1. Introduction
Evidently, rapid social and economic growth is continuing to engender massive demand from governments for investment in public infrastructure projects all over the world [
1,
2]. Such mounting pressures on governments, in addition to worldwide economic decline, mandates innovative project delivery systems, such as public–private partnership, which covers a wide spectrum of infrastructure projects, services, and/or facilities for public use [
3,
4]. At the global level, there is a steadily widening gap between the call for sustainable infrastructure facilities and the resources present for financing these investments from government budgets. Garvin [
5] reported that PPP has been established as the most renowned choice for engaging private sector experience, knowledge, quality, discipline, and arrangements into the delivery of public projects and services, other than private financing. As a result, the possibility of private financing for complex construction projects in relation to PPP projects have become an increasingly crucial factor in international competition. PPP is increasingly adopted by many countries around the world in collaboration with private sector management expertise and discipline and alongside private financing. This public–private cooperation enables accomplishing greater value for investments, fostering more buildable, sustainable, and creative designs, and diminishing capital and operating costs while maintaining higher construction standards [
6,
7]. However, many PPP projects are abandoned or even called off due to several risks, such as the significant difference between public and private sector anticipations and the absence of incentives to draw sustainable financing from private funding sources at reasonable rates [
8,
9]. These risks vary from one project to another [
10,
11].
The quintessence of public–private partnership lies in the compilation of private project execution, private capital, and the delivery of public projects or services. A public–private partnership is an amalgamation of objectives and tasks of the two sectors with varying goals. The goal of the public sector lies in accommodating political or social well-being. On the contrary, the private sector aims to cooperate for commercial and financial purposes, which can be addressed through its rate of return [
12,
13,
14]. Hence, the interaction and linkage between the private and public sectors are prominent for the successful implementation of PPP projects because a deteriorating relationship would easily result in controversy, misunderstanding, conflict, and dispute. Examining which factors can promote or hinder this relationship is quintessential for the success of PPP projects [
15]. Necessitated factors for the successful delivery of PPP projects have been studied and identified by researchers; such factors can be delineated as risk identification and management, strategic financing strategies, procurement protocols, tendering processes, concessionaire selection methods and attributes, and government’s roles and responsibilities [
16,
17]. In general, a PPP project’s private partner is awarded their contract through a public tender, serving the public’s interest by means of competition among contractors [
18]. The most important factor prior to starting the PPP project is the selection process of the private partner. Proper selection of a private-sector partner who is able to adequately perform throughout the PPP development process is strategically important for the project’s success [
19,
20]. The failed delivery of PPP projects induces notable political, economic, and social losses to the government [
21].
In light of the foregoing, the paramount objective of this research paper was to design an integrated model to usher public sector and designated governmental entities in the identification and evaluation of private partners in PPP infrastructure projects. To accomplish the main objective of this study, the following sub-objectives needed to be fulfilled:
Study and determine the selection criteria for analyzing public–private partners in infrastructure projects.
Design an integrated multi-criteria decision-making model for selecting the best private partner.
Validate the developed model using different case studies of infrastructure projects.
The remainder of this research study is outlined in the following manner.
Section 2 enumerates previous research efforts pertaining to the assessment of public–private partnerships.
Section 3 describes the major components and steps of the developed MCDM model.
Section 4 reports the different computational procedures of the MCDM algorithms used in this study.
Section 5 expounds on the data collection processes necessitated to build the developed model.
Section 6 elucidates the results of four real case studies based in Africa.
Section 7 reports a critical analysis of the obtained results in the previous section.
Section 8 highlights the primary conclusions, limitations of this research, and usefulness of the developed model.
3. Research Methodology
The prominent objective of this research paper is to support governmental agencies with a decision-making platform for evaluation and prioritization of potential private partners for public–private partnership infrastructure projects. The framework of the developed research framework is outlined in
Figure 1. The selection criteria are retrieved from reviewing previous literature studies on PPP infrastructure projects [
4,
56,
57,
58,
59,
60,
61,
62,
63,
64,
65,
66,
67,
68,
69,
70,
71,
72,
73,
74,
75] alongside interviews with experts. To this end, a list of 34 selection criteria was defined in this study (9 main criteria and 25 sub-criteria). This list was amended by 12 professional experts in the field of PPP infrastructure projects for their assessment and verification. In this context, face to face interviews with experts were carried out to revise the identified selection factors. To ascertain prolific interviews, a list of questions and key discussion matters were sent before the designated dates of interviews so that interviewees could have sufficient time to prepare and garner related information. In addition, e-mails were sent to a group of practitioners, professionals, consultants and clients. Respondents rendered their feedback and suggestions in relation with the most important criteria. Hence, the criteria list was further tuned, and thus it compiled 23 selection criteria composed of 5 main criteria and 18 sub criteria.
Table 2 reports the compiled list of selection criteria of private partners.
Questionnaire surveys were then created to specify the level of relative importance of the selection criteria of private partners. In this regard, the fuzzy analytical network process is exploited in this research study to interpret relative importance priorities of selection criteria of private partners. Classical ANP and their crisp values are incapable of addressing inherent uncertainties and ambiguities accompanied with human cognition. Hence, FANP is leveraged herein to resolve intrinsic vagueness and imprecision associated with experts’ subjective judgements [
76,
77,
78]. Although analytical hierarchy process is widely used to deal with multi-criteria decision-making problems in real circumstances [
79,
80,
81], it fails to provide satisfactory results in situations that need modelling interrelations and interactions between selection criteria being considered. Additionally, ANP is advised over other methods like full consistency method, best–worst method and level-based weighted assessment in situations that necessitate structured and interdependency-based modeling of complex problems [
82,
83]. To this end, FANP is utilized to accommodate interdependencies and influences between PPP selection criteria [
84,
85,
86,
87]. Furthermore, FANP was successfully and broadly implemented in diverse engineering-related applications. These comprised landfill site location [
88], sustainability assessment of textile wastewater techniques [
89], prioritization of green building materials [
90] and sorting of cyclone readiness activities [
91].
No single MCDM model can guarantee the accuracy and robustness of evaluation processes in engineering-related applications [
92,
93,
94]. Each MCDM model has its own advantages and shortcomings depending on the application [
95,
96]. Moreover, given the different basic rationale of each MCDM algorithm, different algorithms do not compile a single solution when attempting to solve real-world problems even when using the same pool of input data [
97,
98,
99]. This entails the development of hybrid MCDM models that can blend endorse individual strengths and circumvent their weaknesses, inducing a reliable final outcome of multi-criteria decision making [
100,
101,
102]. To this end, the developed HYBD_MCDM model leverages the use of seven multi-criteria decision making algorithms for the sake of prioritizing private partners in PPP infrastructure projects. These algorithms encapsulate CoCoSo, WISP, MARCOS, CODAS, WASPAS, TOPSIS and FANP. In this regard, the developed HYBD_MCDM model accommodates expert-based method (FANP) alongside ranking-based methods (CoCoSo, WISP, MARCOS, CODAS, WASPAS and TOPSIS) for sorting out private partners. This is expected to envision proper integration between human judgment and performance score-based assessment resulting in more efficacious decision. CoCoSo improves the accuracy of decision making, offers proper distinction between alternatives under consideration, and minimally impacted by their removal or addition [
103,
104]. WISP is a recently utility-based MCDM method that is simple and ranks alternatives in a systematized and accurate manner [
105,
106]. MARCOS is another new MCDM method that is accurate in dynamic environment and it is highly stable against rank reversal phenomenon [
107,
108,
109]. CODAS is a relatively advanced distance-based MCDM that ranks alternatives in a methodical scheme based on combining two types of distance: Euclidean and Taxicab [
110,
111]. The main advantage of WASPAS lies in its hybrid nature that can simultaneously combine weighed sum model and weighed product model [
112,
113]. Another advantage is its computational simplicity and robustness against rank reversal [
114,
115,
116]. TOPSIS is acknowledged as one of the most renowned and practical methods in engineering domain [
117,
118]. It is also characterized by its straightforwardness, flexibility and easy interpretation [
119,
120].
The symbol “NUM_MCDM” alludes to the number of used MCDM algorithms used in this research study. Hence, this research paper Copeland algorithm to blend the obtained rankings of seven MCDM algorithms into a final ranking of private partners. Copeland algorithm evinced its efficiency in dealing with complex and practical case studies such as evaluation of stormwater management alternatives [
121], prioritization of road maintenance actions [
122] and sustainability assessment of built environment [
123]. The developed model is validated using four real PPP infrastructure projects in North Africa. The symbol “NUM_EX” alludes to the number of real projects tested in the present research paper. The correctness of the developed HYBD_MCDM is examined through three tiers. The first tier involves adopting Spearman’s rank correlation analysis to measure the similarities between the generated rankings from the investigated MCDM models [
124,
125,
126]. In the second tier, the obtained rankings from the developed HYBD_MCDM model are compared against CoCoSo, WISP, MARCOS, CODAS, WASPAS, TOPSIS, FANP and weighted product model (WPM). The third tier incorporates performing sensitivity analysis to test the stability of the developed model against variations in the changes in the weights of PPP selection criteria.
4. Model Development
This section describes some of the algorithms used in this research paper. Due to space size limitations, FANP, CoCoSo, WISP, MARCOS, CODAS, WASPAS and TOPSIS are only explained herein. More information about the computational procedures of TOPSIS and Copeland algorithms can be found from Wu et al. [
127] and Mohseni et al. [
128].
4.1. FANP
This study adopts a fuzzy interval rather than crisp values for accommodating the intrinsic uncertainties in this multi-criteria decision-making problem. Hence, the developed model relies on FANP that uses Chang’s extent analysis method [
129] to compute the relative importance weights of selection criteria. It is selected herein owing to their ease of computation, low computational requirements, and wide commonality when compared against other FAHP approaches [
130,
131,
132]. Furthermore, it can integrate both quantitative and qualitative data inputs. In Chang’s extent analysis method, a triangular fuzzy number is used for dealing with uncertainties and expressing linguistic evaluations of experts [
133,
134,
135]. In addition, triangular fuzzy numbers are characterized by their wide applicability in similar civil engineering problems [
136,
137,
138,
139] and relative simplicity in modeling and calculation [
140,
141].
The membership of a triangular fuzzy number is denoted by three real numbers of (
). Equation (1) depicts a mathematical representation of the triangular membership function.
A pairwise comparison is performed for every design criterion against other remaining criteria as shown in Equation (2). In this context, the term “
” elucidates the level of relative influence of the i-th element when assessed against the j-th element, with respect to the control criterion. This study uses the triangular fuzzy scale proposed by Kahraman et al. [
142] as expounded in
Figure 2. As can be seen, the linguistic variables are depicted in the form of triangular fuzzy numbers. In this study, there are five linguistic variables of equally important (EI), moderately important (MI), strongly important (SI), very strongly important (VSI) and absolutely important (AI). Their corresponding fuzzy numbers are [1/2, 2, 3/2], [1, 3/2, 2], [3/2, 2, 5/2], [2, 5/2, 3] and [5/2, 3, 7/2], respectively.
The values of fuzzy synthetic analysis with regard to the
-th object are presented in Equation (3).
The degrees of possibility of
are then obtained by using Equations (4) and (5).
where:
and stand for the ordinates associated with the highest intersection point.
The next step involves obtaining the degree of possibility for
to be more than all the other (
) convex fuzzy numbers
as presented in Equation (6).
The final step comprises finding the normalized priority vector
using Equation (7).
The generated eigenvectors from the pair-wise comparison matrices are compiled in an arranged pattern to create a matrix named the supermatrix. The column vector represents the impact, with regard to a base control criterion, of a certain group of elements of a component on a specific element of another or the same component present at the top. If no relationship is present between two elements, the respective entry is zero in the supermatrix. The weighted supermatrix is generated by mapping the influence of all clusters. Then, the limit supermatrix is then obtained by raising the weighted supermatrix to a large power until it converges [
143]. The values stored in the limit supermatrix denote the desired relative importance priorities for selection criteria of PPP.
4.2. CoCoSo
COCOSO is one of the recent MCDM algorithms that was presented by Yazdani et al. [
142]. This algorithm ranks a pre-defined set of alternatives based on merging simple additive weighting and exponentially weighted product models. The steps of the COCOSO algorithm are presented in detail as follows [
144]:
The compromise normalization processes for the cost and benefit attributes are performed using Equations (8) and (9), respectively.
where:
is the normalized measure of performance of the i-th alternative on the j-th criteria.
The weighted compatibility sequence (
) and power weight of compatibility sequences (
) for each alternative are computed using Equations (10) and (11), respectively.
The relative weights of tackled alternative are compiled using three aggregation techniques. Equation (12) demonstrates the sum of arithmetic mean for the sums of scores of weighted sum and weighted product models. Equation (13) demonstrates the sum of relative scores of weighted sum and weighted product models assessed to the best. Equation (14) demonstrates a compromise trade-off between the scores of weighted sum and weighted product models. Usually, the value of the parameter
is assumed as 0.5 or it can be set based on the experts’ judgements.
The final score associated with each alternative is calculated using Equation (15). In this respect, the alternatives are ranked in a descending arrangement, whereas higher values of
indicate more significant alternatives.
4.3. WISP
WISP is a newly developed MCDM algorithm that blends four relationships between the benefit and cost attributes to interpret the overall utility of the alternative. The steps of WISP algorithm are delineated in the following lines [
106]:
The normalized decision matrix is established using Equation (16) in order to convert the input decision matrix into normalized unitless values.
where:
is a normalized dimensionless value of the i-th alternative on the j-th criteria.
The next step comprises computing four utility measures for the benefit and cost attributes using Equations (17)–(20).
where:
and represent the difference between the impact of benefit and cost attributes on the final utility of alternative based on weight sum and weighted product models, respectively. and represent the ratio of benefit and cost attributes on the final utility of alternative based on weight sum and weighted product models, respectively.
The four utility measures are re-calculated using Equations (21)–(24).
where:
, , and stand for the normalized recalculated values of , , and , respectively. The values of and can be negative, positive or zero. Hence, they should be translated into the interval [0, 1] using Equations (21)–(24) before calculating the overall utility of each alternative.
The overall utility of each alternative is computed using Equation (25). In this regard, the tackled alternatives are sorted in a descending manner, whereas the largest overall utility value indicates a more preferred alternative.
4.4. MARCOS
MARCOS is a novel MCDM algorithm that was first presented by Stević et al. [
107]. It comprises the following computational procedures:
The ideal and anti-ideal solutions are defined hinging on the nature of attributes. In this regard, anti-ideal () and ideal () solutions are regarded as the solutions with worst and best characteristics, respectively (see Equations (26) and (27)).
and
and
where:
and represent the cost and benefit attributes, respectively.
The normalization of the elements in the decision matrix is performed by applying Equations (28) and (29), respectively.
The weighted normalized matrix is then obtained through the multiplication of the normalized matrix by the weights of criteria as shown in Equation (30).
In this step, the utility degrees of different alternatives are obtained, such that the utility degrees with respect to anti-ideal and ideal solutions are generated by applying Equations (31) and (32), respectively.
is the summation of elements in the weighted decision matrix.
The next step is to compute the utility functions of each alternative (
), which is a trade-off value of the alternatives with respect to anti-ideal and ideal solutions. The utility functions of alternatives can be obtained using Equation (34).
and stand for the utility functions with respect to anti-ideal and ideal solutions, respectively. In addition, the rankings of alternatives are addressed on the basis of values of final utility functions, whereas higher values of implicate more desirable alternative.
4.5. CODAS
CODAS is a MCDM algorithm that evaluates the desirability level of alternatives counting on two measures, namely Euclidean distance and Taxicab distance from the negative ideal solutions [
145]. In this regard, Euclidean distance is the prominent assessment measures, whereas if the two alternatives sustain the same Euclidean distance. Then, Taxicab distance is utilized as a secondary measure to assess the designated alternatives. The procedures of CODAS algorithms are presented in the following lines [
94]:
The first step incorporates the use of linear normalization to standardize the input decision matrix using Equations (37) and (38).
where:
and stand for the cost and benefit attributes criteria, respectively.
The second step involves computing the weighted normalized decision matrix using Equation (39).
The negative ideal solution is then identified through Equations (40) and (41).
The next step involves interpreting the Euclidean (
) and Taxicab (
) distances of alternatives from negative ideal solutions using Equations (42) and (43), respectively.
The fifth step comprises obtaining the relative assessment matrix as given in Equations (44) and (45).
where:
ϵ {1, 2, 3, 4, 6, …, n}.
is a threshold function that specifies the equality of the Euclidean distances of two alternatives, and it is determined using Equation (46).
where:
stands for a threshold parameter, which usually ranges between 0.01 and 0.05, and it is set as 0.02 in this research paper. In addition, it is worth pointing out that if the difference between the Euclidean distance of the alternative is more than the threshold value. Hence, Taxicab distance is used to compare the design alternatives.
The last step involves computing the overall assessment score as given in Equation (47). In this context, higher values of
induce a better rank for the alternative.
4.6. WASPAS
WASPAS is one of the MCDM algorithms that was first proposed by Zavadskas et al. [
146], whereas it offers a unique blending between the widely-known MCDM algorithms of weighted sum model (WSM) and weighted product model (WPM). In this respect, the basic step-by-step procedures of WASPAS are presented in the following lines [
144]:
The first procedure is the normalization of the input decision matrix. The normalization processes of the benefit and cost indicators are carried out using Equations (48) and (49), respectively.
where:
is the normalized measure of performance of the i-th alternative with regard to the j-th indicator.
The total relative importance of the i-th alternative on the basis of the weighted sum model and weighted product model can be obtained with the help of Equations (50) and (51), respectively.
The joint relative importance of the i-th alternative can be obtained using the general Equation (52) based on compiling the preference scores retrieved from the weighted sum model and weighted product model.
where:
is a coefficient of WAPAS that lies between zero and one. WASPAS behaves as WSM if , and it is transformed to WPM if . In this research paper, is taken as 0.5 to assume a proper tradeoff between WSM and WPM.
4.7. TOPSIS
TOPSIS is a MCDM algorithm that was first proposed by Hwang and Yoon [
147]. The first step is the normalization of performance scores as shown in Equation (53).
The next step involves obtaining the weighted normalized matrix through multiplying the normalized decision matrix by the weights of criteria (see Equation (54)).
The third step is the identification of ideal and negative ideal solutions using Equation (55). They are determined based on the type of criteria under consideration (benefit or cost).
where:
The fourth step is to compute the separate distance of each design alternative from the ideal and negative solutions using Equations (59) and (60), respectively.
The last step is computing the relative closeness for each alternative using Equation (61). The alternatives are sorted in a descending manner, whereas larger values of closeness coefficient denote better design alternatives.
4.8. Sensitivity Analysis
A sensitivity analysis is conducted to evaluate the robustness of the developed MCDM model. The present research study accommodates the one at a time (OAT), which is regarded as one of the most common methods for performing sensitivity analysis in multi-criteria decision analysis [
148,
149,
150]. Herein, the OAT method is carried out by altering the weight of an earmarked PPP selection criteria from 20% to 100% with a percentage increment (step) of 20%. The weights of other PPP selection criteria are then calibrated proportionally to satisfy the additivity constraint. Hitherto, if the weight of the main changing PPP selection criteria under consideration is varied from
to
. Hence, the weights of the remaining PPP selection criteria are adjusted proportionally using Equation (62).
where:
and are the updated and old weights of the remaining PPP selection criteria. and denote the new and original weights of the main changing criteria, respectively.
5. Data Collection
A questionnaire survey was mapped based on the selected criteria, and it was sent to experts and practitioners in the area of public–private partnership infrastructure projects in each project country. Hence, different importance weights are obtained for each case study that can accommodate their different technical, financial and demographic characteristics. Questionnaire surveys were sent to respondents either in hard copy or online format. The pool of respondents comprised personnel from the ministry of transportation in Libya, the African development bank in Morocco, the African development bank in Libya, the Ministry of infrastructure in Ghana, the Ministry of finance in Kenya, and the Ministry of national infrastructure and road safety in Nigeria, among others. For Libya, 35 experts were approached. Yet, a total of 30 responses were gathered, which comprised 8, 16 and 6 from public clients, private companies and academics, respectively. In addition, 26 out of 35, 12 out of 20 and 11 out of 20 were received in Tunisia, Ivory Coast and Senegal, respectively. The respondents were categorized by capitalizing on their position level and number of experience years in infrastructure projects as depicted in
Figure 3.
It can be observed that 16% and 41% of responses were obtained from directors of infrastructure management and project managers, respectively. Additionally, it was found that 16% of responses were from directors of infrastructure management. Moreover, it can be viewed that 32% and 28% of respondents had 6–10 and 11–15 years of experience, respectively. The respondents were asked to fill pairwise comparison matrices for the main criteria and sub-criteria of private partners. In them, the selection criteria were assessed based on nine category scale. Three set of questions (A, B and C) were prepared in order to assess the selection criteria of PPP and determine the appropriate private partner.
Table 3 explicates the set of questions earmarked for the evaluation of private partners in the developed model. The question set C is iterated for other tackled private partners. In the question set C, the expert is asked to identify an answer from a predefined discrete list of available options (absolutely satisfied, extremely satisfied, more than satisfied, satisfied, fairly satisfied, satisfied to some extent and not satisfied at all).
A sample of the pairwise comparison matrix of financial criteria is delineated in
Figure 4. For instance, the expert is asked to specify the degree of importance of equity/debt when compared against foreign financing with respect to their impact on financial criteria.
Each private partner was evaluated by ten experts against each of the five judging attributes. The experts’ evaluation addresses the partner capacity to deliver the best practice of the PPP selection attributes.
Table 4 and
Table 5 explicate a summary of the expert’s assessment for private partners A and B in case study I.
6. Model Implementation
The developed model is deployed on four real projects from four African countries, namely Ivory Coast, Libya, Tunisia, and Senegal. These projects differ with regard to their type, location, cost and concession period. The variability in the project characteristics plays a role in measuring and verifying the efficaciousness of the developed model. The first case study is a toll bridge located in Ivory Coast, and its cost and concession period are USD 282 million and 30 years, respectively. It is composed of two three-lane carriageways with a total length of 1.5 km. The second case study is a highway in Libya extending for approximately 1700 km. Its cost and concession period are EUR 2033 million and 40 years, respectively. The third case study encompasses upgrading a 270 km highway in Tunisia with cost and concession period of EUR 58.1 million and 30 years, respectively. The fourth case study comprises the construction of 20.4 km toll highway in Senegal with cost and construction period of EUR 64 million and 25 years, respectively.
The developed FANP network is designed to capture multiple levels of dependencies such as dependencies between main selection criteria and overall goal, interdependencies of sub-criteria with each other, and dependencies between sub-criteria and private partners’ alternatives.
Table 6 demonstrates the fuzzy judgment matrix for the five selection criteria with respect to the overall goal (case study I). In this regard, the linguistic terms and respective fuzzy intervals are used for computing the weights of selection criteria. Based on Chang’s extent analysis method, the normalized weights of financial criteria, technical criteria, safety and environment criteria, managerial criteria, and political criteria are 26.6%, 23.2%, 10.9%, 22.7% and 16.6%, respectively. It can be noticed that the highest weight is given to the financial criteria followed by technical criteria. This is a result of the due consideration of the huge impact of equity and debt.
Table 7 explicates the sub-criteria of private partner selection (SC1–SC4). It can be observed that equity/debt (37.41%), construction program and ability to meet its targeted milestone (55.25%) and conformance to laws and regulations (35.79%) are the most important sub-criteria under the financial criteria, technical criteria and safety and environment criteria, respectively. In addition, acceptance of risk transfer (42.1%), and understanding of legal requirements (40%) are found to be the most important sub-criteria under the managerial criteria and political criteria, respectively. The priority weights for the sub-criteria regarding the private partners’ alternatives are then determined. The final priorities of the four private partners based on FANP is represented in
Table 8. It should be highlighted that herein private partner 1, private partner 2, private partner 3 and private partner 4 are denoted by P1, P2, P3 and P4, respectively. In addition, the terms I, II, III and IV are used to denote case studies I, II, III and IV, respectively. Priority index (PI) is used to reflect the preference of each private partner alternative. It is interfered that P3I is the optimum private partner to deliver this project and then P2I and P1I were ranked in the second and third places, respectively. In this regard, the PI of P3I, P2I and P1I are 0.361, 0.268 and 0.218, respectively.
Table 9 summarizes the weights of the selection criteria of public private partnerships. It can be noticed that technical and financial aspects are the most important in case study I, political aspect occupies the highest rank in case study II. In addition, financial aspect is the most paramount criteria in case studies III and IV.
Table 10 reports the ranking indices and order of private partners based on CoCoSo, WISP, MARCOS, CODAS, WASPAS, TOPSIS and FANP. A closeness coefficient (
) is used to define the ranking order of private partners based on TOPSIS. It is obvious that CoCoSo, WISP, MARCOS and TOPSIS agree on the ranking of four private partners. Although FANP resembles CoCoSo, WISP, MARCOS and TOPSIS on the ranking of first and fourth private partners, it disagrees with them in the ranking of second and third private partners. It can be also observed that significant discrepancies are obtained in the rankings of CODAS and WASPAS. For instance, P1 and P4 are found to be the best private partners according to CODAS and WASPAS, respectively.
Table 11,
Table 12 and
Table 13 report the ranking indices and order of investigated MCDM algorithms for case studies II, III and IV. In case study II, the alternative denoted as “P2II” is chosen as the top priority private partner based on WISP, MARCOS, CODAS, WASPAS, TOPSIS and FANP. Moreover, the aforementioned algorithms provide the exact same rankings of P3II and P1II as the second and third preferred alternatives, respectively. However, CoCoSo obtained comparatively different results yielding P3II and P2II as the first and second preferred alternatives, respectively. It is worth noting that all seven MCDM algorithms appended P1II as the least preferable private partner in delivering the highway project. With regard to case study III, the majority of MCDM algorithms (CoCoSo, WISP, MARCOS and WASPAS) identified P2III as the top priority private partner. Nonetheless, TOPSIS and CODAS selected P3III as the best private partner while FANP determined P1III as the best private partner. It can also be argued that significant disagreements are obtained by the rankings of MCDM algorithms when attempting to deal with case study III when compared against other case studies. With respect to case study IV, it can be seen that all seven MCDM algorithms dictated P3IV, P4IV, P1IV and P5IV as the first, second, third and sixth most preferred private partners, respectively. Additionally, slight discrepancies are encountered in the rankings of P2IV and P6IV.
Table 14 and
Table 15 elucidate the results of Copeland across the four case studies. It shows the wins, losses, final score and final ranking of each private partner. The consolidated rankings of private partners are determined based on garnering the number of wins and losses of each private partner across the seven investigated MCDM algorithms. It can be seen that P3I is the most superior private partner in case study I with three wins and zero losses while P4I is ranked second with two wins and one loss. It is also indicated that P2II is the best private partner in case study II with two wins and zero losses while P1II is ranked third with two losses and zero wins. P2II outranked the remainder of private partners in case study III with two wins and zero loss. It is also found that P2III is ranked second with one win and one loss. Concerning case study IV, P3IV outperformed other private partners accomplishing five wins and zero losses. In addition to that, P4IV was found to be the second preferred private partner with four wins and one loss.
Figure 5 reports Spearman’s rank correlation coefficients between the investigated seven MCDM models.
Figure 6 shows a Kednall tau correlation matrix for the investigated MCDM models. The values stored herein are obtained based on taking the average of the values of Spearman’s rank correlation coefficient or Kendall tau rank correlation coefficient over the four case studies. It is observed that a perfect correlation exists between the four MCDM models of HYBD_MCDM, WISP, MAROCS and WPM. In addition to that, high Spearman’s correlation and Kendall tau correlation lie between the pairs (HYBD_MCDM, WPM), (WISP, WPM), (MARCOS, WPM) and (WASPAS, WPM). In this context, their Spearman’s rank correlation coefficients were 0.9. Furthermore, Kendall tau rank correlation between the aforementioned pairs of MCDM algorithms is 0.833. It is also derived that weak Spearman’s correlation are exhibited between the pairs (CoCoSo, CODAS), (CODAS, FANP) and (WASPAS, FANP), whereas their correlation coefficients were equal to 0.286 and −0.014, respectively. In addition to that, their Kendall tau rank correlation values are weak and they correspond to 0.217 and −0.033, respectively.
On the grand scheme of things, the average Spearman’s rank correlation coefficients of the investigated MCDM models are calculated to analyze their reliability (see
Table 16). It is shows that the average Spearman’s correlation coefficients of HYBD_MCDM, TOPSIS, CoCoSo, CODAS, WASPAS, WISP, MARCOS, FANP and WPM are 0.83, 0.751, 0.72, 0.383, 0.767, 0.83, 0.83, 0.445 and 0.83, respectively. Besides, the average Kendall tau rank correlation coefficients of HYBD_MCDM, TOPSIS, CoCoSo, CODAS, WASPAS, WISP, MARCOS, FANP and WPM are 0.808, 0.704, 0.662, 0.352, 0.704, 0.808, 0.808, 0.444 and 0.808, respectively. This manifests that the developed HYBD_MCDM, WISP, MARCOS and WPM are the most dominant MCDM algorithms in accordance with conformity and consistency with remainder of MCDM algorithms. This paves the way to draw a conclusion that developed HYBD_MCDM, WISP, MARCOS and WPM are highly applicable and can be used as a reference for the selection and evaluation of private partners in infrastructure projects. It is also noticed that CoCoSo and TOPSIS renders a relatively acceptable agreement with other MCDM models. On the other hand, CODAS and FANP produced rankings that were highly distanced from other MCDM algorithms, which exemplifies their inefficiency in the prioritization of private partners in infrastructure projects.
An automated tool is programmed using C#.net language to facilitate the implementation of the developed private partner selection model by the users.
Figure 7 explicates a visual representation of the capabilities of the developed model. The inputs for the developed automated tool comprise information pertaining to the project and private partners alongside experts’ judgments. The outputs involve a priority index and ranking for each respective private partner.
Figure 8a elucidates the interface where the user is asked to demonstrate his preferences towards the private partner selection criteria. The input and processed data are stored in XML format to simplify and expedite their storage, retrieval and management (see
Figure 8b).
Figure 9 expounds the interface of private partner selection based on ANP. According to the user inputs and experts’ judgments, it is found that the private partner B (57.06%) is ranked first followed by private partner C (27.61%) in the second place and then private partner A (15.31%) in the third place.
7. Analysis and Discussion
Analytical comparison of criteria weights evinces that different relative importance priorities are retrieved for each case study. For instance, financial and technical aspects (35%) share the same level of importance in case study I (Ivory Coast). The political aspect (31.5%) is ranked first in case study II (Libya), while the financial aspect is regarded as the most prominent criteria in case studies III and IV (Tunisia and Senegal) with 36.5% and 26.6%, respectively.
Table 17,
Table 18,
Table 19 and
Table 20 expound a comparison between the rankings of the investigated MCDM models. In the first case study, the developed integrated HYBD_MCDM selected private partner “P3I” as the best alternative. This agrees with the majority of MCDM models like TOPSIS, CoCoSo, WISP, MARCOS and WPM, which suggests that “P3I” is the best private partner. WASPAS and FANP produced a slightly different ranking from the consensus ranking of private partners while CODAS determined a completely distinctive ranking. As for the second study, all MCDM algorithms except CoCoSo are consistent in the full ranking of alternatives and simultaneously identifying “P2II” as the optimum private partner. With regard to the third case study, CoCoSo, WASPAS, WISP, MARCOS and WPM conform with the developed HYBD_MCDM model in identifying “P2III” as the best private partner. On the other hand, private partner “P3III” was determined by TOPSIS and CODAS as the best private partner, and FANP appended private partner “P1III” as the best one. With respect to the fourth case study, the most (P3IV) and least (P5IV) preferable private partners do not change rankings no matter what MCDM model is used. Looking into the four case studies and through careful examination of the obtained rankings, it is reasonable to argue that the rankings rendered by the developed HYBD_MCDM are credible and thus it stands as a reliable reference for decision-making in selection of private partners in transportation projects.
Figure 10,
Figure 11,
Figure 12 and
Figure 13 demonstrate 100 sensitivity analysis simulation runs over the course of four case studies. In this respect, each of the five PPP selection criteria is varied by 20%, 40%, 60%, 80% and 100%, resulting in 25 sensitivity analysis simulation runs for each case study. By visually inspecting the sensitivity analysis runs, it can be observed that the developed MCDM model is minimally affected by the variations in the weightages of PPP selection criteria exemplifying its robustness. In particular, the rankings of the public private partners are altered in 10 simulation runs (10%) over the entire sensitivity analysis. It is also viewed that political aspect is the most sensitive criteria in selecting private partners in case study II while managerial aspect is the most sensitive criteria in selecting private partners in case study III. As for case study IV, technical requirements are found to be most the decisive criteria in evaluation of private partners.
Figure 14 elucidate a comparison between the investigated MCDM models based on sensitivity analysis runs. It can be viewed that the developed HYBD_MCDM, WASPAS, WISP and MARCOS are the most robust MCDM models. In this respect, the ranking or private partners are varied in 10 sensitivity analysis simulation runs. CoCoSo (12) and WPM (13) come in the second and third places, respectively. It is also noted that TOPSIS and CODAS are the most sensitive MCDM models since they were associated with a ranking change in 29 and 44 sensitivity analysis runs, respectively. In order to better measure the robustness of the developed HBD_MCDM model, the average absolute difference in ranking (AADR) is incorporated for doing so (see
Figure 15). In this context, the AADR is computed by dividing the difference in ranking of private partners by the number of simulation runs multiplied by the number of private partners under consideration. It is inferred that the developed HYBD_MCDM model accomplished AADR of 0.067 demonstrating its superior stability over the remainder of MCDM models. Likewise, WASPAS (0.075), WISP (0.075), MARCOS (0.075) and CoCoSo (0.079) attained low values of AADR, exemplifying their low sensitivity to the variation in weightages of PPP selection criteria. The highest values of AADR are linked with TOPSIS (0.213) and CODAS (0.479) signifying their high influence by weight changes.
8. Conclusions
Several countries have adopted public–private partnerships for capital-intense, strategic and complex infrastructure projects as a paradigm to lessen costs, diminish delays and alleviate economic risks. To this end, this research aimed to study, investigate, and design a two-tier multi-criteria decision-making model for selecting the private partners in PPP infrastructure projects. The first tier encompasses studying and analyzing relative importance weights of selection criteria of private partners capitalizing on fuzzy analytical network process. The second tier is envisioned based on the use of seven MCDM algorithms to prioritize private partners in infrastructure projects and identify the best one among them. These algorithms encompassed CoCoSo, WISP, MARCOS, CODAS, WASPAS, TOPSIS and FANP. Analytical results demonstrated that financial and technical aspects are the most influential factors on the selection of private partners with relative importance weights of 26.6% and 23.2%, respectively. It was also deduced that the sub-criteria of equity/debt (37.41%), construction program and ability to meet its targeted milestone (55.25%) and conformance to laws and regulations (35.79%) highly implicate the selection of private partners.
The analysis of case studies illustrated that different MCDM algorithms induced different rankings of private partners, which necessitates the use of the Copeland algorithm for blending the rankings of the tackled MCDM algorithms. For instance, CODAS and WASPAS failed to solve case study I and CoCoSo was not able to select the correct private partner in case study II. TOPSIS, CODAS and FANP were incapable of accurately finding the optimum private partner in case study III. However, all seven MCDM algorithms agreed on the determination of the best private partner in case study IV. It was also interpreted that notable variations are experienced in the rankings of private partners in the highway case study in Tunisia, when compared against other case studies. The correlation analysis suggested that WISP and MARCOS are highly efficient in the selection of private partners in infrastructure projects. On the contrary, CODAS and FANP failed to appropriately handle the ranking process of private partners, urging not using them in similar application. Some limitations still exist in this study. First, more responses need to be garnered that can aid in further validating the obtained results. Second, the collected data were based on projects only located in Africa. Other questionnaire surveys need to be designed for projects in other continents. Third, potential future research can include other selection criteria like expertise and knowledge of public private partners in similar projects besides the concession period of designated projects. Fourth, the developed model can be tedious and more challenging mostly in the presence of large number of criteria and private partners. The developed model is expected to aid governments by equipping them with an efficient decision support system for evaluating and selecting appropriate private partners and thus guaranteeing the successful delivery of PPP projects. The developed model is characterized by its robustness, generalizability and flexibility, which can facilitate the selection of public–private partners in other construction industries. Another advantage of the developed model is that researchers may opt to compare the results of the developed hybrid MCDM model with their other selection models of private partners. The developed model also pinpoints the factors that significantly influence the selection of private partners in PPP transportation projects. In addition, the procedures adopted in this study can be replicated across different infrastructure projects in developing and developed counties.