Risk Allocation Optimization between Owner and Contractor in Construction Projects by Using the UTA-STAR Method

Abstract

Project risk is an uncertain situation or event that, if it occurs, may have a negative or positive effect on one or more project objectives, such as scope, schedule, cost, and quality. Major industrial projects are increasingly facing complexity and uncertainty. The scope of this paper is related to petrochemical projects, in which risks directly affect the approved time, cost, and quality of the project. In such projects, there are risks that neither the owner nor the contractor has the main role in the occurrence or prevention of, and it is not easy to determine who is responsible for them. In such projects, there are risks that neither the owner nor the contractor has the main role in the occurrence or prevention of, and for which it is not easy to determine responsibility. Therefore, predicting, identifying, analyzing, and determining of the optimal allocation of risk responsibility between contracting parties is one of the most important steps before the start of the project. Suppose it is not correctly allocated among project stakeholders, then, in that case, risk responsibility imposes costs on the project that must be paid by the owner, contractor, and partnership, causing, in general, many problems for project management. Therefore, this paper presents a model to calculate the optimal ratio of risk allocation between the project parties in the concluding contract stage, using the UTA-STAR technique to obtain the owner and contractor utility function to create as much of a win-win relationship between them as possible.

1. Introduction

Risk management is a new branch of management science that covers a wide variety of trends, including finance and investment, business, insurance, safety, health and medicine, industrial and construction projects, and even political, social, and military issues [1]. Project risk management includes guiding risk management planning, identification, analysis, response planning, and risk control in a project [2]. The goals of project risk management are to increase the probability and effect of positive events and reduce the probability and impact of negative events on the project. Risks in the project are unknown events or situations that, if they occur, have negative or positive consequences on the project objectives [3,4]. Each event or situation has specific causes and recognizable results and outcomes. These events’ values directly affect the project’s time, cost, and quality [5]. Therefore, identifying the risk and determining the extent of its positive and negative consequences on the project objectives is of particular importance [6,7,8].

Construction projects always involve a high volume of uncertainties and the presence of factors with conflicting interests, which mean that achieving project goals depends on the efficiency of project risk allocation [9,10]. In project environments, project managers must completely understand the concept and nature of risk [11,12]. Experience shows that mega projects contain strategic, technical, economic, and national elements [13,14,15]. Threats and opportunities related to the project’s key features need to be faced to achieve predetermined goals, including time, cost, and quality [16,17]. The roots of these threats and opportunities are detected in unreliability and uncertain situations with different origins, such as technical, management, commercial, and national and international problems [18].

Traditionally, in design and construction contracts, the employer seeks to transfer risk to the contractor [19]. Imposing potential costs entails conservatism in design and legal claims for the contractor. Such defensive strategies increase the duration and cost of projects [20]. According to risk management principles, risk should be allocated to the party that manages and reduces it best [21,22].

Risk can be defined as any unpredictable situation that can hinder the project’s success in achieving time, cost, or quality goals [23,24], and risk allocation is the definition of the division of responsibilities and benefits from possible unplanned conditions [23,24]. Risk allocation provides a framework that specifies in detail which factor should be responsible for managing what risk or, in contrast, determines what incentives or guarantees should be granted to which factor to control a particular risk [24,25]. This framework should assign each identified risk in a construction project to the factor with the most control over that risk, thus ensuring that it does not occur or has the least likely impact [26]. Due to different perceptions and conflicting interests of project actors, risk allocation negotiations to select the optimal risk allocation is a difficult and costly process [27,28]. Contract risk allocation can greatly impact the project’s cost, time, and quality [29]. However, it should be borne in mind that unilateral and unbalanced risk allocation causes the contractor to adopt defensive strategies and ultimately leads to delays in scheduling, increased costs, and financial loss to the employer [30]. The quality of risk management depends largely on how the risks are allocated [31], since risk allocation determines how risk reduction tools in the form of resources, risk owners, and mechanisms are provided and used properly [32,33].

Naturally, each activity faces risks in projects, and doing major projects under various situations has different risks [34,35]. Identifying and analyzing these risks and planning and making proper strategies for dealing with them can determine a project’s success [36,37]. Being aware of the project risks is necessary for a better assessment of the necessary time and cost to do the project [38,39], and, in fact, having access to the aforementioned awareness helps investors and beneficiaries to select and do the project [40,41]. Lack of correct recognition of these threats and lack of proper plans to face them can stop the project from accessing its pre-defined goals [42,43].

Hiyassat et al. (2022) investigated risk allocation in public construction projects in their research. They stated that the correct identification, evaluation, and allocation of risk could reduce the cost and time delay, and their results indicated that the risk factors in the top ranking are delayed customer payments, improper contract forms, competition, delayed approval of permits, subcontractor defaults, unclear specifications, material price fluctuations, different construction standards, and change [29]. In their research, Xu et al. (2018) investigated owner risk allocation and contractor role behavior in a project: a parallel mediation model. They found that risk allocation affects contractor role behavior through the contractor’s sense of trust, not the contractor’s trust in the owner. Feelings of trust partially mediated the effect of risk allocation on the contractor’s in-role (i.e., contractual) behavior and fully mediated the effect on extra-role behavior [44]. In their research, Aminzadeh et al. (2015) examined the identification and prioritization of effective risks in construction projects and their causes. They stated that project risk is an uncertain situation or event that, if it happens, may have an effect. It has a negative or positive impact on project goals, such as scope, timing, cost, and quality. Their research showed that financial risks, timing, executive quality, and environment are among the effective risks in the carrying out of construction projects. Effective risks in construction projects are different, and timing risk and execution quality risk are the most effective risks in construction projects. Internal and external factors significantly affect construction project risk, and external factors are the best predictors of construction project risk [45]. Gido and Clements (2014) defined project risk as an uncertain event or condition that, if it occurs, has a positive or negative effect on one or more project objectives, such as scope, schedule, cost, or quality. Risk management refers to the systematic processes of planning risk management, identifying, analyzing qualitatively and quantitatively, and responding to, monitoring and controlling risks [46].

In their research, Shi et al. (2019) investigated double moral hazard and risk sharing in construction projects. They stated that principal-agent theory has proven to be an effective theoretical device in designing optimal risk-sharing rules and provides contractors with appropriate incentives to improve project efficiency. Offers and construction projects are fraught with risks that owners and contractors must share. According to the principal-agent theory, it is optimal to let the contractor bear all the risk if it is risk neutral [47]. Mock and O’Connor (2019), in their research, stated that owners and contractors identify a distinct set of solution strategies for common industrial startups and startup solution strategies as high value and low effort, with some overlap [48]. Lu (2020), in his research, analyzed the characteristics of owners and contractors and their game relationships and behavioral choices in project safety management, and, based on the problems of owners and contractors in production safety management, he proposed a model in which owners and contractors are jointly committed to the project [49].

Therefore, according to the above, many researchers have used the multi-criteria decision-making technique [50,51,52]. Thus, the necessity of conducting this research can be expressed as follows: in the turbulent environments of developing countries, including Iran, full of rapid changes, uncertainties and risks, the conditions for survival and success are knowledge and understanding of the environment and correct procedure, and also the ability to make quick, effective decisions. Suppose the internal and external risk factors are not identified in the project. In that case, the error of management decisions and time and cost estimation increase. Furthermore, the incorrect allocation of risk can increase the price offered in tenders and, as a result, lead to failure in tenders and even in the case of winning tenders. It increases the cost to the employer. Since contractors try to reduce risk by considering precautionary values in their bids to face possible risks, this raises the project’s price. However, with the correct and fair allocation of risks between the employer and the contractor, it is possible to establish a win-win relationship between them and reduce the resources needed to resolve disputes, reduce project costs and shorten delivery time. Research is clearer than ever before. Therefore, conducting this research can help to solve such problems in projects, especially projects in the petrochemical industry.

Makui and Momeni (2012) investigated the use of CSW weights in the UTA-STAR method in their research. They described tools for solving decision problems. Since the preferences of the decision maker (DM) about alternatives are not considered in classical DEA, some researchers have tried to consider this in DEA. The UTA-STAR method is one of the techniques widely used in multi-criteria decision analysis. In this technique, the decision maker’s preferences regarding the options are considered, and UTA-STAR tries to calculate the most appropriate weights for the criteria and alternatives to obtain the utility function with the minimum deviation from the preferences. The purpose of this paper is to interpret the decision maker’s preferences in the UTA-STAR method in a new way by using the common set of weights (CSW) in DEA [53]. Therefore, there are several techniques for making the best decision in multi-criteria decision making. In this study, we use the UTA-STAR Method to evaluate and analyze the decision utility function. This technique was first introduced in 1980 by YannisSiskos [54]. Since then, this method has been used in various fields. This model makes it possible to estimate the decision-making utility function and requires only an initial ranking of options [55].

Conventional decision models are suitable for situations where comparisons and decisions between options are made by only one decision-maker and are based on several recognizable criteria [56]. Risk allocation negotiations, influenced by several factors, have at least two different decision-makers and give rise to situations in which decisions of one decision-maker require paying attention to the behavior of the other party and necessitate the understanding of both parties [57,58,59]. The UTA-STAR method is used in complex processes with several decision-makers [60,61]. Furthermore, post-optimization sensitivity analysis is performed to make the model more valid [62]. This mode cannot solve the problem of the dependence of indicators on each other while estimating the utility function [63]. The disadvantages of this method are its complex calculations, which can be solved using software designed for this purpose and UTA. The main problem with its derivatives is that one might end up with several optimal answers. In their research, Ghannadpourand Moradi Manesh (2020) investigated the estimation of the utility function of sustainable project selection using a combined approach of SBSC, ANP, and UTA-STAR (Case study: Saipa Company). They stated that the purpose of this study was to design an effective three-step approach for estimating the utility function of project selection. Based on the principles of sustainable development, the results of this study, in addition to identifying the key criteria of sustainable development and classifying them in the form of sustainable, balanced scorecard funds, achieved a prioritization model and optimal project selection for current and future projects [64].

Makubi et al. (2008), in their study, examined the appropriate risk allocation to the employer and the contractor in the implementation of projects. They stated that project implementation is almost impossible in risk-free conditions. The consequences of the risks directly affect the approved time, cost, and quality of the project, so they usually impose charges on the project that the employer and the contractor must pay or pay in partnership. Hence, the issue of determining the risk manager is one of the most challenging issues in contracts. Since the parties to the contract face conflicting goals in allocating risks, each party tries to reduce its risk and transfer it to the other party. In such cases, the art of risk management is the proper and appropriate allocation of risks in a way that best achieves the goals of all stakeholders. If this management is done efficiently, not all risks are unfairly transferred to one party, but, instead, the risks are assigned between the parties, based on their ability to control and guarantee the risk. However, suppose the responsibility is not shared properly. In that case, it can lead to improper and untimely implementation of the project, increase the contract price, cause conflicts, and excessive losses of one of the parties, and, thus, reduce the incentive to invest and participate in the implementation of new projects [65]. Makoei and Momeni (2012), in their research, investigated the use of CSW weight in the UTA-STAR method. Several researchers have considered similarities between Multi-Criteria Decision Making (MCDM) and Data Envelopment Analysis (DEA) tools for solving decision-making problems. As the preferences of the decision-maker (DM) on alternatives are not considered in classical DEA, some researchers have tried to consider them in DEA. The UTA-STAR method is one of the techniques widely used in Multi-Criteria Decision Analysis. In this technique, the preferences of the decision-maker on alternatives are considered, and UTA-STAR tries to compute the most suitable weights for criteria and alternatives to obtain a utility function having a minimum deviation from the preferences. This paper aims to interpret the decision makers’ preferences in the UTA-STAR method in a new manner, using the common set of weights (CSW) in DEA [53].

Issa et al. (2015) concluded, in a study entitled “Risk Allocation Model for Infrastructure Projects in Yemen”, that infrastructure projects are risky due to their complexity and dynamic environment. The risk allocation model is widely used for allocating sensitive risks and accountability of project managers and comparing projects with each other in terms of risk-taking. Fifty-four risks in ten groups were used to develop the model. The most important risks were assigned and ranked (30 risks by Delphi method) between the manager or owner or shared between the parties. The results showed that understanding and using this model is easy for the parties to the contract. It helps decision-makers to make the right decision about choosing between different projects based on risk factors in the bidding phase. The risk allocation model enables risk management [66]. Zhang et al. (2016), in a study entitled “Participatory behavior in construction projects,” assessed the effects of risk allocation on contracts and concluded that risk allocation ultimately leads to value for money and value creation [67]. Mastorakis and Siskos (2016) implemented the UTA-STAR method to assess 192 therapeutic categories for investment purposes in the Greek pharmaceutical market [68]. Grigoroudis et al. (2012) used the UTA-STAR method to aggregate the marginal performance of Key Performance Indicators [69]. Altıntaş (2021) analyzed the G7 rule of law performance using the UTA technique [70]. In their research, Abu Tarabi et al. (2014), presented a model for safety risk assessment in the construction industry using multi-criteria gray decision making. The results showed that the gray multi-criteria decision-making method is a useful and effective tool in risk ranking, compared to other MCDM (multi-criteria decision-making) methods, when the number of samples is low. Conclusion: Due to simple calculations and there being no need to define the membership function, the proposed method is a method that is preferable to the fuzzy method, statistics and probability in conditions of uncertainty, and for small numbers of samples [71]. Specifically, we developed a research model to investigate how a win-win relationship between owner and contractor in petrochemical construction projects affects risk allocation. The research model and findings of an empirical study conducted on construction projects in Iran show the basic mechanisms that lead to the positive effects of risk allocation and trust in the owner–contractor relationship.

The research could be divided into two main groups: preferences in uncertain conditions and preferences in certain conditions. The preferences function is called as a utility function in uncertain conditions and a value function in certain conditions [72].

Due to the probable feature of this paper’s discussion topic, the utility function was used to indicate preference relations.

Within the framework of multi-criteria decision aid under uncertainty, Siskos (1983) [73] developed a specific version of UTA (Stochastic UTA), in which the aggregation model, to infer from a reference ranking, is an additive utility function.

Table 1. Literature review from aspects of both theory and application.

Study	Method	Key Findings
Beuthe and Scannella(2001)	Utility additive (UTA) multi-criteria method	The results of their research state that reference projects should be carefully selected to extract the maximum possible information from the decision maker. It also presents the results of simulations based on utility functions, including the interdependence between criteria, and shows that UTA manages this problem effectively by adjusting its coefficients. [63]
Makui and Momeni (2012)	CSW weights in the UTA-STAR method.	They interpret the decision maker’s preferences in the UTA-STAR method in a new way by using the common set of weights (CSW) in DEA [53].
Grigoroudis et al. (2012).	Based on an MCDA approach, where the UTA-STAR method is used to aggregate the marginal performance of KPIs.	Their results can help the organization evaluate and revise its strategy and adopt modern management approaches in daily practice [69].
Mastorakis and Siskos (2016)	Regression method UTA-STAR	An extreme ranking analysis is implemented, calculating each alternative’s best and worst possible position in the ranking [68].
Xu et al. (2018)	Questionnaire	Their results showed that risk allocation affects the contractor’s role behavior through the contractor’s sense of trust, but not the contractor’s trust in the owner [44].
Mock and O’Connor (2019)	Ratings for the relative values provided by strategies via a survey.	Owners and contractors identify distinct sets of CSU solution strategies as high value and low effort, with some overlap [48].
GhannadpourandMoradiManesh (2020)	Using Sustainable Balanced Scorecard (SBSC), Analyzing Networks Process (ANP) and estimating the utility function of reference projects according to the results created and using the UTA-STAR method.	The results of this research, in addition to identifying the key criteria of sustainable development and classifying them in the form of sustainable, balanced scorecard funds, addressed achieving a prioritization pattern and choosing the optimal project for current and future projects [64].
Lu (2020)	Descriptive and analytical	Providing a model in which owners and contractors are jointly committed to the project [49].
Our study	UTA-STAR Method In uncertainty	In this research, we considered the actual cost to complete the project as a continuous variable (calculation of the expected utility of the employer and the contractor for each deviation from the target cost) and, unlike previous studies, determined the win-win ratio through mathematical reasoning. We used a multi-criteria utility function.

provides a summary of the researches and the proposal of the current research:

Risk management tools are limited due to the complexity and variety of devices, and their data is incomplete. Due to limited resources, risk management information is usually vague, subjective, or even incomplete, and the distribution of samples is unknown. Therefore, in this study, a complex but transparent approach was used in a way that helps the project decision process. This research optimizes the allocation of risk between the owner and the contractor in the search for innovation and developing knowledge frontiers for value creation. So far, no model has been presented in Iran to optimize the risk allocation between the owner and the contractor in construction projects using the UTA-STAR method. Regarding the structure of the article, after the introduction, we deal with problem modeling in the second part, and, in the third part, we conclude.

2. Problem Modeling

2.1. Model Assumptions

In the present research, a model is given for determining the risk-sharing relationship between owner and contractor. So, this paper is based on clear assumptions that there are some risks in the projects that none of the sides are responsible for and cannot control or reduce. Furthermore, each of the two sides wedded in the project tries to reduce their negative risks and transfer them to the opposite side. On the other hand, because using this model of owner and contractor utility functions is based on getting information from the decision-maker, the assumption is that each decision-maker acts logically in his decision-making. This paper is based on the proven assumption that all kinds of multi-attribute utility functions are changeable to additive form. Additive form:

$$u \left(A\right)={\displaystyle \sum }_{j=1}^n{a}_{j }u \left(rj\right)$$

(1)

2.2. Determination of Owner Cost Function and Contractor Profit

By agreement between owner and contractor on the division of the real cost of doing the project from the contracted price, and keeping in mind the risk allocation relationship in owner paid cost formulae and contractors received profit, the formulae could be written as below:

$$C=x+b+p_c \left(x_0-x\right)$$

(2)

$$p=b+p_c \left(x_0-x\right)$$

(3)

where:

• C: the cost that the owner must pay.
• p: contractor received profit
• $x_0$: goal cost (the contract price without undermining the contractor’s profit)
• x: the real cost of doing the project
• b: the profit of the goal
• $p_c$: the contractor’s participation shares in handling the risk

It is clear that $P_0$ is the owner’s participatory share in handling the risk and is equal to $p_0$ = 1 − $p_c$.

2.3. Utility Function Calculation

This dissertation used the UTA-STAR technique utility function that is the developed case of UTA. Although this method was given for the criteria utility additive form, the considerations by BIOUTEH and ESCANELA show that even this method gives good results in cases where there is unity among the criteria. Their considerations showed that when errors survive in the approximation of the utility function, the UTA-STAR looks more trustworthy than other models. Although errors probably show some decision-makers make non-logical judgments [63], their preferences change dynamically during the process, even if the person is logical. The utility of 1 goal and a criterion indicates the degree of possible consenting to that goal (or criterion) for the decision-maker. In multi-attribute decision making (MADM), each Ai choice is shown by n. as each $r_i$ state the value of each criterion, that is:

$$A_i \approx \left\{r_1, r_2, \dots ., r_n\right\}$$

(4)

The decision-maker prefers to select the best solution among the existing choices. As the decision maker’s preference structure is stable based on assumptions, his preferences could be indicated by 1 utility function (or valuable function) as U(Ai) for each choice.

2.4. Model Algorithm

The owner’s decision-making is based on investment, and the contractor’s is made for participation. They are related to criteria determining the attraction of doing the project for each of the 2 sides. The hegemony of the proposed model is that gaining the utility obtained by doing each projected point of view of owner and contractor underpins this criterion as a unit and measures the scale of consenting gained by each project. Of the other benefits of this model, one is the possibility of calculating the real cost of the expected case in every selected stage, and using this quantity, the contractor can predict the share of risk division. Figure 1 indicates the proposed model algorithm.

This model achieves the utility function of owner and contractor using the UTA-STAR technique of doing the project for different quantities of risk allocation. So, in the case of increasing project costs due to agents that neither of the 2 sides are wedded to, and who are not controllable by either party, such as expenses and time postponement due to natural incidents that the insurance companies do not cover or war, the relation of risk allocation must be selected as consent of the two sides to both be absorbed as much as possible. There should then be a winner-winner relation between them. The wrong allocation of the risks could cause increase in the proposed prices in sales and, as a result, an increase in owner costs [74]. On the other hand, if the risks are divided correctly and sincerely the below interests for the project arise:

1-giving rise to a winner -winner relation between the owner and contractor

2-reducing the necessary sources for solving problems

3-shortening the representation time of the project

4-reducingproject costs and improving expectations of project beneficiaries.

The presentation of the proposed model.

Determining the utility function requires information from the decision-maker to determine the utility of choices based on their decision-making structure. This data could be the real information of the previous projects of the organization and/or different scenarios of doing a project. The information of 15 previous projects done in Kangan Petrochemical Development Company is used herein. (

Table 2. Using data in utility function determination.

Project	Time	Cost	Production in Day	Profit
1	60	1771	2000	265.65
2	90	3000	3900	900
3	85	1870	2500	374
4	80	1700	1500	340
5	72	1500	2100	300
6	90	2096	3400	628.8
7	46	800	1000	80
8	66	1600	1500	240
9	38	1000	700	100
10	96	3100	4000	930
11	88	2100	3000	430
12	76	1600	2200	320
13	54	1100	1200	165
14	72	1400	2000	210
15	64	1500	1400	225

Source: Researcher’s findings.

).

The criteria for the owner are cost, time, and the scale of production, and the contractor’s criteria are profit and time. By giving weight to each of the criteria, the priority of that decision-maker’s criterion is determined. The supposition is that the new contract is implemented by the goal cost of 2000 money units and 300 as contractor’s profit units for producing 2300 cargo units and by the time agreed between owner and contractor of 82 months.The goal is to find the proper relation of risk cost allocation between the owner and the contractor.

2.5. Determining the Utility Functions

2.5.1. The Owner’s Utility Function

Time, cost, and project quality are the three important elements for the owner [75]. The project founder’s and time and costs are predictable amounts that are approximately assessed in producing the executive programs of the project. However, project quality is not predictable before starting or during its primary stages. As far as the aim of this model is concerned, these criteria are not usable to determine the proper relation of risk division before beginning the project and at least before completing the project. According to

Table 3. Weight of criteria.

	Time	Cost	Quality
Weight	1	1	1
Normal weight	0.333	0.333	0.333

Source: Researcher’s findings.

, the weight of all three criteria is under the owner’s point of view. It is notable and necessary to say that selecting these criteria does not reduce the model’s generality. The reader can change the owner’s practicality and select other criteria.

As far as there is a need for a primary categorization for determining the utility function, the amounts of each of these criteria are without scale using the linear non-scaling approach and are then categorized using the heavy simple additive way (SAW). The priority of these projects is shown to be based on the owner’s consent. The degrees gained are shown through the heavy simple additive way in

Table 4. Ranking of projects.

Project	Time	Cost	Production in Day	Time	Cost	Production in Day	Rank
7	46	800	1000	0.521	0.742	0.250	0.504
9	38	1000	700	0.604	0.677	0.175	0.486
13	54	1100	1200	0.438	0.645	0.300	0.461
1	60	1771	2000	0.375	0.429	0.500	0.435
14	72	1400	2000	0.250	0.548	0.500	0.433
5	72	1500	2100	0.250	0.516	0.525	0.430
12	76	1600	2200	0.208	0.484	0.550	0.414
6	90	2096	3400	0.063	0.324	0.850	0.412
15	64	1500	1400	0.333	0.516	0.350	0.400
8	66	1600	1500	0.313	0.484	0.375	0.390
11	88	2100	3000	0.083	0.323	0.750	0.385
3	85	1870	2500	0.115	0.397	0.625	0.379
2	90	3000	3900	0.063	0.032	0.975	0.357
10	96	3100	4000	0.000	0.000	1.000	0.333
4	80	1700	1500	0.167	0.452	0.375	0.331

Source: Researcher’s findings.

.

The UTA-STAR technique was selected to obtain the utility function. Indeed, having the utility function, we can measure the utility of each new proposed project. For the rest, using the data in Table 4, the utility of each criterion and the whole utility function can be calculated concerning different steps of the UTA-STAR technique.

Step 1:

The distance between the least and the most amounts of the criteria (3) is divided into equal parts. (The number of parts is determined as favorite).

$$\begin{gathered} [g_{1*},g_1*]=[97,87,77,67,57,47,37] \\ [g_{2*},g_2*]=[3100,2640,2180,1720,1260,800] \\ [g_{3*},g_3*]=[700,1250,1800,2350,2900,3450,4000] \end{gathered}$$

We can use linear evaluation and the value of each project:

$$\begin{array}{cc}\hfill & U\left[g\right(7\left)\right]=0.9u_1\left(47\right)+0.1u_1\left(37\right)+u_2\left(800\right)+\hfill \\ \hfill 0.455& u_3\left(700\right)+0.545u_3\left(1250\right)\hfill \\ \hfill & U\left[g\right(9\left)\right]=0.9u_1\left(37\right)+0.1u_1\left(47\right)+0.435u_2\left(1260\right)+\hfill \\ \hfill 0.565& u_2\left(800\right)+u_3\left(700\right)\hfill \\ \hfill & \dots \hfill \\ \hfill & U\left[g\right(4\left)\right]=0.3u_1\left(87\right)+0.7u_1\left(77\right)+0.956u_2\left(1720\right)+\hfill \\ \hfill 0.044& u_2\left(1260\right)+0.545u_3\left(1250\right)+0.455u_3\left(1800\right)\hfill \end{array}$$

Using normalization conditions, we have the results below:

$$u_1\left(97\right)=u_2\left(3100\right)=u_3\left(700\right)=0$$

We can show the whole value of the projects by W_ij variables:

$$\begin{array}{cc}\hfill & Ug\left(7\right)]=w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+0.1w_{16}+w_{21}+w_{22}+\hfill \\ \hfill w_{23}+& w_{24}+w_{25}+0.545w_{31}\hfill \\ \hfill & U\left[g\right(9\left)\right]=w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+0.9w_{16}+w_{21}+w_{22}+\hfill \\ \hfill w_{23}+& w_{24}+0.565w_{25}\hfill \\ \hfill & U\left[g\right(13\left)\right]=w_{11}+w_{12}+w_{13}+w_{14}+0.3w_{15}+w_{21}+w_{22}+w_{23}+\hfill \\ \hfill w_{24}+& 0.348w_{25}+0.91w_{31}\hfill \\ \hfill & \dots \hfill \\ \hfill & U\left[g\right(4\left)\right]=w_{11}+0.7w_{12}+w_{21}+w_{22}+w_{23}+0.044w_{24}+\hfill \\ \hfill w_{31}+& 0.455w_{32}\hfill \end{array}$$

Steps 2 and 3:

Utilizing formulae and linear planning according to the UTA-STAR algorithm $\mathsf{\sigma }=0.025$

$$\begin{gathered} \mathit{Min}Z={\sigma }_9^{+}+{\sigma }_9^{-}+{\sigma }_7^{+}+{\sigma }_7^{-}+{\sigma }_{13}^{+}+{\sigma }_{13}^{-}+{\sigma }_1^{+}+{\sigma }_1^{-}+{\sigma }_5^{+}+{\sigma }_5^{-}+{\sigma }_{14}^{+}+{\sigma }_{14}^{-}+{\sigma }_{15}^{+}+{\sigma }_{15}^{-}+{\sigma }_{12}^{+}+ \\ {\sigma }_{12}^{-}+{\sigma }_8^{+}+{\sigma }_8^{-}+{\sigma }_6^{+}+{\sigma }_6^{-}+{\sigma }_{11}^{+}+{\sigma }_{11}^{-}+{\sigma }_2^{+}+{\sigma }_2^{-}+{\sigma }_3^{+}+{\sigma }_3^{-}+{\sigma }_{10}^{+}+{\sigma }_{10}^{-}+{\sigma }_4^{+}+{\sigma }_4^{-} \end{gathered}$$

In that ${\sigma }^{+}$ and ${\sigma }^{-}$ are, respectively, the errors of high approximation and low approximation.

$$\begin{gathered} \mathsf{\Delta }\left(7, 9\right)=-0.8w_{16}+0.435w_{25}+0.545w_{31}-{\sigma }_7^{+}+{\sigma }_7^{-}+{\sigma }_9^{+} {\sigma }_9^{-} \ge 0.025; \\ \mathsf{\Delta }\left(9, 13\right)=0.7w_{15}+w_{16}+w_{25} 0.91w_{31}-{\sigma }_9^{+}+{\sigma }_9^{-}+{\sigma }_{13}^{+}-\ge 0.025; \\ \dots \end{gathered}$$

$$\begin{gathered} \mathsf{\Delta }\left(10,4\right)=-0.9w_{11}-0.7w_{12}-w_{21}-w_{22}-w_{23}-0.044w_{24}+0.545w_{32}+w_{33}+ \\ {w}_{34}+w_{35}+w_{36}-{\sigma }_{10}^{+}+{\sigma }_{10}^{-}+{\sigma }_4^{+}-{\sigma }_4^{-} \ge 0.025; \end{gathered}$$

$$\begin{gathered} w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+w_{16}+w_{21}+w_{22}+w_{23}+w_{24}+w_{25}+w_{31}+w_{32}+w_{33}+ \\ {w}_{34}+w_{35}+w_{36}=1 \end{gathered}$$

2.5.2. The Optimum Answer Using Lingo Software Is as Follows

$Z^{*}=0.0238,w_{11}=0,w_{12}=0.0927, w_{13}=0.0522,w_{14}=0.1403, w_{15}=0.0179, w_{16}=0,$ $w_{21}=0.1152, w_{22}=0.044,w_{23}=0, w_{24}=0.1427,w_{25}=0.0575, w_{31}=0, w_{32}=0.1804,w_{33}=0.02, w_{34}=0.0343,w_{35}=0.1027, w_{36}=0$

As far as $Z^{*}\ne 0$,there is no need to do the fourth step of the algorithm, and the mentioned answer is optimum. Now, we calculate the utility amounts for receiving the utility function of each of the criteria.

The amounts of marginal utility of time scale criteria:

$$\begin{gathered} u_1\left[g\left(7\right)\right]=w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+0.1w_{16}=0.30310 \\ {u}_1\left[g\left(9\right)\right]=w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+0.9w_{16}=0.30310 \\ \dots \\ {u}_1\left[g\left(6\right)\right]=0.7w_{11}=0 \end{gathered}$$

The amounts of marginal utility of cost scale criteria:

$$\begin{gathered} u_2\left[g\left(7\right)\right]=w_{21}+w_{22}+w_{23}+w_{24}+w_{25}=0.33444 \\ {u}_2\left[g\left(9\right)\right]=w_{21}+w_{22}+w_{23}+w_{24}+0.565w_{25}=0.35944 \\ {u}_2\left[g\left(6\right)\right]=w_{21}+w_{22}+0.18w_{23}=0.15925 \end{gathered}$$

The amounts of marginal utility of production scale criteria:

$$\begin{gathered} u_3\left[g\left(7\right)\right]=0.545w_{31}=0 \\ {u}_3\left[g\left(9\right)\right]=0 \\ \dots \\ {u}_3\left[g\left(6\right)\right]=w_{31}+w_{32}+w_{33}+w_{34}+0.91w_{35}=0.35132 \end{gathered}$$

The above siding utilities can be normalized by $u_i \left(g_i^j\right)$ dividing on $u_i \left(g_i^{*}\right)$, and, in this case, the whole additive utility function is written as below:

$$\begin{gathered} U\left(g\right)=(w_{11}+w_{12}+w_{13}+w_{14}+w_{15}+w_{16})u_1\left(g_1\right)+\left(w_{21}+w_{22}+w_{23}+w_{24}+ \\ {w}_{25}\right)u_2\left(g_2\right)+(w_{31}+w_{32}+w_{33}+w_{34}+w_{35}+w_{36})u_3\left(g_3\right) \end{gathered}$$

$$u\left(g\right)=0.3031u_1\left(g_1\right)+0.3594u_2\left(g_2\right)+0.3375u_3\left(g_3\right)$$

(5)

In add-on, by inserting the utility amounts of each criterion in statistic software, we can have their siding utility function. Herein Minitab software was used. All the above parts are doing the project by statistics R-Sq and R-Sq (adj) in the Minitab window. Concerning the utility curves and time, we found that the owner was ready for short-time projects and was not ready for projects with a time of 75 months.

So, the owner could be risk-accepting for short term projects and risk-escaping for long term (t ≥ 75) projects.

The Figure 2, Figure 3, Figure 4 and Figure 5 show that the owner’s utility-cost curves were always convex; this showed the owner’s risk-escaping against the cost criterion. Despite the utility curve being more in higher costs, the owner escaped risk for more costs.

Similarly, the utility-production scale curves could be described (Figure 6 and Figure 7). So, the owner was risk-accepting in producing small amounts of products and was risk-escaping in producing large amounts of product.

2.5.3. The Contractor’s Utility Function

The contractor is an institution or company that accepts the responsibility of doing the project or parts of it, versus receiving a certain profit. So, it is clear that the received profit is the most important criterion for each contractor [19]. Conversely, a contractor must know how long his organizational sources are going to be tied up in the project [30]. In other words, the project that gives him or her more profit in less time will be more favorable for him or her. For this reason, herein, the contractor’s utility in participating in the project was measured by two criteria: profit and time. The profit criterion’s importance was two-fold, due to the significance of the time criterion. All the stages are passed for determining the contractor’s utility function, like the owner’s utility function, and by inserting the amounts of the utility of each of the criteria in MINITAB software, the curve, and siding utility function equation were gained as below.

Figure 8 and Figure 9 show that the assumed contractor was risk-escaping for receiving little profit and was risk-accepting for receiving much profit. That is, if a project was offered to this contractor with much profit, he would accept the risks of doing the project and take part in the project. If a project was offered to this contractor with low profit the contractor might accept the project, but if the risk of doing the project was high, the contractor would not take part in the project.

This could be understood from Figure 10, in which this contractor was risk-accepting related to the time, giving less weight to the time criterion in making a project a priority.

2.6. Density Function Determination of Real Cost Probability of Completing the Project

For achieving the expected utility of Equations (1) and (2) in the next part, should density determination of the real cost probability of completing the project be clear? The real cost of the project could be assessed with certain possibilities of cost and has deflection. In other words, the real cost of doing the project could be less or more than its contractive amount, so this could be assumed as a continuous random variable. In this dissertation, expert opinions were used to achieve the probability density function of this random variable.

Table 5. The expert opinions about the real cost probability of completing the project.

Cost	Risk	Cost	Risk	Cost	Risk	Cost	Risk
1650	0.0001	2025	0.67	2500	0.23	3000	0.02
1700	0.001	2050	0.65	2550	0.2	3050	0.01
1750	0.01	2100	0.61	2600	0.17	3100	0.009
1800	0.1	2150	0.56	2650	0.15	3150	0.008
1850	0.25	2200	0.49	2700	0.13	3200	0.0075
1900	0.35	2250	0.46	2750	0.11	3250	0.006
1950	0.55	2300	0.4	2800	0.09	3300	0.005
1970	0.65	2350	0.36	2850	0.07	3350	0.002
2000	0.7	2400	0.32	2900	0.06	3370	0.0001
2015	0.69	2450	0.29	2950	0.04

Source: researcher’s findings.

shows expert opinions about the probability of each amount of the project completion cost.

Using MINITAB software, the accordance of GAMMA statistical dispensation was considered in the data in the Table 5.

As shown in Figure 11, MINITAB drew a confidence limitation on the chart. This confidence limitation was drawn on the existing data as default based on assessments of dispensation parameters. The dispensation parameters, the amount of ANDERSON-DARLING test statistics, and the amount of p-value, were drawn in the chart window, and the number of observations supervised. If the dispensation accorded with the data, then the drawn points were in the confidence distance and close to the right line.

The amount of the ANDERSON-DARLING test was small, and the p-value larger.

Concerning Figure 11 it is observed that the data were in the trust or confidence limitation. The amount of ANDERSON-DARLING (AD = 0.657) was small, and the p-value was larger than $\alpha =0.1$. For this reason, the x goodness approximation had GAMA dispensation with the parameter of figure k = 24, and the scale parameter was $\theta =103$.

2.7. Calculating the Expected Utilities of the Owner Expected Utility

The expectation value of the owner’s final cost-utility could be written as below:

$$EU{V}_c=E\left[u\left(c\right)\right]={{\displaystyle \int }}_{-\infty }^{+\infty }u \left(c\right) f\left(c\right)dc$$

(6)

$$\begin{gathered} u_1\left(c\right)=0.0000001c^2+0.000184c+0.3109 c \le 1700 \\ {u}_{i2}\left(c\right)=0.000000c^2+0.000469c+0.2932 c>1700 \end{gathered}$$

(7)

$$\begin{array}{cc}\hfill EU{V}_c=\underset{}{\overset{\underset{}{\overset{0}{⏞}}}{{{\displaystyle \int }}_{-\infty }^0{u}_1\left(c\right)f\left(c\right)}}& dc+{{\displaystyle \int }}_0^{1700}{u}_1\left(c\right)f\left(c\right)dc\hfill \\ \hfill & +{{\displaystyle \int }}_{1700}^{+\infty }{u}_2\left(c\right)f\left(c\right)dc\hfill \\ \hfill & ={{\displaystyle \int }}_0^{1700}{u}_1\left(c\right)f\left(c\right)dc+(1\hfill \\ \hfill & -{{\displaystyle \int }}_0^{1700}{u}_2\left(c\right))f\left(c\right)dc\hfill \\ \hfill \Rightarrow EU{V}_c={{\displaystyle \int }}_0^{1700}1& \times {10}^{-7}{c}^{2}f\left(c\right)dc\hfill \\ \hfill & +{{\displaystyle \int }}_0^{1700}1.84\times {10}^{-4}cf\left(c\right)dc\hfill \\ \hfill & +{{\displaystyle \int }}_0^{1700}3.109\times {10}^{-1}f\left(c\right)dc+(1\hfill \\ \hfill & -({{\displaystyle \int }}_0^{1700}1\times {10}^{-7}{c}^{2}f\left(c\right)dc\hfill \\ \hfill & +{{\displaystyle \int }}_0^{1700}4.69\times {10}^{-4}cf\left(c\right)dc\hfill \\ \hfill & +2.932\times {10}^{-1}f\left(c\right)dc\hfill \end{array}$$

(8)

It is necessary that the equation be clear by density function and f(c), and what was gained in the previous part was the density function of x, which is the real cost of doing the project. So, C is the function of x.

$$\begin{gathered} x ~ Gamma \left(k,\theta \right) ~ Gamma\left(24,103\right) \\ c=x+b+p_c\left(x_0-x\right) \end{gathered}$$

(9)

It also has the GAMA dispensation with ${\theta }_1, k_1$ parameters. To achieve the parameters of this new dispensation, we use the average and x dispensation variance and its relation to the average and c dispensation variance. We know that the average and variance of GAMA dispensation are received by the relations below:

$$\mu =E\left(x\right)=k\theta ={24}^{*}103=2472$$

(10)

$${\delta }^2=Var\left(x\right)=k{\theta }^2={24}^{*}{103}^2=254616$$

(11)

The relation between c average and variance and x average and variance could be received by the way below:

$$E\left[c\right]=E\left[x\right]+b=p_c{x}_0-p_{c}E\left[x\right]=b+p_c{x}_0+\left(1-p_c\right)E\left[x\right]=k_1{\theta }_1$$

(12)

$$Var\left[C\right]={\left(1-P_C\right)}^{2 }Var\left(x\right)=k_1{\theta }_1{}^2$$

(13)

By placing relations 10 and 11 in relations 12 and 13, the amounts of average and c dispensation variance based on $p_c$ are gained as below:

$$E\left[c\right]=k_1{\theta }_1=2772-472p_c$$

(14)

$$Var\left[c\right]=k_1{\theta }_1{}^2=254616{\left(1-p_c\right)}^2$$

(15)

Table 6. C probability density function parameters for $p_c$.

${\mathit{\rho }}_{\mathit{c}}$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
$k_1$	-	2163.787	562.921	260.148	152.046	101.035	72.799	55.458	43.997	36	30.179
${\theta }_1$	0	1.085	4.254	9.385	16.369	25.1	35.484	47.431	60.858	75.69	91.853

(Source: researcher’s findings).

shows the $p_c$. Probability density function parameters. For example, for $p_c$ = 0.1

$$\Rightarrow {\theta }_1=92.51, k_1=29.454\{\begin{array}{c}{k}_1{\theta }_1=2724.8\\ k_1{\theta }_1{}^2=252069.84\end{array} \Rightarrow$$

(16)

Now we should be able to gain ${\displaystyle \int }{c}^{2}f\left(c\right)dc, {\displaystyle \int }cf\left(C\right)dc,$ in which C has the GAMA dispensation with the probability density function f(c) (Appendix A) and with the parameters of Table 6.

$$\mathrm{relation}13:c~\mathrm{Gamma}\left(k_1,{\theta }_1\right)\Rightarrow f\left(c\right)=\{\begin{array}{ll}\frac{c^{k_1-1}{e}^{\frac{-c}{{\theta }_1}}}{\mathsf{\Gamma }\left(k_1\right){\theta }_1{}^{k_1}}& c\ge 0\\ 0& c<0 \end{array}$$

(17)

So, after the stages mentioned in the reference, Equation (18) is:

$$c^{2}f\left(c\right)~\left(k_1+1\right)k_1{\theta }_1{}^2\mathrm{Gamma}\left(k_1+2, {\theta }_1\right)$$

(18)

Now, with respect to the relations and the given results, we can write the owner’s final cost expected value in Equation (8) as below:

$$\begin{gathered} EU{V}_c=1\times {10}^{-7}{\theta }_1{}^2{k}_1\stackrel{c~\mathrm{Gamma}\left(k_1+2 , {\theta }_1\right)}{\overbrace{\left(k_1+1\right)F}}(c=1700)+1.84\times \stackrel{c~\mathrm{Gamma}\left(k_1+1 , {\theta }_1\right)}{\overbrace{{10}^{-4}{\theta }_1{k}_{1}F}}(c=1700)+\stackrel{c~\mathrm{Gamma}\left(k_1, {\theta }_1\right)}{\overbrace{3.109\times {10}^{-1}F}}\left(C=1700\right)+ \\ (1-(1\times {10}^{-7}{\theta }_1{}^2\stackrel{c~\mathrm{Gamma}\left(k_1+2 , {\theta }_1\right)}{\overbrace{K_1\left(K_1+1\right)F}}\left(C=1700\right)+4.69\stackrel{c~\mathrm{Gamma}\left(k_1+1 , {\theta }_1\right)}{\overbrace{\times {10}^{-4}{\theta }_1{K}_{1}F}}\left(C=1700\right)+\stackrel{c~\mathrm{Gamma}\left(k_1, {\theta }_1\right)}{\overbrace{2.932\times {10}^{-1 }F}}\left(C=1700\right) \end{gathered}$$

(19)

F(c) is the c additive dispensation function in the above relation. In

Table 7. Amounts for $p_c$.

$${\mathit{\rho }}_{\mathit{c}}$$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
EUVc	-	1	1	1	0.999997	0.999957	0.99978	0.999366	0.99868	0.997752	0.996638

(Source: researcher’s findings).

, we see that the ${\theta }_1, k_1$ amounts would be different for different $p_c$ amounts. So, different amounts for $EU{V}_c$ for $p_c$ are gained.

The proposed project’s expected utility is obtained for the owner using Equation (7). That is:

$$EU{V}_{Total}=0.303EU{V}_t+0.3594EU{V}_c+0.3375EU{V}_q$$

(20)

$$q=2300>2000\Rightarrow u\left(q\right)=0.5327+0.00047q+0.000000q^2=1.6137$$

(21)

Since the production scale is set, the predicted production scale by probability 1(P(q) = 1) results:

$$EU{V}_q=1.6137$$

(22)

However, because there is a risk in the project, the time is a continuous random variable and its expected utility scale is calculable as was done with the cost criterion, while making certain contracts after finding the related density function. Paying attention to the goal of this dissertation, gaining the $p_c$ optimum amount is necessary, as $p_c$ determines a winner-winner situation between the owner and the contractor. The whole expected utility quantity is not necessary. Of course, its changes are determined by changes in $p_c$. As far as the changes in $p_c$ are effective in $EU{V}_q$ and $EU{V}_c$ amounts, we aim to gain the t density function and calculate its expected utility for ease and summarization. Comparing the owner’s and contractor’s whole expected utility in $p_c$ amounts, and comparing the expected utility cost for the owner and profit for the contractor.

2.8. The Contractor’s Expected Utility

The value p means that the contractor’s received profit is gained by the below relation, and x is the project’s real cost:

$$p=b+p_c\left(x_0-x\right)$$

(23)

And

$$x ~ \mathrm{Gamma}\left(k,\theta \right)~ \mathrm{Gamma}\left(24,103\right)$$

(24)

So, $p=b+p_c\left(x_0-x\right)$ also has the GAMA dispensation with ${\theta }_2, k_2$ We use the x dispensation average and variance and its relation with the p dispensation average and variance.

$$E\left[p\right]=b+p_c{x}_0-p_{c}E\left[x\right]=k_2{\theta }_2$$

(25)

$$\mathit{Var}\left[p\right]=p_c{}^2\mathit{Var}\left(x\right)=k_2{\theta }_2{}^2$$

(26)

Placing E(x) and Var(x) amounts, the P dispensation average and variance is gained based on $p_c$ by Equations (10) and (11) in the above relation.

$$E\left[p\right]=k_2{\theta }_2=300-472p_c$$

(27)

$$Var\left[p\right]=k_2{\theta }_2{}^2=254616p_c{}^2$$

(28)

The ${\theta }_2, k_2$ amounts have been calculated in

Table 8. C probability density function parameters for different $p_c$ amounts.

$${\mathit{\rho }}_{\mathit{c}}$$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
EUVp	-	-	-	-	0.4915	0.5553	0.7291	1.003	1.3554	1.75	-

(Source: researcher’s findings).

for $p_c$ different amounts.

The profit criterion utility, that is u(p), is described below in 2 parts:

$$\{\begin{array}{c}{u}_1\left(p\right)=-0.000005p^2+0.004339p-0.3230 p\le 400\\ u_2\left(p\right)=0.000001p^2-0.000852p+0.8090 p>400\end{array}$$

(29)

So, the contractor’s expected profit utility is similar to the cost expected utility.

2.9. Determining the Proper Proportion of Risk Cost Allocation

One of the profits of the given model presented in this dissertation is calculating the real cost of completing the project. Using this quantity, the contractor can predict the share of risk losses in the first stages of the project and avoid accepting it. For this work, this is enough to ascertain that the expected value of gained profit of doing the project is bigger than 0 and we have the expected value of the real cost of the project. We calculate the ${\rho }_c$ amounts that make the contractor’s expected profit value negative (

Table 9. EUVp amounts for different $p_c$ amounts.

$${\mathit{\rho }}_{\mathit{c}}$$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
k2	0.116	0.076	0.037	0.007	0.003	0.064	0.304	1.095	4.151	25.1	-
θ2	−1480	−1652	−2099	−4104	5456.05	994.594	366.354	144.668	49.536	10.072	0

(Source: researcher’s findings).

):

$$\begin{gathered} E\left(p\right)=b+p_c{x}_0-p_{c}E\left[x\right] \\ \Rightarrow 300-472p_c>0 \\ \Rightarrow p_c<0.636 \end{gathered}$$

(30)

This point is exactly where the GAMA scale parameter becomes negative, that is, the GAMA dispensation is not definable. This dissertation aims to find the optimum $p_c$ and this is as in the case of deflection, the contractive cost is between the owner and contractor in a winner-winner relation. We can say that the situation is not gained if $p_c>0.6$ because the owner’s profit gets negative, and when $p_c=0$ all the risks are handled by the owner.

Table 10. The cost of expected utilities and the expected benefit of acceptable.

${\mathit{\rho }}_{\mathit{c}}$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
EUVc	-	1	1	1	0.999997	0.999957	0.99978	0.999366	0.99868	0.997752	0.996638
EUVp	-	-	-	-	4915	5533	7291	1.003	1.3554	1.75	-

(Source: researcher’s findings).

shows the cost of expected utilities and the expected benefit of acceptable $p_c$.

Regarding Table 10, if the contractor’s commonness proportion is more, the owner’s expected utility is more, and conversely, the contractor’s expected utility becomes less. As far as we are looking for a proportion that can ensure the consent of the two sides, then, in fact, we can say that our favorite point is the point where the absolute value between the owner and contractor is at a minimum. Related to the data of the above table $p_c=0.3$ and naturally $p_c=0.7$ are the proportion of risk allocation optima between owner and contractor, as these values can ensure the consent of the two sides and make a winner-winner relation between them.

3. Conclusions

Contractual communication and the scope of responsibilities and obligations are the main factors in achieving a project’s goals. The main point in achieving this success is the understanding and knowledge of all the people involved in it regarding the objectives and obligations required to implement the contract and knowing how to acquire them contractually. Concerning the importance of determining responsibility in projects, so as to prevent clashes, increase in costs without reason and in prices of contracts, and/or the loss of one of the two sides, requires the calculative model presented to determine the relation of specialization of non-controllable risk costs. This model can be used to determine the relation of risk between the owner and contractor, as the consent of owner and contractor must be made and a winner-winner relation constructed between them. None of the project members are responsible for non-controllable risks, and no law exists for them. So, this model uses the project’s utility scale for each owner and contractor. The present model can measure the utility-scale of investment or participation in a new project based on different criteria that play important roles in determining the consent of each of the two sides. This model is responsible for each real cost deflection scale of the goal cost (contractive) as one continuous random variable and determines its probability dispensation keeping the real cost in mind. This feature and calculation of expected utility value enables the owner and contractor to easily get ready for determining the relation of risk cost specialization in the contracting stage. Other benefits of this model are the possibility of calculating the real cost of expected completion in every stage of doing the project. By this quantity, the contractor can predict the risk specialization relations likely to result in losses and can avoid accepting them. Therefore, it can be said that risk management is one of the phases of project management [76,77,78]. To formulate and implement the risk management system in construction projects, risks and opportunities (potential and actual) are identified in similar studies [10,17,53,60]. The topic discussed in this research aimed to find the appropriate ratio of risk cost allocation to attract the satisfaction of the contractual parties; in other words, to establish a win-win relationship between them. As seen in the background of the research, the discussion of risk allocation has been seen in contracts from different angles and for different issues. Sometimes the goal is to identify the risks and determine the person responsible for each of them in the contract, which, in most cases, is done through interviews with experts. In such cases, the expert is asked to comment on the determination of the person responsible for the risks according to his work records and subjective criteria or criteria predetermined by the questionnaire. In addition, solutions are also provided by experts. Now, to manage the identified risks and opportunities at the same time, appropriate forecasts and timely dealing with risks instead of risk aversion, as well as proper use of options, are proposed:

• If necessary, officials, contractors, and employers should take financial, timing, executive quality, and environmental risks to reduce costs.
• Employers and contractors should pay more attention to timing risk and execution quality rather than other risks and ensure that these two risks are less dangerous.
• Employers and contractors should be given the necessary training to deal with financial, timing, and environmental risks.
• It is recommended that tendering departments, and others who prepare the tender forms, allocate risks in a balanced and fair way to avoid or minimize claims and disputes.
• There is a need to enhance the efficiency of construction management and contract administration through academic courses and training to improve the managerial skills of managers so that managers can estimate potential risks early in proceedings.

Presenting topics for future research studies

1-

In this model, some discrete amounts for the relation of risk cost sharing given in the development of the model could be solved in a continuous case.

2-

The limit of the proposed model is that it has been limited to the risk allocation between the owner and contractor, so it can be developed to include more members in future research; for instance, different contractors.

3-

The mentioned model only looks at the scale of consent between the owner and contractor for sharing the risk costs, but other factors, like political problems and economical limitations, could be addressed.

4-

In this model, the SAW technique was used. To prioritize attributes, we could use TOPSIS, LINMAP, and MRS.

5-

Categorizing the types of contracting projects and determining the best kind of contract for each category, according to the existing risks, and using decision theory techniques is presented as a suggestion, which could be a good guide for determining the type of contract for employers.

6-

We can propose Fuzzy UTA-STAR, a method for inferring fuzzy utility functions from a partial preorder of options evaluated on multiple criteria.

Although any research is complete and comprehensive, the researcher is faced with problems and limitations in the process of conducting it, which make it impossible to generalize descriptive study in the long-term and all aspects, and this causes the need to conduct other research in the desired direction, which may be repeated until it covers all aspects. In all the research that takes place, limitations are an integral part of the research. Limitations provide the ground for future and new research. This research is not an exception to this rule and has the following limitations:

• There are all kinds of definitions and interpretations of the term “risk allocation between the owner and the contractor,” each of which has a different view of this component. Due to the fact that a specific model was used in this research, less attention may have been paid to some aspects of the mentioned components.
• Due to the multitude of models presented to measure the optimization of risk allocation, the use of different models may lead to different results.
• The results obtained for the research are unique to the defined spatial and temporal territory as well as the considered factors and variables and, therefore, cannot be generalized to situations outside of this framework.
• The findings of this research are only limited to the duration of data collection, and its validity is limited to a short time. The passage of time may change the results obtained. Therefore, it is necessary to conduct similar research in future periods.