The time-cost trade-off problem and its extensions: A state-of-the-art survey and outlook

The time-cost optimization is amongst the most critical fields, which has an extensive range of implementation in project scheduling. Achieving a satisfactory balance between these factors can lead to an efficient construction project by reducing both the length of the project and costs at the same time. An effective balance can be achieved using various methods, depending on the situation. This study aims to incorporate the various algorithms used in the last 15 years to reach a satisfying balance between time and cost, including meta-heuristics, heuristics, and exact algorithms. A comprehensive view of the problems associated with time-cost optimization will be provided throughout this review to assist new and challenging researchers who are interested in this type of research. For this purpose, we have reviewed some objective functions and uncertainty techniques that could be employed in time-cost balancing problems. The literature review tables contain a variety of columns, including uncertainties such as fuzzy, probabilistic, interval, robust, and objective functions, along with cost and time, for the investigation of various types of balance issues. In the conclusion of this article, we will show the results of our literature review table using different types of graphic diagrams. For each main column of the table, we will show various types of diagrams to make the results easier to understand.


Introduction
Project scheduling is a mathematical subject in operation research that involves identifying each activity's start and finish times as a means of optimizing a specific objective, as well as considering resource constraints and priorities between activities in a projection of each industry (Mohagheghi et al., 2017).The preliminary research on time-cost balancing was carried out in 1950 and emphasized the importance of preparing a network using Critical Path Methods (CPM) (Robinson, 1975;Siemens, 1971). and Program Evaluation and Review Techniques (PERT) (Azaron et al., 2005), both of which are essential tools for project management.
In recent years, researchers have done numerous studies on a wide range of topics relating to project scheduling, as well as the extension of project scheduling to time scheduling, budget scheduling, and resource scheduling. Particularly, one of the most prevalent topics in time-scheduling problems is the optimization of both time and costs, which is commonly referred to as TCTP in the industry. Over the years, extensive research has been conducted on TCTPs, which are the most common type of project scheduling problem . A major objective of the TCTP is to define activities' best duration while keeping their total cost to a minimum (Feng et al., 1997). It is also important to mention that TCTP is considered a multi-objective optimization when both duration and costs are to be reduced (Albayrak & Özdemir, 2017). There are various ways we can reduce the implementation time of each activity within a project if we allocate adequate facilities, such as labor, equipment, and procedures, to each one, so we can find an optimal balance of time and cost. To put it another way, each decrease in activity duration increases the cost of the activity .
There is generally a normal time or a crash time that is required for each activity to be completed. Normal time is the amount of time spent on any project activity without interruption. It is important to note that by applying crash time to activities, we will require more direct costs and resources to complete them. In Figure. 1, and show normal duration and crash duration. Aside from direct costs, indirect costs are also present for each activity. The direct costs of the activity include labor, materials, and any type of facility that is required during the activity. As the name implies, indirect costs refer to those costs that do not directly relate to the project's activities, such as organization, public utilities, and the hiring of equipment. Also, and indicate as normal costs and crash costs in Fig. 1.
The relationship between time and cost .

Fig
As discussed previously, the cost slope is the extra direct cost associated with reducing an activity's normal duration by one unit. As well as this, crash costs and indirect costs are also inversely related; an increase in indirect costs will cause the project duration to be extended. We portray the time-cost relation in Fig. 1. It is clear from the total cost curve that a minimum value indicates the best or optimal project duration (Reda & Carr, 1989). This paper refers to a basic mathematical model that describes the TCTP, and we demonstrate its essential components. Indicators, variables, and parameters of the model are described below: Start time of immediate activity k that occurs after activity j, Start time of activity n+1 that occurs after last activity (n),

Variables
Binary variable taking the value of 1 if mode m is selected for executing activity j; otherwise, it is equal to 0, subject to: Based on the mentioned model, all project costs, including direct and indirect costs, are minimized. According to constraints (2), activity j must be linked with only one type of mode. Also, precedence constraints are displayed in the constraints (3). Constraint (4) states that the projects should not exceed the deadline time assigned to them. Finally, constraints (5) and constraints (6) show the type of the parameter and variable.
In the remaining sections of this paper, the purpose of the next section is to briefly outline the types of expansion of TCTP and the approaches that can be used to solve it. Section 3 shows a variety of different types of graphical results for the details of the literature review tables that were presented in part 2. As a final part of this study, we will review the summary and general findings of the study.

Literature Review
As reported in the literature, many models and approaches have been suggested to help project managers to plan and schedule their projects more efficiently. Numerous researchers have worked on this issue in recent decades. During this section, four studies that deal with time-cost trade-off extensions are discussed.

Time-Cost Trade-off Problem (TCTP)
Initially, Kelley Jr, (1961) considered a mathematical model to balance the linear relationship between cost and time. He provided a heuristic algorithm to gain optimal results. After that, many studies were done on this issue. Three mathematical models based on Integer Programming (IP) and Linear Programming (LP) were proposed by  to enhance the performance of TCT. A Mixed Integer Nonlinear Programming (MINLP) was demonstrated by Klanšek and Pšunder, (2012) to identify the optimal duration of each activity. Li et al. (2015) suggested the model based on robust optimization of Mixed Integer Programming (MIP). They also considered time and cost as interval numbers. For discrete types of TCTP, Li et al. (2018) presented two heuristic methods that minimize project durations and costs. A nonlinear TCTP with activity crashing was introduced by Ballesteros-Pérez et al. (2019), and in order to optimize both objectives concurrently, they used a genetic algorithm (GA). Abdel-Basset et al. (2020) incorporated fuzzy uncertainty and neutrosophic numbers to deal with the uncertain conditions, and activity durations are estimated using trapezoidal neutrosophic numbers. Also, Elkalla et al. (2021) assumed fuzzy TCTP, and they used the nearest-symmetric trapezoidal fuzzy numbers (TFN) for converting fuzzy numbers to solve the model. Finally, Dhawan et al. (2021) solved the TCTP by one of the metaheuristic approaches called Simulated annealing (SA).

Time-Cost-Quality Trade-off Problem (TCQTP)
The TCTP problem is extendable by taking into account quality, which is an important factor that is affected by the reduction in time as well. Therefore, in TCTQPs, the overall goal is to implement projects in a short amount of time and with a minimum cost while maintaining a high level of quality. According to the literature, Babu & Suresh, (1996) argued that the failure of critical activities could negatively impact quality. Therefore, there is no doubt that quality should be evaluated along with time and cost in any project. In TCTPs, various approaches were implemented to address quality. To review the TCQTP, a survey study for gas and oil projects was carried out by Wood, (2017). His research was based on fuzzy multiobjective problems, and he solved them with a memetic multi-objective algorithm. Mrad et al. (2019) solved the TCQTP using a Monte-Carlo simulation based on a Mixed Integer Linear Programming (MILP) model.  proposed the TCTQP by considering environmental impact and several execution modes for activities. They utilized the Data Envelopment Analysis (DEA) to trade-off between the objective function and reach the efficient execution mode. After that,  also suggested the TCQTP with environmental effects for industry projects. In their study, the problem was modeled as a single objective to obtain optimal results. Then, they use the Leopold matrix method to assess the environmental objectives. Hamta et al. (2021) assumed the quality function in the TCTP and handled the quality of the project in TCTP through a goal programming model. Finally, Sharma and Trivedi (2022) used a Non-Dominated Sorting Genetic Algorithm (NSGA-II) optimization approach that reduced both time and cost while at the same time increasing quality. They also weighed quality by using the Analytical Hierarchy Process (AHP) approach in TCQTP.

Time-Cost-Quality/Safety Trade-off Problem (TCQSTP)
According to relevant literature, kinds of research have included safety along with other objective functions. As an example, Ning and Lam (2013) suggested a model to trade-off safety along with cost in layout planning. Afshar and Zolfaghar Dolabi, (2014) have presented an analysis of the safety in TCTPs using NSGA-II to illustrate the importance of safety in construction projects. For the project's success, quality and safety are considered together, although there are few studies on this issue. For instance,  developed the TCQSTP by considering resource constraint and multi-mode activities. The real case study was a building construction project. They assumed fuzzy logic for safety parameters and used the NSGA-III. In addition, Panwar & Jha, (2021) integrated quality and safety to handle scheduling in construction projects. Based on this study, an NSGA-III approach was used to demonstrate how to generate an optimal balance.

Time-Cost-Quality-Energy-Environment Trade-off Problem (TCQEETP)
Recent developments have led to dramatic changes in the world. Many construction projects cause the emission of greenhouse gases and increase air pollution . This pollution has a great impact on human life and increases climate change. Due to this, very few studies have included environmental impact reduction within the deadline and cost of the project or in other objective functions related to the project that was mentioned earlier in this article. For large-scale construction systems, Xu et al. (2012) presented multi-mode and fuzzy uncertainty balancing methods for trade-off discrete types of time-cost-environment factors. To determine the results, they used a genetic algorithm. A time-cost-quality-environment trade-off problem (TCQETP) has been proposed by , and he recommends using EBS-based GA for conducting optimizations. As mentioned previously, some other researchers investigated the TCQTP with considering environmental conditions. Tiwari, (2022) also solved the TCETP using NSGA-II. Furthermore, Lotfi, (2022) surveyed a full form of a trade-off as a TCQEETP by considering the resource constraint. They used robust optimization to deal with the uncertainty of the real case study as bridge construction. Also, the Augmented Epsilon Constraint (AUGEPS) was utilized to solve their model. In the other study, by incorporating Block Chain Technology (BCT) and risk into the resource-constrained TCQEETP model, Lotfi, (2022) improved the sustainability, resiliency, and agility of the project. To cope with uncertainty, they assumed robust stochastic programming (Özmen, 2011;Ben-Tal & Nemirovski, 2000), worst case scenarios, and conditional value at risk. Finally, they solved the proposed model with GAMS software for a real healthcare project.
Following the above discussion, the major purpose of this research is as follows: 1. Classification of approaches for solving TCTP based on an exact, heuristic, and meta-heuristic methods, 2. Considering the objective function from a single or a multidimensional perspective, 3. Investigating the types of uncertainties used for objective function in the literature review, 4. Use graphical results to display the results of this review.

Graphical Results
The present section contains several charts that illustrate the graphical results for the different types of columns in the reported tables according to the literature. Therefore, multi-mode and multi-objective strategies have been considered by researchers progressively in recent years (Fig. 2), and time and cost are the two more widespread parameters in multiobjective functions (Fig. 3). Due to solutions methods of TCTP, there has been a surge of interest in using metaheuristic methods (Fig. 6) and more emphasis on Genetic Algorithms (GA). On the other hand, the use of metaheuristic approaches has increased as the models have become more complex. According to Fig. 4, fuzzy uncertainty is more commonplace than different types of uncertainty shown in the tables. Additionally, applied to real world situations, time and cost are two main factors that are less meticulous than they would be in the realworld, and in this way, these factors are more uncertain than others (Fig. 5).

Conclusion
The importance of scheduling has increased in many construction projects. Project managers need to schedule their projects efficiently. Our purpose in this study is to investigate the methods employed to solve the problem of time-cost trade-off through the study of recent studies. According to Fig. 7, in recent years, multi-objective methods have increased over the previous years, while single-objective functions are declining. Over time, considering quality as one of the objective functions has grown due to not ignoring the quality of projects (Fig. 8). Also, as global warming and energy consumption have increased in recent years, it has become more common to consider both energy and environmental criteria in time-cost trade-off issues (Fig. 8). To handle uncertainty in a successful manner, fuzzy uncertainty (Kropat & Weber, 2018), interval number (Allahverdi, 2022) and novel robust formulation  can be considered for all parameters in TCTP. As compared to other uncertainties, fuzzy uncertainties have gained attention over the years, but there has been a decline in the use of probability uncertainty; however, interval uncertainty is rarely used ( Figure. 9). Moreover, heuristic 114 and meta-heuristic algorithms  and other new methods Golab, et al., 2023) are used to solve large-scale TCTP. Based on the radar diagram shown in Figure 10, the use of meta-heuristic methods, to find optimal solutions, in particular for multi-objective problems, is more general than other methods; while TCTPs have rarely been solved by heuristic approaches.