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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.16 No.2 pp.195-204
DOI : https://doi.org/10.7232/iems.2017.16.2.195

# A Sampling Plan for the Selection of Supplier using Process Yield Index based on Linear Profiles

Department of Statistics, GC University, Pakistan
Department of Statistics, Faculty of Science, King Abdul Aziz University, Saudi Arabia
Department of Industrial and Management Engineering, POSTECH, Korea
Department of Statistics and Computer Science, University of Veterinary and Animal Sciences, Pakistan
Corresponding Author, chjun@postech.ac.kr
May 15, 2016 October 25, 2016 March 23, 2016

## ABSTRACT

In this article, a multiple dependent state repetitive (MDSR) sampling plan is developed to choose a more capable process by comparing two suppliers’ processes. The proposed MDSR plan is based on the ratio of two process yield indices based on the estimated linear profiles. The linear profile for both processes is estimated by a regression model with a single explanatory variable having the specified number of levels. The operating characteristic function of the proposed plan will be derived. The plan parameters related to the decision rule as well as the sample size are determined by minimizing the average sample number while satisfying the producer’s and the consumer’s risks.

## 1.INTRODUCTION

In the recent age of global supply chains, products are shipped to several companies and covering the several continents in their way from material suppliers to the final consumer (Shmueli, 2011). Supplier selection is very important when both suppliers have similar processes. Selecting a supplier with more capable process can greatly reduce investing costs, shrink production lead time, enlarge customer satisfaction, and build up business performance. Thus, an efficient approach to relieving the problem of supplier selection is crucial and quality of its process or product is one of the most important components for supplier selection. Process capability indices (PCIs) are mostly used as a standard for the process evaluation. Regression modeling is very useful in linear profile monitoring when most marketable quality characteristic is designated by some functional relationship between a response variable and one or more explanatory variables. Mostly a linear profile exists in a supplier’s process because sample observations collected with that relationship show a straight line on a graph. A review on profile monitoring can be seen in Woodall (2007) and Noorossana et al. (2011).

An acceptance sampling plan can greatly minimize the difference between the required and the real quality level of the product being supplied (Pearn and Wu 2006, 2007). But due to random sampling instead of 100% in- spection, there is always risks of accepting undesired inferior product lots, and of rejecting better items in lots. Acceptance sampling plans provide some levels to both producers and consumers to compromise on the judgment of a consignment of products. A quality level set by the producer is named as acceptable quality level (AQL) with α risk. And the consumer decides a quality level called as the lot tolerance percent defective (LTPD), with β risk. An efficient plan protects the both consumer and producer. An acceptance sampling plan should be efficient and economical to achieve the high quality level (Montgomery, 2009). Sherman (1965) introduced the concept of a repetitive group sampling (RGS) plan for attributes. Balamurali et al. (2005) and Balamurali and Jun (2006) worked on variables RGS plans for one-sided specification. An RGS plan is proved to be more efficient than a single sampling plan. Wortham and Baker (1976) developed a multiple dependent state (MDS) sampling plan for attributes. State refers to as some conditions and the dependent state means a set of past conditions to be dependent. An MDS plan consists of decision rules based on the data collected from the current lot as well from preceding lots making it efficient. Balamurali and Jun (2007) worked on a variables MDS sampling plan for one-sided specification. Aslam et al. (2013a) developed a variables MDS sampling plan based on the process capability index Cpk using normal approximation. To save time and cost of inspection, an RGS plan is combined with the benefits of a multiple dependent state (MDS) sampling plan. Aslam et al. (2013b) proposed multiple states repetitive group sampling plans with process loss consideration. Aslam et al. (2016) developed a multiple dependent state repetitive group sampling plan for Burr XII Distribution. Lee et al. (2015) developed a variables MDS sampling plan using an exact distribution of Cpk. Wang (2014) derived a process yield index $C p u A$ for a simple linear profile with one sided specification when a lower quality characteristic is desirable for the process.

In this paper, a new MDSR plan is proposed based on a ratio of two process yield indices $( C p u A 1 , C p u A 2 )$ based on the estimated linear profiles to select a more capable process. The main objective of this research is to develop an MDSR sampling plan for the comparison of two supplier’s processes by minimizing the sample size in selecting a better supplier and to decide between acceptance and rejection of a lot from a more capable process.

This research work have arranged in six sections. Section 2 contains the description of the process yield index $C p u A$ for a linear profile of the response variable. In Section 3, an MDSR plan is proposed in details. Comparison is discussed in Section 4. To apply the proposed MDSR plan practically, an illustrative example is provided in Section 5.

## 2.PROCESS YIELD INDEX FOR A LINEAR PROFILE

Commonly, a linear regression equation is used to define the linear profile. If a statistically controlled process is designated by a simple linear regression model, then the following relation holds between a predictor (or independent) variable and a response variable.(1)

$y i j = α + β x i + ε i j ( i = 1 , 2 , ⋯ , k , j = 1 , 2 , ⋯ , n )$
(1)

where, xi is the i-th level of an independent variable and yij denote the j-th observation of the response variable at the fixed level xi. Here, k is the number of levels of the independent variable and n is the number of observations. εij ’s are independently identically follows the normal distribution with mean zero and variance σ2.

For a normal process with one-sided specification USL, the index $C p u = ( U S L − μ ) 3 σ$ has been used to measure the process capability (see Yum and Kim, 2011). If there is a one-sided upper specification limit USLi for the response at the i-th level of the explanatory variable, Wang (2014) defines the process yield at the i-th level of explanatory variable for a normal process as(2)

$P u i = P ( y i j < U S L i ) = P ( y i j − μ y i σ y i < U S L i − μ y i σ y i ) P u i = Φ ( 3 C p u i )$
(2)

where Φ (.) is the cumulative distribution function of the standard normal distribution.

The predicted response at level xi will be

$y ^ i = α ^ + β ^ x i$

where $α ^$ and $β ^$ are the estimates of α and β, respectively.

Hence, Wang (2014) proposed a new index $C p u A$ to measure the process capability for a simple linear profile with one sided specification. An average value of process yields is taken for all k levels of independent variables to obtain $C p u A$

$C p u A = 1 3 Φ − 1 [ ∑ i k Φ ( 3 C p u i ) k ]$
(2a)

From a stable process, the following estimator of process yield index is considered for one-sided specification(3)

$C ^ p u A = 1 3 Φ − 1 [ ∑ i k Φ ( 3 C ^ p u i ) k ]$
(3)

where

$C ^ p u i = U S L i − y ^ i 3 σ ^ y i$

Wang et al. (2014) derived the asymptotic distribu tion of $C ^ p u A$ and simplified it as(4)

$C ^ p u A ∼ N ( C p u A , ( 2 + 9 G 2 ) [ Φ ( 3 G ) ] 2 18 k 2 n [ Φ ( 3 C p u A ) ] 2 )$
(4)

where $G = 1 3 Φ − 1 [ k Φ ( 3 C p u A ) − ( k − 1 ) ]$

Since $C ^ p u A$ is an asymptotic normal random variable, Wang et al. (2014) used the ratio of $C ^ p u A 2$ and $C ^ p u A 1$ to compare the two suppliers’ process capabilities.(5)

$R ^ = C ^ p u A 2 C ^ p u A 1$
(5)

## 3.PROPOSED MDSR SAMPLING PLAN BASED ON A RATIO OF TWO PROCESS YIELD INDICES

To compare the two processes from the corresponding suppliers, we propose an MDSR sampling plan based on the ratio of two process yield indices with one sided specification for linear profiles. Both processes require the lower-the-better quality characteristic. It is assumed that the supplier I was already producing the products with index $C p u A 1$ who has been inspected to possess the minimal requirement $( C p u A 1 ≥ c )$. But after the claim of supplier II to have a more capable process, a better selection is required. The optimal values of c are calculated by applying the single acceptance sampling plan based on $C p u A 1$ . Both processes are assumed to be followed the assumptions of a normal distribution.

The proposed MDSR plan consists of following steps:

• Step 1) Choose the producer’s risk and the consumer’s risk. Select the process capability requirements $( C p u A 2 , A Q L a n d C p u A 2 , L T P D )$ at two risks.

• Step 2) Take a random sample of size n from a lot of process provided by supplier II and estimate the linear profile for supplier II

$y ^ 2 i = α ^ 2 + β ^ 2 x i$

Then obtain the estimated process yield index of supplier II by(6)

$C ^ p u A 2 = 1 3 Φ − 1 [ ∑ i k Φ ( 3 C ^ p u 2 i ) k ]$
(6)

where $C ^ p u 2 i = U S L i − y ^ 2 i 3 σ ^ y 2 i$

• Step 3) Calculate the sample estimate of ratio statistic R as follows

$R ^ = C ^ p u A 2 C ^ p u A 1 = C ^ p u A 2 c$

by assuming that the process capability by supplier I is at the level c.

• Step 4) Make a decision on the lot as follows

• 1. If $R ^ > r 1$, then accept the lot selected from supplier II and hence declare it as more capable process (Alternatively, reject supplier I)

• 2. If $R ^ < r 1 ( r 1 > r 2 )$, then reject the lot selected from supplier II (Alternatively, accept supplier I and declare it as more capable process)

• 3. If $r 2 < R ^ < r 1$, then accept the lot selected from supplier II if “m” preceding lots from the process provided by supplier II have been accepted under the condition of $R ^ > r 1$ . Otherwise, move to Step 2 and repeat the process of random sampling for supplier II.

### 3.1.OC Function of Proposed Plan

When acceptance and rejection decision is based on a single random sample, the probability of acceptance will be

$P a 1 = P ( R ^ > r 1 ) + { P ( R ^ < r 1 ) − P ( R ^ < r 2 ) } × [ P ( R ^ > r 1 ) ] m$

Now the ratio $R ^$ follows a normal distribution with the following mean and variance:

$E ( R ^ ) = E ( C ^ p u A 2 c ) = C p u A 2 c$

$V ( R ^ ) = V ( C ^ p u A 2 c ) = ( 2 + 9 G 2 2 ) [ Φ ( 3 G 2 ) ] 2 18 c 2 k 2 n [ Φ ( 3 C p u A 2 ) ] 2$

where

$G 2 = 1 3 Φ − 1 [ k Φ ( 3 C p u A 2 ) − ( k − 1 ) ]$

Therefore,

$P a 1 = [ 1 − Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 1 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) ] + [ Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 1 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) − Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 2 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) ] [ 1 − Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 1 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) ] m$

where

$R = C p u A 2 C p u A 1 = C p u A 2 c$

In case of a repetition of random sampling from supplier II, the probability of that repetition will be

$P 2 = [ Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 1 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) − Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 2 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) ] × [ − { − Φ ( c k [ ( 18 n ) ] Φ ( 3 C p u A 2 ) ( r 1 − R ) ( 2 + 9 G 2 2 ) Φ ( 3 G 2 ) ) } m ]$

Therefore, the OC function of our proposed plan (see Aslam et al., 2013) is given as

$L ( C p u A 2 ) = P a 1 1 − P 2$
(7)

The two points on OC curve, $( C p u A 2 , A Q L , 1 − α ) a n d ( C p u A 2 , L T P D , β )$ are required to satisfy both the producer’s and the consumer’s risks. A producer needs the lot acceptance probability at least to be 1−α at (AQL) and a consumer wants the lot acceptance probability not more than β t LTPD. These two conditions are written in terms of equations as follows

$L ( C p u A 2 L T P D ) ≤ 1 − α$
(8)

and

$L ( C p u A 2 L T P D ) ≤ β$
(9)

where $L ( C p u A 2 , A Q L )$ is the OC function in (7) to be evaluated at $C p u A 2 , A Q L$ and $L ( C p u A 2 , L T P D )$ is the OC function to be evaluated at $C p u A 2 , L T P D$.

The proposed MSR plan is designed by four parameters n, r1, r2and m. Here, m = 1 and m = 2 are usually taken as prefixed parametric values based on the acceptance information of the proceeding lots. e.g. m = 1 means that if one preceding lot by supplier II has accepted, current lot of supplier II will be accepted even with the condition $R ^ < r 2$.

Optimal other parametric values can be obtained by minimizing the average sample number (ASN) subject to the two equations (8) and (9). The probabilistic approach to get ASN to be inspected in proposed MSR plan is defined as follows

$A S N ( C p u A 2 ) = n + n P 2 ( C p u A 2 ) + n [ P 2 ( C p u A 2 ) ] 2 + ⋯ = n 1 − P 2 ( C p u A 2 )$

The ASN function to be minimized is the average value of the ones at two quality levels, AQL and LTPD.

Minimize

$A S N ( C p u A 2 , A Q L , C p u A 2 , L T P D ) = 1 2 { n 1 − P 2 ( C p u A 2 , A Q L ) + n 1 − P 2 ( C p u A 2 , L T P D ) }$

Subject to

$L ( C p u A 2 , A Q L ) ≥ 1 − α L ( C p u A 2 , L T P D ) ≤ β n > 1 , r 1 > r 2 > 0.$

The above optimization problem is solved using R development Core Team (2015) and R-code is available on request. Tables 1-2 show the sampling plan parameters when single sampling instead of repetitive sampling is used. In Table 3-Table 4, the optimal values of the MDSR plan parameters for selecting a more capable process are found for required $C p u A 2 , A Q L = 1.33 , C p u A 2 , L T P D = 1$ with different combinations of α and β risks. The values of m are set in advance as it relates to past information (m = 1, 2). Two and four levels are taken for independent variable in linear profiles for both processes.

For example, when m = 2, k = 2,α = 0.01, β = 0.05 in Table 2, a random sample of size 44 should be drawn from the current lot to compare the sample value of a ratio of two estimated process yield indices with critical values r1 =1.0785, r2 = 0.8998 . That is, if $R ^ ≥ 1.0785$, accept the lot from the process of supplier II and declare it more capable process. If $R ^ ≤ 0.8998$, accept the lot from the process of supplier I and declare it more capable than supplier II. If $R ^$ is between 1.0785 and 0.8998, accept the lot only if past two lots have already accepted. As consumer’s risk or producer’s increases, the ASN values have a decreasing trend. More tolerance of the both risks levels result in less sample size to be inspected. But when the restriction (m value) of preceding accepted lots increases, more sample sizes are required on average for inspection. For example, when α = 0.05, β = 0.05, m =1, k = 2 , ASN is ASN=28.0194 . For m = 2 the ASN value increases to 32.0417. But for β = 0.075 ASN value decreases to 27.8437.

In Table 5-Table 6, there are the calculated values of designed parameters for $C p u A 2 , A Q L = 1.5 , C p u A 2 , L T P D = 1.33$, k = 2, 4, m = 1, 2.

There is a highly increasing trend in the values of ASN as compared to Table 3-Table 4. When both quality levels are needed to be higher, greater sample sizes are required to get quality information from the current lot. Moreover, when the difference between $C p u A 2 , A Q L$ and $C p u A 2 , L T P D$ becomes smaller, both n and ASN will increase because greater number of samples are required to have a good understanding of the quality information and to avoid the errors in assessing the quality charatceristic if good and bad quality levels are same. But there is a decreasing effect on ASN values by the increase in the number of levels of independent variable (k).

## 4.COMPARATIVE STUDY

An efficient and economical sampling plan must have following two qualities:

• 1. The higher lot acceptance probability at AQL and smaller probability of acceptance at LQL

• 2. Shorter sample sizes to be inspected for lower inspection cost

The performance of the proposed MDSR plan is studied by the power curves of the two plans displayed in Figures 1-2, which relates the probability of rejection of a lot from a more capable process with ratio of two process yield indices. It can be seen clearly that power curve of proposed MDSR plan has higher probability of rejecting the lot from a more capable process under the smaller value of ratio of two process yield indices (less than 1) and lower probability of rejecting the lot from a more capable process under the higher value of ratio of two process yield indices (larger than 1). Also, the behavior of the OC curve of the proposed plan is very close to an ideal operating characteristic curve as the slope between $C p u A 2 , A Q L$ and $C p u A 2 , L T P D$ is very sharp.

Table 7-Table 8 clearly show that the proposed MDSR plan also proves to be more economical than an existing plan because it provides the less sample sizes on average to be inspected as compared to the single sampling plan based on ratio of two process yield indices to compare the two processes. For example, the proposed MDSR plan with optimum parameters combination of r1 =1.1063, r2 = 0.8998, n = 20, m =1, under $C p u A 2 , A Q L = 1.33 , C p u A 2 , L T P D = 1$, α = 0.05, β = 0.1, k = 4 provides the minimum ASN = 23.0971 that is 52 % lesser than the sample size n = 44 provided by a single sampling plan based on ratio of two process yield indices.

## 5.AN ILLUSTRATIVE EXAMPLE

The proposed MDSR plan is illustrated with the help of case study of Amiri et al. (2014). Following simple linear profile is provided between the response variable of dissipation factor value yi and the explanatory variable from the soaking stage xi :

$Y i = − 758.92 + 200.81 x i + ε i$

where the variance of the error term is estimated by 2.934. Four levels of x are 3.82, 3.84, 3.86, 3.88.

To illustrate the comparison of two suppliers’ capabilities, 20 profiles for normally distributed error term with standard deviation 1.2274 were simulated to trace supplier II and 44 profiles for normally distributed error term with standard deviation 1.7129 to trace supplier I. Simulation is done using R Development Core Team (2015). Upper specification limits of y values corresponding to each level of explanatory variable (xi) are presented in Table 9. The calculated values of mean and standard deviation at each level of explanatory variable are shown in Table 10.

Simulation gave us the estimated values of indices as $C ^ p u A 1$ = 1.1646 and $C ^ p u A 2$ = 1.3410. The value of m = 1 is chosen. The estimated value of ratio of two process yield indices is obtained as

$R ^ = C ^ p u A 2 C ^ p u A 1 = 1.3410 1.1646 = 1.1515$

The two quality levels of process II are fixed as $C ^ p u A 2 , A Q L = 1.33 and C ^ p u A 2 , L T P D = 1$. Choose α level as 0.05 and β level as 0.10, the acceptance value from table 1 can be found as 1.1063. AsThe two quality levels of process II are fixed as 2 ˆ 1.33 AQL CpuA = and 2 ˆ 1 LTPD C puA = . Choose α level as 0.05 and β level as 0.10, the acceptance value from table 1 can be found as 1.1063. As

$1.1515 > 1.1063$

The supplier II has declared to be more capable as compared to supplier I.

## 6.CONCLUSION

We developed a MDSR sampling plan based on the ratio of two process yield indices for accepting the lot from a more capable process by comparing the two suppliers’ processes for linear profiles. Process yield index CpuA is used to measure the capability of both processes, where the smaller quality characteristic of two processes is more desirable. Competition between two suppliers will enhance the quality of final supply of a company. The optimal values of the proposed MDSR plan parameters are calculated through the R-code.

More efforts are needed for the development of the acceptance sampling plans to compare more than two suppliers. Moreover, nonlinear profiles, polynomial regression and multiple profiles should be taken in account for evaluation.

## ACKNOWLEDGEMENTS

The authors are deeply thankful to reviewers and editor for their valuable suggestions to improve the quality of paper. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, Muhammad Aslam, therefore, acknowledge with thanks DSR technical and financial support.

## Figure

The power curves of sampling plans based on ratio of CpuA1, CpuA2 under k = 2, CpuA2, AQL=1.33, CpuA2, LTPD =1, m = 1, α = 0.05, β = 0.10.

The power curves of sampling plans based on ratio of CpuA1, CpuA2 under k = 4, CpuA2, AQL=1.33, CpuA2, LTPD =1, m = 1, α = 0.05, β = 0.10.

## Table

Parameters of Single Sampling Plan based on Ratio of two process yield indices under CpuA2AQL=1.33,CpuA2LTPD=1

Parameters of Single Sampling Plan based on ratio of two process yield indices under CpuA2AQL=1.5,CpuA2LTPD=1.33

Parameters of MDSR plan based on ratio of two process yield indices under m=1,CpuA2AQL=1.33,CpuA2LTPD=1

Parameters of MDSR plan based on ratio of two process yield indices under m=2,CpuA2AQL=1.33,CpuA2LTPD=1

Parameters of MDSR plan based on ratio of two process yield indices under m=1,CpuA2AQL=1.5,CpuA2LTPD=1.33

Parameters of MDSR plan based on ratio of two process yield indices under m=2,CpuA2AQL=1.5,CpuA2LTPD=1.33

Comparison of sample sizes for sampling plans when CpuA2AQL=1.33,CpuA2LTPD=1

Comparison of sample sizes for sampling plans when CpuA2AQL=1.5,CpuA2LTPD=1.33

The specifications limits

The basic statistics

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