Study on performance evaluation of the production process

Abstract

Improvement of the production process presents a very important management task for both researchers and practitioners and enables a better market position of the enterprise. Key Performance Indicators (KPIs) of the production process can provide useful information on the current state of the ongoing process. In this paper, the relative importance of KPIs and their values at the enterprise level were assessed by the experts and decision-makers. Their estimates are described by the linguistic variables which were modeled by intuitionistic fuzzy numbers. The weights vector of KPIs at the level of the considered enterprise is given by the Fuzzy Analytic Hierarchical Process (FAHP) with Triangular Intuitionistic Fuzzy Numbers (TIFNs). The rank of enterprises with respect to KPIs’ values and their weights was calculated using the modified TOPSIS with TIFNs. The developed model was tested on 30 enterprises from Serbia, belonging to the sector of small and medium-sized (SME) production enterprises. The improvement strategies of KPIs should be proposed at the level of each enterprise, separately, respecting the KPIs’ values of the first-ranked enterprise.

Keywords

Intuitionistic fuzzy numbers fuzzy AHP fuzzy TOPSIS production process benchmarking

1 Introduction

Within business processes activities in each organization, very important task of the manager is continuous improvement by applying appropriate analysis tools, followed by management initiatives required to achieve long-term business goals [24]. Therefore, the improvement of the business processes, primarily the production process in industrial enterprises, has become an important topic of research for both industry and academia in the last decades.

In the literature, as well as in practice, there are models and frameworks that provide guidelines for the development of performance measurement system and review of performance measurement problems from different perspectives [2]. It is a common practice that performance with known current value is the one that can be improved [38], so numerous performance measurement systems have been deployed by using the different logical and mathematical ground [13, 23]. Also, it is worth mentioning that some performances’ values cannot be determined precisely so they should be decomposed to their constituents which are represented by appropriate indicators. Performance indicators provide an important link between strategies and management actions and in this way support the implementation and execution of the improvement actions [38]. The performance indicators are categorised [10] as: Result Indicators-RIs, Key Result Indicators-KRIs, Performance Indicators-PIs and Key Performance Indicators-KPIs. Having that in mind, KPIs may be seen as quantifiable measurements used to help an organization measure the success of critical factors [10]. KPI-based performance measurement is the basis for successful business systems management [3]. Based on their values, it is possible to monitor the realization of the process and determine the management activities focused on improvement. The assessment of the production process performances should be based on determining KPIs [37]. In the literature, there are no rules on how to determine KPIs, so the widely used approaches are the decision maker’s assessment and the results of the best practices [36].

Besides determining the finite number of KPIs within one business process, there is another issue – determining the relative importance of each KPI. This issue may be stated as a fuzzy group decision-making problem [35, 36]. It may be assumed that it is closer to a human way of thinking to consider each pair of KPIs separately during the evaluation of their relative importance. Starting from this assumption, many authors have constructed a fuzzy pair-wise comparison matrix of the relative importance and used different procedures to determine the weights vector [11, 30]. In this paper, the relative importance of each pair of KPIs at the level of each SME is described by TIFNs, and the weights vector is calculated using FAHP [30]. The aggregated weights of KPIs are calculated using the Fuzzy Averaging Operator with TIFNs [19].

It is assumed that the decision-makers in enterprises can make their estimates more precisely by employing linguistic expressions compared to crisp values. Many authors consider that applying the theory of fuzzy sets [7, 17] can be the best and the least complex way to quantitatively describe uncertain variables. The modeling of uncertainties may be based on type-1 fuzzy sets applications [30, 36]. The higher degree of flexibility may be achieved by applying type-2 fuzzy sets while describing uncertainties, but at the same time, type-2 fuzzy sets have more complex calculations [4]. Since the intuitionistic fuzzy sets (IFSs) are characterized by the membership and non-membership functions, they can be designated as the best solution to represent the human way of thinking in solving different MCDM problems [41]. In this research, all existing uncertainties are modeled by TIFNs.

The ranking of enterprises with respect to the relative importance and values of defined KPIs under uncertainties can be stated as the fuzzy multi-criteria decision making (MCDM) problem [11]. The choice of an appropriate method for the ranking of considered items can be viewed as a particular problem. One of the most used MCDM is the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method [39]. In this research, the rank of SMEs with respect to the production processes’ KPIs is determined by using the modified TOPSIS with TIFNs. As a result, the decision-makers may use the obtained rank and KPIs of the first-ranked SME as a benchmark for defining the priorities and appropriate management initiatives. These initiatives should lead to the improvement of the overall efficiency of the production process in other SMEs.

This paper is organized as follows: The literature review is presented in Section 2. The evaluation framework is presented in Section 3. Modeling of existing uncertainties with TIFNs and modified TOPSIS are described in Section 4. In Section 5, data from the Serbian production SMEs were used to verify the developed model. Conclusions are presented in Section 6.

2 Literature review

There are no literature recommendations or standards that designate the most appropriate KPIs for a certain business process. It is common practice to determine KPIs based on the literature data, the results of good practice, knowledge and experience of the management team, etc. In this paper, the KPIs of the production process in SMEs are determined in compliance with the existing literature [37].

In order to have the information necessary for the realization of an enterprise’s vision, decision-makers need to establish an effective method of evaluating the performance of their business processes [38]. To make this evaluation possible, KPIs must be determined, measured and managed for each identified process [36]. It is very important for business managers to identify on time the KPIs with low values and to determine ways to improve them.

Evaluation of business processes performance itself is a very complex task [13]. Some data used in evaluation can be obtained by measuring or reviewing records and thus can be precisely represented by precise numbers. However, there are many ambiguities and uncertainties in decision-making problems. Also, KPIs can have a different nature (be cost-type or benefit-type) and be presented in different measurement units [7]. In multi-criteria decision-making problems, the criteria considered (in this paper, these are KPIs) may not have the same importance or weight. Determining these values and weights is based on the subjective judgment of decision-makers and it is difficult to express them in precise numbers. These are the reasons why the KPIs’ values and weights in this paper were modeled using the fuzzy sets.

The modeling of uncertainties and imprecisions that exist in the considered decision-making problem can be performed more precisely using IFSs than using the type-1 fuzzy sets [28, 33]. These fuzzy sets are characterized by their membership and non-membership functions. The most common shapes of their membership functions are triangular and trapezoidal membership functions, so in this research, existing uncertainties are modeled by TIFNs.

The good practice implies that decision-makers can make their assessments more easily and accurately if they consider each pair of attributes individually rather than all of them at once. A significant number of conducted research describes the extensions of the conventional AHP method with fuzzy sets theory [4] and AHP with TIFNs [22 , 31].

The relative importance of criteria may be described by TIFNs and the domains of these TIFNs may be defined into the interval from 1 to 9 [31], or from 0.8 to 8.2 [30], or from 1 to 10 [26], or from 0 to 9 [22], etc. In this research, the FAHP with TIFNs is used to determine the KPIs’ weights of the production process in SMEs. The domain of used TIFNs is defined on the common measurement scale [40].

It is known that the decision-makers can make unintentional mistakes in the decision making process. Consistency should be checked to determine whether the errors significantly affect the results or they can be ignored. When precise numbers are employed during the group decision-making process, consistency checking may be determined by the appropriate procedure [5]. Consistency checking of the decisionmakers’ fuzzy assessments is usually performed by using the eigenvalue vector method after applying the defuzzification procedure [25 , 31]. Also, it should be mentioned that a consistency check may be performed according to other proposed procedures [22, 42] based on the eigenvalue vector method [40]. On the other hand, some researchers did not perform a consistency check of decision-makers’ assessments [26]. In this research, a consistency check of the stated fuzzy pair-wise comparison matrix is performed according to the conventional AHP.

The weights vector of KPIs can be calculated by using a fuzzy geometric mean [30, 31], extent analyses [26], or some other procedure [42]. In this paper, the weights vector of KPIs is determined according to the existing procedure [30].

As a goal of the research, the rank of alternatives should be obtained. Frequently, in the literature, the rank of different items is calculated by using the modified TOPSIS with IFSs. The combination of TOPSIS and IFSs has been employed in different areas [8 , 39].

In the scope of the research regarding supplier selection [15], the ratings of each alternative, with respect to each criterion and their weights, are given as linguistic terms and modeled by IFSs. In order to aggregate individual decision-makers’ opinions, an intuitionistic fuzzy weighted averaging operator is used. Intuitionistic Fuzzy Positive Ideal Solution (IFPIS) and Intuitionistic Fuzzy Negative Ideal Solution (IFNIS) are calculated according to intuitionistic fuzzy theory and the principle of classical TOPSIS [6]. Determining the distances from IFPIS and IFNIS is based on the Euclidean distance.

The research on personnel selection problem is based on TOPSIS with TIFNs [39]; TIFNs are used to describe the criteria weights and their values for each considered alternative. The domains of TIFNs used are defined on a scale of 0 to 1, i.e. 0 to 10, respectively. The normalized values of the fuzzy decision matrix elements are calculated by using the appropriate procedure [16]. The magnitudes of membership and non-membership function for TIFNs are given by using the existing procedure [32]. IFPIS and IFNIS are obtained by the ranking of the calculated magnitudes. The distances of each alternative from the IFPIS and IFNIS are calculated by existing literature [32].

TOPSIS with IFNs is used for the ranking problem of different types of conventional and alternative passenger vehicles in the US [27]. The criteria weights are described by IFNs and given as normalized values. The assessment of elements’ values in the fuzzy decision matrix is stated as a fuzzy group decision-making problem. The aggregation of fuzzy assessments of the decision-makers is performed using IFWA (Intuitionistic Fuzzy Weighted Averaging) operator. IFPIS and IFNIST are calculated in compliance with existing literature [15]. A fuzzy normalized Euclidean distance formula is used to calculate distance measures.

TOPSIS with IFNs is also used for the portfolio selection problem in [8]. The criteria weights are determined by using the procedure developed in [29]. IFPIS and IFNIS are determined in compliance with the described research (see [15]). The calculation of distances in that paper is based on the Hausdorff metric [29].

TOPSIS with picture fuzzy sets (PFS) is employed for the project selection problem in [20]. It is shown that PFSs are more suitable for this kind of problem than type-1 fuzzy sets since PFSs can be applied to the situation where human opinions include more answer types. The values of alternatives are described by PFSs and the fuzzy decision matrix is constructed by using fuzzy algebra rules. The PIS and NIS are defined according to the procedure developed by Chen [33] and the distances according to the procedure proposed by Ye [18].

In this paper, the IFPIS and IFNIS are defined according to the weighted normalized fuzzy decision matrix by applying a similar procedure as in described research [15]. In order to determine the distances from IFPIS and IFNIS, the Hamming distance [14, 29] is calculated. In all of the analyzed papers, as well as in this research, the closeness coefficient is calculated with respect to conventional TOPSIS. The rank of considered items is determined according to the obtained values of the closeness coefficient.

3 Evaluation framework

In this paper, the problem of ranking the SMEs by respecting the production processes’ KPIs is presented. SMEs can be seen as a set E = {1,..., e,...,E}, where e is the SME index, and E is the total number of SMEs.

KPIs of the production process can be formally represented by the set ς= {1,..., i,...,I}, where i is the KPI’s index, and I is the total number of KPIs considered. In the scope of this research, the expert team that defined the set of KPIs, as well as their relative importance, was consisted of six quality management auditors from a consulting and auditing agency in Serbia. KPIs were adopted according to the existing literature [37] and filtered by the expert team due to the size and business activity of the SMEs: Percentage of customer complaints caused by non-conformities (i = 1), Percentage of non-conformities (i = 2), Percentage of production plan realization (quantity) (i = 3), Percentage of scrap and treatment costs in production (i = 4) and Time of unplanned delays/total production cycle time (i = 5).

It is assumed that the importance of these KPIs is not the same compared to each other. The estimates of the relative importance of KPIs at the level of each concerned SME are given by the expert team and they can be formally presented by the set of indices ψ = {1,..., k,...,K}, where k is the expert’s index, and K is the total number of experts.

In compliance with the existing data in this area, which is not well known, the use of an appropriate mathematical tool should be considered. TIFNs are a good solution for the above mentioned problem since their elements are described with membership function and non-membership function at the same time. The assessments are based on experts’ team knowledge and experience, where experts use the linguistic expressions modeled by TIFNs. The relative importance of KPIs is given by the comparison matrix with TIFNs, by analogy to the conventional AHP method. The elements of the fuzzy pair-wise matrix of the KPIs’ relative importance are: ${\tilde{w}}_{{ii}^{'}}^{k} = {(w_{{ii}^{'} 1}^{k}, w_{{ii}^{'} 2}^{k}, w_{{ii}^{'} 3}^{k}), (w_{{ii}^{'} 1}^{k^{'}}, w_{{ii}^{'} 2}^{k}, w_{{ii}^{'} 3}^{k^{'}})}$

If the relative importance of KPI i’ over the KPI i is higher, then the element in the fuzzy pair-wise -matrix of the relative importance of KPIs is the triangular TIFN:

${\tilde{w}}_{{ii}^{'}}^{k} = {(\frac{1}{w_{{ii}^{'} 3}^{k}}, \frac{1}{w_{{ii}^{'} 2}^{k}}, \frac{1}{w_{{ii}^{'} 1}^{k}}), (\frac{1}{w_{{ii}^{'} 3}^{k^{'}}}, \frac{1}{w_{{ii}^{'} 2}^{k}}, \frac{1}{w_{{ii}^{'} 1}^{k^{'}}})}$

The mapping of the fuzzy pair-wise comparison matrix of KPIs into the pair-wise comparison matrix of KPIs is performed by using the developed defuzzification procedure [1]. The consistency of the assessment is verified by using the eigenvector [40]. The weights vector is calculated by applying the appropriate procedure [30]. The weight of each KPI at the level of each expert is denoted as ${\tilde{w}}_{i}^{k}$ , i = 1,..,I; k = 1,...,K. The overall weights vector ${\tilde{w}}_{i}$ , i = 1,..,I, is calculated by using the Fuzzy Averaging Operator with TIFNs [19].

The KPIs’ values at the level of individual SMEs are estimated by the decision-makers from each SME. Within the scope of business activity, the quality manager in each SME is denoted as a decision-maker who assessed the values of the proposed KPIs. They use seven pre-defined linguistic expressions which are modeled by TIFNs, ${\tilde{v}}_{i}^{e}$ , i = 1,..,I; e = 1,..,E: ${\tilde{v}}_{i}^{e} = {(v_{i 1}^{e}, v_{i 2}^{e}, v_{i 3}^{e}), (v_{i 1}^{e'}, v_{i 2}^{e}, v_{i 3}^{e'})}$

As the KPIs may be benefit-type or cost-type, it is necessary to determine their normalized value, ${\tilde{r}}_{i}^{e}$ , i = 1,..,I; e = 1,...,E. In the scope of this research, the normalization of the values of KPIs is performed by using a linear normalization method [9].

The elements of the weighted normalized fuzzy decision matrix, ${\tilde{d}}_{i}^{e}$ , i = 1,..,I; e = 1,...,E, are calculated as the multiplication of the weight of the KPI, ${\tilde{w}}_{i}$ , and its normalized value, ${\tilde{r}}_{i}^{e}$ . Based on the TIFN multiplication rule [19], weighted normalized values of KPIs are also described by TIFNs.

Fuzzy Positive Ideal Solution with TIFNs (TIFPIS) and Fuzzy Negative Ideal Solution with TIFNs (TIFNIS) under each KPI present the highest and the lowest values of the weighted normalized KPI values, respectively. They are determined by using the procedure for the ranking of TIFNs [34]. Determining the distance from TIFPIS and TIFNIS is based on the existing procedure [18]. The rank of the enterprises is determined according to closeness coefficient values given by the conventional TOPSIS method. It may be assumed that the effectiveness of the production process is the highest in the first-ranked SME. This procedure enables benchmarking, i.e. comparison of each enterprise with the best-ranked one, allowing managers to specify and choose projects to improve the production process in their SMEs [12].

4 Methodology

In this section, the procedure for modeling of uncertainties using IFSs is presented. Also, the proposed Algorithm for ranking KPIs with TIFNs is presented by respecting TIFNs’ algebra (Appendix A).

4.1 Modeling of uncertain variables

Uncertainties that exist in determining the relative importance of KPIs and their values are described by linguistic expressions modeled by TIFNs. The relative importance of production process KPIs at the level of each enterprise is stated by the fuzzy pair-wise comparison matrix with TIFNs, similar to the AHP method [40]. Fuzzy ratings of the relative importance of KPIs at the level of each expert are performed.

Considering the size of the problem, it is assumed that the relative importance of KPIs can be described well enough by using five linguistic expressions, as presented in Table 1.

Table 1
Linguistic expressions for the relative importance of KPIs

Linguistic expression TIFN

Slightly more important ${\tilde{A}}_{1} = {(1, 1.5, 2), (1, 1.5, 2.3)}$

A bit more important ${\tilde{A}}_{2} = {(1, 2, 3), (1, 2, 3.3)}$

More important ${\tilde{A}}_{3} = {(2, 3, 4), (1.8, 3, 4.2)}$

Strongly more important ${\tilde{A}}_{4} = {(3, 4, 5), (2.7, 4, 5)}$

Absolutely more important ${\tilde{A}}_{5} = {(4, 5, 5), (3.7, 5, 5)}$

Linguistic expression	TIFN
Slightly more important	${\tilde{A}}_{1} = {(1, 1.5, 2), (1, 1.5, 2.3)}$
A bit more important	${\tilde{A}}_{2} = {(1, 2, 3), (1, 2, 3.3)}$
More important	${\tilde{A}}_{3} = {(2, 3, 4), (1.8, 3, 4.2)}$
Strongly more important	${\tilde{A}}_{4} = {(3, 4, 5), (2.7, 4, 5)}$
Absolutely more important	${\tilde{A}}_{5} = {(4, 5, 5), (3.7, 5, 5)}$

Domains of these TIFNs are defined on the interval from 1 to 5. Value 1, i.e. value 5, means equal, i.e. extremely important, respectively.

Determining the values of KPIs observed in the production process, at the level of each enterprise, is based on the knowledge, experience and assessment of the decision-makers in SMEs.

The values of these KPIs are described using seven linguistic terms. Those linguistic terms are modeled by TIFNs, as shown in Table 2.

Table 2

Linguistic expressions for the relative importance of KPIs

Linguistic term	TIFN
Very low value	${\tilde{v}}_{1} = {(1, 1, 2), (1, 1, 2.3)}$
Low value	${\tilde{v}}_{2} = {(1, 2, 3), (1, 2, 3.3)}$
Almost low value	${\tilde{v}}_{3} = {(2.5, 3.5, 4.5), (2.3, 3.5, 4.7)}$
Middle value	${\tilde{v}}_{4} = {(4, 5, 6), (3.8, 5, 6.2)}$
Fairly high value	${\tilde{v}}_{5} = {(5.5, 6.5, 7.5), (5.3, 6.5, 7.7)}$
High value	${\tilde{v}}_{6} = {(7, 8, 9), (6.7, 8, 9)}$
Very high value	${\tilde{v}}_{7} = {(8, 9, 9), (7.7, 9, 9)}$

Domains of these TIFNs are defined on the common measurement scale, from 1 to 9. Value 1 indicates that KPI has very low value and value 9 that KPI has very high value.

4.2 The proposed algorithm

In this section, the proposed procedure for evaluation and ranking of SMEs is presented in detail, respecting the values and weights of all KPIs of the production process. The solution of the defined problem can be graphically presented, as shown in Fig. 1.

Fig. 1

The proposed procedure with TIFNs for the production process assessment.

Step 1. Each expert expressed his/her assessment by answering the questionnaire (Appendix B) and the fuzzy pair-wise comparison matrix of the relative importance of KPIs is stated as:

$[{\tilde{w}}_{{ii}^{'}}^{k}]; i, i^{'} = 1, . ., I; k = 1, . ., K$ (1)

Step 2. In order to check the consistency of the matrix, it is necessary to calculate the representative scalars of its elements by using the defuzzification procedure [1]: $defuzz ({\tilde{w}}_{{ii}^{'}}^{k}) = \frac{({\tilde{w}}_{{ii}^{'} 1}^{k} + 2 * {\tilde{w}}_{{ii}^{'} 2}^{k} + {\tilde{w}}_{{ii}^{'} 3}^{k}) + ({\tilde{w}}_{{ii}^{'} 1}^{k^{'}} + 2 * {\tilde{w}}_{{ii}^{'} 2}^{k} + {\tilde{w}}_{{ii}^{'} 3^{'}}^{k})}{8}$ (2)

The consistency index (CI) for this matrix is calculated according to the existing procedure [40]. It is known that if CI <10%, it is considered that the matrix is consistent and that errors made by the decision-makers are insignificant so they do not affect the accuracy of the estimation.

In this way, the fuzzy pair-wise comparison matrices at the level of each expert involved in determining the KPIs’ weights are checked.

Step 3. Using the known methodology [30], the weights of each KPI at the level of each expert are calculated: ${\tilde{w}}_{i}^{k} = \frac{{\tilde{m}}_{i}^{k}}{\sum_{i = 1}^{I} {\tilde{m}}_{i}^{k}}, i = 1, . ., I; k = 1, . ., K$ (3) while ${\tilde{m}}_{i}^{k}$ is calculated as geometric mean: ${\tilde{m}}_{i}^{k} = \sqrt[K]{{\tilde{w}}_{i 1}^{k} \cdot \dots \cdot {\tilde{w}}_{iI}^{k}}; i = 1, . ., I; k = 1, . ., K;$ (4)

Step 4. The aggregated values of the weights of each KPI are calculated by using Fuzzy Averaging Operator with TIFNs [19]: ${\tilde{w}}_{i} = \frac{1}{K} \cdot \sum_{k = 1}^{K} {\tilde{w}}_{i}^{k}; i = 1, . ., I; k = 1, . ., K$ (5)

Step 5. Fuzzy ratings of the KPIs’ values, ${\tilde{v}}_{i}^{e}$ , are given by the decision-makers at the level of each SME (by using the questionnaire in Appendix B).

Step 6. By using the linear normalization procedure [34], all of the f KPIs’ values become comparable:

a) benefit-type of KPIs ${\tilde{r}}_{i}^{e} = {(\frac{r_{i 1}^{e}}{r_{3}^{+}}, \frac{r_{i 2}^{e}}{r_{3}^{+}}, \frac{r_{i 3}^{e}}{r_{3}^{+}}), (\frac{r_{i 1}^{e'}}{r_{3}^{+}}, \frac{r_{i 2}^{e}}{r_{3}^{+}}, \frac{r_{i 3}^{e'}}{r_{3}^{+}})}$ (6)

b) cost-type of KPIs ${\tilde{r}}_{i}^{e} = {(\frac{r_{1}^{-}}{r_{i 3}^{e}}, \frac{r_{1}^{-}}{r_{i 2}^{e}}, \frac{r_{1}^{-}}{r_{i 1}^{e}}), (\frac{r_{1}^{-}}{r_{i 3}^{e'}}, \frac{r_{1}^{-}}{r_{i 2}^{e}}, \frac{r_{1}^{-}}{r_{i 1}^{e'}})}$ (7) where: $\begin{matrix} b_{3}^{+} = \max {b_{i 3}^{e}}; b_{1}^{-} = \min {b_{i 1}^{e}}; \\ i = 1, . ., I; e = 1, . ., E . \end{matrix}$

Step 7. The weighted normalized KPIs values are calculated as: ${\tilde{d}}_{i}^{e} = {\tilde{w}}_{i} \cdot {\tilde{r}}_{i}^{e}; i = 1, . ., I; e = 1, . ., E$ (8)

Step 8. Under each KPI, the TIFPIS, ${\tilde{f}}_{i}^{+}$ , and TIFNIS, ${\tilde{f}}_{i}^{-}$ are determined by using the comparison method for TIFNs [34], so that: $\begin{matrix} {\tilde{f}}_{i}^{+} = \begin{matrix} max_{e} {\tilde{d}}_{i}^{e} \end{matrix} i = 1, . ., I; e = 1, . ., E \\ {\tilde{f}}_{i}^{-} = \begin{matrix} min_{e} {\tilde{d}}_{i}^{e} \end{matrix} i = 1, . ., I; e = 1, . ., E \end{matrix}$ (9) where:

$\begin{matrix} D^{s} ({\tilde{d}}_{i}^{e}) = \frac{(d_{i 1}^{e} + 2 * d_{i 2}^{e} + d_{i 3}^{e}) + (d_{i 1}^{' e} + 2 * d_{i 2}^{e} + d_{i 3}^{' e})}{8} \\ R ({\tilde{d}}_{i}^{e}) = (\frac{d_{i 3}^{e} - d_{i 1}^{e}}{d_{i 3^{'}}^{e} - d_{i 1^{'}}^{e}}) \cdot D^{s} ({\tilde{d}}_{i}^{e}) \end{matrix}$

If $R ({\tilde{d}}_{i}^{e}) > R ({\tilde{d}}_{i}^{e^{'}}) \Rightarrow {\tilde{d}}_{i}^{e} > {\tilde{d}}_{i}^{e^{'}}$ ; i = 1, . . , I; e, e′ = 1, . . , E

In compliance with the calculations, $max_{e} {\tilde{d}}_{i}^{e}$ represents the TIFN which is first-ranked, and $min_{e} {\tilde{d}}_{i}^{e}$ represents the last-ranked TIFN.

Step 9. The total distances from TIFPIS and TIFNIS at the level of each SME are determined using the Hamming distance measure:

$\begin{matrix} ξ_{e}^{+} = \sum_{i = 1}^{I} d ({\tilde{d}}_{i}^{e}, {\tilde{f}}_{i}^{+}) \\ ξ_{e}^{-} = \sum_{i = 1}^{I} d ({\tilde{d}}_{i}^{e}, {\tilde{f}}_{i}^{-}) \end{matrix}$ (10)

The obtained values of these distances are crisps.

Step 10. Closeness coefficients at the level of each SME are calculated as: $c_{e} = \frac{ξ_{e}^{-}}{ξ_{e}^{+} + ξ_{e}^{-}}, e = 1, . ., E$ (11)

Step 11. The closeness coefficients, c_e, e = 1,..,E, are sorted into a monotone non-increasing sequence. The enterprises with the largest and the lowest values of c_e, e = 1,..,E, were the first- and the last-ranked, respectively. The maximum effectiveness of the production process is achieved in first-ranked SME. After the overall calculation, the rank of the production SMEs is obtained.

This information represents the baseline for the activity of benchmarking and proposes the projects [12] for the improvement of the production process in the SMEs that are ranked below the first one.

5 A case study for the production process in Serbian production SMEs

Production SMEs in the Republic of Serbia play a very important role in country’s economy because they are important generators of employment and economic growth. The proposed procedure is tested on the data from 30 production SMEs. At the same time, the values of KPIs are assessed by the decision-makers at the level of each of 30 considered SMEs. All data were obtained through the questionnaires (see Appendix B). The decision-makers personally delivered filled-in questionnaires to researchers who transformed data into Tables shown in Appendix C. The fuzzy pair-wise comparison matrices of the relative importance of KPIs are presented in Appendix C.

5.1 The demonstration of the methodology

The fuzzy pair-wise comparison matrix of the relative importance of KPIs at the level of expert (k = 4) is given and shown (Step 1 of the proposed Algorithm) (Table 3).

Table 3
The fuzzy pair-wise comparison matrix of the relative importance of KPIs at the level of expert (k = 4)

i = 1 i = 2 i = 3 i = 4 i = 5

i = 1 1 ${\tilde{A}}_{1}$ ${\tilde{A}}_{3}$ ${\tilde{A}}_{4}$ ${\tilde{A}}_{4}$

i = 2 ${\tilde{1 / A}}_{1}$ 1 ${\tilde{A}}_{2}$ ${\tilde{A}}_{3}$ ${\tilde{A}}_{3}$

i = 3 ${\tilde{1 / A}}_{3}$ ${\tilde{1 / A}}_{2}$ 1 ${\tilde{A}}_{1}$ ${\tilde{A}}_{1}$

i = 4 ${\tilde{1 / A}}_{4}$ ${\tilde{1 / A}}_{3}$ ${\tilde{1 / A}}_{1}$ 1 1

i = 5 ${\tilde{1 / A}}_{4}$ ${\tilde{1 / A}}_{3}$ ${\tilde{1 / A}}_{1}$ 1 1

	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{3}$	${\tilde{A}}_{4}$	${\tilde{A}}_{4}$
i = 2	${\tilde{1 / A}}_{1}$	1	${\tilde{A}}_{2}$	${\tilde{A}}_{3}$	${\tilde{A}}_{3}$
i = 3	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{2}$	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 4	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1
i = 5	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1

The proposed procedure (Step 2 of the proposed Algorithm) is illustrated as the expert (k = 4) makes an assessment. Defuzzification of the TIFN ${\tilde{A}}_{1}$ . is performed by usin Equation (2): $\begin{matrix} defuzz ({\tilde{A}}_{1}) \\ = \frac{(1 + 2 * 1.5 + 2) + (1 + 2 * 1.5 + 2.3)}{8} = 1.537 \end{matrix}$

In this way, the pair-wise comparison matrix of the relative importance of KPIs can be constructed at the level of expert (k = 4). By applying the method of the Eigen vector, it is calculated that the consistency index (CI) is 0.033. T csistency check of other experts is performed in a similar way (Appendix C). It can be considered that evaluation errors of decision-makers do not affect the accuracy of assessing the relative importance of KPIs.

By using the proposed procedure (Step 3 of the proposed Algorithm), the weights vector of KPIs at the level of expert (k = 4) is calculated as:

$\begin{matrix} {\tilde{w}}_{1}^{4} & = \frac{{\tilde{m}}_{1}^{4}}{\sum_{i = 1}^{I} {\tilde{m}}_{i}^{4}} \\ = {(0.231, 0.392, 0.637), (0.208, 0.392, 0.700)} . \end{matrix}$

where by ${\tilde{m}}_{1}^{4} = {(1.782, 2.352, 2.885), (1.673, 2.352, 2.996)$

In the same way, the weights of the rest of the considered KPIs are calculated.

By applying Step 4 of the proposed Algorithm aggregated values of weights of each KPI are calculated: $\begin{matrix} {\tilde{w}}_{1} = {(0.171, 0.296, 0.479), (0.158, 0.296, 0.528)} \\ {\tilde{w}}_{2} = {(0.164, 0.262, 0.408), (0.151, 0.262, 0.449)} \\ {\tilde{w}}_{3} = {(0.130, 0.195, 0.301), (0.120, 0.195, 0.326)} \\ {\tilde{w}}_{4} = {(0.084, 0.123, 0.196), (0.077, 0.123, 0.207)} \\ {\tilde{w}}_{5} = {(0.084, 0.123, 0.196), (0.077, 0.123, 0.207)} \end{matrix}$

The fuzzy ratings of the KPIs values at the level of each of 30 SMEs are presented in Table b7 (Step 5 of the Algorithm).

The procedure (Step 6 of the proposed Algorithm) is illustrated for KPI (i = 1) at the level of SME (e = 4).

The value of KPI (i = 1) is ${\tilde{v}}_{1}^{4} = {(1, 1, 2), (1, 1, 2.3)} .$

The values of all KPIs are determined in a similar way (see Appendix D).

The normalized value of this cost-type KPI (i = 1) is calculated: $\begin{matrix} {\tilde{r}}_{1}^{4} & = {(\frac{1}{2}, \frac{1}{1}, \frac{1}{1}), (\frac{1}{2.3}, \frac{1}{1}, \frac{1}{1})} \\ = {(0.5, 1, 1), (0.434, 1, 1)} \end{matrix}$

The weighted normalized value of KPI (i = 1) is calculated using the Step 7 of the Algorithm, as: $\begin{matrix} {\tilde{d}}_{1}^{4} = {\tilde{w}}_{1} \cdot {\tilde{r}}_{1}^{4} = {(0.171, 0.296, 0.479), \\ (0.158, 0.296, 0.528)} \cdot {(0.5, 1, 1), (0.434, 1, 1)} \\ = {(0.085, 0.296, 0.479), (0.069, 0.296, 0.528)} \end{matrix}$

The weighted normalized fuzzy decision matrix is presented in Appendix E.

Similarly, weighted normalized values of other KPIs at the level of SME (e = 4) are calculated: $\begin{matrix} {\tilde{d}}_{2}^{4} = {(0.054, 0.131, 0.408), (0.048, 0.131, 0.449)} \\ {\tilde{d}}_{3}^{4} = {(0.058, 0.108, 0.201), (0.051, 0.108, 0.225)} \\ {\tilde{d}}_{4}^{4} = {(0.028, 0.061, 0.196), (0.023, 0.061, 0.207)} \\ {\tilde{d}}_{5}^{4} = {(0.019, 0.035, 0.078), (0.016, 0.035, 0.090)} \end{matrix}$

By applying Step 8 of the proposed Algorithm the fuzzy positive and negative ideal solutions are calculated, with respect to the weighted normalized values of KPIs in all SMEs: $\begin{matrix} {\tilde{f}}_{1}^{+} = {(0.085, 0.296, 0.479), (0.069, 0.296, 0.528)} \\ {\tilde{f}}_{1}^{-} = {(0.038, 0.085, 0.192), (0.034, 0.085, 0.230) \\ {\tilde{f}}_{2}^{+} = {(0.082, 0.262, 0.408), (0.065, 0.262, 0.450)} \\ {\tilde{f}}_{2}^{-} = {(0.036, 0.075, 0.163), (0.032, 0.075, 0.195)} \\ {\tilde{f}}_{3}^{+} = {(0.115, 0.195, 0.301), (0.103, 0.195, 0.326)} \\ {\tilde{f}}_{3}^{-} = {(0.036, 0.075, 0.151), (0.031, 0.075, 0.170)} \\ {\tilde{f}}_{4}^{+} = {(0.042, 0.123, 0.196), (0.033, 0.123, 0.207)} \\ {\tilde{f}}_{4}^{-} = {(0.018, 0.035, 0.078), (0.016, 0.035, 0.090)} \\ {\tilde{f}}_{5}^{+} = {(0.042, 0.123, 0.196), (0.033, 0.123, 0.207)} \\ {\tilde{f}}_{5}^{-} = {(0.014, 0.024, 0.049), (0.012, 0.024, 0.054)} \end{matrix}$

By applying the Algorithm (Step 9) distances from TIFPIS and TNISs are calculated. The proposed procedure ilustrated by using calculated data at the level of SME (e = 4). The proposed procedure is illustrated in the following example: $\begin{matrix} d ({\tilde{d}}_{2}^{4}, {\tilde{f}}_{2}^{+}) = \frac{1}{8} [(| \begin{matrix} 0.054 - 0.082 \end{matrix} |) \\ + 2 (\begin{matrix} | 0.131 - 0.262 | \end{matrix}) + (\begin{matrix} | 0.408 - 0.408 | \end{matrix}) \\ + (\begin{matrix} | 0.048 - 0.065 | \end{matrix}) + 2 (\begin{matrix} | 0.131 - 0.262 | \end{matrix}) \\ + (| \begin{matrix} 0.449 - 0.450 \end{matrix} |)] = 0.038 \end{matrix}$

In the same way, the distances of all KPIs to TIFPIS and TIFNIS are calculated.

By using the Step 10 of the proposed Algorithm, for SME (e = 4) the closeness coefficient is calculated: $c_{4} = 0.58 .$

The rank of SMEs with respect to the obtained closeness coefficients’ values is presented in Fig. 2.

Fig. 2

The rank of SMEs respecting the closeness coefficients

Based on the calculated rank of the considered SMEs it can be seen that the first-ranked SME is SME (e = 10). It can be assumed that the effectiveness of the production process is the highest in this SME, with respect to the considered KPIs and their weights. In compliance with the obtained results and the values of the KPIs in the SME (e = 10), the decision-makers at the level of each considered SME should perform process benchmarking and define activities for the improvement of production process effectiveness.

A comparative view of the crisp weighted normalized KPIs for SME (e = 10) and (e = 4) is shown in Fig. 3.

Fig. 3

Comparative view of the crisp weighted normalized KPIs for SME (e = 10) and (e = 4).

It can be seen that the highest value difference is defined for KPI (i = 3) and KPI (i = 5). In the considered SME (e = 4), it is necessary to define the strategies that should lead to the improvement of the denoted KPIs.

5.2 Manufacturing consequences

By analyzing the KPI (i = 3), it can be said that achieving the planned quantity in production while maintaining low production costs at the same time, is very important. While analyzing KPI (i = 5), which represents the average time of unplanned delays compared to the total production cycle time, it may be useful to determine the reason for the delay (a breakdown of machines, a lack of labor, a delay in the procurement process, etc.). More time to complete the production process means more costs. The short time of unplanned delays leads to more efficient results of the business. In other words, the existence of unnecessary delays leads to the reduction of the profit and to the enterprise’s inability to be successful. One of the management strategies that can be adequate in this case is the implementation of the Lean concept since in enterprises where the Lean concept is applied, delays are almost eliminated.

In the second step, if there are financial resources available, the decision-makers of SME (e = 4) can decide to implement strategies that should lead to the increases of the remaining two KPIs, KPI (i = 2) and KPI (i = 4). One of the improvement strategies that can be applied to KPI (i = 2) is improving the control. By doing this, it is possible to discover products, types of materials and workplaces where the conflict occurs and the causes that lead to deviation of quality. With the increasing deployment of Industry 4.0 procedures, the automation of the entire production process is introduced, and the examination of defects on the products is carried out serially. Using ultrasonic technologies, particle testing, vibration testing and automated resistance testing, rigorous tests are performed in order to ensure that there are no discrepancies between products that pass the control. By implementing these procedures, the production time should be decreased and the product’s series should be in compliance with the planned quality. Also, with the appropriate proactive maintenance of production machines and tools, these problems can be identified and eliminated before affecting the product users. In order to improve KPI (i = 4) it is necessary to make a difference between internal and external costs of the scrap. Internal costs are related to the shortcomings detected before delivery, while external costs are related to the defects discovered after delivery. In the case of a decrease in product quality and increased scrap, corrective maintenance should be applied. The value of KPI (i = 1) is equal in both enterprises, so this KPI does not need to be improved. All other KPIs have lower values in SME (e = 4) than in SME (e = 10).

6 Conclusion and future work

In an environment that is constantly and rapidly changing, very important business goals are oriented toward achieving business competitiveness and sustainability. In order to accomplish these goals, the enterprise should establish stable business processes that should be monitored and improved over time. During this process, it is necessary to define, assess, analyze and improve the business performances. As business performances are not always suitable for direct measurement or assessment, in practice many enterprises analyze the appropriate KPIs that constitute performances.

In the domain of the production process enhancement, it is important to choose the right KPIs that need to be improved by applying the appropriate strategy. The definition of KPIs in this paper was performed by the expert team and the relevant literature. In the presented research, the decision-makers used pre-defined linguistic expressions because it was assumed that, it would have been easier for them to express their assessments than to use precise numbers. The existing linguistic variables were modeled by TIFNs since it was assumed that TIFNs described the treated uncertainty in a far better way than some other fuzzy numbers employed in the scope of other research such as type 1 fuzzy numbers [37]. As business practice indicates, determining the relative importance of KPIs could be stated as a group decision-making problem which was more comprehensive compared to the situation where only one expert was making the decision [26 , 31]. In compliance with the experts’ assessment, the fuzzy pair-wise comparison matrices of the relative importance of KPIs were constructed with TIFNs. As the mistakes made during the process of determining relative importance might significantly impact the final results, the consistency check was performed in contrast to some previous research [26]. Also, it should be noted that there are complex and time-consuming procedures for the consistency check [25] which have not been discussed since the conventional procedure [40] was conducted. As the experts were denoted as equally important, the aggregated KPIs weights were calculated using the Fuzzy Averaging Operator with TIFNs [19]. The rank of SMEs was calculated by applying the modified TOPSIS with TIFNs. The existing procedure for defining TIFPIS and TIFNIS [34] was applied in this research since it was more suitable in the domain of complex mathematical operations of other examined research [27 , 39]. The calculation of distances from TIFPIS and TIFNIS was performed using the Hamming distance. The ranking of SMEs was calculated according to the closeness coefficient by analogy to conventional TOPSIS.

The main contribution of this research provides a solid baseline for the definition of the management initiatives which should lead to the production process improvement. This activity should be performed by applying the benchmarking. In this way, the improvement costs could be significantly reduced. By comparing the calculated values of KPIs of any enterprise and KPIs of the best-ranked enterprise, it is possible to determine both KPIs that require improvement and the appropriate improvement strategies. It can be assumed that the time needed for efficiency improvement of the production process is reduced as well.

The accuracy of input data (the relative importance of KPIs and their values) depends on the knowledge and experience of the decision-makers. Hence, it can be indicated as the main disadvantage of the proposed method. Also, data collection should be further examined to increase the overall accuracy of the solution.

Future research should examine whether the uncertain data in the proposed model could be more suitably and appropriately described by some other intuitionistic fuzzy numbers. The proposed model should be applied to other business processes for their assessment and improvement. It is worth mentioning that obtained data could be further analyzed from the perspective of the strategic regional development area where the treated enterprises operate. This would contribute to a faster and more efficient improvement of the competitiveness of enterprises in the market.

Footnotes

Appendix A. Basic definitions of intuitionistic fuzzy sets

Definition 1. Uncertainty can be defined as a lack of relevant information based on which the decision maker can qualitatively and quantitatively describe the variable [16].

Definition 2. A linguistic variable is a variable that is expressed by linguistic terms [19].

Definition 3. Let X be a nonempty set. An intuitionistic fuzzy set A in X is an object having the form: $\begin{matrix} A = {(x, μ_{A} (x), v_{A} (x)) | x \in X}; \\ μ_{A} (x) \to [0, 1] i v_{A} (x) \to [0, 1]; \\ 0 ⩽ μ_{A} (x) + v_{A} (x) ⩽ 1, \forall x \in X \end{matrix}$

Functions μ_A (x) , v_A (x) ∈ [0, 1] define the degree of membership and the degree of non-membership of the element x ∈ X to the set A, respectively. For every intuitionistic fuzzy set A in X there is hesitation index: $\begin{matrix} π_{A} (x) = 1 - μ_{A} (x) - v_{A} (x); \\ 0 ⩽ π_{A} (x) ⩽ 1,; \forall x \in X \end{matrix};$

Definition 4. Intuitionistic fuzzy set $A = {(x, μ_{A} (x), v_{A} (x)) | x \in X}$

of the real line is called an intuitionistic fuzzy number if:

there are at least two points x₀, x₁ ∈ X such that μA (x₀) = 1 i νA (x₁) = 1,

its membership function μis fuzzy convex and its non-membership function ν is fuzzy concave,

μA is upper semi continuous and νA is lower semi continuous,

suppA = {x ∈ X|νA (x) < 1} is bounded.

Definition 5. TIFN $A = {(a_{1}, a_{2}, a_{3}), (a_{1}^{'}, a_{2}, a_{3}^{'})},$ with parameters $a_{1}^{'} ⩽ a_{1} ⩽ a_{2} ⩽ a_{3} ⩽ a_{3}^{'}$ ∈ R, has the following membership and non-membership functions: $\begin{matrix} μ A (x) = {\begin{matrix} \begin{matrix} \frac{x - a_{1}}{a_{2} - a_{1}}, \end{matrix} & \begin{matrix} \begin{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} a_{1} ⩽ x ⩽ a_{2} \end{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} \frac{x - a_{3}}{a_{2} - a_{3}} \end{matrix} \end{matrix}, \\ 0, \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} a_{2} ⩽ x ⩽ a_{3} \end{matrix} \end{matrix} \\ \begin{matrix} otherwise \end{matrix} \end{matrix} \end{matrix}} \\ ν A (x) = {\begin{matrix} \begin{matrix} \frac{x - a_{2}}{a_{1}^{'} - a_{2}}, \end{matrix} & \begin{matrix} \begin{matrix} a_{1}^{'} ⩽ x ⩽ a_{2} \end{matrix} \\ \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} \frac{x - a_{2}}{a_{3}^{'} - a_{2}}, \end{matrix} \end{matrix} \\ 1, \end{matrix} & \begin{matrix} \begin{matrix} c_{2} ⩽ x ⩽ c_{3} \\ \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} otherwise \end{matrix} \end{matrix} \end{matrix}} \end{matrix}$

Definition 6. If $A = {(a_{1}, a_{2}, a_{3}), (a_{1}^{'}, a_{2}, a_{3}^{'})}$ and $B = {(b_{1}, b_{2}, b_{3}), (b_{1}^{'}, b_{2}, b_{3}^{'})}$ are two TIFNs, and k ∈ R, and y = kA, then arithmetic operations on these numbers are:

$\begin{matrix} A + B = {(a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}), \\ (a_{1}' + b_{1}', a_{2} + b_{2}, a_{3}' + b_{3}')} \\ A - B = {(a_{1} - b_{3}, a_{2} - b_{2}, a_{3} - b_{1}), \\ (a_{1}^{'} - b_{3}^{'}, a_{2} - b_{2}, a_{3}^{'} - b_{1}')} \\ A \cdot B = {(a_{1} \cdot b_{1}, a_{2} \cdot b_{2}, a_{3} \cdot b_{3}), \\ (a_{1}' \cdot b_{1}', a_{2} \cdot b_{2}, a_{3}' \cdot b_{3}')} \\ Y = k \cdot A = {(k \cdot a_{1}, k \cdot a_{2}, k \cdot a_{3}), \\ (k \cdot a_{1}', k \cdot a_{2}, k \cdot a_{3}')}, if k > 0 \\ Y = k \cdot A = {(k \cdot a_{3}, k \cdot a_{2}, k \cdot a_{1}), \\ (k \cdot a_{3}', k \cdot a_{2}, k \cdot a_{1}')}, if k < 0 \\ \frac{A}{B} = {(\frac{a_{1}}{b_{3}}, \frac{a_{2}}{b_{2}}, \frac{a_{3}}{b_{1}}), (\frac{a_{1}'}{b_{3}'}, \frac{a_{2}}{b_{2}}, \frac{a_{3}'}{b_{1}'})} \end{matrix}$

Definition 7. Let $A = {(a_{1}, a_{2}, a_{3}), (a_{1}^{'}, a_{2}, a_{3}^{'})}$ and $B = {(b_{1}, b_{2}, b_{3}), (b_{1}^{'}, b_{2}, b_{3}^{'})}$ be two TIFNs. Hamming distance between A and B is defined analogously to [17]: $\begin{matrix} d (A, B) = \frac{1}{8} (| a_{1} - b_{1} | + 2 | a_{2} - b_{2} | + | a_{3} - b_{3} | \\ + | a_{1}^{'} - b_{1}^{'} | + 2 | a_{2} - b_{2} | + | a_{3}^{'} - b_{3}^{'} |) \end{matrix}$

Appendix B. Questionnaire

Relative importance of KPIs

Respecting Table B1 rate the relative importance of KPIs for production process.

Ratings:

Values of KPIs

Respecting the Table B2 rate values of KPIs for production process.

Ratings:

Appendix C. The fuzzy rating of the relative importance of KPIs

Table C1

Fuzzy pair-wise comparison matrix from (k = 1)

(k = 1), CI = 0.0312	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{3}$	${\tilde{A}}_{3}$
i = 2	1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{3}$	${\tilde{A}}_{3}$
i = 3	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 4	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1
i = 5	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1

Table C2

Fuzzy pair-wise comparison matrix from (k = 2)

(k = 2), CI = 0.0381	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{2}$	${\tilde{A}}_{2}$	${\tilde{A}}_{2}$
i = 2	${\tilde{1 / A}}_{1}$	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 3	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1
i = 4	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1
i = 5	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1

Table C3

Fuzzy pair-wise comparison matrix from (k = 3)

(k = 3), CI = 0.0220	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	${\tilde{A}}_{3}$	${\tilde{A}}_{3}$
i = 2	${\tilde{A}}_{1}$	1	1	${\tilde{A}}_{4}$	${\tilde{A}}_{4}$
i = 3	${\tilde{A}}_{1}$	1	1	${\tilde{A}}_{4}$	${\tilde{A}}_{4}$
i = 4	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{4}$	1	1
i = 5	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{4}$	1	1

Table C4

Fuzzy pair-wise comparison matrix from (k = 4)

(k = 4), CI = 0.033	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{3}$	${\tilde{A}}_{4}$	${\tilde{A}}_{4}$
i = 2	${\tilde{1 / A}}_{1}$	1	${\tilde{A}}_{2}$	${\tilde{A}}_{3}$	${\tilde{A}}_{3}$
i = 3	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{2}$	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 4	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1
i = 5	${\tilde{1 / A}}_{4}$	${\tilde{1 / A}}_{3}$	${\tilde{1 / A}}_{1}$	1	1

Table C5

Fuzzy pair-wise comparison matrix from (k = 5)

(k = 5), CI = 0.0381	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{2}$	${\tilde{A}}_{2}$	${\tilde{A}}_{2}$
i = 2	${\tilde{1 / A}}_{1}$	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 3	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1
i = 4	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1
i = 5	${\tilde{1 / A}}_{2}$	${\tilde{1 / A}}_{1}$	1	1	1

Table C6

Fuzzy pair-wise comparison matrix from (k = 6)

(k = 6), CI = 0.0203	i = 1	i = 2	i = 3	i = 4	i = 5
i = 1	1	1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 2	1	1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 3	1	1	1	${\tilde{A}}_{1}$	${\tilde{A}}_{1}$
i = 4	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	1	1
i = 5	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	${\tilde{1 / A}}_{1}$	1	1

Appendix D. The fuzzy assessment of KPIs’ values

Table D1

Values of all KPIs for all SMEs

	e = 1	e = 2	e = 3	e = 4	e = 5	e = 6	e = 7	e = 8	e = 9	e = 10	e = 11	e = 12	e = 13	e=14	e=15	e=16	e=17	e=18	e=19	e=20	e=21	e=22	e=23	e=24	e=25	e=26	e=27	e=28	e=29	e=30
i=1	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{3}$
i=2	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$
i=3	${\tilde{v}}_{7}$	${\tilde{v}}_{6}$	${\tilde{v}}_{6}$	${\tilde{v}}_{4}$	${\tilde{v}}_{5}$	${\tilde{v}}_{6}$	${\tilde{v}}_{6}$	${\tilde{v}}_{6}$	${\tilde{v}}_{6}$	${\tilde{v}}_{7}$	${\tilde{v}}_{5}$	${\tilde{v}}_{5}$	${\tilde{v}}_{4}$	${\tilde{v}}_{4}$	${\tilde{v}}_{5}$	${\tilde{v}}_{6}$	${\tilde{v}}_{4}$	${\tilde{v}}_{4}$	${\tilde{v}}_{6}$	${\tilde{v}}_{5}$	${\tilde{v}}_{4}$	${\tilde{v}}_{5}$	${\tilde{v}}_{5}$	${\tilde{v}}_{6}$	${\tilde{v}}_{5}$	${\tilde{v}}_{4}$	${\tilde{v}}_{6}$	${\tilde{v}}_{5}$	${\tilde{v}}_{5}$	${\tilde{v}}_{3}$
i=4	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$
i=5	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{1}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{4}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{3}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{2}$	${\tilde{v}}_{3}$

Appendix E The weighted normalized fuzzy decision matrix.

	Weighted normalized value of KPI (i = 1)
e = 1	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 2	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 3	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 4	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 5	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 6	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 7	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 8	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 9	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 10	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 11	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 12	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 13	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 14	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 15	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 16	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 17	{ (0.038, 0.085, 0.192) , (0.034, 0.085, 0.230)}
e = 18	{ (0.038, 0.085, 0.192) , (0.034, 0.085, 0.230)}
e = 19	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 20	{ (0.038, 0.085, 0.192) , (0.034, 0.085, 0.230)}
e = 21	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 22	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 23	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 24	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 25	{ (0.057, 0.148, 0.479) , (0.048, 0.148, 0.528)}
e = 26	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 27	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 28	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 29	{ (0.085, 0.296, 0.479) , (0.069, 0.296, 0.528)}
e = 30	{ (0.038, 0.085, 0.192) , (0.034, 0.085, 0.230)}

	Weighted normalized value of KPI (i = 2)
e = 1	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 2	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 3	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 4	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 5	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 6	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 7	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 8	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 9	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 10	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 11	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 12	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 13	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 14	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 15	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 16	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 17	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 18	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 19	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 20	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 21	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 22	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}
e = 23	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 24	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 25	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 26	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 27	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 28	{ (0.082, 0.262, 0.408) , (0.065, 0.262, 0.450)}
e = 29	{ (0.054, 0.131, 0.408) , (0.048, 0.131, 0.449)}
e = 30	{ (0.036, 0.075, 0.163) , (0.032, 0.075, 0.195)}

	Weighted normalized value of KPI (i = 3)
e = 1	{ (0.115, 0.195, 0.301) , (0.103, 0.195, 0.326)}
e = 2	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 3	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 4	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 5	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 6	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 7	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 8	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 9	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 10	{ (0.115, 0.195, 0.301) , (0.103, 0.195, 0.326)}
e = 11	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 12	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 13	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 14	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 15	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 16	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 17	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 18	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 19	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 20	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 21	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 22	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 23	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 24	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 25	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 26	{ (0.058, 0.108, 0.201) , (0.051, 0.108, 0.225)}
e = 27	{ (0.101, 0.173, 0.301) , (0.089, 0.173, 0.326)}
e = 28	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 29	{ (0.079, 0.141, 0.251) , (0.071, 0.141, 0.279)}
e = 30	{ (0.036, 0.075, 0.151) , (0.031, 0.075, 0.170)}

	Weighted normalized value of KPI (i = 4)
e = 1	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 2	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 3	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 4	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 5	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 6	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 7	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 8	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 9	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 10	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 11	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 12	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 13	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 14	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 15	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 16	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 17	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 18	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 19	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 20	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 21	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 22	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 23	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 24	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 25	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 26	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 27	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 28	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 29	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 30	{ (0.018, 0.035, 0.078) , (0.016, 0.035, 0.090)}

	Weighted normalized value of KPI (i = 5)
e = 1	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 2	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 3	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 4	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 5	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 6	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 7	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 8	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 9	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 10	{ (0.042, 0.123, 0.196) , (0.033, 0.123, 0.207)}
e = 11	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 12	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 13	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 14	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 15	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 16	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 17	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 18	{ (0.014, 0.024, 0.049) , (0.012, 0.024, 0.054)}
e = 19	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 20	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 21	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 22	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}
e = 23	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 24	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 25	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 26	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 27	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 28	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 29	{ (0.028, 0.061, 0.196) , (0.023, 0.061, 0.207)}
e = 30	{ (0.019, 0.035, 0.078) , (0.016, 0.035, 0.090)}

References

Nagoorgani

and Ponnalagu

, A New Approach on Solving Intuitionistic Fuzzy Linear Programming Problem, Applied Mathematical Sciences 6(70) (2012), 3467–3474.

Simeunovic

, Slovic

and Radovic

, GPI Process Performance Mesurement Model. Paper presented at 15th International Symposium of Organizational Sciences - SYMORG 2016: Reshaping the Future through Sustainable Business Development and Entrepreneurship, Zlatibor, Serbia, (2016, June).

Simeunovic

, Development of Process Performance Measurement Model, Ph.D Dissertation, University of Belgrade, Faculty of Organizational Sciences, (2015).

Kahraman

and Tüysüz

, GroupDecision-Making under Uncertainty: FAHP Using Intuitionistic and Hesitant Fuzzy Sets. In: Fuzzy Analytic Hierarchy Process (pp. 125–160), Chapman and Hall/CRC, (2017).

Tian

, Peng

, Zhang

and Wang

, Tourism environmental impact assessment based on improved AHP and picture fuzzy PROMETHEE II methods, Technological and Economic Development of Economy 26(2) (2020), 355–378.

Hwang

C.L.

and Yoon

, Multiple Attribute Decision making: Methods and Applications, Springer, Berlin, (1981).

Dubois

and Prade

, Theory and applications, fuzzy sets and systems, Academic, New York, (1980).

Joshi

and Kumar

, Intuitionistic fuzzy entropy and distance measure based TOPSIS method for multi-criteria decision making, Egyptian Informatics Journal 15(2) (2014), 97–104.

, Nan

and Zhang

, A Ranking Method of Triangular Intuitionistic Fuzzy Numbers and Application to Decision Making, International Journal of Computational Intelligence Systems 3(5) (2010), 522–530.

10.

Parmenter

, Key Performance Indicators: Developing, Implementing, and Using Winning KPIs (Third Ed.), John Wiley & Sons, Inc., Hoboken, (2015).

11.

Tadic

, Aleksic

, Popovic

, Arsovski

, Castelli

, Joksimovic

and Stefanovic

, The evaluation and enhancement of quality, environmental protection and seaport safety by using FAHP, Natural Hazards and Earth System Sciences 17(2) (2017), 261–275.

12.

Tadic

, Arsovski

, Aleksic

, Stefanovic

and Nestic

, A fuzzy evaluation of projects for business processes’ quality improvement, In: Intelligent Techniques in Engineering Management (pp. 559–579), Springer, Cham, (2015).

13.

Kahrovic

, Design and implementation of business process performance measurement system, Facta Universitatis, Series: Economics and Organization 10(3) (2013), 283–299.

14.

Szmidt

and Kacprzyk

, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114(3) (2000), 505–518.

15.

Boran

, Genç

, Kurt

and Akay

, A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method, Expert Systems with Applications 36(8) (2009), 11363–11368.

16.

Jahanshahloo

G.R.

, Lotfi

F.H.

and Izadikhah

, An algorithmic method to extend TOPSIS for decision-making problems with interval data, Applied Mathematics and Computation 175(2) (2006), 1375–1384.

17.

Zimmermann

H.J.

, Fuzzy set theory, Wiley Interdisciplinary Reviews: Computational Statistics 2(3) (2010), 317–332.

18.

, Multicriteria group decision-making method using the distances-based similarity measures between intuitionistic trapezoidal fuzzy numbers, International Journal of General Systems 41(7) (2012), 729–739.

19.

Atanassov

, Intuitionistic fuzzy sets, in: VII ITKR’s Session, V. Sgurev, ed., Sofia, (1983) (in Bulgarian).

20.

Wang

, Zhang

H.Y.

, Wang

J.Q.

and Wu

G.F.

, Picture fuzzy multi-criteria group decision-making method to hotel building energy efficiency retrofit project selection, RAIRO-Operations Research 54(1) (2020), 211–229.

21.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning – 1, Information Sciences 8 (1975), 199–249.

22.

Alipour

, Hafezi

, Ervural

, Kaviani

and Kabak

Ö.

, Long-term policy evaluation: Application of a new robust decision framework for Iran’s energy exports security, Energy 157 (2018), 914–931.

23.

Glykas

, Fuzzy cognitive strategic maps in business process performance measurement, Expert Systems with Applications 40(1) (2013), 1–14.

24.

Shojaei

, Ahmadi

and Shoajei

, Implementation Productivity Management Cycle with Operational Kaizen Approach to Improve Production Performance (Case Study: Pars Khodro Company), International Journal for Quality Research 13(2) (2019), 349–360.

25.

Tavana

, Zareinejad

, Di Caprio

and Kaviani

, An integrated intuitionistic fuzzy AHP and SWOT method for outsourcing reverse logistics, Applied Soft Computing 40 (2016), 544–557.

26.

Julie

and Uthra

, Triangular intuitionistic fuzzy ahp and its application to select best product of notebook computer, International Journal of Pure and Applied Mathematics 113(10) (2017), 253–261.

27.

Onat

, Gumus

, Kucukvar

and Tatari

, Application of the TOPSIS and intuitionistic fuzzy set approaches for ranking the life cycle sustainability performance of alternative vehicle technologies, Sustainable Production and Consumption 6 (2016), 12–25.

28.

Saini

, Bajaj

, Gandotra

and Dwivedi

, Multi-criteria Decision Making with Triangular Intuitionistic Fuzzy Number based on Distance Measure & Parametric Entropy Approach, Procedia Computer Science 125 (2018), 34–41.

29.

Grzegorzewski

, Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems 148(2) (2004), 319–328.

30.

Kaur

, Selection of Vendor Based on Intuitionistic Fuzzy Analytical Hierarchy Process, Advances in Operations Research (2014), 1–10.

31.

Sadiq

and Tesfamariam

, Environmental decision-making under uncertainty using intuitionistic fuzzy analytic hierarchy process (IF-AHP), Stochastic Environmental Research and Risk Assessment 23(1) (2009), 75–91.

32.

Sagaya

and Amirtharaj

, Methods to Find the Solution for the Intuitionistic Fuzzy Assignment Problem with Ranking of Intuitionistic Fuzzy Numbers, International Journal of Innovative Research in Science, Engineering and Technology 4(7) (2015), 10008–10014.

33.

Chen

, Cheng

and Chiou

, Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology, Information Fusion 27 (2016), 215–227.

34.

Bharati

S.K.

, Ranking Method of Intuitionistic Fuzzy Numbers, Global Journal of Pure and Applied Mathematics 13(9) (2017), 4595–4608.

35.

Mousavi

and Vahdani

, Cross-docking Location Selection in Distribution Systems: A New Intuitionistic Fuzzy Hierarchical Decision Model, International Journal of Computational Intelligence Systems 9(1) (2016), 91–109.

36.

Nestic

, Lampón

, Aleksic

, Cabanelas

and Tadic

, Ranking manufacturing processes from the quality management perspective in the automotive industry, Expert Systems (2019).

37.

Nestic

, Stefanovic

, Djordjevic

, Arsovski

and Tadic

, A model of the assessment and optimization of production process quality using the fuzzy sets and genetic algorithm approach, European Journal of Industrial Engineering 9(1) (2015), 77–99.

38.

Kaplan

S.R.

and Norton

P.D.

, The execution premium: linking strategy to operations for competitive advantages, Harvard Business School Publishing Corporation, Boston, (2008).

39.

Gautam

S.S.

, Abhishekh

S.S.

and Singh

S.R.

, TOPSIS for Multi Criteria Decision Making in Intuitionistic Fuzzy Environment, International Journal of Computer Applications 156(8) (2016), 42–49.

40.

Saaty

T.L.

, The modern science of multicriteria decision making and its practical applications: The AHP/ANP approach, Operations Research 61(5) (2013), 1101–1118.

41.

, Song

and Quan

, Evaluating evidence reliability based on intuitionistic fuzzy MCDM model, Journal of Intelligent & Fuzzy Systems 31(3) (2016), 1167–1182.

42.

and Liao

, Intuitionistic Fuzzy Analytic Hierarchy Process, IEEE Transactions on Fuzzy Systems 22(4) (2014), 749–761.

Study on performance evaluation of the production process - fuzzy MCDM approach

Abstract

Keywords

1 Introduction

2 Literature review

3 Evaluation framework

4 Methodology

4.1 Modeling of uncertain variables

5.1 The demonstration of the methodology

6 Conclusion and future work

Footnotes

Appendix A. Basic definitions of intuitionistic fuzzy sets

Appendix B. Questionnaire

Appendix C. The fuzzy rating of the relative importance of KPIs

Appendix D. The fuzzy assessment of KPIs’ values

Appendix E The weighted normalized fuzzy decision matrix.

References