1. INTRODUCTION
In the current time of global sourcing, highly competitive, and interconnected manufacturing environment supplier selection is a critical aspect for any business. It is the process by which firms recognize, assess, and contract with suppliers. The choice of suppliers and the assessment of their capability are among the severe difficulties faced by production managers. There is no single best way to solve the supplier selection problem; therefore, manufacturers apply a variety of approaches to the selection process. Commonly, they assess potential suppliers by employing their selection criteria with assigned relative weights. To solve these difficulties, various studies attempted to furnish practical and capable procedures to meet the manufacturer’s desires. For example, Wang and Tamirat (2016a) presented a case of a computer manufacturer. To improve its production capacities, the computer manufacturer needed to find a new computer cooling fan supplier who offers higher quality than the current supplier. Owing to the nature of the product, both the competitor’s and the existing supplier’s processes possess extremely low fraction defectives; thus, for the decision making the standard selection methods required a considerable sam ple size. The authors utilized the process capability index (PCI) and implemented the ratio test statistic to compare the two suppliers. The result indicated a substantial sample size reduction. However, the case is restricted to two suppliers; in practice, more than two suppliers competing for an order are quite standard. Moreover, evaluating a group of suppliers and choosing at least one of them is difficult (Lin et al., 2018).
Lately, process yield has gained significant importance in the supplier selection processes due to factors such as zero defect, globalization, increased value addition, and rapid technological advances. Process yield evaluates the connection between the design specifications and the natural variability of the process. In this way, both buyers and producers can measure the quality of a product using a single unitless measure. As a result, it has become an essential factor that should be considered in supplier selection (Ebadi and Shahriari, 2013). For instance, Pearn and Wu (2013a) argued that looking at multiple suppliers and selecting the ones that have a fundamentally better process yield is one of the significant factors of supplier selection. PCI can be utilized when sourcing supplies, raw materials, and components for the organization (Pearn et al., 2018).
In the literature, for single or multiple quality characteristics, studies related to supplier evaluation and selection using process yield indices are increasingly common. For instance, Linn et al. (2006) proposed a price comparison chart that integrates the PCI and price information for multiple suppliers, Polansky (2006) utilized a permutation test when there are two or more suppliers, a bootstrap method is employed by Wu et al. (2008), and a group selection among multiple suppliers using the ratio test statistic by Lin and Pearn (2011). For the ratio test statistic, an example of supplier selection on TFTLCD manufacturing processes was provided by Pearn and Wu (2013b). Besides, utilizing multiple comparisons with the best (MCB) based on the yield index, Lin and Kuo (2014) performed a simulation study and found that MCB is more powerful when the number of suppliers is large. Recently, Lin et al. (2018) proposed a subtraction statistics approach with Bonferroni correction to control the overall error rate. Furthermore, Pearn et al. (2018) found that, for the subtraction statistics method, the power increased when the number of suppliers increased. Additional studies addressing the supplier selection problem utilizing PCI can be found in the literature (Hubele et al., 2005;Lin and Pearn, 2010;Wu et al., 2013).
Owing to technological advances, profile monitoring has attracted the attention of many researchers (Kang and Albin, 2000;Tamirat and Wang, 2019;Negash et al., 2020). For instance, in the control chart applications, Adibi et al. (2014) provided a Pvalue method for online monitoring of linear profiles. Zeng and Chen (2015) present the Bayesian hierarchical approach. Moreover, the number of researchers working on process capability analysis for profile data has been increasing recently. For instance, studies on linear profiles, and multivariate profiles can be found in (Ebadi and Amiri, 2012;Ebadi and Shahriari, 2013;Hosseinifard and Abbasi, 2012). For the quality characteristics described by linear profiles and with very low fraction defectives (PPM), the usual supplier selection strategies are challenging to implement because a sample of reasonable size most likely contains zero defective items. Thus, PCIs have been widely used instead. As an example, in the case of two suppliers, Wang and Tamirat (2016a) applied ratio test statistics to solve supplier selection problems for linear profiles, while (Wang, 2016) implemented the ratio and difference test statistics and compared two suppliers.
Furthermore, Wang and Tamirat (2016b) applied the MCB method for linear profiles. However, to the best of our knowledge, there has been no research attempt to implement ratio test statistics to compare multiple suppliers when linear profiles describe the quality characteristic. This study investigated the statistical properties for ratio test statistics and provided easy to implement procedures based on the ratio and MCB methods for supplier selection; a simulation study is conducted to compare the power and sample size requirements between the two methods.
The structure of this study is as follows: in section 2, linear profiles and the process yield index is reviewed. Ratio test statistics are introduced in section 3. In sections 4 and 5, selection procedures that are easy to understand and implement are provided. A simulation study with 100,000 replications is performed to compare the power and the sample size requirement in section 5. In section 6, we cooperated with a laptop computer producer and provided an illustrative example. Conclusions and future directions are provided in section 7.
2. LINEAR PROFILES AND PROCESS YIELD INDEX
In many practical applications, the quality characteristics are described by a linear profile, which is a linear relationship between the response variable or the desired quality characteristics and one or more explanatory variables or the controllable inputs. A process with multiple linear profiles is modeled as follows:
where ${\beta}_{0},\hspace{0.17em}{\beta}_{1},\hspace{0.17em}\mathrm{...},\hspace{0.17em}{\beta}_{p}$ are coefficients of the incontrol profiles, ${x}_{i1,}\hspace{0.17em}{x}_{i2,}\hspace{0.17em}\mathrm{...},\hspace{0.17em}{x}_{ip}$ are the i^{th} level of the predictor variables, m represents that there are m levels of each predictor, n is the sample size, and ${\epsilon}_{ij}s\sim N(0,\hspace{0.17em}{\sigma}^{2})$. When p =1 the multiple profiles became a simple linear model.
A process yield index for simple linear profiles proposed by (Wang, 2014) is given by:
where $P=\frac{1}{m}{\displaystyle \sum _{i=1}^{m}{p}_{i}},{p}_{i}=\left[\Phi \left(\frac{US{L}_{i}{\mu}_{i}}{{\sigma}_{i}}\right)\Phi \left(\frac{LS{L}_{i}{\mu}_{i}}{{\sigma}_{i}}\right)\right],$, Φ(⋅) is the cumulative distribution function (CDF) of the standard normal distribution, Φ^{−1}(⋅) is the inverse of Φ(⋅) , USL_{i} and LSL_{i} are the upper and lower specification limits of the response variable at the i^{th} level of the explanatory variable, μ_{i} and ${\sigma}_{i}^{2}$ are the mean and variance of the i^{th} level of the explanatory variable.
However, p_{i} needs to be estimated from a random sample; hence, the estimator ${\widehat{p}}_{i}$ is applied.
where x_{i} and S_{i} are the sample mean and standard deviation.
To estimate the PCI S_{pkA}, we consider the estimator S_{pkA}, the simple form of the asymptotic normal distribution of the estimator S_{pkA} proposed by (Wang, 2014) is given in Equation (4).
where ${\widehat{S}}_{pki}=\frac{1}{3}{\Phi}^{1}\left[\frac{1}{2}(1+{\widehat{p}}_{i})\right],$ and $G=\frac{1}{3}{\Phi}^{1}\left[\frac{m\left[2\Phi \left(3{S}_{pkA}\right)1\right](m2)}{2}\right]$
3. APPLYING RATIO TEST STATISTICS TO COMPARE MULTIPLE SUPPLIERS
For two suppliers, the ratio test statistic (R) was proposed by (Wang and Tamirat, 2016a) to test the hypotheses given in Equation (5). However, the case is restricted to two suppliers; in practice, more than two suppliers competing for an order are quite standard.
The ratio statistics can be adapted to compare k suppliers where k > 2 following the steps given below.

Apply Equation (4) and determine the process yield index for all suppliers, where ${\widehat{S}}_{pkA[i]}$ denotes the result from the i^{th} supplier.

Rank all the indices in ascending order, where ${\widehat{S}}_{pkA[i]}\le {\widehat{S}}_{pkA[k]}$.

Compute k −1 ratios, ${R}_{i}={\widehat{S}}_{pkA[k]}/{\widehat{S}}_{pkA[i]}$ where ${\widehat{S}}_{pkA[k]}$ is the largest yield index and ${\widehat{S}}_{pkA[i]}$ represents the i^{th} supplier.

For the minimum yield index requirement c , determine cα , where cα is a critical value and can be found as shown in section 3.2.

Make a decision: with a given significance level α, if ${R}_{i}={\widehat{S}}_{pkA[k]}/{\widehat{S}}_{pkA[i]\hspace{0.17em}}\ge {c}_{\alpha},$, reject H_{0} and conclude that there is no adequate information to decide that the i^{th} supplier is among the best.
3.1 Statistical Properties
For the case of only two suppliers, Wang and Tamirat (2016a) apply the Jacobian transformation technique and provide the distribution of R. In the same manner, to compare k suppliers where k > 2, the asymptotic distribution of R_{i} can be found as shown below (detail not included here):
where m_{i} and m_{k} represent the numbers of levels, n_{i} and n_{k} are the numbers of profiles
Hence, to implement the ratio test statistics R_{i} for the comparison of multiple suppliers, there is a need to determine and compare k −1 terms of R_{i}, for example, when there exist five suppliers and the respective yield indices are given as: ${\widehat{S}}_{pkA[1]}$=1.33, [2] ${\widehat{S}}_{pkA[2]}$=1.00, [3] ${\widehat{S}}_{pkA[3]}$=1.50, ${\widehat{S}}_{pkA[4]}$=1.20, ${\widehat{S}}_{pkA[5]}$=1.40. That is, supplier 3 has the largest yield index and the ratios can be calculated as: R_{1}=1.50/1.33, R_{2}=1.50/1.00, R_{4}=1.50/1.20=1.25, and R_{5} = 1.50/1.40 = 1.07. Applying Equation (7) the probability density functions (PDF) of ${f}_{{R}_{1}}(r),\hspace{0.17em}{f}_{{R}_{2}}(r),{f}_{{R}_{4}}(r),\hspace{0.17em}{f}_{{R}_{5}}(r)$ are illustrated by Figure 1 for m_{1} = m_{2} = 6 and n = n_{1} = n_{2} = 40, 50, 100, 150, and 200. From Figure 1, the distribution of R_{i} has a single mode and is symmetric; increasing the number of profiles reduces the variance of R_{i}, so the larger the value of R_{i} the higher the variance and vice versa. The mean and variance of R_{i} can also be obtained using Equation (8).
3.2 Critical Values Determination
Implementing R_{i} requires k −1 tests to determine the best suppliers; however, multiple tests can result in error inflation. Lin and Pearn (2011) suggested that the Bonferroni correction is a practical approach to handle multiple comparison problems; it is applied as an adjustment to control the overall error rate. Assume a total of k −1 tests and an event of wrongly rejecting the i^{th} test is given as E_{i}. By the Bonferroni inequality, controlling each test at a significant level of α/k−1 results in the probability of falsely rejecting any test less than or equal to α.
Hence, not all pairwise comparisons are of interest. Instead, a total of k −1 tests are required. Thus, at a significance level α the critical value c_{α} can be calculated by solving the following equation:
When ${S}_{pkA[1]}={S}_{pkA[2]}=\mathrm{...}={S}_{pkA[k]}=c$, all suppliers are considered equivalent and the value of R_{i} =1 is assumed as the smallest possible value, that is, the selection procedure resulted in the highest typeI error, which is regarded as the least favorable configuration (LFC). Thus, the critical values are determined at the LFC and obtained by solving Equation (11).
For k = 3, 4, 5 the critical values of R_{i} can be found in Tables 13 for fixed values of m_{i} = m_{k} = 4, 6,10 , and n_{i} = n_{k} = 30(10)200 at α =0.05 . To determine the critical values, a computer program was written using R language (R Core Team, 2018). The program can be applied for any number of suppliers and is available from the authors. In practical applications, the decisionmakers might be interested in two different cases:
Case 1: The goal is to select one or more best among competing suppliers; best is defined as a higher yield index at a desired significance level. The critical values are determined by solving Equation (11).
Decision rule: with a given significance level α, if ${R}_{i}={\widehat{S}}_{pkA[k]}/{\widehat{S}}_{pkA[i]\hspace{0.17em}}\ge {c}_{\alpha}$, reject H_{0} and conclude that there is not adequate information to decide if the i^{th} supplier is among the best. For example, if the minimum yield index required is c = 1.33, at α = 0.05 with m_{i} = m_{k} = 6 and n_{i} = n_{k} = 50, from Table 1 and Table 2 it can be found that c_{α} = 1.3160, c_{α} = 1.3547 for three and four suppliers respectively.
Case 2: A supplier is considered best if and only if its yield index outperforms the remaining ones by at least a designated value of h > 0 . Hence, the hypothesis needs to be modified to test H_{0} : ${H}_{0}:{S}_{pkA[k]}\le {S}_{pkA[i]}+h\hspace{0.17em}vs.\hspace{0.17em}$ ${H}_{a}:{S}_{pkA[k]}>{S}_{pkA[i]}+h$S. Thus, at a given significance level α the critical value c_{h} can be calculated by solving Equation (12).
Decision rule: with a given significance level α, if R_{i} ≥ c_{h}, reject H0 and conclude that there is not adequate information to decide if the i^{th} supplier outperforms the remaining ones by a magnitude of h. Similar to the case 1, the critical values are determined at the LFC and obtained by solving Equation (13).
The critical values are presented in Tables 45 at α = 0.05 with the specified values of h = 0.1(0.1)0.4, m_{i} = m_{k} = 4, 6, 10 and n_{i} = n_{k} = 40(20)200. For instance, from Table 5 for five suppliers, if c =1.50 is the minimum yield index requirement, at α =0.05 with m =10, n =100 and h =0.2, it could be find that c_{h} = 1.4259.
3.3 The Number of Profiles Required for a Specified Power
In the previous section, the proposed procedure depends entirely on incorrect rejections of the true null hypotheses (type I error). That is, the method ensures that the probability that the right supplier is considered an error is maintained below α. The power of a test is given as 1−β, where β is the likelihood of failing to reject a false H_{0} (typeII error). To uphold α and decrease β simultaneously, increasing the number of profiles is critical. For a fixed power and significance level, the minimum number of profiles required can be obtained by solving the constraints in Equations (14) and (15) simultaneously.
A search algorithm is applied, and programs are written in R (R Core Team, 2018) to solve the simultaneous Equations, and Table 6 tabulates the required number of profiles for three power values of 0.90, 0.95, and 0.99. For example, if there exist four suppliers to reach the power of 0.95 if the minimum yield index required is c =1.33, m_{w}=10, and h = 0.2 , the required number of profiles for testing becomes 533.
4. MULTIPLE COMPARISONS WITH THE BEST (MCB)
Multiple comparisons with the best method proposed by (Wang and Tamirat, 2016b) can be summarized as follows.

Apply Equation (4) and determine the process yield index for all suppliers, where ${\widehat{S}}_{pkA[i]}$ denotes the result from the i^{th} supplier.

Identify the supplier with the largest yield index ${\widehat{S}}_{pkA[k]}$.

For the minimum yield index requirement C, construct a reference subset S.
$$S=\left\{i:{S}_{pkA[i]}\ge {S}_{pkA[k]}{h}_{\alpha ,k}\sqrt{\frac{{G}_{i}^{2}{\left[\varphi \left(3{G}_{i}\right)\right]}^{2}}{2{m}_{i}{}^{2}{n}_{i}{\left[\varphi \left(3C\right)\right]}^{2}}},\hspace{0.17em}1\le i\le k\right\}$$(16)The critical value h_{α,k} controls the overall confidence level of at least 1−α, and it can be found from:

Compare each supplier yield index with the supplier or suppliers in S. The simultaneous confidence intervals of at least 1−α become:
$$\begin{array}{l}LC{B}_{i}\le {S}_{pkA[i]}\underset{m=1,2,\dots ,k}{\mathrm{max}}{S}_{pkA[m]}\le UC{B}_{i},\\ \text{for}\hspace{0.17em}\hspace{0.17em}i=\text{1},\text{2},\text{}\dots ,\text{k}\end{array}$$(18)where
$$LC{B}_{i}=\text{min}\left(0,\hspace{0.17em}\underset{m\in S}{\text{max}}LC{B}_{i}^{m}\right)$$(19)$$UC{B}_{i}=\text{min}\left(0,\hspace{0.17em}\underset{m\ne i}{\text{min}}UC{B}_{i}^{m}\right)$$(20) 
Decision rule: if LCB_{i} = 0, i is among the best suppliers with a large yield index at a significance level of α. Otherwise, it is the inferior supplier if LCB_{l} < 0 .
5. COMPARISON STUDY OF RATIO VS. MULTIPLE COMPARISONS WITH THE BEST
A simulation study with 100,000 replications is conducted to compare the ratio test statistic and the MCB methods. Simple linear profiles shown in Equation (22) are employed for data generation purposes, and the levels and specification limits are given in Table 7. Two cases are considered for comparison.
5.1 Power Comparison at the LFC Setting
That is, for the ratio test statistics, all suppliers possess the same yield index value, ${S}_{pkA[1]}={S}_{pkA[2]}=\mathrm{...}={S}_{pkA[k]}=c$. Figure 2 illustrates the LFC setting for the MCB method, and the setting has the lowest power at $\lceil k/2\rceil $ of the suppliers are superior to the rest. The power is determined by dividing the successful runs by the number of replications, 100,000 in this case. A run is labeled successful when the following conditions are satisfied.

For MCB, the lower confidence bound given by Equation (19) assumes a value of zero for the best supplier, while it becomes negative for the inferior ones, for instance, for a given h=0.15 and four suppliers with the respective yield index S_{pkA} =1.0, 1.1, 1.3, and 1.4. A successful run should result in LCB_{4} and LCB_{3} as zero, and LCB_{1}, LCB_{2}, less than zero.

For the ratio, a run is successful if all suppliers are selected as the best, for instance for a give h = 0.3 and S_{pkA}=1.15, 1.2, 1.25, 1.3, and 1.4. R_{1}, R_{2}, R_{3}, R_{4} should be less than c_{α} =1.4930(see Table 5) for a successful run.
The highest yield index is set to be 1.5 and S_{pkA[i]} of the secondbest ranges between 1.1(0.025)1.5. Four combinations were analyzed: (1) for K = 3, (S_{pkA[i]} − 0.10, S_{pkA[i]},1.50), (2) for K = 4, (S_{pkA[i]} − 0.20, S_{pkA[i]} −0.10, S_{pkA[i]},1.50) , (3) for K = 5, (S_{pkA[i]} − 0.30, S_{pkA[i]} −0.20, S_{pkA[i]} −0.10, S_{pkA[i]},1.50), and (4) for K = 6, (S_{pkA[i]} −0.40, S_{pkA[i]} − 0.30, S_{pkA[i]} − 0.20, S_{pkA[i]} − 0.10, S_{pkA[i]},1.50).
Figure 3 illustrates that, at a given level and number of profiles, the MCB method possesses higher power. When the number of suppliers increases, the difference in power gets larger and larger. For instance, when k = 3 and the yield indices (1.15, 1.25, 1.5), the power for the MCB and ratio became 0.94 and 0.75 respectively. If the number of suppliers increases to k = 6, (0.85, 0.95, 1.05, 1.15, 1.25, 1.5), the power reduces to 0.83 and 0.32 respectively, by 0.11 for MCB and 0.43 for the ratio.
Furthermore, the simulation study is performed to find the effects of the number of levels of the explanatory variable and the number of profiles. The result indicates that the higher the levels, the higher the power. The result also shows that the higher the number of profiles, the higher the power. For instance, with three suppliers, increasing the number of levels from 4 to 10 improves the power from 0.69 to 0.94 for the MCB. Hence, it can be concluded that both the number of levels and the number of profiles have a substantial impact on the power of the statistical test.
5.2 Power when Having K_{NB} Best Suppliers
For example, when there are four suppliers (K=4), the scenarios considered are: (1) one best supplier (K_{NB} = 1) and three inferior suppliers, (2) two best suppliers (K_{NB} = 2) and two inferior suppliers, and (3) three best suppliers (K_{NB} = 3) and one inferior supplier. The results are summarized in Table 8 and, once again, from this study it can be concluded that MCB is more powerful than the ratio method to deal with a supplier selection problem.
6. ILLUSTRATIVE EXAMPLES
For illustration purposes, we cooperated with a laptop computer producer, referred to for privacy purposes after this as company A. Company A has four suppliers for one of its key components, a central processing unit cooling fan. The quality characteristic of interest is the relationship between the input voltage and speed measured as revolutions per minute. As shown in Table 11, four input levels are considered (m = 4) 2.2V, 2.5V, 4.0V, and 5.0V. The upper and lower specification limits at the four levels are shown in Table 11. From seven randomly selected lots delivered by the suppliers, 120 profiles are collected from each lot, the mean, standard deviation, and the corresponding yield indices are calculated, and the results are summarized in Table 10. Applying MCB from Table 11, the lower confidence bounds for suppliers 1, 3, and 4 are found to be less than zero, indicating inferior suppliers. For the ratio method from Table 2, the critical value for the ratio test method is found to be c_{α} =1.1972 ; compared with the values in Table 11, the ratio method deems only supplier 3 as inferior. Thus, MCB is more conservative.
7. CONCLUSION
In this study, the ratio test statistic and process yield indices are applied to solve the supplier selection problem among multiple (k > 2) suppliers. Decision procedures and rules are provided. The proposed methods allow comparison of multiple suppliers, adopting a yield index S_{pkA}. The Bonferroni correction method is implemented to avoid error inflation due to multiple testing. The result provides useful information to manufacturers in making the best possible decision when selecting among more than two suppliers with superior process capability. Furthermore, for the ratio test statistics, statistical properties are investigated. An illustrative example is included to demonstrate the applicability of the method. For practitioners, the critical value tables were presented at different minimal capability requirements. For a comprehensive decision, the study also provided the number of profiles required for given powers and a significant level. Besides, a simulation study with 100,000 replications is conducted to compare the ratio test statistic and the MCB methods. The result suggests that the MCB method is more powerful. A real data set is collected from a computer producer to illustrate the applicability of the methods. The study can be extended to include more complex profiles such as polynomial or nonlinear structure; the study can also be expanded to other multiple comparison methods such as the difference statistics.