A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project

Sustainable development requires scheduling and implementation of projects by considering cost, environment, energy, and quality factors. Using a robust approach, this study investigates the time-cost-quality-energy-environment problem in executing projects and practically indicates its implementation capability in the form of a case study of a bridge construction project in Tehran, Iran. This study aims to take into account the sustainability pillars in scheduling projects and uncertainties in modeling them. To model the study problem, robust nonlinear programming (NLP) involving the objectives of cost, quality, energy, and pollution level is applied with resource-constrained. According to the results, as time diminished, the cost, energy, and pollution initially decreased and then increased, witha reduction in quality. To make the model close to the real world by considering uncertainties, the cost and quality tangibly improved, and pollution and energy consumption declined. We applied the augmented \begin{document}$ \varepsilon $\end{document} -constraint method to solve the proposed model. According to the result of the research, with regard to the time-cost, time-quality, time-energy, and time-pollution charts, as uncertainty increases, the cost and quality will improve, and pollution and energy will decrease. The proposed model can be employed for all industrial projects, including roads, construction, and manufacturing.

green construction projects. Other studies also discussed the importance of green project management and sustainable development [6,22,27,61].
2. Background and motivation. Numerous studies have been conducted on project scheduling with a time-cost trade-off. For example, Prager [43] and Siemens [50] introduced heuristic algorithms for this problem. Moselhi [39] also presented a heuristic method to solve this problem and compared it to the method introduced by Ahuja et al. [3]. He showed that his proposed method was more efficient and very close to optimization. He also obtained the Pareto optimal border for cost and time.
In recent decades, different methods have been introduced to optimize the time and costs of the project activities, which can be generally classified into three groups: accurate, heuristic, and meta-heuristics ones. Many models have been implemented in mathematical planning for the optimal balance of the three project factors. Babu and Suresh [4] conducted a study to balance the three factors at the same time. In their work, they made a Crashing Hypothesis and assumed that as the time of activity decreases, cost increases linearly, and quality declines linearly. They considered three linear target functions in which the result analysis led to decisionmaking in balancing the mentioned factors. At the end of their paper, they proposed that neither the total quality of the project (whether weighted mean or arithmetic mean) nor the quality calculation as the product of the activities affects their work result procedure. Examples of the proposed mathematical methods are linear programming (LP) method of Hendrickson et al. [23], and Pagnoni [41] and integer programming method of Patterson and Huber [42] where the time-cost trade-off problem is accurately optimized by a mathematical planning model. Since a combination of different options can be selected to do activities at any possible time, there are compound optimization methods that are viewed as difficult optimization problems. With the problems becoming more complicated and dimensions increasing, they become less probable to be solved with common optimization methods or rapid computational methods. Thus, optimal solutions were very difficult to obtain in that situation. The comprehensive and successful development of meta-heuristic optimization algorithms to solve single-objective optimization problems made the researchers consider if these algorithms could be applied to solve multi-objective optimization problems. Various solutions to properly employ these algorithms, classify andapply them indeterministic classes of optimization problems, and their validations were among the problems and complications that would be faced by those who applied these algorithms. In this regard, Feng [12], Li and Love [28], Hegazy [24], and Zheng et al. [60] made efforts to introduce optimal solutions based on genetic algorithm (GA). However, in all these studies, uncertainties werenot considered due to complications, and the studies were conducted in a deterministic space. But in real-world projects, factors such as cost and time of the projects are always affected by many changes due to the uncertainties. Therefore, to solve this problem, Feng et al. [13], Azaron et al. [1], Abbasnia et al. [2], and Zhang and Li [62] studied the bi-objective balance of time and cost in a real-world uncertain space.
Since the 1990s, researchers gradually found that it made no sense to execute a project at the right time with the lowest cost without taking into account the execution quality. As of then, the time-cost-quality balance was brought up, and the researchers started to attempt to find solutions for this problem.
The first study was conducted by Babu and Suresh in 1996. They proposed three LP optimization models as an analytical time-cost-quality balance framework. In 1999, Khang and Myint [26] implemented this model in a real-world project of construction of a cement factory in Thailand. The successful experience of metaheuristic optimization algorithms in solving the two-factor problem of time-cost balance made the researchers focus on solving the three-factor problem of timecost-quality optimization. A number of researchers solved the time-cost-quality optimization problem, including El-Rayes and Kandil [10] by using GA, Zhang et al. [63] and Rahimi and Iranmanesh [47] by employing particle swarm algorithm (PSO), Tareghian and Taheri [53] by implementing electromagnetism meta-heuristic algorithm, Abbasnia et al. [2] by using ant colony optimization (ACO) algorithm, Iranmanesh et al. [25] by using a recently developed version of GA called FAST PGA, Wang and Feng [59] by applying hierarchical PSO, and El Razek et al. [11] by employing GA called automatic execution resource multi-objective optimization system.
In the studies mentioned above, the researchers used the objectives of time, cost, and quality, which are usually considered known and deterministic.
Gap research and one of the initiatives in this study is the expression of the robust optimizationfor time-cost-quality-energy-environment trade-off in construction projects (see Table 1). The addition of the objectives of energy consumption and environmental pollution was done to consider sustainable development goals in the project activities. To consider sustainable development, in addition to the economic aspects of the projects, consideration of the project execution with the lowest social and environmental effects is an important strategic principle for the projects. However, no definite opinion can be made on time, cost, quality, energy, and pollution of the activities during the execution to make the mathematical models close to the real-world conditions as much as possible, and all these are obscure and uncertain. Hence, the robust optimization is applied to take into consideration these uncertainties during the problem-solving process. In this study, the factors time, cost, quality, energy, and CO 2 pollution have uncertainties, which is a new topic in the field of the multi-objective optimization problem of time-cost-qualityenergy-environment trade-off with resource-constrained for the project executions. To solve the mentioned multi-objective model, the augmented ε-constraint method was employed. This method was rarely applied in previous studies.
In Section 2, we state the problem and define a mathematical model for this study. In Section 3, the proposed mathematical model is solved using the augmented εconstraint method. Section 4 provides the case study, and finally, Section 5 is devoted to a conclusion and a prospect to future studies..
3. Problem statement. Moving toward sustainability was started in 1960 when pollution and increasing fuel cost coincided with the prohibition of oil imports and made many organizations review their energy consumption, ways to get energy, and the effects of their activities on earth [48]. Given the factors cost, environment, energy, and quality, the execution and scheduling of the projects are a requirement for the beneficiaries and sustainable development. In this regard, in addition to the economic aspects, it is required to consider the execution of the projects with the lowest negative environmental and social effects.
In this section, we provide a sustainable project management model using a costtime-quality-energy-environment trade-off, which was rarely paid attention in the  Notation and definition:

Indices
I Set of activities i, j ∈ {1, · · · , |I|} ⊂ I, I 1 Set of activities with astart to start the relationship I 1 ⊂ I, I 2 Set of activities with astart to finish the relationship I 2 ⊂ I, I 3 Set of activities with a finish to start the relationship I 3 ⊂ I, I 4 Set of activities with a finish to finish the relationship I 4 ⊂ I, R Set of required resources including renewable and non-renewable items r ∈ R. k Index of the objective function k ∈ {1, · · · , 4} ⊂ K. Nominal daily demand of activity i for resource r (Unit/Day), Cap Available capacity for resource r (Unit), Cap Available capacity for resource r (Unit), SS ij Start to start delay between activities i and j (Day), SF ij Start to finish delay between activities i and j (Day), F S ij Finish to start delay between activities i and j (Day), F F ij Finish to finish delay between activities i and j (Day), Project duration-dependent indirect quality coefficient (Percent), y 3 Project duration-dependent indirect energy coefficient (Mega Joules), y 4 Project duration-dependent indirect pollution coefficient (Ton), f ik A alternative parameter for ci, qi, ei, pi in a simplified form of objective function k, Total time deviation coefficient of the project with respect toT .
Decision Variables z k Value of objective function k, To describe the mathematical model, consider a project based on an Activity on Node (AON) network. This network has i ∈ {1, · · · , |I|} ⊂ I nodes that show the activities. The activity I have a normal time, cost, quality, energy and pollution of t i , c i , q i , e i and p i , while the compacted time, cost, quality, energy, and pollution are denoted as t i , c i , q i , e i and p i . The main assumptions of the proposed model are as follows: 1. No activity is done before providing the prerequisites. 2. Time, cost, quality, energy, and consumption are uncertain for every activity.
Cost and energy consumption increase as time diminishes. 5. It should be noted that the energy concumption of each activity is estimated based on the consumption amount of energy-based resources. 6. Activities have a daily demand for their required resources. 7. Multiple renewable and non-renewable resources are defined. The supply capacity of these resources is restriced and is known at the beginning of the project. 8. Quality and pollution increase as time reduces.
In the following, the mathematical model of the time-cost-quality-energy-environment trade-off is introduced.
In Figure 1, we show that duration (x i ) is between normal time (t i ) and compacted time (t i ). Because of uncertainty in t i and t i , both of them have nominal amount. Nominal normal time is (t i ) and nominal compacted time is (t i ). All uncertainty parmeters applied in this research use this form.

Model 1 Cost-Time-Quality-Energy-Environment Trade-off with
Resource-Constrained: subject to The objective function (1) is the direct and indirect minimization of the total costs for the activities. The objective function (2) is the direct and indirect maximization of the mean quality for the activities. The objective function (3) is the direct and indirect minimization of the project energy consumption. The objective function (4) is the direct and indirect minimization of the total CO 2 pollution for the project. Constraint (5) is the start time for an activity (1) at time 0. Constraint (6) is the finish time for the last activity equal to the mandatory time (T). Constraint (7) is the time of each activity between the compacted time t i and the normal time t i . Constraint (8) represents the capacity limitation of the resources. Constraint (9) denotes that the start time of activity (i) plus the time of activity (i) equals to the finish time of activity (i). Constraints (10)-(13) represent the dependency degree and the prerequisite between activities (i) and (j). Constraint (14) represents decision variables, involving a time of each activity as well as the start and finish time of it.

3.2.
Robust optimization of the model. Since a long time ago, it has been an important subject of how to deal with uncertainties in mathematical planning problems, or in other words, system optimizations. Different approaches have been developed to deal with uncertainties in mathematical planning problems, including fuzzy and robust planning (in ambiguity cases) and random planning (in case of historical data). The robust planning approach is one of the latest methods to deal with uncertainties. It is a popular method due to its significant capabilities. As a prerequisite in dealing with uncertainties, Soyster [49] developed a pessimistic planning method for inaccurate NLP models. A few decades later, in 2000, Ben-Tal and Nemirovski [5] developed Soysters method for uncertain NLP models with different convex uncertainty sets and took a significant step forward in developing a robust planning theory.
According to them, an uncertain robust optimization problem involving a set of linear optimization problems is defined.
Assume the following certain linear optimization model with Objective Function (15) and Constraint (16).
Model 2 Deterministicform of the LP Model: subject to where b, A, d, and c change in the given uncertainties set, converting into Eqs. (17)- (19). Model 3 Uncertain form of the LP Model: subject to A vector x is a robust solution for Model (3) if it satisfies all the constraints obtained from uncertainties set U . Ben-Tal and Nemirovski [5] defined the robust counterpart problem as Model (4).

Model 4 Robust Problem of the LP Model:
subject to An optimal solution for Model (4) is a robust optimal solution for Model (3). Such a solution satisfies all the constraints for all the uncertain data and ensures that the value of the optimal objective functionĉ(x * ) is not worse than any value [31,44,64]. It means that Model (4) solvesthe model in the worst case. Model (4) is a semi-finite linear optimization problem and seems to be computationally strong. However, it seems that for a wide range of convex uncertainties sets, a convex mathematical problem can be solved (in the form of a solvable polynomial) -it is usually a linear optimization or a conic quadratic problem.
For ease of solution, the compact form of Model (1) can be expressed as Model (5). (1):

Model 5 Compact form of Model
subject to where f ik has replaced normal parameters and f ik has replaced normal parameters (t i , c i , q i , e i and p i ) and f ik has replaced compact parameters (t i , c i , q i , e i and p i ) for objective function k. To develop the above robust counterpart model, all the parameters are treated as having uncertainties. Each of the uncertainty parameters is assumed to change in a box range [49]: whereξ t is the nominal value of vector ξ t (of n dimensions). A positive value of G t represents uncertainty scale, and ρ > 0 represents uncertainty level. The most widely applied case is G t =ξ t which belongs to a simple case, that is, a box containing ξ t whose maximum relative deviation from the nominal data is ρ .
According to the above statements, the robust counterpart model can be expressed as:
Ben-Tal and Nemirovski [5] indicated that for this case (i.e., close box), the robust counterpartproblem could be solved as an equivalent problem such that u box is replaced by a finite set u ext and u ext involves the radical points of u box . To demonstrate the solvable form of compacted robust Model (5), Eq. (20) addressed in the following robust solvable form: The left side of (28) contains uncertainty parameters. Thus, the solvable form of the semi-finite inequality of the above objective function can be rewritten as:

Solution technique.
Given that the presented model is nonlinear and multiobjective, the augmented ε-constraint method is utilized to calculate the objectives of the above model. In the following, the ε-constraint method is first defined and, then, the augmented ε-constraint method is described.
The main limitations of the study are the scale of the problem. When the scale of the problem is large, the time of solving is exponential growth and NP-hard. Metaheuristic algorithms are the best choice to show the best possible solutions for these large-scaled models within a reasobnable computational time. However, the proposed approach of this research considers an exact solution algorithm. 4.1. ε-constraint method. This method is based on transforming the multiobjective problem into a single-objective optimization problem. In this method, one of the objectives is optimized as the main objective [36,14]. The advantages of this method are • By changing ε value, different solutions can be obtained.
• Unlike the weighting method, differences in the scales of the objectives do not make any problem. This method can obtain a more diverse set of Pareto optimal solutions. On the other hand, the disadvantages of this method are • The solutions obtained considerably depend on the values selected for ε. These values have to be selected such that they fall between the maximum and minimum values of each fixed objective function. • As the number of objectives increase, more information has to be given by the user. Assume that it is decided to minimize Objective Function (53) subject to Constraints (54) and (55).
subject to According to the method, one of the objective functions is selected as the main objective function, according to Eq. (56). Other objective functions are treated as Constraint (57), and the problem is solved with respect to one of the objective functions each time, calculating the optimal and peer value for each objective function. The range between the two optimal and peer values is divided into a predefined number, and t values table is determined for ε j . Ultimately, Pareto solutions are obtained [58].
subject to However, despite its advantages of ε-constraint over the weighting method, it has three points that need attention in its implementation: (a) the calculation of the domain of the objective functions over the efficient set, (b) the guarantee of efficiency of the acquired solution and (c) the increased solution time for problems with several (more than two) objective functions. We try to address these three issues with a novel version of the e-constraint method that is presented in the next section [37].
The ε-constraint method graph is drawn in Figure 2.

4.2.
Augmented ε-constraint method (AUGMECON). Here, the augmented ε-constraint method (AUGMECON) Method is employed for multi-objective optimizations. It produces efficient optimal Pareto solutions and avoids inefficient solutions. In this method, one of the objective functions is optimized as the main Figure 2. Schematic performance of the ε-constraint algorithm [38].
objective function, while other objective functions appear as constraints [37]. An innovative addition to the algorithm is the early exit from the nested loop when the problem becomes infeasible, and this significantly accelerates the algorithm in the case of several (more than three) objective functions. AUGMECON is defined in Objective Function (59): subject to The optimal solutions of the model are added by changes in the right side of ε j (Constraint (60)). Optimal Pareto solutions are obtained where r is the changing scope of objective function i, δ is a very small value between 0.000001 and 0.001, and s i is a non-negative surplus variable. The minimum and maximum values of objective function j are calculated as N IS j and P IS j , respectively. Then, the changing scope r j of objective function j is calculated as Eq according to (61): Then, r j is divided into an equal number l j . Then, l j + 1 network points are calculated by Eq. (62) based on the value of ε j : where n is the number of network divisions. The augmented ε-constraint model has to be solved for each vector ε. Thus, n j=2 (l j + 1) optimization sub-problems should be solved.
To solve the mentioned multi-objective robust resource-constrained time-costquality-energy-environment trade-off (RRCTCQEPTP) model, the augmented εconstraint method is used. In the following, the importance of simultaneous consideration of all the objectives is highlighted by presenting a bridge construction project.

5.
Case study and sensitivity analysis. Here, the proposed methodology of the research is validated using a real case study. The case study of this research is an underpass bridge construction project in downtown Tehran, Iran. For this project, it is required to consider time, cost, quality, energy, and environment specifically since the consideration of time, cost, and quality enhances the employers satisfaction and consideration of energy consumption, and the environment is a legislation requirement. The underpass bridge project has an abutment, west and east ramps, and a column at the deck (cf. Figure 3).  Table 3 gives a list of activities and prerequisites along with time, cost, quality, energy, and pollution of each activity at normal and compact conditions for the underpass bridge project. Figure 4 represents the network graph based on the AON of the case study. Every Box represents activity i and has id-code, duration (cf. Table 3), early start and finish, late start and finish in nominal normal. Every activity has a dependency on other activity that was shown with an arrow. Activities 1 to 24 have been shown in Figure 4. The model was solved by GAMS software on a computer with 1.7GHz and 6GB of CPU and RAM, which is implemented by BONMIN solver of GAMS software. Table 4 shows the computation results obtained using the augmented ε-constraint method. Figures 5(a)-(d) demonstrate augmented ε-constraint method results and the time-cost, time-quality, time-energy, and time-pollution charts. In this figure, uncertainty is equal to zero (ρ = 0), and as time decreases, cost, energy, and pollution first decline and then increase, while quality reduces as time reduces. On the one hand, this shows the correction of the modeling, and on the other hand, for decisionmakers, it shows what would happen to quality, cost, pollution, and energy as time decreases. As it is seen, at the time about 305, the cost is at the minimum level, and quality, energy, and pollution are acceptable.      6 show that as uncertainty increases and becomes closer to real-world situations, sustainability can be improved in every aspect of project execution. As can be seen, with respect to the consideration of sustainable development, in addition to the time and cost of project execution, environment and pollution are considered. This study makes project managers pay attention to sustainable development in executing projects. All the beneficiaries of the projects are required to manage the project execution in a way that it is executed in a sensible period and cost, energy, and pollution are minimized. So, considering these items provides a sustainable development to the project.  6. Conclusion and outlook. This study investigated the robust problem of costtime-quality-energy-environment trade-off with resource-constrained and provided a case study of a bridge construction project in Tehran. The goal of this study was to consider sustainable development in scheduling projects and simultaneously taking into account all sustainability factors, including cost, environment, energy, and quality in executing projects. Nonlinear programming (NLP) model with four objectives (i.e., cost, quality, energy, and pollution) was applied for formulating the problem, which was solved by BONMIN solver of GAMS software. Time affected all the objectives, both directly and indirectly. It usually first declines and then increases cost, energy, and pollution, while it only reduces quality. The robust optimization technique proposed by Ben-Tal and Nemirovski [5,40] was utilized to include uncertainties in the model efficiently. As it is obvious in the results section, uncertainty is very low (almost zero). When uncertainty is equal to zero, and as time decreases, cost, energy, and pollution first decline and then increase, while quality reduces as time reduces. Moreover, in time-cost, time-quality, time-energy, and time-pollution charts, as uncertainty increases, cost and quality improvement and pollution and energy reduction. This model can be employed for all industrial projects, including roads, construction, manufacturing, etc. The main limitations of the study arethe scale of the problem. When the scale of the problem is large, the time of solving is exponential growth and NP-hard. Future works can investigate resource constraintsand inventory [32,33] into the model, using fuzzy uncertainty and robust stochastic programming [34]. Moreover, given the type problem (i.e., nonlinearity of the model and NP-hardness of the problem), heuristics [45,46] and meta-heuristic algorithms [51,55,65,66] can be developed to solve large-sized problems in project management.