1. Introduction

The present scenario of total quality management demands the effective use of statistical tools for analyzing quality problems and improving the performance of production process. Measuring, monitoring and maintaining several quality characteristics to achieve further improvements are required in every production process. Many of the quality characteristics are measurable in nature and can be expressed in terms of numerical measurement by measuring them using micrometer, vernier calipers etc. For example, length, diameter, weight, pocket width, distance and groove diameter of a shaft are some of the quantifiable quality characteristics encountered during the production of shafts. When dealing with measurable quality characteristics, it is usually necessary to monitor the behaviour of the mean value of the quality characteristics and its variability. To study the behaviour of the production process and to take a necessary action on the process, control charts for variables are very much used which requires to collect random samples from the production process and to compute the estimate of the quality characteristics. Further one has to show that the process is under statistical control before computing the process capability indices. Nowadays process capability analysis is a mandatory requirement and to have a minimum prescribed value for the process capability indices C_p and C_pk to be a supplier for original manufacturing companies, particularly in the field of automobile industries. Process capability indices (PCI) have proliferated in both use and variety during the last three decades. Their widespread and often uncritical use led to improvements in quality.

Process capability indices are considered as a practical tool by many advocates of the statistical process control industry. They are used to determine whether the manufacturing process is capable of producing units within the product or process specifications. A capability index is a dimensionless measure based on the process parameters and the process specifications, designed to quantify, in a simple and easily understood way, the performance of the process. Several process capability indices have been developed among which the basic and widely used indices are C_p and C_pk. (see Kane[8]). A detailed review of the process capability indices can be seen in Kotz and Johnson[9].

The process capability indices are unit less measures showing the capability of a manufacturing process whether it is capable of operating within its specifications limits. If the value of the PCI exceeds one, then the process is considered to be satisfactory or capable. However this approach ignores the fact that these PCIs are random variables with distributions. Confidence limits play a major role for the correct interpretation of PCIs. Recently, techniques and tables were developed to construct confidence limits for the process capability indices. These techniques are based on the assumption that the underlying process is normal. Moreover, since many processes can frequently have skewed or heavily tailed distributions. In practice, the interval estimation technique that is free from the assumption of distribution is desirable. The various other properties and the behaviour of process capability indices at different type of distributions were studied by many authors. For example, the 95% confidence limits for the process capability indices C_p and C_pk were constructed by Chou, Owen and Borrego[5]. As their limits on C_pk can produce 97% or 98% lower confidence limits (instead of 95%) making them conservative, an approximation presented by Bissell[4] is recommended. Franklin and Wasserman[6] presented an initial study of the properties of these three bootstrap confidence intervals for C_pk. Franklin and Wasserman[7] constructed the confidence limits for some basic capability indices and examined their behaviors when the underlying process is normal, skewed or heavily tailed distributions. Balamurali andKalyanasundaram[3] constructed bootstrap confidence limits for the capability indices C_p, C_pk and C_pm based on lognormal and chi-squared distributions. Balamurali[2] considered the above capability indices under short run production processes and constructed the confidence limits.

However the problems one has to face are the following: There is a considerable time taken for establishing and monitoring the process through control charts for variables; inadequate technical personnel to correctly interpret the outcome of the control charts; unnecessary inventory and delay due to the computation of process capability indices and to take a decision on the process based on the resulting values of process capability indices. Hence it is felt to have an alternative method to achieve the benefit of both control charts for variables and the process capability analysis. As a result we have proposed new type of control charts for variables based on the required process capability indices C_p and C_pk, called as “control charts for variables with specified value for process capability indices”. Further we have presented the control charts constants to construct the proposed control charts and are explained with the help of a numerical example. For a more detailed discussion on the control charts for variables and process capability analysis, the readers are referred to Montgomery[10], Spiring[11], Sarkar and Pal[1], Subramani[12, 13], Subramani and Balamurali[14] and the references cited therein. In this paper, we propose a capability based control chart for on line process control. The relative performance of this control charts for variables is assessed with that of existing usual control charts for variables. It is observed that the proposed method is simple to apply and does not warrant any tedious computations both for control charts and for computing process capability indices. We are also presenting the control charts constants for constructing capability based control charts. The proposed method is also illustrated with a numerical example.

2. Control Charts and Process Capability Analysis

2.1. Usual Method

Figure 1. Present M anufacturing System

For the sake of convenience to the readers, we have presented the control charts for variables for checking whether the process is under statistical control or not. That is, to take a decision whether the process can be allowed further without making any adjustment or to take corrective actions if any, to bring back the process under statistical control. Once the process has been brought under the state of statistical control by both then the process capability indices can be computed. If the values of the process capability indices are at the satisfactory level, the process can be allowed further without making any change or adjustment in the process. Otherwise suitable adjustments have to be made so that the process shall satisfy both the requirements of control charts and the process capability indices. The present manufacturing process can be explained in the flow chart as shown in Figure 1.

The following are the various computational formulae involved in constructing the control charts and for the computation of process capability indices.

LSL: Lower Specification Limit of the Quality Characteristics

USL: Upper Specification Limit of the Quality Characteristics

When the sample size n=5, the values for the control chart constants obtained from the table are as given below:

D₄ = 2.114, D₃ = 0, d₂ = 2.326 and A₂ = 0.577

The Control limits for constructing and are as given below:

Control limits for:

Control limits for :

2.2. Proposed Method- Control Charts for Variables with Specified Value

Suppose that a customer wants to have products produced in a process with a specified value for the process capability index. To achieve this, the manufacturer has to establish a process under statistical control and then the process capability index has to be computed. If the computed value of is more than the specified value given by the customer, the process can be run without any adjustment. Otherwise suitable actions have to be taken to meet the required process capability values. The drawback of this method is that the manufacturer has to keep all the produced items in the inventory until the computation of the process capability index and to take a decision on the process based on the process capability index. It is a time consuming procedure as well as requires additional skilled personnel. To avoid this, we have proposed control charts for variables with specified process capability indices. In the proposed method, the process is under statistical control means that it satisfies the required process capability index. Further it is not necessary to compute the process capability indices separately. The proposed manufacturing process can be explained in the flow chart shown in Figure 2.

Figure 2. Present M anufacturing System

The following are the derivation of various computational formulae involved in constructing the proposed control charts with specified process capability index.

If value is given then value can be obtained from the above formula as

Similarly one can obtain the lower control limit (LCL) and upper control limit (UCL) of the as given below:

In the similar manner one can compute the control limits to construct with specified value as given below.

Lower control limit for the proposed is obtained as given below:

One can use the above control limits to construct with specified value.

Similarly one may obtain the upper control limit for the proposed as

Table 1. Control Charts Constants for specified      value

Thus the control limits for the control charts with specified value are as given below:

Control limits for proposed:

Control limits for proposed:

One can use the above control limits to construct the andwith specified value. For the sample size, we have presented the values of the control charts constants in the Table 1 given below.

The advantageous of the proposed control charts is that the observed range values are not required for computing the control limits. Further if the process is under the state of statistical control means that it automatically satisfies the required conditions regarding the values of process capability index.

2.3. Proposed Method- Control Charts for Variables with Specified value

In the section 2.2, we have introduced a method for constructing control charts for variables with specified value for the process capability index. In the similar manner we have developed a method for constructing control charts

for variables with specified value for the process capability index. The following are the derivation of various computational formulae involved in constructing control charts for variables with specified process capability index

After a little algebra, one can rewrite the as

If value is given then value can be obtained from the above formula as

Similarly one can obtain the lower control limit (LCL) and upper control limit (UCL) of the as given below:

One can use the above control limits to construct with specified value.

In the similar manner one can compute the control limits to construct with specified value as given below.

Lower control limit for the proposed is obtained as given below:

Similarly one may obtain the upper control limit for the proposedas

Thus the control limits for the proposed control charts with specifiedvalue are as given below:

Control limits forwith specifiedvalue

Control limits for the proposed with specifiedvalue:

The above control limits can be further simplified by substituting the values of and, which lead to the following two cases:

Case 1: When

Control limits forwith specifiedvalue

Control limits for the proposed with specifiedvalue:

Table 2. Control Charts Constants for specified      value

Case 2: When

Control limits forwith specifiedvalue

Control limits for the proposed with specifiedvalue:

One can use the above control limits to construct the andwith specified value. For the given sample size, we have presented the values of the control charts constants in the Table 2 given below.

The advantageous of the proposed control charts with specified and values, is that the observed range values are not required for computing the control limits. Further if the process is under the state of statistical control means that it automatically satisfies the required conditions regarding the values of process capability indices and .

3. Numerical Example

Consider the data given below in Table 3 is the Example 5.1 (Montgomery[10], page 213). The data is pertaining to the manufacturing of Piston Rings for an automotive engine produced by a forging process. Twenty five samples, each of size five have been taken and the inside diameter is measured. The resulting data together with the sample means and sample range values are given below in Table 3.

From the above values we have obtained the following:

Case 1: Usual Method of Control Charts and the Computation of Process Capability Indices

Control Limits for R-Chart are obtained as:

Control Limits for are obtained as:

Table 3. Data of Inside Diameter of Piston Rings (Spec: 74.000±0.05 mm) Sample Number x1 x2 x3 x4 x5 Sample Mean Range R 1 74.030 74.002 74.019 73.992 74.008 74.0102 0.038 2 73.995 73.992 74.001 74.011 74.004 74.0006 0.019 3 73.988 74.024 74.021 74.005 74.002 74.0080 0.036 4 74.002 73.996 73.993 74.015 74.009 74.0030 0.022 5 73.992 74.007 74.015 73.989 74.014 74.0034 0.026 6 74.009 73.994 73.997 73.985 73.993 73.9956 0.024 7 73.995 74.006 73.994 74.000 74.005 74.0000 0.012 8 73.985 74.003 73.993 74.015 73.988 73.9968 0.030 9 74.008 73.995 74.009 74.005 74.004 74.0042 0.014 10 73.998 74.000 73.990 74.007 73.995 73.9980 0.017 11 73.994 73.998 73.994 73.995 73.990 73.9942 0.008 12 74.004 74.000 74.007 74.000 73.996 74.0014 0.011 13 73.983 74.002 73.998 73.997 74.012 73.9984 0.029 14 74.006 73.967 73.994 74.000 73.984 73.9902 0.039 15 74.012 74.014 73.998 73.999 74.007 74.0060 0.016 16 74.000 73.984 74.005 73.998 73.996 73.9966 0.021 17 73.994 74.012 73.986 74.005 74.007 74.0008 0.026 18 74.006 74.010 74.018 74.003 74.000 74.0074 0.018 19 73.984 74.002 74.003 74.005 73.997 73.9982 0.021 20 74.000 74.010 74.013 74.020 74.003 74.0092 0.020 21 73.982 74.001 74.015 74.005 73.996 73.9998 0.033 22 74.004 73.999 73.990 74.006 74.009 74.0016 0.019 23 74.010 73.989 73.990 74.009 74.014 74.0024 0.025 24 74.015 74.008 73.993 74.000 74.010 74.0052 0.022 25 73.982 73.984 73.995 74.017 74.013 73.9982 0.035 Average 74.00118 0.02324

By plotting the control limits, sample ranges and sample means, one may get and as given in Figure 3 and Figure 4. From the , we observe that all the plotted sample range values are falling within the control limits. Hence we may conclude that the variations are under control. Similarly from the, we observe that all the plotted sample means are falling inside the control limits. Hence we may conclude that the averages are also under the statistical control.

Figure 3. R-Chart- Usual Method

Figure 4. X-Bar Chart- Usual Method

The manufacturing capability of a process can normally be evaluated in terms of process capability indices and_. The process capability indices obtained from the above values are given below:

If the customer’s requirement of the process capability index is 1.5, then one may conclude from the above capability indices that the process is an efficient one. However If the customer’s requirement of the process capability index is 2.0, then one may conclude from the above capability indices that the process is not an efficient one and hence the process has to be adjusted so as the resulting process capability index is at least 2.0.

Case 2: Control Charts with specified

In this case we construct the control limits for the given value of process capability index and hence separate computation of the process capability index is not required. Further the proposed control charts states that the process is under the statistical control means that the process satisfies the requirement of the process capability index. Hence one can construct several control limits for different values of process capability index. If any plotted points falls outside the control limits with specified means that the process does not satisfy the required process capability index.

Let the given process capability index . Then the Control limits for with specifiedare obtained as given below:

Control limits forwith specified

By plotting the control limits, sample ranges and sample means, one may get and with specified as given in Figure 5 and Figure 6. From the, we observe that all the plotted sample range values are falling within the control limits. Hence we may conclude that the variations are under control. Similarly from the, we observe that all the plotted sample means are falling inside the control limits. Hence we may conclude that the averages are also under the statistical control. Further we conclude that the process capability index for the given process is at least 1.5 and satisfies the customer’s requirements.

Figure 5. R- Chart with Specified Cp=1.5

Figure 6. X-Bar Chart with Specified Cp=1.5

Case 3: Control Charts with specified

In this case we construct the control limits for the given value of process capability index and hence separate computation of the process capability index is not required. Further the proposed control charts states that the process is under the statistical control means, the process satisfies the requirement of the process capability index. Hence one can construct several control limits for different values of process capability index. If any plotted points falls outside the control limits with specifiedmeans that the process does not satisfy the required process capability index.

Let the given process capability index . Further the control limits with specified process capability index are obtained as given below:

Control limits forwith specifiedvalue

Control limits for the proposed with specifiedvalue are obtained as given below:

Figure 7. R-Chart with Specified Cpk=1.5

Figure 8. X-Bar Chart with Specified Cpk=1.5

By plotting the control limits, sample ranges and sample means, one may get and with specified as given in Figure 7 and Figure 8. From the, we observe that all the plotted sample range values are falling within the control limits. Hence we may conclude that the variations are under control. Similarly from the, we observe that all the plotted sample means are falling inside the control limits. Hence we may conclude that the averages are also under the statistical control. Further we conclude that the process capability index for the given process is at least 1.5 and satisfies the customer’s requirements.

4. Conclusions

In this paper, we have proposed a control chart which is based on the process capability indices C_p and C_pk for on line process control, which have the benefit of the usual control charts and the process capability indices. That is, the proposed control charts combines the two stage controlling mechanism namely, Control Charts and the process capability indices into a single stage controlling mechanism to monitor the process online and to assess the suitability of the manufacturing process. It has been shown that the relative performance of the proposed control charts for variables can be assessed with that of usual variable control charts. It has also been shown that the proposed control chart is simple to apply and does not warrant any tedious computations both for control charts and for computing process capability indices. Moreover, we have also presented tables for the control charts constants for computing the control limits of the capability based control charts. The proposed method has also been illustrated with a numerical example. This study can also be extended to develop control charts based on other process capability indices.

ACKNOWLEDGEMENTS

The authors are thankful to the editor and the referees for their useful comments, which have helped to improve the presentation of the paper.