AN IMPROVED TIME-COST TRADE-OFF MODEL WITH OPTIMAL LABOR PRODUCTIVITY

Optimization of the time-cost trade off (TCT) has received considerable attention for several decades. However, few studies 
 have considered improving performance/productivity of existing crews. To shorten the gap to real-world applications, this study presents an improved 
 TCT model that considers variable productivity using genetic algorithms (GAs). Through an illustrative case and a real world case, the results 
 demonstrate that improving labor productivity of selected activities by allocating existing crews and management can yield an optimized solution. 
 As such, a decision maker can implement a better optimized technique to reduce a project duration under budget while reducing the risk of liquidated 
 damages. The main contribution of this study is to apply managerial improvement of labor productivity to TCT optimization, the project duration can be 
 reduced owing to improved productivity of existing crews rather than inefficient overmanning, overlapping or costly substitution. In the end, three 
 important managerial insights are presented and future research is recommended.


Introduction
It is an established fact that projects are almost always behind schedule (Gerk & Qassim, 2008). Therefore, optimizing the performance of complex construction projects has received considerable attention. Determining how to complete projects on time and under budget has become the most important goal in the optimization field. Furthermore, as well as minimizing project costs, to decrease the risk of contract-specified liquidated damages (LD), contractors attempt to complete projects earlier than the stipulated duration because LD will impact total project costs if the contractor fails to meet the contract completion date.
Various studies have considered time-cost trade off (TCT) optimization using a variety of methods. Most researchers consider the straight relationship between time and cost no matter minimizing cost within granted duration or compressing schedule under budget. It is intuitively assumed that direct costs will increase with schedule acceleration. However, such an assumption is deficient because labor productivity as a critical factor, which generally influences project time and cost is rarely considered when evaluating optimization of project execution. Furthermore, even though current TCT optimizations could provide instant effect on improving project performance, side effects such as rework, fatigue or costly pay might be inevitable. Accordingly, the critical factor should be further considered in TCT optimizations.
Previous researches have revealed that labor productivity can be improved through management such as training, process improvement or incentive etc. To address this issue, the objective of this study is to presents an improved TCT model that considers variable productivity relative to the working environment and management. The proposed model addresses the influence of variable productivity on project duration and direct cost by establishing the interaction among labor resource, deliverable, duration and direct cost. Genetic algorithms (GAs) is used to search for and identify optimal/near optimal productivity and scheduling. Therefore, the model can reduce the project duration without side effect and thereby decrease the risk of LD without going over the original direct cost. This study presents an illustrative case and a real world case to validate the proposed optimization model.
The remainder of this paper is organized as follows. Studies related to the TCT problem and labor produc-tivity are discussed in Section 1. The formulation of the proposed model is given in Section 2, and the selection and use of implementation methodology are explained in Section 3. An illustrative case and a real world case are described in Section 4, and analysis results are provided and discussed in Sections 5 and 6, respectively. Final section presents conclusions and recommendations for potential future research.
Almost all these studies considered that reducing a project's duration (i.e., schedule acceleration) will inevitably incur cost increase under a TCT framework. However, for a multi-objective problem, decision makers can employ techniques that can quantify and control the decision variables to find an optimization solution that incurs minimum cost.
Based on previous studies, optimization techniques for the TCT can be classified as (1) crashing (acceleration by overmanning, overtime, or shift time); (2) fast-tracking (concurrent engineering by evaluating the magnitude of evolution and sensitivity from upstream to downstream); and (3) substitution (improvement by utilizing better resources and advanced technologies). These techniques can reduce time; however, each of them has side effects. For example, crashing might cause fatigue, fast-tracking might accompany rework, and substitution could be costly. Furthermore, although previous studies have provided a variety of robust optimization models applicable to manufacturing and construction industries, few have investigated labor productivity.
Peña-Mora and Li (2001) established a feedback model to fast-track construction projects using system dynamics.
They altered the labor productivity assumption, which has been considered a constant in previous studies. In their dynamic model, productivity determined by the function of schedule pressure, experience level with a phase, the effect of fatigue, and the normal productivity will eventually affect both project duration and cost; Roemer and Ahmadi (2004) indicated that means to increase work intensities (i.e., crashing) throughout a project include working overtime, adding staff, or utilizing more experienced staff or better equipment. El-Rayes and Kandil (2005) developed a multi-objective optimization model and analyzed the time-cost-quality trade-off between two crew formations comprised of eight resource utilization configurations with different levels of productivity. Senouci and El-Rayes (2009) considered crew performance relative to a time-profit trade-off model. To optimize project execution, the performance generated by eight crew formations are assumed in the present model. Hossain et al. (2012), Hossain and Chua (2014) identified redesign and rework as productivity loss that is incurred by overlapping. They also evaluated the trade-off between overlapping and redesign/rework (i.e., productivity loss) and quantified the probability of redesign/rework. Kuo (2013) searched for an optimal schedule by substituting three resources with different efficiencies. Float reallocation can be more appropriate after considering float consumption and cost in the optimized schedule.
Even though these studies considered productivity change, optimal productivity cannot be determined via simulated dynamic models. In optimization, changeable productivities are primarily utilized from the perspective of external resource substitution. It does not contribute to the improvement of productivity nor accumulation of competence for the existing resource although worker's sustainability has been continuously emphasized (Florez et al., 2013). Furthermore, the influence of the environment and management processes on labor productivity has not been considered relative to the TCT. Typically, in construction projects, labor costs are more than 40% of the total project budget (Hanna, Russell, Gotzion, & Vandenberg, 1999). If the productivity of existing crew can be improved, it is possible to reduce a project's duration without increasing costs or reduce project costs within the scheduled duration, particularly for large construction projects with long durations. Therefore, this study presents an integrated technique to directly improve the labor productivity of existing resources and reduce time through management initiatives, such as improving processes, decreasing waste, and providing training and incentives.
Labor productivity is calculated as the man hours (mh) per unit of work (Thomas & Yiakoumis, 1987). For example, labor productivity for a concrete foundation is 20 mh/m 3 . In practice, labor productivity is primarily influenced by the environment and management (Thomas & Yiakoumis, 1987;Thomas & Raynar, 1997), and such influences affect project time and cost. The environment can be classified as external (e.g., weather) (Thomas, Riley, &  Sanvido, 1999) and working environment (e.g., crowded working space) (Thomas, 2000). Previous studies identified management as a critical factor influencing labor productivity (Rojas & Aramvareekul, 2003;Mojahed & Aghazadeh, 2008;El-Gohary & Aziz, 2014). Consequently, time or duration can be reduced effectively by improving the work environment and management in order to improve labor productivity. Thomas, Sanvido, and Sanders (1989) suggested that contractors should consider the trade-offs between investing cost and disruptions saving. It would be beneficial to improve labor productivity and reduce time and cost through a management system that involves process improvement and information feedback (Sanvido, 1988), material control during construction (Thomas et al., 1989), shortening the decision-making process in the design and construction phase (Peña-Mora & Li, 2001), and incentives (Chokor, Asmar, & Paladugu, 2017).

Model formulation
Improving productivity through management has been proved a feasible and fundamental method for large construction projects with long durations in previous researches. However, improvement takes time and incurs costs; thus, all possible improvement activities may not be implemented under time and budget constraints. Considering multi-objective decisions with TCTs, decision makers will prioritize the most efficient alternative (e.g., improvement with shorter time or lower cost). Consequently, the objective of the proposed model is to help decision makers find an improvement alternative that provides optimal productivity and minimizes time and direct costs.
This section presents a generic formulation and objective function for the proposed model, which provides an optimal improvement strategy to reduce the project duration.
According to the critical path method (CPM), the original contract duration (CD) is computed as follows: The optimal objective is to minimize project duration (PD), the objective function is as follows: Subject to < CD; and is the original duration of activity (i) on the critical path (CP) using labor resource (U) with productivity (P) to complete the quantity of work (Q) , the original CP includes l activities. After optimization, a new CP composed of l′ activities might generate according to their logic relationships and minimized duration. Where is the new duration of activity (j) on the CP using labor resource (U′) with improved productivity (P″) to complete the quantity of work (Q″), and is the improvement duration of activity (j) on the CP using labor resource (U′) with productivity (P′) to complete the quantity of work (Q′). Generally, productivity (P″) after improvement is better than productivity (P) before improvement and productivity (P′) during improvement (i.e. P″< P < P′). Total quantity of work (Q) is the quantity of work (Q′) completed during improvement period plus the quantity of work (Q″) completed after improvement.
Referring to the factor model presented by Thomas and Raynar (1997), the various factors that influence labor productivity can be categorized into four variables.
Labor productivity (P) is expressed as follows: where mh q is the input man hours (mh) per output work quantity (q), E is the environment identified as working environment allowance to constraint the resource usage, M is the management method identified as improvement of labor productivity, L is the skill required to fulfil the work, and W is the work content such as work scope or work complexity. In this study, L and W are constants because it is assumed that the labor is qualified to complete the work and the work content is specified as a contract requirement.
In this study, congestion factor (C) is a function of actual labor usage (U) versus labor allowance (A) in a work environment (E). When actual labor usage is more than labor allowance in a work environment, congestion factor is more than 1. The improvement factor (I) is an index of measuring the improvement effect through management.
The labor productivity (P″) after improvement through management equals improvement factor (I) multiplies labor productivity (P). The optimal labor productivity (O) is defined as the value considering both congestion factor (C) and labor productivity after improvement (P″). The value is determined by GAs.
The relations among congestion factor (C), improved labor productivity (P″), improvement factor (I), and the optimal labor productivity (O) are presented as follows: The duration (D) of an activity is expressed as follows: where Q is work to be done, DMH is daily manpower multiplied by daily straight time (default: eight hours). When Q is constant, D will decrease if labor productivity (P) decreases or MH increases (i.e., daily manpower increases). The project direct labor cost (PC) with optimal duration can be computed as follows: where DC x is direct labor cost, MC x is labor mobilization/ demobilization cost, IC x is improvement cost, and CB is direct labor cost of contract budget.
To sum up the notations used in the proposed model, actual labor usage (U) and improvement factor (I) are defined as 2 sets of key decision variables which jointly decide the optimization result. The range of labor usage and improvement factor with time and cost will be further evaluated under resource availability and manager's expertise. In other words, if over-crowding in a work environment is not possible, labor usage might be allocated as much as resource availability after optimization. The other notations such as normal labor productivity (P), work to be done (Q), working environment labor allowance (A), direct labor cost (DC x ), labor mobilization/demobilization cost (MC x ) and improvement cost (IC x ) are regarded as activity's properties which should be predetermined in a project.
Note that the following considerations and assumptions should be made in advance: 1. The cost of improving productivity through management (IC x ) is identified as one of direct cost because crew should be involved in the improvement process. 2. An overtime policy is excluded because it will increase costs and decrease productivity. Moreover, overtime will not be considered if the project duration can be reduced by improving productivity. 3. Since weather is unpredictable and uncontrollable, its impact is not considered in the model. 4. Decision makers are assumed to be risk-averse. To avoid LD, they attempt to complete a project as quickly as possible and under budget rather than minimize the cost within the contract completion date. 5. Construction is a labor-intensive industry; therefore, to simplify the model, the only resource considered in this study is labor. 6. The existing crew in the proposed model is sufficient, therefore, substitution of external crew is not considered.

Methodology
Several methodologies have been used to evaluate the optimization problem relative to the construction projects, including nonlinear integer programming (Gerk & Qassim, 2008;Klansek, 2016), 0-1 integer programming (Cristobal, 2009;Florez et al., 2013), the heuristic method/rule of thumb (Hazini et al., 2013), GAs (Chan et al., 1996;Li & Love, 1997), and dynamic programming (Peña-Mora & Li, 2001). Although heuristic methods can determine optimum degree of activity accelerating and overlapping in schedule compression, they might not guarantee a global optimum (Hazini et al., 2013). In addition, mathematical programming (i.e. integer programming) could provide the optimal solution but it could not be applicable to industry practitioners owing to difficulties of formulating construction projects with complicated schedule. Hegazy (1999) compared the heuristic method, mathematical programming model, and GAs for TCT analysis and noted that having mechanisms of simulating natural evolution and survival-of-the-fittest, GAs have been used to solve several engineering and construction management problems. Furthermore, GAs can solve problems with discrete time-cost relationships. The results obtained by GAs model do not indicate an exponential growth in the computational time required for larger problems (Chan et al., 1996). In terms of scheduling optimization, GAs have provided a more efficient way to search for optimal/near optimal solutions compared to traditional methods evaluated by schedulers (Dehghan et al., 2015). Referring to previous researches, the comparison of three major techniques is shown in Table 2. Therefore, considering the multiobjective TCT problem in this study, GAs as a mature method are used for 2 cases' optimization. It can further assist project scheduler for optimizing large real projects.
In the process of the TCT with optimal productivity, three steps are taken by the decision maker: (1) determine the magnitude of improving productivity depending on how much time and cost are necessitated for improvement of each activity; (2) allocate optimal resources in a specific environment for each activity to carry out the work; and (3) search for an optimal productivity plan from possible solutions as an optimization benchmark for project execution.
Representation of problem in GAs can be solved by a finite-length string which is analogous to a chromosome in a biological system. Each individual chromosome represents a random solution that encompass many genes. The optimized solution is generated through objective function and GAs procedure. Generally, the GAs procedure is as follows: (1) generate an initial population of random solutions in a parent generation; (2) search for solution with excellent fitness; (3) regenerate an offspring population of solutions by crossover and mutation operators; and (4) re-evaluate and search for optimized solution within the child generation. This iterative process is terminated when the optimized solution is found, i.e., all termination criteria in the GAs model are satisfied. The GAs procedure is shown in Figure 1. The detailed implementation and description of GAs can be found in the literature (Goldberg, 1989).
In this study, each chromosome represents a management improvement strategy, and a gene represents 2 sets of key decision variables: actual labor usage and improvement factor, as shown in Figure 2.
Generally, two major modules exist in GAs to solve the TCT. The time module minimizes time under the current budget, and the cost module minimizes cost within a limited time. In this study, the time module is analyzed to search for the optimization acceleration and the cost module is used to illustrate the optimization curve of TCT.
The most popular programs used to analyze GAs include the MATLAB GAs Module, the Microsoft Excel Evolutionary Solver, and the Palisade Evolver. This study uses Microsoft Excel Evolutionary Solver and Palisade Evolver add-in for Microsoft Excel which both are suitable programs for applying a wide variety of variables and conducting complex and iterative computations. Microsoft Excel used as a GAs platform provides flexibility to the researcher to easily change variables and review and analyze the results, and it is quite compatible with Microsoft Project (Dehghan et al., 2015). Generate an initial population of random solution s in parent generation (g = 1) Project properties: (1) Activity Duration (2)

Case validation
In this section, 2 cases are examined involving working environment and variable productivity in TCT optimization problem to demonstrate the effectiveness of the proposed model. For better understanding the nature of proposed model, the first case is illustrative and simplified. The second case is a real construction project considering more activities with complicated relationships.

Case study I
The first case comprises 10 activities with different properties depicted in Table 3. The relationships are all assumed to be Finish-to-Start with zero lag time because they are determined by characteristics of each activity, which do not vary during optimization process and affect the final result. The only 4 paths in this model is easier validated by scheduler as a traditional method instead of GAs optimization ( Figure 3). Considering the working environment constraint, the appropriate amount of available labor is identified according to normal conditions for each activity, i.e., the working environment allowance (A x ). The normal labor productivity (P x ) for each activity is identified respectively. The actual labor usage (U x ) is initially equal to the working environment allowance (A x ). The manager should allocate labor with different skills to implement the given activity. Moreover, labor cannot be substituted among different activities, e.g., the labor working on activity 01 cannot be allocated to activity 02. To simplify the model, the direct labor cost is assumed to be $2,000 per day, and the cost of mobilization and demobilization for each labor is also assumed to be $2,000.
It is practical that the minimum time fraction is 0.5 day in the construction industry (Li & Love, 1997). However, any decimal value reduction of duration in the developed model is acceptable to reflect optimal productivity. Two kinds of results are presented in the next section.
The duration of each activity is computed based on Eqn (7). For example, the duration of activity 01 is 100 days (i.e., 320×25÷10÷8). According to the CPM, the initial CP of the case project is 01→03→06→09→10, and its duration is 381.25 days. The project direct labor cost is $17,513,500 (direct labor cost $17,287,500 + mobilization/ demobilization cost $226,000). If the minimum unit of 0.5 days is considered, the original duration of the project is 383 days. The project direct labor cost is $17,596,000 (direct labor cost $17,370,000 + mobilization/demobilization cost $226,000).
To reduce the project duration, the decision maker may allocate more manpower to accelerate the schedule if labor availability is sufficient. However, as more manpower is allocated, the working environment becomes increasingly crowded. As a result, labor productivity may deteriorate. Previous studies have revealed that overmanning causes inefficiencies due to high labor density and congestion (Thomas, 2000). In this study, the ratio of actual labor usage (U x ) to working environment allowance (A x ) is defined as congestion factors (C x ), which influence the normal labor productivity (P x ). The congestion factors (C x ) are assumed as follows:  The decision maker will become more familiar with each activity after reviewing its details, such as working environment constraint, and execution process. The time and cost expenses can be evaluated, and the magnitude of improving productivity through management can be further identified as improvement factor (I x ). In previous empirical investigations, the reduced labor productivity losses, reduced schedule delays and reduced waste costs ranged from 5% to 32% (Sanvido, 1988;Thomas et al., 1989Thomas et al., , 1999Thomas et al., , 2003Chokor et al., 2017).
As a result, improvement factor (I x ) is assumed to increase the output of work to the maximum 20% in this illustrative case. The magnitude ranges from 1/(1+20%) to 1/(1+10%), 1/(1+10%) to 1 and 1. Note that better improvement results in both increased work performance and longer improvement duration and higher cost. Because labor would spend more time learning or being trained instead of simply working during improving period, the work done with labor productivity (P x ′) during the period is conservatively assumed to be 80% of the work done with the original productivity (P x ). In other words, completing a quantity of work would take longer during the improving period, the labor productivity becomes worse accordingly. For example, if labor productivity (P x ) of an activity is 25 mh/q, labor productivity (P x ′) during improving period is 31.25 mh/q (25 mh/0.8q). Besides, the improvement cost (IC x ) is assumed 20% higher than direct labor cost (DC x ).
The above assumptions for each activity in next real world case were further evaluated by experienced practitioners through an interview. After improvement, the remaining work will be done at labor productivity (P x ″) in consideration of improvement factor (I x ). The improvement factor, duration and cost are assumed as shown in Table 4.

Case study II
In the second case, the proposed model is used to ana-lyze a real world case which is an ultra-supercritical boiler construction project implemented by a general contractor in Taiwan. The construction period was from 2013/9/3 to 2016/9/1 (1,094 days). Owing to the confidentiality of the case project, the project direct labor cost (budget) was US$ 57,637,738 (direct labor cost US$ 57,520,338 + mobilization/demobilization cost US$ 117,400) through a factor transformation. The project primary schedule involved 9 major disciplines comprises 22 activities with 55 relationships and 3 relationship types with lag time, i.e. FS, SS and FF (Figure 4).
There are total 220 paths in the schedule. In practice, the primary schedule is developed for overview and integration of comments from each discipline. Once the schedule is configured, each activity can be further developed to detailed schedule. The project experienced practitioners including project control manager, construction manager and subcontractor's superintendent were invited to evaluate the possibility of improving overall schedule. In the construction period, acceleration methods such as crashing, fast tracking and substitution were also jointly used to reduce the schedule, but only the improvements relative to labor productivity were separated to demonstrate the individual effect. After explaining the TCT proposed model to project practitioners, the parameters of congestion factor in case I can be applied to case II. The work done ratio with labor productivity (P x ′) during the improving period was concluded as 80% of the work done with the original productivity (P x ). However, the improvement factor could be optimized to maximum 10% after reviewing the all feasible improvement and identifying the improvement durations for each activity. The improvement cost (IC x ) is 10% higher than direct labor cost (DC x ). The project basic information and major improvements of activities are shown in Tables 5 and 6.

Extreme data test
If the decision maker determines to improve labor productivity through management on all activities. The results show that when improvement factor (I x ) performs the maximum output valued at 0.833 (1/1.2), the project duration will be reduced from 381.25 days to 361.38 days; however, the project direct labor cost increases from $17,513,000 to $17,591,183. On the other hand, if the decision maker decides to mobilize all available labor to reduce the schedule, the project duration will become 302.34 days, but the project direct labor cost increases to $24,658,750. Note that both decisions cannot reduce the project duration and stay under budget simultaneously.

Optimization results
The optimization results analyzed using Palisade Evolver are shown in Table 7. As can be seen, the project duration is reduced from 381.25 days to 342.49 days (i.e., a reduction of 38.76 days; 10.2%). The project direct labor cost is reduced from $17,513,500 to $17,499,482 (12,721,482 + 230,000 + 4,548,000).
If the optimization is further evaluated using the minimum unit of duration, i.e., 0.5 days, the project duration is reduced from 383 days to 347 days (i.e., a reduction of 36 days; 9.4%). The project direct labor cost is reduced from $17,596,000 to $17,570,400.
The duration of each activity is reduced sequentially in the optimization process, and after optimization, the final CP remains on 01→03→06→09→10.

Extreme data test
Following the test in case I, decision maker might improve labor productivity through management on all activities as the maximum improvement or mobilize all available labor to accelerate the schedule. The two extreme strategies are both accelerated and schedules (i.e. 1071 and 992 days) are less than 1,094 days. However, the costs (i.e. US$ 57,796,274 and 73,594,450) are over budget (US$ 57,637,738).

Optimization results
The optimization results analyzed using Palisade Evolver are shown in Table 8. As can be seen, the project duration under 0.5 days' minimum unit is reduced from 1094 days to 1047.5 days (i.e., a reduction of 46.5 days; 4.3%). The project direct labor cost is reduced from US$57,637,738 to US$57, 619,505 (43,970,770 + 106,200 + 13,542,535).
To sum up the results, either the simplified test case or the real world case can apply the improved TCT model to find the optimal solution. The first illustrative case is further discussed in the next section.

Discussion
It is worth improving productivity if (1) the reduced duration of the project schedule is greater than the increased duration of improvement on the CP, and (2) the increased Congestion factor (C x ) Improvement factor (I x ) Labor productivity (P   cost of improvement is less than the reduced project direct labor cost. The optimization results show that the reduction of the original project duration, i.e., 144.76 days, is greater than the increase of improvement, i.e., 106 days. In addition, the increased improvement cost of $4,548,000 is still less than the decreased project direct labor cost of $4,562,018. Consequently, the feasibility of the proposed optimization technique is corroborated.

Resource allocation trade-off
Furthermore, from a resource allocation perspective, when the actual labor usage (U x ) is greater than the working environment allowance (A x ), labor productivity deteriorates because of congestion in the working environment. Therefore, the actual labor usage in most activities is less than the working environment allowance, e.g., activities 01-08. However, labor usage in some activities is still greater than the working environment allowance, e.g., activities 09 and 10. This reveals that bearing the productivity loss due to congestion to reduce the project duration can become an optimized solution if the project direct labor cost after improvement is still under the original budget. The similar result can be validated in case II (e.g., activities 41 and 91).

Selective improvement
Although improving productivity through management can feasibly reduce the project duration, not all activities are worth improving in consideration of the TCT. The optimized result from the previous section demonstrates that the improvement factors for activities 02, 03, 04, 08, and 10 are approximately 0.91 (1/1.1), which does not maximize improvement. The similar result can be validated in case II (e.g., activities 43). It is concluded that selective improvement of activities can achieve optimization using GAs.
In the beginning of the project, the contractor may not be familiar with the working environment for each activity. The amount of the project direct labor cost in the contract may not be estimated according to optimum resource allocation and labor productivity. The optimized project duration with labor usage and optimal productivity obtained via GAs can be achieved without going over budget while reducing the risk of LD, as shown in Figure 5 (diagonal area). Table 9 presents the original resource allocation and six sets of optimization without going over budget. Although the optimal set 6 with acceleration rate (10%) is the best option, the decision maker can still arrange flexible labor usage as other optimization sets depending on actual acceleration goal (acceleration rate %) as long as the cost is still under budget.

Comparison with previous techniques
In previous TCT optimization techniques, the project total cost is generally composed of direct and indirect cost which are negatively and positively related to the project duration. When the duration is compressed, the indirect cost is reduced, but the direct cost is inevitably increased owing to inefficiency by crashing or rework by fasttracking. As a result, the total cost presented as a concave could achieve the minimum with the optimal reduction of time. However, the direct cost might not be necessarily increased with the schedule acceleration if the efficiency of resource utilization can be improved as the proposed model. Consequently, the total cost can be further reduced once considering indirect cost simultaneously.
To illustrate the advance of proposed model and major difference from previous researches on direct cost, the variation rate on time and direct cost optimization using major TCT techniques and methodologies are presented for a comparative review (As shown in Table 10). It reveals that in consideration of optimal labor productivity, the direct cost can even reduce after optimization.   Comparing with previous TCT optimization techniques and proposed model in this study, even though the effect of previous techniques is instant, relevant side effects are major concerns as shown in Figure 6. Although time acceleration through management might take longer time such as improving process, decreasing waste, training or incentive, they can not only fulfil the optimization without side effect, but also accumulate the competence of the existing crews.

Expert judgement
In order to further emphasize the superiority and robustness of the proposed model using GAs, referring to Dehghan et al. (2015) research, two schedule professionals (one has 15 years of work experience and the other has 30 years) were invited in the experiment to optimize the results. After explaining the purpose of optimization and required information such as maximum resource usage, maximum improvement of each activity, they were requested to maximum the schedule acceleration under budget for case I. For experiment time, there is half hour for the illustrative case. The best results provided by two professionals were as follows: One schedule professionals reduced 16.27 days with $107,315 saving, the other reduced 22.28 days with $22,892 saving when the time is up. However, comparing with optimization result using GAs, GAs performed a better optimization result (i.e. 38.76 days with 14,018 saving) with better efficiency (i.e. within 10 minutes). The comparison between GAs and expert experiments is shown in Table 11. Detail information and several rounds of computation are provided in Appendix 1.

Conclusion and recommendation
In most previous studies, the TCT was typically executed by optimization techniques of crashing by increasing external resources, overlapping activities or substitution of external resources. However, improving the productivity of existing crews was not considered among those techniques. After reviewing previous critical studies, this paper has presented an optimization technique for reducing a project duration. The improved TCT model considers variable productivity influenced by the working environment and management to search for an optimized solution using GAs on the most popular program platform (i.e. Microsoft Excel) instead of a specific program. Therefore, applying the proposed model to real projects will be easier for researcher and practitioner.
Through an illustrative case and a real world case, the results demonstrate that the decision maker can reduce a project duration under the original budget, while avoiding side effects presented in current optimization techniques, such as rework or fatigue. The optimized solutions with variable productivity under different working environment constraints can be considered a benchmark and direction for further improvement.
Although improving productivity though management seemed to be commonplace in industry, few studies have examined the factor in the TCT optimization. Ultimately, the effect of improvement is feasible in the long term, and even in a given period as long as the increased duration of the improvement is less than the decreased project duration.
The main contribution of this study is to apply managerial improvement of labor productivity to TCT optimization which has not been addressed in previous researches. As a result, the project duration can be reduced owing to improved productivity of existing crews rather than inefficient overmanning, overlapping or costly substitution. Furthermore, it would be beneficial for organizational learning and competitiveness in the long term development. This study also recommends that the decision maker shall employs the proposed optimization technique as their initial consideration to reduce a project duration and decrease the risk of LD when addressing the TCT problem under limited budget.
This study has also presented three important managerial insights from the proposed model: 1. Rather than thorough improvement, selective improvement among the project activities is considered critical in the process of the optimization. GAs   can help practitioners optimize improvements in a large construction project. 2. Although labor productivity deteriorate when the allocated labor usage is greater than the given working environment allowance, the allocation will be still adopted to reduce the project duration under TCT optimization. 3. Reducing the project duration does not necessarily increase the project direct labor cost. In consideration of tight schedule pressure, the cost the contractor pays for acceleration through the proposed TCT model with optimal productivity may not exceed direct labor cost of contract budget. This insight can be shared with owners and contractors when evaluating reasonable duration and compensation relative to project acceleration. This study contributes to the body of knowledge and bridges the proposed model to a real-world application. However, the following limitations shall be considered: 1. The effectiveness of improvement through management necessitates time delay. Because of the uniqueness of a construction project, this study cannot identify how long the duration of a project can be applied into the model. 2. The evaluation of the improvement factor depends on schedulers with different experience. It may incur additional burden and time in the beginning of construction works. Future research can expand the proposed model to be more complete, such as combining other acceleration techniques or resources, considering fuzzy logic to reduce bias when evaluating the improvement factor, or distinguishing environment and management into more specific variables to examine the influence of optimal productivity.