Evaluation methods for completed Six Sigma projects through an interval type-2 fuzzy ANP

Abstract

We propose project evaluation criteria by considering not only the Six Sigma approach, but also European Foundation for Quality Management (EFQM) model features, and introduce an evaluation model for completed Six Sigma projects. The proposed model uses seven main criteria: leadership, policies/strategies, main performance results, social results, employees, cooperation, and resources. These main criteria are enhanced by 18 sub-criteria. Due to the evaluation scale being based on human judgments, fuzzy set theory is an obvious methodology choice to account for uncertainty in the evaluation data. Type-2 fuzzy sets can provide flexibility for uncertainty by considering membership functions and their footprints. We also propose an evaluation methodology for weighting the main and sub-criteria and for ranking completed Six Sigma projects through a fuzzy analytic network process (ANP) method with interval type-2 fuzzy sets. From our analysis, performance results were found to have the highest weight among the seven main criteria, while achieving project objectives was found to be the most effective criteria.

Keywords

Six Sigma methodology project evaluations fuzzy sets the EFQM model interval type-2 fuzzy ANP

1 Introduction

Implemented in a variety of areas from manufacturing to service environments, Six Sigma is a disciplined, project-oriented, data-driven approach and methodology for the elimination of defects in a process [1]. The Six Sigma approach was developed at Motorola in the 1980 s after they added many new tools and methods for total quality management (TQM). Leading companies, such as General Electric, Sony, Caterpillar, and Honeywell then used the Six Sigma approach to improve their processes, causing it to gain wide acceptance as a way to increase organizational performance [2]. Currently, the Six Sigma approach is implemented as a management philosophy that aids organizations in improving their organizational efficiency and decreasing variation within processes, and is heavily driven by an understanding of customer requirements.

A review of the Six Sigma project evaluation literature reveals the fact that former studies on this topic have mainly focused on project selection. Although selection of the right projects is crucial, especially during the deployment of quality improvement approaches, it is also important to evaluate and assess completed projects to determine their relative success and guide superior future implementations. Very few researchers have measured the success of improvement methodologies, such as Six Sigma, by evaluating the effectiveness of completed projects [3]. By defining the right criteria and evaluation method the completed Six Sigma project is a guide for new Six Sigma projects that are effective and have comprehensive impact for improvement quality of company. In addition, Six Sigma leaders can have knowledge about the main effective features for the best projects. Therefore, the top Six Sigma project can be lead to implement the new effective projects by using the example in the training of Six Sigma. Considering this scarcity and the significant gap in the literature, our research focuses on the evaluation of completed Six Sigma projects and contributes to the project evaluation literature in a unique way.

The completed Six-Sigma project evaluation problem is a multiple criteria decision-making problem. Fuzzy set theory is used to account for the use of a linguistic variable for expressing pair-wise comparisons. Therefore, fuzzy multiple criteria decision-making (FMCDM) methods have been developed due to the imprecision in assessing the relative importance of criteria and the performance ratings of alternative techniques (projects) [4]. In order to overcome this difficulty, we adopt fuzzy set theory in this study. We use an interval type-2 fuzzy ANP approach for the evaluation of completed Six Sigma projects. We also extend the criteria by combining EFQM model components with the traditional DMAIC steps criteria for Six Sigma project evaluation to examine project impact on society and the environment. After determining criteria and sub-criteria for evaluating completed Six Sigma projects based on expert opinions, the weights of the main criteria and sub-criteria are calculated using an interval type-2 fuzzy ANP method. The fuzzy approach is perfectly suitable for reflecting and modelling uncertainty and expert linguistic terms for the pairwise comparison of various criteria. Therefore, after the criteria are determined, a comparison of the criteria in linguistic terms is conducted by a Sigma Six Master Black Belt from a household and appliances company. Then, the views of others Six Sigma Master Black Belts from vehicle production factory and also from factory in defense sector are taken into account for determining the criteria. The team was conducted mentioned Six Sigma Master Black Belts and related Six Sigma Black Belts who are from factory and also from universities. After obtaining weights for the main and sub-criteria, 15 Six Sigma projects selected from the pool of completed projects from the same company are evaluated, ranked, and compared based on the proposed criteria through the interval type-2 fuzzy ANP method.

The main contributions of this study can be summarized as follows:

Criteria and sub-criteria are extended by considering not only DMAIC steps, but also EFQM model components. Therefore, a project that contributes to company’s strategic goals, society, and environment is a good model for other projects and can provide valuable training for project team members. Additionally, accurate evaluation and ranking of projects can reflect team performance in the form of salary increases or other rewards. Thus, it is important to evaluate completed projects using true criteria.

The weights of criteria are determined by using a fuzzy ANP method with interval type-2 fuzzy sets. Therefore, the linguistic expressions of experts can be represented as fuzzy numbers. Pairwise comparisons and the interactions between criteria are also considered.

The proposed approach is compared to a real-word ranking of projects and the results are discussed. The Sigma Six Master Black Belt from the company confirmed that the proposed approach ranked projects more accurately than their evaluation methods and approved the approach.

The rest of this paper is organized as follows. A literature review on the Six Sigma approach and project selection is presented in Section 2. The background and methodology for the proposed evaluation criteria for completed Six Sigma projects FANP method with interval type-2 fuzzy sets are given in Section 3. The application of the proposed approach are discussed in Section 4. Further discussion regarding the results is provided in Section 5. Our conclusions and future research directions are discussed in Section 6.

2 Literature review on Six Sigma and project selection

Six Sigma is a project-guided approach to quality improvement. These improvements are translated into financial benefits through Six Sigma projects [5]. According to [6], ‘Projects are one of the primary means by which organizations improve their operations, quickly respond to opportunities, develop innovative technologies and products, and finally, manage challenges. The immediate and direct impacts of projects on the organization, including whether or not the objectives have been met and benefits have been realized, are important to take into account’. Therefore, project evaluation tools were developed by users of various continuous improvement methodologies, such as TQM and Six Sigma, and these tools have been used commonly in project management [7]. Previous research on Six Sigma has utilized project-selection tools to determine the relative priority of potential projects and selecting only those that will have the greatest impact [8 –10]. Project selection tools simply create a matrix in which weights are assigned to criteria based on the relative importance of each criterion. Each project is then evaluated based on the criteria and ranked based on its assigned score. Project selection and evaluation have become critical components of successful business operations. These two terms are frequently used interchangeably in literature. Project evaluation is mainly defined as being ‘a systematic method of scoring a project, strategic initiative, or other form of proposed work based on specific criteria’ [7].

Prior researchers that have dealt specifically with Six Sigma project selection have utilized and proposed various project evaluation approaches, including an approach using both balanced score cards and a prioritization matrix [3], a hybrid multiple criteria decision-making model combining the DEMATEL technique, ANP, and the VIKOR method [10]. ANP is firstly introduced by Saaty [11] for considering innerdependency among criteria and many paper are published to define the weights of criteria. An analytic hierarchy process (AHP) model to assess the benefits of each project, a hierarchical failure mode and effects analysis (FMEA) model to calculate risk and determine a rating for each project [12], a weighted scorecard approach [13], ANP and DEMATEL-ANP [9, 14], real option analysis for assessing the value of a project and a zero–one integer linear programming model for selecting an optimal project portfolio [15], AHP and FMEA to rank potential Six Sigma projects [16], data envelopment analysis [17, 18], fuzzy AHP and fuzzy TOPSIS for the selection of construction projects [19], intuitionistic fuzzy MCDM based on an axiomatic design methodology for the supplier selection problem [20], and an ELECTRE I-based MCDM method using interval type-2 fuzzy numbers [21]. Six Sigma project selection studies have also utilized different types of fuzzy approaches, including the Delphi FMCDM method [4], an integrated model based on ANFIS and fuzzy goal programming [8], and a fuzzy set theory with linguistic variables and fuzzy numbers for determining project effectiveness [3].

Papers on Six Sigma project selection have also utilized various criteria for evaluation. These include the national quality award criteria [4], business excellence, productivity, revenue growth, benefits, opportunities, risks, and costs as factors [9], business and customer criteria, financial criteria, and process and technological criteria [8], and business benefits, feasibility, and organization impact criteria [22]. The input criteria are project duration, project cost, and the number of Sigma Six Black Belts and Green Belts. The output criteria are customer satisfaction, increase in sigma level, impact on business strategy, financial impact (impact of poor quality on costs), and increase in productivity [17]. Additional criteria include project cost, labour hours, financial gains, increase in sigma level, and increase in customer satisfaction [18].

3 Background and proposed methodology

The implementation of the proposed methodology has six main steps, as shown in Fig. 1. Initially, we determine the completed project selection criteria and sub-criteria by obtaining expert opinions. Next, criteria weights are determined by using an interval type-2 fuzzy ANP method. In order to obtain defuzzified values, we adopt the approach used in [23]. The completed projects are then evaluated, ranked, and compared based on the proposed criteria through the ANP method.

Fig. 1

The implementation steps of the proposed methodology.

The following subsections discuss the determination of criteria, introduce basic type-2 fuzzy theory, and provide a detailed discussion of the step-by-step process.

3.1 The criteria for evaluating a completed Six Sigma project based on an EFQM model

In the literature, different criteria have been introduced for project selection, but no studies have been performed regarding completed project evaluation. In this study, we concentrate on a comprehensive range of criteria that should be considered as discussion points for completed Six Sigma project evaluation and assessment. We base our proposed criteria largely on EFQM excellence model. The EFQM model divides nine major criteria into two categories: enablers and results. Enablers contain information regarding what an organization does and how it does it. Results contain information regarding what an organization achieves. The first category contains the criteria of leadership, strategy, people-staff, partnership and resources, and processes-products-services. The second category contains the criteria of people results, customer results, society results, and key results [24].

Tabari et al. [25] showed that the business excellence model of EFQM is related to Six Sigma projects and discussed the similarities between these two systems and the ways in which they complement each other. In their study, criteria were classified in accordance with the method used in [26] and are based on the excellence model of EFQM, which has significant similarities with Six Sigma in terms of operational excellence.

In addition to these studies, Brun’s study in [27] reveals the fact that the most-highlighted critical success factors (CSFs) in a sample of 18 papers include: management involvement, communication, cultural change, training, organization infrastructure, linking Six Sigma to business strategy, connecting Six Sigma to customers, binding Six Sigma to human resources, linking Six Sigma to suppliers, project management skills, understanding tools and techniques within Six Sigma, and project prioritization and selection. In [28], Zu et al. used a TQM/Six Sigma strategy that implements top management support, customer relationships, workforce management, supplier relationships, process management, quality information, and product/service design. Six Sigma improvement procedures and metrics are linked to group study, developmental culture, and rational culture within the company. In [29], De Mast & Lokkerbol determined that DMAIC is suitable for experimental problems ranging from well-structured to semi-structured settings, and that it is applicable for extensive problem solving tasks, problem definition, diagnosis, and the design of products. Finally, in [5], Sin et al. stated that socialization, externalization, combination, and internalization have positive effects on knowledge in a Six Sigma DMAIC project, as well as the success of Six Sigma projects.

In this study, people-result, which is one of the major criteria of EFQM, is altered, and instead, the results of the projects on the people are investigated through the sub-criterion category of people-employees (Six Sigma Project Team). Similarly, customer results are also investigated through the criterion of compatibility with the methodology (i.e., whether or not to conduct voice of customer (VOC) analysis) and the criterion of meeting project goals, which is based on main performance results (i.e., reducing customer complaints).

After the determination of the primary criteria for assessment and the evaluation of completed Six Sigma projects based on the EFQM excellence model, we interviewed the team for Six Sigma project evaluation to verify our proposed criteria. Attaining expert opinions on the project evaluation criteria helped us to integrate and harmonize the theoretical criteria with the practical criteria. Additionally, in order to finalize the criteria, suggestions from experienced Six Sigma project leaders at a company that has been implementing Six Sigma for a long time were obtained. Finally, seven main criteria (leadership, policies and strategies, performance results, social results, methodology compliance, and personnel and cooperation-resources) and 18 sub-criteria were defined [30]. The main criteria and their related sub-criteria are presented in Table 1 and explained below.

Table 1
Proposed criteria and related sub-criteria for completed Six Sigma project evaluation

Criteria Sub-Criteria

Leadership (Senior management commitment) ✓ Involvement, support and control of senior management during selection of project team members, application and evaluation of the project

Policies and Strategies ✓ Contribution of the project to the organization policies and strategies

Main Performance Results ✓ The extent to which the project has been able to meet the project objectives

✓ Financial return and impact of the project

✓ Contribution of the project to the innovative development

Social Results ✓ Social benefit

✓ Impact on the environmental performance improvement

Employees (Six Sigma Project Team) - Personnel ✓ Contribution of the project to the employee (team members) motivation

✓ Contribution of the project to the employee (team members) training and skills development

✓ Contribution of the project to the employee (team members) and interdepartmental communication improvement

Cooperation and resources ✓ Contribution of the project to the cooperation with suppliers, supplier motivation for utilization of statistical tools and enhancement supplier quality level (reflection of Six Sigma philosophy to the supplier)

✓ Utilization of proper and necessary resources

Project Execution Compatibility with DMAIC Methodology - (Methodology Compliance) ✓ Execution compatibility with Define phase requirements

✓ Execution compatibility with Measure phase requirements

✓ Execution compatibility with Analyze phase requirements

✓ Execution compatibility with Improve phase requirements

✓ Execution compatibility with Control phase requirements

✓ Ability to meet the specified deadlines for each DMAIC phases

Criteria	Sub-Criteria
Leadership (Senior management commitment)	✓ Involvement, support and control of senior management during selection of project team members, application and evaluation of the project
Policies and Strategies	✓ Contribution of the project to the organization policies and strategies
Main Performance Results	✓ The extent to which the project has been able to meet the project objectives
	✓ Financial return and impact of the project
	✓ Contribution of the project to the innovative development
Social Results	✓ Social benefit
	✓ Impact on the environmental performance improvement
Employees (Six Sigma Project Team) - Personnel	✓ Contribution of the project to the employee (team members) motivation
	✓ Contribution of the project to the employee (team members) training and skills development
	✓ Contribution of the project to the employee (team members) and interdepartmental communication improvement
Cooperation and resources	✓ Contribution of the project to the cooperation with suppliers, supplier motivation for utilization of statistical tools and enhancement supplier quality level (reflection of Six Sigma philosophy to the supplier)
	✓ Utilization of proper and necessary resources
Project Execution Compatibility with DMAIC Methodology - (Methodology Compliance)	✓ Execution compatibility with Define phase requirements
	✓ Execution compatibility with Measure phase requirements
	✓ Execution compatibility with Analyze phase requirements
	✓ Execution compatibility with Improve phase requirements
	✓ Execution compatibility with Control phase requirements
	✓ Ability to meet the specified deadlines for each DMAIC phases

Leadership (senior management commitment) criteria examine the involvement, support, and control of senior management during the selection of the Six Sigma project team members, project implementation, and evaluation of the completed project. The direct involvement of the senior management in Six Sigma projects, rather than acting as an outside authority, is questioned within this criteria.

Policies and strategies criteria help to assess Six Sigma projects from the perspective of their contribution to organizational policies and strategies, such as an increase in market share.

Main performance results criteria directly focus on the results of the Six Sigma project. The project’s ability to meet the project objectives is the rate at which project goals, such as a decrease in customer complaints, defect rates, and process variability. The financial return and impact of the project sub-criterion is the Six Sigma project’s ability to rework and devalue added process costs, and reduce resource utilization, energy consumption costs, waste disposal costs, etc. The impact on innovative development criterion questions whether or not the completed Six Sigma project contributes to product and material innovation, process innovation, marketing, and managerial system innovation areas.

Social results criteria assess Six Sigma projects in terms of their impact on society and on the improvement in environmental performance. These criteria examine whether or not the completed project can be a social model project for other Six Sigma projects or other organizations. The measures for social dimensions mainly consider the project’s impact on employment rate, water/energy consumption, waste disposal, and the usage of hazardous materials.

Employees (Six Sigma Project Team) criteria examine the contribution of the Six Sigma project to the employee’s (team member’s) motivation, training, and skills development by using statistical tools and systematic methodology during the implementation of the project, as well as its contribution to employee and interdepartmental communication improvement.

Cooperation and resources criteria focus on the Six Sigma project’s contribution to collaboration with suppliers, supplier motivation for utilization of statistical tools, and the enhancement of the supplier’s quality level. Therefore, these criteria primarily address the project’s impact on the linkage between the Six Sigma philosophy and the supplier. Successful utilization of appropriate and necessary organizational resources for projects are is investigated under these criteria.

Project Execution Compatibility with DMAIC methodology criteria look at project execution compatibility with the systematic define, measure, analyze, improve, and control phase requirements. Additionally, the project’s ability to meet the specified deadline for each DMAIC phase is questioned.

Compatibility with define phase requirements address how well problem and project scope is defined, process maps are created, voice-of-customer is obtained and analysed, and measurable objectives are set within the project of interest.

Compatibility with measure phase requirements address how carefully the potential root causes of problems are stated, necessary data is compiled, and how accurately measurement systems analysis has been conducted.

Compatibility with analyze phase requirements address the depth of analysis and how well analysis is conducted, appropriate statistical tools are utilized, and root causes are determined.

Compatibility with improve phase requirements address how well improvement suggestions are made, potential risks related with improvement suggestions are assessed, appropriate improvements are selected and pilot-tested, and new applications are evaluated.

Compatibility with control phase requirements address how well project objectives are monitored and verified, and standardization and documentation are performed. Finally, the project’s ability to meet the determined timeline is evaluated within this category.

3.2 Interval type-2 fuzzy sets

Type-1 fuzzy logic is not capable of directly accounting for uncertainties, because it uses type-1 fuzzy sets that are described by only one membership function [31]. Thus, type-1 fuzzy set membership functions have no means to represent uncertainty [23]. Additionally, type-2 fuzzy logic systems are useful in situations where it is difficult to determine an exact numeric membership function [31]. Therefore, type-2 fuzzy sets have the potential to provide better performance than type-1 fuzzy sets in this application [32]. According to [32], ‘most researchers use interval type-2 fuzzy sets because of the extreme computational complexity of using general type-2 fuzzy sets’. Several studies have used interval type-2 fuzzy sets, such as [33 –35]. In this section, we first introduce interval type-2 fuzzy sets and some important definitions, which are adapted from Mendel [32] and Chen’s studies [34]. We then explain defuzzification methods in detail. The advantages of interval type-2 fuzzy sets is to consider uncertainty of the membership function. The interval can be defined for membership function by setting degree for it as seen in Fig. 2.

Fig. 2

The upper and lower trapezoidal membership function.

A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can be represented by a type-2 membership function $μ_{\tilde{\tilde{A}}}$ presented as follow [35]; $\tilde{\tilde{A}} = {((x, u), μ_{\tilde{\tilde{A}}} (x, u)) | x \in X, u \in [0, 1]}$ (1) $I_{x} = {u \in [0, 1] | μ_{\tilde{\tilde{A}}} (x, u) > 0}$ (2)

An interval type-2 fuzzy set (IT2 FS) is a type-2 fuzzy sets $\tilde{\tilde{A}}$ such that: $I_{x} = {u \in [0, 1] | μ_{\tilde{\tilde{A}}} (x, u) = 1}$ (3)

Then interval type-2 fuzzy set is called a closed interval type-2 fuzzy set (CIT2 FS) if I_x is closed interval for every x ∈ X.

In this paper, we refer to Chen and Lee’s study in [37] to model interval type-2 fuzzy sets where the heights of the upper membership functions and lower membership functions are used to define interval type-2 fuzzy sets. Figure 2 presents a trapezoidal interval type-2 fuzzy set: $\begin{matrix} {\tilde{\tilde{A}}}_{i} = ({\tilde{A}}_{i}^{U}, {\tilde{A}}_{i}^{L}) = (({\tilde{a}}_{i 1}^{U}, {\tilde{a}}_{i 2}^{U}, {\tilde{a}}_{i 3}^{U}, {\tilde{a}}_{i 4}^{U}; H_{1} ({\tilde{A}}_{i}^{U}), \\ (H_{2} ({\tilde{A}}_{i}^{U})), ({\tilde{a}}_{i 1}^{L}, {\tilde{a}}_{i 2}^{L}, {\tilde{a}}_{i 3}^{L}, {\tilde{a}}_{i 4}^{L}; (H_{1} ({\tilde{A}}_{i}^{L}), H_{2} ({\tilde{A}}_{i}^{L}))) \end{matrix}$ (4)

where ${\tilde{A}}_{i}^{U}$ and ${\tilde{A}}_{i}^{L}$ are type-1 fuzzy sets, ${\tilde{a}}_{i 1}^{U}$ , ${\tilde{a}}_{i 2}^{U}$ , ${\tilde{a}}_{i 3}^{U}$ , ${\tilde{a}}_{i 4}^{U}$ , ${\tilde{a}}_{i 1}^{L}$ , ${\tilde{a}}_{i 2}^{L}$ , ${\tilde{a}}_{i 3}^{L}$ , and ${\tilde{a}}_{i 4}^{L}$ are the reference points of the interval type-2 fuzzy set and H_i( ${\tilde{A}}_{i}^{U}$ ), H_i( ${\tilde{A}}_{i}^{L}$ ) denote the membership values [37]. In Fig. 2, ${\tilde{A}}_{i}^{U}$ is the upper and ${\tilde{A}}_{i}^{L}$ is the lower trapezoidal membership function of the interval type-2 fuzzy set ${\tilde{\tilde{A}}}_{i}$ [37].

3.3 Defuzzification methods for type-2 fuzzy numbers

Although there are many existing defuzzification methods for obtaining a crisp number from type-1 fuzzy numbers, few defuzzification methods have been published for type-2 fuzzy numbers. First, type-2 fuzzy numbers are defuzzified into type-1 fuzzy numbers, which are then redefuzzified to a crisp value.

Three defuzzification methods for type-2 fuzzy number are introduced in this paper that are: ‘the centroid of a type-2 fuzzy set’, ‘type reduction indices methods’, and ‘Kahraman et al.’s methods’ [23 , 38–41].

The centroid of a type-2 fuzzy set is described by the following equations, where is the centroid of an interval type-2 fuzzy set $\tilde{\tilde{A}}$ : $C_{\tilde{\tilde{A}}} = \frac{1}{[C_{l}, C_{r}]}$ (5) $C_{l} = \min (C_{\tilde{\tilde{A}}}) = \frac{\int_{- \infty}^{L} x . {\bar{μ}}_{\tilde{\tilde{A}}} (x) d_{x} + \int_{L}^{\infty} x . {\underline{μ}}_{\tilde{\tilde{A}}} (x) . d_{x}}{\int_{- \infty}^{L} {\bar{μ}}_{\tilde{\tilde{A}}} (x) d_{x} + \int_{L}^{\infty} {\underline{μ}}_{\tilde{\tilde{A}}} (x) . d_{x}}$ (6) $C_{r} = \max (C_{\tilde{\tilde{A}}}) = \frac{\int_{- \infty}^{R} x . {\bar{μ}}_{\overset{\begin{array}{l} \approx \end{array}}{A}} (x) d_{x} + \int_{R}^{\infty} x . {\underline{μ}}_{\tilde{\tilde{A}}} (x) . d_{x}}{\int_{- \infty}^{L} {\bar{μ}}_{\tilde{\tilde{A}}} (x) d_{x} + \int_{L}^{\infty} {\underline{μ}}_{\tilde{\tilde{A}}} (x) . d_{x}}$ (7)

C_l and C_r are the minimum and maximum points in the centroid of $\tilde{\tilde{A}}$ , respectively. L and R are the switch points that define the change from ${\bar{μ}}_{\tilde{\tilde{A}}} (x)$ to ${\underline{μ}}_{\tilde{\tilde{A}}} (x)$ and ${\underline{μ}}_{\tilde{\tilde{A}}} (x)$ to ${\bar{μ}}_{\tilde{\tilde{A}}} (x)$ , respectively. In order to solve Equations (6 and 7), the L and R points must be known [38 , 41].

Type reduction indices methods were developed by Niewiadomski et al. [39]. They introduced optimistic, pessimistic, realistic, and weighted average indices. These indices can be described by the equations below: ${TR}_{opt} (\tilde{\tilde{A}}) = {\bar{μ}}_{\tilde{\tilde{A}}} (x), x \in X$ (8) $T R_{o p t} (\tilde{\tilde{A}}) = {\underline{μ}}_{\tilde{\tilde{A}}} (x), x \in X$ (9) $T R_{r e} (\tilde{\tilde{A}}) = 0.5 {\underline{μ}}_{\tilde{\tilde{A}}} (x) + {\underline{μ}}_{\tilde{\tilde{A}}} (x), x \in X$ (10) $T R_{w a} (\tilde{\tilde{A}}) = w_{1} {\bar{μ}}_{\tilde{\tilde{A}}} (x) + w_{2} {\underline{μ}}_{\tilde{\tilde{A}}} (x), x \in X$ (11)

where w₁ and w₂ are coefficients that satisfy w₁ + w₂ = 1.

A modified version of Kahraman et al.’s method for defuzzification. By using a best non-fuzzy performance (BNP) value, defuzzification and ranking methods for both triangular and trapezoidal type-2 fuzzy sets were developed by Kahraman et al. [23]. This defuzzified method adjusts the center of area method for interval type-2 fuzzy set. The method is for trapezoidal type-2 fuzzy numbers is shown below: $\begin{matrix} TraT & = & [\frac{(u_{U} - l_{U}) + (β_{U} . m_{1 U} - l_{L}) + (α_{U} . m_{2 U} - l_{U})}{4} + l_{U}] / 2 \\ + \dots + [\frac{(u_{L} - l_{L}) + (β_{L} . m_{1 L} - l_{L}) + (α_{L} . m_{2 L} - l_{L})}{4} + l_{L}] / 2 \end{matrix}$ (12)

where α and β are the lower membership function’s maximum membership degrees for the considered type-2 fuzzy set, u_U is the upper membership function’s largest possible value, l_U is the upper membership function’s smallest possible value, m_1U and m_2U are the upper membership function’s second and the parameters, u_L is the lower membership function’s largest possible value, l_L is the lower membership function’s smallest possible value, and m₁L and m₂L are the lower membership function’s second and third parameters. In this paper, Kahraman et al.’s method for defuzzification is used.

4 Interval type-2 fuzzy Analytic Network Process and application

In the current study, the interval type-2 fuzzy analytic network process (FANP) method is used for criteria weighting and evaluating completed Six Sigma projects. The theoretical structure of the interval FANP was proposed by Şentürk et al. in [42]. The steps for analysis are as follows:

Step 1. Comparing the criteria

Seven main and 18 sub-criteria were determined in Section 3.1 and the main frame of the evaluation for completed Six Sigma projects is presented in Fig. 3. After the determination of criteria, a pairwise comparison matrix for the main criteria is constructed using linguistic variables. The definitions of the linguistic variables and their trapezoidal interval type-2 fuzzy scales are provided in Table 2. Linguistic comparisons between the criteria are made by a team based on the definitions in Table 2. The fuzzy pairwise comparison matrix for the main criteria using linguistic terms is presented in Table 3.

Fig. 3

The main frame of the evaluation of completed Six Sigma projects.

Table 2

Definitions and the corresponding interval type-2 fuzzy scalesof the linguistic variables (adapted from Kahraman et al. [23])

Linguistic Variables Definition	Trapezodial Interval Fuzzy Scales
Equally important (EI)	(1,1,1,1;1,1) (1,1,1,1;1,1)
Slightly important (SI)	(1,2,4,5;1,1) (1.2,2.2,3.8,4.8;0.8,0.8)
Fairly important (FI)	(3,4,6,7;1,1) (3.2,4.2,5.8,6.8;0.8,0.8)
Very strongly important (VSI)	(5,6,8,9;1,1) (5.2,6.2,7.8,8.8; 0.8,0.8)
Absolutely important (AI)	(7,8,9,9;1,1) (7.2, 8.2, 8.8, 9; 0.8, 0.8)

Table 3

Pairwise comparison matrix for main criteria in linguistic terms

	Leadership	Policies and strategies	Performance results	Social results	Methodology Compliance	Personnel	Cooperation-Resources
Leadership	EI	VSI	1/FI	VSI	VSI	SI	SI
Policies and strategies	1/VSI	EI	1/VSI	VSI	FI	VSI	VSI
Performance results	1/FI	VSI	EI	FI	SI	FI	VSI
Social results	1/VSI	1/VSI	1/FI	EI	1/VSI	1/SI	1/FI
Methodology Compliance	1/VSI	1/FI	1/SI	VSI	EI	VSI	VSI
Personnel	1/SI	1/VSI	1/FI	SI	1/VSI	EI	SI
Cooperation-Resources	1/SI	1/VSI	1/VSI	FI	1/VSI	1/SI	EI

After the construction of the fuzzy pairwise comparison matrix in linguistic terms (Table 3), the fuzzy pairwise comparison matrix for the main criteria using interval type-2 fuzzy numbers is created by given in Equation 12. $\tilde{\tilde{A}} = | \begin{matrix} 1 & {\tilde{\tilde{a}}}_{12} & \dots & {\tilde{\tilde{a}}}_{1 n} \\ {\tilde{\tilde{a}}}_{21} & 1 & \dots & {\tilde{\tilde{a}}}_{2 n} \\ \dots \\ {\tilde{\tilde{a}}}_{n 1} & {\tilde{\tilde{a}}}_{n 2} & \dots & 1 \end{matrix} | = | \begin{matrix} 1 & {\tilde{\tilde{a}}}_{12} & \dots & {\tilde{\tilde{a}}}_{1 n} \\ 1 / {\tilde{\tilde{a}}}_{21} & 1 & \dots & {\tilde{\tilde{a}}}_{2 n} \\ \dots \\ 1 / {\tilde{\tilde{a}}}_{n 1} & 1 / {\tilde{\tilde{a}}}_{n 2} & \dots & 1 \end{matrix} |$ (13)

where: $1 / \tilde{\tilde{a}} = (\begin{matrix} (\frac{1}{a_{14}^{U}}, \frac{1}{a_{13}^{U}}, \frac{1}{a_{12}^{U}}, \frac{1}{a_{11}^{U}}; H_{1} (a_{12}^{U}), H_{2} (a_{13}^{U})) \\ (\frac{1}{a_{24}^{L}}, \frac{1}{a_{23}^{L}}, \frac{1}{a_{22}^{L}}, \frac{1}{a_{21}^{L}}; H_{1} (a_{22}^{L}), H_{2} (a_{23}^{L})) \end{matrix})$ (14)

The fuzzy pairwise comparison matrix is constructed through the above operation.

The fuzzy pairwise comparison matrix for the main criteria using interval type-2 fuzzy numbers is calculated. Following this operation, the consistency of the pairwise comparison matrix is analysed. Similar to a classical fuzzy ANP, the consistency is checked by using a defuzzified matrix where the consistency ratio of the defuzzified pairwise comparison matrix is smaller than 0.1.

To check consistency ratio, trapezodial fuzzy numbers applied defuzzification by the graded mean integration method: $A = \frac{l + 2 m_{1} + 2 m_{2} + u}{6}$ (15)

for validating the pairwaise comparision matrix, the consistency index proposes Saaty [11] and Dong and Cooper [43] improved the consistency with pairwise comparisons matrices for AHP method by reducing the number of needed comparisons significantly. Also Zangand and Guo [44] proposed an algorithm based on a linear programming to check and improve the consistency of an intuitionistic multiplicative preference relation. In addition, Wu and Liao [45] improve the multiplicativeconsistency-based multi-objective programming model that has and advantages not losing information. Wu et al. [46] calculated three levels of consensus degree with distributed linguistic trust functions for identifying the inconsistent users. Wu et al. [47] developed a trust based recommendation mechanism to achieve higher levels of consensus for inconsistency. Liu et al. [48] introduced interval-valued trust functions, interval-valued trust score and interval-valued knowledge to generate personalised advices for the inconsistent experts to reach higher consensus level.

Step 2. Calculation of geometric means and fuzzy weights The geometric mean of each row is computed after the fuzzy weights are calculated through normalization. The geometric mean of each row ${\tilde{\tilde{r}}}_{i}$ is calculated as: ${\tilde{\tilde{r}}}_{i} = {({\tilde{\tilde{a}}}_{i 1} \otimes \dots \otimes {\tilde{\tilde{a}}}_{in})}^{1 / n}, \forall I$ (16)

The geometric means of all rows (i.e., the main criteria) are computed in a similar manner. The fuzzy weight of the ith criterion is computed as: ${\tilde{\tilde{w}}}_{i} = {\tilde{\tilde{r}}}_{i} \otimes {({\tilde{\tilde{r}}}_{i} \oplus \dots \oplus {\tilde{\tilde{r}}}_{n})}^{- 1}$ (17)

The interval type-2 fuzzy weights for all rows (i.e., the main criteria) are computed in a manner. These calculated weights are presented Table 4. Finally, fuzzy utilities are calculated as follows:

Table 4

Interval Type-2 fuzzy weights for the main criteria with respect to goal

Main Criteria	Fuzzy Weights
Leadership	(0.10,0.16,0.34,0.50;1,1)(0.12,0.18,0.31,0.46;0.8,0.8)
Policies and strategies	(0.8,0.12,0.21,031;1,1)(0.9,0.12,0.20,0.28;0.8,0.8)
Performance results	(0.17,0.27,0.54,0.77;1,1)(0.19,0.29,0.50,0.71;0.8,0.8)
Social results	(0.01,0.02,0.03,0.05;1,1)(0.01,0.02,0.03,0.05;0.8,0.8)
Methodology Compliance	(0.06,0.08,0.16,0,25;1,1)(0.06,0.09,0.15,0.22;0.8,0.8)
Personnel	(0.02,0.03,0.07,0.12;1,1)(0.02,0.04,0.07,0.11;0.8,0.8)
Cooperation-Resources	(0.02,0.03,0.06,0.09;1,1)(0.02,0.03,0.05,0.08;0.8,0.8)

{\tilde{\tilde{U}}}_{i} = \sum_{l = 1}^{n} {\tilde{\tilde{w}}}_{j} {\tilde{\tilde{r}}}_{ij}

(18)

where ${\tilde{\tilde{U}}}_{i}$ is the fuzzy utility of alternatives.

Step 3. Calculation of defuzzified weights Deffuzzified weights are calculated using the DTraT method for trapezoidal type-2 fuzzy numbers proposed by Kahraman et al. in [23], presented in equation (17). These weights are presented inTable 5.

Table 5

Defuzzified weights for the main criteria with respectto goal by using DTraT method

Leadership	0.4495
Policies and strategies	0.2725
Performance results	0.6966
Social results	0.0442
Methodology Compliance	0.2129
Personnel	0.1056
Cooperation-Resources	0.0793

Step 4. Calculation of the global weights via Super Decisions^® software After the defuzzified weights attained, the weights for each pairwise comparison matrix are used directly in the Super Decisions^® software in accordance with the proposed model. Through the use of the Super Decisions^® software, an unweighted supermatrix, weighted supermatrix, and limit matrix are computed. The calculated global weights for the main and sub-criteria, which indicate their relative priorities, are presented in Table 6. By observing this table, it can be seen that the highest weight among the seven main criteria belongs to the performance results criterion, while the achieving the project objectives criterion is found to be the most effective criteria among the sub-factors.

Table 6

The ranking of the completed Six Sigma projects

Factors and local weights	Sub Factors and local weights	Global weights
Leadership (Senior management commitment) (0.242)	- Involvement, support and control of senior management during selection of project team members, application and evaluation of the project (1)	(0.242)
Policies and strategies (0.146)	- Contribution of the project to the organization policies and strategies (1)	(0.146)
Main Performance Results (0.374)	- Meet the project objectives (0.698)	(0.261)
	- Financial return and impact of the project (0.206)	(0.077)
	- Contribution of the project to the innovative development (0.096)	(0.036)
Social results (0.023)	- Social benefit (0.210)	(0.005)
	- Impact on the environmental performance improvement (0.790)	(0.018)
Project Execution Compatibility with DMAIC Methodology - (Methodology Compliance) (0.114)	- Execution compatibility with Define phase requirements (0.072)	(0.0008)
	- Execution compatibility with Measure phase requirements (0.222)	(0.0253)
	- Execution compatibility with Analyze phase requirements (0.230)	(0.0260)
	- Execution compatibility with Improve phase requirements (0.280)	(0.0320)
	- Execution compatibility with Control phase requirements (0.141)	(0.0160)
	- Ability to meet the specified deadlines for each DMAIC phases (0.053)	(0.0060)
Employees (Six Sigma Project Team) - Personnel (0.056)	- Contribution of the project to the employee (team members) motivation (0.114)	(0.006)
	- Contribution of the project to the employee (team members) training and skills development (0.593)	(0.033)
	- Contribution of the project to the employee (team members) and interdepartmental communication improvement (0.292)	(0.016)
Cooperation-Resources (0.053)	- Utilization of proper and necessary resources (0.573)	(0.030)
	- Reflection of Six Sigma philosophy to supplier (0.427)	(0.023)

Step 5. Evaluation of completed Six Sigma projects The weights for the main and sub-criteria were obtained in step 4. Next, 15 Six Sigma projects, which were selected from a pool of completed projects within the household and appliances company, were suggested for evaluation by the Six Sigma Master Black Belt at the same company. A scale of 1–10, where 1 is the worst and 10 is the best, is used for evaluation. Finally, the projects are ranked by using weighted sum methods. Table 7 presents the completed project rankings. It is suggested that the best completed project is project 15, with project 4 being the second best, and project 5 being the third best. This evaluation yielded a different ranking for the projects when compared to the company’s listed ranking. In real-world practice, the company evaluates completed Six Sigma projects by examining only the compatibility of the project with the theoretical DMAIC steps. It is appropriate but not enough methodology. Because one completed project can be meet all of the steps of DMAIC procedure but can be a little effect of all over the company. With the proposed project performance evaluation system, it is possible to evaluate the project’s effect on the company in wide spread and compare diverse project types with different backgrounds and goals by taking care of also EFQM features’.

Table 7

Global and sub-weights for the main and sub-criteria

Project weighted sum	The Fuzzy ANP Ranking	Type 2 compatibility ranking	DMAIC
Project 15	8.643	1	4
Project 4	8.376	2	3
Project 5	8.166	3	7
Project 11	7.696	4	2
Project 2	7.683	5	5
Project 3	7.649	6	1
Project 6	7.373	7	9
Project 9	7.264	8	8
Project 13	7.055	9	10
Project 14	6.591	10	11
Project 1	6.458	11	6
Project 12	4.947	12	14
Project 10	4.350	13	15
Project 8	4.035	14	12
Project 7	3.714	15	13

5 Discussion

The completed Six-Sigma project evaluation problem falls within the fuzzy multiple criteria decision-making (FMCDM) problem domain. FMCDM methods have been developed due to imprecision in assessing the relative importance of criteria and the performance ratings of alternative techniques [4]. In order to overcome the difficulty of conducting a pairwise comparison between alternatives using linguistic variables, fuzzy set theory has been adopted in this study. We have utilized an interval type-2 fuzzy ANP approach for evaluation of completed Six Sigma projects and have extended the comparison criteria by combining EFQM model components with the traditional DMAIC steps criteria for Six Sigma project evaluation to a larger extent, including evaluation of project impact on society and the environment.

As previously mentioned, in real-world practice, the company considered in this study evaluates completed Six Sigma projects by examining only the compliance of the project with the theoretical DMAIC steps. However, other aspects and contributions of projects should be considered for superior project evaluations. The results of this study reveal the fact that the main performance results criterion (with a global weight of 0.374) is the most significant evaluation criterion, followed by the leadership (0.242) and policies and strategies (0.146) criteria. Although the project execution compatibility with DMAIC methodology criterion is also important, it is ranked below the main performance results, leadership, and policies and strategies criteria. Furthermore, the degree of meeting the project objectives criterion is the most effective criterion among the sub-factors, followed by the involvement, support and control of senior management during selection of project team members, application and evaluation of the project (0.242), and contribution of the project to the organization policies and strategies (0.146) criteria. These results indicate that although execution compatibility with DMAIC steps requirements is seen as being important in real-world practice, it is also important to consider different aspects of projects, as well as project outcomes, and to evaluate completed projects accordingly.

6 Conclusions and future research

The main objective of this study was to identify a new method for evaluating the effectiveness of completed Six Sigma projects. A review of Six Sigma project evaluation literature revealed the fact that former studies on the topic have largely focused on project selection. Although selection of the right projects is crucial, especially during the deployment of quality improvement approaches, it is also important to evaluate and assess completed projects in order to determine their relative success for guiding superior future implementations. The evaluation of completed projects should be thorough, unbiased, straightforward, and objective. Therefore, we have developed measurement structure by combining traditional and novel criteria for Six Sigma, expert experience gained over many years, and the concept of fuzzy set theory. The main contribution of proposed methodology is to presents new evaluation criteria that come from both DIMAC steps and EFQM components. Therefore, completed Six Sigma projects can be evaluated more true and widen criteria. In addition, the scores of Six Sigma projects can show the best example to other project leaders which projects are more valuable for company. Other contribution is to provide real and unbiased evaluation method that can be used for salary increases or rewards. Also the linguistic variables are modelled with fuzzy numbers in this study.

Fuzzy set approach is an excellent way to deal with linguistic variables and provide reasonable interpretations of Six Sigma project evaluations. The novel evaluation structure formulated in this study is based on an interval type-2 fuzzy ANP approach and considers the following main criteria for Six Sigma project implementation: leadership, policies and strategies, main performance results, social results, employees (Six Sigma project team), cooperation and resources, and project execution compatibility with DMAIC methodology.

We determine appropriate weights for the main and sub-criteria through an interval type-2 fuzzy ANP and ultimately assign suitable performance ratings for each criteria for the effective implementation of completed Six Sigma projects. Although an interval type-2 fuzzy AHP was previously used in the literature, using an interval type-2 fuzzy ANP is a novel approach to multiple criteria decision making. The suggested Six Sigma project evaluation methodology was tested on 15 completed projects from one company. An effective project performance measurement system should be able to measure and compare projects of various types. The proposed rating system makes it possible for management to compare different projects with diverse requirements and objectives.

The horizon of this study can be extended by adding additional levels of evaluation criteria. The completed project performance measurement system can also be modified for different types of organizations, such as manufacturing and services, as well as for different industries. Furthermore, a different fuzzy sets approach and using another defuzzification techniques could be tested. Future work in this area could be to identify specific criteria weights, create particular project evaluation structures for different industries, and enhance the use of fuzzy logic within project evaluation structures.

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