand of product 4 and the processing time per unit on type B. Hence, the processing requirements of product 4 using machine type B is 60,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 4}=\text{Annual demand}\times \text{Processing time}\\ =30,000\times 2\text{ minutes}\\ =60,000\text{ minutes}\end{array}$

Determine the total processing requirement using machine type B:

It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type B is 208,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 1}+\\ \text{Processing requirement of product 2}+\\ \text{Processing requirement of product 3}+\\ \text{Processing requirement of product 4}\end{array}\right)\\ =64,000+48,000+36,000+60,000\\ =208,000\text{ minutes}\\ \simeq \text{3,466}\text{.67 hours}\end{array}$

Determine the needed number of machine type B:

It is calculated by dividing the total processing requirement and available minutes for machine type B. Hence, the needed number of machine type B is 2 machines.

$\begin{array}{c}\text{Number of machine type B needed}=\frac{\text{Processing time needed}}{\text{Available minutes for machine type B}}\\ =\frac{208,000}{150,000}\\ =1.38\\ \simeq 2\text{ machines}\end{array}$

Determine the processing requirement of product 1 using machine type C:

It is calculated by multiplying annual demand of product 1 and the processing time per unit on type B. Hence, the processing requirements of product 1 using machine type B is 32,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 1}=\text{Annual demand}\times \text{Processing time}\\ =16,000\times 2\text{ minutes}\\ =32,000\text{ minutes}\end{array}$

Determine the processing requirement of product 2 using machine type C:

It is calculated by multiplying annual demand of product 2 and the processing time per unit on type B. Hence, the processing requirements of product 2 using machine type B is 36,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 2}=\text{Annual demand}\times \text{Processing time}\\ =12,000\times 3\text{ minutes}\\ =36,000\text{ minutes}\end{array}$

Determine the processing requirement of product 3 using machine type C:

It is calculated by multiplying annual demand of product 3 and the processing time per unit on type B. Hence, the processing requirements of product 3 using machine type B is 24,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 3}=\text{Annual demand}\times \text{Processing time}\\ =6,000\times 4\text{ minutes}\\ =24,000\text{ minutes}\end{array}$

Determine the processing requirement of product 4 using machine type C:

It is calculated by multiplying annual demand of product 4 and the processing time per unit on type B. Hence, the processing requirements of product 4 using machine type B is 30,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 4}=\text{Annual demand}\times \text{Processing time}\\ =30,000\times 1\text{ minutes}\\ =30,000\text{ minutes}\end{array}$

Determine the total processing requirement using machine type C:

It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type C is 122,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 1}+\\ \text{Processing requirement of product 2}+\\ \text{Processing requirement of product 3}+\\ \text{Processing requirement of product 4}\end{array}\right)\\ =32,000+36,000+24,000+30,000\\ =122,000\text{ minutes}\\ \simeq \text{2,033}\text{.33 hours}\end{array}$

Determine the needed number of machine type C:

It is calculated by dividing the total processing requirement and available minutes for machine type C. Hence, the needed number of machine type C is 1 machine.

$\begin{array}{c}\text{Number of machine type C needed}=\frac{\text{Processing time needed}}{\text{Available minutes for machine type C}}\\ =\frac{122,000}{150,000}\\ =0.81\\ \simeq 1\text{ machines}\end{array}$

Determine the total cost considering purchase cost of machine type A:

It is calculated by multiplying number of machines and the cost of the machine. Hence, the total cost is $80,000.

$\begin{array}{c}\text{Total cost}=\text{Number of machines}\times \text{Cost of the machine}\\ =2\times 40,000\\ =80,000\end{array}$

Determine the total cost considering purchase cost of machine type B:

It is calculated by multiplying number of machines and the cost of the machine. Hence, the total cost is $60,000.

$\begin{array}{c}\text{Total cost}=\text{Number of machines}\times \text{Cost of the machine}\\ =2\times 30,000\\ =60,000\end{array}$

Determine the total cost considering purchase cost of machine type C:

It is calculated by multiplying number of machines and the cost of the machine. Hence, the total cost is $80,000.

$\begin{array}{c}\text{Total cost}=\text{Number of machines}\times \text{Cost of the machine}\\ =1\times 80,000\\ =80,000\end{array}$

Hence, Buying 2 of the B machines at a total cost of $60,000 have lower total cost.

b)

Summary Introduction

To determine: The machine that would have lower total cost considering operating cost.

Introduction: Capacity planning is the process of planning the required production output based on the requirement or the demand that is predicted.

Answer to Problem 11P

Buying two B machines would have lower total cost.

Explanation of Solution

Given information:

A company should decide whether to buy machine of type A, type B, or type C. The cost of type A machine is $40,000, cost of type B machine is $30,000, and the cost of type C machine is $80,000.

Annual demand and processing unit is given as follows for each product:

Product	Annual demand	Processing time per unit
		A	B	C
1	16,000	3	4	2
2	12,000	4	4	3
3	6,000	5	6	4
4	30,000	2	2	1

Operating cost for machine A is $10 per hour, operating cost for machine B is $11 per hour, and operating cost for machine C is $12 per hour.

Calculate the total cost considering both purchasing cost and operating cost for machine A:

It is calculated by adding the total cost considering purchasing cost and the operating cost which can be attained by multiplying operating cost per hour and the total processing hours.

$\begin{array}{c}\text{Total cost}=\text{Total cost considering purchasing cost}+\left(\begin{array}{l}\text{Total processing hours}\times \\ \text{Operating cost per hour}\end{array}\right)\\ =\$80,000+\left(3,100\text{ hours}\times \text{\$10}\right)\\ =\$80,000+\$31,000\\ =\$111,000\end{array}$

Calculate the total cost considering both purchasing cost and operating cost for machine B:

It is calculated by adding the total cost considering purchasing cost and the operating cost which can be attained by multiplying operating cost per hour and the total processing hours.

$\begin{array}{c}\text{Total cost}=\text{Total cost considering purchasing cost}+\left(\begin{array}{l}\text{Total processing hours}\times \\ \text{Operating cost per hour}\end{array}\right)\\ =\$60,000+\left(3,466.67\text{ hours}\times \text{\$11}\right)\\ =\$60,000+\$38,133\\ =\$98,133\end{array}$

Calculate the total cost considering both purchasing cost and operating cost for machine C:

It is calculated by adding the total cost considering purchasing cost and the operating cost which can be attained by multiplying operating cost per hour and the total processing hours.

$\begin{array}{c}\text{Total cost}=\text{Total cost considering purchasing cost}+\left(\begin{array}{l}\text{Total processing hours}\times \\ \text{Operating cost per hour}\end{array}\right)\\ =\$80,000+\left(2,033.22\text{ hours}\times \text{\$12}\right)\\ =\$80,000+\$24,400\\ =\$104,400\end{array}$

Hence, buying two B machines would have lower total cost.