A generalised fuzzy TOPSIS with improved closeness coefficient
Introduction
Multi-Criteria Decision Making (MCDM) methods are widely used to assist in decision making when there are different criteria and the best alternative is to be selected. Often one needs to make the best compromise choice from the available options, since finding the best alternative may not be practically feasible. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the most widely used MCDM methods (Zyoud & Fuchs-Hanusch, 2017) in such situations. TOPSIS works on the principle of finding the best compromise solution when compared to an Ideal Solution. However, in contemporary business situations such as technology selection, decisions are often taken in uncertain environments, and evaluators may feel more confident in expressing the ratings of alternatives for given criteria in fuzzy sets or interval-values (Durbach & Stewart, 2012). To address this challenge, researchers have presented different versions of Fuzzy-TOPSIS method for specific decision-making environments (Behzadian et al., 2012, Joshi and Kumar, 2016, Mardani et al., 2015, Walczak and Rutkowska, 2017). Further, it is also possible that there could be different types of fuzzy or interval-valued numbers, with or without subjective weights of criteria by evaluators, and the current approaches do not incorporate these flexibilities and uncertainties in a single method.
We therefore seek to enhance the decision making approach to incorporate all the above problem variants. Developing a generalised and flexible selection model is important, since an organisation or decision-maker may not be willing to invest unduly high time or money in the development of different types of selection models. Accordingly, in this paper we propose a Generalised-Fuzzy-TOPSIS (GFTOPSIS) method, a versatile evaluation model capable of incorporating different kinds of flexibilities and uncertainties in the decision making process.
The proposed approach is a generalised, flexible and intelligent fuzzy MCDM method, using Interval-Valued Intuitionistic Fuzzy Set (IVIFS) for preference rating, suitable for use in uncertain environments. TOPSIS method is modified to incorporate IVIFS preference rating along with Degree of Optimism (DOpt). The proposed GFTOPSIS method uses DOpt to derive the expected IFS matrix, and subsequent calculations are performed based on the expected IFS matrix and distance between the two IFS. This method is different from the work of F. Ye (2010), who recommended TOPSIS based on distances between two IVIFS. The benefit of using DOpt is to help include the individual biases of an evaluator in the method. Thus, the evaluator may decide whether the interval values are inclined towards higher limits or lower limits or in-between. Additionally, weights of criteria are obtained by a combination of subjective weights given by evaluators, and fuzzy entropy or uncertainty weights. The entropy weights are derived based on input variability suggested by J. Ye (2010) for fuzzy decision making, so as to increase the intelligence in the model.
One limitation of the TOPSIS method is that it does not take care of different weights of Negative Ideal Solution (NIS) and Positive Ideal Solution (PIS) distances (Opricovic & Tzeng, 2004). We extend the TOPSIS method to include a situation where a decision-maker may decide final ranking with more focus either on PIS or on NIS. This is achieved by extending the final ranking step of TOPSIS method, which is the calculation of closeness coefficient, based on the separation from NIS and proximity to PIS. We also demonstrate that with the same focus on PIS and NIS distances, our proposed ranking is identical to TOPSIS, and also performs excellently when distance weights are varied. We also compare the experimental results of GFTOPSIS with earlier research on ranking steps.
The degree of generalisability of the model allows its application with several types of Fuzzy Set (FS) or Interval-Valued Fuzzy Set (IVFS), with different focuses on NIS and PIS distances. Specifically, GFTOPSIS method uses expert ratings in the form of IVIFS, and use DOpt to convert IVIFS into IFS. Further, the model is intelligent since it assigns weights based on the uncertainty levels in the ratings, and can work with or without subjective weights of criteria by evaluators. We also demonstrate the flexibility of the model with relevant examples and sensitivity analyses based on criteria with subjective weights, DOpt, evaluators’ weights in group decision making, and distance weights.
We illustrate our proposed method using two cases of technology selection and material selection in the context of Additive Manufacturing (AM). AM, colloquially three-dimensional printing or ‘3D printing’, is defined as “a process of joining materials to make objects from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing methodologies” (ASTM Standard, 2012). AM is considered to have potential to disrupt the production and supply chain (D'Aveni, 2015, Jiang et al., 2017). Recent developments in the AM technologies and AM materials have evoked interest among technologists to look for the best alternatives out of available processes and materials (Wohlers, 2016). Often, properties of the new technology's processes and materials are not available in crisp form, and this necessitates the expression of preference ratings as an appropriate fuzzy set. We also demonstrate how to use GFTOPSIS with different fuzzy inputs, and sensitivity analyses are presented to assist managers in choosing the best alternatives along with assessing the change in ranking with the change in the weights or DOpt. This provides users with a tool to facilitate making more informed decisions, while considering any particular condition which may have been missed by the evaluators while choosing input preference ratings. Such sensitivity analysis enhances the users’ confidence in the results.
In Section 2 we present pertinent literature, basic concepts of fuzzy sets, and define DOpt. Detailed steps of GFTOPSIS are explained in Section 3. The extension in the final ranking step is described in Section 4. The proposed method is used to demonstrate some instances for Additive Manufacturing (AM) technology and material selection in Section 5, followed by the conclusions and future directions in Section 6.
Section snippets
Background
Considerable literature is available on applications of MCDM methods in the selection of technology, suppliers, and materials. Different MCDM methods have been proposed in the literature to assist in decision making with multiple criteria and uncertain situations. TOPSIS, proposed by Hwang and Yoon (1981) is one of the widely used methods to derive ranks of the candidate alternatives (Zyoud & Fuchs-Hanusch, 2017). Preferences or rating of alternatives are often vague or underspecified, and
GFTOPSIS method
The proposed GFTOPSIS methodology is summarised in Fig. 1. Inputs need to be specified before calculation steps. First, evaluators explore available alternatives and criteria on which the alternatives will be ranked. From the available candidates or alternatives, we select the alternatives which pass the minimum requirement for evaluation. Initial screening is necessary to reduce the assessment efforts later. Further, criteria are defined for evaluation. It is also preferred to define the
New CCGFTOPSIS
In TOPSIS, CCTOPSIS is calculated as follows:
Here,, thus rank is implicitly decided based on the reverse order of ratio .
The most obvious way to add the relative importance of the distances from PIS and NIS is to take distance multiplied by weights, as follows:
Here
Application examples
We have explored applications of the GFTOPIS approach in solving real life problems. In a practical situation, managers or technologists are often comfortable in expressing the preference rating of alternatives on chosen criteria in different forms. For example, supplier or technology selection decisions require more information, and an individual may have both positive inclination as well as negative inclination for a particular alternative, so IVIFS is most appropriate for this type of
Discussions and conclusions
This paper, to the best of the authors’ knowledge, presents a novel generalisable GFTOPSIS method with improved closeness coefficient to select the best alternative in an uncertain environment. Different input preference numbers such as IFS, IVFS, FS are shown to be the special cases of our proposed method which is based on IVIFS. We have combined the uncertainty entropy weights with subjective weights, to present an intelligent and flexible model. Further, the degree of optimism has been
Acknowledgements
The authors thank the two industry experts with extensive experience in automotive design and familiarity with AM technologies and material, for their initial suggestions to develop a generalised selection method and subsequently providing inputs for the application examples. We also express our gratitude towards the two anonymous reviewers for their valuable suggestions which helped in improving the manuscript to its present form.
References (25)
- et al.
Interval valued intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1989) Intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1986)- et al.
A state-of the-art survey of TOPSIS applications
Expert Systems with Applications
(2012) - et al.
Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets
Fuzzy Sets and Systems
(1996) Extensions of the TOPSIS for group decision-making under fuzzy environment
Fuzzy Sets and Systems
(2000)- et al.
Computing with words to assess the sustainability of renewable energy options
Expert Systems with Applications
(2010) - et al.
Modeling uncertainty in multi-criteria decision analysis
European Journal of Operational Research
(2012) - et al.
An algorithmic method to extend TOPSIS for decision-making problems with interval data
Applied Mathematics and Computation
(2006) - et al.
Predicting the future of additive manufacturing: A Delphi study on economic and societal implications of 3D printing for 2030
Technological Forecasting and Social Change
(2017) - et al.
Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making
European Journal of Operational Research
(2016)