A new weighted distance-based approximation methodology for flow shop scheduling group decisions under the interval-valued fuzzy processing time
Graphical abstract
Introduction
Flow shop scheduling problems (FSSP) have received extensive attention from various researchers and scholars due to the manifold industrial applications. The standard FSSP is constructed of jobs with the equal manufacturing flow on a collection of machines. The purpose is to figure out the sequence of jobs that convinces one or a collection of criteria [1], [2], [3]. The flow shop problem is formed from jobs (i {1, 2, …, n}), which are processed by a set of machines (j {1, 2, …, m}). Every job moves across the machines in the identical technological sequence (i.e., it begins on machine 1, next moves to machine 2, up to machine m) [4]. The best sequence of jobs on machines should be chosen for achieving the minimum makespan. There are four categories to solve FSSPs [3], [5]:
- (1)
Exact methods.
- (2)
Heuristic approaches.
- (3)
Meta-heuristic techniques.
- (4)
Hybrid heuristic/meta-heuristic solutions.
When the number of machines exceeds three in the FSSP, the problem becomes an NP-completeness problem [6], [7]. The exact method fails to achieve a high-quality solution because of too much computational time. Thus, researchers have concentrated on heuristic methods for solving FSSPs by a large number of jobs [8]. Ribas et al. [9] proposed a heuristic method based on the iterated greedy algorithm and local search. Dios et al. [10] introduced an efficient heuristic method for the FSSP under a missing operation.
Then, looking for the solutions nearer to the optimum, meta-heuristics have been used for FSSPs [11]. Shao et al. [12] introduced a meta-heuristic algorithm based on the variable neighborhood search (VNS) and iterated local search (ILS) methods for the FSSP. Ruiz et al. [13] proposed a meta-heuristic method based on the iterated greedy algorithm for the FSSP by using the distributed permutation. Pan et al. [14] presented a meta-heuristic method for the FSSP based on the distributed assembly permutation.
Regarding the hybrid heuristic or meta-heuristic techniques, Chen et al. [15] applied a hybrid genetic algorithm to solve flow shop problems. Mou et al. [16] proposed a combined heuristic algorithm for flow shop problems under a dynamic environment. Govindan et al. [17] minimized the makespan by a combination of decision tree and scatter search algorithms. Engin and Güçlü [18] presented an ant colony optimization method to solve flow shop scheduling problems. Abdel-Basset et al. [19] introduced a whale optimization algorithm using the local search for FSSPs.
Although heuristic, meta-heuristic, and hybridization of them can be caused to improve the solution quality, these algorithms are very sophisticated, require advanced programming skills, and need much computing time [5]. In these conditions, Gupta and Kumar [5] applied the techniques for order preference by similarity to an ideal solution (TOPSIS), which is a simple and well-known multi-criteria decision-making (MCDM) method to the FSSP successfully. The weighted distance-based approximation (WDBA) method as similar to the TOPSIS method is classified in the field of compromise solution methods. Jain et al. [20] proved that the WDBA method is much effective than the TOPSIS method. They also indicated that the WDBA method produces better results in less time as compare to TOPSIS. The WDBA method involves simple mathematical formulations and easy to understand; hence it is much better than TOPSIS. The WDBA method has been successfully applied in many MCDM problems, such as evaluation of E-learning website [20], [21], selection of a COTS component [22], software effort estimation model evaluation [23], emergency decision problems [24], [25].
Bansal et al. [23] applied the WDBA method in the selection of the software effort estimation model. Peng and Garg [26] extended the WDBA method under the interval-valued fuzzy soft sets for decision-making problems. Peng and Li [25] used the WDBA method for emergency decision problems. Peng [24] introduced an MCDM method, which adopted the WDBA method. In this paper, to use advantages of the WDBA method, a new version of the WDBA method is developed for job-sequence determination. Moreover, in group decision-making problems, the weighting of experts is necessary.
Yue [27] introduced the concept of the average ideal solution to determine the experts’ weight. The expert with the opinions nearer to the average ideal solution, catch the higher importance. Mohagheghi et al. [28] presented a new expert’s weight method to decide internet companies. Gitinavard et al. [29] introduced a new multicriteria weighting and ranking method for selection problems. In this paper, the WDBA method is extended by using the average ideal solution for specifying the expert’s weight. Furthermore, the processing time has many uncertainties in practical situations gathered from experts.
In most of the recent studies, the deterministic processing time is considered. However, in reality, there are many inherent uncertainties in the processing time owing to the loading of batch, the capacity of processing unit, operator skills, transformation quality of raw materials in the production systems, imperfect information regarding systems, transportation lag, traffic jam, machine disablements, arrival of new jobs, and resources deficiencies [30], [31], [32], [33]. In these cases, the fuzzy sets theory introduced by Zadeh [34] can be used.
Classic fuzzy sets have many disadvantages in coping with the uncertainty. It is easier for the decision-makers (DMs) to reflect their opinions by an interval instead of deterministic values for lower and upper values of fuzzy numbers [35], [36]. It is mostly tough for the DM to entirely quantify his/her judgment as a number in the interval [0, 1] [37], [38]. Thus, it is appropriate to demonstrate this grade of actuality by a range [39], [40]. Hence, in this paper, to better address, the uncertainty of the practical situation, the interval-valued fuzzy sets (IVFSs) are used.
By taking the above statements into account, to use the advantages of the WDBA method, a new version of WDBA is presented to determine the sequence of the job in FSSPs. Furthermore, an extended version of WDBA based on the average, negative, and positive ideal solutions is introduced to specify the experts’ weights. Moreover, the processing time of each job on each machine is considered by using the IVTFSs. Finally, two new versions of the WDBA method are developed under the IVTF environment. To better illustrate the novelties of this paper, the literature review is summarized in Table 1.
Regarding the benefits of the WDBA method, it is regarded as the well-known MCDM method. In this method, the best and worst desirable cases are demonstrated by the ideal and anti-ideal points, respectively (e.g., [50]). The WDBA method is more efficacious in comparison with other available MCDM methods owing to the involvement of uncomplicated matrix operations and has some significant benefits, such as less complication, natural fulfillment, and intellect [20], [24], [26]. The proposed WDBA method is relatively more impressive and efficient as it involves easy and straightway mathematical operations, such as matrix operations [23], [26].
As can be seen, regarding the uncertainty of processing time, limited researchers have concentrated on the classic fuzzy processing time, and no one has considered IVF processing time. IVFSs are more fruitful than traditional fuzzy sets for addressing the uncertainty. In conjunction with the solution methods, heuristic, meta-heuristic, and hybridization of them can be caused to improve the solution quality; however, these algorithms are very sophisticated, require advanced programming skills, and need much computing time [5]. In this situation, the WDBA method with the simple mathematical procedure and easy to understand process are applied for job sequence determination. Note that Jain et al. [20] proved that the WDBA method is much effective than the TOPSIS method. Moreover, concerning the modeling, a group decision model is applied for the first time of the literature. Also, in a group decision-making process, the weight of experts was taken into account equally in the previous studies. Considering the equal importance for experts are incorrect because the expert with the higher experience and knowledge should be got higher weight. In this paper, a new method is introduced to the weight determination for the first time of literature.
Concluding that the novelties of this paper are explained below:
- •
Considering the interval-valued fuzzy processing time for FSSP decisions to better address the uncertainty of real-world scheduling problems.
- •
Introducing a new version of the WDBA method for determining the weight of each DM by adding the concept of average, negative, and positive ideal solutions to the WDBA method.
- •
Extending the WBDA method for the FSSP because this method does not require any particular calculation and complex mathematics.
- •
Developing two new versions of the WDBA for ranking and weighting under an interval-valued fuzzy number to tackle the uncertainty of real-world FSSPs.
This paper consists of the following sections. Section 2 presents the necessary information about IVFSs. Section 3 introduces the proposed methodology for the FSSP. Section 4 expresses an illustrative example to show computation trends. The paper is concluded in Section 5.
Section snippets
Preliminary
In this section, the IVFSs and their operations are expressed. The IVFSs are introduced by Gorzałczany [51]. Afterwards, the interval-valued trapezoidal fuzzy number (IVTFN) is explained by Yao and Lin [52], as shown in Fig. 2. An IVTFN is expressed by: It encompasses two segments, namely the lower bound () and the upper bound (), where (see Fig. 1).
Two IVTFNs can be specified by:
Proposed methodology
In this section, at first, a new version of WDBA is presented for specifying the sequence of jobs for the FSSP. Then, WDBA is extended using negative, positive, and average ideal solutions for determining the experts’ weights. Furthermore, in a practical situation, the processing time has an inherent uncertainties owing to the loading of batch, the capacity of processing unit, operator skills, transformation quality of raw materials in the production systems, imperfect information regarding
Application
In this section, an application that is adopted in the literature [5] is solved to show the performance of the proposed methodology better. The decision matrices based on the opinions of three DMs are constructed.
Step 1: The decision matrices based on Eq. (8) are formed and depicted in Table 2, Table 3, Table 4.
Step 2: All decision matrices are normalized by using Eqs. (9), (10).
Step 3: Weighted normalized decision matrices are constructed by using Eq. (11).
Step 4: Weight of each DM is
Conclusion
In reality, it is mostly tough for the decision-maker (DM) to entirely quantify his or her judgment as a number in the interval [0, 1]. Thus, it is appropriate to demonstrate this grade of actuality by an interval. That is why, in this paper, the uncertainty of the processing time has been considered by using the interval-valued fuzzy sets (IVFSs). Then, an extended version of the weighted distance-based approximation (WBDA) method has been used to determine the sequence of jobs in flow shop
CRediT authorship contribution statement
Y. Dorfeshan: Conceptualization, Writing - original draft. R. Tavakkoli-Moghaddam: Methodology, Supervision. S.M. Mousavi: Project administration, Resources. B. Vahedi-Nouri: Data curation, Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the Editor-in-Chief andanonymous referees for their valuable comments on the initial version of this study for the improvements.
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