Applying a Hybrid MCDM Model for Six Sigma Project Selection

Abstract

Six Sigma is a project-driven methodology; the projects that provide the maximum financial benefits and other impacts to the organization must be prioritized. Project selection (PS) is a type of multiple criteria decision making (MCDM) problem. In this study, we present a hybrid MCDM model combining the decision-making trial and evaluation laboratory (DEMATEL) technique, analytic network process (ANP), and the VIKOR method to evaluate and improve Six Sigma projects for reducing performance gaps in each criterion and dimension. We consider the film printing industry of Taiwan as an empirical case. The results show that our study not only can use the best project selection, but can also be used to analyze the gaps between existing performance values and aspiration levels for improving the gaps in each dimension and criterion based on the influential network relation map.

1. Introduction

Six Sigma is a business strategy that seeks to identify and eliminate the causes of errors or defects, which are defined as any factors that could lead to customer dissatisfaction or failures in business processes, by focusing on outputs that are critical to customers [1–3]. Six Sigma has been widely implemented in the manufacturing sector to reduce product costs, improve quality, shorten delivery times, and increase customer satisfaction. Moreover, the Six Sigma movement is also gaining acceptance in healthcare, marketing, engineering, and financial and legal service organizations in addition to achieving major benefits in the manufacturing sector [4].

Project selection (PS) plays a vital role in the effective introduction and implementation of Six Sigma [5]. Pande et al. [6] suggested the following mantra for PS: meaningful and manageable. However, this then raises the question of how to create a meaningful and manageable PS, which will become the key issue facing the organization when conducting Six Sigma projects. This question is the essential reason for our focus on PS in Six Sigma. Also, PS is a type of multiple-criteria decision-making (MCDM) problem. Banuelas et al. [7] reported that several approaches including cost-benefit analysis, cause and effect matrices, Pareto analysis, priority indices, analytic hierarchy process (AHP), the theory of constraints (TOC), and quality function deployment (QFD) have been applied to the selection of Six Sigma projects by various organizations in the UK. Other methods such as the Balanced Scorecard method [8], data envelopment analysis (DEA) [9], a multiobjective model [10], a fuzzy-AHP with goal programming approach [11], and a hierarchical criterion evaluation process based on national quality award criteria [12] were used to select Six Sigma projects. Recently, Büyüközkan and Öztürkcan [13] developed a novel approach based on decision-making trial and evaluation laboratory (DEMATEL) technique and an analytic network process (ANP) to help logistics companies to identify and prioritize Six Sigma projects. Perçin and Kahraman [14] integrated three methods including a modified Delphi method, an AHP, and a fuzzy technique for the selection of Six Sigma projects. Tkáç and Lyócsa [15] proposed a new model based on a real options approach for evaluating Six Sigma projects, which involves the stochastic nature of project outcomes, costs, and uncertainty regarding payoffs and managerial options. Vinodh et al. [16] and Boran et al. [17] applied a fuzzy-ANP method for the selection of agile concept in a manufacturing company. Padhy and Sahu [18] proposed a two-stage methodology based on (i) real options analysis for evaluating the value of the project to improve the managerial flexibility and (ii) a zero-one integer linear programming model for selecting and scheduling an optimal project portfolio. Saghaei and Didehkhani [19] applied a fuzzy-weighted additive goal programming model for the evaluation and selection of Six Sigma projects. Unfortunately, these researches only provide the prioritization of Six Sigma projects. To evaluate the difference among Six Sigma projects is an important task for quality improvement. Thus, a new approach for Six Sigma project selection is needed.

We present a hybrid MCDM model to assess performance not only in ranking and selection, but also in improving and making Six Sigma project strategies for reducing gaps of each dimension/criterion to be perfect, and promote Six Sigma projects for satisfying the user needs to be number one. These processes not only can help Six Sigma project managers to understand users’ wants and needs, but also can assist them with how to improve Six Sigma projects to reduce performance gaps for achieving aspiration level (see the appendix) of user satisfaction by building effective Six Sigma project strategies. An empirical study of film printing industry of Taiwan as an empirical Six Sigma projects case is illustrated to show the hybrid MCDM model for selection and improvement.

The remainder of this paper is organized as follows. Section 2 presents the PS criteria in Six Sigma. A hybrid MCDM model is developed in Section 3. Section 4 highlights the managerial implications generated by the case analysis. Finally, we offer a conclusion.

2. Research Problem

A project is defined as a unique set of coordinated activities, with defined starting and finishing points, which is undertaken by an individual or an organization to meet specific performance objectives within a defined schedule and within defined cost and performance parameters [20]. The objectives of a project must be clear, succinct, achievable, realistic, and measurable, with a high probability of success [6, 21]. Antony and Fergusson [21] described how the project selection process may begin by creating a simple customer expectations-process matrix that focuses on critical business performance characteristics including quality, cost, delivery, and responsiveness.

Projects must be aligned with a strategic business plan and with organizational goals. Kendrick and Saaty [8] illustrated a set of 17 project alternatives for a manufacturing company, using AHP to determine the priorities of these projects. Project alternatives can be suggested by a bottom-up process, such as by Black or Green Belts, which are familiar with operational problems and flaws, or by a top-down process, when senior managers raise strategic issues [22]. We modified the AHP developed by Kenderick and Saaty [8] to establish a framework for project selection that includes the dimensions of feasibility, impact on customers, impact on finance, impact on operations, and impact on employees and that aligns with the strategy of the company. There are four levels of hierarchy, as well as different dimensions and criteria (see Table 1 and Figure 1).

Table 1 

The influence dimensions and criteria of comprehensive PS in Six Sigma.

Figure 1 

The network of project selection in Six Sigma.

We consider Corporation J, which has dedicated its efforts since 1993 to the development of soft packaging material for electronics and food packaging in Taiwan. Corporation J is one of the largest soft packaging material (SPM) manufacturers in Taiwan. The products of Corporation J include polycello and aluminum; in particular, it produces easy-peel top film and heat-sealable aluminum foil for food packaging and antistatic pouches for electronic components packaging. The quality of its products is very important to the food and electronics industries. The SPM manufacturers in Taiwan are promoting Six Sigma activities at this time. PS is the key management action required for the success of Six Sigma. There are eight alternative projects (see Table 2) that could be used to improve the manufacturing processes of Corporation J. We denote the above 6 dimensions, 17 criteria, and 8 alternatives to form a hierarchy for PS in Six Sigma as shown in Figure 1. In order to avoid “select the best among inferior projects/alternatives (i.e., pick the best apple among a barrel of rotten apples),” we compared the performance of projects and applied methods to improve the performance of dimensions and criteria in reducing the gaps remaining to achieve the aspiration levels in this study.

Table 2 

The alternatives of PS in Six Sigma.

To assess the interinfluence of the PS criteria for the DEMATEL technique calculation, we designed a questionnaire to collect data from experts in the SPM industry and consultants for Six Sigma. These sixteen knowledge-based experts were the corporation general manager, the plant assistant general manager, the R&D manager, the purchasing manager, the vice plant manager, and the section managers.

3. A Hybrid MCDM Model

A hybrid MCDM model combines the DEMATEL technique [23] with the basic concept of ANP [24] to find the influential weights of DANP and the “VIseKriterijumska Optimizacija I Kompromisno Resenje” (VIKOR, which translates as “Multi-criteria Optimization and Compromise Solution”) method [25] for evaluating the gaps of performance in each criterion and dimension. The DEMATEL technique was designed to determine the degrees of influential relationship matrix of the PS criteria and apply them to build supermatrix (including the unweighted and weighted supermatrices) in the basic concept of ANP. The ANP handles dependence within a dimension (inner dependence) and among different dimensions (outer dependence). The ANP is a nonlinear structure, whereas the AHP is hierarchical and linear, with goals at the top and alternatives at lower levels [26]. A hybrid MCDM model combining DEMATEL with the basic concept of ANP (referred to as DANP) has been widely applied in various fields for finding influential weights, such as innovation policy portfolios for Taiwan’s silicon/semiconductor intellectual property mall, airline safety measurements, e-learning evaluations, and exploring stock selection [27–33]. Using the VIKOR method to rank and improve the prioritization of dimensions/criteria can be found in [34–38].

A hybrid MCDM model contains three main phases (see Figure 2): (1) constructing the influential network relation map (INRM) among the criteria by the DEMATEL technique, (2) calculating the influential weights of each criterion by using the basic concept of ANP based on the total-influence matrix by the DEMATEL technique, and (3) ranking and improving the prioritization of dimensions/criteria or projects through the VIKOR method for reducing gaps in each criterion and dimension to achieve the aspiration levels.

Figure 2 

The process of a hybrid MCDM model combining DANP and VIKOR.

3.1. Phase I

The three steps of DEMATEL and INRM are summarized as follows.

Step 1. Calculate the direct-influence matrix 𝐀 by scores. An assessment of the relationship between each criterion of mutual influence is made according to the opinions of knowledge-based experts in Six Sigma and maganers in the printing firms of Taiwan in real situation, using a scale ranging from 0 to 4, with scores represented by natural language: “absolutely no influence (0),” “low influence (1),” “medium influence (2),” “high influence (3),” and “very high influence (4).” The knowledge-based experts are required to indicate the direct-influence by a pairwise comparision, and if they believe that criterion 𝑖 has an effect/influence on criterion 𝑗, they should indicate this by 𝑎_𝑖𝑗. Thus, average matrix 𝐀=[𝑎_𝑖𝑗]_𝑛×𝑛 of direct relationships can be obtained as𝑎𝐀=¹¹⋯𝑎_1𝑗⋯𝑎_1𝑛𝑎⋮⋮⋮_𝑖1⋯𝑎_𝑖𝑗⋯𝑎_𝑖𝑛𝑎⋮⋮⋮_𝑛1⋯𝑎_𝑛𝑗⋯𝑎_𝑛𝑛. (1)

Step 2. Calculate the normalized direct-influence matrix 𝐗. The normalized direct-influence matrix 𝐗 is derived by 𝐗=𝑧×𝐀, (2) where 𝑧=min_𝑖,𝑗{1/max_𝑖∑^𝑛_𝑗=1𝑎_𝑖𝑗,1/max_𝑗∑^𝑛_𝑖=1𝑎_𝑖𝑗}, for all 𝑖,𝑗=(1,2,…,𝑛). Its diagonal is zero, and the maximum sum of rows or columns is one.

Step 3. Derive the total-influence matrix 𝐓. The continuous decrease in the indirect effects of problems can be determined using the powers of 𝐗, for example, 𝐗²,𝐗³,…,𝐗^ℎ, with lim_ℎ→0𝐗^ℎ=[0]_𝑛×𝑛, where 𝐗=[𝑥_𝑖𝑗]_𝑛×𝑛, 0≤𝑥_𝑖𝑗<1, 0≤∑_𝑖𝑥_𝑖𝑗≤1, 0≤∑_𝑗𝑥_𝑖𝑗≤1, and at least one column or one row of summation, but not every column or row, equals one; then lim_ℓ→∞𝐗^ℓ=[0]_𝑛×𝑛 can be guaranteed. Thus, the total-influence matrix 𝐓 is derived by 𝐓=𝐗+𝐗²+⋯+𝐗^ℎ=𝐗(𝐈−𝐗)⁻¹, (3) where lim_ℎ→0𝐗^ℎ=[0]_𝑛×𝑛 and 𝐈denotes the identity matrix.
The sum of the rows and the sum of the columns of 𝐓 are expressed separately as vector 𝐫 and vector 𝐬. That is, we have 𝐫=𝑟_𝑖𝑛×1=^𝑛_𝑗=1𝑡_{𝑖𝑗𝑛×1},𝐬=𝑠_𝑗𝑛×1=^𝑛_𝑖=1𝑡_𝑖𝑗^′_1×𝑛, (4) where the superscript ′ denotes the transpose. Also, 𝑟_𝑖 shows the sum of the direct and indirect effects of criteria 𝑖 on the other criteria and 𝑠_𝑗 shows the sum of the direct and indirect effects that criteria 𝑗 has received from the other criteria. Furthermore, (𝑟_𝑖+𝑠_𝑖) shows the degree that the criteria 𝑖 plays in the problem, and the difference (𝑟_𝑖−𝑠_𝑖) shows the net effect that criteria 𝑖 contributes to the problem. If (𝑟_𝑖−𝑠_𝑖) is positive, then criteria 𝑖 is affecting other criteria, and if (𝑟_𝑖−𝑠_𝑖) is negative, then criteria 𝑖 is being influenced by other criteria. These results are used to generate the INRM. The INRM can provide ideas for improvement.

3.2. Phase II

The five steps of DANP influential weights are described as follows.

Step 1. Find the normalized matrix 𝐓^𝛼_𝑐 by using dimensions. After normalizing the total-influence matrix 𝐓_𝑐 by dimensions, we can obtain a new normalized matrix 𝐓^𝛼_𝑐 by dimensions and is given by𝐓^𝛼_𝑐=𝐷₁𝐷_𝑗𝐷_𝑛𝑐₁₁⋯𝑐_1𝑚1𝑐_𝑗1⋯𝑐_𝑗𝑚𝑗⋯𝑐_𝑛1⋯𝑐_𝑛𝑚𝑛𝐷₁𝑐₁₁𝑐₁₂⋮𝑐_1𝑚1𝐷⋮⋮_𝑖𝑐_𝑖1𝑐_𝑖2⋮𝑐_𝑖𝑚𝑖𝐷⋮⋮_𝑛𝑐_𝑛1𝑐_𝑛2⋮𝑐_𝑛𝑚𝑛𝐓_𝑐^𝛼11⋯𝐓_𝑐^𝛼1𝑗⋯𝐓_𝑐^𝛼1𝑛𝐓⋮⋮⋮_{𝑐𝛼𝑖1}⋯𝐓_𝑐^𝛼𝑖𝑗⋯𝐓_𝑐^𝛼𝑖𝑛𝐓⋮⋮⋮_{𝑐𝛼𝑛1}⋯𝐓_𝑐^𝛼𝑛𝑗⋯𝐓_𝑐^𝛼𝑛𝑛. (5)For instance the normalization 𝐓_𝑐^𝛼11 is shown as (5) and (6).
Consider the following:𝑑¹¹_𝑐𝑖=^𝑚1_𝑗=1𝑡¹¹_𝑐𝑖𝑗,𝑖=1,2,…,𝑚₁,𝐓_𝑐^𝛼11=𝑡¹¹_𝑐11𝑑¹¹_𝑐1⋯𝑡¹¹_𝑐1𝑗𝑑¹¹_𝑐1⋯𝑡¹¹_𝑐1𝑚1𝑑¹¹_𝑐1𝑡⋮⋮⋮_11𝑐𝑖1𝑑¹¹_𝑐𝑖⋯𝑡¹¹_𝑐𝑖𝑗𝑑¹¹_𝑐𝑖⋯𝑡¹¹_{𝑐𝑖𝑚1}𝑑¹¹_𝑐𝑖𝑡⋮⋮⋮_11𝑐𝑚11𝑑¹¹_𝑐𝑚1⋯𝑡¹¹_{𝑐𝑚1𝑗}𝑑¹¹_𝑐𝑚1⋯𝑡¹¹_{𝑐𝑚1𝑚1}𝑑¹¹_𝑐𝑚1=𝑡^𝛼11_𝑐11⋯𝑡^𝛼11_𝑐1𝑗⋯𝑡^𝛼11_𝑐1𝑚1𝑡⋮⋮⋮_{𝛼11𝑐𝑖1}⋯𝑡^𝛼11_𝑐𝑖𝑗⋯𝑡^𝛼11_{𝑐𝑖𝑚1}𝑡⋮⋮⋮_{𝛼11𝑐𝑚11}⋯𝑡^𝛼11_{𝑐𝑚1𝑗}⋯𝑡^𝛼11_{𝑐𝑚1𝑚1}. (6)(7)

Step 2. Find the unweighted supermatrix 𝐖. Let the total-influence matrix match and be filled into the interdependence dimensions. It is based on transposing the normalized influence matrix𝐓^𝛼_𝑐 by using dimensions, that is, 𝐖=(𝐓^𝛼_𝑐)^′, and is derived by𝐖=𝐓^𝛼_𝑐^′=𝐷₁𝐷_𝑖𝐷_𝑛𝑐₁₁⋯𝑐_1𝑚1𝑐_𝑖1⋯𝑐_𝑖𝑚𝑖⋯𝑐_𝑛1⋯𝑐_𝑛𝑚𝑛𝐷₁𝑐₁₁𝑐₁₂⋮𝑐_1𝑚1𝐷⋮⋮_𝑗𝑐_𝑗1𝑐_𝑗2⋮𝑐_𝑗𝑚𝑖𝐷⋮⋮_𝑛𝑐_𝑛1𝑐_𝑛2⋮𝑐_𝑛𝑚𝑛𝐖¹¹⋯𝐖^𝑖1⋯𝐖^𝑛1𝐖⋮⋮⋮_1𝑗⋯𝐖^𝑖𝑗⋯𝐖^𝑛𝑗𝐖⋮⋮⋮_1𝑛⋯𝐖^𝑖𝑛⋯𝐖^𝑛𝑛. (8)

Step 3. Obtain the normalized supermatrix 𝐓^𝛼_𝐷 by dimensions. Total-influencematrix 𝐓_𝐷 is obtained by 𝐓_𝐷=𝑡_𝐷¹¹⋯𝑡_𝐷^1𝑗⋯𝑡_𝐷^1𝑛𝑡⋮⋮⋮_𝐷𝑖1⋯𝑡_𝐷^𝑖𝑗⋯𝑡_𝐷^𝑖𝑛𝑡⋮⋮⋮_𝐷𝑛1⋯𝑡_𝐷^𝑛𝑗⋯𝑡_𝐷^𝑛𝑛. (9) Each column of total-influence matrix 𝐓_𝐷 by dimensions can be summed for normalization. Then we normalize the total-influence matrix 𝐓_𝐷 and obtain a new normalized matrix 𝐓^𝛼_𝐷. A new normalized matrix 𝐓^𝛼_𝐷 is derived by 𝐓^𝛼_𝐷=𝑡_𝐷¹¹𝑑₁⋯𝑡_𝐷^1𝑗𝑑₁⋯𝑡_𝐷^1𝑛𝑑₁𝑡⋮⋮⋮_𝐷𝑖1𝑑_𝑖⋯𝑡_𝐷^𝑖𝑗𝑑_𝑖⋯𝑡_𝐷^𝑖𝑛𝑑_𝑖𝑡⋮⋮⋮_𝐷𝑛1𝑑_𝑛⋯𝑡_𝐷^𝑛𝑗𝑑_𝑛⋯𝑡_𝐷^𝑛𝑛𝑑_𝑛=𝑡_𝐷^𝛼11⋯𝑡_𝐷^𝛼1𝑗⋯𝑡_𝐷^𝛼1𝑛𝑡⋮⋮⋮_{𝐷𝛼𝑖1}⋯𝑡_𝐷^𝛼𝑖𝑗⋯𝑡_𝐷^𝛼𝑖𝑛𝑡⋮⋮⋮_{𝐷𝛼𝑛1}⋯𝑡_𝐷^𝛼𝑛𝑗⋯𝑡_𝐷^𝛼𝑛𝑛, (10) where 𝑡_𝐷^𝛼𝑖𝑗=𝑡_𝐷^𝑖𝑗/𝑑_𝑖 and 𝑑_𝑖=∑^𝑛_𝑗=1𝑡_𝐷^𝑖𝑗.

Step 4. Calculate the weighted supermatrix. The weighted supermatrix 𝐖^𝛼 is derived by 𝐖^𝛼=𝐓^𝛼_𝐷𝐖=𝑡_𝐷^𝛼11×𝐖¹¹⋯𝑡_𝐷^𝛼𝑖1×𝐖^𝑖1⋯𝑡_𝐷^𝛼𝑛1×𝐖^𝑛1𝑡⋮⋮⋮_{𝐷𝛼1𝑗}×𝐖^1𝑗⋯𝑡_𝐷^𝛼𝑖𝑗×𝐖^𝑖𝑗⋯𝑡_𝐷^𝛼𝑛𝑗×𝐖^𝑛𝑗𝑡⋮⋮⋮_{𝐷𝛼1𝑛}×𝐖^1𝑛⋯𝑡_𝐷^𝛼𝑖𝑛×𝐖^𝑖𝑛⋯𝑡_𝐷^𝛼𝑛𝑛×𝐖^𝑛𝑛. (11)

Step 5. Limit the weighted supermatrix by raising it to a sufficiently large power 𝑘, until the supermatrix has converged and become a long-term stable supermatrix to obtain the global vector weights, called the DANP influential weights, such that lim_𝑔→∞(𝐖^𝛼)^𝑔, where 𝑔 represents any number of powers when 𝑔→∞.

3.3. Phase III

The development of the VIKOR method began with the following form of 𝐿_𝑝 metric: 𝐿^𝑝_𝑘=^𝑛_𝑗=1𝑤_𝑗𝑓^∗_𝑗−𝑓_𝑘𝑗𝑓^∗_𝑗−𝑓⁻_𝑗^𝑝1/𝑝, (12) where 𝑓_𝑘𝑗 is the performance score of the 𝑗th criterion on 𝑘th alternative, 𝑤_𝑗 is the influential weight of the 𝑗th criterion from DANP, and 𝑛 is the number of criteria and 1≤𝑝≤∞ [39]. Two measures 𝐿_𝑘^𝑝=1 (as 𝑆_𝑘 for minimal average gap or degree of regret) and 𝐿_𝑘^𝑝=∞ (as 𝑄_𝑘 for priority improvement) are established as 𝑆_𝑘=𝐿_𝑘^𝑝=1=^𝑛_𝑗=1𝑤_𝑗𝑓^∗_𝑗−𝑓_𝑘𝑗𝑓^∗_𝑗−𝑓⁻_𝑗,𝑄_𝑘=𝐿_𝑘^𝑝=∞=max_𝑗𝑓^∗_𝑗−𝑓_𝑘𝑗𝑓^∗_𝑗−𝑓⁻_𝑗∣𝑗=1,2,…,𝑛. (13)

The compromise solution min_𝑘𝐿^𝑝_𝑘 shows the synthesized gap to be minimized, which will be selected or improved such that its gap values in each criterion and dimension will be improved for the closest to the aspiration level. Regarding INRM, the group utility is emphasized when 𝑝 is small (such as 𝑝=1); in contrast, if 𝑝 grows toward infinity, the individual maximal gaps attain greater importance for priority improvement in each dimension or criterion. Consequently, min_𝑘𝑆_𝑘 stresses the maximum group utility; however, min_𝑘𝑄_𝑘 focuses on selecting the minimal value from the maximum individual gaps for priority improvement.

The VIKOR method in this study has four steps.

Step 1. Obtain an aspired/desired and tolerable level. We calculated the best 𝑓^∗_𝑗 values (the aspiration level) and the worst 𝑓⁻_𝑗 values (the tolerable level) of all criterion functions, 𝑗=1,2,…,𝑛. In traditional approach we suppose that the 𝑗th function denotes benefits: 𝑓^∗_𝑗=max_𝑘𝑓_𝑘𝑗 and 𝑓⁻_𝑗=min_𝑘𝑓_𝑘𝑗, but in this research we use the performance scores from 1 to 9 (very poor ?1, 2, …, 8, 9? the best) in questionnaires (see the appendix), so the aspiration level can be set at 9 score and the worst value at 1 score. Therefore, in this research, we set 𝑓^∗_𝑗=9 as the aspiration level and 𝑓⁻_𝑗=1 as the worst value, which differs from traditional approach. This approach can avoid “choose the best among inferior choices/options/alternatives (i.e., avoid pick the best apple among a barrel of rotten apples).” Furthermore, an original rating matrix can be converted into the following matrix: 𝑟_𝑘𝑗=𝑓^∗_𝑗−𝑓_𝑘𝑗𝑓^∗_𝑗−𝑓⁻_𝑗. (14)

Step 2. Calculate the mean of group utility and maximal regret. The values can be computed by 𝑆_𝑘=∑^𝑛_𝑗=1𝑤_𝑗𝑟_𝑘𝑗 (the synthesized gap for all of the criteria) and 𝑄_𝑘=max_𝑗{𝑟_𝑘𝑗∣𝑗=1,2,…,𝑛} (the maximal gap in criterion 𝑘 for priority improvement), respectively.

Step 3. Rank or improve the alternatives for a compromise solution. We order the alternatives from the calculated values, which are obtained as 𝑅_𝑘=𝑣𝑆_𝑘−𝑆^∗(𝑆⁻−𝑆^∗)+(1−𝑣)𝑄_𝑘−𝑄^∗(𝑄⁻−𝑄^∗),∀𝑘=1,2,…,𝑚, (15) where 𝑆^∗=min_𝑖𝑆_𝑖 or 𝑆^∗=0 (when all of the criteria have been achieved to the aspiration level); 𝑆⁻=max_𝑖𝑆_𝑖 or 𝑆⁻=1 (in the worst case); 𝑄^∗=min_𝑖𝑄_𝑖 or can be set as 𝑄^∗=0; 𝑄⁻=max_𝑖𝑄_𝑖 or 𝑄⁻=1; and 𝑣 is presented as the weight of the strategy of the maximum group utility. Conversely, 1−𝑣 is the weight of individual regret. When 𝑆^∗=0, 𝑆⁻=1, 𝑄^∗=0, and 𝑄⁻=1, we can rewrite (15) as 𝑅_𝑘=𝑣𝑆_𝑘+(1−𝑣)𝑄_𝑘. (16)

Step 4. Decreasing order of the values of 𝑆_𝑘, 𝑄_𝑘, and 𝑅_𝑘, then, we can use them for a compromise solution for all alternatives.

4. Results and Discussions

The significant confidence of the 16 experts’ group consensus in the questionnaires, the mean of sample gap equals only 1.795%; that is, the significant confidence is obtained as 98.205% which is greater than 95% (see note in Table 3). There are eight projects that were submitted by the Six Sigma team of Corporation J. Utilizing the aforementioned six dimensions and 17 criteria, we evaluated the performance of each project based on the opinions of sixteen knowledge-based experts and the consultants in SPM manufacturing and the Six Sigma technique. We evaluated performances on a scale of 1 to 9, with 1 indicating very poor performance (i.e., 𝑓⁻_𝑗=1is called the worst value) and 9 indicating the best performance (i.e., 𝑓^∗_𝑗=9is called the aspiration level). Then, we used the average performance scores of each project and applied the VIKOR method to obtain the performances and the gaps to the aspiration levels of the alternative projects.

Table 3 

The initial influence matrix 𝐀 for criteria.

Using (2) and (3), the normalized direct-influence matrix and the total-influence of 𝐓_𝐶 and 𝐓_𝐷 can be easily derived. The INRM was constructed using the vectors 𝐫 and 𝐬 fromthe total direct-influence matrix 𝐓_𝐷 (see Table 4). The sum of influences given and received on criteria is shown in Table 5. In addition, the INRM maps are shown in Figure 3.

Table 4 

The total-influence matrix 𝐓_𝐷 and influence given/received for dimensions.

Table 5 

The sum of influences given and received on criteria.

Figure 3 

The influential network relation maps.

In Figure 3, we found that six dimensions are influenced by each other such as strategy (𝐷₁) will be influenced by impact on customers (𝐷₃), impact on finance (𝐷₄), and impact on operations (𝐷₅); feasibility (𝐷₂) will be influenced by strategy (𝐷₁), impact on customers (𝐷₃), impact on finance (𝐷₄), and impact on employees (𝐷₆). These influential relationships will help the Six Sigma team to perform the decision-making. To increase the feasibility of projects, the first step is to improve their impact on finance. In particular, the Six Sigma team can first refer to 𝐷₄ in Figure 3 to improve the contribution of ROI (𝐶₁₀) for the project. For the same reason, the projects need the support of the top management of strategy (𝐷₁), but strategy (𝐷₁) depended on the impact on finance (𝐷₄), impact on customers (𝐷₃), and impact on employees (𝐷₆). In conclusion, the managers of Six Sigma can refer to the INRM to prioritize their improvements to the dimensions and criteria in PS.

We used the DANP method to obtain the influential weights and priorities of the dimensions and criteria in the empirical case of Corporation J. The performances and the gaps to the aspiration levels of the alternative projects are shown in Table 6. We found that the priority in global weights of the first dimension is feasibility (𝐷₂), followed by strategy (𝐷₁), impact on finance (𝐷₄), impact on customers (𝐷₃), impact on operations (𝐷₅), and impact on employees (𝐷₆), in that order. In addition, we extended the priority of the criteria in each dimension from the local weights in Table 6. For instance, feasibility (𝐷₂) is the first priority in dimensions of global weights; when extended to the local weight dimensions, however, we know that the technical feasibility (𝐶₄) will be the first priority of feasibility (𝐷₂). All of these local and global weights will be helpful in selecting and improving the best alternatives in MCDM problems with the VIKOR method.

Table 6 

The performance and gap of desired level of all alternative projects.

In analyzing the empirical case of an SPM manufacturing company using a hybrid MCDM model, we successfully obtained the influential relationship matrix as well as the map of the PS dimensions and criteria in interdependent and feedback problems. We also, compared the performance of projects and obtained how to improve the performance of dimensions and criteria in reducing the gaps remaining to achieve the aspiration levels. We conclude the following.

(1) Based on the DANP method, the global influential weights and local influential weights of the dimensions and criteria can be obtained, and then by combining DANP with the VIKOR method the average scores of the projects from project 𝑃₁ to project 𝑃₈, in performance, are 6.423, 6.191, 6.363, 6.880, 6.281, 6.613, 6.173, and 6.099, respectively (see Table 6). We found that project 𝑃₄ has the best performance among the eight projects, with a score of 6.88, which means that the training and practice of employees (𝑃₄) are the first priority for the Six Sigma projects. In accordance with the scores of the projects, the subsequent priorities are the establishment of a lean production system (𝑃₆), decision support (𝑃₁), operational process improvement (𝑃₃), the upgrading of supply chain management (𝑃₅), the improvement of the material/products inspection system (𝑃₃), the promotion of quality function deployment (𝑃₇), and the investment in inspection equipment/instruments (𝑃₈).

(2) The traditional PS approach is to rank the project alternatives and then only select the best of the project. Our proposed model can not only use the best project selection but can also analyze which gaps in the dimensions and criteria should be the priorities for improvement. From the perspective of the VIKOR method, project 𝑃₄ is the first priority of Six Sigma projects in this SPM case; however the gap of 0.292 in this case should also be reduced to zero (the aspiration level, no gap) questions regarding how to minimize this gap as much as possible. We can propose several improvement strategies and suggestions for the managers of Six Sigma to promote the project’s performance based on the INRM. For example, in Table 6 and Figure 3, the largest gap to the aspiration level in project 𝑃₄ is 0.291 (impact on finance, 𝐷₄); next is 0.255 (impact on customers, 𝐷₃), then 0.233 (impact on operations, 𝐷₅), 0.227 (feasibility, 𝐷₂), 0.204 (impact on employees, 𝐷₆), and 0.201 (strategy, 𝐷₁). This information can assist managers in improving the project performance of Six Sigma to reduce the gaps in question and achieve the aspiration level quickly and directly.

5. Conclusion

PS is one of the most critical and challenging activities faced by companies,and selecting the right project in Six Sigma is a major factor in its early success and long-term acceptance within any organization. Based on the definition of a project, the objectives must be clear, succinct, achievable, realistic, and measurable, with a high probability of success. We developed dimensions and criteria that align with the strategy of a SPM manufacturing company to evaluate and select Six Sigma projects.

We present a hybrid MCDM model combining the influential weights of DANP with the VIKOR method for an empirical case in the SPM industry to integrate each criterion performance into each dimension and overall performance. So a DANP model can overcome the problems of interdependence and feedback among dimensions and criteria in the real world. Based on the global and local influential weights of DANP and combined with VIKOR’s calculation (see Table 6), mangers can make decisions which take into account the integration of performances and the improvement of the gaps to the aspiration level of each dimension. In this case study, we not only designated the prioritization of the projects in the order of training and practice of employees (P4) ? establish lean production system (P6) ? decision support (P1) ? operational process improvement (P3) ? upgrade supply chain management (P5) ? improve material and products inspection system (P2) ? promote quality function deployment (P7) and invest in inspection equipment and instrument (P8), but we also determined how to improve the gaps to achieve the aspiration level of performance in each project, from low value to high value.

To reduce the gaps and achieve the aspiration level (zero gaps) in the best or most suitable areas, other methods, such as the dominance-based rough set approach (DRSA) and PROMETHEE III, should be addressed in the future studies. In order to deal with uncertainty or fuzziness in the decision making process, a fuzzy-ANP can be considered in our proposed model.

Appendix

Liou and Tzeng [40] found that the traditional MCDM ignored some important new concepts and trends (some assumptions limit/defects) for solving actual problems in the real world. First, the traditional model assumes that the criteria are independent and hierarchical in structure; however, criteria are often interdependent in real-world problems; because “Statistics and Economics are unrealistic in the real world,” DEMATEL technique can be used to find the influence matrix and build an INRM for solving this problem. Second, the relative good solution from the existing alternatives is replaced by the aspiration levels. Note: Herbert A. Simon, in his lecture given in Stockholm upon receiving the Nobel Prize in Economics, 1978; the Scottish word “satisficing” (=satisfying) has been revived to denote problem solving and decision making that sets an aspiration level, searches until an alternative is found that is satisfactory by the aspiration level criterion, and selects that alternative [41]. In this research, if 𝑓^∗_𝑗 is the aspiration level and 𝑓⁻_𝑗 is the worst value, we use the performance scores from 0 to 10 (very dissatisfaction ←0,1,2,…,9,10→ very satisfaction) in questionnaires, so the aspiration level can be set at 10 score (i.e., 𝑓^∗_𝑗=10) and the worst value at zero score (i.e., 𝑓⁻_𝑗=0); this approach can avoid “Choose the best among inferior/choices/alternatives,” that is, avoid “Pick the best apple among a barrel of rotten apples.” Third, the trends have shifted from how to carry out the “ranking” or “selection” of the most preferable alternatives, to how to “improve” their performances based on INRM, because “we need a systematic approach to problem-solving; instead of addressing the systems of the problem, we need to identify the sources of the problem [42].” These ideas and concepts can offer major contributions in this type of research problem.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors gratefully acknowledge assistant editor and two reviewers of this paper who helped clarify and improve this presentation.