The construction industry plays a critical role in the economic growth of any country, with approximately $10 trillion being spent on construction goods and services annually [1]. The industry employs 18 million people in the European Union and contributes approximately 9% to the gross domestic product (GDP) within the European Union [2]. Because of that, construction project management is crucial as a slight improvement in this sector can have a significant positive impact on the national and global economy ^[2][3][2,3]. However, construction project management is complex due to the uncertainties involved in the construction processes, the complexity of construction projects, and the large number of activities involved [4]. One of the primary tasks of a project manager is to manage the trade-offs among the constraints of time, cost, and scope, based on the nature and priorities of the project, through establishing a project priority matrix ^[5][6][7][8][5,6,7,8]. 

Time-cost trade-off (TCT) problems are considered complicated combinatorial problems that require effort and careful analysis to develop and evaluate different alternative schedules in order to determine the optimum schedule that optimizes the project duration with the minimum total project cost ^[9][10][9,10]. Some of the most important negative risks that influence TCT include poor planning and design, poor estimation, contract modifications, increases in material prices, equipment breakdown, project location and high transportation costs, shortage of materials, labor shortage, and natural disasters ^{[11][12][13][14][15][16]}[11,12,13,14,15,16]. On the other hand, early completion of projects within the budgeted cost and specified quality can lead to savings in overhead cost, a bonus for early completion, and may provide an opportunity to move resources to other projects for early commencement [17]. Thus, time-cost problems have been the focus of research since the 1960s, and various formulations have been developed to optimize these problems.

Although some researchers conducted a review study of time and cost optimization of construction projects in 2019, their study focused on the software used to improve the efficiency of optimization tools [17]. Some researchers have categorized the models into two categories, which are mathematical models and heuristics models ^[18][19][20][18,19,20]. The first category utilizes linear programming, integer programming, or dynamic programming to find the exact solution of the optimization problem. The second category lacks mathematical rigor and provides a good and approximate solution, but it is not guaranteed to be an optimal solution.

2. Exact Models: Linear Programming

Traditionally, the exact mathematical models that were used to solve time-cost optimization problems were mainly based on LP models ^[21][23]. In fact, the earliest reference to activity crashing dates back to 1961, when Fulkerson used linear programming to compute the least cost curve for a project ^[22][24]. Similarly, Islam ^[23][25] used an LP model to minimize the cost of crashing the total project time, subject to crash time, unfolding the network, and project completion constraints. The model helped to reach optimality with flexibility provided by sensitivity analysis in a controlled computational effort and cost. Additionally, Karmaker and Halder used a linear programming approach to crash the activities of a construction project. The objective function was to minimize the total cost of crashing activities, taking into account the maximum reduction constraints, start time constraints, and project duration constraints. When the developed optimization model was applied, it was able to reduce the time by 17% while only increasing the cost by 3.73% ^[24][26]. However, these models had their own deficiencies such as compressing unnecessary activities and computing an inaccurate earliest start time of the activities if they were not on the critical path ^[25][27]. As time progressed, many studies attempted to challenge the linearity of the relationship between time and cost, and more complex models were developed to overcome the deficiencies of previous models, such as linear integer, mixed integer, non-linear, and dynamic programming models. Chassiakos and Sakellaropoulos ^[26][28] developed a linear integer programming model to generate all time-cost combination alternatives for a construction project with generalized activity constraints to generate the optimal time-cost curve at minimum project cost. The authors concluded that the results were very accurate, but this comes at the expense of a large amount of computational time. Other researchers have used dynamic programming models to decompose and schedule the project network into several subnetworks in order to reduce the computational effort ^[27][29].

3. Exact Models: Non-Linear Programming

Al-Haj and El-Sayegh ^[28][30] presented a non-linear integer programming model to solve time-cost optimization problems by taking into account the effect of total float loss. The authors claimed that such a model can provide managers with more flexibility and new trade-offs between time and cost, thereby increasing the success rate of construction projects. Moreover, Tatar et al. ^[29][31] utilized a mixed integer programming model with a delay penalty for discrete time-cost optimization in a total of 600 problem instances with 1000 activities, and the majority of the instances were solved to optimality. Among the generated instances, there were large-scale instances, which reflect real-life construction projects. A follow-up study by Moussourakis and Haksever presented a flexible mixed integer programming model to solve the time-cost trade-off problem in construction ^[30][32]. The model is flexible as it explores different possibilities and answers “what if” questions, which is very helpful to the decision maker. Similarly, in an attempt to reflect more real-life characteristics of construction projects, Ammar ^[31][33] presented a non-linear programming time-cost optimization model that takes into account discounted cashflows. The author concluded that the model guarantees an optimum solution while using a precise discrete activity time-cost relationship and considering the value of money over time. Klansek and Psunder ^[32][34] used a non-linear programming algorithm, where a continuous total project cost function was subjected to activity precedence relationship constraints, as well as activity duration and project duration constraints. The model yielded the optimal start times, durations, and direct costs of the project activities, which provides invaluable insight to contractors before tender submission and proved to surpass methods such as CPM and PERT analysis. Other researchers who also implemented non-linear models include Diaby et al. ^[33][35], who incorporated negative exponential curves, as well as Goh and Hall ^[34][36], who used convex piecewise linear functions. The composite objective function of Goh and Hall was to minimize the total project cost through three cost components: contract penalty, overhead cost component, and sum of crashing cost of each activity. The first cost component was modeled as a piecewise linear non-decreasing convex function, while the second component is linear, and the third component was assumed to be individually piecewise linear, non-decreasing, and convex ^[34][36]. Moreover, in a recent study by Ballesteros-Pérez et al., two non-linear models were proposed that assumed either collaborative or non-collaborative resources to allow for both discrete and continuous as well as deterministic and stochastic configurations ^[35][37]. Despite the wide use of exact mathematical methods to optimize time-cost problems and their obvious strengths in guaranteeing optimality, they are not very efficient in solving multi-objective time-cost optimization problems. These models may also not be compatible with very large-scale problems and discontinuous decision space ^[36][38]. That is why approximate, metaheuristic, and evolutionary models have started gaining more attention in the literature of this field ^[37][39].

4. Approximate Models

Usually, reducing the time for the project or some of the critical activities in the project is associated with an increase in the direct cost. However, there are some circumstances or reasons that require a reduction in time, for instance, when the imposed completion date of the project cannot be met with the normal duration of the activities. Thus, one or more of the activities in the critical path must be compressed, which will increase its direct cost. The total cost of the project consists of direct cost (cost of materials, labor, equipment, etc.) and indirect cost (overhead cost of supervision, administration, etc.). The direct cost is associated with the duration of each activity, so it increases as the project time is reduced from its planned time. However, the indirect cost is directly related to the project’s duration, and it is not associated with any activity. Thus, it will decrease as the duration of the project is reduced. Thus, the project manager faces a TCT problem in which a decision has to be made: Is the reduction in project time worth the additional cost? Which activity or activities should be shortened? And to what extent should it be shortened? As this is a complicated problem and involves many uncertainties about the durations and cost of the activities, approximate models can be used to give near-optimal solutions of time and cost problems ^[26][28].

5. Hybrid Algorithm Models

Interest in hybrid algorithms to solve optimization problems has grown significantly in recent years as they exploit the benefits of different algorithms and ultimately provide high-quality solutions ^[38][86]. Among the famous hybrid genetic algorithm models that have been adopted in the literature to optimize TCT in construction projects is the combination of genetic algorithms with simulated annealing as it enables an improved hill-climbing ability and has a better fining capability ^[39][40][87,88]. In fact, Sonmez and Bettemir ^[41][89] presented a hybrid strategy for a time-cost trade-off problem that incorporated genetic algorithm (GA), simulated annealing (SA), and quantum simulated annealing (QSA) techniques. SA is a stochastic search algorithm that uses temperature to determine the probability of moving from one state to another. Meanwhile in QSA, the tunneling field strength determines the distance between the current and neighboring states ^[42][90]. The results showed that the local search capability was enhanced due to the quantum simulated annealing techniques, and the adopted hybrid strategy significantly improved the convergence of the genetic algorithm ^[41][89]. In fact, when the authors compared the results of the hybrid algorithm with GAs, they concluded that, on average, the genetic algorithm has 4.63% deviation from the optimal solution, while the hybrid algorithm only has 1.12% deviation. Alavipour and Arditi ^[43][91] proposed a hybrid algorithm that combines both a genetic algorithm and linear programming for time-cost trade-off, which considered several financing alternatives rather than the conventional one line of credit for contractors. The results concluded that an optimum combination of different financing alternatives leads to an optimum financing schedule, which increases the negotiating power of the contractors in front of the lender. Another hybrid genetic algorithm time-cost optimization model applied a combination of GA and dynamic programming. In a study by Ezeldin and Soliman ^[44][92], they proposed this hybrid technique to solve time-cost trade-off problems of project schedules with repetitive non-serial subprojects. The authors first generated a mathematical model to explore the factors that affect cost and duration relationships at activity and project levels. Then, a genetic algorithm was used to locate optimum and near-optimum solutions from the hyperplane developed by the mathematical code. Finally, the dynamic programming model searched the vicinity of each of the near-optimum solutions found by the genetic algorithm to converge on the global optimum ^[44][92]. This type of hybridization provides effective optimal solutions to real-life construction problems. Additionally, in an attempt to account for labor and equipment allocation during time-cost optimization in construction projects, Shen et al. ^[45][93] implemented a hybrid algorithm that combined a genetic algorithm and the Cobb–Douglas production function (CDPF). The function relates technology to labor input and capital input to compute the total production. Indeed, CDPF is a feasible tool that has very significant features that can explain the origin of crashing cost under diverse scenarios. The results indicated that in all the ten cases where the hybrid algorithm was implemented, the outcome was the same as the optimum solution, which confirms the efficiency of the model. A novel hybrid genetic algorithm that has recently been gaining attention in the literature of time-cost optimization in construction projects is the hybridization of PSO and GA. In fact, Albayark ^[46][94] developed this hybrid algorithm for resource-constrained construction projects. The author concluded that this hybrid strategy is very helpful in global optimization as it exploits the social component of PSO and the local search capability of genetic algorithms. Similar results were found by Ashuri and Tavakolan ^[47][95], who stated that a fuzzy-enabled GA-PSO hybrid strategy had a faster processing time for optimizing TCT problems in construction management than other existing algorithms. On a different note, Albayrak and Özdemir ^[48][96] hybridized a firefly algorithm with PSO to optimize time-cost trade-off in construction projects. The authors had two objective functions: minimize total project cost and minimize total project duration with constraints such as non-violation of precedence constraints, one mode per activity, initial project start time, and maximum project completion time. The outcomes were proven to be shorter and more economical than other metaheuristic algorithms ^[48][96]. Moghayedi and Windap ^[49][97] proposed a novel method for modeling uncertainty in construction projects. They identified three sources of uncertainties, namely, variability, correlation, and disruptive events, that can affect the time and cost of highway projects. To address these uncertainties, they developed a hybrid intelligent tool based on a neuro-fuzzy inference system (NFIS), which combines neural networks and fuzzy logic. The proposed tool was able to effectively model the different sources of uncertainty, leading to improved project planning and decision making. The advantages of the algorithm include more accurate cost and time estimations, better risk management of cost and time overruns, and enhanced project performance. The potential limitations and weaknesses of the proposed hybrid intelligent tool for modeling uncertainty in construction projects include the need for more sensitivity analysis to validate its effectiveness in capturing the impact of various sources of uncertainties on estimating the cost and duration of infrastructure projects. Additionally, the tool may require significant data preparation efforts and may not be easily applicable to smaller-scale projects due to the complexity of the model. Furthermore, the accuracy of the model’s predictions depends on the quality and quantity of available data, and the model may require frequent updates as new data becomes available. Moreover, the tool relies heavily on historical data, which may not always be available or accurate.