Chapter 10, Problem 29P

a)

Summary Introduction

To determine: The standard time for the operation

Introduction: Time study is the basis that helps set the standard time; it would time a sample of the performance of the worker.

Answer to Problem 29P

Thestandard time is calculated as 85.56 seconds.

Explanation of Solution

Given information:

The allowance factor is given as 23%. The observed time and performance rating for each element is given as follows:

Element	Performance rating	Observation (seconds)
		1	2	3	4	5
Grasp and place bag	110%	8	9	8	11	7
Fill bag	85%	36	41	39	35	112a
Seal bag	105%	15	17	13	20	18
Place bag on conveyor	90%	8	6	9	30b	35b

^a refers to the bag breaking open

^b refers to the conveyor getting jammed

Formulae to calculate standard time:

$\text{Average time}=\frac{\text{Observations}}{n}$

$\text{Normal time}=\text{Average time}\times \text{Performance rating}$

$\text{Standard time}=\frac{\text{Normal time}}{1-\text{Allowance factor}}$

Calculate normal time:

Element	Performance rating	Observation (seconds)					Average time	Normal time
		1	2	3	4	5
Grasp and place bag	110%	8	9	8	11	7	8.6	9.46
Fill bag	85%	36	41	39	35	112a	37.75	32.0875
Seal bag	105%	15	17	13	20	18	16.6	17.43
Place bag on conveyor	90%	8	6	9	30b	35b	7.67	6.9

Calculate the average time to grasp and place bag:

The average time is calculated by taking an average of the observation given, which is the summation of 8, 9, 8, 11, and 7 and dividing the result by 5. Hence, the average is 8.6 seconds.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{8+9+8+11+7}{5}\\ =\frac{43}{5}\\ =8.6\text{ seconds}\end{array}$

Calculate the average time to fill bag:

The average time is calculated by taking an average of the observation given, which is the summation of 36, 41, 39, and 35 and dividing the result by 4. Hence, the average is 37.75 seconds.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{36+41+39+35}{4}\\ =\frac{151}{4}\\ =37.75\text{ seconds}\end{array}$

Note: The observation with (X^a) should not be considered while calculating the average time.

Calculate the average time to seal bag:

The average time is calculated by taking an average of the observation given, which is the summation of 15, 17, 13, 20, and 18 and dividing the result by 5. Hence, the average is 16.6 seconds.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{15+17+13+20+18}{5}\\ =\frac{83}{5}\\ =16.6\text{ seconds}\end{array}$

Calculate the average time to place bag on conveyor:

The average time is calculated by taking an average of the observation given, which is the summation of 8, 6, and 9 and dividing the result by 3. Hence, the average is 7.67 seconds.

$\begin{array}{c}\text{Average time}=\frac{\text{Observation}}{n}\\ =\frac{8+6+9}{3}\\ =\frac{23}{3}\\ =7.67\text{ seconds}\end{array}$

Note: The observations with (X^a and X^b) should not be considered while calculating the average time.

Calculate the normal time to grasp and place bag:

The normal time is calculated by multiplying average time and performance rating. Average time is calculated as 8.60 seconds, and performance rating is given as 110%. Hence, the normal time is 9.46 seconds.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =8.60\times 110\%\\ =8.60\times 1.1\\ =9.46\text{ seconds}\end{array}$

Calculate the normal time fill bag:

The normal time is calculated by multiplying average time and performance rating. Average time is calculated as 37.75 seconds, and performance rating is given as 85%. Hence, the normal time is 32.09 seconds.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =37.75\times 85\%\\ =37.75\times 0.85\\ =32.09\text{ seconds}\end{array}$

Calculate the normal time to seal bag:

The normal time is calculated by multiplying average time and performance rating. Average time is calculated as 16.60 seconds, and performance rating is given as 105%. Hence, the normal time is 17.43 seconds.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =16.60\times 105\%\\ =16.60\times 1.05\\ =17.43\text{ seconds}\end{array}$

Calculate the normal time to place bag on conveyor:

The normal time is calculated by multiplying average time and performance rating. Average time is calculated as 7.67 seconds, and performance rating is given as 90%. Hence, the normal time is 6.90 seconds.

$\begin{array}{c}\text{Normal time}=\text{Average time}\times \text{Performance rating}\\ =7.67\times 90\%\\ =7.67\times 0.90\\ =6.90\text{ seconds}\end{array}$

Calculate the total normal time:

The normal time is calculated by adding the normal time to grasp and place bag, fill bag, seal bag, and place bag on conveyor.

$\begin{array}{c}\text{Normal time}=\text{Sum of normal times}\\ =9.46+32.09+17.43+6.90\\ =65.88\text{ seconds}\end{array}$

Hence, the total normal time is 65.88 seconds.

Calculate the standard time:

The standard time is calculated by dividing normal time with the value attained by subtracting allowance factor from 1. Normal time is calculated as 65.88 seconds and allowance factor is calculated as 23%

$\begin{array}{c}\text{Standard time}=\frac{\text{Normal time}}{1-\text{Allowance factor}}\\ =\frac{\text{65}\text{.88 seconds}}{1-0.23}\\ =\frac{\text{65}\text{.88 seconds}}{0.77}\\ =85.56\text{ seconds}\end{array}$

Hence, standard time is 85.56 seconds.

b)

Summary Introduction

To determine: The number of sample sizes required.

Introduction: Thesample size is required to determine the number of observations that are necessary to find the true cycle time.

Answer to Problem 29P

Thesample size to grasp and place bag is 83, to fill bag is 14, to seal bag is 70, and to place bag on conveyor is 107 samples.

Explanation of Solution

Given information:

Confidence level is 99% and accuracy is ±5%.

Element	Performance rating	Observation (seconds)
		1	2	3	4	5
Grasp and place bag	110%	8	9	8	11	7
Fill bag	85%	36	41	39	35	112a
Seal bag	105%	15	17	13	20	18
Place bag on conveyor	90%	8	6	9	30b	35b

Formula to determine the required number of observations:

$n={\left(\frac{zs}{h\overline{x}}\right)}^2$

n refers to the number of observations required

z refers to the table value for confidence level

s refers to the standard deviation

h refers to accuracy level

x(bar) refers to the average observed time

Determine the proper number of observations required:

Element	Mean observed time	s	Sample
Grasp and place bag	8.6	1.52	8
Fill bag	37.75	2.75	14
Seal bag	16.6	2.7	70
Place bag on conveyor	7.67	1.54	107

Calculate the standard deviation to grasp and place bag:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the result should be square rooted. The standard deviation to grasp and place bag is 1.52.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(8-8.6\right)}^2+{\left(9-8.6\right)}^2+{\left(8-8.6\right)}^2+{\left(11-8.6\right)}^2+{\left(7-8.6\right)}^2}{5-1}}\\ =\sqrt{\frac{9.2}{4}}\\ =1.52\end{array}$

Calculate the standard deviation to fill bag:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the resultant should be square rooted. The standard deviation to fill bag is 2.75.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(36-37.75\right)}^2+{\left(41-37.75\right)}^2+{\left(39-37.75\right)}^2+{\left(35-37.75\right)}^2}{4-1}}\\ =\sqrt{\frac{22.75}{3}}\\ =2.75\end{array}$

Calculate the standard deviation to seal bag:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the resultant should be square rooted. The standard deviation to seal bag is 1.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(15-17.43\right)}^2+{\left(17-17.43\right)}^2+{\left(13-17.43\right)}^2+{\left(20-17.43\right)}^2+{\left(18-17.43\right)}^2}{5-1}}\\ =\sqrt{\frac{32.6445}{4}}\\ =2.70\end{array}$

Calculate the standard deviation to place bag on conveyor:

The standard deviation can be calculated by dividing two values. The first value can be calculated by adding the square of the value attained by subtracting the mean observed value from each observation. The second value can be attained by subtracting 1 from the number of samples and the resultant should be square rooted. The standard deviation to place bag on conveyor is 1.54.

$\begin{array}{c}s=\sqrt{\frac{{{\displaystyle \sum \left(\text{Sample observation}-\overline{x}\right)}}^2}{n-1}}\\ =\sqrt{\frac{{\left(8-7.67\right)}^2+{\left(6-7.67\right)}^2+{\left(9-7.67\right)}^2}{3-1}}\\ =\sqrt{\frac{4.6667}{2}}\\ =1.54\end{array}$

Calculate the sample size to grasp and place bag:

It is calculated by multiplying the z value of 2.58, the standard deviation that is given as 1.52, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 8.6, respectively. Hence, the required number of observations is 83 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{2.58\times 1.52}{0.05\times 8.6}\right)}^2\\ ={\left(\frac{3.9216}{0.43}\right)}^2\\ \simeq 83\text{ samples}\end{array}$

Calculate the sample size to fill bag:

It is calculated by multiplying the z value of 2.58, the standard deviation that is given as 2.75, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 37.75, respectively. Hence, the required number of observations is 14 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{2.58\times 2.75}{0.05\times 37.75}\right)}^2\\ ={\left(\frac{7.095}{1.8875}\right)}^2\\ \simeq 14\text{ samples}\end{array}$

Calculate the sample size to seal bag:

It is calculated by multiplying the z value of 2.58, the standard deviation that is given as 2.7, and dividing the result with the multiple of the accuracy level and average observed time that is given as 0.05 and 16.6, respectively. Hence, the required number of observations is 70 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{2.58\times 2.7}{0.05\times 16.6}\right)}^2\\ ={\left(\frac{6.966}{0.83}\right)}^2\\ \simeq 70\text{ samples}\end{array}$

Calculate the sample size to place bag on conveyor:

It is calculated by multiplying the z value of 2.58, the standard deviation that is given as 1.54, and dividing the resultant with the multiple of the accuracy level and average observed time that is given as 0.05 and 7.67, respectively. Hence, the required number of observations is 107 samples.

$\begin{array}{c}n={\left(\frac{zs}{h\overline{x}}\right)}^2\\ ={\left(\frac{2.58\times 1.54}{0.05\times 7.67}\right)}^2\\ ={\left(\frac{3.9732}{0.3835}\right)}^2\\ \simeq 107\text{ samples}\end{array}$

Hence, the standard time for the process is 85.56 seconds. Sample size to grasp and place bag is 83, to fill bag is 14, seal bag is 70, and to place bag on conveyor is 107 samples.