Question

The accounts receivable department at Rick Wing Manufacturing has been having difficulty getting customers to pay the full amount of thier bills. Many customers complain that the bills are not correct and do not reflect the materials that arrived at thier receiving docks. The department has decided to implement SPC in its billing process. To set up control charts, 10 samples of 50 bills each were taken over a month's time and the items on the bills checked off against the bill of lading sent by company's shipping department to determine the number of bills that were not correct. The results were:

sample # # of incorrect bills

1 6
2 5
3 11
4 4
5 0
6 5
7 3
8 4
9 7
10 2

a) Determine the value of the p-bar, the mean fraction defective. Then determine the control limits for the p-chart using 99.73% confidence level (3 standard deviations). Is this process in control? If not, which samples were out of control?
b) How might you use quality tools to determine the source of the billing defects and where might you start your improvement efforts to eliminate the causes?

Answer 1

Answer:

a) $\bar p = \frac{\sum_{i=1}^{10} \hat p_i}{n}=\frac{0.12+0.10+0.22+0.08+0+0.1+0.06+0.08+0.14+0.04}{10}=0.094$

Now we can find the deviation with the following formula:

$\sigma =\sqrt{0.094(1-0.094)}{50}=0.0413$

And we can find the control limits like this:

$Lower= 0.094- 3(0.0413)=-0.0298$ since a negative value not makes sense we can assume as lower limit 0.

$Upper = 0.094+3(0.0413)=0.218$

b) If we use Total quality management (TQM) tools help organizations to analyze and assess qualitative and quantitative data that is relevant to the producer. Is important to mention that that tools should be used in conjunction with other tools to understand the full scope of the issue with the bills.

We can use the Pareto principle that suggests that most effects come from relatively few causes.  We can use for example Pareto charts in order to compare 'before and after' situations with the bills

Some sessions could result in a fish-bone chart can identify the possible causes of incorrect bills. Then a check sheet could be designed to collect data on the types of defects that occurred most frequently .

Explanation:

The population proportion have the following distribution  

$p \sim N(p,\sqrt{\frac{p(1-p)}{n}})$

Part a

For this case we can calculate the proportion of defectives for each sample, and we take in count that each sample have a size of 50.

$\hat p_1 = \frac{6}{50}=0.12$

$\hat p_2 = \frac{5}{50}=0.10$

$\hat p_3 = \frac{11}{50}=0.22$

$\hat p_4 = \frac{4}{50}=0.08$

$\hat p_5 = \frac{0}{50}=0$

$\hat p_6 = \frac{5}{50}=0.10$

$\hat p_7 = \frac{3}{50}=0.06$

$\hat p_8 = \frac{4}{50}=0.08$

$\hat p_9 = \frac{7}{50}=0.14$

$\hat p_{10} = \frac{2}{50}=0.04$

And in order to estimate $\bar p$  we can use the following formula:

$\bar p = \frac{\sum_{i=1}^{10} \hat p_i}{n}=\frac{0.12+0.10+0.22+0.08+0+0.1+0.06+0.08+0.14+0.04}{10}=0.094$

Now we can find the deviation with the following formula:

$\sigma =\sqrt{0.094(1-0.094)}{50}=0.0413$

And we can find the control limits like this:

$Lower= 0.094- 3(0.0413)=-0.0298$ since a negative value not makes sense we can assume as lower limit 0.

$Upper = 0.094+3(0.0413)=0.218$

So then the control limits are Lower =0 and Upper= 0.218.

And if the proportion of samples is higher than 0.218 we have the process out of control.

Part b

If we use Total quality management (TQM) tools help organizations to analyze and assess qualitative and quantitative data that is relevant to the producer. Is important to mention that that tools should be used in conjunction with other tools to understand the full scope of the issue with the bills.

We can use the Pareto principle that suggests that most effects come from relatively few causes.  We can use for example Pareto charts in order to compare 'before and after' situations with the bills

Some sessions could result in a fish-bone chart can identify the possible causes of incorrect bills. Then a check sheet could be designed to collect data on the types of defects that occurred most frequently .