Z-Delphi: A Z-Number-Based Delphi Technique for Technological Forecasting to Reduce Optimism/Pessimism Bias in Experts’ Convergent Opinions

The Delphi method is a formal, comprehensive, systematised, qualitative, and interactive methodology of technology forecasting developed by a research group at the RAND Corporation in 1950 [1]. The technique aims to abbreviate the diversity of opinions within a small group, so they converge on a single viewpoint [2]. Since it results in an effective convergent decision by a large group of experts, the Delphi method is widely used in construction engineering [3], blockchain [4], web development and computing [5], and smart-city development [6].

In order to reduce the variance of the group’s opinions by defining the quantification of the responses obtained, the Delphi method entails circular stages of questions, answers, and analysis that are returned to the subject-matter experts [7]. The first round of Delphi focuses on ideation. The second round of questionnaires measures the viewpoint dispersion. In the third round, the focus shifts to obtaining expert consensus. At the end of the cycle, experts weigh the pros and cons of the group members’ and their own points of view. The cycle continues until the required questionnaires or opinions/ideas are collected. At the conclusion of these rounds, opinions with convergence and divergence are compiled [8]. Convergent views describe the majority-supported premise. Divergent perspectives help formulate alternative future hypotheses.

The Delphi technique has its limitations while being deployed extensively by researchers in fields such as engineering, innovation, and futuristic forecasting. Goodman [9] and Hirschhorn [10] have detailed the Delphi method’s limitations. The Delphi method has the following drawbacks, according to the literature:

1. 1.

   Dependency of reliability on expertise: The credibility of the study is compromised if the researcher is unable to recruit genuine experts for the experiment

2. 2.

   Repetition: The experiment is repeated with the same group of experts, which is likely to bore them.

3. 3.

   Count of experts: Occasionally, the sample of selected experts for a given problem can be small. A handful of experts may not provide the correct opinion on all issues under consideration by a researcher.

4. 4.

   Natural language processing (NLP): During the preliminary rounds, experts express their opinions in natural language. Consequently, it is challenging to derive a clear concept or the variable of interest from the opinions of experts.

5. 5.

   Ambiguity aversion: This is a common issue in decision-making. Experts’ opinions typically mirror the familiar over the unfamiliar because it is human nature to endorse the status quo.

6. 6.

   Unknown statistical optimism/pessimism bias: Statistical optimism bias may arise from either a distortion of personal estimates, which represents personal optimism, or a distortion for others, which expresses personal pessimism. Statistical pessimism bias is a psychological phenomenon in which people overestimate the likelihood of negative outcomes. Both biases are antagonistic to one another.

Researchers across the globe modify the Delphi method in order to circumvent its limitations. Ishikawa et al. [11] proposed the max–min Delphi method and the fuzzy Delphi method in order to sidestep the first two limitations listed previously. Li et al. [12] strengthened reliability by combining the Delphi method with the artificial neural network (ANN). Chen et al. [13] surmounted the first three limitations described above by combining the fuzzy Delphi method with the Kano two-dimensional quality model and the theory of innovative problem-solving. To avoid the third limitation described in the preceding list, Ahmad et al. [14] merged the fuzzy Delphi method and the nominal group technique. By implementing fuzzy clustering, Di Zio et al. [15] were able to eschew the first four limitations mentioned above. The pairing of fuzzy logic with the Delphi technique yielded favourable outcomes. In light of this, we intend to implement the fuzzy logic theory in the current work.

The Delphi technique introduced the fuzzy theory to save time and clearly articulate expert opinions. The fuzzy Delphi method reduces costs and time while reducing redundancy. However, challenges concerning behavioural fuzziness, ambiguity aversion, and statistical optimism/pessimism bias persist. Further, the classical fuzzy Delphi method is incapable of handling inconsistent or conflicting expert opinions. Following the initial round of Delphi, some variations of Delphi methods collect experts’ responses employing Likert scales. These rounds suffer from extreme response style (ERS) and non-extreme response style (NERS) psychological issues [16]. ERS respondents prefer to mark only the extreme ends of the Likert scale. NERS respondents, on the other hand, avoid systematically selecting the extreme ends of the Likert scale. In the present work, we have addressed ERS and NERS concerns along with the six disadvantages mentioned above.

Prof. Zadeh [17] and Aliev et al. [18] argued that Z-numbers outperform fuzzy numbers because they manage behavioural fuzziness and ambiguity aversion. Aliev et al. [18] offered an in-depth explanation on Z-number utilisation to eliminate ambiguity aversion. Leveraging Z-numbers allows for the avoidance of ambiguity aversion since it takes into consideration both the value of an element or variable and its reliability. Z-numbers may be exploited to tackle NLP challenges since they rank among the most effective tools for managing opinions expressed in natural human language [19]. Anjaria [20] demonstrated how Z-numbers can help overcome psychological issues like ERS and NERS related to the Likert scale. Given these merits, Z-numbers became the obvious option for this study.

In the present study, we opted for Z-numbers over alternative uncertainty representation models, such as D-numbers and R-numbers, due to the superior ability of Z-numbers to emulate human thought processes. Z-numbers effectively encapsulate both the reliability and uncertainty inherent in information, rendering them an ideal choice for representing expert viewpoints [21]. As an extension of fuzzy numbers, Z-numbers amalgamate a real number (representing a quantity) with a fuzzy number (depicting the degree of certainty or reliability) [22]. This composition enables Z-numbers to concurrently represent an expert's opinion and the confidence or reliability they attribute to that perspective. By incorporating the reliability component, Z-numbers adeptly tackle the intrinsic ambiguity and uncertainty present in expert evaluations, which is of paramount importance within the Delphi technique. To surmount the aforementioned limitations of the Delphi method, we integrated it with Z-numbers in this investigation. Our decision to employ Z-numbers in the proposed fuzzy Delphi technique underscores our commitment to devising a methodology that closely adheres to the intricacies and subtleties of human cognition in expert appraisals, ultimately culminating in more accurate technology forecasting outcomes.

The combination of the Z-number with the Delphi technique is, without question, exceedingly promising, yet, it does present some practical hurdles. The generation of Z-numbers from expert opinions is one of the challenges. The generation of a fuzzy number, including a Z-number, from real-world data, is extremely complicated [20, 23]. The Delphi procedure necessitates the processing of natural language views and the calculation of consensus with the help of a statistical method. There are several instances in literature where Z-numbers addressed linguistic variables or when Z-numbers have been constructed with the help of real-world linguistic variables [19, 24, 25]. However, with the Delphi approach, Z-numbers based only on linguistic factors will not provide the intended results for two reasons. (1) The experts will not limit their responses to specific linguistic phrases; (2) with linguistic Z-numbers, one needs to execute sophisticated fuzzy statistical computations following each round. Therefore, we not only merged Z-numbers with the Delphi approach in this study, but we also addressed the practical implementation challenges mentioned above.

The motivations behind the present research can be summarised as follows. Primarily, our objective is to confront the constraints inherent in the Delphi technique, specifically those pertaining to ambiguity aversion, uncertainty, and the biases of statistical optimism and pessimism. In addressing these limitations, our aspiration is to elevate the field of technology forecasting by delivering a more effective and precise methodology that duly considers the subjective evaluations of experts. Moreover, our interest lies in devising an approach that can wield considerable influence on decision-making processes within vital domains such as sustainable production, where precise forecasting is indispensable for strategic planning and judicious decision-making.

In the present work, we performed arithmetic operations on discrete Z-numbers to determine the degree of consensus after integrating a conventional discrete Z-number with the Delphi technique [26]. We calculated Z-numbers by taking statistical optimism/pessimism parameters into account. Currently, no Delphi variant is capable of overcoming all six of the aforementioned restrictions. This paper proposes a Z-number-based variant of the Delphi approach, which addresses most of its shortcomings for technology forecasting. We applied the Delphi technique to generate ideas and predict dairy technologies in Gujarat, India. We leveraged the Delphi approach to predict computational and AI-driven water-saving solutions for Gujarat’s dairy facilities. Environment-friendly dairy operations and cleaner milk processing demand interdisciplinary expertise, as demonstrated by the case study. Applying the Delphi approach, we examined the role of AI in water conservation policy advocacy and a standard implementation.

In a nutshell, the current work makes the following contributions:

1. 1.

   A novel Z-number-based fuzzy Delphi technique: We propose a new method that combines the Delphi technique with Z-numbers to simulate human thinking and reduce limitations associated with the traditional Delphi technique.

2. 2.

   Integration of basic probability assignments (BPAs) and grey clustering: We develop a method that generates BPAs from expert responses and considers statistical dispersion using grey clustering, which enhances the performance of the proposed fuzzy Delphi technique.

3. 3.

   Application to water-saving technology forecasting: We demonstrate the applicability of our proposed method in a real-world scenario by forecasting water-saving technology for dairy plants, involving 11 experts.

4. 4.

   Comparison with other fuzzy Delphi approaches: The proposed method is compared to other fuzzy Delphi approaches using entropy, demonstrating its superiority in terms of reduced uncertainty.

5. 5.

   Discussion of future research directions: The paper concludes by discussing potential avenues for future research, providing insights for researchers interested in further exploring and expanding the Z-number-based Delphi technique.

The structure of this paper is organised as follows: Sect. 2 presents the foundational concepts relevant to the topic. Section 3 delves into the proposed approach for integrating Z-numbers with the Delphi technique. Section 4 applies this method to a real-world case study involving the dairy industry in Gujarat. Section 5 offers a comparative analysis between the proposed technique and existing methods, illustrating that the suggested approach exhibits lower uncertainty than its counterparts. Lastly, Sect. 6 concludes the paper and explores potential directions for future research.

Since Ishikawa et al. [11] developed the fuzzy Delphi method with the aid of the fuzzy membership function, we initiated this section by explicitly defining it. A fuzzy membership function is a mathematical function that assigns a degree of membership to an element in a set, in the context of fuzzy set theory. It maps an element of the set to a value between 0 and 1, where 0 represents non-membership and 1 represents full membership. The membership function allows for modelling uncertain or imprecise data, as elements can have partial membership in a set. For example, a membership function for the fuzzy set “hot” might assign a value of 0.8 to an element “tea” indicating it is very hot and a value of 0.3 to an element “water” indicating it is not very hot. In fuzzy logic, a membership function is used to evaluate how well an input value maps to a particular output value. In other words, it determines whether or not a given input belongs to a particular set. The fuzzy membership function has the following formal definition:

Definition 1

Fuzzy membership function [27]. A fuzzy set \(A\) is one that is defined by a mathematical function known as the fuzzy membership function, which is denoted by \({\mu }_{A}\in [\mathrm{0,1}]\). If \({\mu }_{A}=0\), it implies that \(x\notin A\). On the other hand, if \({\mu }_{A}=1\), then \(x\in A\).

Impulsive fuzzy membership functions are of the most prevalent type:

A triangular fuzzy membership function with \(a, b>0\), is expressed as

A right-sided trapezoidal fuzzy membership function with \(a>0\) is defined as

A left-sided trapezoidal fuzzy membership function with \(a>0\) is defined as

The Gaussian fuzzy membership function with \(\sigma\) standard deviation is defined as

The aforementioned fuzzy membership function facilitates the creation and incorporation of Z-numbers in real-world circumstances. In the current effort, we chose to blend the Z-number concept with the Delphi approach. Next, we came to grips with the Z-numbers. Prof. Zadeh advocated applying Z-numbers to simulate real-world uncertainty and ambiguity aversion [17].

Definition 2

Z-number [17]. A Z-number is an ordered pair of fuzzy numbers, denoted as \(Z (A, B)\), where \(A\) determines the restriction on the values of some real-valued uncertain variables expressed via \(X\), while \(B\) is a measure of reliability for the first component.

Both elements \(A\) and \(B\) of the Z-number \(Z (A, B)\) may be represented by any of the fuzzy membership function types described in Definition 1. For example, in \(Z (A, B)\), \(A=\mathrm{Gaussian}[0.6, 0.2]\) and \(B=\mathrm{Gaussian}[0.8, 0.05]\). In the example mentioned, \(\mathrm{Gaussian}[0.6, 0.2]\) is a function in which the first element is \(c\) and the second \(\sigma\) of the function \({\mu }_{A}\equiv {e}^{-\frac{{(x-c)}^{2}}{2{\sigma }^{2}}}\). The Z-number can also be represented as \(Z = Z(A, B) = (A, {\mu }_{A}.{P}_{{X}_{A}}is B)\). Under the second representation, \({X}_{A}\) for \(A\) is a random variable, \({P}_{{X}_{A}}\) is the probability involved with the random variable \({X}_{A}\). \(A\) and \(B\) are fuzzy sets, \({\mu }_{B}(x)\) is the membership function for the fuzzy set \(A\) where \(x\in {X}_{A}\) and \(x\in R,\) where \(R\) is the real value. Finally, the term “\({\mu }_{B}.{P}_{{X}_{A}}is B\)” may be represented as \({\mu }_{B}(\int {\mu }_{A}(x).{P}_{{X}_{A}}(x)\mathrm{d}x)\). We adopted the second method of Z-number representation in the present work. Mathematically, the set of fuzzy Z-numbers can be defined as \({Z}^{*}=\{(A,{\mu }_{A})|A\in Z, {\mu }_{A}:[0, 1] \to [0, 1]\}\)

In the above equation, \(A\) is an integer from the set of integers \((Z)\), and \({\mu }_{A}\) is the membership function associated with \(A\), mapping a value between \(0\) and \(1\) to another value between \(0\) and \(1\), representing the degree of uncertainty or membership of the value in the fuzzy set. An instance of a fuzzy Z-number would be a specific fuzzy Z-number from this set. For example, let us consider the fuzzy Z-number \((3, {\mu }_{3})\), where the integer \(A = 3\) and the membership function \({\mu }_{3}\) can be given by: \({\mu }_{3}\left(x\right)={e}^{({-x}^{2})}\)

This specific fuzzy Z-number, shown in the above equation, is an instance of a fuzzy Z-number from the set Z*. Let us consider an example to understand uncertainty handling in a better way via a Z-number. An investor purchases stocks in the company. Based on the stock market study, an investor concludes that the acquired equities are likely to provide high returns expressed as: < High return, likely > . This prediction may be expressed formally as a Z-number via Definition 2 as “\(X\) is \(Z = (A, B)\).” Here, \(X\) is the variable that describes the high return, \(A\) is a fuzzy set used to describe the soft constraint on a high return, and \(B\) is a fuzzy number to describe a soft constraint on partial reliability, which is expressed as ‘likely’ of \(A\). The Z-number is commonly utilised in the literature to build strategies and approaches that may be applied in research methodology. Anjaria [20] constructed a Likert scale based on the Z-number, while Jiang et al. [28] deployed a Z-number to design a Z-network-based Bayesian model. Aliev et al. [29] published a Z-number-based linear programming model, while Shen and Wang [66] suggested the VIKOR approach using a Z-number. Finally, Azadeh et al. [10] came up with a Z-number-driven AHP process, while Jia et al. [30] proposed a multicriteria decision-making process using Z-numbers. Figure 1 depicts the instance of a Z-number.

An abstract example of a Z-number

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In the current effort, we intended to obtain the Z-number using real-world data. The Z-number may be obtained from real-world data using basic probability assignment (BPA) and interval-value logic [31, 32]. In this paper, we integrated clustering, BPA, and statistical optimism/pessimism bias with interval-valued logic to construct Z-numbers. Consequently, we examined the fundamentals of BPA in this section. The frame of discernment and BPA function form the basis of classical BPA theory. It serves as the underlying structure for assigning probabilities to the individual outcomes within the frame. The frame of discernment is typically defined as a set of mutually exclusive and exhaustive events, which means that each outcome in the frame belongs to exactly one event and no other outcomes exist outside of the frame. For example, the frame of discernment for a coin toss would be the set of outcomes \(\{\mathrm{heads}, \mathrm{tails}\}\), which are mutually exclusive (cannot happen at the same time) and exhaustive (together they represent all possible outcomes). Once the frame of discernment is defined, a BPA can be assigned to each outcome, which is a measure of the degree of belief that the outcome will occur. Definitions 3 and 4 provide explicit definitions for these concepts.

Definition 3

Frame of discernment [23]. Let \(\theta\) be the complete set of all possible values of the variable \({X}_{A}\). The elements of \(\theta\) are mutually exclusive. In this case, the frame of discernment has been defined as follows:

The power set of \(\theta\) is represented as \({2}^{\theta }\) which can be described as \({2}^{\theta }=\{\varnothing , \left\{{H}_{1}\right\}, \dots , \left\{{H}_{N}\right\}, \left\{{H}_{1}, {H}_{2}\right\}, \dots , \left\{{H}_{1}, {H}_{2},{H}_{i}\right\}, \dots , \theta \}\).

The powerset plays a significant role in BPA function formation described in Definition 4.

Definition 4

BPA function [23]. A BPA function \(m\) is a mathematical mapping of \({2}^{\theta }\) to the probability interval [0,1], which is expressed by the following equation: \(m: {2}^{\theta }\to \left[\mathrm{0,1}\right].\)

The BPA equation, which encapsulates the fuzziness [33] in Definition 4, adheres to the subsequent criteria:

Under the aforementioned criteria, \(m\left(A\right)\) reflects the degree to which the evidence supports proposition \(A\). The BPAs may, additionally, support commutative and associative rules via Definitions 3 and 4. Using the preliminaries stated in this section, we have outlined the recommended methodology in the next section.

As illustrated in Fig. 1, a Z-number is a mathematical concept designed to capture both uncertainty and imprecision by amalgamating two distinct elements: membership degree and probability. Within the realm of fuzzy sets, membership degree signifies the extent to which an element belongs to a specific fuzzy set, with values ranging from 0 to 1. Fuzzy sets are especially advantageous in situations where category boundaries are ambiguous, allowing for the portrayal of uncertain and imprecise characteristics inherent in real-world information. Conversely, probability is a numerical metric that gauges the likelihood of an event or outcome transpiring, with values also spanning from 0 to 1. Probability functions as an instrument for quantifying and managing uncertainty across various fields, including decision-making and risk evaluation. By integrating both membership degree and probability, Z-numbers offer a holistic representation of uncertainty and imprecision in intricate scenarios. The dual structure depicted in Fig. 1 fosters a more refined comprehension of uncertain circumstances and a more precise representation of relationships among diverse elements.

In this section, we launched our experiment by asking the experts open-ended questions in the first round of the Delphi technique. A key property of an open-ended question is that it cannot be answered with a binary response, such as ‘yes’ or ‘no.’ We ordered the experts’ responses depending on their frequencies after the first round of Delphi. Assume that 4 of the 15 experts in the Delphi survey believe that one specific issue can be handled via Artificial Intelligence (AI), while 3 experts offer the Internet of Things (IoT) as a solution. Two experts recommend the Petri net analysis (PNA). The rest of the experts have different points of view. AI, IoT, and PNA turned out to be high-frequency solutions in the present Delphi survey. The researchers tapped high-frequency solutions aided by text-mining algorithms [34], Delphi scenarios [35], fuzzy fractional solution [36], fuzzy clustering [15], neuro-fuzzy system [37] and distributed order derivatives [38]. In the second round, participating experts were required to evaluate every solution with a high frequency. In the above fictitious scenario, 15 experts would have been asked to rank AI, IoT, and PNA techniques based on their usefulness in tackling a single issue. The second round began with our recommended technique. How a group of experts may appropriately rank solutions has been discussed in the next subsection.

3.1 Receiving Ranks from the Experts After the First Round of Delphi

Expert rankings are clearly futuristic projections. The experts provide their ‘best estimates’ about the future. Expert solutions are never accurate. Mathematically, there will always be a gap expressed as \(\Delta x = |\overline{x} - x|\) between the experts’ rank of the solution represented as \(\overline{x }\) and the actual rank as \(x\) of the solution [39]. Measurement and opinion-recording operations are replete with uncertainties [1]. Manufacturers of measuring instruments set upper and lower boundaries for calculating the absolute value of the measurement error such that \(\Delta x \to \left| {\Delta x} \right| \le \Delta\) in order to reduce measurement uncertainty. In light of the measurement result \(\overline{x}\), the boundaries convey minimal information about measurement uncertainty. Furthermore, the only conclusion we can infer about the unknown value \(x\) using boundaries is that \(x \in \left[ {\underline {x} ,\overline{x}} \right]\mathop = \limits^{{{\text{def}}}} \left[ {\overline{x} - \Delta , \overline{x} + \Delta } \right]\). The relationship of \(x\) with \(\left[ {\underline {x} ,\overline{x}} \right]\) is known as interval uncertainty [40]. For the relationship \(x\) with \(\left[ {\underline {x} ,\overline{x}} \right]\), if \(x < \left[ {\underline {x} ,\overline{x}} \right]\) and \(x^{\prime} < x\), it follows that \(x^{\prime} < \left[ {\underline {x} ,\overline{x}} \right]\). Similarly, if \(\left[ {\underline {x} ,\overline{x}} \right] < x\) and \(x < x^{\prime}\), then \(\left[ {\underline {x} ,\overline{x}} \right] < x^{\prime}\). It is evident that if \(x < \underline {x}\), then whatever be the actual value from the interval \(\left[ {\underline {x} ,\overline{x}} \right]\), it will be greater than \(x\). Based on this discussion, we may mathematically infer that \({\text{sup}}\left\{ {x:x < \left[ {\underline {x} ,\overline{x}} \right]} \right\} = {\text{inf}}\{ x:\left[ {\underline {x} ,\overline{x}} \right] < x\}\).

In the current study, we sought to manage interval uncertainty by employing range sliders. The range sliders comprise a custom range-type HTML5 input control [41] that allows users to adjust pointers at both ends of the scale. Current research solely employs range sliders for filtering, not data collection. Many e-commerce platforms allow consumers to filter items by price range [42]. According to the literature, range sliders are simple and do not give insights into the underlying data since the attribute values obtained using range sliders are not uniformly distributed. Nonetheless, the objective of the current effort is to accurately gather the upper and lower ranges of the rank without concern for the underlying distribution. The underlying distribution in the present study is handled by implementing the distance function and BPA. Figure 2 depicts the range sliders for the aforementioned AI, IoT, and PNA examples:

Range sliders as data collecting tool

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Figure 2 depicts the experts’ opinions in the form of \(A[a,b]\), with \(a\) representing the lower bound and \(b\) representing the upper bound. We set the range sliders’ minimum and maximum limits to 0 and 10 in that order. Range sliders can be useful for collecting responses for “AI”, “IoT”, and “PNA” by allowing users to indicate their preferences, interest levels, or ratings for each option on a scale ranging from 0 to 10. To collect responses using the range sliders for “AI”, “IoT”, and “PNA”, one can follow these steps:

1. 1.

   Set up the range sliders as shown in the previous code examples, with labels indicating each option (AI, IoT, and PNA).

2. 2.

   Encourage users to move the sliders to reflect their preferences, interest levels, or ratings for each option.

3. 3.

   Collect the slider values using JavaScript or a form submission. For example, you could add a “Submit” button to the form, and process the responses on the server-side (using a backend language such as PHP and Node.js) or in the browser using JavaScript (e.g. sending the data to a database or API).

4. 4.

   Analyse the collected responses to gain insights into user preferences, interest levels, or ratings for each option. You can use various data analysis techniques to find patterns, trends, or correlations between the responses.

The utility of range sliders depends on the context and the specific questions one wants to ask. They may not be the best choice for every situation, but they can be a valuable tool for collecting user feedback. The subsequent section leverages the interval number A to calculate one of the fuzzy measurements, namely the Z-number.

3.2 Generating Z-Number from Real-World Interval Numbers Using BPA

As previously established, the actual rank of a specific solution proposed by a group of experts is indeterminate. The rank is a type \(A[a,b]\) interval value. In the literature on fuzzy decision-making, interval values are determined by Hurwicz’s optimism–pessimism parameter [43]. Hurwicz’s optimism–pessimism parameter meets the monotonicity and additivity criteria. The maximum value of the following combination of interval values and Hurwicz’s optimism–pessimism parameter is as follows: \(\beta .b+\left(1-\beta \right).a=a+\beta .\left(a-b\right)\). According to this equation, each interval number attempts to maximise the following equations:

In Eqs. (1) and (2), \(\beta\) represents Hurwicz’s optimism–pessimism parameter. The value \(\beta =1\) represents absolute optimism, while \(\beta =0\) denotes complete pessimism. Thus, the parameter \(\beta\) should be determined in such a way that the outcome reflects the balanced optimism and pessimism of the expert. To achieve balanced outcomes, the researchers integrated Hurwicz’s optimism–pessimism parameter with optimisation theory [44,45,46,47]. Hurwicz’s criterion multiplies the best payoff by \(a\) and the worst payoff by \((a-b)\) for each decision, then selects the optimal outcome. To balance the expert’s optimism and pessimism, we assigned the values \(\beta =1\) and \(\beta =0\) to both Eqs. (1) and (2). Hence, by substituting values \(\beta =1\) and \(\beta =0\) in Eqs. (1) and (2), we derived two interval values for each interval number. For \(A[a,b]\), we generate \({A}_{1}[{a}_{1},{b}_{1}]\) and \({A}_{2}[{a}_{2},{b}_{2}]\) by substituting values \(\beta =1\) and \(\beta =0\) for Eqs. (1) and (2).

Next, we sought to balance Hurwicz’s optimism–pessimism parameter using the distance between interval numbers \({A}_{1}[{a}_{1},{b}_{1}]\) and \({A}_{2}[{a}_{2},{b}_{2}]\). To determine the distance between the interval numbers, we followed Tran and Duckstein’s approach [40]. The distance between \({A}_{1}\) and \({A}_{2}\) is represented as \(D({A}_{1}, {A}_{2})\) and defined as [40]

The simplification of the above equation provides

For the distance expressed in Eq. (3), \(D({A}_{1}, {A}_{2})\ge 0\). If \(D\left({A}_{1}, {A}_{2}\right)=0\), then \({A}_{1}={A}_{2}\). The integration in the preceding equation expresses the integration of the square of the Euclidean distance. The basic principle behind applying interval numbers to construct BPA is to establish the model interval number from the collected samples and then calculate the distance between the sample to be tested and the model interval number. Using this information, we calculated the reciprocal of the interval number to derive similarity. The attribute value similarities were then determined using the distance algorithm given above. \(S({A}_{1}, {A}_{2})\) denotes attribute value similarity, which has been formalised as follows:

In Eq. (4), \(\gamma >0\) is a support coefficient, and \(D({A}_{1}, {A}_{2})\) is the distance function defined in Eq. (3). The next step involves calculating evidence leveraging the distances and similarities between interval numbers. We needed a frame of discernment for evidence calculation, as mentioned in Definition 3. To determine the frame of discernment from the interval data, we employed the Grey Clustering Approach (GCA) [48]. GCA is regarded as one of the most efficient techniques for clustering interval numbers [48]. To address the challenges of faulty data and inadequate information in the prediction of the environment, the GCA was developed and deployed to estimate the effectiveness of a water resource project on wild animals [49]. GCA’s main steps [48] involve the following:

1. 1.

   Finding the sample interval matrix: Let there be \(n\) clustering object. For these objects, decide a complete set of all possible values \(p\) of the variable \(X\), where \(\theta\) is the full set of all possible values of the variable \(X\).

2. 2.

   Determining the clustering function: Determine the clustering function \({f}_{i,j}\left(x\right)\in [\mathrm{0,1}]\) such that ith clustering index belongs to jth clustering criteria. The clustering criteria may be presented as \({\lambda }_{i,j}\).

3. 3.

   Computing clustering weight: A clustering weight is denoted as \({w}_{i,j}\) where the weight of cluster index \(i\) belongs to jth criterion in such a way that \({w}_{i,j}=\frac{{\lambda }_{i,j}}{\sum_{i=1}^{p}{\lambda }_{i,j}}\). In the present work, we have implemented the normalised \({\lambda }_{i,j}\).

4. 4.

   Computing clustering matrix based on the clustering coefficient: The clustering coefficient \({\sigma }_{k,j}\) was computed as \({\sigma }_{k,j}=\sum_{i=1}^{p}{f}_{i,j}\left(x\right){w}_{i,j}\)

5. 5.

   Cluster generation: Finally, the cluster may be identified as \({\sigma }_{k,j}=\mathrm{max}\{{\sigma }_{k,j}|k=1, 2, \dots , l\}\).

We next computed each cluster’s average similarity \(\mathrm{Avg}(S)\). Delphi implementers may be utilised to determine the number of required clusters. For each \(n\) clustering item, we estimated (1 − |\({n}_{i}-\mathrm{Avg}(S)|)\). Finally, we normalised the values of (1 − |\({n}_{i}-\mathrm{Avg}(S)|)\) and obtained BPAs. As outlined in the previous section, input data gathered via the range sliders was not going to be dispersed uniformly. We implemented the multiscale probability transformation (MPT) to distribute expert responses evenly [50]. The MPT is a pignistic probability transformation (PPT) generalisation. In decision theory, there are two terms that are used: MPT and PPT. Both phrases refer to the likelihood that a reasonable person will choose one alternative over another when forced to make a choice. In addition, MPT and PPT adhere to the insufficient reasoning principle [51]. Once the idea of MPT is attributed to the gathered replies, the accompanying uncertainty can be documented during the collection and computation of responses. The formal definition of MPT is provided as follows:

Definition 5

(Multiscale probability transformation (MPT)) [50]. Let m be BPA on the frame of discernment \(\theta\). Then, the associated multiscale probability function \(\mathrm{Mul}P\left(m\right):\theta \to [\mathrm{0,1}]\) on \(\theta\) is defined as \({\mathrm{Mul}P}_{m}\left(\rho \right)={\sum }_{A\subseteq P\left(\theta \right),\rho \in A}\left(\frac{{\left(Pl\left(\rho \right)-Bel(\rho )\right)}^{q}}{{\sum }_{\beta \in A}^{|A|}{\left(Pl\left(\beta \right)-Bel(\beta )\right)}^{q}}*\frac{m(A)}{1-m(\varnothing )}\right)\).

Under Definition 5, \(|A|\) is the cardinality of set \(A\), q is the entropic factor that is based on the Tsallis entropy [52] to amend the proportion of the interval, while \(\mathrm{Pl}\left(\rho \right)\) and \(\mathrm{Bel}(\rho )\) are plausibility and belief functions. The formal definitions of these functions are given as follows:

Definition 6

Belief (\(\mathrm{Bel}\)) and plausibility (\(\mathrm{Pl}\)) functions [52]: For a \({T}_{2}\subseteq \theta\), (where \(\theta\) is the frame of discernment as described in Definition 3) the belief function \(\mathrm{Bel}: {2}^{\theta }\to [0, 1]\) is defined as

Further, plausibility function \(\mathrm{Pl}:{2}^{\theta }\to [0, 1]\) is defined as

Evidence spacing helps rank choices and responses. In addition, the computation of distance and similarity between evidence helps to comprehend the differences in expert opinion. The distance between two bodies of evidence \({m}_{i}\) and \({m}_{j}\), denoted as \(d({m}_{i}, {m}_{j}\)), is calculated as \(d\left({m}_{i}, {m}_{j}\right)=\sqrt{\frac{1}{2}{(\overrightarrow{{m}_{i}}-\overrightarrow{{m}_{j}})}^{T}\overline{D }(\overrightarrow{{m}_{i}}-\overrightarrow{{m}_{j}})}\). In the equation, \(D\) is a scalar value measuring the similarity of fuzzy sets \(A\) and \(B\) as \(D(A,B)=\frac{|A\cap B|}{|A\cup B|}\). In contrast, the concept of similarity opposes the concept of distance. The similarity between two bodies of evidence \(d({m}_{i},{m}_{j})\) is defined as \(\mathrm{Sim}\left({m}_{i},{m}_{j}\right)=1-d\left({m}_{i},{m}_{j}\right)\). Let us assume there exist \(t\) evidence aggregations. After determining the degrees of similarity between bodies of evidence, a Similarity Measurement Matrix (SMM) may be constructed. It comes as no surprise that it might give us the impression that the various bodies of evidence are in agreement. The SMM looks like this:

We acquire equidistant evidence after calculating MPT. However, if there is a disparity, the credibility degree concept may be implemented [53]. The credibility degree concept is formalised as follows:

Definition 7

Credibility degree [53]: The credibility degree \({\mathrm{Crd}}_{i}\) of the body of evidence\({m}_{i}\), where \((i = 1, 2, ...,k)\), in that case

\(S({m}_{i}, {m}_{j})\) is the similarity between the evidence, according to the concept described under Definition 6. The credibility degree is a weight expressing the relative importance of the collected evidence. The discounted coefficient may be used to determine the relative relevance of the BPA [54].

Definition 8

Discounted Co-efficient [54]: The discount coefficient \(\alpha\) is formally defined as

Here, \(\mathrm{Max}({\mathrm{Crd}}_{i})\) is the maximum credibility derived using the equation described in Definition 7. Generally, \(\mathrm{Max}({\mathrm{Crd}}_{i})\) for the known ith evidence may be considered as 1. However, an unknown or uncertain evidence \(i\) may be discounted with the coefficient \(\alpha\). Following that, all pieces of evidence are fused using the Dempster fusion rule, deploying the discount coefficient α to assign various weights to each item of evidence. The general equation for evidence fusion given two pieces of evidence \(X\) and \(Y\) is \(\frac{{\mathop \sum \nolimits_{X \cap Y = C, \forall X,Y \subseteq \theta } m_{i}^{\alpha } \left( X \right) \times m_{i}^{\prime \alpha } \left( Y \right)}}{{1 - \mathop \sum \nolimits_{X \cap Y = \emptyset , \forall X,Y \subseteq \theta } m_{i}^{\alpha } \left( X \right) \times m_{i}^{\prime \alpha } \left( Y \right)}}\) if \(X \cap Y \ne \emptyset\). The fusion equation represents the resolution of any inconsistencies in the available data. The distance of all the original pieces of evidence is determined from the fused evidence the distance equation \(d\left({m}_{i}, {m}_{j}\right)=\sqrt{\frac{1}{2}{(\overrightarrow{{m}_{i}}-\overrightarrow{{m}_{j}})}^{T}\overline{D }(\overrightarrow{{m}_{i}}-\overrightarrow{{m}_{j}})}\). The greater the distance from fused evidence m, the lower the credibility of the actual evidence. Consequently, we applied the function \({\omega }_{i}=\frac{{e}^{-{d}_{i}}}{\sum {e}^{-{d}_{i}}}\) to calculate the discount coefficient \({\omega }_{i}\) of each piece of evidence, which amounts to the weight of the evidence, i.e. the importance of each piece of evidence. Finally, for the variable \({X}_{A}\), we calculated \(X\)-axis values using the equation \(X=\sum_{i=1}^{t}{p}_{i}\times {\mu }_{i}\). The Z-number has been represented by plotting \(X\) versus \({\omega }_{i}\) graph. The complete algorithm for generating Z-number from the collected expert opinion is described as follows:

In the algorithm detailed herein, we initially logged interval figures utilising a range slider—a graphical user interface component that enables users to choose a range of values. An exemplar of a comparable range slider employed during this procedure is depicted in Fig. 2. Following that, we implemented Hurwicz’s optimism–pessimism parameter, a decision-making standard that amalgamates optimistic and pessimistic perspectives when assessing alternatives, to compute the interval figures, taking into account the optimal and least favourable outcomes. Thereafter, we employed GCA to determine clusters. GCA represents a clustering method that assembles analogous data points according to their distances and resemblances. To ascertain these distances and similarities among the data points, Eqs. (3) and (4) were applied. Subsequently, we adopted the normalisation technique proposed by Kang et al. [31] to derive a normalised BPA. Normalisation constitutes a vital step, ensuring the total probability assigned to all potential outcomes equals one.

Post-normalisation, the MPT was computed using the formula delineated in Definition 5. MPT is a value signifying the most plausible outcome amongst all possible alternatives. We then examined the Z-numbers’ characteristics, determining if they were triangular, trapezoidal, or exhibited other shapes. This analysis is crucial for grasping the uncertainty and imprecision affiliated with the Z-numbers. During the algorithm’s concluding stages, we ascertained the Z-number by calculating the credibility degree and the discounted coefficient. These calculations were conducted by implementing Definitions 7 and 8. Both the credibility degree and discounted coefficient are critical for appraising the uncertainty and dependability of the Z-numbers in decision-making contexts. In summary, we delineate a comprehensive algorithm addressing interval numbers, uncertainty, clustering, and Z-numbers, integrating diverse techniques and definitions to precisely portray and assess the uncertainty and imprecision within decision-making scenarios.

Assume that in the first round of Delphi, \(n\) number of ideas generated from \(m\) number of questions. Consequently, \(n\times m\) Z-numbers would be generated using the Algorithm 1. The subsection that follows outlines how we determined the consensus for the options collected via the Z-number.

3.3 Calculating Consensus Using Z-Numbers

After gathering ideas in the first Delphi round, a consensus-building exercise was conducted in subsequent rounds. The traditional Delphi approach calculates consensus via statistical dispersion. The statistical dispersion indicates how far a distribution is stretched or compressed. Statistical dispersion assesses other entities’ distance from the centre. Some examples of statistical dispersion approaches include the range method, inter-quartile range (IQR) [55], and the mean deviation method [56]. IQR and range methods help calculate consensus strength. The absolute value of the difference between the 75th and 25th percentiles is represented by IQR, with lower values indicating elevated consensus levels. These statistical dispersion approaches all have flaws. Extreme values, for instance, impact range-based dispersion. They do not work, though, for open-ended series. IQR is impacted by sampling fluctuations and fails for series with high variations. We previously addressed statistical dispersion in this study, having implemented the GCA based on similarities and distances between the numbers aided by Hurwicz’s optimism–pessimism parameter. Since dispersion and bias had previously been accounted for, it was necessary to compute the rank of the Z-numbers. The ranking system was expected to reflect the consensus of experts.

We obtained a bag of Z-numbers {\({Z}_{1}\), \({Z}_{2}\), …, \({Z}_{n}\)} after having applied algorithm 1. In the proposed algorithm, we applied the phrase “bag” (or list) as a mathematical term representing an unordered collection of items that could incorporate duplicates unlike sets [56]. Picking two random Z-numbers from the list, we assessed their ranks. We picked another number from the list and compared it to the higher figure acquired in the prior stage. We proceeded in this fashion until we found the highest number on the list. We converted the Z-numbers into generalised fuzzy numbers for ranking purposes [57]. Assume we have \({Z}_{1}({A}_{1}, {B}_{1})\) where \({A}_{1}=\{{a}_{11}, {a}_{12}, \dots , {a}_{1n}\}\), \({B}_{1}=\{{b}_{11}, {b}_{12}, \dots , {b}_{1n}\}\). For \({Z}_{1}\), \(\alpha\) for \(\alpha\)-cut may be calculated as follows:

In Eq. (5), integration refers to algebraic integration. With the help of \(\alpha\) value, we combined the reliability and restriction components of the Z-numbers by applying the following equation [55]:

The \(\alpha\) value was calculated using Eq. (6). Following the calculation of Eq. (6), the generalised fuzzy number took the form \(M=\left\{{Z}^{\alpha };\alpha \right\}\). We transformed all Z-numbers into generalised fuzzy numbers at this point. The generalised fuzzy number was then translated into a standardised generalised fuzzy number [58]. The generalised fuzzy number \(M=\left\{{Z}^{\alpha };\alpha \right\}\) was subsequently converted into a standardised generalised fuzzy number as \(\widetilde{M}=\left\{\frac{{Z}^{\alpha }}{k};\alpha \right\}\), where \(k=\mathrm{max}({Z}^{\alpha },1)\). Continuing in this vein, we applied the following equation to quantify the defuzzified value described as \(\widetilde{X}\) for each standardised generalised fuzzy number: \(\widetilde{X}=\frac{{Z}^{\alpha }/k}{|{Z}^{\alpha }|}\). For the equation, \(\widetilde{X}\in [-1, 1]\) and \(|{Z}^{\alpha }|\) define the number of elements in \({Z}^{\alpha }\). The next step calculated the standard deviation \(\mathrm{STD}\) of a standardised generalised fuzzy number in order to determine the rank of two Z-numbers. Equation (7) depicts the \(\mathrm{STD}\) calculation of a standardised generalised fuzzy number \({\widetilde{M}}_{i}\):

The maximum value determined by Eq. (7) was derived from the biggest conceivable shape of \(\widetilde{M}\). The fuzzy score of each standardised generalised fuzzy number was calculated as the concluding step of ranking the Z-number. The fuzzy score could be determined with the help of Eq. (8) [57] as depicted as follows:

The higher value of the fuzzy score calculated using Eq. 8 corresponds to a higher rank of the matching Z-number. In the literature, researchers have proposed many approaches for calculating fuzzy scores. Mohamad et al. [57] presented a standard deviation-based approach, Ezadi et al. [59] offered sigmoid and sign methods, while Jiang et al. [28] demonstrated a superior ranking method. Chutia [60] presented an ambiguity-driven ranking method, while Qiao et al. [61] reported a likelihood-based Z-number ranking approach. Since the standard deviation-based method involves fewer computations, we adopted it for our investigation [57]. Algorithm 2 describes the phases of consensus computation using Eqs. (5) through (8).

In the first two steps of Algorithm 2, we indirectly evaluated the restriction component and responded to each Delphi question calculated in Sect. 3.2. Based on the various answers, we aggregated Z-numbers. For aggregation, we applied the equation \(Z_{1} \oplus Z_{2} = \left( {\frac{{A_{1} + A_{2} }}{2}, {\text{min}}\left( {B_{1} ,B_{2} } \right)} \right)\) based on an extension principle presented by Chen and Chen [62]. Steps 3 to 8 of Algorithm 2 were straightforward based on the Eqs. (5)–(8) discussed above. This section demonstrates how to implement the Z-number-based Delphi approach. Before Z-numbers are generated, GCA addresses the statistical optimism/pessimism bias and statistical dispersion. In addition, BPA-based Z-number generation manages ambiguity aversion [26], ERS, and NERS [20].

The merits of the current paper’s methodology are manifold. Foremost, it adeptly integrates expert opinions by compiling their rankings following the initial round of the Delphi process, ensuring the outcomes are deeply informed by the expertise and experience of field specialists. Furthermore, the methodology skillfully transforms real-world data by generating Z-numbers from interval numbers using BPAs, encapsulating both the quantitative value and the uncertainty linked to expert opinions. This holistic representation of data augments the consensus calculation [63] through the ranking of Z-numbers, yielding a more precise evaluation of expert concurrence in contrast to traditional Delphi methods. A pivotal strength of this methodology lies in its capacity to address the intrinsic ambiguity and uncertainty associated with expert judgments, engendering more robust and dependable results. As a consequence, this enhanced methodology holds the potential to improve decision-making processes across various domains, particularly those in which accurate forecasting is crucial, such as sustainable production. In addition, the methodology’s applicability and adaptability render it a versatile approach for researchers and practitioners in technology forecasting and related disciplines, given its suitability for a broad array of real-world scenarios and its adaptability to diverse forecasting tasks. The subsequent section delves into a case study to exemplify the practical applications of the proposed method.

Dairy has always been an important sector in India. According to the data, since 2003, a significant portion of the processed liquid milk marketed in India has been accounted for by dairy cooperatives. The National Milk Grid delivers milk to 750 cities [64]. According to a 2018 study, most dairy plants in India consume 1 m³ of water per 0.3 L of milk production compared to the global average of 1 m³ of water per 1.1 L of milk. Thus, India’s average is thrice the global total [65]. This published research inspires us to implement Delphi in dairy plants with water conservation as the key objective and AI and IoT as the primary enablers.

A dairy plant is a specialised facility where raw milk sourced from dairy farms is processed, transformed, and packaged into a variety of dairy products for consumption [66]. At a dairy plant, the raw milk is first received and stored in refrigerated silos or tanks at the appropriate temperature to maintain freshness and prevent spoilage. The milk then undergoes standardisation, where the fat and protein content are adjusted to meet the desired specifications for different products, ensuring consistent quality and taste. Pasteurisation follows, which is a heat treatment process that destroys harmful microorganisms in milk, ensuring safety and extending the shelf life of dairy products. The milk is then separated into its two main components: cream and skim milk, using centrifugal separators. This allows for the production of various dairy products with different fat contents. Homogenisation is another crucial step, which involves breaking down fat globules in milk into smaller, more uniformly distributed particles, preventing the cream from separating and rising to the top, and resulting in a smoother and creamier texture in dairy products.

For the production of fermented dairy products [67] such as yogurt and cheese, specific starter cultures containing lactic acid bacteria are added to the milk. These bacteria convert lactose into lactic acid, resulting in a characteristic tangy flavour and thicker consistency. Dairy plants process milk into various products, such as butter, cheese, yogurt, ice cream, and milk powders, with each product requiring a specific set of processing steps, equipment, and techniques to achieve the desired characteristics. Once processed, dairy products are packaged into appropriate containers [68], such as bottles, cartons, tubs, or bags, depending on the product type. Packaging helps protect the products from contamination and ensures they maintain their freshness and quality during transport and storage. Dairy plants have stringent quality control measures in place to ensure the safety, quality, and consistency of their products, including regular testing of raw materials, in-process samples, and finished products.

Since the White Revolution in India began in Gujarat, we picked the state for the study [69]. The state has a population of more than 60 million people, and its capital is Gandhinagar. It is also the home of Amul, which is the world’s largest producer of milk.

Purposive sampling, a non-probabilistic sampling technique, was applied to pick Delphi participants for this study. The purpose of purposive sampling is to create a study sample that is representative of the population being studied. It involves selecting participants who are likely to have knowledge or opinions on the topic at hand, such as experts in the field and people who are likely to be affected by the results of the study. According to the literature, this sampling approach is best suited for semi-quantitative research [70]. On the basis of the sample size provided by Jones and Twiss [71], we selected 11 experts as our sample. In the current study, a homogenous sample was analysed. We designated our sample as homogenous since each Delphi method participant in the current study had a minimum of 10 years of work experience in the dairy plant. In addition, we prioritised participation from individuals with strong communication and feedback skills and impeccable work histories (no disciplinary actions in the previous two years) when making our final selections. Figure 3 depicts a full profile of the participants, together with the location of their dairy facilities in Gujarat.

Workplaces of the experts who participated in the Delphi

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We emailed each Delphi participant an invitation letter detailing the study’s aim. The questionnaire was written entirely in English. We verified the questions with three independent experts at each level of the Delphi process. Participants received a comprehensive update on the outcomes of all stages of the Delphi technique. Details about the invitation letters, experts, and questionnaires are contained in the Appendices. We gave each participant 2 weeks to complete and respond to the questionnaire. Consequently, we maintained a 3-week hiatus between each round of the Delphi survey, which is hailed as the standard interval specified in the literature [72]. In the first round, we solicited expert suggestions for water conservation in dairy facilities.

In the second round, we gathered expert responses in the form of interval numbers. We utilised range sliders to capture interval numbers, as explained in the preceding section. Here, we consider the first question: “Which dairy product requires the highest amount of water in the dairy plant?” In response to the preceding query, we offered three options predicated on the ideation phase. These included: buttermilk, ghee, and cream. Table 1 displays the replies we received from 11 experts for the first option, buttermilk.

Table 1 demonstrates that three experts, namely Expert-2, Expert-5, and Expert-7, assign a perfect score to the buttermilk option. In instances where the lower and upper bounds are identical, the optimistic and pessimistic parameters β do not wield considerable influence. Expert-4, on the other hand, presents a response exhibiting considerable statistical dispersion, as the pessimistic score is 0, and the optimistic score for the buttermilk option is 10. Experts 2, 5, and 7 exhibit the greatest similarity and the smallest distance between their pessimistic and optimistic responses, which are, in fact, indistinguishable. Conversely, Expert-4 displays the lowest similarity and the largest distance between their pessimistic and optimistic responses. The response from Expert-4 shows high statistical dispersion. To address this situation, wherein a balance between the low and high discrepancies in experts’ pessimistic and optimistic responses is required, we employ Z-numbers in our current analysis.

Table 1 displays Hurwicz’s optimism–pessimism parameter \(\beta\). We derived the response based on two extreme values of Hurwicz’s optimism–pessimism parameter \(\beta\) by applying Eqs. (1) and (2) based on steps 2 and 3 of algorithm 1. We also deployed Expected Opportunity Loss (EOL) to calculate \(\beta\). EOL is a statistical technique used to estimate optimum business actions [73]. Applying Eq. (3), we determined the gap between optimistic and pessimistic values. Subsequently, the similarity was calculated with the help of Eq. (4). As previously established, a higher distance between optimistic and pessimistic values indicates greater statistical dispersion. In other words, experts are unsure about their values. The similarity is the inverse of the distance since it indicates a confidence level between 0 and 1. The greater is the similarity, the greater the respondents’ confidence. Following that, we sought evidence based on response distances. We implemented GCA to gather evidence based on step 4 of algorithm 1. In the current study, three clusters were generated: one for high confidence, one for medium confidence, and one for low confidence. Table 2 describes the clustering of experts based on their opinions stated in Table 1. By employing the distance equation delineated in Eq. (3), we ascertained the geometric centroid of the data cluster under consideration. Upon examination of the data furnished in Table 1, the calculated centroid equates to 4.6. Subsequently, we partitioned the data into clusters contingent upon their proximity to this centroid. The respective centroids for the first, second, and third clusters are 1.1, 4.2, and 8.9.

We gathered evidence by taking the experts’ level of confidence into account. Hence, each expert’s evidence was annotated \(m\left(c\right), m\left(n\right),\) and \(m(i)\). In the annotation, \(m\left(c\right)\) depicts the evidence of the expert’s confidence level, \(m(n)\) shows the evidence when an expert is not confident, and \(m(i)\) shows evidence when an expert is neutral about the confidence level. We normalised evidence using algorithm 1, step 5. Based on these annotations, the following evidence is provided for each expert:

Expert-1: \(m\left(c\right)=0.2955, m\left(i\right)=0.3747\) and \(m\left(n\right)=0.3297\)

Expert-2: \(m\left(c\right)=0.6772, m\left(i\right)=0.2169\) and \(m\left(n\right)=0.1058\)

Expert-3: \(m\left(c\right)=0.2153, m\left(i\right)=0.3954\) and \(m\left(n\right)=0.3892\)

Expert-4: \(m\left(c\right)=0.1935, m\left(i\right)=0.3806\) and \(m\left(n\right)=0.4258\)

Expert-5: \(m\left(c\right)=0.6772, m\left(i\right)=0.2169\) and \(m\left(n\right)=0.1058\)

Expert-6: \(m\left(c\right)=0.2955, m\left(i\right)=0.3747\) and \(m\left(n\right)=0.3297\)

Expert-7: \(m\left(c\right)=0.6772, m\left(i\right)=0.2169\) and \(m\left(n\right)=0.1058\)

Expert-8: \(m\left(c\right)=0.2525, m\left(i\right)=0.3954\) and \(m\left(n\right)=0.3519\)

Expert-9: \(m\left(c\right)=0.2955, m\left(i\right)=0.3747\) and \(m\left(n\right)=0.3297\)

Expert-10: \(m\left(c\right)=0.1987, m\left(i\right)=0.3826\) and \(m\left(n\right)=0.4186\)

Expert-11: \(m\left(c\right)=0.3972, m\left(i\right)=0.3256\) and \(m\left(n\right)=0.2771\)

We calculate the distance matrix based on steps 6 and 7 of Algorithm 1. The distance matrix is as follows:

Based on the distance matrix, we calculate the similarity matrix described in step 8 of Algorithm 1.

To calculate the credibility and discount coefficient discussed in Definitions 7 and 8, we quantified the support level of each row of the similarity matrix. The support level of each piece of evidence has been presented as follows:

\(\mathrm{Sup}({m}_{1})\) = 9.1425.

\(\mathrm{Sup}({m}_{2})\) = 7.0922.

\(\mathrm{Sup}({m}_{3})\) = 8.8398.

\(\mathrm{Sup}({m}_{4})\) = 8.6241.

\(\mathrm{Sup}({m}_{5})\) = 7.0922.

\(\mathrm{Sup}({m}_{6})\) = 9.1425.

\(\mathrm{Sup}({m}_{7})\) = 7.0922.

\(\mathrm{Sup}({m}_{8})\) = 8.9276.

\(\mathrm{Sup}({m}_{9})\) = 9.1425.

\(\mathrm{Sup}({m}_{10})\) = 8.7072.

\(\mathrm{Sup}({m}_{11})\) = 8.8996.

Based on the support values, we executed step 9 of Algorithm 1. The credibility is shown in Table 3.

The discount coefficient for all pieces of evidence is described in Table 4.

Next, we applied Dempster’s evidence combination rule, formally represented as \(\frac{{\mathop \sum \nolimits_{X \cap Y = C, \forall X,Y \subseteq \theta } m_{i}^{\alpha } \left( X \right) \times m_{i}^{\prime \alpha } \left( Y \right)}}{{1 - \mathop \sum \nolimits_{X \cap Y = \emptyset , \forall X,Y \subseteq \theta } m_{i}^{\alpha } \left( X \right) \times m_{i}^{\prime \alpha } \left( Y \right)}}\) for two pieces of evidence \(X\) and \(Y\), where, \(X\cap Y\ne \varnothing\). If we combine all 11 pieces of evidence, we get the BPA \(m(c)=0.6092\), \(m(i)=0.3644,\) and \(m(n)=0.0263\). Table 5 displays the distances between the individual evidence and the combined evidence, as well as the discount coefficient.\({d}_{0}\) represents the distance between the central evidence \(m\) and itself in Table 5. Hence, its value is 0.0. The same rationale applies to the discounted coefficient \({\omega }_{0}\). The \({X}_{B}\) component of the Z-number was generated by combining steps 13 and 14. We then calculated the membership function for triangular Z-numbers. The membership function for the triangular Z-number is \(\mu =\{0.5, 1, 0.5\}\). We calculated the value of \({X}_{B}\) using formula \({X}_{B}=\sum_{i=1}^{k}{p}_{i}*{\mu }_{i}\). Table 5 displays the \({X}_{B}\) calculation.

Finally, combining \({X}_{B}\) and \({\omega }_{i}\) yields the Z-number for the first option of the first question (Table 6). The resulting Z-number is

The graphical representation of \({Z}_{1}\) is shown in Fig. 4.

The graphical representation of \({Z}_{1}\)

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For the remaining two options, i.e. ghee and cream, we repeated the above steps, narrated in algorithm 1, to generate Z-numbers. For ghee, we calculated Z-number \({Z}_{2}\). \({Z}_{2}\) is as follows:

For cream, we calculated Z-number \({Z}_{3}\). \({Z}_{3}\) is as follows:

We generated Z-numbers using algorithm 1. Next, we applied algorithm 2 to calculate consensus. We created a bag of Z-numbers \(\left\{{Z}_{1}, {Z}_{2}, {Z}_{3}\right\}\) to apply algorithm 2. Based on the initial step of algorithm 2, we selected the first two Z-numbers \({Z}_{1}\) and \({Z}_{2}\) from the bag. As described in steps 2 and 3 of Algorithm 2, we calculated \(\alpha -\) cut. For \({Z}_{1}, {\alpha }_{1}=0.0834,\) and for \({Z}_{2}, {\alpha }_{2}=0.077\). Next, as suggested in step 4 of Algorithm 2, we combined the reliability and restriction components of the Z-numbers using Eq. 6. Thus, \({{Z}^{\alpha }}_{1}=0.667\) and \({{Z}^{\alpha }}_{2}=0.642.\) Next, we calculated the standard deviation as suggested in step 5 of Algorithm 2. We achieved the following values: \({\mathrm{STD}}_{{Z}_{1}}=0.038\) and \({\mathrm{STD}}_{{Z}_{2}}=0.026\). Finally, we calculated the fuzzy score as suggested in step 6 of algorithm 2. For \({Z}_{1}\), the fuzzy score is \(S\left({Z}_{1}\right)=0.053,\) and for \({Z}_{2}\), the fuzzy score is \(S\left({Z}_{2}\right)=0.047\). Thus, the opinion denoted by \({Z}_{1}\) reported a higher consensus than \({Z}_{2}\). Based on Algorithm 2, we compared \({Z}_{1}\) and \({Z}_{3}\). For \({Z}_{3}\), \({\alpha }_{3}=0.09\), \({{Z}^{\alpha }}_{3}=0.674\) and \({\mathrm{STD}}_{{Z}_{3}}=0.0114\). For \({Z}_{3}\), the fuzzy score is \(S\left({Z}_{3}\right)=0.0599\). Thus, for the question “Which dairy product requires/wastes the highest amount of water in the dairy plant?” the cream option had the highest consensus.

We also posed additional questions to the same set of experts. According to their responses, cream wastes the maximum water in Gujarat compared to other products. Gujarat’s dairies require water flow monitoring policies via the intelligent dashboard [74] and S-ERP [75] to effectively implement the ISO14046 standard [76]. Although most dairies in Gujarat are fully digital, scientists predict IoT and AI may play a key role in water conservation in dairy facilities. Our study also recommends that wastewater be handled and recycled with care. Most dairies in Gujarat are in the planning stages of implementing water-recharging strategies. All the participating dairies in Gujarat retrieved vapour water from drying facilities. This study recommends that dairies in Gujarat consider implementing an Internet of things (IoT) solution to facilitate water conservation. The current status of IoT implementation in the dairy sector is low, with only 30% of dairies in Gujarat currently using an IoT solution. The section that follows compares the suggested approach to current methods.

In the previous section, we acquired BPA and Z-numbers. We compared the projected and current methodologies in this section. We exploited entropy as a tool [77] to compare several strategies to the proposed one. Leveraging entropy, we were able to quantify the uncertainty associated with several fuzzy Delphi approaches. Higher levels of unpredictability or entropy could induce erroneous results through the Delphi approach. Higher levels of unpredictability or entropy could induce erroneous results through the Delphi approach. For example, if we used the same set of experts to forecast the value of BPA and Z-number in both methods (current method and proposed), we would have a difference between uncertainties of the two forecasts. One has to repeat the Delphi to get accurate results. The repetition of fuzzy Delphi takes more time and resources. The present section demonstrates that the outcome achieved using the proposed approach can yield minimum uncertainty with higher accuracy. For comparison purposes, we considered the fuzzy Delphi method based on the intuitionistic fuzzy analytical hierarchy process [78], cloud Delphi method [79], Intuitionistic fuzzy Delphi [80], Fuzzy-rough based Delphi method [81], interval-valued fuzzy Delphi [82], and the Delphi method with a triangular fuzzy number [83].

We compared the proposed approach with the Delphi method based on the intuitionistic fuzzy number [84]. Intuitionistic fuzzy numbers (IFN) were calculated using an intuitionistic fuzzy set (IFS). IFS is one of the extensions of fuzzy sets theory [85]. The following is the formal definition of IFS:

Definition 9

Intuitionistic fuzzy set (IFS) [86]. Let \(X\ne \varnothing\) is a set. IFS based on \(X\) is an object formally described as \(A=\{<x, {\mu }_{A}\left(x\right), {v}_{A}\left(x\right)>:x\in X\}\) where \({\mu }_{A}\left(x\right):X\to [\mathrm{0,1}]\) and \({v}_{A}\left(x\right):X\to [\mathrm{0,1}]\) satisfy the condition \(0\le {\mu }_{A}\left(x\right)+{v}_{A}\left(x\right)\le 1\), for every \(x\in X\).

According to Definition 9, \({\mu }_{A}\) and \({v}_{A}\) are membership and non-membership degrees, correspondingly. In addition, set \(X\) is known as the universe of discourse. IFN is an extension of fuzzy numbers based on IFS. The formal definition of IFN is provided as follows:

Definition 10

Intuitionistic Fuzzy Number (IFN) [86]. Intuitionistic fuzzy subset \(A=\{<x, {\mu }_{A}\left(x\right), {v}_{A}\left(x\right)>:x\in R\}\) on real number R is known as IFN if it satisfies the following:

1. a.

   There exist \(b\in R\) such that \({\mu }_{A}\left(b\right)=1\) and \({v}_{A}\left(b\right)=0\).

2. b.

   \({\mu }_{A}\) is a continuous mapping from \(R\to [\mathrm{0,1}],\) and for every \(x\in R,\) the correlation \(0\le {\mu }_{A}\left(x\right)+{v}_{A}\left(x\right)\le 1\).

3. c.

   The membership and non-membership functions should be structured as follows:

Here, \(p, q, r, s\) are functions of \(R\to [0, 1]\) \(, p\) and \(s\) are mathematically expressed, strictly increasing functions, while \(q\) and \(r\) are strictly decreasing functions. According to Definition 10, IFN elements are represented as\(<\mu \left(x\right), v\left(x\right)>\). Both \(\mu \left(x\right)\) and \(v\left(x\right)\) depict the opposite end of the entity. Anjaria [7] leveraged this information to link LIFN to pieces of evidence. If the frame of discernment is \(\theta =\{\mathrm{confident} (c), \mathrm{not confident}(n)\}\) then,\(P\left(\theta \right)=\{\varnothing ,\mathrm{confident}(c), \mathrm{not confident}(n), \mathrm{neutral}(i)\}\). Next, if we link \(P(\theta )\) and IFN, then BPA \(m\left(\varnothing \right)=0, m\left(c\right)=\mu \left(x\right), m\left(n\right)= v\left(x\right)\) and \(f\left(i\right)=(1-\mu \left(x\right)- v\left(x\right))\). One may also calculate belief and plausibility functions based on the quantified BPA. As a result, IFN and BPA are linked. Thus, in the proposed work, based on data described in Table 1, IFN for each expert is expressed as < 0.2955, 0.3297 > , < 0.6772, 0.1058 > , < 0.2153, 0.3892 > , < 0.1935, 0.4258 > , < 0.6772,0.1058 > , < 0.2955, 0.3297 > , < 0.6772, 0.1058 > , < 0.2525, 0.3519 > , < 0.2955, 0.3297 > , < 0.1987, 0.4186 > , and < 0.3972, 0.2771 > . Thus, the IFN from each expert creates an intuitionistic fuzzy set. Next, we calculated the uncertainty of IFN. The entropy of intuitionistic fuzzy sets is the ratio of the biggest cardinalities. The cardinality is defined as \(\mathrm{Count}\left({L}_{i}\right)= {\mu }_{{L}_{i}}+(1-{\mu }_{{L}_{i}}-{v}_{{L}_{i}})\). A generalised entropy measure of an intuitionistic fuzzy number \(L=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\mathrm{Max Count}({L}_{i}\cap {L}_{i}^{c})}{\mathrm{Max Count}({L}_{i}\cup {L}_{i}^{c})}\right)\), where \({L}_{i}\cap {L}_{i}^{c}=<\mathrm{min}\left({\mu }_{{L}_{i}}, {\mu }_{{L}_{i}}^{c}\right),\mathrm{max}\left({v}_{{L}_{i}}, {v}_{{L}_{i}}^{c}\right)>\) and \({L}_{i}\cap {L}_{i}^{c}=<\mathrm{max}\left({\mu }_{{L}_{i}}, {\mu }_{{L}_{i}}^{c}\right),\mathrm{min}\left({v}_{{L}_{i}}, {v}_{{L}_{i}}^{c}\right)>\) [87]. We computed the uncertainty of IFN and the Z-number. We compared the Delphi methodologies based on the results. The entropy of the IFN is 2.33 units. To put in differently, when we apply the Intuitionistic fuzzy Delphi [80] in the proposed case, the uncertainty involved in the procedure yields 2.33 units.

We then calculated the uncertainty involved in the cloud Delphi method [79]. The formal definition of the cloud model is as follows:

Definition 11

Cloud model [79]. Let \(X\) be a quantitative domain and \(U\) be the qualitative concept on \(X\). An element \(x\in X\) is a single random realisation on \(U\). The certainty of \(x\) to \(U\) is \(y\left(x\right)\in [\mathrm{0,1}]\). Here, \(x\) is a random number with a stable tendency, the distribution of \(x\) over \(X\) is known as a cloud model, and each \((x, y\left(x\right))\) is referred to as cloud drop.

We leveraged the method proposed by Xiang et al. [88] to derive the cloud model from BPA. Suppose one has \(n\) bodies of evidence\(X=\left\{{m}_{1}, {m}_{2}, \dots , {m}_{n}\right\}\). Based on the method proposed by Xiang et al. [88], the membership degree \({\mu }_{i}={e}^{-\frac{({{x}_{i}-{E}_{{x}_{i}})}^{2}}{2{E}_{{n}_{i}}^{2}}}\). In the equation, \({\mu }_{i}\) is a membership degree of the \(i\)th evidence, \({E}_{{x}_{i}}\) is the expectation value of the \(i\)th evidence, and \({E}_{{n}_{i}}\) is a standard random number. The uncertainty of the cloud model is calculated as \(\mathrm{entropy}=\frac{({C}_{i,\mathrm{ max}}-{C}_{i,\mathrm{ min}})}{6}\) where, \({C}_{i, \mathrm{max}}\) and \({C}_{i, \mathrm{min}}\) comprise the range of values of the evaluation interval corresponding to the \(i\)th evidence [88]. First, we obtained a cloud membership degree from BPA by applying the equation \({\mu }_{i}={e}^{-\frac{({{x}_{i}-{E}_{{x}_{i}})}^{2}}{2{E}_{{n}_{i}}^{2}}}.\) Later, we applied the entropy equation to the data described in Table 1; we derived the entropy in the form of 0.56 units for the present case. When we use cloud Delphi [79] in the proposed case, the uncertainty involved in the procedure amounts to 0.56 units.

After computing the cloud-based Delphi and IFN-based Delphi, we calculated the uncertainty associated with the interval-valued fuzzy Delphi technique [82, 89]. As had been mentioned in Sect. 3.1, the association of \(x\) with \(\left[\underline{x},\overline{x}\right]\) is known as interval fuzziness [40]. The entropy of interval fuzziness was measured with the help of the equation \(\mathrm{Entropy}=1-\frac{1}{n}\sum_{i=1}^{n}|\underline{x}+\overline{x}-\mathrm{range}|\) [90]. In the present work, we had 11 fuzzy interval numbers with the maximum value of \(\mathrm{range}=10,\) as represented in Table 1. Entropy is calculated as 0.41 units in this context using the entropy equation. If we use the interval-valued fuzzy Delphi approach [82] in the presented scenario, the procedure’s uncertainty is 0.41 units.

We determined the entropy of the Delphi method driven by a triangular fuzzy number [75]. The entropy of triangular fuzzy number \([a,b,c]\) has been expressed as \(\mathrm{entropy}={\int }_{a}^{b}\mu \left(x\right)*p\left(x\right)\mathrm{d}x+{\int }_{b}^{c}\mu \left(x\right)*p\left(x\right)\mathrm{d}x\) [42]. Based on the entropy equation and data in Table 1, we calculated the entropy of the triangular fuzzy number-based Delphi approach. The triangular fuzzy number has an uncertainty of 2.05 units.

Following that, we determined the entropy of the Z-number. Kang et al. [91] established the discrete Z-number uncertainty measure. We used the same approach in the present work. The uncertainty measure calculation takes into account the first component \(A\) of the Z-number \(Z= (A, B)\) as a restriction involving the real-valued uncertain random variable \({X}_{A}\), and the second component \(B\) as a measure of reliability or uncertainty for the first component \(A\). The uncertainty measure \(H\) of the discrete Z-number is \(H=\frac{1}{2}\left[\frac{M\left(A\cap {A}^{c}\right)+\left|{X}_{A}\right|-1}{M\left(A\cup {A}^{c}\right)+\left|{X}_{A}\right|-1}+\frac{1}{2}\left(\frac{M\left(B\cap {B}^{c}\right)+\left|{X}_{B}\right|-1}{M\left(B\cup {B}^{c}\right)+\left|{X}_{B}\right|-1}+1-{x}_{B}^{*}\right)\right]\). In the entropy equation, \(A\) is the first component of the Z-number, \({A}^{c}\) is the complementary set of \(A\), \(M(.)\) is the sum function, \(A\cap {A}^{c}=\mathrm{min}\{A, {A}^{c}\}\), \(A\cup {A}^{c}=\mathrm{max}\{A, {A}^{c}\}\), \({x}_{B}^{*}=\begin{array}{c}\mathrm{Sup}\\ x\in {X}_{B}\end{array}\left\{x\in {X}_{B}|{\mu }_{B}\left(x\right)=hgt(A)\right\}\), and \(hgt(A)\) expresses the maximum height in the fuzzy set \(A\). The same annotations apply to \(B\), the second component of the Z-number. The entropy of \({Z}_{1}\) shown in Fig. 4 amounts to 0.3813 units. As a result, we establish that the uncertainty associated with the proposed technique is 0.3813 units. Table 7 highlights the proposed technique’s comparison with current methodologies.

Table 7 compares the proposed method to current approaches and demonstrates that the proposed approach has the least amount of uncertainty associated with the Delphi verdict. The reduction in uncertainty is due to two factors: (1) the proposed method accounts for both expert opinion and decision-making under uncertainty, which is not accounted for by current approaches. (2) The proposed method uses a grey clustering and Hurwicz’s optimism–pessimism parameter to combine expert opinion and decision-making under risk, which accounts for the prior probability of each case.

The outcome of the aforementioned comparison, which reveals that the entropy of the proposed Z-number-based Delphi method is the lowest, carries several significant implications. First, the reduced entropy indicates that this method excels in managing and quantifying uncertainty in expert opinions, leading to more reliable and accurate consensus outcomes. In addition, the Z-number-based approach seems to offer a more comprehensive representation of expert opinions by capturing both the value of the opinion and the associated reliability, which may contribute to the decreased entropy. Moreover, the reduced entropy suggests that the Z-number-based Delphi method is more resilient to optimism/pessimism biases, resulting in more balanced and objective forecasting. This comparison demonstrates that the proposed method surpasses the other methods in handling uncertainty and ambiguity, indicating that the Z-number-based Delphi method may be a more fitting choice for technology forecasting and other applications where expert opinions are crucial. The diminished entropy of the Z-number-based Delphi method also implies that it could be effectively applied across various fields where precise forecasting and decision-making are essential, such as sustainable production, public policy, and strategic planning. Ultimately, the promising results of the Z-number-based Delphi method lay a strong foundation for continued investigation and refinement of this approach, potentially opening doors for further advancements and applications in the realm of technology forecasting and expert opinion management. The subsequent section concludes the current work and examines its limitations and potential scopes.

In conclusion, the current research presents a significant contribution to addressing the limitations associated with ambiguity aversion, uncertainty, and statistical optimism/pessimism bias in conventional fuzzy Delphi approaches. By incorporating Z-numbers, which capture both the probabilistic and fuzzy aspects of expert opinions, the proposed method provides a more comprehensive and nuanced representation compared to Type-2 fuzzy sets. As a result, the Z-number-based Delphi technique effectively reduces ambiguity aversion and optimism/pessimism bias, leading to more objective and realistic consensus measurements. A real-world case study demonstrates the applicability, objectivity, and effectiveness of the proposed technique. The findings indicate that the Z-number-based Delphi method successfully overcomes the limitations of existing fuzzy Delphi approaches by integrating expert opinions with statistical dispersion, ultimately yielding a more accurate and objective Z value. However, the study does have some limitations, such as not addressing group size thresholds, information overload measurement, or the psychological disposition of the expert group. In addition, the research does not investigate the impact of varying levels of expertise or the effect of varying communication modes between experts, which may influence the consensus outcomes. Furthermore, the study does not consider the potential biases that may arise from the selection and composition of the expert panel, which may affect the quality and reliability of the results. Lastly, the research does not explore the sensitivity of the proposed method to different weighting schemes, aggregation techniques, or iterative processes, which may provide further insights into the robustness and adaptability of the Z-number-based Delphi approach.

Future research could delve into the integration of machine learning and Bayesian updating methods for automatically generating Z-numbers from expert opinions. This approach has the potential to expand the applicability of the proposed method across various domains, including medical services, industrial wastewater management, social impact assessments, and energy options. Furthermore, future work could investigate the combination of Z-scores with the analytic hierarchy process (AHP) and the analytic network process (ANP) for ranking expert opinions more effectively. Another promising research direction involves the development of a novel Delphi approach based on Z-numbers for estimating respondents’ propensities. In summary, this research makes a valuable contribution to the understanding and prediction of expert opinions, with the potential to inform public policy decisions by incorporating Bayesian updating methods alongside Z-numbers. The proposed Z-number-based Delphi technique addresses the limitations of existing methods while offering meaningful insights for future research and practical applications in technology forecasting and expert opinion management.