1 Introduction

Today, along with population growth and contemporary living, the use of traditional fossil fuel-powered vehicles for transportation, which are responsible for almost 25% of worldwide CO2 emissions and 20% of greenhouse gas (GHG) releases, contributes significantly to the increase in environmental issues involving air pollution, noise concerns and traffic congestion [1, 2]. Consequently, a smart transportation system based on electric energy also known as ‘electric mobility or e-mobility’ is seen as a superior option for a sustainable and smart city infrastructure, lowering the carbon footprint of transit systems [3,4,5]. In view of the above, public transport buses are especially well-suited for electrification because of their predetermined routes, on-time operation schedules and shared infrastructure [6]. From a global perspective, EBs have the potential to be the most energy efficient and sustainable option when combined with electricity produced from renewable sources including wind, solar, hydraulic, bio- and geothermal energy [6,7,8]. EBs have been increasingly popular in recent years, and urban transportation companies are gradually incorporating them into their fleets [9]. However, operating bus services is a complicated process that entails a multitude of considerations varying from strategic to operational. Specifically, the siting of charging stations, charging strategies and technologies, rechargeable battery type, and charging frequency are all critical factors that must be evaluated for financial sustainability when aiming for full fleet electrification of bus system operations [10]. When managing a public EB fleet, it is critical to deploy charging stations in strategic locations across the city since the short driving range and poor charging rate of battery EBs necessitate periodic recharges during the day in bus operations [9, 11,12,13]. Therefore, the best placement of charging stations along EB lines will have a great impact on the development of a solid bus operation control system in terms of efficiency and service quality [13]. To this end, the subject of selecting the optimal site for EBCSs via a multi-criteria decision-making (MCDM) problem involving urban fleet planners will be evaluated from a smart and sustainable standpoint.

Modeling and managing uncertainty become crucial when public bus operators review criteria weights and alternatives, due to the importance of ambiguous information in the nature of the EBCS location selection problem. Nonetheless, (i) hardly any research has been conducted to discuss numerous options for particularly deploying EBCSs; (ii) there has been no previous research that has stated the criteria for sustainable and smart charging systems to solve the EBCS issue. Consequently, presenting an enhanced solution procedure for the multi-criteria EBCS site selection problem to assist bus operators in effectively determining the optimal sites is the major aim of this research.

Distinct from earlier research, this research project develops a hybrid procedure for the MCDM framework by combining the BWM and MULTIMOORA approaches in a spherical fuzzy setting to fulfill the aforementioned major goal and cover the significant research gaps within the extant literature. The key contributions of this paper from a practical and theoretical viewpoint are outlined as follows: (1) a brand-new BWM with a unique aggregation operator in a spherical fuzzy context is proposed; (2) an entirely novel spherical fuzzy-based BWM-integrated MULTIMOORA methodology for the EBCS site selection problem is developed; (3) advanced SFSs are incorporated to deal with location-related data that are ambiguous, vague and uncertain; (4) a comprehensive list of service area, technological, economic, environmental, and social criteria from sustainable and smart perspectives is presented; and (5) the proposed methodology is used to tackle the challenge of locating the best EB fast charging stations in Istanbul, Turkiye.

The remaining part of the article is arranged as follows: In Sect. 2, a thorough analysis of the germane literature is presented. The core concepts of fuzzy set theory are briefly outlined and a framework for the combined SF-BWM and SF-MULTIMOORA methodology is suggested in Sect. 3. In Sect. 4, a real-life case study is introduced for validation of the developed approach. The findings of the research and a sensitivity analysis executed to attest that the designed procedure is coherent and robust are depicted in Sect. 5. Ultimately, Sect. 6 provides inferences and recommendations for further study.

2 Literature review

Some criteria can only be measured with fuzzy values, due to the ambiguity and intangibility resulting from experts’ qualitative assessments, as well as the vagueness stemming from information deficiency in the real-world matters. The fuzzy set theory, which Zadeh [14] initially described in 1965, can be employed in problems of this nature. Following the exposition of regular fuzzy sets by Zadeh, over 20 improved variations have emerged in published works, including type-2, Pythagorean, and fermatean fuzzy sets [15]. Aside from these prominent fuzzy systems, Gündoğdu and Kahraman [16] have lately presented the conception of a ‘spherical fuzzy set’ to the academic community. In this sense, SFSs defined by three dimensions are a solid alternative because they can competently alleviate information loss by dealing with the uncertain data inherent in this evolving EBCS site selection problem.

While most of the literature studies have concentrated primarily on the private electric vehicle charging station (EVCS) location selection problem, comparatively few research has addressed EBCSs. The majority of these analyses highlight the development and operation of EBCSs, their technological specifications, cost–benefit analysis, energy management strategies, and environmental concerns. Additionally, numerous preceding studies have addressed the subject of optimal location selection for EBCSs by employing optimization techniques such as mixed integer linear programming [17], multi-objective linear programming [18], bi-objective optimization [19], bi-level programming models [20], two-stage heuristic algorithms [21] and artificial fish swarm algorithms [12].

On the other hand, EBCS site selection from multiple alternatives may be conceived as a complicated MCDM problem, because of the incorporation of numerous contradictory and complex attributes including service area, economic, technological, environmental, and social factors. Guo and Zhao [22] designed an evaluation index system with 11 sub-criteria and rated location possibilities for EVCSs from a sustainability viewpoint using the fuzzy TOPSIS technique. Erbaş et al. [23] merged fuzzy AHP and TOPSIS approaches to figure out the best EVCS sites in Ankara through employing Geographic Information Systems (GIS). Cui et al. [24] presented a Pythagorean fuzzy VIKOR strategy for tackling EVCS site selection problems in the city of Shanghai. Liu et al. [25] suggested an incorporated MCDM procedure exploiting Grey DEMATEL with Uncertain Linguistic MULTIMOORA to determine the ideal EVCS positions in Shanghai. Ju et al. [26] proposed an extensive framework combining AHP and GRP techniques for selecting the best EVCS locations in a picture fuzzy environment from a sustainability standpoint. Lin et al. [27] calculated criteria weights with a method based on knowledge measures and an expanded version of the MULTIMOORA approach was devised to rank the potential places for vehicle sharing stations. Karaşan et al. [28] integrated DEMATEL, AHP and TOPSIS in the intuitionistic fuzzy surrounding for the selection of EVCS sites in Istanbul. Ghosh et al. [29] combined MCDM methods with GIS. The variable weights were established by the fuzzy AHP scheme, while fuzzy TOPSIS and fuzzy COPRAS algorithms were used to rank the chosen locations. Rani and Mishra [30] suggested a method utilizing Fermatean fuzzy MULTIMOORA and the maximizing deviation technique. Mishra et al. [31] established a neutrosophic-based ARAS model to rate and assess sustainable EVCS sites. The attribute weights were computed by a single-valued subjective and objective weighted integrated approach (SVNSOWIA). Liu et al. [32] proposed a three-stage fuzzy MCDM framework for positioning sharing EVCSs in China. The criteria weights were derived by means of the fuzzy BWM and fuzzy EWM with a distance basis, and the alternatives were ranked via the fuzzy GRA. Even though SFSs have been employed to address several distinct site selection problems, there are just a few studies that are pertinent to the location of electric charging stations. Furthermore, Gündoğdu and Kahraman [33] upgraded the standard TOPSIS model under the spherical fuzzy context to pinpoint the ideal places for EVCSs in Izmir, Turkiye. Regarding the selection of optimal EBCS sites by the MCDM framework, Krawiec [34] discussed the PMP for multi-criteria analysis, which used simulation findings as input data. Türk et al. [11] recently suggested an MCDM method enhanced with simulated annealing algorithm exploiting interval type-2 fuzzy sets to identify the ideal placement of electric charging stations for Istanbul’s Bus Rapid Transit (BRT) operations. In the latest study, Sang et al. [35] presented a framework that united the fuzzy DEMATEL, PROMETHEE and PT to analyze potential locations for EBCSs in China. The assessment criteria weights were computed by fuzzy DEMATEL scheme and PROMETHEE and PT techniques were fused to prioritize the options for the selection of the EBCS locations. Table 1 summarizes several MCDM approaches relevant to EBCS site selection.

Table 1 Literature review on some MCDM models associated with EBCS site selection
Full size table

The literature evaluation listed above exhibits that EVCS site selection problems have employed a number of decision-making strategies, whereas problems related to EBCS site selection have just been briefly mentioned. Sang et al. [35] dealt with the problem of choosing the best positions for EBCSs in a non-fuzzy environment. Furthermore, Türk et al. [11], the only research that conducted a case study for EBCSs in Istanbul, considered charging solely at the main stations for BRT operations, which have a smooth route and never stuck in traffic. On the other hand, this research will examine combined overnight and opportunity charging at the intermediate stations for public transport EBs whose remaining charge is unpredictable due to exposure to heavy traffic congestion, passenger load and bumpy road routes and therefore needs to be recharged quickly and frequently.

Information aggregation for the EBCS site selection problems is significantly influenced by the criteria weights. AHP, ANP and DEMATEL, which are the most common criteria weight determination methods, may not allow decision-makers to make consistent choices over a wide range of criteria or alternatives. In this context, the BWM, which follows a structured pairwise assessment scheme that compares merely the best and worst criteria with other criteria, was first presented by Rezaei [36] as a solution to this problem. Likewise, compared to the notable AHP and ANP, the BWM is deemed a more efficient and robust approach utilized in several phases of tackling a variety of MCDM applications. Hence, it has been applied to many fields from green supplier selection to electric vehicle charging station (EVCS) site selection and utilized in various research as an effective weight computation technique due to its benefits of fewer pairwise comparisons, a greater consistency, and a higher capability to represent expert judgments [37]. In addition to these qualities, the BWM has the distinct advantage of integrating with other MCDM techniques quite productively. A literature review on BWM-related techniques with an emphasis on BWM–MULTIMOORA hybrids under diverse fuzzy environments is summarized in Table 2. Accordingly, the BWM has been essentially used to locate sharing EVCSs and has produced acceptable results [32]; however, no previous study has addressed the selection of an EBCS location in a spherical fuzzy surrounding. Hence, this paper seeks to apply a novel SF-BWM approach to analyze the priority weights of the EBCS site selection problem.

Table 2 Literature review on fuzzy BWM-integrated MCDM methodologies
Full size table

The MULTIMOORA approach outperforms AHP, TOPSIS, VIKOR, and TODIM in terms of simplicity of mathematical expressions, speed of calculation, robustness, and stability [30]. In fact, it is among the most reliable and robust methods for multi-objective optimization, giving broad rankings in the decision-making process [25]. Furthermore, it is more methodical than other MCDM techniques as it comprises three distinct subordinate modules: the ratio system (RS), the reference point (RP), and the full multiplicative form (FMF) [52]. The MULTIMOORA model has recently been employed across a variety of fields and fuzzy settings and has provided consistent outcomes. To this end, the SF-MULTIMOORA approach will be utilized to rate the available alternatives and identify the top suited EBCS locations in this research.

3 Proposed hybrid methodology

3.1 Spherical fuzzy set theory

SFSs are an expanded version of three-dimensional (membership, non-membership, and hesitancy) intuitionistic, Pythagorean and picture fuzzy sets that attempt to elicit expert opinions in a more informative and clear manner.

Given that X is a fixed set, \({\upmu }_{\tilde{{\text{F}}}}\left({\text{x}}\right):\mathrm{X }\,\mapsto\, [\mathrm{0,1}]\) and \({{\text{v}}}_{\tilde{{\text{F}}}}\left({\text{x}}\right):\mathrm{X }\,\mapsto\, [\mathrm{0,1}]\) describe the membership and non-membership levels of \({\text{x}}\in {\text{X}}\) in the fuzzy set \(\tilde{{\text{F}}}\), respectively. The hesitancy level (π) may be determined for both intuitionistic and Pythagorean fuzzy sets as follows:

$${\uppi }_{\tilde{{\text{I}}}}\left({\text{x}}\right)= 1-{\upmu }_{\tilde{{\text{I}}}}\left({\text{x}}\right)-{{\text{v}}}_{\tilde{{\text{I}}}}\left({\text{x}}\right)$$
(1)
$${\uppi }_{\tilde{{\text{P}}}}\left({\text{x}}\right)= \sqrt{1-{\upmu }_{\tilde{{\text{P}}}}{\left({\text{x}}\right)}^{2}-{{\text{v}}}_{\tilde{{\text{P}}}}{({\text{x}})}^{2}}$$
(2)

In contrast to intuitionistic and Pythagorean fuzzy sets, SFSs allow for both the squared sum of the parameters representing membership, non-membership, and hesitancy levels and independent specification of each of these parameters to be in the interval of 0 and 1 [16].

Definition 1

Suppose that universe U is a fixed set. A single-valued spherical fuzzy set (SVSFS) belonging to U is expressed as [33]:

$$\tilde{{\text{S}}}\cong \left\{{\text{x}}, {\tilde{{\text{S}}} (\upmu }_{\tilde{{\text{s}}}}\left({\text{x}}\right), {{\text{v}}}_{\tilde{{\text{s}}}}\left({\text{x}}\right), {\uppi }_{\tilde{{\text{s}}}}\left({\text{x}}\right));\mathrm{ x}\in {\text{U}}\right\}$$
(3)

where, respectively, \({\upmu }_{\tilde{{\text{s}}}}\left({\text{x}}\right):\mathrm{X }\mapsto [\mathrm{0,1}]\), \({{\text{v}}}_{\tilde{{\text{s}}}}\left({\text{x}}\right):\mathrm{X }\mapsto [\mathrm{0,1}]\) and \({\uppi }_{\tilde{{\text{s}}}}\left({\text{x}}\right):\mathrm{X }\mapsto [\mathrm{0,1}]\) symbolize the membership, non-membership and hesitancy levels of \({\text{x}}\in {\text{U}}\) to the spherical fuzzy set \(\tilde{{\text{S}}}\).

$$0\le {\upmu }_{\tilde{{\text{s}}}}{\left({\text{x}}\right)}^{2}+{{\text{v}}}_{\tilde{{\text{s}}}}{\left({\text{x}}\right)}^{2}+{\uppi }_{\tilde{{\text{s}}}}{\left({\text{x}}\right)}^{2}\le 1;\mathrm{ x }\in {\text{U}}$$
(4)

Definition 2

Basic operations on two single-valued spherical fuzzy numbers (SVSFNs) \(\tilde{\mathrm{\alpha }}=\tilde{{\text{S}}}\left({\upmu }_{\mathrm{\alpha }}, {\upnu }_{\mathrm{\alpha }}, {\uppi }_{\mathrm{\alpha }}\right)\) and \(\tilde{\upbeta }=\tilde{{\text{S}}}\left({\upmu }_{\upbeta }, {\upnu }_{\upbeta }, {\uppi }_{\upbeta }\right)\) are represented as [16]:

  1. 1.
    $$\tilde{\mathrm{\alpha }}\oplus \tilde{\upbeta }=\tilde{{\text{S}}}\left\{\sqrt{{{\upmu }_{\tilde{\mathrm{\alpha }}}^{2}+{\upmu }_{\tilde{\upbeta }}^{2}-\upmu }_{\tilde{\mathrm{\alpha }}}^{2}{\upmu }_{\tilde{\upbeta }}^{2}}, {{\text{v}}}_{\tilde{\mathrm{\alpha }}}{{\text{v}}}_{\tilde{\upbeta },} \sqrt{\left(1-{\upmu }_{\tilde{\mathrm{\alpha }}}^{2}\right){\uppi }_{\tilde{\upbeta }}^{2}+\left(1-{\upmu }_{\tilde{\upbeta }}^{2}\right){\uppi }_{\tilde{\mathrm{\alpha }}}^{2}-{\uppi }_{\tilde{\mathrm{\alpha }}}^{2}{\uppi }_{\tilde{\upbeta }}^{2}}\right\}$$
    (5)
  2. 2.
    $$\tilde{\mathrm{\alpha }}\otimes \tilde{\upbeta }=\tilde{{\text{S}}}\left\{{\upmu }_{\tilde{\mathrm{\alpha }}}{\upmu }_{\tilde{\upbeta },} \sqrt{{{{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2}+{{\text{v}}}_{\tilde{\upbeta }}^{2}-{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2}{{\text{v}}}_{\tilde{\upbeta }}^{2}} , \sqrt{\left(1-{{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2}\right){\uppi }_{\tilde{\upbeta }}^{2}+\left(1-{{\text{v}}}_{\tilde{\upbeta }}^{2}\right){\uppi }_{\tilde{\mathrm{\alpha }}}^{2}-{\uppi }_{\tilde{\mathrm{\alpha }}}^{2}{\uppi }_{\tilde{\upbeta }}^{2}}\right\}$$
    (6)
  3. 3.
    $$\uplambda \tilde{\mathrm{\alpha }}=\tilde{{\text{S}}} \left\{\sqrt{1-{(1-{\upmu }_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }}, {\mathrm{ v}}_{\tilde{\mathrm{\alpha }}}^{\uplambda }, \sqrt{{(1-{\upmu }_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }-{(1-{\upmu }_{\tilde{\mathrm{\alpha }}}^{2}-{\uppi }_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }}\right\}\quad\mathrm{where}\quad \lambda \ge 0$$
    (7)
  4. 4.
    $${\tilde{\mathrm{\alpha }}}^{\uplambda }=\tilde{{\text{S}}} \left\{{\upmu }_{\tilde{\mathrm{\alpha }}}^{\uplambda }, \sqrt{1-{(1-{{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }}, \sqrt{{(1-{{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }-{(1-{{\text{v}}}_{\tilde{\mathrm{\alpha }}}^{2}-{\uppi }_{\tilde{\mathrm{\alpha }}}^{2})}^{\uplambda }}\right\}\quad\mathrm{where}\quad \lambda \ge 0$$
    (8)

3.2 Integrated SF-BWM and SF-MULTIMOORA methodology

There are two primary phases in the recommended single-valued spherical fuzzy BWM-integrated spherical fuzzy MULTIMOORA approach: i) The BWM uses a spherical fuzzy setting to specify the main and sub-criteria weights. ii) The optimal candidate EBCS sites are ranked via inserting the obtained criteria weights in the MULTIMOORA method.

Let\({{\text{C}}}_{{\text{j}}}=\left\{{{\text{C}}}_{1}, {{\text{C}}}_{2}, ..., {{\text{C}}}_{{\text{n}}}\mid \mathrm{ j}=1, ...,\mathrm{ n \,and \,n}\ge 2\right\}\), \({{\text{A}}}_{{\text{i}}}=\left\{{{\text{A}}}_{1}, {{\text{A}}}_{2}, ..., {{\text{A}}}_{{\text{m}}} \mid \mathrm{ i}=1, ...,\mathrm{ m \,and \,m}\ge 2\right\}\) and \({{\text{E}}}_{{\text{k}}}=\left\{{{\text{E}}}_{1}, {{\text{E}}}_{2}, ..., {{\text{E}}}_{{\text{p}}} \mid \mathrm{ k}=1, ...,\mathrm{ p \,and \,p}\ge 2\right\}\) represent a finite criteria set, an m feasible alternatives set and a finite experts set, respectively. Additionally, let \({{\text{W}}}_{{\text{j}}}=\left\{{{\text{W}}}_{1}, {{\text{W}}}_{2}, ..., {{\text{W}}}_{{\text{n}}} \mid \mathrm{ j}=1, ...,\mathrm{ n}\right\}\) be a criteria weight vector satisfying \(\sum_{{\text{j}}=1}^{{\text{n}}}{{\text{W}}}_{{\text{j}}}=1\) and \({{\text{W}}}_{{\text{j}}}\ge 0\mathrm{ \,as \,well \,as \,}{\Theta }_{{\text{k}}}=\left\{{\Theta }_{1}, {\Theta }_{2}, ..., {\Theta }_{{\text{p}}} \mid \mathrm{ k}=1, ...,\mathrm{ p}\right\}\) be an experts weight vector satisfying \(\sum_{{\text{k}}=1}^{{\text{p}}}{\Theta }_{{\text{k}}}=1\mathrm{ \,and \,}{\Theta }_{{\text{k}}}\ge 0\), correspondingly. Figure 1 depicts the stages of the formulated novel methodology [36, 62].

Fig. 1
figure 1

Flowchart for the proposed BWM-integrated MULTIMOORA methodology

Full size image

3.2.1 Phase 1: assessment

Step 1. Alternatives and relevant assessment criteria are established by constructing a four-level hierarchical structure.

3.2.2 Phase 2: SF-BWM

Step 2. The best or most significant criterion \({{\text{C}}}_{{\text{B}}}\) and the worst or least significant criterion \({{\text{C}}}_{{\text{W}}}\) are chosen by the expert panel using the Delphi method.

Step 3. The spherical fuzzy best-to-others vector \({\tilde{{\text{V}}}}_{{\text{Bk}}}\) is derived for each expert through comparing the best criterion with other criteria. \({\tilde{{\text{V}}}}_{{\text{Bk}}}=\left({\tilde{{\text{v}}}}_{{\text{B}}1{\text{k}}}, {\tilde{{\text{v}}}}_{{\text{B}}2{\text{k}}},\dots ,{\tilde{{\text{v}}}}_{{\text{Bnk}}}\right)\) where \({\tilde{{\text{v}}}}_{{\text{Bjk}}}=({\mu}_{{\text{Bjk}}}, {\upnu }_{{\text{Bjk}}}, {\uppi }_{{\text{Bjk}}})\) denotes the significance level of the best criterion \({{\text{C}}}_{{\text{B}}}\) in regard to criteria \({{\text{C}}}_{{\text{j}}}\) by expert \({{\text{E}}}_{{\text{k}}}\) and \({\tilde{{\text{v}}}}_{{\text{BBk}}}=1\).

Step 4. The spherical fuzzy others-to-worst vector \({\tilde{{\text{V}}}}_{{\text{Wk}}}\) is generated for each expert via comparing other criteria with the worst criterion. \({\tilde{{\text{V}}}}_{{\text{Wk}}} ={\left({\tilde{{\text{v}}}}_{1{\text{Wk}}},{\tilde{{\text{v}}}}_{2{\text{Wk}}} ,\dots ,{\tilde{{\text{v}}}}_{{\text{nWk}}}\right)}^{{\text{T}}}\) where \({\tilde{{\text{v}}}}_{{\text{jWk}}}=({\mu}_{{\text{jWk}}}, {\upnu }_{{\text{jWk}}}, {\uppi }_{{\text{jWk}}})\) represents the significance level of the criteria \({{\text{C}}}_{{\text{j}}}\) in accordance with the worst criterion \({{\text{C}}}_{{\text{W}}}\) by expert \({{\text{E}}}_{{\text{k}}}\) and\({\tilde{{\text{v}}}}_{{\text{WWk}}}=1\).

Step 5. The individual criteria weight vectors \({\tilde{{\text{V}}}}_{{\text{Bk}}}\) and \({\tilde{{\text{V}}}}_{{\text{Wk}}}\) are defuzzified to derive defuzzified criteria weight vectors \({{\text{V}}}_{{\text{Bk}}}= \left({{\text{v}}}_{{\text{B}}1{\text{k}}}, {{\text{v}}}_{{\text{B}}2{\text{k}}},\dots ,{{\text{v}}}_{{\text{Bnk}}}\right)\) and\({{\text{V}}}_{{\text{Wk}}}={\left({{\text{v}}}_{1{\text{Wk}}},{{\text{v}}}_{2{\text{Wk}}} ,\dots ,{{\text{v}}}_{{\text{nWk}}}\right)}^{{\text{T}}}\). Score values \({{\text{v}}}_{{\text{Bjk}}}\) and \({{\text{v}}}_{{\text{jWk}}}\) are computed utilizing Eq. (9).

$${{\text{v}}}_{{\text{Bjk}}}=\mathrm{Score }\left({\tilde{{\text{v}}}}_{{\text{Bjk}}}\right)=\tilde{{\text{S}}}\left\{\frac{10 {\mu}_{{\tilde{{\text{v}}}}_{{\text{Bjk}}}}\left(1-{\upnu }_{{\tilde{{\text{v}}}}_{{\text{Bjk}}}}\right)\left(1-{\uppi }_{{\tilde{{\text{v}}}}_{{\text{Bjk}}}}\right)}{\mathrm{1,5}}\right\}$$
(9)

Step 6. The optimal local criteria weights \({{{\text{w}}}_{{\text{jk}}}}^{*}=\left({{\text{w}}}_{1{\text{k}}}^{*},{{\text{w}}}_{2{\text{k}}}^{*},\dots ,{{\text{w}}}_{{\text{nk}}}^{*}\right)\) and the optimal objective function values \({\upzeta }_{{\text{k}}}^{*}\) are obtained for both main and sub-criteria by solving the following nonlinear mathematical model of the BWM for each expert separately with corresponding \({{\text{V}}}_{{\text{Bk}}}\) and \({{\text{V}}}_{{\text{Wk}}}\) vectors using Eqs. (10–14):

$$\mathrm{min }\,\,{\upzeta }_{{\text{k}}}$$
(10)
$$\left|\frac{{{\text{w}}}_{{\text{Bk}}}}{{{\text{w}}}_{{\text{jk}}}}-{{\text{v}}}_{{\text{Bjk}}}\right|\le {\upzeta }_{{\text{k}}}\quad \forall {\text{j}}$$
(11)
$$\left|\frac{{{\text{w}}}_{{\text{jk}}}}{{{\text{w}}}_{{\text{Wk}}}}-{{\text{v}}}_{{\text{jWk}}}\right|\le {\upzeta }_{{\text{k}}} \quad\forall {\text{j}}$$
(12)
$$\sum_{{\text{j}}=1}^{{\text{n}}}{{\text{w}}}_{{\text{jk}}}=1$$
(13)
$${{\text{w}}}_{{\text{jk}}}\ge 0\quad \forall {\text{j}}$$
(14)

here \({{\text{w}}}_{{\text{Bk}}}\) and \({{\text{w}}}_{{\text{Wk}}}\) signify weights of the best and the worst criteria by expert \({{\text{E}}}_{{\text{k}}}\), respectively.

Step 7. The consistency of the defuzzified criteria weight vectors \({{\text{V}}}_{{\text{Bk}}}\) and \({{\text{V}}}_{{\text{Wk}}}\) are checked for all experts.

Step 7.a. The highest value of \({\upzeta }_{{\text{k}}} ({\text{max}}{\upzeta }_{{\text{k}}}\)) is estimated by solving Eq. (15) for various values of \({{\text{v}}}_{{\text{BWk}}}\) as displayed in Table 5.

$${{\upzeta }_{{\text{k}}}}^{2}- \left(1+2{\mathrm{ v}}_{{\text{BWk}}}\right) {\upzeta }_{{\text{k}}}+\left({{{\text{v}}}_{{\text{BWk}}}}^{2}- {{\text{v}}}_{{\text{BWk}}}\right)=0$$
(15)

in which \({{\text{v}}}_{{\text{BWk}}}\) denotes the significance level of the best criterion \({{\text{C}}}_{{\text{B}}}\) in relation to the worst criterion \({{\text{C}}}_{{\text{W}}}\) by expert \({{\text{E}}}_{{\text{k}}}\).

Step 7.b. The consistency of the individual criteria weight vectors is checked using Eq. (16).

$$\text{Consistency}\, \text{ratio }({{\text{CR}}}_{{\text{k}}})=\frac{{\upzeta }_{{\text{k}}}^{*}}{\mathrm{Consistency\, Index\, }({{\text{CI}}}_{{\text{k}}})}$$
(16)

where \({{\text{CI}}}_{{\text{k}}}={\text{max}}{\upzeta }_{{\text{k}}}\). \({{\text{CR}}}_{{\text{k}}}\in [\mathrm{0,1}]\) and for consistency checks, \({{\text{CR}}}_{{\text{k}}}<0.1\) is usually acceptable. The \({{\text{CR}}}_{{\text{k}}}\) decreases as the \({\upzeta }_{{\text{k}}}^{*}\) value approaches 0, increasing the reliability and consistency of the comparisons.

Step 8. The aggregated local criteria weights \({{{\text{w}}}_{{\text{j}}}}^{*}= \left({{\text{w}}}_{1}^{*},{{\text{w}}}_{2}^{*}, \dots ,{{\text{w}}}_{{\text{n}}}^{*}\right)\) of main and sub-criteria are computed using Eq. (17) with the criteria values obtained for each expert from BWM optimization.

$${{{\text{w}}}_{{\text{j}}}}^{*}=\frac{\sum_{{\text{j}}=1}^{{\text{n}}}\sum_{{\text{k}}=1}^{{\text{p}}}{{\text{w}}}_{{\text{jk}}}}{{\text{p}}}$$
(17)

Step 9. Aggregated global criteria weights \({{{\text{W}}}_{{\text{j}}}}^{*}=\left({{\text{W}}}_{1}^{*},{{\text{W}}}_{2}^{*}, \dots ,{{\text{W}}}_{{\text{n}}}^{*}\right)\) are calculated for all criteria by multiplying each aggregated local sub-criteria weight with the associated aggregated local main criteria weight.

3.2.3 Phase 3: SF-MULTIMOORA

Step 10. The individual spherical fuzzy evaluation matrix \({\tilde{{\text{M}}}}_{{\text{k}}}\) is created for each expert by assigning spherical fuzzy importance weights via Eq. (18).

$${\tilde{{\text{M}}}}_{{\text{k}}}={\left({\tilde{{\text{X}}}}_{{\text{ijk}}}\right)}_{{\text{mxn}}}=\begin{array}{c}\begin{array}{ccc}{{\text{C}}}_{1} & \dots & {{\text{C}}}_{{\text{n}}}\end{array}\\ \begin{array}{c}{{\text{A}}}_{1}\\ \vdots \\ {{\text{A}}}_{{\text{m}}}\end{array}{\left( \begin{array}{ccc}{\tilde{{\text{x}}}}_{11{\text{k}}}& \dots & {\tilde{{\text{x}}}}_{1{\text{nk}}}\\ \vdots & \ddots & \vdots \\ {\tilde{{\text{x}}}}_{{\text{m}}1{\text{k}}}& \dots & {\tilde{{\text{x}}}}_{{\text{mnk}}}\end{array} \right)}_{{\text{mxn}}}\end{array}$$
(18)

here \({\tilde{{\text{x}}}}_{{\text{ijk}}}=({\mu}_{{\text{ijk}}}, {\upnu }_{{\text{ijk}}}, {\uppi }_{{\text{ijk}}})\) is a SFN that represents the significance level of the alternative \({{\text{A}}}_{{\text{i}}}\) in regard to sub-criterion \({{\text{C}}}_{{\text{j}}}\) by expert \({{\text{E}}}_{{\text{k}}}\).

Step 11. The spherical weighted arithmetic mean (SWAM) operator is employed to aggregate individual spherical fuzzy evaluation matrices using Eq. (19) and the aggregated evaluation matrix \(\tilde{{\text{M}}}\) is derived via Eq. (20).

$$ {\text{SWAMw}}\left( {{{\tilde{x}}}_{{{\text{ij}}1}} , \ldots ,{{\tilde{x}}}_{{{\text{ijp}}}} } \right) = {{\tilde{\text{S}}}}\left\{ {\begin{array}{*{20}c} {\sqrt {1 - \mathop \prod \limits_{{{\text{k}} = 1}}^{{\text{p}}} \left( {1 - \mu _{{{{\tilde{x}}}_{{{\text{ijk}}}} }}^{2} } \right)^{{{{\Theta }}_{{\text{k}}} }} } ,} \\ {\mathop \prod \limits_{{{\text{k}} = 1}}^{{\text{p}}} {{\nu }}_{{{{\tilde{x}}}_{{{\text{ijk}}}} }}^{{{{\Theta }}_{{\text{k}}} }} \sqrt {\mathop \prod \limits_{{{\text{k}} = 1}}^{{\text{p}}} \left( {1 - \mu _{{{{\tilde{x}}}_{{{\text{ijk}}}} }}^{2} } \right)^{{{{\Theta }}_{{\text{k}}} }} - \mathop \prod \limits_{{{\text{k}} = 1}}^{{\text{p}}} \left( {1 - \mu _{{{{\tilde{x}}}_{{{\text{ijk}}}} }}^{2} - {{\pi }}_{{{{\tilde{x}}}_{{{\text{ijk}}}} }}^{2} } \right)^{{{{\Theta }}_{{\text{k}}} }} } } \\ \end{array} } \right\} $$
(19)
$$\tilde{{\text{M}}}={\left({\tilde{{\text{X}}}}_{{\text{ij}}}\right)}_{{\text{mxn}}}=\begin{array}{c}\begin{array}{ccc}{{\text{C}}}_{1} & \dots & {{\text{C}}}_{{\text{n}}}\end{array}\\ \begin{array}{c}{{\text{A}}}_{1}\\ \vdots \\ {{\text{A}}}_{{\text{m}}}\end{array}{\left( \begin{array}{ccc}{\tilde{{\text{x}}}}_{11}& \dots & {\tilde{{\text{x}}}}_{1{\text{n}}}\\ \vdots & \ddots & \vdots \\ {\tilde{{\text{x}}}}_{{\text{m}}1}& \dots & {\tilde{{\text{x}}}}_{{\text{mn}}}\end{array} \right)}_{{\text{mxn}}}\end{array}$$
(20)

in which \({\tilde{{\text{x}}}}_{{\text{ij}}}=({\mu}_{{\text{ij}}}, {\upnu }_{{\text{ij}}}, {\uppi }_{{\text{ij}}})\) is a SFN that signifies the aggregated evaluation of the alternative \({{\text{A}}}_{{\text{i}}}\) in accordance with the sub-criterion \({{\text{C}}}_{{\text{j}}}\).

Step 12. Scalar multiplication is operated in Eq. (21) and the aggregated weighted spherical fuzzy evaluation matrix \({\tilde{{\text{M}}}}_{{{\text{W}}}^{*}}\) is determined via Eq. (22).

$${\tilde{{\text{X}}}}_{{{\text{ijW}}}^{*}}={\mathrm{ W}}_{{\text{j}}}^{* } {\tilde{{\text{X}}}}_{{\text{ij}}}=\tilde{{\text{S}}}\left\{\sqrt{1-{(1-{\upmu }_{{\tilde{{\text{x}}}}_{{\text{ij}}}}^{2})}^{{{\text{W}}}_{{\text{j}}}^{*}}}, {\upnu }_{{\tilde{{\text{x}}}}_{{\text{ij}}}}^{{{\text{W}}}_{{\text{j}}}^{*}}, \sqrt{{(1-{\upmu }_{{\tilde{{\text{x}}}}_{{\text{ij}}}}^{2})}^{{{\text{W}}}_{{\text{j}}}^{*}}-{(1-{\upmu }_{{\tilde{{\text{x}}}}_{{\text{ij}}}}^{2}-{\uppi }_{{\tilde{{\text{x}}}}_{{\text{ij}}}}^{2})}^{{{\text{W}}}_{{\text{j}}}^{*}}}\right\}$$
(21)
$${{\tilde{{\text{M}}}}_{{{\text{W}}}^{*}}=\left({\tilde{{\text{X}}}}_{{{\text{ijW}}}^{*}}\right)}_{{\text{mxn}}}=\begin{array}{c}\begin{array}{ccc}{{\text{C}}}_{1} & \dots & {{\text{C}}}_{{\text{n}}}\end{array}\\ \begin{array}{c}{{\text{A}}}_{1}\\ \vdots \\ {{\text{A}}}_{{\text{m}}}\end{array}{\left(\begin{array}{ccc}{\tilde{{\text{x}}}}_{11{{\text{W}}}_{1}^{*}}& \dots & {\tilde{{\text{x}}}}_{1{{\text{nW}}}_{{\text{n}}}^{*}}\\ \vdots & \ddots & \vdots \\ {\tilde{\mathrm{ x}}}_{{\text{m}}1{{\text{W}}}_{1}^{*}}& \dots & {\tilde{{\text{x}}}}_{{{\text{mnW}}}_{{\text{n}}}^{*}}\end{array} \right)}_{{\text{mxn}}}\end{array}$$
(22)

where \({\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}=({\mu}_{{{\text{ijW}}}^{*}}, {\upnu }_{{{\text{ijW}}}^{*}}, {\uppi }_{{{\text{ijW}}}^{*}})\) is a SFN that denotes the aggregated weighted evaluation of the alternative \({{\text{A}}}_{{\text{i}}}\) according to the sub-criterion \({{\text{C}}}_{{\text{j}}}\).

Step 13. The ratio system method

Step 13a. The relative significance \(({\tilde{{\text{U}}}}_{{\text{i}}})\) of the alternatives is figured with the SWAM operator through Eq. (23).

$${\tilde{{\text{U}}}}_{{\text{i}}}=\tilde{{\text{S}}} \left\{\sqrt{1-\prod_{{\text{j}}=1}^{{\text{n}}}{\left(1-{{\mu}_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}}^{2}\right)}^{\frac{1}{{\text{n}}}}, } \prod_{{\text{j}}=1}^{{\text{n}}}{{\upnu }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}}^{\frac{1}{{\text{n}}}}, \sqrt{\prod_{{\text{j}}=1}^{{\text{n}}}{\left(1-{{\mu}_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}}^{2}\right)}^{\frac{1}{{\text{n}}}}-\prod_{{\text{j}}=1}^{{\text{n}}}{\left(1-{{\mu}_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}}^{2}-{{\uppi }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}}^{2}\right)}^{\frac{1}{{\text{n}}}}}\right\}$$
(23)

here n denotes the overall sub-criteria number.

Step 13b. \({\tilde{{\text{U}}}}_{{\text{i}}}\) values are defuzzified to derive \({{\text{u}}}_{{\text{i}}}\) score degrees by applying the score function in Eq. (24).

$${{\text{u}}}_{{\text{i}}}=\mathrm{Score }\left({\tilde{{\text{U}}}}_{{\text{i}}}\right)=\tilde{{\text{S}}} \left\{{\left(3{\mu}_{{\tilde{{\text{U}}}}_{{\text{i}}}}-\frac{{\uppi }_{{\tilde{{\text{U}}}}_{{\text{i}}}}}{2}\right)}^{2}-{\left({\upnu }_{{\tilde{{\text{U}}}}_{{\text{i}}}}-\frac{{\uppi }_{{\tilde{{\text{U}}}}_{{\text{i}}}}}{2}\right)}^{2}\right\}$$
(24)

Step 13c.The alternatives are arranged by their \({{\text{u}}}_{{\text{i}}}\) value in which the alternative with the greatest \({{\text{u}}}_{{\text{i}}}\) degree is deemed to be the finest.

Step 14. The reference point method

Step 14a. The reference points \({(\tilde{{\text{X}}}}_{{\text{j}}}^{*})\) are defined utilizing Eq. (25).

$${\tilde{{\text{X}}}}_{{\text{j}}}^{*}=\tilde{{\text{S}}}\left\{{{\text{C}}}_{{\text{j}}}\left(\underset{{\text{imax}}}{\mu}_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}},\underset{{\text{imin}}}{\upnu }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}, \underset{{\text{i min}}}{\uppi }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}\right)<{\tilde{{\text{M}}}}_{{{\text{W}}}^{*}}={\left({\tilde{{\text{X}}}}_{{{\text{ijW}}}^{*}}\right)}_{{\text{mxn}}}>\right\}$$
(25)

Step 14b. The distance between every alternative and reference point is computed by Eq. (26).

$$ {{d}}\left( {{{\tilde{X}}}_{{{{ijW}}^{{{*}}} }} {{~\tilde{X}}}_{{{j}}}^{{{*}}} } \right) = {{\tilde{S}}}\left\{ {{{~}}\frac{1}{2}\left( {{{~}}\left| {\mu _{{{{\tilde{x}}}_{{{{ijW}}^{{{*}}} }} }}^{2} - \mu _{{{{\tilde{x}}}_{{{{j}}^{{{*}}} }} }}^{2} } \right| + \left| {{{\nu }}_{{{{\tilde{x}}}_{{{{ijW}}^{{{*}}} }} }}^{2} - {{\nu }}_{{{{\tilde{x}}}_{{{{j}}^{{{*}}} }} }}^{2} } \right| + \left| {{{\pi }}_{{{{\tilde{x}}}_{{{{ijW}}^{{{*}}} }} }}^{2} - {{\pi }}_{{{{\tilde{x}}}_{{{{j}}^{{{*}}} }} }}^{2} } \right|} \right)} \right\} $$
(26)

Step 14c. The deviation values \({{{y}}}_{{{i}}}\) for each alternative are determined with Eq. (27).

$$ y_{i} = \frac{{\sum\nolimits_{{j = 1}}^{n} {\text{d}} \left( {\tilde{X}_{{{\text{ijW}}^{*} }} ,\tilde{X}_{j}^{*} } \right)}}{n} $$
(27)

Step 14d. The alternatives are ranked by their \({{\text{y}}}_{{\text{i}}}\) value where the alternative with the lowest \({{\text{y}}}_{{\text{i}}}\) ratio is regarded to be the highest rated.

Step 15. The full multiplicative form method

Step 15a. The \({\tilde{{\text{Z}}}}_{{\text{i}}}\) ratio values are calculated with the spherical weighted geometric mean (SWGM) operator using Eq. (28).

$${\tilde{{\text{Z}}}}_{{\text{i}}}=\tilde{{\text{S}}}\left\{{\prod }_{{\text{j}}=1}^{{\text{n}}} {\mu}_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}^{\frac{1}{{\text{n}}}}, \sqrt{1-{\prod }_{{\text{j}}=1}^{{\text{n}}}{\left(1-{\upnu }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}^{2}\right)}^{\frac{1}{{\text{n}}}}}, \sqrt{{\prod }_{{\text{j}}=1}^{{\text{n}}}{\left(1-{\upnu }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}^{2}\right)}^{\frac{1}{{\text{n}}}}-{\prod }_{{\text{j}}=1}^{{\text{n}}}{\left(1-{\upnu }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}^{2}-{\uppi }_{{\tilde{{\text{x}}}}_{{{\text{ijW}}}^{*}}}^{2}\right)}^{\frac{1}{{\text{n}}}}}\right\}$$
(28)

where n represents the total sub-criteria number.

Step 15b. \({\tilde{{\text{Z}}}}_{{\text{i}}}\) ratios are defuzzified to estimate score values \({{\text{z}}}_{{\text{i}}}\) via Eq. (29).

$${{\text{z}}}_{{\text{i}}}=\mathrm{Score }\left({\tilde{{\text{Z}}}}_{{\text{i}}}\right)=\tilde{{\text{S}}}\left\{{\left(3{\mu}_{{\tilde{{\text{Z}}}}_{{\text{i}}}}-\frac{{\uppi }_{{\tilde{{\text{Z}}}}_{{\text{i}}}}}{2}\right)}^{2}-{\left({\upnu }_{{\tilde{{\text{Z}}}}_{{\text{i}}}}-\frac{{\uppi }_{{\tilde{{\text{Z}}}}_{{\text{i}}}}}{2}\right)}^{2}\right\}$$
(29)

Step 15c. Alternatives are prioritized based on their \({{\text{z}}}_{{\text{i}}}\) values where the alternative with the maximum \({{\text{z}}}_{{\text{i}}}\) score is regarded to be the finest.

Step 16. The final ranking is performed by merging these subordinate rankings obtained from all three modules by utilizing dominance theory (DT).

4 Case study

4.1 Problem statement

Istanbul serves as Turkey's primary commercial and cultural hub, with a population of approximately 16 million today. It features a well-connected urban mobility network that includes mainly public buses, BRT, light rail, metro, tram, and ferry. The primary public bus operator, the Istanbul Electricity Tramway Tunnel (IETT), currently operates a total of 3212 public buses, with the existing IETT bus fleet consisting primarily of diesel buses (79%) and CNG buses (21%). However, as part of its mission to pioneer sustainable and smart bus services in Turkiye, IETT intends to upgrade its fleet, progressively switching out diesel buses for electric buses [63].

Congested traffic, air conditioner use, a steep route, and an excessive number of passengers on board the buses are the major factors that reduce range by increasing the charge consumption in EBs. Particularly in a metropolitan area such as Istanbul, where virtually all these elements exist, for EBs to overcome the range issue and fulfill their daily operations uninterruptedly, they must swiftly recharge while waiting for their departure at the beginning or end of the line. Following consultation with IETT authorities and electric bus manufacturers, the combined overnight and opportunity charging idea is the most effective option in Istanbul's particular circumstances. In addition, since relatively smaller batteries can be used with this charging strategy, it allows both more space for passengers on the bus and less energy consumption by making the bus lighter. For this purpose, if solo EBs with a battery capacity of 300 kWh are added to the IETT fleet and charged from a 75 kWh power cabinet, it will be fully charged in almost 4 h and will have a range of approximately 180 km under Istanbul conditions. In this respect, it is critical that EB fast charging facilities are initially deployed in moderately small existing bus terminals in areas close to the city center of Istanbul. However, in the replacement process of a bus fleet, choosing the optimal bus stations for EB fast charging is a challenging task as it entails multiple interrelated criteria from a sustainable and smart view. In the sections below, we discuss in detail the criteria considered in our method and alternative EBCSs based on actual data with the purpose of resolving this major issue.

4.2 Data collection for evaluation criteria

Regarding the issue of assessing and selecting the optimal EBCS positions, a four-level hierarchical structure consisting of five main and 22 qualitative and quantitative sub-criteria was developed using existing literature, IETT expert input, and feasibility research reports from EBCS installers. In addition, six stations with the most comprehensive service capability out of the 28 small garages in operation on the European side of Istanbul were selected as suitable candidate sites for EB fast charging stations. The alternative locations are shown in Fig. 2 and more information about these prospective sites is given in Table 3.

Fig. 2
figure 2

Geographical locations of the alternative EBCSs [64]

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Table 3 Information about alternative EBCS locations [63]
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The candidate locations were appraised by a team of seven experts with extensive knowledge and expertise in the field for deploying fast charging stations as displayed in Table 4.

Table 4 Information about the expert panel
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4.3 Evaluation criteria for EBCS site selection

Sustainable and smart charging stations focus more on operational efficacy and service quality compared to conventional electric charging stations. In this study, the overall efficiency and financial sustainability of charging stations were taken into account while creating an evaluation index system that included the main criteria of service area, technological, economic, environmental, and social as outlined in the hierarchical structure in Fig. 3.

Fig. 3
figure 3

Hierarchical design of EB charging station site selection

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4.3.1 Service area criteria

(C11) Garage space availability: This highlights the significance of having sufficient garage capacity to host EBs and charging stations [11].

(C12) Comprehensive service capability: This emphasizes the maximum quantity of charge an EBCS can deliver daily with regard to the total number of EBs and bus routes.

(C13) Capacity expansion possibility: In the event that the electrified bus fleet expands in the future, the existing garage must be flexible and capable of accommodating additional space [11].

(C14) Traffic convenience of site accessibility: This usually refers to locating EBCSs close to main arteries and easily accessible points, to maximize the number of EBs using the service [35].

(C15) Distance to traffic congestion: Service zones with reduced traffic jams are much more desirable because traffic congestion increases the amount of electricity consumed by EBs, causing range problems [11].

(C16) Sloping topographical features: A steep land slope has a detrimental influence on both construction cost and operation feasibility as higher slopes cause EBs to consume more electricity, triggering range issues [11, 65].

4.3.2 Technological criteria

(C21) Power substation capacity (Power grid load): This specifies the maximum power and charge load that the charging station is capable of handling. Once an EBCS is installed in a specific area, it may be essential to replace the existing electrical grid infrastructure, as this may jeopardize the safe functioning of the power grid [35].

(C22) Proximity to power substation (Closeness to power grid): The EBCS should be located near enough to the energy grid to provide enough input of energy and minimize power loss and degradation [11].

(C23) Power supply reliability: This refers to the capacity of the power distribution grid to offer a consistent and continuous power supply to the EBCSs. To achieve this, EBCSs ought to be positioned away from high energy consumption lines and regular power outages to ensure the energy stability of the network, as the simultaneous usage of multiple charging units can degrade the power stability and quality of these stations [11].

(C24) Accessibility of renewable energy resources: Providing access to renewable energy resources such as wind turbines, solar panels, hydraulic or biofuel is crucial to maintaining a sustainable power supply in EBCSs [11].

(C25) Compatibility with different charging strategies: This expresses that the EBCS is appropriate for both fast charging of EBs throughout the daytime while waiting for their departure and slow charging during night parking when the substation's capacity is adequate to power all vehicles simultaneously.

(C26) Adaptability to smart charging technologies: Smart charging works in tandem with operational strategies to efficiently allocate energy for charging based on operational priorities and scheduling, ensuring that every EB is completely charged for its subsequent route. That is, smart charging systems allow the charging of buses in a large fleet with limited capacity to be prioritized according to the criticality of their charging status and time of departure. In that way, certain buses may postpone the start of full charge to reduce the power consumption peak which often happens when several buses enter the EBCS at once with smart charging.

(C27) Power system security: This states the capability of the EBCSs to withstand an emergency such as network security, fire prevention and resistance to natural catastrophes. The EBCSs must also have a solid security system including security guards and CCTV cameras to be safeguarded against theft, vandalism, and other forms of threat [11].

4.3.3 Economic criteria

(C31) Total project investment cost: This includes construction, installation and equipment purchase costs. Land purchase and site leveling fees are not included as these garages belong to the municipality [11, 35].

(C32) Annual operating & maintenance cost: Operating costs chiefly consist of labor costs, power loss costs throughout the charging station operation, charging fees, and battery amortization costs. Maintenance expenses include equipment repair and routine maintenance [11, 35].

(C33) Investment pay-back duration: This signifies the amount of time it takes for the total investment cost of the electric charging station to be recouped via total revenue. The total profits of an EBCS may be assessed by utilizing savings from using electricity instead of fuel and passenger revenues from those EBs.

4.3.4 Environmental criteria

(C41) Emissions and noise reduction: EBCSs should be positioned particularly in locations with considerable quantity of releases as lowering GHG (CO2 and CH4) and fine particle (PM10, PM2.5) emissions as well as noise is critical to urban sustainability.

(C42) Waste disposal: This tracks construction waste, sewage released during EBCS building, and wastewater discharged from vehicle cleaning plus battery disposal throughout EBCS service [25].

4.3.5 Social criteria

(C51) Integration capacity with urban power grid design: An EBCS should be proximate to the power load while also taking into account the existing position of the power network, policies for urban design and development, local population, and the surrounding natural environment [35].

(C52) Integration capacity with urban road network design: This refers to the complete compatibility of EBCSs with the urban transportation system such as the city's highways, main artery intersections, and urban primary operational regions [35].

(C53) Impact on life quality in service area: This alludes to the detrimental impacts of radiation and electromagnetic fields on the surrounding population that are brought on by EBCS construction and operation activities [35].

(C54) Promotion of EBs potential: Continuous improvements to EB charging infrastructure can help to foster accelerated growth in the emerging energy bus sector by driving significant demand for EBs.

5 Results and discussions

In this part, empirical work from Istanbul was implemented to demonstrate the practicality and competence of the formulated EBCS site selection strategy centered around a group decision-making framework. All calculations necessary for both criteria weight calculations including solver tool for BWM optimization and alternative ranking were done in Microsoft Office Excel.

5.1 Obtaining criteria weights by SF-BWM

First, the best and worst criteria were chosen via experts with equal decision-making capacity. Experts reviewed the potential alternatives in line with the performance of the criteria to create individual best-to-others and others-to-worst criteria weight vectors and evaluation matrices using the linguistic expressions for SFNs. The scale for SF-BWM linguistic terms with the possible values that \({{\text{v}}}_{{\text{BWk}}}\) can take and their corresponding.

\({{\text{CI}}}_{{\text{k}}} ({\text{max}}{\upzeta }_{{\text{k}}})\) are displayed in Table 5.

Table 5 Scale for SF-BWM linguistic phrases
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Linguistic importance assessments of best-to-others \({(\tilde{{\text{V}}}}_{{\text{Bk}}})\) and others-to-worst \({(\tilde{{\text{V}}}}_{{\text{Wk}}})\) criteria weight vectors by the expert committee on the main criteria are depicted in Tables 6 and 7.

Table 6 Spherical fuzzy best-to-others vector regarding main criteria
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Table 7 Spherical fuzzy others-to-worst vector regarding main criteria
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Aggregated local criteria weights of the main criteria (\({{{\text{w}}}_{{\text{jk}}}}^{*})\) values and the corresponding optimal \(\upzeta \) \({(\upzeta }_{{\text{k}}}^{*})\), consistency index \({({\text{CI}}}_{{\text{k}}})\) and consistency ratio \({({\text{CR}}}_{{\text{k}}})\) values derived from BWM optimization are exhibited in Table 8. According to the table, the \({{\text{CR}}}_{{\text{k}}}\) values of every expert are less than the threshold rate of 0.1, implying reliable results. It has also been determined that all \({{\text{CR}}}_{{\text{k}}}\) values of expert assessments for the service area, technology, economic, environmental, and social criteria are below 0.1.

Table 8 Aggregated local weights of the main criteria (Excel SOLVER)
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Aggregated local \({{({\text{w}}}_{{\text{j}}}}^{*})\) and global \({{({\text{W}}}_{{\text{j}}}}^{*})\) criteria weights were created by the BWM model utilizing the defuzzified best-to-others \({({\text{V}}}_{{\text{Bk}}})\) and others-to-worst \({({\text{V}}}_{{\text{Wk}}})\) criteria weight vectors, as exposed in Table 9. Accordingly, the three most significant criteria are "(C41) Emissions and noise reductions”, followed by “(C21) Power substation capacity” and “(C31) Total project investment cost”.

Table 9 Aggregated local and global criteria weights in the EBCS site selection problem
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5.2 Ranking of alternatives by SF-MULTIMOORA

The ranking of potential locations in accordance with the criteria was established by the expert committee and the aggregated weighted SF-evaluation matrix is displayed in Table 10. This table describes the ultimate phase before rating the alternatives by the SF-MULTIMOORA method, indicating the significance of fuzzy values, i.e., which alternative is the best based on the assessment criteria. For instance, when evaluating the criterion C41, alternative A6 is the best while alternative A3 is the worst in terms of spherical fuzzy values. The final rankings are then determined by applying Eqs. (23–29) as exhibited in Table 11.

Table 10 Aggregated weighted SF-evaluation matrix \(({\tilde{{\text{M}}}}_{{{\text{W}}}^{*}})\)
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Table 11 The final alternatives ranking based on the SF-MULTIMOORA framework
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The ranking of the prospective EBCS positions according to the RS, RP and FMF techniques was performed, and the ultimate preference ranking was conducted via DT as depicted in Table 11 where the final ordering was A6 ≻ A2 ≻ A3 ≻ A5 ≻ A4 ≻ A1. Hence, “(A6) Topkapi” is the best alternative, followed by “(A2) Sultangazi” while “(A1) Atakent” is the worst candidate for situating the EB fast charging station in Istanbul, based upon the presented spherical fuzzy BWM-weighted MULTIMOORA model.

5.3 Sensitivity analysis

A sensitivity analysis was executed to find out the susceptibility degree of the generated results to the alterations in the criteria weights of the suggested procedure. In this regard, a one-by-one sensitivity analysis grounded in modifying each main criterion weight was carried out to express the effects of the sub-criteria on the outcomes of the hybridized SF-BWM and SF-MULTIMOORA methodology as demonstrated in Table 12 and Fig. 4. The analysis results unveiled that “(A6) Topkapi” was the finest option and “(A1) Atakent” was the worst in all scenarios including the current situation. Moreover, the ranking order was not susceptible to the criteria weight changes in scenario 2 while scenario 5 was the most responsive one.

Table 12 Scenario description for sensitivity analysis of the proposed approach
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Fig. 4
figure 4

Final ranking of the prospective EBCS sites according to each scenario

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6 Conclusions

The research offers several methodological and theoretical contributions to the existing literature by merging the BWM and MULTIMOORA approaches in a spherical fuzzy domain. One of the key contributions of this paper is that, through an inclusive literature analysis, it uncovers that both the EBCS distribution problem and the BWM have not previously been investigated in the spherical fuzzy setting. Another major contribution is the provision of a comprehensive evaluation index system that may be utilized as a reference in other large cities for the positioning of EBCSs from a sustainable and smart perspective. Sensitivity analysis findings exploiting scenarios that involve changes in criteria weights further supported the plausibility and robustness of the approach. A final contribution of this research is that public bus operators may find the developed spherical fuzzy BWM–MULTIMOORA method useful as it diminishes information loss when placing EBCSs with the inclusion of SFSs into the decision-making procedure.

Utilizing the newly presented spherical fuzzy BWM–MULTIMOORA technique was amply demonstrated via the delivered real-world case study of identifying EB charging station points. This comprehensive SFS-based technique for multi-criteria EBCS site selection revealed “(C41) Emissions and noise reduction” as the most crucial factor, followed by “(C21) Power substation capacity”, and “(A6) Topkapi” as the finest alternative out of six potential locations.

Consequently, the findings of this research can benefit public bus operators in choosing the ideal sites for electric charging stations. Furthermore, the formulated generic methodology is also easily applicable to diverse and complex multiple-criteria problems in the spherical fuzzy domain and also can be easily adapted to other cases, in terms of other MCDM problems or EBCS site selection problem in different regions, provided that the criteria in the applied problem are modified and updated suitably.

This study also includes certain shortcomings that may open up new research possibilities in the future. First, since the interrelationships between the criteria were disregarded, an upcoming study may concentrate on enhancing the suggested methodology to address this drawback. Second, as the criteria weights were defined via the BWM which follows the subjective weighting technique, the presented method might be hybridized with one of the adequate objective weighting techniques (i.e., CRITIC or fuzzy entropy) to unbiasedly assess the significance of all criteria.

As for future studies, several MCDM techniques such as DEMATEL, CRITIC, and WASPAS based on SFSs or heuristics including simulated annealing and particle swarm algorithms may be incorporated into the suggested approach to assure a more comprehensive and integrative assessment. Finally, the SFS-based BWM-integrated MULTIMOORA technique has the potential to excel in a variety of enterprises including green supplier selection, material selection, site selection, performance evaluation and risk assessment to confirm the consistency of the introduced approach.