# Statistical Process Control Chart for a Grocery Shopping Process

﻿Statistical Process Control Chart for a Grocery Shopping Process

A statistical process control (SPC) was carried out for the grocery store shopping process as identified in Week One. The importance of individual observations, in this case the daily amount of time spent at the grocery is underscored in this work. Consequently, it was necessary to use the Individual X-Moving Range (XMR) charts. Chary (2009) observed that it is often more convenient to employ this SPC tool when each subgroup is viewed as consisting of a single observation, or expressly stated, processes with a sample size/subgroup of one. This approach is a special form of the variable control charts. Variable control charts are based on a process characteristic that is continuous and can be measured. Time, the measurement metric used in this process, is a measurable variable. The following data, collected over two consecutive weeks was used in the SPS procedures and represents the total time in minutes spent on a daily basis at the grocery. As earlier identified, the main bottleneck in the process is the waiting time to check out at the grocery.

Figure SEQ Table * ARABIC 1 Data for Week One and Two

Day/Period Time in minutes MR1

Week one Mon 1 40 Tue 2 33 7

Wed 3 36 3

Thurs 4 30 6

Fri 5 25 5

Sat 6 47 22

Sun 7 42 5

Week two Mon 8 38 4

Tue 9 35 3

Wed 10 27 8

Thurs 11 43 16

Fri 12 32 11

Sat 13 33 1

Sun 14 42 9

Mean ( HYPERLINK “http://en.wikipedia.org/wiki/Xbar_and_s_chart” o “Xbar and s chart” INCLUDEPICTURE “http://upload.wikimedia.org/math/0/f/2/0f2dc2ebb7b41329b4f7b41635c64f8b.png” * MERGEFORMATINET ) = 35.9 Mean (MR) 7.69

The following steps were followed in completing the SPC according to the X-MR tool (Chary, 2011).

Calculation of the average moving range, MR

This involved summing the absolute values of the differences between consecutive values, then dividing the sum by n minus 1. MR was determined as 7.69 as observed in figure 1.

Calculation of the upper and lower control limits

The formulae below were used:

Upper control limit, UCL =   HYPERLINK “http://en.wikipedia.org/wiki/Xbar_and_s_chart” o “Xbar and s chart” INCLUDEPICTURE “http://upload.wikimedia.org/math/0/f/2/0f2dc2ebb7b41329b4f7b41635c64f8b.png” * MERGEFORMATINET + E * MR

= 35.9 + 0.881 * 7.69

= 42.7

Lower control limit, LCL=  HYPERLINK “http://en.wikipedia.org/wiki/Xbar_and_s_chart” o “Xbar and s chart” INCLUDEPICTURE “http://upload.wikimedia.org/math/0/f/2/0f2dc2ebb7b41329b4f7b41635c64f8b.png” * MERGEFORMATINET – E * MR

= 35.9 – 0.881 * 7.69

= 29.1

Where HYPERLINK “http://en.wikipedia.org/wiki/Xbar_and_s_chart” o “Xbar and s chart” INCLUDEPICTURE “http://upload.wikimedia.org/math/0/f/2/0f2dc2ebb7b41329b4f7b41635c64f8b.png” * MERGEFORMATINET represents the average of all daily time units, E is the correction factor and depends on the number of periods, 14 periods in this case and is equal to 0.881 (Chary, 2011).

Plotting the control chart

The graph was plotted for the daily amount of time spent at the grocery in minutes against the period, which is numbered successively for respective days of the fortnight over which the data was collected.

This is shown in figure 2 below:

Process Control Chart for Grocery Wait Times

The upper control limit was found to be 42.7 minutes while the lower control limit was is 29.1 minutes as indicated above. The variations that fall within the two control limits are due to random chance events whereas it is unlikely that those observations that fall outside the above control limits are caused by random chance events. On one hand, the variations exhibited by the points that are found within the upper and lower control limits are often common causes inherent in a process and can be eliminated only through improvements in the system (Chary, 2009). On the other hand, those that fall outside the upper and lower control limits are due to identifiable factors and can be modified through action of the operator or management. Even so, Porter (2011) noted that the fact there exist variations even within the upper and lower limits call for consideration. There may be some special factors that can be addressed as the non-random causes although this may fall within the control limits. In the grocery shopping process, four observations fall outside the control limits. These are represented by the shopping events with the durations of 25 and 27 minutes for the case of the lower control limit, and 43 and 47 minutes for the case of the upper limit.

The confidence level achieved in this SPC process is fairly favorable for a number of reasons. As a general observation, variable control charts are more sensitive than attribute control charts and therefore offer a generally higher percent of the data fall within the upper and the lower control limits (Porter, 2011). This is a favorable indicator for the confidence level and the validity of the control chart with respect to the process. Despite this, the fact that only 72% of the data lie within the upper and lower control limits is of concern as regards the confidence interval.

Some of the important seasonal factors in the process relate to both the cashiers and the shopper. A notable factor here is the length of queues. It was evident that on some occasions, the queues were unusually long. This is indicated by the observations of Friday in Week One data and Thursday in Week Two data. This occurrence was not common to all days but when it happened, the waiting time to check out was longer than usual. However, this factor may as well be considered as one of the most significant since it often ended up with frustrations on the part of some shoppers. Equally important is the fact that the attendants were sometimes slow than on an average occasion. This could be perhaps due to the reason that they may have been working for long hours and were therefore fatigued. As a shopper, the experience at the grocery is not always pleasant enough to encourage one to spend long spells of time in the process. It happened that when some of the products were out of stock or at least not up to expectations, less time was spent at the grocery than usual as indicated by the observation represented by 25 minutes.

References

Chary, N. S. (2009) Production & Operations Management. New York: McGraw Hill.

Porter, A. (2011) Operations Management. New Jersey: Albert Porter & Ventus Publishing APS.