Enhanced Fuzzy Delphi Method in Forecasting and Decision-Making

Abstract

The Delphi method is a process where subjective data are transformed into quasi-objective data using statistical analysis and are converged to stable points. The Delphi method was developed by the RAND Corporation at Santa Monica, California, and is widely used for long-range forecasting in management science. It is a method by which the subjective data of experts are made to converge using some statistical analyses. This article proposes a variation of the Delphi method using triangular fuzzy numbers, where the communication method with the experts is the same, but the estimation procedure is different. The utility of the method is illustrated by a numerical example.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In the Delphi method, experts with high qualifications are first requested to give their opinions separately, without any intercommunication, regarding predicting and realizing certain events in science and technology. Then, a statistical analysis is conducted to assess these subjective data, and their first, second, and third quartiles are computed. This statistical information is transmitted to the experts. Furthermore, these new estimates are then analyzed, and the first, second, and third quartiles are recalculated. The new information is sent out to the experts once again, and this process of restoration is continued until the process converges to a reasonable stable solution. Grisham [1] provided project management students with an example of the Delphi research approach applied to a recent doctoral research thesis, reviewed the literature on the Delphi approach, and explained the research method tool. Many authors have dealt with the Delphi method such as Story et al. [2], Kauko and Palmroos [3], Skinner et al. [4], and Beiderbeck et al. and Lawnik and Banasik [5, 6]. Piecyk and McKinnon [7] reviewed applications of the method to the field of freight transport and logistics, assessed the usualness of Delphi-derived predictions of long-term freight, and associated environmental trends. They introduced a case study, the experience of a large two-round Delphi survey undertaken in the UK to elicit projections of long-term trends in a number of road freight and logistics variables. Revez et al. [8] demonstrated the transformative potential of combining participatory action research approaches with a modified Delphi method to understand energy transition issues, particularly beyond forecasting instruments.

In the literature, first of all, Zadeh [9] proposed the philosophy of fuzzy sets. Decision-making in a fuzzy environment, developed by Bellman and Zadeh [10], has been improved, enhancing the management of decision problems. Zimmermann [11] introduced fuzzy programming and linear programming with multiple objective functions. Then, several researchers have worked on fuzzy set theory. The theory and applications of fuzzy sets, systems, and fuzzy mathematical models were studied by many authors [12–19]. Cheng et al. [20] investigated a new fuzzy Delphi method (FDM) to fit membership functions and examined the stability of the process of the technique. Roy and Garai [21] used a triangular intuitionistic fuzzy number to introduce a new and improved version of the FDM. Moreover, Habibi et al. [22] applied the FDM to the single round for screening criteria. Hsu et al. [23] summarized the key courses in the education of construction engineers, where the evaluation aspects and index are established using the FDM. Ciptono et al. [24] classified and identified scientific research and analyzed the current literature on the FDM to gain a broad and detailed understanding of education.

Furthermore, He et al. [25] proposed a method to select the proper green supplier successfully. Huang et al. [26] used the FDM to identify the key input variables with the deepest influence on the prediction of peak particle velocity (PPV) based on the expert’s opinions and applied the effective parameters on PPV in hybrid artificial neural network-based models, i.e., genetic algorithm, particle swarm optimization, imperialism competitive algorithm, artificial bee colony, and firefly algorithm, for the PPV prediction. Kabir and Sumi [27] proposed a new forecasting mechanism modeled by integrating the FDM with an artificial neural network technique to manage the demand with incomplete information. Many researchers have used the FDM in their works [28–30].

This article proposed a variation of the Delphi method using triangular fuzzy numbers, where the communication method with experts is the same, but the estimation procedure is different.

Having motivation from the above literature, in this study, a variation of the Delphi method using triangular fuzzy numbers, where the communication method with the experts is the same, but the estimation procedure is different, is studied.

The rest of the paper is outlined as shown in Figure 1.

[figure omitted; refer to PDF]

This information is given to each expert, and each reconsiders his/her previous forecast and gives a new TFN, thus obtaining a new sheaf. This process is continued until a stable solution according to certain criteria is reached. Of course, the number of such iteration phases can be limited a priori. Many variations of this procedure are possible. For instance, the experts can be advised not to increase the divergence, but this is only a suggestion because an expert must give their own unbiased opinion.

Definition 6.

The normalized distance between two TFNs is defined as follows:$$\begin{array}{}\text{(14)}& \delta M_i,M_j=\frac{1}{2{\gamma }_2-{\gamma }_1}{\Delta }_l{M}_i,M_j+{\Delta }_r{M}_i,M_j,\end{array}$$where $M_i$ and $M_j$ are the TFNs given by experts $i$ and $j$; ${\mathrm{\Delta }}_l$ and ${\mathrm{\Delta }}_r$ are the left and right distances, respectively; and ${\gamma }_2$ and ${\gamma }_1$ are arbitrary values at the right and left, respectively, chosen with the condition $\delta \in \mathrm{0,1}.$

Using formula (14), the results of the computation are inserted in Table 2 for the arbitrary values of ${\gamma }_1=2021$ and ${\gamma }_2=2049.$

As shown in Table 2, we see that the minimum distance is $\delta \mathrm{7,11}=0.03$, and the maximum distance is $\delta \mathrm{4,11}=\mathrm{0.87.}$

Let us assume that we are interested in a pair of experts for whom the distance $\delta M_i,M_j\le 0.15$. The matrix in Table 3 gives these pairs. The corresponding pairs are shown in Figure 4.

A direct examination of Figure 4 gives the maximum subrelations of similarity. For example, we have the following:$$\begin{array}{}\text{(15)}& \mathrm{1,5,8},\mathrm{1,8,9},\mathrm{1,10},\mathrm{2,4},\mathrm{3,5,8},\mathrm{3,8,9},\mathrm{5,6,8},\mathrm{7,11},\mathrm{9,12}.\end{array}$$

If we consider, for instance, experts $\mathrm{1,8,9}$ and $\mathrm{3,8,9}$, we find that $\delta \mathrm{1,3}=0.16$. We can assume, therefore, that experts $\mathrm{1,3,8,9}$ have given almost the same estimation. A similar situation appears with $\mathrm{1,5,8}$, $\mathrm{3,8,9}$, and $\mathrm{1,3,8,9}$ that have almost the same estimation. Therefore, we can aggregate experts $\mathrm{1,3,5,8,9}.$

Table 2

Distances of $\delta M_i,M_j$.

$\delta M_i,M_j$	1	2	3	4	5	6	7	8	9	10	11	12
1	$0$	$0.36$	$0.16$	$0.43$	$0.12$	$0.16$	$0.4$	$0.13$	$0.13$	$0.13$	$0.43$	$0.19$
2		$0$	$0.35$	$0.07$	$0.4$	$0.44$	$0.76$	$0.37$	$0.23$	$0.58$	$0.8$	$0.16$
3			$0$	$0.42$	$0.04$	$0.017$	$0.41$	$0.10$	$0.12$	$0.23$	$0.44$	$0.18$
4				$0$	$0.47$	$0.51$	$0.83$	$0.46$	$0.30$	$0.60$	$0.87$	$0.24$
5					$0$	$0.13$	$0.34$	$0.09$	$0.16$	$0.18$	$0.40$	$0.23$
6						$0$	$0.33$	$0.07$	$0.21$	$0.26$	$0.35$	$0.27$
7							$0$	$0.39$	$0.53$	$0.17$	$0.03$	$0.59$
8			Symmetry			$0$	$0.14$	$0.21$	$0.44$	$0.20$
9									$0$	$0.35$	$0.57$	$0.06$
10										$0$	$0.21$	$0.41$
11											$0$	$0.72$
12												$0$

Table 3

Pair of experts with arbitrary $\delta \le 0.15$.

${\delta }_{ij}\le 0.15$	1	2	3	4	5	6	7	8	9	10	11	12
1	$1$				$1$			$1$	$1$	$1$
2		$1$		$1$
3			$1$		$1$			$1$	$1$
4				1
5					$1$	$1$		$1$
6						$1$		$1$
7							$1$				$1$
8								$1$	$1$
9									$1$			$1$
10										$1$
11											$1$
12												$1$

[figure omitted; refer to PDF]

6. Discussion of the Results

If we consider $\delta \le 0.20$, we find this aggregation directly. In fact, using a convenient algorithm, the problem can be studied with various values of $\delta ,$ that is, $\delta \le 0.2,0.25,\dots$. If we wish to consider the similarity problem, then a value smaller than $1-\delta$ gives the level of similarity, which is an important point in any a posteriori discussion between experts. The maximal subrelations of similarity in a resemblance relation play the same role as classes in a similarity relation. The second one is always disjointed, but this is not the case for the first one.

7. Conclusions and Future Works

A variation of the Delphi method using TFNs has been proposed, in which the method of communication with experts is the same, but the estimation procedure is different. Of note, in many cases, a forecasted success requires various steps of operations; therefore, using the critical path method for forecasting is more efficient in such cases. In fact, many other ways can be used for forecasting using fuzzy numbers, which we may investigate in the future. Future work may include the further extension of this study to other fuzzy-like structures, i. e., interval-valued fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, spherical fuzzy set, picture fuzzy set, neutrosophic set, etc., with more discussion on real-world problems.

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