1. Introduction
The public–private partnership (PPP) is an alternative procurement method that has gained popularity in many countries over recent decades [
1]. Governments have adopted it widely worldwide to provide quality public goods and services [
2]. Generally speaking, PPPs are partnerships between the public and private sectors, with varying responsibilities, to deliver public services [
3]. The public sector holds expertise in administrative matters and the safety and well-being of its citizens, while the private sector can adapt skills, innovate, assess risks, and use technology. Therefore, public and private expertise must be combined, as neither sector can deliver projects independently without collaborating [
4]. Investing in PPPs allows the private sector to increase long-term returns and improve the reputation of companies.
In contrast, the public sector utilizes capital, technologies, and management skills to achieve sustainable development [
5]. Notwithstanding, according to the literature, PPP failures have not been rare [
6]. Because of the long-term agreement and high levels of uncertainty, various failures are inevitable in PPP projects [
7]. In addition, the lack of government capacity and incorrect interference in managing PPP projects may lead to failure [
8]. Moreover, selecting partners is often challenging and poses a significant risk of PPP failures [
4]. Selection of the optimum private partner with the necessary qualities and capabilities is critical to successful completion. Failure to choose the right partner hinders the execution of the project, resulting in substantial negative economic impacts [
9]. The preliminary selection of partners, which often leads to the failure of the concluded agreement and is fraught with loss of state budget funds, is confirmed by much statistical data [
10]. Researchers have explored the current state of the research in PPP and identified partner selection as a research gap and future research trends in the PPP literature [
11].
In a PPP, the public and private sectors are parties to a long-term contract. Since PPP contracts are long-term, they are susceptible to the environment [
12]. One of the ways to deal with environmental uncertainty is to choose a robust partner [
13]. While selecting the right private-sector partner is a crucial problem in PPP projects, a few researchers have addressed the decision-making problem [
10]. Choosing partners can be challenging, resulting in conflict and failure of PPP relationships [
4]. In such projects, the public sectors are essential in determining suitable private sectors [
14]. Therefore, it is believed that there needs to be a focus on well-structured and practical decision methods obligatory to improve the performance of PPPs [
15]. Decision-making methods can facilitate partner selection decisions partially intuitively [
16]. An important question is evaluating conflicting targets and criteria when estimating alternatives [
17].
Unlike the supplier selection problems [
18], the literature focusing on partner selection in PPP projects is limited [
9]. Recently, researchers have proposed a novel integrated private partner selection framework applying the best–worst method (BWM) and the technique of order preference similarity to the ideal solution (TOPSIS). They evaluated partners’ performance considering economic, social, technological, and environmental aspects [
11]. Others applied the analytic hierarchy process (AHP) and multi-attribute utility theory (MAUT) to develop an approach for selecting private partners in the housing industry. Their findings categorized criteria into four categories: financial, technical, managerial, and safety/environmental [
9]. Grey rational analysis-VIekriterijumsko KOmpromisno Rangiranje (GRA-VIKOR) is proposed in a paper that considers the combined weights of the improved CRITIC-entropy weight method (EWM). In conjunction with the GRA, the VIKOR approach considers the inherent correlation between evaluation indicators, thus improving the validity of selection assessment results [
19]. A two-phase framework is used in another paper to select partners. The first phase integrates data envelopment analysis (DEA) and differential evolution (DE) algorithms to compute efficiency scores. In phase two, those efficiency scores are utilized to allocate orders using a multi-objective model [
20]. An integrated model was developed using the BWM for evaluating and ranking the selection criteria and the VIKOR for selecting a final partner [
21]. Another study uses the AHP and VIKOR to choose the right partner [
22].
Some limitations need further improvement when investigating the private-sector partner selection from various perspectives using various multi-criteria decision-making (MCDM) methods. The following two aspects illustrate the need for further improvement: (
1) Evaluation of the private sector is usually sophisticated, which can lead to vague and uncertain judgments by decision-makers [
23]. (
2) Collaborating requires identifying the partner’s characteristics most relevant to problem owners’ needs [
24].
For the first limitation, scholars have used fuzzy decision-making methods to deal with the ambiguity of assessment language [
25,
26,
27]. The TOPSIS method using interval-valued intuitionistic fuzzy sets (IVIFSs) was developed to choose a partner for PPP projects [
28]. According to researchers, the fuzzy sustainable supplier index is a result of combining the triple bottom-line criteria of sustainable development with performance and using MCDM technology to determine the most sustainable supplier. To control uncertainty, researchers used the trapezoidal fuzzy membership function [
29]. Multi-attribute group decision-making (MAGDM) problems were solved using the q-rung orthopair fuzzy entropy-based gained and lost dominance score (GLDS) method as an alternative to the traditional GLDS method. Based on the Hamacher operation laws, the q-rung orthopair fuzzy Hamacher weighting average (q-ROFHWA) and q-rung orthopair fuzzy Hamacher weighting geometric (q-ROFHWG) operators were presented to fuse q-rung orthopair fuzzy information effectively [
30].
Moreover, the attribute weights are also determined by q-rung orthopair fuzzy entropy (q-ROFE). A MAGDM model with q-rung orthopair fuzzy information was then constructed based on the q-ROFHWA operator, the q-ROFE, and the traditional GLDS method [
30]. A study focused on the Pythagorean fuzzy environment. Partner selection problems with criteria weights are solved using Pythagorean fuzzy sets and TOPSIS methods. A new similarity measure was developed based on the trigonometric function for PFSs to calculate criteria weights [
31]. Researchers identified selection criteria for private partners in PPP projects using an extended multi-criteria operation and compromise solution (VIKOR) method. A comprehensive method for selecting an optimal private partner using VIKOR-based tools with an intuitionistic fuzzy set was developed. The factors were divided into five packages: essential ability, management ability, previous performance, credit performance, project performance, and sustainable development [
32]. The fuzzy-based approach dealt with vague, uncertain, and qualitative information to present a practical analytical method for selecting a suitable partner. The fuzzy decision-making trial and evaluation laboratory (DEMATEDL) method has been combined with anti-entropy weighting (AEW) and FVIKOR operations to select the most suitable candidate based on the combined weighting technique [
33]. An integrated subjective or objective fuzzy group decision-making (FGDM) method and a factor risk scoring system were applied to select private sector partners under the PPP model [
34]. Therefore, it can be seen that using neutrosophic sets (NSs) that can express the uncertainty in experts’ judgments more appropriately [
35] has not been considered in the literature on private-sector partner selection.
For the second limitation, the development of criteria to select private partners in PPP contracts has been undertaken in multiple prior studies. Still, these studies either pay attention to risk factors or success factors. Some researchers identified five risk groups: financial, political, project-specific, social, and uncontrollable [
36]. Scholars have highlighted that the social risks extend to land acquisition, environmental pollution, and demolition [
37]. Risk analysis revealed seven critical risk groups. In addition to institutional capacity and the local economy, public sector maturity, project finance, project planning, implementation, and project revenue achieved a high-impact linguistic assessment [
38]. Inflation and change in interest rates are some of the financial risks. In contrast, political risks, including unstable policies; economic risks, such as high operating and maintenance costs; technical risks, especially charging technology; and risks posed by the project and the project participant, such as the PPP experience, cannot be ignored [
39]. Other researchers have shown that “government intervention” is one of the crucial risk groups, with “government maturity risk” being the second and “economic viability risk” the third [
17].
In contrast, other researchers have attempted to recognize CSFs, increasing the possibility of success in PPP projects [
40]. It has been shown that the government must implement specific PPP policies, has well-organized and committed public agencies, and provide a stable political and social environment, favorable legal frameworks, and good governance to implement PPP projects effectively. For PPP projects, sector-specific laws, guidelines, standard bidding documents, and contract models should also be used to regulate the PPP procurement process suggested in the PPP proclamation [
8]. Researchers surveyed 27 stakeholders to test CSFs identified through a literature review empirically. Results showed that acceptance and support given by the community, project feasibility, the laws, regulations, guidelines, the available financial market, and having a well-organized and committed public agency were the high-priority CSFs [
41]. An analysis of 42 CSFs yielded six primary categories: public sector clusters, private sector clusters, procurement process clusters, project information clusters, and external clusters. In implementing PPP infrastructure projects, the private sector, project information, and procurement process clusters had the most significant influence [
42]. In developing countries, investigating the case studies indicated that a transparent bidding process, good partnering, and risk allocation are the CSFs in PPP projects [
3].
Accordingly, three crucial points have not been considered in previous research, including the use of neutrosophic sets to convert the qualitative judgments of experts into quantitative data, the consideration of CRFs and KPIs in the evaluation of partners simultaneously, and the investigation of the performance of alternatives in case of changes in environmental conditions in the future. Furthermore, there is another criticism about the previous approaches focusing on the problem of selecting a private sector partner: they complicate the process of obtaining the answer (computationally and interpreting it by the decision maker). On the contrary, this research is intended to simplify this process for decision-makers who lack sufficient operations research (OR) expertise. Therefore, the current study aims to provide a practical and straightforward approach that helps decision-makers consider the above characteristics when choosing the best partner. For this purpose, the other sections of the article are organized as follows: in the next section, the proposed methodology is introduced, and in
Section 3, by presenting a numerical example, the method of implementing the approach is mentioned. Finally, the last part summarizes the contents and provides suggestions for future research.
2. Methodology
In general, decision-making involves selecting the best option from among a set of other alternatives based on the judgment of a decision-maker or a group of decision-makers [
43]. In recent years, decision making has become increasingly complex due to the growing amount of decision information and alternatives, the inherent uncertainty in decision-making, and the fuzzy nature of human reasoning [
44]. It is impossible to express the decisions made by people with crisp numerical values [
45]. Practical multi-attribute decision-making (MADM) problems are characterized by fast-increasing complex uncertainties requiring effective fuzzy tools to express judgments and preferences [
46]. A top feature of fuzziness is its ability to solve engineering and statistical problems efficiently. Several realistic situations can be solved by applying uncertainty theory, including networking problems, decision-making problems, and the impact of uncertainty on social science [
47]. With the help of Zadeh’s linguistic variables, Xu’s uncertain linguistic variables proved helpful in dealing with decision-making problems with qualitative attributes [
23]. The intuitionistic fuzzy set (IFS) model, with membership μ
A(x
i) ∈ [0, 1] and nonmembership ν
A(x
i) ∈ [0, 1] functions, such that μ
A(x
i) + ν
A(x
i) ≤ 1 for each x ∈ X [
48], was introduced to overcome the weakness of fuzzy sets (FS). However, IFSs or FSs cannot manage partial or incomplete information, whereas neutrosophic sets can manage inconsistent and indeterminate information very well [
23].
A three-dimensional NS based on three fundamental elements, truth, indeterminacy, and falsity, was developed to handle incomplete information [
49]. An intuitionistic fuzzy set theory cannot make a proper decision without the indeterministic part of uncertain data, which appears in NS theory [
50]. The most exciting point is that all these three functions are entirely independent, and one function is unaffected by another [
50]. NS has the potential to be an investigation tool when dealing with uncertain data sets; Consequently, many different applications have been developed around it [
31]. Ratings of alternatives the decision-maker provides can be expressed with NSs in the MADM context. SVNSs were developed to ease the application of neutrosophic sets to real-life scientific and engineering studies [
51].
Additionally, as the SVNS model’s membership functions assume values within the standard interval [0, 1], it is compatible with other fuzzy-based models, which can be applied to real-life decision-making problems using actual datasets more efficiently [
52]. SVNS has proven helpful in handling various real-life applications due to its flexibility in describing indeterminate and inconsistent information. Several studies have been conducted and applications made in several fields since its appearance [
51].
To determine the best alternative in a decision set, we can use the concept of the ideal point. Although the ideal alternative does not exist in the real world, it does provide a sound theoretical construct against which to evaluate options [
53]. The similarity measure (SM) measures the similarity between an asset and other entities whose properties are known [
54]. Therefore, NS and SVNS theories must include SM, quantifying the similarity between two objects [
55]. Researchers and scholars have continually proposed new similarity measures for fuzzy-based models, including the SVNS model, applied to various practical MCDM problems [
56]. This section provides the related definitions to compute SVNS in the application.
Definition 2.1.: Ref. [
57]
Let be a universe of discourse. A neutrosophic set is an object having the farm characterized by truth-membership indeterminacy-membership and falsity-membership functions These functions are real standard or nonstandard subsets of with the condition Definition 2.2.: Ref. [
58]
Let be a universe of discourse. A single-valued neutrosophic set is an object having a farm where the functions are real standard subsets with the condition .
To make things easier, we apply to characterize an element in SVNS and is a single-valued neutrosophic number [59]. Definition 2.3.: Ref. [
53]
Let and be two SVNSs and be a similarity measure for SVNSs. satisfies the following properties: Definition 2.4.: Ref. [
56]
Let and be two SVNSs. We can calculate applying Equation (1) as follows: Figure 1 shows this practical, straightforward method includes seven steps.
The proposed algorithm begins with listing alternatives (private-sector partners Pi, I = 1, …, n) in the first step. In step 2, after identifying CRFs and KPIs, we should choose some of the most important ones (Cj, j = 1, …, m). Identifying CRFs and KPIs can be accomplished by consulting the literature, as we did in this study. However, it is imperative to remember that these indicators vary for each project. In light of this, it would be better to combine the views of decision-makers (problem owners) with those of the literature. Step 3 defines the ideal Partner (PI) with the perfect performance in all the criteria. The next step will be to design future scenarios (Snf, f = 1,…,g). To develop future scenarios, we can apply different approaches to scenario planning, such as three-scenario (optimistic, pessimistic, and probable), four-scenario (taking into account two environmental variables), or robustness analysis (no limit on the number of scenarios). We use the second approach in this research by defining four scenarios according to the considerations that should be considered when using the last one. Suppose that the two indicators, including the interest rate (with two states of stability or decrease) and the floating investment costs (with two states of stability or increase), are the most significant indicators that determine the future. In this case, we will face four scenarios f1: stability of interest rate and investment floating costs, f2: decrease in the interest rate and stability of investment floating costs, f3: stability of the interest rate and increase in investment floating costs, and f4: decrease in the interest rate and increase in investment floating costs. In step 5, we should construct decision matrices Df using experts’ viewpoints. Therefore, at this step, a survey of decision-makers (problem owners) should be conducted in as many as the designed scenarios. In step 6, applying Equation (1), we must compute the similarity measure for all partners in all scenarios. Finally, we rank the partners based on their similarity scores.
6. Conclusions
The government’s funds alone cannot meet the significant investment needs for many massive projects [
37]. PPPs combine outsourcing, privatization, and government partnership to utilize private sector resources [
4]. In addition to developing public facilities, PPPs create a system of public service provision [
64]. There are considerable challenges associated with PPP projects for both the public and private sectors [
65]. Governments have been actively promoting PPP projects, increasing their number. However, many PPP projects are still unsuccessful because of implementation difficulties. A PPP project’s large-scale investment, long concession contract duration, and complex technologies create many potential risk factors during implementation [
2]. Furthermore, identifying critical success factors before implementing PPP projects is essential if the project is to run smoothly [
66]. Simultaneous consideration of these two categories of factors, risk and success, is one of the points not considered in previous research.
PPP projects’ success depends on selecting the right private partners, but few studies have been conducted on the subject [
67]. Information uncertainty is a standard paradigm in modern decision-making because perfect information is seldom available to decision-makers [
68]. As shown in the research background, previous approaches either have not addressed uncertainty or only have used fuzzy sets. While neutrosophic sets can more effectively represent the uncertainty in judgments [
56], they were not considered. In this study, we introduced an uncomplicated, practical approach to address the problem of selecting the best private-sector partner. This approach considers the most critical risk and key performance factors affecting PPP projects. Our proposed methodology copes with uncertainty by using SVNSs.
The current study has several weaknesses that researchers in future studies can address. In this study, only qualitative factors are considered, while the analysis of quantitative factors also significantly impacts the partner selection process. Another weakness is related to not examining the relationships between factors. Researchers can consider the interdependency criteria by applying the analytic network process (ANP) or DEMATEL-based ANP. Additionally, a fuzzy clustering method can identify the internal relations among the indicators [
69]. Not thoroughly studying the uncertainty of the future is another shortcoming of this approach, which we plan to address in our future research. Since it is reductionist to consider only two indicators to define future scenarios, we intend to carry out this development while keeping the approach simple.
Moreover, neutrosophic hesitant fuzzy sets (NHFS) represent a practical and general solution to the problem of hesitancy in decision-making. Researchers can keep this point in mind in future studies. In addition, considering the weights of indicators in a multi-criteria decision-making approach has always been emphasized, but this point is not considered in the proposed approach. Researchers can pay attention to this matter in future studies. Finally, when the number of alternatives is large, judgment becomes complicated for decision-makers. Using the DEA model to eliminate inefficient options can be practical. We suggest the study of this subject to other researchers.