A DEA Game Cross-Efficiency Model with Loss Aversion for Contractor Selection

Abstract

Evaluating and selecting appropriate contractors is critical to the success of specific construction projects in the building industry. Existing approaches for addressing this problem are unsatisfactory due to the ignorance of the multi-dimensional nature of the evaluation process and inappropriate consideration of existent risks. This study presents a DEA game cross-efficiency model with loss aversion for evaluating and selecting specific contractors. The competitiveness of the evaluation process is modeled using game theory with respect to the adoption of the cross-efficiency model. The attitude of the decision maker toward risks is tackled with the use of loss aversion, which is a phenomenon formalized in prospect theory. As a result, the proposed approach can adequately screen available contractors through prequalification and adequately consider the attitude of the decision maker toward risks, leading to effective decisions being made. An example is presented to demonstrate the applicability of the proposed model in evaluating and selecting appropriate contractors for specific construction projects. The results show that the proposed model is effective and efficient in producing a unique solution for contractor selection through appropriate modeling of the multi-dimensional contractor selection process and adequate consideration of the competition between the contractors and the attitude of the decision maker toward risks in practical situations.

1. Introduction

Effectively evaluating and selecting appropriate contractors for a specific project is becoming increasingly critical for the overall success of such a project [1]. This is due to the increasing complexity of individual projects, the changing requirements and expectations of specific organizations, the presence of multiple, often conflicting criteria, the existence of project risks, and the attitude of the decision maker toward such risks [2,3,4]. Furthermore, there are a growing number of contractors who are actively competing for individual projects in practical situations. This often requires a rigorous screening of individual contractors in the contractor evaluation and selection process [5]. To ensure that the most appropriate contractor is selected for a specific project, structured approaches capable of comprehensively considering all the critical factors as described above are required in the contractor evaluation and selection process.

The contractor evaluation and selection process is complex and challenging. On the one hand, the process of contractor evaluation and selection is uncertain, imprecise, and often multi-dimensional [5]. Three stages are often involved including preparation, bidding, and tendering. Decisions must be made in each and every stage with a significant amount of uncertainty and imprecision [6]. On the other hand, the attitude of the decision maker toward risk, which is always present, often plays a major role in the contractor evaluation and selection process [5]. Such attitudes, which are different in various situations, are usually influenced by specific factors such as personal experience, educational background, economic environment, and risk preference [7]. As a result, selecting the most suitable contractor poses a significant challenge for both government and private enterprises.

Tremendous research has been conducted, and numerous approaches have been developed, for facilitating the contractor evaluation and selection process. An examination of such approaches shows that there are two types of approaches in addressing this problem from different perspectives, including the subjective method and the objective method [7]. The subjective method focuses on evaluating and selecting appropriate contractors through adequately considering the subjectivity and fuzziness of the human decision-making process [8,9]. The objective method concentrates on exploring the interactivity between different criteria in evaluating and selecting the most appropriate contractor in a given situation [10]. These methods have provided useful solutions from different perspectives for evaluating and selecting the appropriate contractor for a specific project in a given situation.

Despite the approaches discussed above, the problem of adequately evaluating and selecting contracts in the building industry has not been satisfactorily addressed. There are several issues and concerns that need to be appropriately addressed in the contractor evaluation and selection process. There are, for example, numerous contractors present in the evaluation and selection process This requires an adequate consideration of the competition between the contractors in the evaluation and selection process. This usually leads to subjective and imprecise assessments being used in the evaluation and selection process [1]. Furthermore, the attitude of the decision maker toward the contractor evaluation and selection process always exists. Effectively evaluating and selecting the most appropriate contractor for a specific project in a given situation requires a comprehensive consideration of all these issues and concerns through the adoption of structured approaches.

This paper proposes a DEA game cross-efficiency model with loss aversion for adequately evaluating and selecting the most suitable contractor in a given situation. The competitiveness of the evaluation process is modeled using game theory with respect to the adoption of the cross-efficiency model. The attitude of the decision maker toward risks is tackled with the use of the loss aversion concept. As a result, the proposed model can adequately screen available contractors through prequalification and appropriately consider the attitude of the decision maker toward risks, leading to effective decisions being made. A numerical application of evaluating and selecting contractors in an infrastructure project is presented, which demonstrates the applicability of the proposed model for solving the contractor evaluation and selection problem in the real world due to the comprehensibility of the concept underlying the model and the efficiency of the computation involved.

The rest of this paper is structured as follows: Section 2 reviews the related literature for justifying the need for the development of a structured model in solving the contractor evaluation and selection problem. Section 3 describes the contractor evaluation and selection process. Section 4 details the development of the DEA game cross-efficiency approach with loss aversion. Section 5 presents a numerical application for demonstrating the applicability of the proposed model in solving a real contractor evaluation and selection problem. Section 6 discusses the conclusions and further research.

2. Literature Review

Contractor evaluation and selection is a complex problem that needs to be adequately solved [6]. Such a problem is usually multi-dimensional [1]. There are numerous factors that need to be properly considered in the contractor evaluation and selection process. Numerous approaches have been developed for solving the problem of contractor evaluation and selection, including analytic hierarchy process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), and DEA, among the others [7].

AHP is a commonly used approach for contractor evaluation and selection in construction. Almohassen et al. [11], for example, apply AHP to help evaluate contractors at the pre-tendering stage. Wang et al. [12] use the AHP to weight the decision criteria for selecting the best-value contractors in two construction projects in Taiwan. These studies show that AHP is simple in concept and easy to use in practice that make it popular in addressing contractor evaluation and selection problem. The use of AHP, however, requires precise assessments in the pairwise comparison process, and there are always inconsistencies present in the comparison process. Furthermore, subjective, imprecise, and ambiguous preferences always exist in the contractor evaluation and selection process. This makes the use of AHP in evaluating and selecting contracts difficult and challenging.

Fuzzy AHP was developed through the utilization of fuzzy sets theory for better modeling the subjective and imprecise assessments in the pairwise comparison process. This leads to the use of fuzzy AHP for contractor evaluation and selection under uncertainty. Thus, Afolayan et al. [13] propose an integrated fuzzy analytic hierarchy process (FAHP) model for ranking decision criteria for contractor selection. Jaskowski et al. [14] present an extended fuzzy AHP method to solve the problem of contractor evaluation and selection in the context of group decision making. And this method could express relative preferences and uncertainty of opinions. Furthermore, the extended fuzzy AHP method does not need to construct a reciprocal matrix, which would avoid errors caused by its use. Zhou et al. [15] adopted FAHP to address credit assessment issues for power infrastructure contractors, they believed that the FAHP could improve evaluation accuracy. But there are also problems with this approach, namely that the minimum mean deviation was not found between the groups’ weights and weights assigned by each expert. Therefore, the application of fuzzy set theory in multi-criteria decision analysis is one of the more common methods. Nieto-Morote [16] propose a fuzzy multi-criteria decision-making approach for contractor evaluation and selection. Afshar et al. [17] develop an interval type-2 fuzzy set-based approach for assessing and selecting contracts in projects through better handling the impressions and subjectivity inherent in the selection process. These methods have proved their applicability in addressing the problem of contractor evaluation and selection under uncertainty in various situations.

Recognizing the need for better modeling the uncertainty and imprecision inherent in the contractor evaluation and selection process, many fuzzy multi-criteria decision-making approaches have been developed. Naghizadeh Vardin et al. [18] propose a new contractor evaluation and selection approach based on the best-worst method (BWM) and well-known fuzzy VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) techniques that can be used to ameliorate the problems with the traditional “minimum bid price” rule. Chang [19] propose the PROMETHEE II method to solve the problem of contractor selection for better handling the missing and non-existent data. The method can handle information about the relative importance of criteria and subjective preferences of experts. To address the complexity and uncertainty of contractors, Karami et al. [20] propose a new approach using the interval-valued fuzzy step-wise weight assessment ratio analysis and the interval-valued fuzzy combined compromise solution for contractor selection in construction projects. Rivero Gutiérrez et al. [21] apply the ELmination Et Choice Translating REality (ELECTRE) approach for evaluating and selecting project contractors. In addition, Chen et al. [6] propose a novel ELECTRE III-based approach capable of adequately considering the indetermination, imprecision, and uncertainty in the contractor evaluation and selection process. In order to select the most valuable contractor, Lee et al. [22] present a value estimation approach using the service dominant logic capable of selecting the best contractor by considering the changes in value at both the construction and operational stages. Mahmoudi et al. [23] develop an approach to evaluate the qualifications of subcontractors using the ordinal priority approach (OPA) method for performance assessment. These approaches have demonstrated their respective use in solving the contractor evaluation and selection problem due to their ability to better handle the subjectiveness and imprecision inherent in the human decision-making process.

The approaches discussed above usually require the use of subjective judgment for determining the criteria weights in the decision-making process. This is sometimes undesirable as the criteria weight process is often complex and challenging and may be influenced by the subjective opinion of the decision maker. As a result, DEA is proposed as an alternative approach for contractor evaluation and selection. It is becoming one of the fastest growing approaches to solving management problems for the multi-criteria evaluation of efficiency [10]. For instance, Wu et al. [24] use DEA to evaluate sustainable manufacturing performance when wastes are recycled and re-used while measuring the energy and environmental performance of transportation systems [25,26]. Similarly, as a decision-making tool, DEA is proposed to differentiate efficient from inefficient contractors [27]. With the use of the DEA, individual contractors can be assessed for their appropriateness in a given project without the need for setting specific weights and scoring functions [28,29,30]. Yang et al. [31] apply DEA to facilitate the criteria evaluations for each bidder. DEA, in general, however, is a non-parameter approach for evaluating the performance of a set of decision-making units with multiple inputs and outputs. This makes the use of DEA in contractor evaluation and selection difficult and challenging.

To better handle such complexity and challenges, traditional DEA has been extended to better address the problem of contractor evaluation and selection. Luo et al. [32] propose a three-stage network DEA approach for performance evaluation. This approach decomposes the whole system into multiple sub-processes, and evaluates the efficiency of each stage, respectively. However, the three-stage network DEA approach has strict requirements on the input and output data. Another improved DEA method is cross-efficiency DEA, which differs in conducting peer evaluations for each alternative solution. Falagario et al. [33] propose the cross-efficiency evaluation approach for evaluating different offers in a public tender, and selecting the best supplier among the eligible candidates. These approaches show potential in addressing the problem of contractor evaluation and selection without the involvement of the decision maker.

Overall, DEA-based approaches are common in which contractors can be evaluated and ranked without the need for subjective judgments from the decision maker. This makes the decision-making process transparent and fair, therefore leading to better decisions being acceptable to all the stakeholders in the tendering process. There are, however, still several issues and concerns in the use of the improved DEA for contractor evaluation and selection, including the lack of the consideration of the attitude of the decision maker toward risks and the existence of sub-optimal solutions in assessing the average cross efficiency of the contractors in the evaluation and selection process.

Table 1. A summary of existing approaches for contractor evaluation and selection.

Methods	Descriptions	Characteristics	Ref.
VIKOR	Compromises ranking of alternatives by maximizing group utility and minimizing individual regret.	Advantages: the method has more stability in the ranking results; the aggregation index of ranking is based on the distance from the ideal solution, which is more compatible with human choice rationale. Disadvantages: it relies on individual subjective preferences to determine weights, which can influence the decision outcomes.	Naghizadeh Vardin, A. et al., 2021 [18]
PROMETHEEⅡ	Based on the pairwise comparison of alternatives for each criterion to solve MCDM problems	Advantages: it can handle incomplete assessment criteria information; considers the relative importance of criteria; considers the subjective preferences of experts. Disadvantages: the method is influenced by subjective preferences; when there are inconsistencies in preferences among different decision makers, PROMETHEE II may struggle to effectively handle such situations.	Chang, K., 2021 [19]
AHP	After pairwise comparisons of the importance of standards, the relative weights of indicators are determined.	Advantages: it decomposes complex issues into a hierarchical structure, making decision problems easier to manage and comprehend. Disadvantages: this method cannot capture the uncertainty of the preference ratings; and did not consider the objective weight of the evaluation data.	Almohassen, A.S. et al., 2023 [11] Wang, W., 2013 [12]
FAHP	Merging the strength of Fuzzy methods with AHP.	Advantages: it includes a few more preferences inside uncertain, imprecise, and obscure settings than AHP. Disadvantages: large amounts of data are required to support the construction of fuzzy sets and the modeling of logic. Insufficient or low-quality data might impact the accuracy and reliability of the results.	Afolayan, A.H. et al., 2020 [13] Jaskowski, P., 2010 [14]
BWM	A new pairwise comparison-based method.	Advantages: uses less comparison data and provides more consistency in the results. Disadvantages: for situations involving a large number of alternative options, BWM can become complex and time-consuming, making it unsuitable for large-scale decision-making problems.	Naghizadeh Vardin, A. et al., 2021 [18]
ELECTRE	ELECTRE III ranks these alternatives based on a predefined set of criteria, determining their superiority or inferiority by considering the preference relationships among them.	Advantages: being able to generate a ranking order of alternative solutions aids decision makers in better understanding the hierarchy among these alternatives. Disadvantages: when dealing with large-scale decision systems, efficiency might be lower, and the applicability range could be restricted.	Rivero Gutiérrez, L. et al., 2022 [21] Chen, Z. S., 2021 [6]
DEA	A quantitative analysis method for the relative efficiency evaluation of comparable units, based on multiple input and output indicators.	Advantages: there is no need to assign weights to the different inputs and outputs; the measurement units of the different inputs and outputs need not be congruent. Disadvantages: it is sensitive to the selection of input and output data as well as their weights. Different choices of data and weight assignments may affect the results of efficiency evaluations.	Ratner, S.V. et al., 2023 [10] Ratner, S. et al., 2021 [26] de Boer, L. et al., 2001 [27] El-Mashaleh, M. S. et al., 2010 [28] Toloo, M. et al., 2014 [29] Cheaitou, A. et al., 2019 [30] Yang, J.B. et al., 2016 [31]

summarizes the discussion above.

Construction projects often face much uncertainty, leading to various risks factors throughout the entire project [2]. The most common risks include political risk, economic risk, natural disasters, and other force majeure risks. There are also risks arising from the differences between a contractor’s own conditions and the risk appetite of the project organizer. For instance, even for the high-quality contractors, there are risks such as financial risk, management risk, procurement risk, operational risk, environmental risk, and so on. They are, however, often criticized due to the ignorance of the attitude of the decision maker toward risks and the lack of consideration of the efficiency of the contractor in a given situation. Decision makers perceive risk differently in different risk situations. In general, personal characteristics and experience, as well as their education and environmental background, have a certain impact on people’s risk appetite [30]. Thus, this leads some people to make completely different decisions even in the same decision-making environment. Furthermore, the risk appetite of the decision maker involves more subjective factors, which play a very important role in the contractor evaluation and selection process.

However, there is little literature on the consideration of the attitude of the decision maker toward risks in bidding in the construction industry. The attitude of the decision maker toward risks, however, plays an important role in construction project management, which includes contractor evaluation and selection [34]. Individuals try to avoid risks when making risk-based decisions based on personal characteristics and experience to avoid risk when making risk-based decisions. To improve risk-based decision making, Wang and Yuan [34] investigate the critical factors affecting the risk attitudes of contractors in construction projects. It reveals that there are four main categories of factors that influence the decision makers, including knowledge and experience, characteristics of contractor, personal perception, and economic environment. Cheaitou et al. [30] believe that the risk is usually very high during the bidding process. In order to reduce the risk of the bidding process, Cheaitou et al. [30] propose a three-stage-based approach to assist the decision maker in selecting the most appropriate construction contractors, and use a fuzzy logic-based approach to identify the risk factors in the second stage. Despite these studies, the attitude of the decision maker toward risk in the contractor evaluation and selection has been tackled appropriately in the existing literature.

Table 2. A summary of risks for contractor evaluation and selection.

Sources of Risks	Definition	Risk Causes	Ref.
Construction projects	Risks arising from the external conditions of the construction project	Policy environment Market instability War Adverse weather conditions	Hu, J. et al., 2011 [35] Shibani, A. et al., 2022 [4]
Contractors	Risks associated with choosing the wrong contractor	Time delay Cost overruns Quality problems Safety incidents Enterprise management capability	Cheaitou, A. et al., 2019 [30]
Decision makers	Risks arising from the risk attitude of decision makers	Education background Character traits Decision motivation Boldness Psychological endurance	Wang, J. et al., 2011 [34]

summarizes the risks for contractor evaluation and selection.

To better consider the attitude of the decision maker toward risk in contractor evaluation and selection, the concept of loss aversion for reflecting the attitude of the decision maker toward risks in the prospect theory of Kahneman and Tversky [36] can be used. With the use of the prospect theory along the lines of the loss aversion concept, it is well recognized that people are more sensitive to loss than to gain, relative to a given reference point [37]. This means that during the contractor evaluation and selection process, the decision maker perceives the prospect utility in gains and losses relative to the reference efficiency level.

The selection of contractors in construction projects is often influenced by multiple evaluation indicators. The evaluation index system is the standard and basis for the comprehensive evaluation of contractors. And the index system is different for different types of projects, so the evaluation committee should establish the evaluation index system according to the principles of conciseness, scientific rigor, and overall consistency. The selection of a contractor for a construction project is traditionally dependent on the lowest bid price [38]. However, using price as the sole evaluative indicator would cause an inappropriate selection of a contractor. This decision may result in poor quality, delay, accident, claims, and disputes. Consider that the price should not be the only index to evaluate contractors, multiple selection criteria have been addressed by more and more experts over the last years. Antoniou et al. [39,40] believed that the selection criteria of contract types should include cost, scope, process, value for money, criticality of schedule, performance criticality, etc. Cheaitou et al. [30] considered that the selection of the most appropriate contractor should be based on a set of criteria such as technical capability, financial stability, risk, safety, etc. Jaskowski et al. [14] selected manpower and equipment, financial capacity, past project performance, organization experience, and certified management systems as the prequalification criteria. Nieto-Morote and Ruz-Vila [16] regarded technical capacity, experience, management capability, financial stability, past performance, past relationships, reputation, and occupational health and safety as the selection criteria. Borujeni and Gitinavard [41] chose 15 conflict criteria for the contractor evaluation, such as tender price, financial statement, quality, human resources, safety performance, etc. Afshar et al. [17] believed that the contractor’s prequalification is the key to the delivery of a construction project. They took financial resources, equipment, management and technical ability, experience, historical non-performance, personnel, quality, health and safety, environmental aspects, and current workload as contractor prequalification criteria. Through a literature review, the authors found the most cited criteria in the process of contractor evaluation and selection, and these are demonstrated in

Table 3. Most cited criteria in the process of contractor evaluation and selection.

Criteria	Ref.
Experience and Past Performance	Nieto-Morote, A. et al., 2012 [16] Afshar, M. R. et al., 2017 [17] Borujeni, M.P. et al., 2017 [41] Zhao, L. et al., 2020 [42]
Financial stability	Nieto-Morote, A. et al., 2012 [16] Borujeni, M.P. et al., 2017 [41]
Management Capability	Nieto-Morote, A. et al., 2012 [16] Afshar, M. R. et al., 2017 [17]
Reputation	Nieto-Morote, A. et al., 2012 [16] Xie, H. 2016 [43]
Technology	Nieto-Morote, A. et al., 2012 [16] Cheaitou, A. et al., 2019 [30]
Bid price	Cheaitou, A. et al., 2019 [30] Borujeni, M.P. et al., 2017 [41]
Human resources	Mahmoudi, A. et al., 2022 [23]
Time and cost overrun in past projects	Safa, M. et al., 2016 [44]

.

The discussion above has demonstrated the applicability of various approaches in tackling the multi-dimensional nature of the contractor evaluation and selection problem in the real world from different perspectives. It has revealed there is little research on contractor evaluation and selection that has adequately considered the attitude of the decision maker toward risk. Furthermore, there are no DEA-based approaches that can comprehensively evaluate and select contractors while trying to reduce risks using the DEA cross-efficiency model and the loss aversion model. To address this issue, this study aims to develop a DEA game cross-efficiency model with loss aversion for evaluating and selecting specific contractors in a given project. The competitiveness of the evaluation process is modeled using the game theory with respect to the adoption of the concept of cross efficiency. The attitude of the decision maker toward risks is tackled with the use of the loss aversion concept. As a result, the proposed approach can effectively screen available contractors through prequalification and adequately consider the attitude of the decision maker toward risk. This leads to effective selection decisions being made in various contract assessment situations. Based on this, this paper solves the problem of contractor selection through three steps: above all, the DEA game cross-efficiency approach is used to rank the contractors’ capabilities in order to screen the most capable contractor(s). Secondly, the risk preference generated by the decision maker is taken into account. And this paper analyzes the risk preference of decision makers by using loss aversion psychology. Finally, an objective optimization model is established.

3. Contractor Selection Procedure

This contractor evaluation and selection is an indispensable part of project service outsourcing. Choosing the wrong contractor can lead to the degradation of project quality, an increase in cost, and the delay of the project schedule. Selecting the right contractor is conducive to the saving and rational use of procurement funds and ensuring the quality of the procurement projects. Therefore, as the decision maker, they procure a project through public bidding and adopt a general contract with a fixed total price and predetermined construction period. Most contractor selection processes are mainly divided into the prequalification of bids, obtaining bids, the opening and evaluation of bids, and signing of the contract. Therefore, without loss of generality, the contractor selection procedure is defined as the following three stages. Each of these stages is detailed below (see Figure 1):

3.1. The Preparation Stage

The first is the preparatory stage. In this stage, there are three parts, which are to determine the bidding method and the bidding organization form, and to prepare the bidding announcement. There are usually three ways to facilitate the bidding method in a given situation, including open tendering, restricted tendering, and negotiation-based tendering [16]. In the open tendering process, the tenderer must specify the criteria and requirements for selection, and every individual contractor can apply for the project after a strict tender evaluation process, with the optimal bid being awarded the contract. In restricted tendering, contractors are usually screened with respect to specific requirements before their evaluation and selection in the first round. The contractors entering the next round of bidding generally have considerable strength, which is conducive to ensuring the success of public service outsourcing activities. With respect to negotiation-based tendering, specific contractors are often approached by the decision maker with specific requirements and expectations. Overall, open tendering is the most frequently used method for contractor evaluation and selection. Such a method, however, is often criticized due to the possibility of crowding out the appropriate contractor under some circumstances. Selecting the correct bidding method determines the contractor’s screening range. Then, there is a need to determine the bidding organization form. Self-bidding and commissioned bidding are two ways of organizing bidding activities. Self-bidding refers to the decision maker handling the entire bidding process, including publishing bidding notices, preparing prequalification documents and bidding documents, clarifying prequalification documents and bidding documents, organizing prequalification and bidding, establishing evaluation committees, evaluating and selecting winning bidders, etc. Commissioned bidding refers to the decision maker entrusting the entire or partial bidding activities to an agent, usually a professional bidding agency, who carries out the bidding activities according to the requirements of the client. The final step in the pre-preparation stage is to prepare and publish the bidding announcement, which is then followed by the bidding stage.

3.2. The Tendering Stage

In the tendering stage, the decision maker publishes a bidding announcement, and inspects the contractor’s qualifications. The qualifications include a range of selection criteria, including but not limited to, economic benefits, credit record, management skills, and engineering level. The contractor’s qualification determines whether the contractor has the ability to undertake public service outsourcing. Afterward, the decision maker will sell bidding documents to pre-qualified bidders and put forward specific requirements for the project to be bid on. The contractor’s objectives need to be aligned with those of the decision maker, if not, it will be difficult to maintain a cooperative relationship between the contractor and decision maker.

3.3. The Bidding Stage

In the bidding stage, the contractors shall decide whether to bid or not, and the bidding strategy according to the bidding documents, relevant construction standards, bill of quantities quota, equipment and materials market price, and other data information, combined with the bidder’s own construction cost, quality, and construction period. Afterward, by screening the bidding contractors, the decision maker has to select the contractors that meet the requirements. Then, the decision maker opens bids for contractors, and keeps track of the quotations from each contractor. Then, the candidate contractor needs to be evaluated [6]. In order to evaluate contractors comprehensively and scientifically, the decision maker should establish an evaluation committee, which should have professional skills and a teamwork spirit. According to the evaluation criteria, the decision maker selects suitable contractors under the guidance of the evaluation committee. Finally, if the previous steps are successfully completed, the decision maker will negotiate with the candidate contractor. And, if the two sides reach an agreement, they will sign the contract. After completing the contracted project, the candidate contractor should hand over the project rights after it passes the check and acceptance process with the approval of the municipal government.

The selection of contractors discussed in this paper belongs to BOT (Build-Operate-Transfer) projects. The BOT pattern is a cooperative model between the government and private contractors. It might be said that the BOT model is a typical principal–agent model, and the ownership and management are separated. This mode of operation makes the interests of the contractor and the contractor inconsistent. In the process of contractor selection, the potential contractors can use their own information advantage to hide their defects in funds or management ability. Some contractors save on production costs and increase their profit by cutting corners, which could cause great risks to the project. Therefore, each step of the contractor selection is essential; if an improper contractor is chosen, the project might not be completed on time and could cause huge rework costs. More seriously, it may become a jerry-built project because of the selection of the unqualified contractor.

4. The DEA Game Cross-Efficiency Model with Loss Aversion

4.1. The DEA Game Cross-Efficiency Model

The DEA method is a non-parametric method used to evaluate and compare the efficiency between multiple inputs and outputs [45]. However, the DEA method can only distinguish evaluated units into efficient and inefficient units, but cannot rank decision-making units (DMUs) in terms of superiority or inferiority. In order to solve the problem, Sexton et al. [46] proposed the cross-efficiency method based on the CCR model, which is an extended form of the DEA method, which is further improved by Doyle and Green [47]. In the DEA cross-efficiency model, not only is the efficiency between evaluated units considered, but also their relative efficiency compared to other units. This model allows each evaluated unit to be benchmarked against all other units, providing a more comprehensive efficiency assessment.

However, in many applications of DEA cross-efficiency models, there exists a certain competitive relationship among decision-making units. For example, during the bidding process of a project, contractors compete with each other to win the sole project, creating a certain competitive relationship among them. Then, Liang et al. [48] proposed a DEA game cross-efficiency model to address the issue of competitive peer-efficiency evaluation. In the DEA game cross-efficiency model, decision-making units are viewed as players in a game, and they take a non-cooperative stance, aiming to maximize their own scores (cross-efficiency values) as much as possible.

The contractors are in a competitive relationship, aiming to win bids while maximizing their own interests. Therefore, considering that previous DEA studies have performed few model competitions among peer decision-making units (DMUs) or contractors, this paper introduces the model of game cross efficiency proposed by Liang et al. [48] to select contractors in a competitive relationship.

In the DEA game cross-efficiency model, the decision-making units (DMUs) are viewed as players in a game. The DEA cross-efficiency scores are regarded as payoffs, whereas each DMU makes great efforts to maximize (worst possible) its payoff. We define the following parameters and decision variables:

• $n$: total number of optional contractors (alternatives) in the construction projects;
• $m$: quantity of contractor’s input indicators;
• $s$: quantity of contractor’s output indicators;
• x_ij: the input index i of contractor j, with i = 1, 2, …, m, and j = 1, 2, …, n;
• $y_{rj}$: the output index $r$ of contractor j, with $r=1,2,\dots ,s$, and j = 1, 2, …, n;
• $v_i$: the weight of input indicators $i$, with i = 1, 2, …, m;
• $u_r$: the weight of output indicators $r$, with $r=1,2,\dots ,s$.

In the conventional DEA, the contractor efficiency score of the whole system for the contractor d can be calculated as follows,

$$\max \hspace{1em}\frac{{\displaystyle {\sum }_{r=1}^s{u}_r^d{y}_{rd}}}{{\displaystyle {\sum }_{i=1}^m{v}_i^d{x}_{id}}}$$

(1)

$$s.t.\hspace{1em}\frac{{\displaystyle {\sum }_{r=1}^s{u}_r^d{y}_{rj}}}{{\displaystyle {\sum }_{i=1}^m{v}_i^d{x}_{ij}}}\le 1,\hspace{1em}j=1,2,\dots ,n,$$

$$v_i^d, u_r^d\ge 0,\hspace{1em}i=1,2,\dots ,m; r=1,2,\dots ,s.$$

In order to simplify the calculation, model (1) can be transformed into the LP model as follows,

$$\max \hspace{1em}{\displaystyle \sum _{r=1}^s{\mu }_r^d{y}_{rd}}={\theta }_d$$

(2)

$$s.t.{\displaystyle \sum _{i=1}^m{\omega }_i^d{x}_{ij}}-{\displaystyle \sum _{r=1}^s{\mu }_r^d{y}_{rj}}\ge 0,\hspace{0.33em}\hspace{1em}j=1,2,\dots ,n,$$

$${\displaystyle \sum _{i=1}^m{\omega }_i^d{x}_{id}}=1,$$

$${\omega }_i^d, {\mu }_r^d\ge 0,\hspace{1em}i=1,2,\dots ,m; r=1,2,\dots ,s.$$

For each contractor $d$ under the evaluation, we obtain a set of weights ${\omega }_i^{d*}$ $\left(i=1,2,\dots ,m\right)$, ${\mu }_r^{d*}$ $\left(r=1,2,\dots ,s\right)$. Using this set of weights, the $d$-cross efficiency for any contractor $j$ can be calculated as,

$$E_{dj}=\frac{{\displaystyle {\sum }_{r=1}^s{\mu }_r^{d*}{y}_{rj}}}{{\displaystyle {\sum }_{i=1}^m{\omega }_i^{d*}{x}_{ij}}},d,j=1,2,\dots ,n.$$

(3)

For each contractor $j$, the average of the $d$-cross efficiency is as follows,

$${\overline{E}}_j=\frac{1}{n}{\displaystyle \sum _{d=1}^n{E}_{dj}}.$$

(4)

However, the cross-efficiency model yields non-unique solutions. According to Liang et al. [48] and Sun et al. [49], we assume the contractors are players of a non-cooperative game, and compete with each other. Suppose that in the sense of a non-cooperative game, a player contractor $d$ obtains an efficiency score ${\gamma }_d$ (${\gamma }_d\le 1$), which cannot be decreased when another contractor $j$ tries to maximize its own efficiency. Then, the game $d$-cross efficiency of the overall system for contractor $j$ relative to contractor $d$ is defined as

$${\gamma }_{dj}=\frac{{\displaystyle {\sum }_{r=1}^s{\mu }_r^d{y}_{rj}}}{{\displaystyle {\sum }_{i=1}^m{\omega }_i^d{x}_{ij}}},d,j=1,2,\dots ,n,$$

(5)

where ${\mu }_r^d$ $\left(r=1,2,\dots ,s\right)$, ${\omega }_i^d$ $\left(i=1,2,\dots ,m\right)$ are the optimal weights of the following game d-cross efficiency model (6).

For each contractor $j$, the game d-cross efficiency model is formulated as follows,

$$\max \hspace{1em}{\displaystyle \sum _{r=1}^s{\mu }_{rj}^d{y}_{rj}}$$

(6)

$$s.t.{\displaystyle \sum _{i=1}^m{\omega }_{ij}^d{x}_{il}}-{\displaystyle \sum _{r=1}^s{\mu }_{rj}^d{y}_{rl}}\ge 0,\hspace{0.33em}\hspace{1em}l=1,2,\dots ,n,$$

$${\gamma }_d\times {\displaystyle \sum _{i=1}^m{\omega }_{ij}^d{x}_{id}}-{\displaystyle \sum _{r=1}^s{\mu }_{rj}^d{y}_{rd}}\le 0,\hspace{0.33em}$$

$${\displaystyle \sum _{i=1}^m{\omega }_{ij}^d{x}_{ij}}=1,$$

$${\omega }_{ij}^d, {\mu }_{rj}^d\ge 0,\hspace{1em}i=1,2,\dots ,m; r=1,2,\dots ,s.$$

assume that ${\mu }_{rj}^{d*}\left({\gamma }_d\right)$ is an optimal solution to model (6), for each contractor $j$, the average game cross-efficiency is defined as ${\gamma }_j=\left(1/n\right){\displaystyle {\sum }_{d=1}^n{\displaystyle {\sum }_{r=1}^s{\mu }_{rj}^{d*}\left({\gamma }_d\right)y_{rj}}}$.

By referencing the algorithm proposed by Liang et al. [48], we can find the final game cross efficiency.

Step 1: solve model (2), and obtain a set of original DEA cross-efficiency scores defined in Equation (4). Let $t=0$ and ${\gamma }_j={\gamma }_j^0={\overline{E}}_j$.

Step 2: solve model (6), let ${\gamma }_j^1=1/n{\displaystyle {\sum }_{d=1}^n{\displaystyle {\sum }_{r=1}^s{\mu }_{rj}^{d*}\left({\gamma }_d^0\right)}}{y}_{rj}$, or in a general format, ${\gamma }_j^{t+1}=1/n{\displaystyle {\sum }_{d=1}^n{\displaystyle {\sum }_{r=1}^s{\mu }_{rj}^{d*}\left({\gamma }_d^t\right)y_{rj}}}$, where ${\mu }_{rj}^{d*}\left({\gamma }_d^t\right)$ represents the optimal value of in model (6) when ${\gamma }_d={\gamma }_d^t$.

Step 3: If $\left|{\gamma }_j^{t+1}-{\gamma }_j^t\right|\ge \epsilon$ for some $j$, where $\epsilon$ is a specified small positive value, then let ${\gamma }_d={\gamma }_d^{t+1}$ and go to step 2. If $\left|{\gamma }_j^{t+1}-{\gamma }_j^t\right|<\epsilon$ for all $j$, then stop. Then, the ${\gamma }_j^{t+1}$ is the best average game cross efficiency to $DM{U}_j$.

The solution of the DEA game cross-efficiency model is a Nash equilibrium solution, which solves the non-unique problem of cross-efficiency solutions.

4.2. The Model with Loss Aversion

In all countries, the decision maker pays a lot of attention to the contractor selection of public projects. The wrong decision may lead to disruptions to the development of the economy [36]. Considering the bounded rational psychology and choice risk of decision makers in real life, and inspired by Tversky and Kahneman [50], loss aversion is introduced into a DEA game cross-efficiency model. According to the descriptive foundations of value function proposed by Tversky and Kahneman [50], the objective optimization model is defined as follows:

$$v\left({\gamma }_j\right)=\left\{\begin{array}{cc}{\left({\gamma }_j-{\gamma }_j^0\right)}^{\alpha },& if{\gamma }_j-{\gamma }_j^0>0\\ -\lambda {\left({\gamma }_j^0-{\gamma }_j\right)}^{\beta },& if {\gamma }_j-{\gamma }_j^0\le 0\end{array}\right.\begin{array}{cc},& j=1,2,\dots ,n\end{array}$$

(7)

where the average game cross efficiency ${\gamma }_j$$\left(j=1,2,\dots ,n\right)$ is the outcome of the value function. Solve model (2) and obtain a set of original average DEA cross-efficiency scores defined in (4), and regard ${\gamma }_j^0$ as the reference point of the prospect utility. The parameter $0<\alpha <1$ and $0<\beta <1$ represent the risk preference coefficient for gain and loss, respectively, which means the higher the coefficient of risk preference, the more risks the decision maker would tend to take. The parameter $\lambda$ is the index of loss aversion, and it has been proven that the index of loss aversion is estimated to be between 1.75 and 2.5 [51], measuring people’s relative distaste for loss while comparing very small losses to very small gains [52,53]. And when $\lambda >1$, the decision maker is more sensitive to the loss. And they are loss-neutral when $\lambda =1$.

The integrated approach presented above can be summarized as follows:

Step 1: determine a set of contractors for evaluation; that is to say, select a set of decision-making units (DMUs).

Step 2: determine a set of evaluation indices that could influence the operating efficiency of a construction project.

Step 3: select appropriate input indicators and output indicators for calculating the DEA game cross-efficiency approach, as expressed in Equation (6).

Step 4: regard the average game cross efficiency as the reference point of the objective function and calculate the objective function value, as expressed in Equation (7).

Step 5: find the most appropriate contractor for the decision maker.

5. Implementation and Validation of Methods

In order to validate the proposed methodology, a numerical example is used to evaluate and select contractors. As far as a construction project is concerned and, in particular, under a build–operate–transfer (BOT) project, research on infrastructural projects has attracted considerable attention in the area of public management. It is noted that infrastructural projects have been an important influence on the development of the economic and social environment.

5.1. Indicators Selection and Data Source

The evaluation and selection of a contractor for a construction project are subject to various factors and involve various challenges for the decision makers. Therefore, selecting a contractor based solely on the lowest bid price can never be a wise decision. According to Table 3 and the characteristics of BOT projects, the contractor selection criteria are divided into two aspects:

–

For the contractor: refers to the indicators that can illustrate the properties of the contractors, such as manpower, financial stability, operation time, and bid price. These indicators are considered as the input indicators of the model.

–

For the project: refers to the indicators that can describe the properties of the contract project, such as schedule and franchise period. These indicators are considered as output indicators of the model.

This paper takes the example that 10 contractors are involved in the competition of a construction project. without loss of generality, using the following evaluation criteria for the case $\left(j=1,2,\dots ,10\right)$, the following indicators were as follows:

• Manpower (defined with), with the corresponding performance value $x_{1j}$ measured in people for the $j$th contractor and $j=1,2....,10$, which can be used to measure workforce level.
• Financial stability (defined with $i=2$), with the corresponding performance value $x_{2j}$ measured in million dollars for the $j$th contractor and $j=1,2....,10$, which is defined as the registered capital of the enterprise, and it represents the ability to take risks.
• Operation time (defined with $i=3$), with the corresponding performance value $x_{3j}$ measured in years for the $j$th contractor and $j=1,2....,10$. The operation time criterion refers to the capability to undertake the project of the contractor; here, this paper uses the operation time of the contractor to represent the ability of an enterprise to operate.
• Bid price (defined with $i=4$), with the corresponding performance value $x_{4j}$ measured in million dollars. The bid price is offered by the contractors in accordance with the conditions of the articles of association of the bidding documents to complete the bidding policy plan, start construction, complete it, and repair any defects.
• Schedule (defined with $r=1$), with the corresponding performance value $y_{1j}$ measured in years for the $j$th contractor and $j=1,2....,10$. The schedule criterion is mainly concerned with the completion time of the infrastructural project.
• Franchise period (defined with $r=2$), with the corresponding performance value $y_{2j}$ measured in years. The decision maker allows the contractor to operate after completion of the project.

Table 4. Input–output data for 10 contractors.

$\mathit{j}$	$\mathbf{Manpower} {\mathit{x}}_{1\mathit{j}}$	$\mathbf{Financial} \mathbf{Stability} {\mathit{x}}_{2\mathit{j}}$	$\mathbf{Operation} \mathbf{Time} {\mathit{x}}_{3\mathit{j}}$	$\mathbf{Bid} \mathbf{Price} {\mathit{x}}_{4\mathit{j}}$	$\mathbf{Schedule} {\mathit{y}}_{1\mathit{j}}$	Franchise Period ${\mathit{y}}_{2\mathit{j}}$
1	822	710	10	40	10	25
2	1212	1465	7	53	5.5	25
3	1350	1176	21	55	8	30
4	735	395	15	47	6.5	25
5	967	954	13	48	6	30
6	1187	856	10	51	8	25
7	1345	1267	30	60	7	25
8	807	1457	20	62	5.5	30
9	560	665	18	50	7	25
10	850	987	10	46	9	25

shows the input–output data for 10 contractors. According to Table 4, the manpower ranges from 560 people (see contractor 9) to 1350 people (see contractor 3), the financial stability ranges from USD 395 million (see contractor 4) to USD 1465 million (see contractor 2), and the operation time ranges from 7 years (see contractor 2) to 30 years (see contractor 7). The bid price ranges from USD 40 million (see contractor 1) to USD 62 million (see contractor 8). The schedule ranges from 5.5 years (see contractor 2 and contractor 8) to 10 years (see contractor 1). Finally, franchise period ranges from 25 years to 30 years.

In the case of the DEA game cross-efficiency model, manpower, financial stability, operation time, and bid price are the input variables, which are the fundamental indicators of choosing a contractor for the infrastructure project. The schedule and franchise period are output variables, which are used to evaluate the final result after completing the project.

5.2. Contractor Evaluation and Selection Result

The input–output data of performance indicators for contractors are given in Table 4; the proposed DEA game cross-efficiency model (6) and objective function model (7) are applied to the data, and the results are calculated for each contractor (implementd in Matlab R2023b, The MathWorks, Inc., Natick, MA, USA). The obtained results are presented in

Table 5. The DEA cross-efficiency scores (${\gamma }_j^0$) and ranks, the DEA game cross-efficiency scores (${\gamma }_j$) and ranks.

Contractors	${\mathit{\gamma }}_{\mathit{j}}^0$	Ranking	${\mathit{\gamma }}_{\mathit{j}}$	Ranking
1	0.7821	3	0.8902	3
2	0.6729	6	0.7466	7
3	0.5819	8	0.6734	8
4	0.8310	1	0.8944	2
5	0.7104	5	0.8359	4
6	0.7365	4	0.7996	5
7	0.4186	10	0.4959	10
8	0.5545	9	0.6686	9
9	0.6394	7	0.7597	6
10	0.8284	2	0.9061	1

. From Table 5, it can be observed that the DEA cross-efficiency method considers the fourth contractor as the highest-scoring contractor, while the DEA game cross-efficiency model identifies the first contractor as the highest-scoring one. And Table 5 shows that there are differences in the evaluation and ranking of contractors between the DEA cross-efficiency model and the DEA game cross-efficiency model.

The DEA game cross-efficiency scores in the fourth column of Table 5 are the results obtained through the algorithm after three iterations. Figure 2 shows that after three iterations, the results tend to be stable and a unique solution is obtained, which solves the non-unique problem of the cross-efficiency model.

Finally, we compare the ranking difference between the DEA game cross-efficiency model and the model with loss aversion. The ranking differences between the results of the DEA game cross-efficiency model and the DEA game cross-efficiency model with loss aversion are given in

Table 6. The differences in ranking between the two methods.

Contractors	Ranking		Ranking Difference
	Without Loss Aversion	With Loss Aversion
1	3	4	1
2	7	8	1
3	8	5	−3
4	2	9	7
5	4	1	−3
6	5	10	5
7	10	7	−3
8	9	3	−6
9	6	2	−4
10	1	6	5

. And the ranking differences between the two methods are listed in the last column of Table 6. The results in Table 6 show that decision makers’ risk preferences affect the results of contractor selection.

In order to analyze the influence of the parameters, we perform a sensitivity analysis of loss aversion with respect to the values of parameters of $\alpha$, $\beta$, and $\lambda$, respectively. According to the definition of the loss aversion model and the value of the game cross efficiency, it can be found that $\beta$ and $\lambda$ make no difference to the results. Thus, only sensitivity analysis is needed for the $\alpha$ parameter.

Without loss of generality, when the risk coefficient $\alpha$ for gains can be chosen in the set {0.1, 0.3, 0.5, 0.7, 0.9}, the other parameters $\beta =0.5$ and $\lambda =2$ remain unchanged. The consistency of these efficiency results based on different $\alpha$ and values is shown in Figure 3.

For each value of $\alpha$, we have solved the DEA game cross-efficiency model with loss aversion and obtained the optimally selected fifth contractor. The sensitivity of the loss aversion model to parameters $\alpha$ is shown in Figure 3. It is apparent that the contractor’s final selection remains unchanged as the risk coefficient for gains increases.

The research method proved in this experiment that risk appetite will have an impact on decision makers’ choice results. In the meantime, the sensitivity analysis shows that the result is not affected by the risk coefficient. Therefore, in the process of contractor selection, it is very important to consider the risk appetite of decision makers.

6. Conclusions

The issue of contractor evaluation and selection is a common one in leadership decision making, as highlighted in numerous studies. However, the approach taken in this paper toward contractor evaluation and selection sets it apart from others, as it takes into account the competitive dynamics among contractors and the decision-makers’ risk preferences. To ensure fair and impartial competition among contractors, decision makers must consider various indicators, making the selection process more complex. This paper introduces a novel approach to assist decision makers in selecting contractors, specifically a DEA game cross-efficiency model with loss aversion. This method not only accounts for the competitive dynamics between contractors but also incorporates the risk preferences of decision makers.

The proposed method first utilizes a DEA game cross-efficiency model to comprehensively evaluate contractors engaged in competition with each other, thereby avoiding subjective selection biases. Subsequently, the paper addresses decision makers’ risk preferences through the incorporation of the loss aversion concept. Unlike previous studies, this utility evaluation approach can rank contractors using both objective data and psychological factors. By integrating the characteristics of the cross-efficiency model and the loss aversion concept of DEA game theory, this paper offers a combined solution to the contractor selection problem.

The effectiveness of our method was assessed through a numerical simulation involving ten contractors. The findings indicate discrepancies in the assessment and ranking of contractors when comparing the DEA cross-efficiency model to the DEA game cross-efficiency model, suggesting that game theory influences contractor evaluation and selection. Furthermore, the results demonstrate that the DEA game cross-efficiency model serves as a Nash equilibrium solution, addressing the issue of non-uniqueness in cross-efficiency solutions. Moreover, the case study highlights that apart from external risks and competition among contractors, decision makers’ risk preferences also play a role in contractor evaluation and selection. These outcomes underscore the utility of the proposed method as a valuable decision-making tool for aiding decision makers in selecting the most suitable contractor.

While this study introduces a novel DEA game cross-efficiency model with loss aversion, it is important to note that it focuses primarily on risk preference and a limited number of evaluation indicators. A key limitation of this research is the exclusive consideration of risks faced by decision makers, whereas contractors also encounter various risks during project execution, such as construction accidents and inexperienced workers. Additionally, the relationship between efficiency and the loss aversion model could benefit from more robust methodologies. These identified limitations offer opportunities for future research endeavors, which should aim to employ more rigorous methodologies in exploring the connection between the DEA game cross-efficiency and loss aversion model.

A more rational way should be found to evaluate contractors for construction clients and for public officers in the case of public–private partnerships. In addition, the proposed method should be compared with the traditional DEA method, AHP, and other traditional methods, while it is also necessary to identify the advantages and disadvantages among these methods.