Evaluation of Flexible Manufacturing Systems Using a Hesitant Group Decision Making Approach

1 Introduction

In today’s global market, there is a great competition between companies for producing the highest quality products with the lowest cost and having the best customer satisfaction. In this highly competitive environment, the survival of a company depends on its ability to tackle variations and adapt to changing conditions. Hence, developing new strategies in order to provide high quality customer service is significant to survive in the global market. In this respect, flexible manufacturing systems (FMS) are capable of processing various parts, styles, and quantities of production in a manufacturing system [23]. FMS is easily adapted to changing market conditions and gives some advantages in terms of efficient and quick response to customer needs. FMS is a manufacturing technology; most companies follow this strategy to be flexible in their operations, and then they can satisfy various market segments effectively. The components of the FMS are workstations, material handling and storage systems, computer control system, and system control operators. Flexibility in manufacturing means the ability of producinghigh quality products with low cost considering restricted delivery times. Browne et al. [3]’s study is one of the earliest works that define flexibility in FMS with eight subclasses such as machine flexibility, product flexibility, process flexibility, production flexibility, volume flexibility etc., to show an overall system flexibility.

How to determine the optimal FMS among a number of feasible alternatives is a critical issue. Selecting a suitable FMS is important for manufacturing companies when making capital investment decisions to improve their manufacturing performance [5]. To increase the manufacturing flexibility, manufacturing companies are looking at FMS as a practicable alternative to enhance their competitive superiority [8].

In the literature, manufacturing technology attributes are divided into two subclasses: objective and subjective [15, 21, 27]. Objective attributes are the quantitative effects of manufacturing technology and are determined using some numerical scales. On the other hand, subjective attributes are defined by decision makers (DMs) using different linguistic scales, and the nature of these attributes is defined as qualitative [5].

The analytic hierarchy process (AHP) is a mostly preferred method to evaluate advanced manufacturing technology. Wabalickis [33] proposed a justification methodology based on AHP to evaluate FMS investment. Stam and Kuula [28] developed an AHP-based multi-objective programming model to choose an FMS. AHP is utilized firstly by combining objective criteria and subjective criteria, then the multi-objective mathematical programming is used to find the best option. Shang and Sueyoshi [26] introduced a decision making procedure in FMS employing AHP, simulation, and data envelopment analysis (DEA). Although AHP is mostly used for FMS selection problems owing to its simplicity, it has some drawbacks such as difficulties when the number of criteria/alternatives increase rank, reversal problems, and inappropriateness of the crisp ratio representation [2]. In Bayazit [1]’s study AHP is used for the decision by a tractor manufacturing plant to implement FMS. Yurdakul [38] utilized a combination of AHP and goal programming for selection of computer-integrated manufacturing technologies. Jain and Raj [7] also presented a method ranking of flexibility in FMS by using combined multiple attribute decision making methods, which are AHP and technique for order preference by similarity to ideal solution (TOPSIS). Tseng [32] applied a game theoretical model for a technology selection in FMS. Kulak and Kahraman [16] proposed axiomatic design principles for multiple attribute comparison of advanced manufacturing systems. Rao [22] proposed a decision making model for FMS selection using digraph and matrix methods. Liu [19] presented a DEA/AR (assurance region) approach for selection of FMS. Rao and Parnichkun [24] proposed a methodology based on a combinatorial mathematics-based decision making method for evaluation of alternative FMS. In another study of Rao [23], a procedure is based on a combined multiple attribute decision making method using TOPSIS and AHP methods together that is proposed to evaluate alternative FMS for a given industrial application. Chatterjee and Chakraborty [4] considered the application of six preference ranking methods for selecting the best FMS for a given manufacturing organization. Talebanpour and Javadi [29] presented a decision making model that includes the decision making trial and evaluation laboratory and simple additive weighting method in order to evaluate FMS.

Under ambiguous decision environment, fuzzy set theory is a helpful tool to incorporate subjective attributes like linguistic terms which are mostly based on imprecise judgments. Many researchers have utilized fuzzy multiple attribute decision making (FMADM) models for FMS selection problems. Liang and Wang [18] used the fuzzy set theory for robot selection problem. Karsak [12] considered an integrated framework that includes DEA and fuzzy robot selection algorithm. Perego and Rangone [20] proposed multi-criteria decision making (MCDM) that employs fuzzy set theory for determining advanced manufacturing technologies. Karsak and Tolga [15] proposed an FMADM procedure for evaluating advanced manufacturing system investments which consist of economic and strategic criteria. Karsak and Kuzgunkaya [14] presented a fuzzy multi-objective programming approach to evaluate FMS alternatives under economic and strategic aspects.

FMS has a multi-dimensional framework, and it is difficult to solve such complex systems under different aspects [25]. Most of the FMS evaluation problems include objective and subjective criteria together. Experts’ evaluations of the subjective criteria result with imprecise and vague information to be considered. Fuzzy set theory is a very effective tool to make a decision considering imprecise judgment under different criteria in such complex models [14]. Additionally, the fuzzy set theory provides an acceptable way to evaluate qualitative and subjective criteria in unstable conditions like dynamic manufacturing market.

According to Kabak and Ervural [9], fuzzy set theory along with its extensions has been effectively used for solving real life group decision making problems, and especially the use of the hesitant and intuitionistic fuzzy sets is an open future study area. Although fuzzy set theory in general exhibits a satisfactory performance to deal with vague and incomplete data, ordinary fuzzy sets may be insufficient to achieve the expected results in different ambiguous situations. Hence, several extensions of fuzzy sets have been developed to overcome this shortcoming. Recently, a novel extension of fuzzy sets, hesitant fuzzy sets (HFS), has been introduced by Torra [30], which allows the membership degree of an element to a set to be represented by several possible values. Besides, HFS received growing interest among most scholars nowadays since it reflects the human’s hesitancy more objectively [17].

Due to its complex nature, FMS evaluation problem also includes some hesitation in the judgments of the experts. Therefore; HFS could provide good results to handle vagueness caused by hesitation in FMS evaluation problem. In this study, we aim to deal with the assessment of the suitability of FMS technology by utilizing HFS owing to some of its advantages, as mentioned above. According to our best knowledge, there has been no work using HFS to solve this complex multi-dimensional FMS evaluation problem in the fuzzy environment.

The purpose of this paper is to implement hesitant fuzzy sets in the selection of the most appropriate alternative for an FMS in a MCDM framework. The rest of this paper is organized as follows. Section 2 presents hesitant fuzzy sets concept and a fuzzy MCDM approach for selecting a more suitable flexible manufacturing technology. Section 3 provides an illustrative example. The results of the study are discussed with comparative analysis and sensitivity analysis in Section 4, and finally, some concluding remarks and future research directions are given in Section 5.

2 An MAGDM Approach with Hesitant Fuzzy Information

Many group decision making problems often require the examination of objective criteria as well as subjective criteria [6]. Therefore, methods to solve these problems should be capable of integrating the objective criteria scores and subjective evaluations of experts. Moreover, the experts may hesitate between several numerical or linguistic values in assessing the criteria. Therefore, in this study, we propose a new FMADM methodology under hesitant fuzzy information capable of handling subjective and objective criteria for the evaluation and selection of FMS.

3 An Illustrative Example

In this section, an FMS evaluation problem in a manufacturing company adapted from Karsak [13] and Chuu [5] is used to illustrate the proposed approach.

An expert group was formed to determine the best FMS among three FMS alternatives a_i(i=1, 2, 3). Three experts e_k(k=1, 2, 3) form a decision making committee with the weights vector λ=(0.4, 0.3, 0.3)^T. In our problem, we consider both objective and subjective criteria. Six criteria are under consideration: (1) product flexibility (c₁); (2) product quality (c₂); (3) volume flexibility (c₃); (4) required floor space (m²) (c₄); (5) investment cost (×$100,000) (c₅); and (6) lead time (h) (c₆). Among the considered criteria, c₁, c₂, and c₃ are the benefit and subjective criteria, and c₄, c₅, and c₆ are the cost and objective criteria. Importance weight of criteria is c_j(j=1, 2,…, 6), w=(0.15, 0.20, 0.15, 0.20, 0.20, 0.10)^T. The experts evaluate the FMS alternatives with respect to subjective criteria c_j (j=1, 2, 3) and construct hesitant fuzzy decision matrices, D(k)=(hij(k))m×n (k=1, 2, 3) as seen in Table 1, where hij(k)∈H is a HFE that indicates all of the possible values for the alternative with respect to criterion c_j. The performance ratings for three alternatives under each objective criterion (D^O) are shown in Table 2.

To select the best FMS alternative, the following steps are given:

Step 0 Normalization.

In this problem, since all of the subjective criteria are benefit criteria, decision matrix D^(k) is equal to normalized matrix R^(k). Then the normalized values for objective criteria are calculated using Eq. (3) as given in Table 3. For instance, for alternative 1 and criterion 4 normalized value is calculated as follows:

r1,4=1/x1,4∑i=13|1/xi,4|=1/5001/500+1/600+1/750=0.400

Step 1 Calculate the weighted support for individual decision matrices.

Calculate the supports Sup(rij(k), rij(t))=Supkt, which refers to supports between R^(k) and R^(t) (i, j=1, 2, 3, t, k=1, 2, 3and t ≠ k) (Table 4).

For instance, supports between R⁽²⁾ and R⁽³⁾ for alternative 1 and criterion 3 is calculated as follows:

Sup(r1,3(2), r1,3(3))=Sup1,323=1−d(r1,3(2), r1,3(3))=1−d({0.6, 0.4}, {0.3})=1−|0.6−0.3|+|0.4−0.3|2=0.8

Then utilize Eq. (4) to calculate the weighted support T(rij(k)) of the evaluated value rij(k) by the other evaluated values rij(t) (t=1, 2, 3 and t ≠ k). T_k denotes (T(rij(k)))3×3 (k=1, 2, 3) in Table 5. For instance, for weighted support T(r3, 2(2)) of the evaluated value r3,2(2) by the other evaluated values rij(t) is calculated as follows:

T(r3,2(2))=∑t=1(t≠2)3λtSup(r3,2(2), r3,2(t))=0.4∗0.7000+0.3∗0.6000=0.4600

After that by Eq. (5), calculate the weights ξij(k) (i, j=1, 2, 3, k=1, 2, 3). V_k denotes (ξij(k))3×3 (k=1, 2, 3) in Table 6. For example, weight of expert 1 under alternative 2 and criterion 3 is calculated as follows:

ξ2,3(1)=λ1(1+T(r2,3(1)))∑k=13λk(1+T(r2,3(k)))=0.4∗(1+0.55)0.4∗(1+0.55)+0.3∗(1+0.615)+0.3∗(1+0.635)=0.3895

Step 2 Calculate the collective decision matrix.

By the WGHFPA operator given in Eq. (6), aggregate all individual hesitant fuzzy decision matrices R(k)=(rij(k))m×n (k=1, 2, 3) into the collective hesitant fuzzy decision matrix.

For instance, for alternative 1 and criterion 1 collective hesitant value r_1,1 is calculated as follows:

r1, 1=WGHFPA(r1, 1(1), r1, 1(2))=∪γ1,1(1)ϵr1,1(1),γ1,1(2)ϵr1,1(2){(1−((1−0.52)0.3937)∗((1−0.52)0.307)∗((1−0.72)0.2992))1/2,(1−((1−0.52)0.3937)∗((1−0.32)0.307)∗((1−0.72)0.2992))1/2}={0.5760 , 0.5393 }

In this step, the standardized performance ratings obtained by Eq. (2) under each objective criterion are added to decision matrix R=(r_ij)m×s. The collective hesitant fuzzy decision matrix is shown in Table 7.

Step 3 Calculate the weighted support for collective decision matrix.

Calculate the supports Sup(r_ij, r_ip)=Sup_jp, which refer to supports between rows of collective decision matrix R (Table 8). For instance, support between R⁽¹⁾ and R⁽⁶⁾ for alternative 1 is calculated as follows:

Sup(r1,1, r1,6)=Sup1,6=1−d(r1,1, r1,6)=1−d({0.5760, 0.5393}, {0.171})=1−|0.5760−0.171|+|0.5393−0.171|2 =0.6131

Then utilize Eq. (7) to calculate the weighted support T(r_ij) of the evaluated value r_ij by the other evaluated values r_ip (p=1, 2,…, 6 and p ≠ j). T denotes (T(r_ij))_3×6 in Table 9. For instance, for weighted support T_(3,6) of the evaluated value r_3,6 by the other evaluated values r_ip is calculated as follows:

T(r3,6)=∑p=1(p≠6)6wp Sup(r3,6,r3,p)=0.15∗0.7605+0.2∗0.9015+0.15∗0.9381+0.2∗0.7789+0.2∗0.7816=0.7472 

After that by Eq. (8), calculate the weights η_ij (i=1, 2, 3, j=1, 2,…, 6). V denotes (η_ij)_3×6 in Table 10. For example, weight of alternative 3 and criterion 1 is calculated as follows:

η3,1=w1(1+T(r3,1))∑j=16wj(1+T(r3,j))=0.15∗(1+0.591)0.15∗(1+0.591)+0.2∗(1+0.637)+0.15∗(1+0.706)+0.2∗(1+0.603)+0.2∗(1+0.605)+0.1∗(1+0.747)=0.1457

Step 4 Obtain the collective overall preference.

Utilize the WGHFPA operator given in Eq. (9) to aggregate all of the preference values r_ij and then provide the collective overall preference value r_i of the alternatives a_i (i=1, 2, 3). It is shown in Table 11. For instance, for alternative 1 collective overall preference value r₁is calculated as follows:

r1=WGHFPA(r1,1, …, r1,s)=∪γi1ϵri1, ,…,γisϵris{(1−((1−0.5762)0.1509)∗((1−0.4542)0.2014)∗((1−0.592)0.1505)∗((1−0.42)0.2006)∗((1−0.4122)0.2011)∗((1−0.1712)0.0955))1/2, …}={0.468, … }

Step 5 Rank the alternatives.

Use Eq. (10) to calculate the score functions s(r_i) of r_i, and rank the alternatives according to s(r_i) (i=1, 2, 3) and then select the best alternative.

s(r1)=0.453   s(r2)=0.478   s(r3)=0.517

For a group of DMs, based on score function, the ranking order of three alternatives is given as a₃>a₂>a₁.

The third alternative (a₃) appears to be the most appropriate FMS alternative as a result of the hesitant fuzzy multi-attribute decision procedure.

4 Comparison and Sensitivity Analysis

In this section, we compared the results of our methodology with Xia and Xu [34]‘s method. And in order to show the robustness of the final decision, sensitivity analyses are conducted.

4.2 Sensitivity analysis

In order to examination the robustness of the proposed algorithm, it is necessary to estimate the rate of change in the output of a model caused by the changes in the model inputs. In the proposed model, the ranking of alternatives may depend on both criteria and DMs’ weights. For this reason, sensitivity analysis was employed by changing the weights of the criteria and DMs separately.

First, a sensitivity analysis was performed by changing the criteria weights. In this analysis, Case 0 refers to the current situation. In each case, the weight of a criterion is decreased or increased by 0.10 and the weights of other criteria are updated proportionally. For instance, in Case 1, the weight of c₁ was decreased by 0.10 (from 0.15 to 0.05), and 0.10 weight is distributed to the other criteria proportionally. The other cases of the analysis are designed, and the related weights are calculated in the same way as given in Table 13.

Table 13:

Sensitivity Analysis for Criteria Weights.

		c1	c2	c3	c4	c5	c6
Case 0	Current situation	0.15	0.20	0.15	0.20	0.20	0.10
Case 1	c1 decreased by 0.10	0.05	0.22	0.17	0.22	0.22	0.11
Case 2	c1 increased by 0.10	0.25	0.18	0.13	0.18	0.18	0.09
Case 3	c2 decreased by 0.10	0.17	0.10	0.17	0.23	0.23	0.11
Case 4	c2 increased by 0.10	0.13	0.30	0.13	0.18	0.18	0.09
Case 5	c3 decreased by 0.10	0.17	0.22	0.05	0.22	0.22	0.11
Case 6	c3 increased by 0.10	0.13	0.18	0.25	0.18	0.18	0.09
Case 7	c4 decreased by 0.10	0.17	0.23	0.17	0.10	0.23	0.11
Case 8	c4 increased by 0.10	0.13	0.18	0.13	0.30	0.18	0.09
Case 9	c5 decreased by 0.10	0.17	0.23	0.17	0.23	0.10	0.11
Case 10	c5 increased by 0.10	0.13	0.18	0.13	0.18	0.30	0.09
Case 11	c6 decreased by 0.10	0.17	0.22	0.17	0.22	0.22	0.00
Case 12	c6 increased by 0.10	0.13	0.18	0.13	0.18	0.18	0.20

Twelve additional cases are used for analyzing, and the ranks of the alternatives are recalculated. The sensitivity results of ranking alternatives are shown in Figure 1. As a result of the analysis, although there were slight changes in the scores of the alternatives, the final rankings remained the same. These findings obtained by sensitivity analysis at different cases show that the result of the proposed approach in the illustrative example is robust to alterations in the weights of the criteria.

Figure 1:

Sensitivity Analysis Results when the Weights of Criteria are Changed.

The same process is applied for the experts’ weights in order to study the effect on the results. In the analysis, expert weights were also increased and decreased by 0.10, which was proportionally reflected to the weight of the other experts. Table 14 shows the current situation and the corresponding weights of experts for the six cases. As can be seen in Figure 2, although there were minor changes in the scores of the alternatives, the final rankings remained the same.

Table 14:

Sensitivity Analysis for Experts’ Weights.

		e1	e2	e3
Case 0	Current situation	0.4	0.3	0.3
Case 1	e1 decreased by 0.10	0.3	0.35	0.35
Case 2	e1 increased by 0.10	0.5	0.25	0.25
Case 3	e2 decreased by 0.10	0.46	0.2	0.34
Case 4	e2 increased by 0.10	0.34	0.4	0.26
Case 5	e3 decreased by 0.10	0.46	0.34	0.2
Case 6	e3 increased by 0.10	0.34	0.26	0.4

Figure 2:

Sensitivity Analysis Results when the Weights of Experts are Changed.

The sensitivity analysis results presented in Figures 1 and 2 show that the model adopted for analysis is free from any biases and results are robust to changes in the weights of the criteria and experts.

5 Conclusion

One of the most important issues of manufacturing systems is to ensure sustainability and thus respond to customer demands according to expected quality with minimum cost and adapt to market conditions immediately. Under current conditions of competition, FMS has emerged as an important concept to satisfy all expected qualifications. Development of hesitant fuzzy sets is a recent approach to cope with vague and imprecise information in complex systems under the MCDM framework.

In this paper, we have analyzed an FMS in order to determine the most appropriate FMS alternative under the hesitant fuzzy environment. One of the important properties of the proposed method, which distinguishes it from traditional or hesitant versions of classical methods (e.g. AHP, TOPSIS, etc.), is that it can aggregate the objective criteria and subjective evaluations of the experts. The validity of the proposed methodology is presented using an illustrative example and a comparative study. The robustness of the results is tested through a sensitivity analysis. Consequently, the proposed methodology is shown to be effective to deal with problems including objective and subjective criteria that are presented as hesitant fuzzy sets. The results are proven to be robust to the changes of weights related to criteria as well as experts.

For further research, the proposed MAGDM method can be applied to different complex evaluation problems. For simplicity and ease of computation, a decision support software can be developed for the proposed method, and any regular aggregation operator can be used instead of the used power aggregation operator in this study. We also plan to use a method for aggregating expert evaluations given in any kind of format such as rating, interval, linguistic evaluations, fuzzy sets and their extensions like HFS and intuitionistic fuzzy sets, etc. Cumulative belief degree approach [10, 11] can be an appropriate method for this.