Control chart (x̅ and R chart)

X-bar and R charts help determine if a process is stable and predictable.



The X-bar chart shows how the mean or average changes over time.

The R chart shows how the range of the subgroups changes over time. It is also used to monitor the effects of process improvement theories





Xbar Control Chart:

Upper Control Limit (UCL) = X double bar + A2 * R bar.
Lower Control Limit (LCL) = X double bar - A2 * R bar.

R Control Chart:

Upper Control Limit (UCL) = D4 * R bar.
Lower Control Limit (LCL) = D3 * R bar.


X double bar  = Average of average
R bar  = Average of  the mean

Question:

The management of steel bar manufacturer is concerned about the production of it's steel bars. The diameter is a critical factor. Data from 5 samples are shown in the table below:

1. Construct an x chart.
2. Construct an R chart.
3. Comment on the process average and the process variability.

(1) UCL = 10.27 + (0.729)(0.21) = 10.42

      LCL = 10.27 - (0.729)(0.21) = 10.12

The x-bar chart indicates that the production process is out of control. This is so because the last value exceeds UCL. Therefore, production must stopped and the cause of this abnormal variation must be identified and rectified.

(2) UCL = (2.282)(0.21)

       LCL = (0)(0.21)

The R-bar chart indicates that the process variability is in statistical control, This is so because range value lie between the UCL and the LCL. 

3 sigma control limits

Question:

Construct an appropriate control control chart with 3 sigma control and comment on the result.

Mean/average P = 0.8/20 = 0.04

σp = √{[0.04(1 - 0.04)]/100} = 0.0196

UCL = 0.04 + 3(0.0196) = 0.0988

LCL = 0.04 - 3(0.0196) = -0.0188

University of Mauritius past papers Question and Answer

May 2012 Q.3
(b) The operations manager of Desbro Ltd has been concerned about  machine No. 3 in the assembly line in the factory.  In order to make sure that the machine is operating correctly, samples are taken, and the average and range for each sample is computed.  Each sample consists of 10 items produced from that machine. Recently, 12 samples were taken and the sample range and average computed for each are as follows :

During installation, the supplier sets an average of 46 for the process with an average range 0.9.
(i) Draw the appropriate control charts for machine No. 3. [8 marks]
(ii) What is your conclusion? [2 marks]
Ans: 

•  = 46
•  = 0.9
• X-bar chart
  UCL = 46 + (0.308 x 0.9) = 46.2772
  LCL = 46 - (0.308 x 0.9) = 45.7228
• R-bar chart
  UCL = 1.777 x 0.9 = 1.5993
  LCL = 0.223 x 0.9 = 0.2007

• The x-bar chart indicates that the production process is out of control. Therefore, production must stopped and the cause of this abnormal variation must be identified and rectified.
• The R-bar chart indicates that the process variability is in statistical control, This is so because range value lie between the UCL and the LCL. 

(c) A garment manufacturer wants to set up a control chart for irregularities. Each week a random sample of 25 garments is collected from production and the number of irregularities is recorded. The table below gives the results for the past 20 weeks :

(i) Using the data set up an appropriate control chart using a three sigma control limits.  Comment on the process. [7 marks]  

(ii) Suppose that the next five samples had 15, 13, 12, 2.5 and 2 irregularities, what would you conclude? 

Ans:

Σ(irregularities) = 203

P = 203/100 = 2.03

Mean/average P = 2.03/20 = 0.1015

σp = √{[0.1015(1 - 0.1015)]/100} = 0.0302

UCL = 0.1015 + 3(0.0302) = 0.1921

LCL = 0.1015 - 3(0.0302) = 0.0109

Nov 2011 Q.1
(a)  Discuss the use of statistical process control in operations management.
Statistical Process Control (SPC) Analysis: Control Charts
What
One major challenging in maintaining service quality is identifying when quality has drifted away from that which is acceptable.  When a service is heterogeneous, each incident of service can present different standards of quality.  Further, these quality standards can be very subjective, measured by the opinions of the customers.

When service quality standards (a) are expected to be consistent over time, and (b) are quantitatively measured, then statistical techniques can be used to help identify service quality drift.  Examples of quality measurements and standards that may fit this criteria include:

• transaction error rates at banks
• on-time departure for airlines
• recurrence rates for repair services
• scaled service ratings on customer comment cards

If we look at the quality measurements over time we may find that they may be mostly consistent, with only occasional random variation.  In other situations, the measurements may change with a pattern that results from a fundamental flaw in the service delivery process.  Unfortunately, it can be quite difficult to tell the difference between a flaw in the service delivery process and a mere random variation.

How
One way to investigate this is with Statistical Process Control, or SPC.  SPC allows us to make assumptions about data to help separate simple random variation–also called natural variation–from variation that is caused by changes in the process–called assignable variation.

The foundation of Statistical Process Control is the central limit theorem, which states that the sum (or average) of a number of measurements from any single given probability distribution will approximate a normal distribution.  This holds true regardless of the distribution of the individual measurements.  How close the sums (or averages) approximates the normal distribution depends on how many measurements are in a sample, which number is called the sample size.

The two general types of quantitative quality measures are attributes and variables.  An attribute is dichotomous, meaning that it takes on the values of “acceptable” or “unacceptable.”  Unacceptable represents a quality defect.  An example is a bank transaction, which is either correct or is not correct.


(b) The management of a steel bar manufacturer is concerned about the production of its steel bars.  The diameter is a critical factor.  Data from 5 samples are shown in the table below.



(i) Draw an X Chart.
(ii)  Draw an R Chart.
(iii) Comment on the process average and                     process variability.