# An economic Shewhart control chart adjustment strategy for the twenty-first century

## PMBOK® quality management education

**David X. Zou, Ph.D., PMP, Sr. Operations Research Analyst, Praxair, Inc.**

Shewhart control charts are widely used to display sample data form a production process. They have also been found valuable in evaluating process capability, in estimating process parameters, and in monitoring the behavior of a production process. A control chart is maintained by taking samples from a process and plotting in time order on the chart some statistic computed form the samples. Control limits on the chart represent the limits within which the plotted points would fall with high probability if the operating in control. A point outside the control limits is taken as an indication that something, sometimes called a special cause of variation, has happened to change the process. When the chart signals that a special cause is present, rectifying action is taken to remove the special cause and bring the process back into control. In addition to the common causes, which produce random variation, special causes can individually produce a substantial amount of variation. When a special cause of variation is present the distribution of the quality metric is indexed by one or more parameters and the effect of the presence of a special cause is to change the values of these parameters. The purpose of a control chart is to detect special causes of variation so that these causes can be found and eliminated. Because a special cause is assumed to produce a parameter change, the problem for which a control chart is used can be formulated as the problem of monitoring a process to detect any changes in the parameters of the distribution of the quality variable.

**Exhibit 1. The Generalized Shewhart Control Chart**

Duncan (1956) indicates that the usual practice in maintaining a control chart is to plot the sample form the process relative to constant width control limits, say three-sigma limits. In this paper, a modification to standard practice in which the sampling control limits are not fixed but instead can vary after the process has operated for a period of time is investigated. The basis of choice of control limit width is a model for the cost of operating the chart. Cost model is developed to describe the total cost per unit of time of monitoring the mean of a process using both the standard and the generalized Shewhart control chart. The cost model is developed under the assumption that the quality characteristic of interest is normally distributed with known and constant variance.

The definition of the cost model for the standard Shewhart control chart proceeds in two steps as defined by Zou & Nachlas (1993). First, the uniform lifetime distribution is employed to describe the random variable t, the time until a process shift. It is assumed that the process is subject to a shift from the in-control value of the process mean, μ_{1}, to an-out-of-control value, μ_{2}, at a random point in time. Then, the cost of operating a standard Shewhart control chart is defined using four cost terms. They are (1) Inspection cost; (2) False alarm cost; (3) True signal cost; and (4) Cost of producing additional non-conforming items when the process is out of control. In addition, the expected cycle length is determined. Then the expected total cost per unit time is constructed as the inspection cost plus the ratio of the sum of the three expected costs to the expected cycle length. The definition of the corresponding cost model for the generalized Shewhart control chart proceeds in a similar manner. Assume we plan to start the chart with one set of control limits and to change the limits to be tighter after the process has operated for a period of time that is determined. Specifically, we assume the process is samples every h hours and after m^{th} sample the control limits are changed. This is illustrated in Exhibit 1. The objective is to choose the economic values of the cost parameter to minimize the expected total cost. The cost model is constructed to allow the optimal choice of change-over time and the best values for the initial and adjusted control limits and therefore can increase the control chart sensitivity to small but anticipated shifts in the process average so that the chart is able to rapidly detect a special cause and bring the process into control. The cost model is also used to provide a comparison with conventional implementation of the Shewhart control chart for the PMBOK^{®} quality management education purpose.

## Model Development

Assume a process is monitored using a Chart and the process is subject to a shift from the in-control value of the process mean, μ_{1}, to an-out-of-control value, μ_{2}, at a random point in time. Assume the time until a process shift is a random variable with F(t) = t/θ, (0 < θ < ∞). Let N be the maximum value of t, then N = θ/h, and suppose N is as integer. To construct the expected total cost per unit time the following cost categories are considered:

1. C* _{i}* = sampling and inspection cost, unit cost per event = c

2. C* _{f}*= false alarm cost, unit cost per event = c

3. C* _{t}* = true signal and process correction cost, unit cost per event = c

4. C* _{d}* = cost of producing substandard product while out-of-control, unit cost per item = c

5. C* _{T}* = total cost per unit of time

The expected total cost per unit time function is then defined as:

Where E[t] is the expected cycle length (time to signal). The following notations are used:

μ_{1} = in-control value of the process mean

μ_{2} = out-of-control value of the process mean

σ_{x} = the known and constant population standard deviation

UCL = upper control limit = μ_{1} + k^{σ}_{x} / n^{1/2}_{2}

LCL = lower control limit = μ_{1} - k^{σ}_{x} / n^{1/2}

U_{x} = upper specification limit

L_{x} = lower specification limit

p1 = proportion non-conforming when μ = μ_{1}

p_{2} = proportion non-conforming when μ = μ_{2}

p = p_{1} – p_{2}

h = time between samples

r = production rate in units/hour

n = number items inspected per sample

m = number samples before changing the control limits

δ = number of units of σ_{x} from μ_{1} to μ_{2}

k_{1} = number σ_{x} /n^{1/2} of from μ_{1} to UCL before sample mh

k_{2} = number σ_{x} /n^{1/2} of from μ_{1} to UCL after sample mh

α = the type I error probability

β = the type II error probability

The decision variables are n, h, m, k1 and k2. The optimal values for the decision variables are chosen to minimize the expected total cost per unit time function.

(1) Inspection cost = C_{i} = {fixed cost + (unit cost)(number inspected)} / {time between samples}, therefore:

**Exhibit 2. Time Intervals Involving T and t _{p}**

Note that the inspection cost is the same for both the standard and the generalized Shewhart control chart.

(2) False alarm cost = C_{f} = (unit cost)(probability of false alarm) = c_{f} P[false alarm].

Let A = “false alarm”, A1 = “false alarm on sample I,” A2 = “no process shift before sample I,” then the probability of false alarm is constructed as:

Thus the false alarm cost is:

The probability of false alarm for the generalized Shewhart control chart is quite different from that for the standard control chart. We must consider t ≤ mh or t > mh separately. Thus:

Therefore:

(3) True signal cost = C_{t} = (unit cost)(probability of a true signal) = c_{t}P[true signal].

Let B = “true signal,” B_{1}= “process shift in interval j”, B_{2} = “no false alarm on proceeding j-1 samples,” then the expression for P[B] is:

Thus, the true signal cost has the following form:

The probability of true signal for the generalized Shewhart control chart is defined as:

Thus:

(4) Cost of producing nonconforming items when the process is out of control = Cd = (unit cost)(production rate)(increase in proportion non-conforming)(expected time out of control).

The time intervals at this step can be reviewed in Exhibit 2.

The E[time out of control] = E[length of partial interval after shift and before sample] + E[time comprised of full intervals until a true signal] = E[t_{p}] + E[t_{s}]. Note that the part of interval before the process shift can be written as T = t – jh, therefore:

Then:

Finally:

The E[t_{p}] is the same for the generalized Shewhart control chart but the E[t_{s}] is a bit different as the identify of the interval in which the shift occurs affects the signal probability. Thus:

Therefore:

(4) Let E_{1} = “false alarm on sample j and no process shift before sample j,” E_{2} = “process shift during interval s, no false alarm before intervals, and true signal on sample j (j-s+1^{st} after shift).” Then the expression for the expected cycle length is:

The expected cycle length for the generalized Shewhart control chart must also reflect differences in signal events before and after mh. E[t]_{(g)} can be written as:

Therefore:

## Model Analysis

Refereeing to the cost model developed earlier, the cost terms are functions of the decision variables, cost parameters and the distribution parameter. Two of the decision values of m and n are constrained to be integers, while k_{1} and k_{2} may take real values. As Montgomery (1980) indicates that a sampling frequency of one hour is common for many control charts, h = one unit of time is used. The behavior of the cost model is analyzed numerically. GINO (Lasdon & Warren, 1985) is used to examine the behavior of the cost model over reasonable parameter sets and the generalized reduced gradient (GRG) algorithm is used to attempt to minimize the expected total cost per unit time function for those parameter sets. The parameter ranges evaluated are listed below.

(1) θ ∈ (8, 200)

(2) δ = 0.522, the magnitude of the shift in the mean when a shift occurs. This value is selected because is corresponds to an increase in the proportion nonconforming from 0.01 to 0.02.

(3) c_{i} = 1.0; 5.0

(4) c_{d} ∈ (1, 10)

(5) c_{f} = 100

(6) r = 200, the rate of production

(7) c_{t} = 10

The above parameter ranges define the scenarios under which the economic performance of the standard and the generalized Shewhart control chart are investigated. The numerical analysis of the behavior of the expected total cost per unit time function with respect to the decision variables for a family of the parameter ranges is examined.

The expected total cost per unit time function is convex in k for all ranges of the other parameters. Small values of k create large expected total cost because an excessive number of false alarms is given. This may dominate any cost saving due to rapid shift detection. Intermediate values of k produce the smallest expected total cost because they balance costs of nonconforming production against the false alarm cost. Large values of k provide reduced shift detection probabilities and thus increasingly large nonconforming production cost. The total effect is that the expected cost decrease to a minimum and then rises again as k increases.

The expected total cost function is also convex in n for all ranges of the other parameters. Small values of n imply low sampling costs but high nonconforming costs since shifts are not rapidly detected. Intermediate values of n balance the sampling cost against the nonconforming product cost to achieve the lowest expected total cost. Large values of n imply large sampling costs, which may dominate the savings in nonconforming product costs achieved through greater detection probabilities. These interpretations vary depending upon the relative importance of each cost categories but the overall effect is that the expected total cost function is convex in n.

The above results for n and k are anticipated for the standard Shewhart control charts in general and confirmed for the generalized Shewhart control charts. The generalized Shewhart control chart has features that the standard Shewhart control chart does not. The properties that result from these additional features are now explored.

Model behavior in terms of the decision variable m, k1, and k2 is characterized by three cases. The relative magnitudes of the cost parameters in each case determine which behavior is observed. In case one, the expected total cost per unit time function C_{T}, displays convex behavior in each of the decision variables m, k_{1}, and k_{2} and a minimum occurs in the interior of the convex feasible region. This means that the minimum cost control chart is some form of the generalized Shewhart control chart. In case two, C_{T} is still convex but it has a minimum corresponding to a boundary of m = 0 and k_{2} = k_{1} and it increases strictly in each of those variables. This means that the minimum cost control chart is a standard Shewhart control chart with no control limits changes. In case three, C_{T} strictly decrease in both m and k2 and has a minimum at the boundary k_{1} = k_{2} and m = ∞. This implies that the minimum cost control chart is a standard Shewhart control chart with no control limits changes.

## Conclusion

The analysis presented above yields several interesting points. The first of these is that the analysis of the cost of operating any type of control chart should be treated very carefully, as the cost function may not always have the commonly assumed regularity. The choice of cost coefficients, the time of shift distribution and distribution parameters have a direct influence on the performance of the expected total cost per unit time function. The important results of the analysis performed show that the generalized Shewhart chart for means may be economically attractive when the inspection cost, the true signal cost, and the nonconforming cost together balance the expected cycle length and the false alarm cost. When this is the case, the expected total cost per unit time function is convex with an interior minimum and an opportunity for optimization of the generalized Shewhart control chart. When one or more of the model terms dominates the others, the expected total cost per unit time will display the same increasing or decreasing behavior as the dominating factor and the generalized cost model as studied in this paper will be unattractive.

The second conclusion is that all model parameters and variables are important to the expected total cost per unit time function. The control limits k_{1} and k_{2} have a great effect than do the distribution parameter θ and k_{2} has a greater effect than does k_{1}. It is also true that the sample size, n, and the time of the change in the width of the control limits, m, enhance the effect of the distribution parameter, K_{1}, and k_{2}.

The final conclusion is that there are control chart applications for which the cost model is useful. Values of the production process parameters that display more commonly encountered relationships leads to the generalized Shewhart control chart having lower cost than the corresponding standard Shewhart control chart. For the example case analyzed above, the optimal saving is $0.22 per item produced. Since the production rate assumed is 200/hour, the saving $44 per hour. This saving is dramatic and therefore the generalized Shewhart control chart is worth pursuing.

### Reference

Duncan, A.J. (1956). The Economic Design of -Chart Used to Maintain Current Control of a Process. *Journal of American Statistical Association,* 51, 274, 228–242.

Lasdon, L., & Warren, A. (1985). *General Interactive Optimizer—GINO.* Palo Alto: Scientific Press.

Montgomery, D.C. (1980). The Economic Design of Control Charts: A Review and Literature Survey. *Journal of Quality Technology,* 12 (2), 75–87.

Reynolds, M.R., Amin, R., Arnold, J.C. & Nachlas, J.A. (1988). -Charts with Variable Sampling Intervals Between Samples. *Technometrics,* 30 (2), 181–192.

Zou, X., & Nachlas, J.A. (1993). A Robust Shewhart Control Chart Adjustment Strategy. *Unpublished Ph.D. Dissertation,* Department of ISE, Virginia Tech.

Proceedings of the Project Management Institute Annual Seminars & Symposium

September 7–16, 2000 • Houston, Texas, USA

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