Applying a Hybrid MCDM Model for Six Sigma Project Selection

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Research Article

Applying a Hybrid MCDM Model for Six Sigma Project Selection

Fu-Kwun Wang,

1

Chen-Hsoung Hsu,

1

and Gwo-Hshiung Tzeng

2,3

1_{Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan} 2_{Graduate Institute of Urban Planning, College of Public Affairs, National Taipei University, New Taipei, Taiwan} 3_{Institute of Management of Technology, National Chiao Tung University, Hsinchu, Taiwan}

Correspondence should be addressed to Fu-Kwun Wang; [email protected]

Received 27 September 2013; Revised 17 November 2013; Accepted 19 November 2013; Published 26 February 2014 Academic Editor: Hao-Chun Lu

Copyright © 2014 Fu-Kwun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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, engineer-ing, 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. Perc¸in 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

manufactur-ing 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

Volume 2014, Article ID 730934, 13 pages http://dx.doi.org/10.1155/2014/730934

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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 inSection 3.Section 4highlights

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 activ-ities, 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 (seeTable 1andFigure 1).

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. Corpora-tion 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 (seeTable 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 inFigure 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.

To assess the interinfluence of the PS criteria for the DEMATEL technique calculation, we designed a question-naire 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 inno-vation 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

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Table 1: The influence dimensions and criteria of comprehensive PS in Six Sigma.

Dimensions Influence criteria Statements

Strategy (𝐷₁)

Meaningful to organization (𝐶₁) The projects should contribute to process improvement within the_{organization.} Critical to quality (𝐶₂) The selected projects should be critical for upgrading the quality of_{the product.} Variable is measurable (𝐶₃) The response variables in the projects should be easily measurable for

implementation. Feasibility

(𝐷₂)

Technical feasibility (𝐶₄) Any project in Six Sigma should be technically feasible for the_{organization.} Resources are available (𝐶5) The resources of the organization should be available to support the_project. Time schedule (𝐶₆) Project completion within four to six months should be feasible. Impact on customers

(𝐷₃)

Customer satisfaction (𝐶₇) The projects for process improvement should be aimed at satisfying_{customer needs.} Customer complaints (𝐶₈) The projects should reduce the complaints of the customers and_{increase the reliability of the products.} New business (𝐶₉) The new project should not only consider present customers but also

provide benefits to new customers in the future.

Impact on finance (𝐷4)

ROI (𝐶10)

Many projects in Six Sigma require the investment of equipment or human power to improve the process. The return of investment (ROI) of the project must be considered.

Cost reduction (𝐶11) The project for process improvement should upgrade the quality of_{products and reduce the cost of manufacturing.} Profit generation (𝐶12) The projects selected should increase the generation of profit for the_{organization.} Impact on operations

(𝐷₅)

Reduction in cycle time (𝐶₁₃)

The projects for process improvement should not only improve the quality of products but also reduce the cycle time of relevant processes.

Upgrade operational performance (𝐶14)

The projects should increase the unit time production quantity and upgrade the equipment’s utilization.

Improved in compliance and controls (𝐶15)

The compliance and controls in operational systems should be improved by discussion and communication regarding project selection.

Impact on employees (𝐷6)

Retaining rate (𝐶₁₆)

The retaining rate refers to the rate at which key employees were kept within the organization. The project selection in Six Sigma should support employee retention for the company.

Improved capability (𝐶₁₇) Employees should be educated and trained to improve the capability of manufacturing processes and the performance of the organization.

by using the basic concept of ANP based on the

total-influence matrix by the DEMATEL technique, and(3)

rank-ing and improvrank-ing the prioritization of dimensions/criteria or projects through the VIKOR method for reducing gaps in each criterion and dimension to achieve the aspiration levels.

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 criterion𝑖 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

as A = [ [ [ [ [ [ [ [ [ [ 𝑎₁₁ ⋅ ⋅ ⋅ 𝑎_1𝑗 ⋅ ⋅ ⋅ 𝑎_1𝑛 .. . ... ... 𝑎_𝑖1 ⋅ ⋅ ⋅ 𝑎_𝑖𝑗 ⋅ ⋅ ⋅ 𝑎_𝑖𝑛 .. . ... ... 𝑎_𝑛1 ⋅ ⋅ ⋅ 𝑎_𝑛𝑗 ⋅ ⋅ ⋅ 𝑎_𝑛𝑛 ] ] ] ] ] ] ] ] ] ] . (1)

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Table 2: The alternatives of PS in Six Sigma.

Alternatives Purposes Statements of alternatives in Six Sigma

(1) Decision support (𝑃1) Easy implementation of Six Sigma The projects implemented always need support from top_management. (2) Improve material and

products inspection system (𝑃₂)

Ensure the quality of materials and products

An inspection system for quality is the foundation of total quality management and a key successful factor of Six Sigma. (3) Operational process

improvement (𝑃₃) Upgrade operational performance

Operations management techniques for the manufacturing or services provided by the company should be improved. (4) Training and practice of

employees (𝑃₄)

Promote improvement of employees’ abilities

Training of employees includes projects operation and management in Six Sigma.

(5) Upgrade supply chain management (SCM) (𝑃5)

Shorten lead time and improve inventory control

SCM includes material requirement planning, logistics, and inventory control.

(6) Establish lean

production system (𝑃6) Waste reduction and variability reduction

Lean production supplies the customer with exactly what the customer wants when the customer wants it, without waste, through continuous improvement.

(7) Promote quality function deployment (QFD) (𝑃₇)

Satisfaction of the customer and deployment of quality efforts

QFD refers to both (1) determining what will satisfy the customer and (2) translating those customer desires into the target design.

(8) Invest in inspection equipment and instruments (𝑃8)

Upgrade the correctness of inspection results for Six Sigma

improvement

The key equipment and instruments for polyester film printing for the packaging industry are gas chromatographs and color difference inspectors.

Overall performances of Six Sigma projects

Strategy D1 Feasibility D2 _{customers D}Impact on₃ _{finance D}Impact on₄ _{operations D}Impact on₅ _{employees D}Impact on₁

Project (P1) Project (P2) Project (P3) Project (P4) Project (P5) Project (P6) Project (P7) Project (P8)

M ea n in gf ul t o o rga niza tio n ( C1 ) Cri tical t o q u ali ty ( C2 ) Va ri ab le is me asur ea ble ( C3 ) T echnica l f easib il it y ( C4 ) Res o ur ces a re a va ila b le ( C5 ) T ime s che d u les ( C6 ) C u sto m er s at isfac ti o n ( C7 ) C u st o m er co m p la in ts ( C8 ) N ew b u siness ( C9 ) R O I ( C10 ) C o st r ed u ct io n ( C11 ) Profit ge n er at ion ( C12 ) Re d u ct io n in c yc le time ( C13 ) U p gr ade o p era tio na l p er fo rma nce (C 14 ) Im p ro ve d inco m p lia n ce an d co n tr o ls ( C15 ) Ret ainin g ra te ( C16 ) Im p ro ve d ca pa b ili ty ( C17 )

Figure 1: The network of project selection in Six Sigma.

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, . . . , Xℎ, with

lim_{ℎ → 0}Xℎ = [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

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Criteria questionnaire

Solve Tand

INRM

Brain storming and Experts opinion

DEMATEL

Solve the influential weights

of DANP

Solve projects ranking, improve and make

strategies VIKOR based on INRM Performance questionnaire Tc TD W DANP limg→∞(W𝛼 W𝛼 )g 󳰀

Figure 2: The process of a hybrid MCDM model combining DANP and VIKOR.

guaranteed. Thus, the total-influence matrix T is derived by

T = X + X2+ ⋅ ⋅ ⋅ + Xℎ= X(I − X)−1, (3)

where lim_{ℎ → 0}Xℎ= [0]_𝑛×𝑛and I denotes 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_𝑖]_𝑛×1= [ [ 𝑛 ∑ 𝑗=1 𝑡_𝑖𝑗] ]𝑛×1 , s = [𝑠_𝑗]_𝑛×1= [∑𝑛 𝑖=1 𝑡_𝑖𝑗] 󸀠 1×𝑛 , (4)

where the superscript 󸀠denotes the transpose. Also,𝑟_𝑖shows

the sum of the direct and indirect effects of criteria𝑖 on the

other criteria and𝑠_𝑗shows 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 criteria𝑖 contributes to the problem. If (𝑟_𝑖− 𝑠_𝑖) is

positive, then criteria𝑖 is affecting other criteria, and if (𝑟_𝑖−𝑠_𝑖)

is negative, then criteria𝑖 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 T𝛼_𝑐 by using

dimen-sions. After normalizing the total-influence matrix T_𝑐 by

dimensions, we can obtain a new normalized matrix T𝛼_𝑐 by

dimensions and is given by

T𝛼_𝑐 = 𝐷1 𝐷𝑗 𝐷𝑛 𝑐₁₁ ⋅ ⋅ ⋅ 𝑐_1𝑚₁ 𝑐_𝑗1 ⋅ ⋅ ⋅ 𝑐_𝑗𝑚_𝑗 ⋅ ⋅ ⋅ 𝑐𝑛1 ⋅ ⋅ ⋅ 𝑐𝑛𝑚𝑛 𝐷₁ 𝑐₁₁ 𝑐₁₂ .. . 𝑐1𝑚1 .. . ... 𝐷_𝑖 𝑐_𝑖1 𝑐𝑖2 .. . 𝑐_𝑖𝑚_𝑖 .. . ... 𝐷𝑛 𝑐_𝑛1 𝑐_𝑛2 .. . 𝑐_𝑛𝑚_𝑛 [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ T𝛼11_𝑐 ⋅ ⋅ ⋅ T𝛼1𝑗 𝑐 ⋅ ⋅ ⋅ T 𝛼1𝑛 𝑐 .. . ... ... T𝛼𝑖1 𝑐 ⋅ ⋅ ⋅ T 𝛼𝑖𝑗 𝑐 ⋅ ⋅ ⋅ T 𝛼𝑖𝑛 𝑐 .. . ... ... T𝛼𝑛1 𝑐 ⋅ ⋅ ⋅ T 𝛼𝑛𝑗 𝑐 ⋅ ⋅ ⋅ T 𝛼𝑛𝑛 𝑐 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . (5)

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For instance the normalization T𝛼11_𝑐 is shown as (5) and

(6).

Consider the following:

𝑑11_𝑐𝑖 =∑𝑚1 𝑗=1𝑡 11 𝑐𝑖𝑗, 𝑖 = 1, 2, . . . , 𝑚1, (6) T𝛼11_𝑐 = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ 𝑡_𝑐1111 𝑑11 𝑐1 ⋅ ⋅ ⋅ 𝑡11 𝑐1𝑗 𝑑11 𝑐1 ⋅ ⋅ ⋅ 𝑡11 𝑐1𝑚1 𝑑11 𝑐1 .. . ... ... 𝑡11 𝑐𝑖1 𝑑11 𝑐𝑖 ⋅ ⋅ ⋅ 𝑡 11 𝑐𝑖𝑗 𝑑11 𝑐𝑖 ⋅ ⋅ ⋅ 𝑡 11 𝑐𝑖𝑚1 𝑑11 𝑐𝑖 .. . ... ... 𝑡11_𝑐𝑚₁₁ 𝑑11 𝑐𝑚1 ⋅ ⋅ ⋅ 𝑡 11 𝑐𝑚1𝑗 𝑑11 𝑐𝑚1 ⋅ ⋅ ⋅ 𝑡 11 𝑐𝑚1𝑚1 𝑑11 𝑐𝑚1 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] = [ [ [ [ [ [ [ [ [ [ 𝑡𝛼11 𝑐11 ⋅ ⋅ ⋅ 𝑡𝛼11𝑐1𝑗 ⋅ ⋅ ⋅ 𝑡𝛼11𝑐1𝑚1 .. . ... ... 𝑡𝛼11 𝑐𝑖1 ⋅ ⋅ ⋅ 𝑡𝛼11𝑐𝑖𝑗 ⋅ ⋅ ⋅ 𝑡𝛼11𝑐𝑖𝑚1 .. . ... ... 𝑡_𝑐𝑚𝛼11₁₁ ⋅ ⋅ ⋅ 𝑡𝛼11_𝑐𝑚₁_𝑗 ⋅ ⋅ ⋅ 𝑡𝛼11_𝑐𝑚₁_𝑚₁ ] ] ] ] ] ] ] ] ] ] . (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

influ-ence matrix T𝛼_𝑐 by using dimensions, that is, W= (T𝛼_𝑐)󸀠, and

is derived by W = (T𝛼_𝑐)󸀠= 𝐷₁ 𝐷_𝑖 𝐷_𝑛 𝑐₁₁ ⋅ ⋅ ⋅ 𝑐_1𝑚₁ 𝑐_𝑖1 ⋅ ⋅ ⋅ 𝑐_𝑖𝑚_𝑖 ⋅ ⋅ ⋅ 𝑐𝑛1 ⋅ ⋅ ⋅ 𝑐𝑛𝑚𝑛 𝐷1 𝑐₁₁ 𝑐₁₂ .. . 𝑐_1𝑚₁ .. . ... 𝐷𝑗 𝑐𝑗1 𝑐_𝑗2 .. . 𝑐𝑗𝑚𝑖 .. . ... 𝐷_𝑛 𝑐_𝑛1 𝑐_𝑛2 .. . 𝑐_𝑛𝑚_𝑛 [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ W11 _{⋅ ⋅ ⋅} _W𝑖1 _{⋅ ⋅ ⋅} _W𝑛1 .. . ... ... W1𝑗 _{⋅ ⋅ ⋅} _W𝑖𝑗 _{⋅ ⋅ ⋅} _W𝑛𝑗 .. . ... ... W1𝑛 ⋅ ⋅ ⋅ W𝑖𝑛 ⋅ ⋅ ⋅ W𝑛𝑛 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . (8)

Step 3. Obtain the normalized supermatrix T𝛼_𝐷 by

dimen-sions. Total-influencematrix T_𝐷is obtained by

T_𝐷= [ [ [ [ [ [ [ [ [ [ [ [ 𝑡11 𝐷 ⋅ ⋅ ⋅ 𝑡1𝑗𝐷 ⋅ ⋅ ⋅ 𝑡1𝑛𝐷 .. . ... ... 𝑡𝑖1 𝐷 ⋅ ⋅ ⋅ 𝑡𝑖𝑗𝐷 ⋅ ⋅ ⋅ 𝑡𝑖𝑛𝐷 .. . ... ... 𝑡𝑛1 𝐷 ⋅ ⋅ ⋅ 𝑡𝑛𝑗𝐷 ⋅ ⋅ ⋅ 𝑡𝑛𝑛𝐷 ] ] ] ] ] ] ] ] ] ] ] ] . (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

normal-ized matrix T𝛼_𝐷. A new normalized matrix T𝛼_𝐷 is derived

by T𝛼_𝐷= [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ 𝑡11 𝐷 𝑑1 ⋅ ⋅ ⋅ 𝑡1𝑗_𝐷 𝑑1 ⋅ ⋅ ⋅ 𝑡1𝑛 𝐷 𝑑1 .. . ... ... 𝑡𝑖1_𝐷 𝑑_𝑖 ⋅ ⋅ ⋅ 𝑡𝑖𝑗_𝐷 𝑑_𝑖 ⋅ ⋅ ⋅ 𝑡𝑖𝑛_𝐷 𝑑_𝑖 .. . ... ... 𝑡𝑛1 𝐷 𝑑_𝑛 ⋅ ⋅ ⋅ 𝑡𝑛𝑗_𝐷 𝑑_𝑛 ⋅ ⋅ ⋅ 𝑡𝑛𝑛 𝐷 𝑑_𝑛 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]

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= [ [ [ [ [ [ [ [ [ 𝑡𝛼11 𝐷 ⋅ ⋅ ⋅ 𝑡𝛼1𝑗𝐷 ⋅ ⋅ ⋅ 𝑡𝛼1𝑛𝐷 .. . ... ... 𝑡𝛼𝑖1_𝐷 ⋅ ⋅ ⋅ 𝑡𝛼𝑖𝑗_𝐷 ⋅ ⋅ ⋅ 𝑡𝛼𝑖𝑛_𝐷 .. . ... ... 𝑡𝛼𝑛1 𝐷 ⋅ ⋅ ⋅ 𝑡𝛼𝑛𝑗𝐷 ⋅ ⋅ ⋅ 𝑡𝛼𝑛𝑛𝐷 ] ] ] ] ] ] ] ] ] , (10) where𝑡𝛼𝑖𝑗_𝐷 = 𝑡𝑖𝑗_𝐷/𝑑_𝑖and𝑑_𝑖= ∑𝑛_𝑗=1𝑡𝑖𝑗_𝐷.

Step 4. Calculate the weighted supermatrix. The weighted

supermatrix W𝛼is derived by W𝛼= T𝛼_𝐷W = [ [ [ [ [ [ [ [ [ 𝑡𝛼11 𝐷 × W11 ⋅ ⋅ ⋅ 𝑡𝛼𝑖1𝐷 × W𝑖1 ⋅ ⋅ ⋅ 𝑡𝛼𝑛1𝐷 × W𝑛1 .. . ... ... 𝑡𝛼1𝑗_𝐷 × W1𝑗 _{⋅ ⋅ ⋅ 𝑡}𝛼𝑖𝑗 𝐷 × W𝑖𝑗 ⋅ ⋅ ⋅ 𝑡𝛼𝑛𝑗𝐷 × W𝑛𝑗 .. . ... ... 𝑡𝛼1𝑛_𝐷 × W1𝑛 ⋅ ⋅ ⋅ 𝑡𝛼𝑖𝑛_𝐷 × W𝑖𝑛 ⋅ ⋅ ⋅ 𝑡𝛼𝑛𝑛_𝐷 × W𝑛𝑛 ] ] ] ] ] ] ] ] ] . (11) Step 5. Limit the weighted supermatrix by raising it to a

suf-ficiently large power𝑘, 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 → ∞}(W𝛼)𝑔, where𝑔 represents any number of

powers when𝑔 → ∞.

3.3. Phase III. The development of the VIKOR method began

with the following form of𝐿_𝑝metric:

𝐿𝑝_𝑘={_{ { 𝑛 ∑ 𝑗=1 [ [ 𝑤_𝑗_{(󵄨󵄨󵄨󵄨󵄨𝑓}_𝑗∗− 𝑓_𝑘𝑗󵄨󵄨󵄨󵄨_󵄨) (󵄨󵄨󵄨󵄨󵄨𝑓∗ 𝑗 − 𝑓𝑗−󵄨󵄨󵄨󵄨_󵄨) ] ] 𝑝 } } } 1/𝑝 , (12)

where𝑓_𝑘𝑗is the performance score of the𝑗th criterion on 𝑘th

alternative,𝑤_𝑗 is the influential weight of the𝑗th criterion

from DANP, and𝑛 is the number of criteria and 1 ≤ 𝑝 ≤ ∞

[39]. Two measures𝐿𝑝=1_𝑘 (as𝑆_𝑘 for minimal average gap or

degree of regret) and𝐿𝑝=∞_𝑘 (as𝑄_𝑘for priority improvement)

are established as 𝑆_𝑘 = 𝐿𝑝=1_𝑘 =∑𝑛 𝑗=1 [ [ 𝑤_𝑗_{(󵄨󵄨󵄨󵄨󵄨𝑓}∗ 𝑗 − 𝑓𝑘𝑗󵄨󵄨󵄨󵄨_󵄨) (󵄨󵄨󵄨󵄨󵄨𝑓∗ 𝑗 − 𝑓𝑗−󵄨󵄨󵄨󵄨_󵄨) ] ] , 𝑄_𝑘= 𝐿𝑝=∞_𝑘 = max 𝑗 { { { (󵄨󵄨󵄨󵄨󵄨𝑓𝑗∗− 𝑓𝑘𝑗󵄨󵄨󵄨󵄨_󵄨) (󵄨󵄨󵄨󵄨󵄨𝑓∗ 𝑗 − 𝑓𝑗−󵄨󵄨󵄨󵄨_󵄨) | 𝑗 = 1, 2, . . . , 𝑛}_} } . (13)

The compromise solution min_𝑘𝐿𝑝_𝑘shows 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 when𝑝 is small (such

as𝑝 = 1); in contrast, if 𝑝 grows toward infinity, the

indi-vidual maximal gaps attain greater importance for priority

improvement in each dimension or criterion. Consequently,

min_𝑘𝑆_𝑘stresses 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 best𝑓_𝑗∗ values (the aspiration level) and the

worst𝑓_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_𝑘𝑗and𝑓_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 − 𝑓𝑘𝑗󵄨󵄨󵄨󵄨_󵄨) (󵄨󵄨󵄨󵄨󵄨𝑓∗ 𝑗 − 𝑓𝑗−󵄨󵄨󵄨󵄨_󵄨) . (14)

Step 2. Calculate the mean of group utility and maximal

regret. The values can be computed by𝑆_𝑘 = ∑𝑛_𝑗=1𝑤_𝑗𝑟_𝑘𝑗(the

synthesized gap for all of the criteria) and𝑄_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= V (𝑆𝑘− 𝑆∗)

(𝑆−_{− 𝑆}∗₎ +

(1 − V) (𝑄_𝑘− 𝑄∗₎

(𝑄−_{− 𝑄}∗₎ , ∀𝑘 = 1, 2, . . . , 𝑚 ,

(15)

where𝑆∗ = min_𝑖𝑆_𝑖or𝑆∗ = 0 (when all of the criteria have

been achieved to the aspiration level);𝑆−= max_𝑖𝑆_𝑖or𝑆− = 1

(in the worst case);𝑄∗ = min_𝑖𝑄_𝑖or can be set as𝑄∗ = 0;

𝑄− = max𝑖𝑄𝑖or𝑄− = 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. When𝑆∗= 0, 𝑆−= 1, 𝑄∗ = 0,

and𝑄−= 1, we can rewrite (15) as

𝑅_𝑘= V𝑆_𝑘+ (1 − V) 𝑄𝑘. (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

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Table 3: The initial influence matrix A for criteria. Criteria 𝐶₁ 𝐶₂ 𝐶₃ 𝐶₄ 𝐶₅ 𝐶₆ 𝐶₇ 𝐶₈ 𝐶₉ 𝐶₁₀ 𝐶₁₁ 𝐶₁₂ 𝐶₁₃ 𝐶₁₄ 𝐶₁₅ 𝐶₁₆ 𝐶₁₇ C₁ 0.00 3.38 3.00 3.13 3.31 3.06 3.19 3.13 2.94 3.00 3.06 3.06 3.00 3.25 2.56 2.63 2.88 C2 3.25 0.00 3.00 3.19 3.13 3.13 3.56 3.44 2.56 2.81 2.69 3.00 2.69 3.06 2.56 2.50 2.81 C3 2.81 3.06 0.00 2.88 2.81 2.81 2.81 2.56 2.38 2.69 2.69 2.75 2.94 3.00 2.31 2.13 2.19 C₄ 2.88 3.25 3.19 0.00 3.00 3.06 3.06 2.88 2.75 3.31 2.88 3.25 3.13 3.31 2.44 2.19 2.69 C5 3.00 3.06 2.94 3.19 0.00 3.13 2.88 2.75 2.44 3.06 3.13 3.13 3.00 3.13 2.38 2.69 2.56 C6 2.63 2.88 3.00 3.00 2.88 0.00 3.06 2.75 2.06 2.69 2.94 2.88 3.00 3.06 2.31 2.50 2.50 C₇ 3.38 3.38 3.00 3.19 3.06 3.06 0.00 3.38 2.81 2.94 2.63 3.06 2.38 2.81 2.69 2.94 2.63 C8 3.31 3.56 2.94 3.13 2.88 2.88 3.44 0.00 2.50 2.63 2.44 2.69 2.38 2.50 2.75 2.81 2.63 C9 3.19 2.56 2.69 3.19 3.13 2.56 3.19 2.56 0.00 3.13 2.63 3.13 2.38 2.63 2.69 2.44 2.69 C₁₀ 2.94 3.06 3.19 3.25 3.38 2.81 3.25 2.94 3.00 0.00 3.00 3.25 3.19 3.31 2.56 2.69 2.94 C₁₁ 3.00 3.00 3.06 3.19 3.13 3.06 3.19 3.13 2.81 3.00 0.00 3.31 2.75 3.19 2.75 2.56 2.81 C12 3.13 3.25 2.94 3.44 3.38 3.13 3.00 3.00 3.13 3.25 3.44 0.00 3.06 3.25 2.81 3.06 2.81 C₁₃ 3.00 2.75 3.13 3.31 3.13 3.31 2.94 2.81 2.63 2.81 3.25 3.19 0.00 2.94 2.75 2.44 2.94 C₁₄ 3.13 3.31 3.00 3.25 3.25 3.31 3.25 3.00 2.63 2.94 2.94 3.13 3.31 0.00 2.69 2.94 3.06 C15 2.69 2.88 2.50 2.75 2.69 2.63 2.88 2.81 2.50 2.56 2.50 2.56 2.75 2.81 0.00 3.00 2.75 C₁₆ 2.50 2.75 2.50 2.38 2.44 2.13 2.81 2.81 2.25 2.63 2.69 2.63 2.69 2.88 3.00 0.00 2.81 C₁₇ 2.88 3.06 2.63 3.06 2.88 2.69 3.13 2.94 2.94 2.88 2.88 3.06 3.31 3.50 2.88 2.69 0.00

Note: the mean of sample gap =(1/𝑛(𝑛 − 1)) ∑𝑛_𝑖=1∑𝑛_𝑗=1(|𝑎𝑝_𝑖𝑗− 𝑎𝑝−1_𝑖𝑗 |/𝑎_𝑖𝑗𝑝) × 100% = 1.795% < 5%, where 𝑛 is the number of criteria and 𝑝 is the sample of 16

experts.

Table 4: The total-influence matrix T_𝐷and influence given/received for dimensions.

T_𝐷 𝐷₁ 𝐷₂ 𝐷₃ 𝐷₄ 𝐷₅ 𝐷₆ 𝑟_𝑖 Dimensions 𝑟_𝑖 𝑠_𝑖 𝑟_𝑖+ 𝑠_𝑖 𝑟_𝑖− 𝑠_𝑖 𝐷₁ 0.78 0.80 0.77 0.78 0.76 0.71 4.60 Strategy (𝐷₁) 4.60 4.76 9.36 −0.16 𝐷₂ 0.79 0.78 0.76 0.77 0.75 0.71 4.57 Feasibility (𝐷₂) 4.57 4.80 9.36 −0.23 𝐷3 0.79 0.79 0.75 0.77 0.75 0.71 4.55 Impact on customers (𝐷3) 4.55 4.61 9.16 −0.05 𝐷₄ 0.84 0.85 0.81 0.80 0.80 0.75 4.85 Impact on finance (𝐷₄) 4.85 4.65 9.49 0.20 𝐷₅ 0.80 0.81 0.77 0.78 0.74 0.72 4.62 Impact on operations (𝐷₅) 4.62 4.54 9.16 0.09 𝐷6 0.77 0.77 0.74 0.75 0.74 0.67 4.43 Impact on employees (𝐷6) 4.43 4.28 8.71 0.16 𝑠_𝑖 4.76 4.80 4.61 4.65 4.54 4.28

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− = 1 is called the worst value) and 9 indicating the

best performance (i.e.,𝑓_𝑗∗ = 9 is 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.

Using (2) and (3), the normalized direct-influence matrix

and the total-influence of T_𝐶and T_𝐷can be easily derived.

The INRM was constructed using the vectors r and s fromthe

total direct-influence matrix T_𝐷 (seeTable 4). The sum of

influences given and received on criteria is shown inTable 5.

In addition, the INRM maps are shown inFigure 3.

InFigure 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

influen-tial 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 to𝐷₄inFigure 3to 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 inTable 6. We found that the priority in global

weights of the first dimension is feasibility (𝐷₂), followed by

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0.000 0.200 −0.200 0.400 0.600 0.800 1.000 1.200 23.500 24.000 24.500 25.000 25.500 26.000 D₅ D₁ D₂ D4 D3 D6 ri+ si ri+ si ri+ si ri+ si ri+ si ri −s i ri −s i ri −si ri −s i ri −s i −0.800 −0.600 −0.400 −0.200 0.000 0.200 0.400 0.600 0.800 24.500 25.000 25.500 26.000 26.500 27.000 27.500 (C6) (C4) (C5) (C3) _(C₂₎ (C1) (C15) (C13) (C14) (C11) 10) (C12) −0.25 −0.20 −0.15 −0.10 −0.05 Retaining Improved capability rate (C16) (C17) (23.85, −0.07) ri+ si ri+ si ri −si ri −s i (25.749, 1.01) Customer satisfaction New business (C9) (C7) (27.306, −0.6) Customer complain (C8) (26.08, −0.41) (24.691, 0.6) 0.00 0.05 0.10 0.15 0.20 0.25 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 Strategy (D1) (9.401, −0.159) Feasibility (D2) (9.412, −0.223) Impact on consumer (D3) Impact on finance (D4) Impact on process (D5) Impact on employee (D6) (9.198, −0.050) (8.696, 0.144) (9.538, 0.203) (9.504, 0.084) Reduction in cycle time Upgrade operation performance Improved compliance and controls 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 24.000 24.500 25.000 25.500 26.000 26.500 27.000 27.500 28.000 (26.339, 0.37) (27.535, 0.13) (24.178, 0.31) Technical feasibility Available resources Time schedule −0.800 −0.700 −0.600 −0.500 −0.400 −0.300 −0.200 −0.100 0.000 25.600 25.800 26.000 26.200 26.400 26.600 26.800 27.000 27.200 27.400 (27.306, −0.589) (26.796, −0.53) (25.746, −0.72) ROI (C Cost reduction Revenue generation 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 26.400 26.600 26.800 27.000 27.200 27.400 27.600 27.800 Meaningful to organization Critical to quality Measureable variable −1.200 −1.000 −0.800 −0.600 −0.400 −0.2000.000 0.200 0.400 25.000 25.500 26.000 26.500 27.000 27.500 (26.464, 0.586) (26.835, 0.655) (27.640, 0.527) (27.156, 0.230) (27.246, −0.501) (25.37, −1.043)

Figure 3: The influential network relation maps.

Table 5: The sum of influences given and received on criteria.

Criteria 𝑟_𝑖 𝑠_𝑖 𝑟_𝑖+ 𝑠_𝑖 𝑟_𝑖− 𝑠_𝑖

Meaningful to organization (𝐶₁) 13.69 13.46 27.16 0.23

Critical to quality (𝐶2) 13.37 13.87 27.25 −0.50

Variable is measurable (𝐶3) 12.16 13.21 25.37 −1.04

Technical feasibility (𝐶₄) 13.36 13.95 27.31 −0.59

Resources are available (𝐶5) 13.13 13.66 26.80 −0.53

Time schedule (𝐶6) 12.51 13.24 25.75 −0.73 Customer satisfaction (𝐶₇) 13.34 13.97 27.31 −0.63 Customer complaints (𝐶₈) 12.84 13.25 26.08 −0.41 New business (𝐶9) 12.68 12.01 24.69 0.67 ROI (𝐶₁₀) 13.74 13.09 26.84 0.65 Cost reduction (𝐶₁₁) 13.53 12.94 26.46 0.59 Profit generation (𝐶12) 14.08 13.56 27.64 0.53

Reduction in cycle time (𝐶₁₃) 13.36 12.98 26.34 0.37

Upgrade operational performance (𝐶₁₄) 13.83 13.70 27.53 0.13

Improved in compliance and controls (𝐶15) 12.25 11.93 24.18 0.32

Retaining rate (𝐶₁₆) 11.89 11.96 23.85 −0.07

Improved capability (𝐶₁₇) 13.38 12.37 25.75 1.01

(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.

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T a b le 6 :Th e p er for m an ce an d ga p of d es ire d le ve lof al la lt er n at ive proj ec ts . Loc al we ig h t Glob al we ig h t Pe rf o rm an ce G ap o f d es ir ed le ve l 𝑃1 𝑃2 𝑃3 𝑃4 𝑃5 𝑃6 𝑃7 𝑃8 𝑃1 𝑃2 𝑃3 𝑃4 𝑃5 𝑃6 𝑃7 𝑃8 St ra te gy (D 1 ) 0.17 2 (2 ) 7. 03 1 6.5 49 6.7 94 7.1 92 6.4 62 6.6 48 6.4 37 6.17 5 0.21 9 (1 ) 0.27 2 (2 ) 0.24 5 (1 ) 0.2 01 (6 ) 0.2 82 (4 ) 0.26 1 (4 ) 0.2 85 (2 ) 0.3 14 (3 ) M ea nin gf ul to o rga niza tio n (𝐶1 ) 0.3 32 (2 ) 0.057 7.87 5 5.7 50 6.87 5 7. 12 5 6.5 63 6.6 88 6.2 50 5.7 50 0.12 5 0.3 61 0.2 36 0.208 0.2 71 0.2 57 0.3 0 6 0 .3 61 Cri tical to q u ali ty (𝐶 2 ) 0.3 4 2 (1 ) 0.05 9 7.3 13 7. 31 3 6.87 5 7. 50 0 6.6 25 6.7 50 6.5 0 0 6.8 13 0.188 0.188 0.2 36 0.167 0.26 4 0.2 50 0.2 78 0.2 43 V ar iabl e is me asu rabl e (𝐶 3 ) 0.3 25 (3 ) 0.05 6 5.87 5 6.5 63 6.6 25 6.9 38 6.188 6.5 0 0 6.5 63 5.9 38 0.3 47 0.2 71 0.26 4 0.229 0.3 13 0.2 78 0.2 71 0.3 4 0 Fe as ib ili ty (D 2 ) 0.17 3 (1 ) 6.4 67 6.3 82 6.7 09 6.9 59 6.47 3 6.7 04 6.421 6.21 2 0.2 81 (3 ) 0.291 (3) 0.2 55 (2 ) 0.227 (4) 0.2 81 (2 ) 0.2 55 (3 ) 0.2 87 (3 ) 0.3 10 (2 ) T ec hnical fe asib ili ty (𝐶 4 ) 0.3 41 (1 ) 0.05 9 6.7 50 6.9 38 6.8 13 7. 12 5 6.2 50 6.5 0 0 6.6 25 6.3 75 0.2 50 0.229 0.2 43 0.208 0 .3 0 6 0.2 78 0.26 4 0.29 2 Res o ur ces ar e ava il ab le (𝐶5 ) 0.3 34 (2 ) 0.058 6.8 13 6.12 5 6.5 63 6.7 50 6.2 50 6.6 25 6.4 38 6.3 13 0.2 43 0.3 19 0.2 71 0.2 50 0.3 0 6 0.26 4 0.2 85 0.29 9 T ime sc he d u le (𝐶 6 ) 0.3 24 (3 ) 0.05 6 5.8 13 6.0 63 6.7 50 7. 0 0 0 6.9 38 7. 0 0 0 6.188 5.9 38 0.3 54 0.3 26 0.2 50 0.222 0 .229 0.222 0.3 13 0.3 4 0 Im pa ct on cu st om er s( D3 ) 0.16 6 (4 ) 6.27 1 6.6 45 6.3 12 6.7 01 6.4 69 6.2 85 6.6 65 6.6 16 0.3 03 (4 ) 0.26 2 (1 ) 0.299 (4) 0.2 55 (2 ) 0.2 81 (3 ) 0.3 02 (5 ) 0.2 59 (1 ) 0.26 5 (1 ) C u st o m er sa ti sfac ti o n (𝐶7 ) 0.3 56 (1 ) 0.05 9 6.5 63 7. 25 0 6.6 88 7. 188 6.813 6.5 63 7. 0 0 0 6.9 38 0.2 71 0.1 94 0.2 57 0.201 0.2 43 0.2 71 0.222 0 .229 C u st o m er co m p la in ts (𝐶 8 ) 0.3 37 (2 ) 0.05 6 6.4 38 6.87 5 6.3 13 6.9 38 6.2 50 6.2 50 6.6 88 7. 0 63 0.2 85 0.2 36 0.29 9 0.229 0 .3 0 6 0.3 0 6 0.2 57 0.215 Ne w b u si n es s (𝐶9 ) 0.3 0 6 (3 ) 0.05 1 5.7 50 5.6 88 5.87 5 5.87 5 6.3 13 6.0 0 0 6.2 50 5.7 50 0.3 61 0.3 6 8 0.3 47 0.3 47 0.29 9 0.3 33 0.3 0 6 0.3 61 Im pa ct on fin an ce (D 4 ) 0.16 8 (3 ) 6.5 62 6.0 21 6.29 0 6.3 77 6.6 05 6.8 50 5.6 43 5.7 94 0.27 1 (2 ) 0.3 31 (4 ) 0.3 01 (5 ) 0.291 (1) 0.26 6 (1 ) 0.23 9 (2 ) 0.3 73 (6 ) 0.3 56 (5 ) ROI (𝐶10 ) 0.3 30 (2 ) 0.05 6 6.5 0 0 6.7 50 6.0 0 0 6.188 6.6 25 6.3 13 5.6 25 5.9 38 0.2 78 0.2 50 0.3 33 0.3 13 0.26 4 0.29 9 0.3 75 0.3 4 0 Co st re d u ct io n (𝐶 11 ) 0.3 26 (3 ) 0.05 5 6.6 25 5.4 38 6.6 25 6.3 75 6.5 63 7. 50 0 5.87 5 5.5 63 0.26 4 0.3 9 6 0.26 4 0.29 2 0.2 71 0.167 0 .3 47 0.3 82 Re ve n u e genera tio n (𝐶12 ) 0.3 4 2 (1 ) 0.058 6.5 63 5.87 5 6.2 50 6.5 63 6.6 25 6.7 50 5.4 38 5.87 5 0.2 71 0.3 47 0.3 0 6 0.2 71 0.26 4 0.2 50 0.3 9 6 0.3 47 Im pa ct on op er at io ns (D 5 ) 0.16 4 (5 ) 6.15 0 5.8 85 6.42 5 6.9 01 6.22 5 6.9 06 6.0 67 6.159 0.3 17 (5 ) 0.3 46 (5 ) 0.2 86 (3 ) 0.23 3 (3 ) 0.3 08 (5 ) 0.23 3 (1 ) 0.3 25 (4 ) 0.3 16 (4 ) Red u ctio n in cy cle tim e (𝐶13 ) 0.3 36 (2 ) 0.05 5 6.12 5 5.188 6.4 38 6.0 63 5.6 88 6.7 50 5.5 63 6.12 5 0.3 19 0.4 24 0.2 85 0.3 26 0 .3 6 8 0.2 50 0.3 82 0.3 19 Up gr ad e o p er at io n al pe rf o rm an ce (𝐶14 ) 0.3 54 (1 ) 0.058 6.2 50 6.5 0 0 7. 0 0 0 7. 50 0 6.87 5 7. 62 5 6.87 5 6.87 5 0.3 0 6 0.2 78 0.222 0.167 0.2 36 0.15 3 0.2 36 0.2 36 Im p ro ved co m p lia n ce an d co n tr o ls (𝐶15 ) 0.3 0 9 (3 ) 0.05 1 6.0 63 5.9 38 5.7 50 7. 12 5 6.0 63 6.2 50 5.6 88 5.3 75 0.3 26 0.3 4 0 0.3 61 0.208 0.3 26 0.3 0 6 0 .3 6 8 0.4 03 Im pa ct on em pl oy ee (D 6 ) 0.15 4 (6 ) 6.0 04 5.6 00 5.5 68 7. 16 6 5.3 75 6.2 58 5.7 62 5.6 03 0. 333 (6) 0.3 78 (6 ) 0.3 81 (6 ) 0.2 04 (5 ) 0.4 03 (6 ) 0.3 05 (6 ) 0.3 59 (5 ) 0.3 77 (6 ) Ret ainin g ra te (𝐶16 ) 0.4 91 (2 ) 0.07 6 5.7 50 5.2 50 5.2 50 6.5 63 5.3 75 5.7 50 5.0 63 5.0 63 0.3 61 0.41 7 0.41 7 0.2 71 0.4 03 0.3 61 0.4 38 0.4 38 Improv ed ca p abi li ty (𝐶 17 ) 0.5 08 (1 ) 0.079 6.2 50 5.9 38 5.87 5 7. 75 0 5.3 75 6.7 50 6.4 38 6.12 5 0.3 0 6 0.3 4 0 0.3 47 0.13 9 0 .4 03 0.2 50 0.2 85 0.3 19 To ta lp er fo rm an ce 6.423 (3) 6.1 91 (6 ) 6.3 63 (4 ) 6.8 80 (1 ) 6.2 81 (5 ) 6.6 13 (2 ) 6.17 3 (7 ) 6.0 99 (8 ) To ta lg ap 0.2 86 (3 ) 0.3 12 (6 ) 0.29 2 (4 ) 0.23 5 (1 ) 0.3 02 (5 ) 0.26 5 (2 ) 0.3 14 (7 ) 0.3 22 (8 ) N o te :t he n u m b ers in the ( ) d eno tes th e ra n ks o f lo ca lw eig h ts in d imen sio n s and cr it er ia.

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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 perfor-mance of projects and obtained how to improve the per-formance 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 project𝑃₄has 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

estab-lishment 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

perspec-tive of the VIKOR method, project𝑃₄is 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, inTable 6andFigure 3, the largest gap to the

aspiration level in project𝑃₄is 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 interde-pendence 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 (seeTable 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

func-tion deployment (P7) and invest in inspecfunc-tion 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 and𝑓_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

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“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.

References

[1] J. Antony and R. Banuelas, “Key ingredients for the effective implementation of six sigma program,” Measuring Business

Excellence, vol. 6, no. 4, pp. 20–27, 2002.

[2] J. Antony, “Some pros and cons of six sigma: an academic perspective,” The TQM Magazine, vol. 16, no. 4, pp. 303–306, 2004.

[3] R. D. Snee, “Six sigma: the evolution of 100 years of business improvement methodology,” International Journal of Six Sigma

and Competitive Advantage, vol. 1, no. 1, pp. 4–20, 2004.

[4] A. Y. T. Szeto and A. H. C. Tsang, “Antecedents to successful implementation of six sigma,” International Journal of Six Sigma

and Competitive Advantage, vol. 1, no. 3, pp. 307–322, 2005.

[5] Y. H. Kwak and F. T. Anbari, “Benefits, obstacles, and future of six sigma approach,” Technovation, vol. 26, no. 5-6, pp. 708–715, 2006.

[6] P. S. Pande, R. P. Neuman, and R. R. Cavanagh, The Six Sigma

Way, McGraw-Hill, New York, NY, USA, 2000.

[7] R. Banuelas, C. Tennant, I. Tuersley, and S. Tang, “Selection of six sigma projects in the UK,” The TQM Magazine, vol. 18, no. 5, pp. 514–527, 2006.

[8] J. D. Kendrick and D. Saaty, “Use analytic hierarchy process for project selection,” Six Sigma Forum Magazine, vol. 6, no. 4, pp. 22–29, 2007.

[9] U. D. Kumar, H. Saranga, J. E. Ram´ırez-M´arquez, and D. Nowicki, “Six sigma project selection using data envelopment analysis,” The TQM Magazine, vol. 19, no. 5, pp. 419–441, 2007. [10] G. Hu, L. Wang, S. Fetch, and B. Bidanda, “A multi-objective

model for project portfolio selection to implement lean and six sigma concepts,” International Journal of Production Research, vol. 46, no. 23, pp. 6611–6625, 2008.

[11] C. Kahraman and G. Büyüközkan, “A combined fuzzy AHP and fuzzy goal programming approach for effective six-sigma project selection,” Journal of Multiple-Valued Logic and Soft

Computing, vol. 14, no. 6, pp. 599–615, 2008.

[12] T. Yang and C.-H. Hsieh, “Six-Sigma project selection using national quality award criteria and Delphi fuzzy multiple crite-ria decision-making method,” Expert Systems with Applications, vol. 36, no. 4, pp. 7594–7603, 2009.

[13] G. Büyüközkan and D. Öztürkcan, “An integrated analytic approach for six sigma project selection,” Expert Systems with

Applications, vol. 37, no. 8, pp. 5835–5847, 2010.

[14] S. Perc¸in and C. Kahraman, “An integrated fuzzy multi-criteria decision-making approach for six sigma project selection,”

International Journal of Computational Intelligence Systems, vol.

3, no. 5, pp. 610–621, 2010.

[15] M. Tkáç and ˇS. Lyócsa, “On the evaluation of six sigma projects,”

Quality and Reliability Engineering International, vol. 26, no. 1,

pp. 115–124, 2010.

[16] S. Vinodh, S. G. Gautham, R. Anesh Ramiya, and D. Rajanayagam, “Application of fuzzy analytic network process for agile concept selection in a manufacturing organisation,”

International Journal of Production Research, vol. 48, no. 24, pp.

7243–7264, 2010.

[17] S. Boran, H. R. Yazgan, and K. Goztepe, “A fuzzy ANP-based approach for prioritising projects: a six sigma case study,”

International Journal of Six Sigma and Competitive Advantage,

vol. 6, no. 3, pp. 133–155, 2011.

[18] R. K. Padhy and S. Sahu, “A real option based six sigma project evaluation and selection model,” International Journal of Project

Management, vol. 29, no. 8, pp. 1091–1102, 2011.

[19] A. Saghaei and H. Didehkhani, “Developing an integrated model for the evaluation and selection of six sigma projects based on ANFIS and fuzzy goal programming,” Expert Systems

with Applications, vol. 38, no. 1, pp. 721–728, 2011.

[20] M. Kumar, J. Antony, and B. R. Cho, “Project selection and its impact on the successful deployment of six sigma,” Business

Process Management Journal, vol. 15, no. 5, pp. 669–686, 2009.

[21] J. Antony and C. Fergusson, “Six sigma in the software industry: results from a pilot study,” Managerial Auditing Journal, vol. 19, no. 8, pp. 1025–1032, 2004.

[22] J. Van den Heuvel, R. J. M. M. Does, A. J. J. C. Bogers, and M. Berg, “Implementing six sigma in the Netherlands,” Joint

Commission Journal on Quality and Patient Safety, vol. 32, no.

7, pp. 393–399, 2006.

[23] E. Fontela and A. Gabus, “The DEMATEL observe,” Battelle Institute, Geneva, Switzerland, 1976.

[24] T. L. Saaty, Decision Making with Dependence and Feedback: The

Analytic Network Process, RWS, Pittsburgh, Pa, USA, 1996.

[25] S. Opricovic and G.-H. Tzeng, “Multicriteria planning of post-earthquake sustainable reconstruction,” Computer-Aided Civil

and Infrastructure Engineering, vol. 17, no. 3, pp. 211–220, 2002.

[26] T. L. Saaty, “Rank from comparisons and from ratings in the analytic hierarchy/network processes,” European Journal of

Operational Research, vol. 168, no. 2, pp. 557–570, 2006.

[27] C.-Y. Huang, J. Z. Shyu, and G.-H. Tzeng, “Reconfiguring the innovation policy portfolios for Taiwan’s SIP mall industry,”

Technovation, vol. 27, no. 12, pp. 744–765, 2007.

[28] F. Jin, P. Liu, and X. Zhang, “The evaluation study of human resources based on intuitionistic fuzzy set and TOPSIS method,”

Journal of Information and Computational Science, vol. 4, no. 3,

pp. 1023–1028, 2007.

[29] J. J. H. Liou, G.-H. Tzeng, and H.-C. Chang, “Airline safety measurement using a hybrid model,” Journal of Air Transport

[30] G.-H. Tzeng, C.-H. Chiang, and C.-W. Li, “Evaluating inter-twined effects in e-learning programs: a novel hybrid MCDM model based on factor analysis and DEMATEL,” Expert Systems

with Applications, vol. 32, no. 4, pp. 1028–1044, 2007.

[31] J. J. H. Liou, L. Yen, and G.-H. Tzeng, “Building an effective safety management system for airlines,” Journal of Air Transport

(13)

[32] W.-S. Lee, G.-H. Tzeng, J.-L. Guan, K.-T. Chien, and J.-M. Huang, “Combined MCDM techniques for exploring stock selection based on Gordon model,” Expert Systems with

Appli-cations, vol. 36, no. 3, part 2, pp. 6421–6430, 2009.

[33] L. Pei-de, “A novel method for hybrid multiple attribute decision making,” Knowledge-Based Systems, vol. 22, no. 5, pp. 388–391, 2009.

[34] J. J. H. Liou, C.-Y. Tsai, R.-H. Lin, and G.-H. Tzeng, “A mod-ified VIKOR multiple-criteria decision method for improving domestic airlines service quality,” Journal of Air Transport

[35] P. Liu and X. Zhang, “Research on the supplier selection of a supply chain based on entropy weight and improved ELECTRE-III method,” International Journal of Production Research, vol. 49, no. 3, pp. 637–646, 2011.

[36] C.-H. Hsu, F.-K. Wang, and G.-H. Tzeng, “The best vendor selection for conducting the recycled material based on a hybrid MCDM model combining DANP with VIKOR,” Resources,

Conservation and Recycling, vol. 66, no. 1, pp. 95–111, 2012.

[37] C.-H. Liu, G.-H. Tzeng, and M.-H. Lee, “Improving tourism policy implementation—the use of hybrid MCDM models,”

Tourism Management, vol. 33, no. 2, pp. 239–488, 2012.

[38] P. Liu and X. Wu, “A competency evaluation method of human resources managers based on multi-granularity linguis-tic variables and VIKOR method,” Technological and Economic

Development of Economy, vol. 18, no. 4, pp. 696–710, 2012.

[39] S. Opricovic and G.-H. Tzeng, “Compromise solution by MCDM methods: a comparative analysis of VIKOR and TOP-SIS,” European Journal of Operational Research, vol. 156, no. 2, pp. 445–455, 2004.

[40] J. H. Liou and G. H. Tzeng, “Comments on multiple cri-teria decision making (MCDM) methods in economics: an overview,” Technological and Economic Development of

Econ-omy, vol. 18, no. 4, pp. 672–695, 2012.

[41] H. A. Simon, “A behavioral model of rational choice,” The

Quarterly Journal of Economics, vol. 69, no. 1, pp. 99–118, 1955.

[42] J. H. Liou, “New concepts and trends of MCDM for tomorrow— in honor of professor Gwo-Hshiung Tzeng on the occasion of his 70th birthday,” Technological and Economic Development of

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