1 Introduction

Hydrogen serves as a versatile energy medium with numerous applications in various industrial processes. With a substantial global production of 70 million tonnes per year, it is primarily used in petroleum extraction and refining, as well as ammonia manufacturing. Nevertheless, the rising interest in electrochemically generated hydrogen by alkaline and PEM electrolysers presents promising opportunities for electricity storage and as an eco-friendly energy medium for various utilisations including the transportation industry. A key benefit of these hydrogen production methods is its ability to provide a clean and efficient means of storing surplus electricity generated by power plants including renewable sources. In particular renewable electricity plants based on Solar PV and wind are well indicated for the production of hydrogen by the net of the sun and the wind respectively. Accordingly, excess of electricity of a grid or dedicated solar or wind power plants can be harnessed to produce hydrogen, which can then be storage for future electricity generation when required [1]. The International Energy Agency highlights that water electrolysis is a promising technique for hydrogen production, with "green" hydrogen increasingly generated through this method to lower greenhouse gas emissions and foster sustainable energy generation. Moreover, hydrogen fuel cell vehicles, which only emit water vapour, serve as a zero-emission alternative to conventional gasoline-powered vehicles. The IEA reports [1]. That out of the hydrogen produced globally, approximately 4% or 2.8 million tons were derived from water electrolysis by alkaline and PEM electrolysers. Hydrogen can also be employed in fuel cells to produce heat and power for residential, commercial, and industrial uses. With the expansion of hydrogen refuelling infrastructure, fuel cell vehicles have the potential to significantly transform the transportation industry [2].

The production of green hydrogen based on alkaline and PEM electrolysers is accelerating, as they offer an environmentally friendly and sustainable substitute during their utilisation to conventional hydrogen production methods which involve hydrocarbon fuel processes [3]. Although the upfront investment for these electrolysers can be substantial, their decentralized functionality makes them appropriate for various applications. Nevertheless, for PEM electrolysers, the expense of materials employed in their assembly, like platinum-based electrodes at the cathode and or iridium based electrodes and others at the anode represent parts of the limiting factors responsible for the high cost of the hydrogen production. The iridium charge per kW power of the electrolyser is 1.3 to 2 g/kW [4]. The platinum estimated charge per kW power of the electrolyser is 0.5–1 g/kW [4]. The commercial PEM electrolyser system of 5 MW needs at least 2.5 kg of Pt and 6.5 kg of iridium. The annual production of platinum in 2022 was 190 tons. The annual production of iridium in 2022 was 7.7 tons [5]. The platinum cost per kilogram was 33 755$/kg in 2022 [6]. The iridium price in 2022 was between 80 700 $/kg to 161 400 $/kg [7]. The specific energy of platinum extraction is 67.5 kWh/kg of Pt and involves the production of 12,5 tons of CO2/kg of Pt and that of iridium is 10 tons of CO2/kg of Ir [4]. Other critical materials used in alkaline or PEM electrolysers are: Cobalt for PEM and Nickel for Alkaline Electrolyser. These costly materials can notably elevate the total production cost [8, 9]. Addressing this issue, ongoing research aims to identify cost-effective non nobles and available for materials used in PEM electrolysers manufacturing. Multiple aspects contribute to the high cost of green hydrogen production. Firstly, the price of renewable electricity, which serves as the primary input for electrolysis, must keep dropping. Secondly, technological advancements, scaled-up production, and intensified competition among providers must drive down electrolyser costs. Lastly, the expenses associated with producing and transporting water for electrolysis for alkaline, PEM electrolysers must be minimized [10]. A thorough techno-economic assessment of water electrolysis can help identify the best strategy for hydrogen production, considering factors such as capital and operational costs, electrolysis cell performance, and the accessibility of inexpensive electricity. By closely analyzing these elements, the most economically viable approach for producing hydrogen through electrolysis can be established. As stated by the U.S. Department of Energy's (DOE) Hydrogen and Fuel Cells Program Record, the mean cost of electrolyser stack and balance of plant (BOP) components, as provided by manufacturers in the United States in 2013, stood at $5.14/kg for a daily production capacity of 1,500 kg and $5.12/kg for a daily production capacity of 50,000 kg [11]. A comparative analysis of hydrogen production techniques, such as AWE (Alkaline Water Electrolyser), PEMEC (PEM Electrolyser), and SMR(Steam Methane Reforming), was performed [12] for production capacities of 30, 100, and 300 Nm3/hr. The findings revealed that PEM Electrolyser was the most cost-effective option at $16.54 per kgH2 for a 30 Nm3/hr capacity.

Later, an investigation was carried out to assess the cost-effectiveness of four distinct water electrolysis technologies including alkaline (AWE) [13], proton exchange membrane (PEM). The study determined that the per-unit hydrogen production costs were $7.60, $8.55, per kgH2 for AWE, PEMEC (PEM Electrolyser), respectively. Nevertheless, the research does not offer suggestions for reducing capital expenses or implementing subsidies and tax cuts to enhance competitiveness with other hydrogen production methods. An evaluation of the costs and benefits of using solar power in Tunisia [14] for electricity from PV and hydrogen production via PEM electrolysis was conducted. They showed that the initial cost analysis of a PV-HRS (Hydrogen Refueling Station) reveals that the major expenses are the PV system (48.5%) and electrolysers (41%), while the storage system cost is relatively low at 3.2%. Producing 1 kg of hydrogen costs 3.32€/kg, and the project's total cost is 2.34 million €.

These various considerations of parameters effect on the hydrogen cost do not indicate which parameters among others may limit the cost of the green hydrogen produced from water electrolysis using alkaline and PEM electrolysers.

However, to compare the cost-effectiveness of green hydrogen production from PEM or alkaline electrolysers taking into account several aspects, criteria and sub criteria, Multi-Criteria Decision Making (MCDM) or Multi-Criteria Decision Analysis (MCDA) methods are practical tools for this techno-economic assessment.

During the last decades different types of MCDM methods have been introduced and developed. The differences between the various methods are related on the level of algorithms complexity, the criteria of the weighting, how to represent the evaluation of the criteria, how to aggregate the data and to consider or not possible uncertain data [15].

Previously our group used MCDA methods as ELECTREI, ELECTRE III with Fuzzy, TOPSIS, VIKOR, ELECTRE IV and TOPSIS for materials selection for various applications including sensitive components for aerospace applications [16, 17] or bipolar plates for Polymer Electrolyte Fuel Cell Applications [18,19,20] and sensitive materials election [21]. It was shown that the optimum value of each criterion is independent of other criteria values (i.e., no interaction is allowed). This allowed a good ranking of materials considered for this application. In all cases, it was found good agreement between the results of the MCDC methods being used for the material selection and available experimental data the data bases of the Cambridge Engineering Selector (CES). It was concluded that the proposed approaches may be applied to other problems of material selections for energy applications.

In the case of hydrogen energy, by considering factors such as technological parameters, cost-effectiveness, environmental impact, technical efficiency, the offer of hydrogen and the intensity of the hydrogen demand or the utilisation sector, MCDM may help to forecast the green hydrogen development and stakeholders to take appropriate decision by identifying the most suitable cost effective of the technology for green hydrogen production in the future for the user’s specific needs.

The study employed [22] AHP and Fuzzy AHP methods to assess and compare the cost-effectiveness and benefits of various hydrogen production technologies. Eight technologies were evaluated using five criteria: greenhouse gas emissions, raw material and utilities consumption, energy efficiency, scalability, and waste disposal and atmospheric emissions. While fossil fuel-based processes were more cost-effective, renewable-based processes offered greater benefits. The cost–benefit analysis indicated that water splitting by chemical looping was the most promising among renewable approaches [23]. The study presents a novel multi-criteria decision making (MCDM) approach for assessing the sustainability of hydrogen production technologies. The two-stage framework aims to identify the most promising technologies based on sustainability potential and then evaluate the remaining options, considering trade-offs among economic, environmental, and social factors. In the first stage of the MCDM framework, the authors use fuzzy best–worst and fuzzy TOPSIS methods to determine the top alternatives, considering decision-makers' preferences and information uncertainty. In the second stage, they introduce a new method that combines preference data on attributes and alternatives for evaluating the remaining technologies. This method applies a multi-objective optimization approach using the NNC method to address varying priorities and information uncertainty. The two-stage MCDM method is used to assess and compare the sustainability of hydrogen production technologies, such as steam methane reforming, partial oxidation, and electrolysis. The study's findings reveal that electrolysis is the most sustainable choice due to its low greenhouse gas emissions and high potential for utilizing renewable energy sources. The study [24] employs a hybrid MCDM method that combines AHP and TOPSIS to evaluate various hydrogen production options based on criteria such as economic feasibility, environmental impact, technical feasibility, and safety. It examines steam methane reforming (SMR), partial oxidation, and electrolysis, considering different scenarios and uncertainties like fluctuating natural gas prices and carbon taxes.

The findings reveal that steam methane reforming is the most suitable short-term hydrogen production option due to its lower cost and high technical feasibility. In the long term, however, electrolysis emerges as the most sustainable choice, as it uses renewable energy sources and produces low greenhouse gas emissions. The study highlights the hybrid MCDM approach's usefulness in assessing sustainable hydrogen production options, offering valuable insights for policymakers and industry professionals in their decision-making processes. Until now, no MCDM method has integrated TOPSIS, WASPAS, and BWM with fuzzy logic by incorporating economic. Technical, technological, environmental and social aspects for the techno-economic analysis of electrolytic hydrogen production using alkaline and PEM Electrolysers.

The growing demand for sustainable energy solutions calls for advanced methods in evaluating hydrogen production technologies. This study introduces a pioneering approach by applying a combination of MCDM techniques to not only assess but also compare the techno-economic viability of Alkaline and PEM electrolysers for hydrogen production. By incorporating environmental, technical, technological, economic, and social considerations into a comprehensive framework, this work sets a new standard in the field, offering stakeholders invaluable insights for informed decision-making. This novel application of MCDM techniques, particularly the integration of TOPSIS, WASPAS, and BWM with fuzzy logic, represents a significant advancement in the techno-economic analysis of electrolytic hydrogen production using alkaline and PEM Electrolyser technologies.

2 Objectives and methodology

The general objective of this work is to use a new MCDM method to analyse the techno-economic performances of commercially mature electrolyser technologies (Alkaline and PEM electrolysers) which are the two alternatives. Five main criteria (technical, technology, economic, environmental and social factors) which are linked to more than 30 sub-criteria to develop of the tow alternatives are used to build this new MCDM approach. The validity of this new approach will be used in the next future to analyse the techno-economic performance of other electrolyser technologies in development. In thus sub-chapter we will describe the alkaline and PEM technologies considered in this work followed by its detailed objectives and methodology.

2.1 Why in this work PEM and Alkaline electrolysers are only considered as alternatives among the others?

The process of selecting the most suitable water electrolyser for techno-economic analysis using the Multi-Criteria Decision Making (MCDM) method is based on their maturity for commercial mass production and the rate of their implementation worldwide presently and in the next future. During the year 2020, 176 MW of commercial alkaline electrolysers were installed worldwide and those of the PEM electrolysers technologies where 89 MW [25] the other not specified technologies (probably mainly hydrogen production from chloro-alkaline) were 21 MW. Furthermore, electrolyser total capacity of 175 GW is announced by 2030 and only 40% of this capacity, eg. 70GW will be built with PEM, [26]. Until now, these data indicate that the alkaline technology is the most mature technology which is must used. The alkaline water electrolysis was used in the chlorine or fertilizer industries since the 1920’s. This technology is supposed to have minimal capital expenses because its components involve less cost materials.

Accordingly, Proton Exchange Membrane (PEM) and Alkaline electrolysers have emerged as prominent options, each with its own unique set of advantages and challenges. By thoroughly evaluating and comparing the technical, economic, and environmental aspects of these alternatives, decision-makers can effectively identify the most appropriate technology to adopt, ensuring long-term benefits and sustainability in the energy sector. Principles and parameters of both technologies are described in the followings.

2.1.1 Alkaline electrolyser

Alkaline electrolysers, a well-established and dominant technology, operate at continuous current densities of 2000–4000 A/m2 and temperatures between 80 and 90 °C [27]. Their theoretical efficiency can reach 80%, but in practice, it ranges from 63 to 70% due to the lower heating value of gaseous hydrogen [28].

Cathode: 2H2O + 2e− → H2 (g) + 2HO-

Anode: 2HO− → ½ O2 (g) + H2O + 2e−

Net reaction: H2O + Electricity → ½ O2 + H2 + heat:

Their theoretical efficiency can reach 80%, but in practice, it ranges from 63 to 70% due to the lower heating value of gaseous hydrogen [29]. The electrolytic cell features a permeable membrane allowing water and alkaline electrolyte to pass through. The zero-gap cell concept, where both electrodes press against the membrane, has been developed for efficient electrolysis [30].

Advancements in electrode design and the zero-gap system have enhanced alkaline electrolysis performance. The process uses a diaphragm, made of composite materials like ceramic or microporous materials, to separate the anode and cathode. Temperature control is crucial for maintaining optimal electrolyte conditions [31]. Proper electrolyser function requires additional components, such as power electronics, DC power distribution systems, gas-exit pipes, cooling systems, and possibly hydrogen and oxygen drying systems (Fig. 1). The block diagram of the Alkaline electrolyser involves a lot of components which may impact the maintenance of the system.

Fig. 1
figure 1

Schematic block of the Alkaline electrolysis system [4]

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2.1.2 PEM electrolyser

A common configuration of a PEM electrolyser stack consists of multiple cells connected in series electrically, with parallel water and gas flow. Thick metal end plates hold these cells together. Each cell features a catalyst-coated membrane (CCM), comprising an acidic proton conductor polymer membrane which separates the anode and the cathode catalyst layers coated on their respective electrodes. A porous transport layer (PTL) enhances water diffusion and the water splitting reaction on the membrane surface. Bipolar plates separate the cells and include channels for water, hydrogen, and oxygen transport [32]. When the stack is connected to an electric power (grid our renewable energy sources), the reactions at the electrodes surfaces are:

Anode: H2O → ½ O2 (g) + 2H +  + 2e-

Cathode: 2H +  + 2e- → H2 (g)

Net reaction: H2O + Electricity → ½ O2 + H2 + heat:

In comparison to alkaline electrolysers, PEM electrolysers are less commercially developed because the technologies emerges in the 1980’s and the alkaline technology was used since the 1920’s. In addition PEM electrolyser use more expensive materials like platinum, iridium or cobalt as catalysts and costly solid proton conducting membrane materials [33]. Than alkaline technology which uses less expensive Ni catalyst electrodes and diaphragms. PEM electrolysers is fed with deionized water. As indicated on the above equation reactions, during the process, water is oxidized at the anode, producing protons and gaseous oxygen. These protons then move to the cathode via the solid acidic proton exchange membrane. The electron provided by the electricity trough the external circuit of the stack arrive at the surface of the cathode and discharge the proton to allow hydrogen gas production). The operating current density is between 10 000 and 20 000 Am−2. The operating temperature is between 60 and 80 °C. The electrolyser design allows an efficient gas separation with a higher hydrogen purity and higher energy efficiency compared to alkaline electrolysers [34].

PEM electrolyser plants also comprise major subsystems, including power supply, deionized water circulation, hydrogen processing, cooling, and miscellaneous components like ventilation and safety features [32]. These subsystems collectively ensure the efficient operation of the PEM electrolyser system (Fig. 2). The block diagram of the PEM Electrolyser system exhibits less component than the Alkaline system. On the other hand, it is fed by deionised water rather than a chemical such as KOH in the case of alkaline Electrolyser which involves specific handlings. Thus, operating and maintenance costs are different. The environment impacts will be also different.

Fig. 2
figure 2

Block diagram of industrial hydrogen production with PEM technology [4]

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2.2 Objectives and methodology

As summerised in the introduction, MCDM methods have been employed for the techno economic the electrolytic hydrogen production and water electrolysis projects in existing literature [23, 24]. But no MCDM method which integrates TOPSIS, WASPAS with Vikor index and BWM with fuzzy logic by incorporating economic, technical, technological, environmental and social aspects has been used for the techno-economic analysis of electrolytic hydrogen production using alkaline and PEM Electrolysers.

Accordingly, there remains significant needs to develop new relevant knowledge in the electrolysers’ business for the main commercial technologies, which can be summarized as follows:

  • The development and utilisation of the new MCDM method which integrates TOPSIS, WASPAS with Vikor index and BWM with fuzzy logic by incorporating economic, technical, technological, environmental and social aspects for the techno-economic analysis of electrolytic hydrogen production using alkaline and PEM Electrolysers which are the two alternatives;

  • The utilisation of at least thirty criteria and sub-criteria which are distributed through the above five aspects to identify the key factors that influence the techno-economic cost and efficiency of hydrogen production through water electrolysis from the two alternatives alkaline and PEM electrolysers;

  • The utilisation of the new MCDM results to identify the key factors that influence the techno-economic cost and efficiency of hydrogen production through water electrolysis from alkaline end PEM electrolysers;

  • The utilisation of these MCDM results to present a hierarchic comparison of alkaline and PEM electrolysers, in terms of economic, technical, technological, environmental and social aspects.

  • The determination of the ability of MCDM methods to guide decision-makers to identify the most appropriate hydrogen production technology.

To succeed in the development of such new knowledge using MCDM, some fundamental aspects are assumed and defined and can be summarized into five main aspects which are not yet hierarchized:

  1. 1.

    Technical feasibility which considers factors such as energy consumption, production capacity, conversion efficiency, and durability of the system;

  2. 2.

    Economic viability which considers factors such as capital costs, operating costs, and the revenue generated from hydrogen sales;

  3. 3.

    Environmental sustainability which assesses the technology's environmental impact, including its carbon footprint, the use of renewable energy sources, and the potential for waste and emissions;

  4. 4.

    Technological factors which include the availability of the technology, the level of technological development, and the potential for improvement and innovation;

  5. 5.

    Social factors which include public acceptance, regulatory and policy frameworks, and the potential for social impact.

To obtain the best weight for each criterion, MCDM Fuzzy Best–Worst Method (FBWM) is used. Accordingly, the following steps are achieved:

  • Identify the criteria like the efficiency, cost, durability, reliability, and safety of the electrolyser.

  • Establish the decision-making team with experts in the field of water electrolysers.

  • Determine the linguistic terms like for example: "very high", "high", "moderate", "low", and "very low" for the electrolyser efficiency.

  • Pairwise comparison like the importance of efficiency with respect to cost, durability, reliability, and safety in a form of a matrix A, where aij is the degree of importance of criterion i with respect to criterion j. The matrix A should satisfy certain consistency requirements, such as the transitivity property. The decision-makers can use their expertise to provide the values for aij.

  • Calculate the best–worst values by using the pairwise comparisons, calculate the best and worst values for each criterion.

  • Calculate the degree of optimism by subtracting the worst value from the best value.

Figure 3 shows the schematic representation of the MCDM evaluation system used in this work. This diagram represents a criterion system that includes economic, technical, environmental, and technological aspects, social factors) linked to an alternative system (PEM electrolyser, alkaline electrolyser) by sub-criteria such as CAPEX, OPEX, Electricity cost, hydrogen gas diffusion, voltage efficiency, CO2 emissions, land use, catalyst cost, lifetime, BOP (Balance Of the Plant), social acceptability.

Fig. 3
figure 3

Schematic representation of the link between the Criterion system and the alternative system

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Each category of criterion encompasses various other criteria and sub criteria to produce a number of criteria known as a criterion system. This paper features a selection of 30 criteria across these five aspects to create a comprehensive criterion system (Table 1). The chosen criterion system encompasses nearly all commonly used criteria for assessing water electrolysers, providing a solid foundation for evaluating alternative water electrolysis technologies. This approach presented here for the first time is also based on the combination of MCDM methods (see Sect. 3.1) which has never been used before and ensure more reliability of the evaluation process.

Table 1 Criterion sytem formed from iteraction between the 5 main aspects and the various criteria and subcriteria to give 30 criteria
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3 MCDM methods used to operate the schematic representation of figure

3.1 Novelty of the work

In this study, a new approach of a Multi-Criteria Decision Making (MCDM) or Multi criteria Decision Analysis (MCDA) framework has been established to assess the sustainability of hydrogen production technologies through water electrolysis from a techno-economic standpoint, in which. The Fuzzy Best–Worst Method (FBWM) is applied within the MCDM framework to establish the weight of different criteria. Following this interval grey, TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and the interval grey WASPAS (Weighted Aggregated Sum Product Assessment) are utilized to conduct a techno-economic analysis of hydrogen production via water electrolysis. In the final stage, the most suitable MCDM method is determined, and the limiting parameters are identified for further investigation. A short introduction of each of those MCDM methods are shown in the followings.

3.2 Fuzzy best–worst MCDM method

The Best–Worst Method (BWM) was introduced by [35] as an extension to the Analytic Hierarchy Process (AHP) [36] and Analytic Network Process (ANP) [37] methodologies. BWM is a pairwise comparison-based method that simplifies the decision-making process by reducing the number of comparisons required. It involves comparing the best (most desirable) and worst (least desirable) criteria or alternatives in a decision problem.

Roberts in 1986 [38] introduced the fuzzy set theory as a mathematical approach to handle uncertainty, imprecision, and vagueness that often occur in real-world situations [39]. A fuzzy set is a set whose elements have degrees of membership between 0 and 1, as opposed to crisp sets, where elements have a binary membership of either 0 or 1. In fuzzy set theory, an element can belong to a set with a certain degree of truth, which is represented by a membership function.

A fuzzy set A is defined as a collection of ordered pairs of elements (x) from the universe of discourse (X) and their corresponding membership values (µ_A(x)):

$$ {\mathbf{A}}\, = \,\left[ {\left( {{\mathbf{x, \mu \_A}}\left( {\mathbf{x}} \right)} \right)\,|\,{\mathbf{x}} \in {\mathbf{X}}} \right] $$
(1)

The membership function (µ) maps the elements of the universe of discourse (X) to their membership values in the interval [0, 1]:

$$ {{\varvec{\upmu}}}\left( {\mathbf{x}} \right):\,{\mathbf{X}}\, \to \,\left[ {{\mathbf{0, 1}}} \right] $$
(2)

The Fuzzy Best–Worst Method (FBWM) was developed as an extension of the BWM to handle the uncertainty and vagueness often encountered in real-world decision-making scenarios. By incorporating fuzzy set theory, FBWM allows decision-makers to express their preferences using linguistic variables, thereby capturing the inherent uncertainty and imprecision in their judgment [40].

The use of linguistic variables in (Table 2) and the FBWM method provide a more flexible and intuitive way for decision-makers to express their preferences and opinions, and the transformation rules allow for the conversion of these linguistic variables into numerical values that can be used in mathematical operations. This approach helps to handle imprecise and uncertain information in real-world decision-making situations, leading to more accurate and effective decision-making.

Table 2 fuzzy transformation rules of linguistic variables
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The subjective assessments of the decision-makers are required [40] (Table 3):

Table 3 Transformation rules of linguistic variables of decision-makers
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The steps for determining the fuzzy weights of alternatives in the fuzzy best worst method (FBWM) in Multi-Criteria Decision Making (MCDM) are as follows:

Step 1. The decision criteria system provides a framework for decision-makers to assess the performance of different alternatives. It helps to ensure that all relevant factors are considered and evaluated consistently and comprehensively, leading to more informed and effective decision-making.

Step 2. Construct a comparison matrix for each criterion, which contains the linguistic variables and the degree of preference or importance of one variable over another and for each criterion, determine the best and worst values of each alternative based on the comparison matrixWe represent the best criterion with CB and the worst criterion with CW.

Step 3. In this step, decision-makers compare the best criterion with other criteria using a predefined linguistic scale, such as (Table 1). The comparison is between the best criterion (CB) and the other criteria (j), with the best criterion's comparison with the worst criterion being the highest among all. The comparison of CBB with itself is equal to (1,1,1). The comparison being evaluated in this step is represented as a1B, and the comparison of the worst criterion CWW with itself is also equal to (1,1,1) the comparison relations (3) and (4) are as follow [40]:

$$ \tilde{\varvec{A}}_{{\varvec{w}}} = \left( {\tilde{\varvec{a}}_{{1{\varvec{w}}}} ,\tilde{\varvec{a}}_{{2{\varvec{w}}}} , \ldots ,\tilde{\varvec{a}}_{{3{\varvec{w}}}} } \right) $$
(3)
$$ \tilde{\varvec{A}}_{{\varvec{B}}} = \left( {\tilde{\varvec{a}}_{{{\varvec{B}}1}} ,\tilde{\varvec{a}}_{{{\varvec{B}}2}} , \ldots ,\tilde{\varvec{a}}_{{{\varvec{B}}3}} } \right) $$
(4)

where.

\({\widetilde{{\varvec{A}}}}_{{\varvec{w}}} {\varvec{a}}{\varvec{n}}{\varvec{d}}\) \({\widetilde{{\varvec{A}}}}_{{\varvec{B}}}\) represents the fuzzy others-to-worst vector and the fuzzy best to others vector respectively \( \varvec{\tilde{a}}_{{{\mathbf{i}}\varvec{j}{\text{~}}}} \;\varvec{is}\;\varvec{defined}\;\varvec{as}\;\varvec{a}\;\varvec{fuzzy}\;\varvec{reference}\;\varvec{comparison}(\varvec{preference}\;\varvec{of}\;\varvec{criterion}\;\varvec{i}\;\varvec{to}\;\varvec{criterion}\;\varvec{j} \)

Which is a triangular fuzzy number.\(.\)

\({\widetilde{{\varvec{a}}}}_{\mathbf{i}{\varvec{w}}} \, {\varvec{and}} \, {\widetilde{{\varvec{a}}}}_{{\varvec{B}}\mathbf{j}}\text{respectively}.\text{r}\) represents the fuzzy preference of criterion i over the worst criterion and the fuzzy preference of the best criterion over criterion j,

consider CW, i = 1,2,…,n \({\widetilde{{\varvec{a}}}}_{{\varvec{B}}\mathbf{j}}\), \({\widetilde{{\varvec{a}}}}_{{\varvec{w}}{\varvec{w}}}\) = (1,1,1) and \({\widetilde{{\varvec{a}}}}_{{\varvec{B}}{\varvec{B}}}\) = (1,1,1).

Step4. Determining the Optimal Fuzzy Weights, The weight in which, for each pair \(\frac{{\widetilde{w}}_{B}}{{\widetilde{w}}_{j}}\) and \(\frac{{\widetilde{w}}_{j}}{{\widetilde{w}}_{w}}\)

Where \({\widetilde{w}}_{w} and {\widetilde{w}}_{j} and {\widetilde{w}}_{B}\) are triangular fuzzy number and are different from that in BWM the following relation (5) and (6) [39] will be as follows:

$$ \frac{{\tilde{\varvec{w}}_{{\varvec{b}}} }}{{\tilde{\varvec{w}}_{{\varvec{j}}} }} = \tilde{\varvec{a}}_{{{\varvec{Bj}}}} $$
(5)

and

$$ \varvec{\frac{{\tilde{w}_{j} }}{{\tilde{w}_{w} }}} = \varvec{\tilde{a}_{jw}} $$
(6)

To establish this condition for all j, we need to find a solution where the maximum absolute differences, |\(\frac{{\widetilde{w}}_{b}}{{\widetilde{w}}_{j}}-{\widetilde{a}}_{Bj}\)| and |\(\frac{{\widetilde{w}}_{j}}{{\widetilde{w}}_{w}}-{\widetilde{a}}_{jw}\)| be minimal for all j. Considering the non-negativity of the counters and the conditions of the sum of the weights, the following problem (7) [39] is obtained.

$$ \begin{aligned} & {\mathbf{Min}},{\mathbf{MaxJ }}\left[ {\left| {\frac{{\varvec{\tilde{w}}_{\varvec{b}} }}{{\varvec{\tilde{w}}_{\varvec{j}} }} - \varvec{\tilde{a}}_{{\varvec{Bj}}} } \right|,\left| {\frac{{\varvec{\tilde{w}}_{\varvec{j}} }}{{\varvec{\tilde{w}}_{\varvec{w}} }} - \varvec{\tilde{a}}_{{\varvec{jw}}} } \right|} \right] \\ & \quad \quad \quad \quad \quad \quad \mathop \sum \limits_{{\mathbf{j}}} {\mathbf{R}}({\mathbf{\tilde{w}}}_{\varvec{j}} ) = {\mathbf{1}} \\ & \varvec{l}_{\varvec{j}}^{\varvec{w}} \le \varvec{m}_{\varvec{j}}^{\varvec{w}} \le \varvec{u}_{\varvec{j}}^{\varvec{w}} ,\quad \varvec{l}_{\varvec{j}}^{\varvec{w}} \ge {\mathbf{0}}\quad {\mathbf{For}}\;{\mathbf{all}}\;{\mathbf{j}} \\ & \quad \quad \quad \varvec{R}\left( {\varvec{\tilde{a}}_{\varvec{i}} } \right) = \frac{{\varvec{l}_{\varvec{i}} + {\mathbf{4}}\varvec{m}_{\varvec{i}} + \varvec{u}_{\varvec{i}} }}{{\mathbf{6}}} \\ \end{aligned} $$
(7)

where I, m and u respectively represent the lower,modal and upper value of the support of \(\widetilde{{\varvec{a}}}\) all of which are crisp numbers which is \(\widetilde{{\varvec{a}}}\) represent the ranking of triangular fuzzy number. A Triangular Fuzzy Number can be expressed in the form of a triplet, represented as (l, m, u) [41]. The problem of the above relation model can be transformed into the following problem (8) [40]:

$$ \begin{aligned} &{\mathbf{min}}\;{\boldsymbol{\tilde{\xi }}}\\&\quad{\mathbf{s.t.}}\\&\left|{\frac{{\varvec{\tilde{w}}_{\varvec{b}} }}{{\varvec{\tilde{w}}_{\varvec{j}} }} - \varvec{\tilde{a}}_{{\varvec{Bj}}} } \right| \le {\tilde{a}}\quad \quad {\mathbf{For}}\,{\mathbf{all}}\,{\mathbf{j}} \\ & \left| {\frac{{\varvec{w}_{\varvec{j}} }}{{\varvec{w}_{\varvec{w}} }} - \varvec{\tilde{a}}_{{\varvec{jw}}} } \right| \le \varvec{\tilde{a}}\quad \quad {\mathbf{For}}\,{\mathbf{all}}\,{\mathbf{j}} \\ & \quad \quad \quad \quad \quad \mathop \sum \limits_{{\mathbf{j}}} {\mathbf{R}}({\mathbf{\tilde{w}}}_{\varvec{j}} ) = {\mathbf{1}} \\ & \varvec{l}_{\varvec{j}}^{\varvec{w}} \le \varvec{m}_{\varvec{j}}^{\varvec{w}} \le \varvec{u}_{\varvec{j}}^{\varvec{w}} ,\;~\varvec{l}_{\varvec{j}}^{\varvec{w}} \ge {\mathbf{0}}\quad {\mathbf{Wj}} \ge {\mathbf{0}},{\mathbf{for}}\,{\mathbf{all}}\,{\mathbf{j}}\quad \quad {\mathbf{For}}\,{\mathbf{all}}\,{\mathbf{j}} \\ \end{aligned} $$
(8)

min \(\widetilde{{\boldsymbol{\upxi}}}\)where \({\widetilde{w}}_{j}\) =(\({{\varvec{l}}}_{{\varvec{j}}}^{{\varvec{w}}}\), \({{\varvec{m}}}_{{\varvec{j}}}^{{\varvec{w}}} ,\) \({{\varvec{u}}}_{{\varvec{j}}}^{{\varvec{w}}})\) represent the fuzzy weight of criterion j,

By solving the above problem, the optimal weights \(\left({\widetilde{{W}_{1}}}^{*},{\widetilde{{W}_{2}}}^{*},\dots ,{\widetilde{{W}_{n}}}^{*}\right)\) and \(\widetilde{\upxi }\)*are obtained.

The Consistency Ratio (CR) is a vital indicator used to assess the degree of consistency in pairwise comparisons within the Fuzzy Best–Worst Method (FBWM). The comparisons in the Fuzzy Best–Worst Method (FBWM) are considered fully consistent when the following relationship (9) [40] holds for all j's:

$$ {\text{a}}_{{{\text{BW}}}} = {\text{a}}_{{{\text{Bj}}}} \times {\text{a}}_{{{\text{jw}}}} [{\varvec{a}}_{{{\varvec{BW}}}} \in ] $$
(9)

where aBW, aBj, and ajw are the fuzzy preference of the best criterion over the worst criterion, the fuzzy preference of the best criterion over the criterion j, and the fuzzy preference of the criterion j over the worst criterion, respectively. When the consistency ratio relation (10) is closer to zero, the comparisons are more consistent and stable, and when it is closer to one, the comparisons are less consistent and stable.

Ensuring a low consistency ratio in the FBWM is crucial to maintaining the reliability of decision-makers' judgments in multi-criteria decision-making problems.

$$ \varvec{\left( {{\text{CR}}} \right)}\, = \, \boldsymbol{\frac{{\upxi }}{{{\text{CI}}}}} $$
(10)

where \({{\boldsymbol{\upxi}}}\) represents the optimized inconsistency index, and CI represents the Consistency Index, a measure to check the consistency of the pairwise comparisons. Measure of inconsistency, known as the optimized inconsistency index (ξ^), is calculated by taking the geometric mean of the individual inconsistency indices for all the pairwise comparisons. This overall measure is then used to calculate the consistency ratio (CR), which is a key indicator of the reliability of the decision-making process [40].

The values of the consistency index show clearly that the worst situation for a technology is the equally importance (Table 4).

Table 4 Values of the Consistency Index (CI) for fuzzy BWM for various languistic terms [39]
Full size table

3.3 TOPSIS method

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is an old multi-criteria decision-making method, originally developed by Ching-Lai Hwang and Yoon K.in 1981 [42, 43]. The alternative Interval TOPSIS extends the original method to handle interval data. The following steps to use for the Interval TOPSIS method and its relations which allow its execution were presented by [44, 45].

Step1. The decision matrix for this method, as in other methods, is composed of columns representing the criteria and rows representing the alternatives under consideration. In this step, the method's decision matrix needs to be established. The VIKOR, the acronym, in Serbian of “Multicriteria Optimization and Compromise Solution’’ [9] decision matrix in relation (11) includes a matrix with columns for criteria and rows for research alternatives. To address qualitative criteria in this method, a 7-point linguistic scale, as shown in Table 5, is utilised [45]

$$ \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left[ {\underline {x}_{11} ,\overline{x}_{11} } \right]} \\ {\left[ {\underline {x}_{21} ,\overline{x}_{21} } \right]} \\ \end{array} } & \cdots & {\begin{array}{*{20}c} {\left[ {\underline {x}_{1n} ,\overline{x}_{1n} } \right]} \\ {\left[ {\underline {x}_{2n} ,\overline{x}_{2n} } \right]} \\ \end{array} } \\ \vdots & \ddots & \vdots \\ {\left[ {\underline {x}_{m1} ,\overline{x}_{m1} } \right]} & \cdots & {\left[ {\underline {x}_{mn} ,\overline{x}_{mn} } \right]} \\ \end{array} } \right] $$
(11)
Table 5 Linguistic scale and interval numbers [44]
Full size table

We assume A1, A2, …, Am our m possible alternatives that decision-makers have to select from, and C1, C2, …, Cn are the criteria employed to measure the performance of these alternatives. Here, xij represents the rating of alternative Ai in terms of criterion Cj. This value is not known precisely, and we can only affirm that xij falls within the interval \(\left[{\underset{\_}{{\varvec{x}}}}_{\mathbf{i}\mathbf{j}},{\overline{{\varvec{x}}}}_{{\varvec{i}}{\varvec{j}}}\right]\) (Table 5).

Step2. Normalize the decision matrix. Normalize the interval-valued decision matrix by dividing each element by the square root of the sum of squares of its corresponding column elements and use of the relations (12) and (13). This step ensures comparability across different criteria.

$$ {\overline{\text{x}}}_{{{\text{ij}}}}^{*} = \frac{{{\overline{\text{x}}}_{{{\text{ij}}}} }}{{\sqrt {\mathop \sum \nolimits_{{{\text{j}} = 1}}^{{\text{m}}} \left( {\underline {\text{x}}_{{{\text{ij}}}}^{2} + {\overline{\text{x}}}_{{{\text{ij}}}}^{2} } \right)} }} $$
(12)
$$ \underline {\text{x}}_{{{\text{ij}}}}^{*} = \frac{{\underline {\text{x}}_{{{\text{ij}}}} }}{{\sqrt {\mathop \sum \nolimits_{{{\text{j}} = 1}}^{{\text{m}}} \left( {\underline {\text{x}}_{{{\text{ij}}}}^{2} + {\overline{\text{x}}}_{{{\text{ij}}}}^{2} } \right)} }} $$
(13)

where \({\underset{\_}{\mathbf{x}}}_{\mathbf{i}\mathbf{j}}^{\mathbf{*}}\) and \({\overline{\mathbf{x}}}_{\mathbf{i}\mathbf{j}}^{\mathbf{*}}\) are the normalized interval valued. now interval [\({\underset{\_}{\mathbf{x}}}_{\mathbf{i}\mathbf{j}}^{\mathbf{*}}\), \({\overline{\mathbf{x}}}_{\mathbf{i}\mathbf{j}}^{\mathbf{*}}\)] is normalized of interval \(\left[{\underset{\_}{{\varvec{x}}}}_{\mathbf{i}\mathbf{j}},{\overline{{\varvec{x}}}}_{{\varvec{i}}{\varvec{j}}}\right]\)

Step3. Calculate the weighted normalized in the decision matrix. Multiply each element in the normalized decision matrix, by the corresponding criterion's weight \({\mathbf{w}}_{\mathbf{j}}\) in relation (14) and (15). The weights represent the relative importance of each criterion in the decision-making process.

$$ \mathbf{\underline {\text{x}}_{{\text{i}}}^{\prime}} = \mathbf{{\text{w}}_{{\text{j}}} \underline {\text{x}}_{{{\text{ij}}}}^{*}} $$
(14)
$$ \mathbf{{\overline{\text{x}}}_{{\text{i}}}^{\prime}} = \mathbf{{\text{w}}_{{\text{j}}}} \mathbf{{\overline{\text{x}}}_{{{\text{ij}}}}^{*}} $$
(15)

Step4. Determine the interval ideal and negative-ideal solutions for each criterion; identify the interval-valued ideal solution (the best performance among alternatives) and the interval-valued negative-ideal solution (the worst performance among alternatives). Positive and negative ideals are obtained from the following relationships:

In these relationships (16) and (17), I represents criteria with a positive aspect, and J represents criteria with a negative aspect. \({\mathbf{v}}_{1}^{+}\) and \({\mathbf{v}}_{1}^{-}\) are the ideal value for the ith criterion and anti-ideal value for the ith criterion respectively.

$$ \overline{{\mathbf{A}}} ^{ + } = \left\{ {{\mathbf{v}}_{{\mathbf{1}}}^{{\mathbf{ + }}} , \ldots ,{\mathbf{v}}_{{\mathbf{n}}}^{ + } } \right\} = \left\{ {\left( {\mathop {{\mathbf{max}}}\limits_{{\mathbf{j}}} \overline{{\mathbf{x}}} _{{\mathbf{i}}} ^{\prime } |{\text{i}} \in {\text{I}}} \right),\left( {\mathop {{\mathbf{min}}}\limits_{{\mathbf{j}}} \underline{{\mathbf{x}}} ^{\prime } _{{\mathbf{i}}} |{\mathbf{i}} \in {\mathbf{J}}} \right)} \right\} $$
(16)
$$ \overline{{\mathbf{A}}} ^{ - } = \left\{ {{\mathbf{v}}_{{\mathbf{1}}}^{ - } , \ldots ,{\mathbf{v}}_{{\mathbf{n}}}^{ - } } \right\} = \left\{ {\left( {\mathop {{\mathbf{min}}}\limits_{{\mathbf{j}}} \underline{{{\mathbf{x^{\prime}}}}} _{{\mathbf{i}}} |{\mathbf{i}} \in {\mathbf{I}}} \right),\left( {\mathop {{\mathbf{max}}}\limits_{{\mathbf{j}}} \overline{{\mathbf{x}}} ^{\prime } _{{\mathbf{i}}} |{\mathbf{i}} \in {\mathbf{J}}} \right)} \right\} $$
(17)

Step5. Determining the distance of alternatives from positive and negative ideals. Using the following relationships (18) and (19), the distance of alternatives (d+ distance between the jth alternative from the positive ideals and d− distance between the jth alternative from the negative ideals) is calculated.

$$ {\mathbf{d}}_{{\mathbf{j}}}^{ + } = \left\{ {\sum\nolimits_{{{\mathbf{i}} \in {\mathbf{I}}}} {\left( {\overline{{{\mathbf{x^{\prime}}}_{{{\mathbf{ij}}}} }} - {\mathbf{v}}_{{\mathbf{i}}}^{ + } } \right)^{{\mathbf{2}}} } + \sum\limits_{{{\mathbf{i}} \in {\mathbf{I}}}} {\left( {\underline{{{\mathbf{x^{\prime}}}_{{{\mathbf{ij}}}} }} - {\mathbf{v}}_{{\mathbf{i}}}^{ + } } \right)^{{\mathbf{2}}} } } \right\}^{{\frac{{\mathbf{1}}}{{\mathbf{2}}}}} \varvec{j} = {\mathbf{1}},{\mathbf{2}}, \ldots ,\varvec{m} $$
(18)
$$ {\mathbf{d}}_{{\mathbf{j}}}^{ - } = \left\{ {\sum\nolimits_{{{\mathbf{i}} \in {\mathbf{I}}}} {\left( {\overline{{{\mathbf{x^{\prime}}}_{{{\mathbf{ij}}}} }} - {\mathbf{v}}_{{\mathbf{i}}}^{ - } } \right)^{2} } + \sum\nolimits_{{{\mathbf{i}} \in {\mathbf{I}}}} {\left( {\underline{{{\mathbf{x^{\prime}}}_{{{\mathbf{ij}}}} }} - {\mathbf{v}}_{{\mathbf{i}}}^{ - } } \right)^{{\mathbf{2}}} } } \right\}^{{\frac{{\mathbf{1}}}{{\mathbf{2}}}}} \varvec{j} = {\mathbf{1}},{\mathbf{2}}, \ldots ,\varvec{m} $$
(19)

Step6. Calculating the similarity index and ranking alternatives by using the following relationship (20), we calculate the similarity index (CL).This index is a metric used to rank different alternatives based on their proximity to the ideal solution and their distance from the anti-ideal solution of the alternatives and rank them accordingly. CL for each alternative is a value between 0 and 1. A higher value of CL indicates that the alternative is closer to the ideal solution and further from the anti-ideal solution, Therefore, with the help of the closeness coefficient, we can figure out the order of all options and choose the best one from the list of possible choices.

$$ {\mathbf{CL}} = \frac{{{\mathbf{d}}_{{\mathbf{j}}}^{ - } }}{{\left( {{\mathbf{d}}_{{\mathbf{j}}}^{ - } + {\mathbf{d}}_{{\mathbf{j}}}^{ + } } \right)}}{\text{ }}{\mathbf{j}} = {\mathbf{1}},{\mathbf{2}}, \ldots ,{\mathbf{m}} $$
(20)

3.4 Weighted Aggregated Sum Product Assessment with Grey Values (WASPAS-G)

The WASPAS (Weighted Aggregated Sum Product Assessment) method is a modern multi-criteria decision-making technique, introduced by Zavadskas et al. [9]. This method combines two models: WSM (Weighted Sum Model) and WPM (Weighted Product Model), providing higher accuracy compared to independent methods. The steps of the grey model and all relation and equation for this method are outlined below [46].

Step1. The decision matrix in (21) for this method, like other methods, known as the VIKOR decision matrix, consists of columns representing various criteria and rows signifying the different alternatives under evaluation. In this phase, the formation of the decision matrix is essential. It includes a matrix containing columns dedicated to criteria and rows for the research alternatives being assessed. To incorporate qualitative criteria within this method, a 7-point linguistic scale is utilized, as demonstrated in (Table 5). ⨂X reperesents the grey evaluations of the i-th alternative with respect to the j-th attribute.

$$ \otimes {\varvec{X}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\left[ {\varvec{\underline {x} }_{11} ,\overline{\varvec{x}}_{11} } \right]} \\ {\left[ {\varvec{\underline {x} }_{21} ,\overline{\varvec{x}}_{21} } \right]} \\ \end{array} } & \cdots & {\begin{array}{*{20}c} {\left[ {\varvec{\underline {x} }_{{1{\varvec{n}}}} ,\overline{\varvec{x}}_{{1{\varvec{n}}}} } \right]} \\ {\left[ {\varvec{\underline {x} }_{{2{\varvec{n}}}} ,\overline{\varvec{x}}_{{2{\varvec{n}}}} } \right]} \\ \end{array} } \\ \vdots & \ddots & \vdots \\ {\left[ {\varvec{\underline {x} }_{{{\varvec{m}}1}} ,\overline{\varvec{x}}_{{{\varvec{m}}1}} } \right]} & \cdots & {\left[ {\varvec{\underline {x} }_{{{\varvec{mn}}}} ,\overline{\varvec{x}}_{{{\varvec{mn}}}} } \right]} \\ \end{array} } \right] $$
(21)

Step2. In this step, we need to normalize the decision matrix. If the criteria have a positive aspect, we use Eq. (22), and if the criteria have a negative aspect, we use Eq. (23) [9], where \(\otimes \overline{\varvec{x}}_{{{\varvec{ij}}}}\) represents normalized decision matrix and \({\overline{{\varvec{x}}}}_{{\varvec{i}}{\varvec{j}}\boldsymbol{\alpha }}\) represents normalized maximal values.The initial stage involves pinpointing both the favorable and unfavorable criteria. For every positive criterion assign the highest value within its range to the respective column and divide by the maximum value. Conversely, for each negative criterion, allocate the lowest value within its range to the corresponding column, divide by the minimum value, and subsequently invert the sequence of values.in the (Table 6) Normalized matrix is calculated.

$$ \otimes \overline{\varvec{x}}_{{{\varvec{ij}}}} = \frac{{ \otimes {\varvec{x}}_{{{\varvec{ij}}}} }}{{\mathop {{\mathbf{max}}}\limits_{{\mathbf{i}}} \otimes {\varvec{x}}_{{{\varvec{ij}}}} }}, $$
$$ \varvec{i.e.\,\bar{x}_{{ij\alpha }}} = \varvec{\frac{{x_{{ij\alpha }} }}{{\mathop {\max }\limits_{i} {\mkern 1mu} \otimes x_{{ij\beta }} }}}\,\varvec{and}\,\varvec{\bar{x}_{{ij\beta }}} = \varvec{\frac{{x_{{ij\beta }} }}{{\mathop {\max }\limits_{i} {\mkern 1mu} \otimes x_{{ij\beta }} }}} $$
(22)
$$ \otimes \overline{x}_{ij} = \frac{{\mathop {\min }\limits_{i} \otimes x_{ij} }}{{ \otimes x_{ij} }} $$
(23)
$$ {\varvec{i.e.}}\ \overline{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\alpha} } = \frac{{\mathop {\varvec{min} }\limits_{{\varvec{i}}} \boldsymbol{\otimes}{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\alpha} } }}{{{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\beta} } }} \ {\varvec{and}} \ \overline{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\beta} } = \frac{{\mathop {\varvec{min} }\limits_{{\varvec{i}}} \boldsymbol{\otimes }{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\alpha} } }}{{{\varvec{x}}_{{\varvec{ij}}\boldsymbol{\alpha} } }} $$
Table 6 Norrmalized matrix (interval WASPAS) for the technology subcriteria of PEM and Alkaline Electrolysers
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Step3. we need to calculate the values of the Weighted Sum Model (WSM), which is calculated using the following formula (24).In this formula, \(\otimes {\hat{\text{x}}}_{{{\text{ij}}}}\) represents the normalized weighted value, which is obtained from the following relationship (25), In this relationship, ⦻w_j is the weight (normalized) of the criteria obtained from the Gray AHP method [46]. \(\otimes {\overline{\mathbf{x}}}_{{{\mathbf{ij}}}}\) represents the matrix of normalized decision values. The grey value \(\otimes {\mathbf{S}}_{{\mathbf{i}}}\) represents the additive optimality function (WSM) for i the alternative.

$$ \otimes {\mathbf{S}}_{{\mathbf{i}}} = \sum\nolimits_{{{\mathbf{j}} = {\mathbf{1}}}}^{{\mathbf{n}}} \otimes \widehat{{\mathbf{x}}}_{{{\mathbf{ij}}}} ,{\mathbf{j}} = {\mathbf{1}}, \ldots ,{\mathbf{m}},{\mathbf{or}} $$
(24)
$$ \otimes {\mathbf{S}}_{{\mathbf{i}}} = 0.5\mathop \sum \limits_{{{\mathbf{j}} = 1}}^{{\mathbf{n}}} \left( {{\hat{\mathbf{x}}}_{{{\mathbf{ij\alpha }}}} + {\hat{\mathbf{x}}}_{{{\mathbf{ij\beta }}}} } \right) $$
$$ \otimes \widehat{{\mathbf{x}}}_{{{\mathbf{ij}}}} = \otimes \overline{{\mathbf{x}}} _{{{\mathbf{ij}}}} \times \otimes {\mathbf{w}}_{{\mathbf{j}}} $$
(25)

Step4. In this step, we need to calculate the values of the Weighted Product Model (WPM), which is calculated using the following formula (26) [46]. The geometric mean (P) is also obtained through the multiplication of the normal distance numbers, raised to the power of the criteria weights. Finally, we compute the WASPAS index. The grey value \(\otimes {\text{P}}_{{\text{i}}}\) represents optimality function according to WPM, where \(\otimes {\mathbf{w}}_{{\mathbf{j}}}\) is the grey weight of the j attribute and \(\otimes {\hat{\mathbf{x}}}_{{{\mathbf{ij}}}}\) is the grey normalized rating of the of the i-th alternative with respect to the j-th attribute. \(\otimes {\overline{\mathbf{x}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }}\) the weight exponent to the corresponding grey normalized decision value, representing the weighted power of each decision value in the final computation of ⨂P_i,.

$$ \otimes {\mathbf{P}}_{{\mathbf{i}}} = \prod\nolimits_{{{\mathbf{j}} = {\mathbf{1}}}}^{{\mathbf{n}}} { \otimes \overline{{\mathbf{x}}} ^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }} } ,{\mathbf{j}} = {\mathbf{1}}, \ldots ,{\mathbf{m}},{\mathbf{or}} $$
(26)
$$ \otimes {\mathbf{P}}_{{\mathbf{i}}} = \mathop \prod \limits_{{{\mathbf{j}} = 1}}^{{\mathbf{n}}} 0.5( \otimes {\overline{\mathbf{x}}}_{{{\mathbf{j\alpha }}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }} \, + \, \oplus {\overline{\mathbf{x}}}_{{{\mathbf{j\beta }}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }} ) ,{\mathbf{j}} = 1, \ldots ,{\mathbf{m}} $$

where α refers to the lower bound of the j-th decision criterion under uncertainty. \(\otimes {\overline{\mathbf{x}}}_{{{\mathbf{j\alpha }}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }}\) is raised to the power of the weight assigned to that criterion. \(\otimes {\overline{\mathbf{x}}}_{{{\mathbf{j\beta }}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }}\) represents the upper bound of the j-th decision criterion raised to the power of the weight of that criterion.

Step5. In this relationship, the VIKOR index is obtained using the following formula (27) and (28). The introduction of compromise solution in MCDM was done by Yu [47] and Zeleny [48]. Vikor is in serbian and means Multicriteria optimisation and Compromise Solution. It is a MCDM method which was developed by Seraphin Opricovic, firstly in his Ph.D dissertation in 1979 and published in 1980 [9] to find solutions for conflicting and criteria with different units. In 1998, the real applications were shown [49]. This method was internationally recognised from the publication in 2004 [50].

In formula (27), the value of λ is obtained from the following relationship (28) [9]. Lambda (λ) (4.26) serves as a parameter that establishes the relative significance of the WSM and WPM in the comprehensive evaluation. Altering λ consequently modifies the influence of the WSM and WPM on the overall assessment. Specifically, increasing λ amplifies the WSM's impact on the evaluation while diminishing the WPM's influence. In contrast, decreasing λ enhances the WPM's impact while reducing the WSM's influence. The value of λ can be ascertained based on the objectives and priorities of the decision-making process and the attributes of the alternatives under comparison. The alternative exhibiting the highest overall score according to the WASPAS method is deemed the most optimal choice.

$$ \otimes {\mathbf{Q}}_{{\mathbf{i}}} = \uplambda \sum\nolimits_{{{\mathbf{j = 1}}}}^{{\mathbf{n}}} \otimes \widehat{{\mathbf{x}}}_{{{\mathbf{ij}}}} + \left( {{\mathbf{1}} - \uplambda } \right)\prod\nolimits_{{{\mathbf{j}} = {\mathbf{1}}}}^{{\mathbf{n}}} { \otimes \overline{{\mathbf{x}}} ^{{ \otimes {\mathbf{w}}_{{\mathbf{j}}} }} } $$
(27)
$${\varvec{\uplambda}}=0,\dots ,1$$
$$ \uplambda = {\mathbf{0}}{\mathbf{.5}}\frac{{\sum\nolimits_{{{\mathbf{i}} = {\mathbf{1}}}}^{{\mathbf{m}}} {{\mathbf{P}}_{{\mathbf{i}}} } }}{{\sum\nolimits_{{{\mathbf{i}} = {\mathbf{1}}}}^{{\mathbf{m}}} {{\mathbf{S}}_{{\mathbf{i}}} } }} $$
(28)

where \(\otimes {\mathbf{Q}}_{{\mathbf{i}}}\) the overall score for the i-th alternative, computed based on the aggregation of the weighted normalized values for each criterion (⦻x ̂_ij) and the geometric mean of the grey interval normalized values raised to their corresponding weights \(\left( { \otimes {\overline{\mathbf{x}}}^{{ \otimes {\mathbf{w}}_{{\mathbf{i}}} }} } \right)\). This score provides a comprehensive evaluation of the i-th alternative according to all the criteria and their weights.

The values of P_i and S_i are converted to definite numbers using the following relationships (29) and finally, using the VIKOR index value, we can rank the alternatives.

$$ {\mathbf{P}}_{{\mathbf{i}}} = {\mathbf{0.5}}\left( {{\mathbf{P}}_{{{\mathbf{i}}\mathbf{\upalpha} }} + {\mathbf{P}}_{{{\mathbf{i}}\mathbf{\upbeta} }} } \right) $$
(29)
$${\mathbf{S}}_{\mathbf{i}}={\mathbf{0.5}}\left({\mathbf{S}}_{\mathbf{i}{\varvec{\upalpha}}}+{\mathbf{S}}_{\mathbf{i}{\varvec{\upbeta}}}\right)$$

Using the above methods applied to the schematic representation of (Figure) allow us to generate data which are used to make decision on the ability of an electrolyser technology to be economically viable. To complete this method of analysis, we may integrate the parameters of the determination of the Levelized cost of hydrogen.

3.5 Parameters used to determine levelized cost of Hydrogen for 5 MW PEM and Alkaline Electrolysers

The levelized cost of hydrogen (LCOH) is a metric used to estimate the total cost of producing and delivering hydrogen over the lifetime of a hydrogen production system. The LCOH takes into account all of the costs associated with producing hydrogen, including the initial capital cost, operation and maintenance costs, and the cost of the energy used to produce the hydrogen.

We develop MATLAB code for calculating the characteristics of a 5 MW water electrolyser and the corresponding LCOH can be achieved by following these steps:

  • The first step is to develop a MATLAB code to calculate the LCOH for a 5 MW water electrolyser and to define the inputs required for the calculations. These inputs may include the efficiency of the electrolyze, its cost of electricity, and its expected lifetime.

  • The second step is to use the result of the first step and to calculate the characteristics of the electrolyse using equations, such as the Nernst equation, which relates the potential difference between the electrodes to the equilibrium potential of the cell. Other relevant properties, such as the enthalpy and entropy changes of the electrochemical reactions, can also be calculated using standard data of reactions

  • The third step is to calculate in kg/kWh the hydrogen produced per unity electricity consumed.

  • The fourth step is to obtained or estimated the cost of electricity from energy market data or using data on the cost of energy in the region where the electrolyser is located.

  • Finally, we estimate the lifetime of the electrolyse based on the expected lifespan of the materials used in the electrolyser and the anticipated maintenance requirements.

Using all these factors we determine the LCOH for a 5 MW proton exchange membrane (PEM) or alkaline water electrolysis system and make comparison of data of the two technologies.

A chart is used to represent the LCOH values for the two types of water electrolysers.

3.6 Criteria weight

In this section the fuzzy weight is directly obtained from solving the model in Lingo software, then these fuzzy weights are calculated by the relation (9) [40] and after that it has become to final weight as described in Sect. (2.1). By following this procedure, the fuzzy weights of criteria are initially determined, and the overall weight of each criterion is subsequently obtained. The weights are presented in (Table 7). It is important to note that crisp numbers are used to represent all weights in the subsequent calculations.

Table 7 Overall Fuzzy Weigh of each of the main criteria or aspects, their Crysp Weight and their final rank
Full size table

For instance, the economic criteria weight is (0.314, 0.326, 0.372), resulting in a final weight of 0.332. Therefore, the economic aspect has the highest weight of 0.332 among all aspect (rank 1). The economic aspect is the highest one because it is the most important aspect for industrial applications of a technology as PEM electrolysers. The technical aspect is the second important aspect (rank 2) to be considered because the technical performance defines the rate of hydrogen production kg of H2/day, applied current density, voltage efficiency, etc. which indicates which technic parameters are the best for this application. This why the technical criterion, with a weight of 0.226, ranks second. The technology (catalyst, lifetime, electrode, electrolyte, bipolar plates, diaphragm, technology maturity, etc.) rank is dependant of the technic parameters. This is why the technology criterion, with a weight of 0.206, ranks third. Hydrogen production from PEM or alkaline electrolysers typically results in CO2 emissions in the range of 1–2 kg CO2 per kg of hydrogen produced [51]. These values are proportional to the carbon content of the electricity used for water electrolysis: These values are smaller than those of CO2 emissions from hydrogen produced gas using Steam Methane Reforming (SMT (8–10 kg of CO2/kg of produced hydrogen) or coal gasification (14–15 kg/kg of produced hydrogen). Due to the small values of the CO2 emissions to electrolytic hydrogen production, the environment aspect is ranked The social factors have the last rank for the Crisp Weight because the cost and the technology efficiency are the most important sub criteria considered in this analysis.

(Fig. 4) and (Table 8) display the final weights obtained from the software after formulating the linear optimization model and solving it.

Fig. 4
figure 4

Examples of the process of the criteria weighting (This work)

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Table 8 Global weight of the criterion system of the 30 criteria: This work
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3.7 Multi-criteria assessment

The assumptions of the assessment of this work are summarised in the followings. The first assumption states that using a high number of aspects in MCDM analysis may give more reliable technic-economic analysis. Five aspects (economic, technical, technology, environment, social) are used here. From our knowledge it is the highest number of aspects used until now for this study. The second assumption considered that using a number of valuable criteria and sub-criteria may enhance the accuracy of the MCDM analysis. In this work, a total of thirty criteria have been used (Table 1) for the five aspects plus six sub-criteria related to the three criteria of the economic aspect. The third assumption considers that the economic aspect is the most limiting factor for the mass production of electrolytic hydrogen (rank 1). For a stack cost of 1500 $/kW, a factor capacity of 90%, an electricity cost of 7 cents/kWh, the produced hydrogen cost using PEM electrolysers is the range of 5–6 $/kH2 [52] This is significantly higher than those of the hydrogen produced by SMR (steam Methane Reforming) [53] and whch is less that 2$/kgH2. This economic rank is followed respectively by the technical (rank 2), the technology (rank 3), the environment (rank 4) and the socal (rand 5) (Table 1). This section aims to determine the final order of preference for cost and factors related to technology, configuration, and technical aspects, which play a crucial role in reducing costs or increasing the efficiency of each water electrolysis technology, considering all criteria defined in Sect. 4. To simplify the evaluation of MCDM methods, (Table 8) summarizes the relevant values of each technology concerning all criteria. The next step involves constructing the decision matrix. It is important to note that the ranges shown in (Table 8) are defined by a specific interval number. Therefore, using an interval grey approach in this case is appropriate. Consequently, in this study, the ranges in (Table 8) are based on cost and efficiency, each criterion is labeled as negative if a higher value is detrimental, or positive if a higher value is beneficial. Since most selected MCDM methods have a mathematical structure, qualitative values in (Table 9) must be converted to numerical ones before creating the decision matrix. This can be accomplished using linguistic conversions [45]. The decision matrix is then constructed, as displayed in (Table 10) a structured comparison of two technologies is conducted using the TOPSIS and WASPAS interval methods of MCDM. Closeness coefficient and ranking the alternatives.

Table 9 Data from the litterature of quantitative and quality parameters of the water electrolysis technologies attending to all factors shown in (Table 6) [4, 33, 54]
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Table 10 Decision matrix of the technology aspects for combined TOPSIS and WAPAS
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For TOPSIS, the similarity index (CL) is fist calculated by the determination of the similarity of each alternative solution to the ideal solution (Table 11). Calculation of weighted average(S), geometric mean(P) and Index of WASPAS(Weighted Aggregated Sum Product Assessment) for ranking the alternatives is done (Table 12). As λ is a parameter that determines the relative importance of the WSM (Weighted Sum Model) and WPM (Weighted Product Model) in the overall assessment. When λ is changed, the influence of the WSM and WPM on the overall assessment changes as well.

Table 11 weighted average(S),geometric mean(P) and Index of WASP
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Table 12 Ranking alternatives in TOPSIS
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4 Analysis results and discussion

4.1 Analysis for MCDM

The TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and WASPAS (Weighted Aggregated Sum Product Assessment) MCDM (Multi-Criteria Decision Making) methods have been applied for ranking), and the Fuzzy BWM (Best–Worst Model) method has been used for weighting criteria to determine the most appropriate water electrolysis technologies for hydrogen production. These methods assessed various factors, including economic, environmental, technical, technology, and social aspects, which are crucial for prioritizing technologies in sustainable hydrogen production.

The distribution of importance assigned to economic, environmental, and social factors is shown in (Fig. 5), with the economic factor being the most important. In (Fig. 5), the economic factor is divided into Investment costs and O&M for better comprehension. The remaining share is evenly distributed among social, environmental, technology, and technical factors. Technical and technology factors correspond to the criteria with the highest weight values in (Table 6).

Fig. 5
figure 5

Distribution of importance for water electrolysers assigned to economic, Technical, environmental, technological, and social factors, with the economic factor being the most important

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Figure 6 shows that the maintenance costs and the lifetime (in hours) of Alkaline Electrolyser (SE) are higher than the PEM Electrolyser. In contrary the cold of PEM technology is better than the AE technology. With a high social acceptance, a less land use, a high potential of technology development, a low operating and maintenance and others the results in Table 9 also indicate that PEM Electrolyser is the preferred method for sustainable hydrogen production, although AE ranks highest based on costs. This can be attributed to AE scoring well on almost all economic-related criteria, but performing relatively poorly on hydrogen flow rate, voltage efficiency, current density, and gas purification, as shown in (Fig. 7).

Fig. 6
figure 6

Distribution of importance of criteria comparison in Alkaline and PEM

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Fig. 7
figure 7

Distribution of importance of some technological subcriteria comparison for Alkaline and PEM Electrolyser sytems

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In this study, two different MCDM methods were employed to rank technologies while focusing on costs and considering 30 criteria and sub-criteria for detailed assessment of each electrolyser. This approach ensured coherent and consistent rankings. All methods determined that, given the specifications of the two water electrolyser technologies and the specific goal of producing low-cost hydrogen, Alkaline Electrolyser(AE) is the best technology. However, for producing more and purer hydrogen, PEM is considered more reliable.

Alkaline electrolyser presents the best scores in economic criteria, where PEM Electrolyser does not perform as well.

The λ parameter determines the relative importance of WSM (Weighted Sum Model) and WPM (Weighted Product Model) which are related to the effect of the alternatives in the overall assessment. The value of λ ranges between 0 and 1, with λ = 0.5 often used as a starting point, indicating an equal weighting between WSM and WPM.

When λ is closer to 1, the WASPAS method is more aligned with the Weighted Sum Model (WSM). WSM is an additive model, which means that it's better suited for criteria that can be compensated, i.e., a poor performance in one criterion can be offset by a good performance in another criterion.

When λ is closer to 0, the WASPAS method aligns more with the Weighted Product Model (WPM). WPM is a multiplicative model, meaning it's better suited for criteria that are non-compensatory, i.e., a poor performance in one criterion cannot be made up by a good performance in another criterion. When λ is increased, WSM has a greater influence, while WPM has a lesser influence, and vice versa.

The λ value can be determined based on the specific goals and priorities of the decision-making process and the characteristics of the alternatives being compared. The alternative with the highest overall score according to the WASPAS method is considered the best overall alternative (Fig. 8). Shows the sensitivity analysis of the WASPAS method to different λ, if the λ for PEM is smaller than for Alkaline, it implies that the decision-making for PEM electrolysers leans more towards the non-compensatory model (WPM), meaning the weaknesses in one criterion cannot be compensated by strengths in another.

Fig. 8
figure 8

WASPAS sensivity analysis with the change of λ value.0

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This could imply that while PEM might have advantages in certain aspects (perhaps technical or environmental), its economic performance (a key consideration) can't be compensated by these strengths, hence it might not be the optimal choice if economic factors weigh heavily in the decision-making process.

On the other hand, a higher λ value for Alkaline electrolysers suggests a greater alignment with the compensatory model (WSM), meaning good performance in one area could offset poorer performance in another. Thus, even if Alkaline electrolysers might not perform as well as PEM electrolysers in some areas, their superior economic performance compensates for these weaknesses, making them a better choice under the given criteria and weights. Although both the WASPAS and TOPSIS methods are widely used MCDM techniques, and we obtain similar results for the economic mission of PEM and Alkaline electrolysers, WASPAS offers greater flexibility and may provide more accurate and robust results by combining the strengths of both additive and multiplicative aggregation techniques. Furthermore, its sensitivity analysis capabilities can enhance the overall decision-making process by offering insights into the impact of changes in the λ parameter.

4.2 Cases summary and cost data

To assess the economic characteristics of a grid-connected hydrogen production plant, two, system cases were analyzed—the AE system and the PEM Electrolyser system. The numerical models of each water electrolysis system are depicted in Table 13 both system cases were considered as 5 MWel for low-temperature electrolysis class plant technologies. The operating point was calculated using a MATLAB code which was developed to model the thermodynamic characteristics of PEM and Alkaline electrolysis for hydrogen production (Table 14). Technical results of the system thermodynamic modeling. The data for this analysis was obtained from the research studies by [55] and [56].

Table 13 Set of economic criteria used for levelized cost of hydrogen
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Table 14 Technical results of the system thermodynamic modeling
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The cost of hydrogen was determined using the method presented in the studies by [12, 57]. The cost data in (Tables 13 and 15) was based on the reference year 2019 [32, 58].

Table 15 Input data for economic evaluation [56]
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The plant's operating lifetime was set to 25 years, and the stacks were completely replaced every 10 years throughout the lifespan The capacity factor was set as 97% as a reference condition, with the system degradation rate decreasing by 10% over the stack's lifetime, taking into account the variable capacity factors. The electric cost was calculated based on the cost in Quebec, Canada, which is approximately CAD$0.097/kWh.The use of 10 kg of water to produce 1 kg of hydrogen was assumed, considering some losses incurred during the hydrogen purification process, and the total cost was calculated based on the price presented. KOH was only utilized in the AE system to create electrolyte solutions.

The other main assumption made for the economic analysis in this study are as follows (Table 15):

  • 5 MWel hydrogen production plants for each electrolysis type are compared.

  • The discount rate (r) and tax rate (TR) are applied at 8% and 30% respectively for the net present value calculation.

  • The selling price for O2 is 0.054 $/kg O2 and 9$/kg H2 [13].

4.3 Analysis results and discussion for cost Hydrogen

The hydrogen production rate was determined by taking into account the power consumption and operating efficiency of each plant when a 5 MW power supply was provided. The rate was higher in the order of PEMEC > AWE. The highest rate was recorded for the PEM plant, at 780042.96 kgH2/year, due to its high system efficiency. On the other hand, the Alkaline plant produced 578149.488 kgH2/year. The Life Cycle Cost of Hydrogen (LCOH) also followed the same order. The LCOH was calculated by comparing the cost incurred over the plant's lifespan with the amount of hydrogen produced during that time.

AWE systems were found to be the most economical for hydrogen production, with an investment cost of CAD$7.60 per kg, compared to CAD$8.66 per kg for PEM. The main differences between the electrolysis types were observed in maintenance and stack replacement costs. Electric costs alone accounted for at least half of the LCOH formation. The cost of the stack replacement was largely influenced by the stack price, and since the PEM stack was more expensive, it had a significant impact on both the AWE and PEMEC cases.

A profitability analysis of the project was conducted using the Net Present Value (NPV) and the Internal Rate of Return (IRR). The payback period (PBP) was defined as the point where NPV becomes positive. If NPV is negative at the end of the project, it indicates that profitability cannot be expected. The IRR was calculated to be 9.85% for the Alkaline electrolyser, which is higher than the set discount rate of 8%, indicating a net income.

5 Conclusion

In summary, this study utilized The TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) and WASPAS (Weighted Aggregated Sum Product Assessment), and Fuzzy BWM (Best–Worst Model) MCDM (Multi-Criteria Decision Making) methods to evaluate water electrolysis technologies for hydrogen production. The economic, environmental, technical, technology, and social factors where considered as main aspects. The alternatives were AE and PEM Electrolyser, Although Alkaline (AE) technology emerged as the most cost-effective, PEM Electrolyser was identified as the preferred choice for sustainable hydrogen production due to its superior performance in other important criteria.

The WASPAS method's sensitivity analysis provided valuable insights into the decision-making process, while the assessment of hydrogen production rates and LCOH confirmed AE systems as the most economical option. This comprehensive evaluation of water electrolysis technologies offers crucial guidance for decision-makers, emphasizing the need to consider a range of factors to ensure well-rounded and robust decision-making in sustainable hydrogen production.