Chapter 5, Problem 10P

A company must decide which type of machine to buy, and how many units of that type, given the following information:

Product demand and processing times for the equipment are:

a. How many machines of each type would be required to handle demand if the machines will operate 8 hours a day, 250 days a year, and what annual capacity cushion in processing time would result for each?

b. With high certainty of annual demand, which type of machine would be chosen if that was an important consideration?

c. If purchasing and operating costs are taken into account, which type of machine would minimize total costs, given your answer for part a? Operating costs are $6/hr for type 1 and $5/hr for type 2.

Summary Introduction

To determine: The number of machines of each type required to handle the demand and its capacity.

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 10P

The number of machines of type 1 required is 2 machines and the number of machines of type 2 required is 2 machines.

Explanation of Solution

Given information:

A company should decide whether to buy machine of type 1 or type 2. The cost of type 1 machine is $10,000 and the cost of type 2 machine is $14,000.

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

Product	Annual demand	Processing time per minute
		Type 1	Type 2
001	12,000	4	6
002	10,000	9	9
003	18,000	5	3

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

It is given that the machines will operate 60 minutes per hour, 8 hours per day and 250 days per year.

Determine the minutes available for machine type 1:

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 1 is 120,000 minutes per year.

$\begin{array}{c}\text{Available minutes}=\text{Number of days per year}\times \text{Hours per day}\times \text{Minutes per hour}\\ =250\times 8\times 60\\ =120,000\text{ minutes per year}\end{array}$

Determine the processing requirement of product 001 using machine type 1:

It is calculated by multiplying annual demand of product 001 and the processing time per unit on type 1. Hence, the processing requirements of product 001 using machine type 1 is 48,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 001}=\text{Annual demand}\times \text{Processing time}\\ =12,000\times 4\text{ minutes}\\ =48,000\text{ minutes}\end{array}$

Determine the processing requirement of product 002 using machine type 1:

It is calculated by multiplying annual demand of product 002 and the processing time per unit on type 1. Hence, the processing requirements of product 002 using machine type 1 is 90,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 002}=\text{Annual demand}\times \text{Processing time}\\ =10,000\times 9\text{ minutes}\\ =90,000\text{ minutes}\end{array}$

Determine the processing requirement of product 003 using machine type 1:

It is calculated by multiplying annual demand of product 003 and the processing time per unit on type 1. Hence, the processing requirements of product 003 using machine type 1 is 90,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 003}=\text{Annual demand}\times \text{Processing time}\\ =18,000\times 5\text{ minutes}\\ =90,000\text{ minutes}\end{array}$

Determine the total processing requirement using machine type 1:

It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type 1 is 228,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 001}+\\ \text{Processing requirement of product 002}+\\ \text{Processing requirement of product 003}\end{array}\right)\\ =48,000+90,000+90,000\\ =228,000\text{ minutes}\end{array}$

Determine the needed number of machine type 1:

It is calculated by dividing the total processing requirement and available minutes for machine type 1. Hence, the needed number of machine type 1is 2 machines.

$\begin{array}{c}\text{Number of machine type 1 needed}=\frac{\text{Processing time needed}}{\text{Available minutes for machine type 1}}\\ =\frac{228,000}{120,000}\\ =1.9\\ \simeq 2\text{ machines}\end{array}$

Determine the capacity of the machines:

It is calculated by multiplying the number of machine type 1 needed and the available minutes for machine type 1. Hence, the capacity of the machine is 240,000 minutes.

$\begin{array}{c}\text{Capacity of machines}=\text{Number of machines}\times \text{Available minutes for machine type 1}\\ =2\times 120,000\\ =240,000\text{ minutes}\end{array}$

Determine the capacity cushion:

It is calculated by subtracting total processing requirement from capacity of machines.

$\begin{array}{c}\text{Capacity cushion}=\text{Capacity}-\text{Processing requirement}\\ =240,000-228,000\\ =12,000\text{ minutes}\end{array}$

Determine the minutes available for machine type 2:

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 2 is 120,000 minutes per year.

$\begin{array}{c}\text{Available minutes}=\text{Number of days per year}\times \text{Hours per day}\times \text{Minutes per hour}\\ =250\times 8\times 60\\ =120,000\text{ minutes per year}\end{array}$

Determine the processing requirement of product 001 using machine type 2:

It is calculated by multiplying annual demand of product 001 and the processing time per unit on type 2. Hence, the processing requirements of product 001 using machine type 2 is 72,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 001}=\text{Annual demand}\times \text{Processing time}\\ =12,000\times 6\text{ minutes}\\ =72,000\text{ minutes}\end{array}$

Determine the processing requirement of product 002 using machine type 2:

It is calculated by multiplying annual demand of product 002 and the processing time per unit on type 2. Hence, the processing requirements of product 002 using machine type 2 is 90,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 002}=\text{Annual demand}\times \text{Processing time}\\ =10,000\times 9\text{ minutes}\\ =90,000\text{ minutes}\end{array}$

Determine the processing requirement of product 003 using machine type 2:

It is calculated by multiplying annual demand of product 003 and the processing time per unit on type 2. Hence, the processing requirements of product 003 using machine type 2 is 54,000 minutes.

$\begin{array}{c}\text{Processing requirement of product 003}=\text{Annual demand}\times \text{Processing time}\\ =18,000\times 3\text{ minutes}\\ =54,000\text{ minutes}\end{array}$

Determine the total processing requirement using machine type 2:

It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type 2 is 216,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 001}+\\ \text{Processing requirement of product 002}+\\ \text{Processing requirement of product 003}\end{array}\right)\\ =72,000+90,000+54,000\\ =216,000\text{ minutes}\end{array}$

Determine the needed number of machine type 2:

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

$\begin{array}{c}\text{Number of machine type 2 needed}=\frac{\text{Processing time needed}}{\text{Available minutes for machine type 1}}\\ =\frac{216,000}{120,000}\\ =1.8\\ \simeq 2\text{ machines}\end{array}$

Determine the capacity of the machines:

It is calculated by multiplying the number of machine type 2 needed and the available minutes for machine type 2. Hence, the capacity of the machine is 240,000 minutes.

$\begin{array}{c}\text{Capacity of machines}=\text{Number of machines}\times \text{Available minutes for machine type 2}\\ =2\times 120,000\\ =240,000\text{ minutes}\end{array}$

Determine the capacity cushion:

It is calculated by subtracting total processing requirement from capacity of machines.

$\begin{array}{c}\text{Capacity cushion}=\text{Capacity}-\text{Processing requirement}\\ =240,000-216,000\\ =24,000\text{ minutes}\end{array}$

Summary Introduction

To determine: The machine type that can be chosen for the high and low certainty of demand.

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 10P

Machine type 2 can be chosen when there is high certainty of annual demand. Machine type 1 can be chosen when there is low certainty of annual demand.

Explanation of Solution

Given information:

A company should decide whether to buy machine of type 1 or type 2. The cost of type 1 machine is $10,000 and the cost of type 2 machine is $14,000.

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

Product	Annual demand	Processing time per minute
		Type 1	Type 2
001	12,000	4	6
002	10,000	9	9
003	18,000	5	3

Determine the machine type that can be chosen for the high and low certainty of demand:

The machine with the high capacity cushion should be selected, if the firm faced high certainty of annual demand. hence, machine type 2 should be selected.

The machine with the low capacity cushion should be selected, if the firm faced low certainty of annual demand. hence, machine type 1 should be selected.

Summary Introduction

To determine: The machine type that would minimize the total cost, if purchasing and operating costs are considered.

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 10P

Machine type 1 would minimize the total cost.

Explanation of Solution

Given information:

A company should decide whether to buy machine of type 1 or type 2. The cost of type 1 machine is $10,000 and the cost of type 2 machine is $14,000.

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

Product	Annual demand	Processing time per minute
		Type 1	Type 2
001	12,000	4	6
002	10,000	9	9
003	18,000	5	3

Operating cost is given as $6 per hour for type 1 and $5 per hour for type 2.

Determine the purchase cost for machine type 1:

It is calculated by multiplying the number of machines and cost of type 1 machine. Hence, the purchase cost for machine type 1 is $20,000.

$\begin{array}{c}\text{Purchase cost}=\text{Number of machines}\times \text{Machine cost}\\ =2\text{ machines}\times \$10,000\\ =\$20,000\end{array}$

Determine the total operating time for machine type 1:

Total operating time is the total processing requirement. It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type 1 is 228,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 001}+\\ \text{Processing requirement of product 002}+\\ \text{Processing requirement of product 003}\end{array}\right)\\ =48,000+90,000+90,000\\ =228,000\text{ minutes}\\ \simeq \text{3,800 hours}\end{array}$

Determine the total operating cost:

Total operating cost is calculated by multiplying the total operating time and the operating cost per hour. Hence, the total operating cost is $22,800.

$\begin{array}{c}\text{Total operating cost}=\text{Total operating time}\times \text{Operating cost per hour}\\ =3,800\text{ hours}\times \text{\$6 per hour}\\ =\$22,800\end{array}$

Determine the total cost:

It is calculated by adding the purchasing cost of machine type 1 and total operating cost. Hence, the total cost is $42,800.

$\begin{array}{c}\text{Total cost}=\text{Total purchase cost}+\text{Total operating cost}\\ =\$20,000+\$22,800\\ =\$42,800\end{array}$

Determine the purchase cost for machine type 2:

It is calculated by multiplying the number of machines and cost of type 2 machine. Hence, the purchase cost for machine type 2 is $28,000.

$\begin{array}{c}\text{Purchase cost}=\text{Number of machines}\times \text{Machine cost}\\ =2\text{ machines}\times \$14,000\\ =\$28,000\end{array}$

Determine the total processing requirement using machine type 2:

Total operating time is the total processing requirement. It is calculated by adding the processing requirement of all the products. Hence, the total processing requirement using machine type 2 is 216,000 minutes.

$\begin{array}{c}\text{Total processing requirements}=\left(\begin{array}{l}\text{Processing requirement of product 001}+\\ \text{Processing requirement of product 002}+\\ \text{Processing requirement of product 003}\end{array}\right)\\ =72,000+90,000+54,000\\ =216,000\text{ minutes}\\ \simeq \text{3,600 hours}\end{array}$

Determine the total operating cost:

Total operating cost is calculated by multiplying the total operating time and the operating cost per hour. Hence, the total operating cost is $18,000.

$\begin{array}{c}\text{Total operating cost}=\text{Total operating time}\times \text{Operating cost per hour}\\ =3,600\text{ hours}\times \text{\$5 per hour}\\ =\$18,000\end{array}$

Determine the total cost:

It is calculated by adding the purchasing cost of machine type 2 and total operating cost. Hence, the total cost is $46,000.

$\begin{array}{c}\text{Total cost}=\text{Total purchase cost}+\text{Total operating cost}\\ =\$28,000+\$18,000\\ =\$46,000\end{array}$

Hence, Machine type 1 would minimize the total cost.