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It is the goal of project planners to complete a project at a minimal time and cost. Additional demand for resources which result in increased cost is required to complete the project within a shorter time period. This problem of time-cost trade-off (TCT) has led to several studies conducted to find an optimal solution using different optimization techniques such as the heuristic method and other optimization techniques. However, in these studies using the heuristic method, the researchers mainly use deterministic activity durations to model the TCT problem. TCT analysis using deterministic durations will only produce deceiving results which are unrealistic. With activity durations likely to deviate from actual durations, this study presents a heuristic-based approach for time-cost optimization by taking into account the impact of risk and uncertainty of activity duration. This was done using a probabilistic approach through simulation to develop realistic durations and use these durations as input to reschedule the critical path method network. A numerical example problem was applied and the results were compared to existing studies to understand the impact of risk and uncertainty on TCT analysis. This approach will guide decision-makers in making efficient and effective decisions in TCT optimization problems.

Completing projects at a minimal time and within a minimum cost are the main objectives of construction projects. The need for work from contractors has resulted in clients tightening their requirements. Project managers are under intense pressure to complete the project under tight deadlines subject to huge penalties if they failed to complete the project within the stipulated deadline. One of the best solutions to meeting deadline projects has been the use of crashing technique to shorten the project duration. However, as the project’s duration is reduced, the total project cost increases and the more the project’s duration is shortened, the more the total cost increases. This means that there is a trade-off between time and cost which requires an optimum solution to complete the project. Feng

Time-cost relationship of an activity

TCT concept has been defined and explained in many ways by different authors. The TCT is defined by Abbasnia

The heuristic method has been and still remains one of the most useful techniques to solve TCT problems. Siemens[

Despite the useful application of the heuristic method to solve TCT problems, these models mainly use deterministic durations to model the TCT problem. However, in reality, activities may not be completed as planned. Some activities may take a longer time to complete, while others may take a shorter time to complete. The uncertainties in the activity duration may affect the project duration, as well as the cost. Using deterministic activity durations may lead to the wrong selection of construction options (resource allocation), thus leading to selecting the wrong optimum solution to TCT problem. Uncertainty variables may include productivity rate, on-time availability of resources on site before starting an activity, and weather conditions. Hence, to improve the accuracy of optimum solution in TCT analysis, this study presents a heuristic-based approach for time-cost optimization by taking into account the impact of risk and uncertainty of activity duration. This was done by using a probabilistic approach through simulation to develop realistic durations and use these durations as input to reschedule the critical path method (CPM) network. Simulation deals with the uncertainties of activity duration in a project.

This study developed a multiobjective time-cost optimization problem in CPM-PERT network using realistic (random) durations through simulation as input. Developing a heuristic-based time-cost optimization problem by considering the risk and uncertainty of activity duration will guide decision-makers in making efficient and effective decisions in TCT optimization problems.

In this section, we will formulate the heuristic model that accounts for risk and uncertainty in the activity durations by generating simulation durations and reschedule the CPM network. The next subsections will explain how the simulation durations are calculated and computational steps involved in solving a heuristic-based time-cost optimization problem.

In almost all cases, the previous research papers on time-cost optimization through crashing of activity duration have been applied on construction projects. Therefore, this paper will also focus on applying the model on a simple construction project.

To account for the uncertainty of activity duration, Monte Carlo simulation is applied. Activity duration can be simulated using probability distributions. In this study, we used triangular distribution to model random durations. In the triangular distribution technique, the expected duration of activity i (eD_{i}) can be calculated using Eqn. (1)

Where, _{i}_{i}_{i}

Where, _{i}_{i}

These generated random durations are further simulated many times, and in this study, we used 10,000 iterations to find realistically eD_{i}. Visual basic for application (VBA) was used to perform this iteration, and these generated durations become the actually expected durations to be used to prepare the network diagram (with activity uncertainty taken care of).

The cost-slope method is the main heuristic method for TCT analysis deployed to reduce the project duration based on the fundamental assumption that the relationship between time and cost is linear. With this common assumption, cost slope of activity i is defined by Hegazy[

Where, _{i}_{i}_{i}_{i}_{i}

The objective here is to minimize the total cost of shortening the project duration to meet specific deadline such that the project’s profit is not much affected:

Use the simulation durations for all activities

Develop the CPM network

Calculate and identify all the paths in the network (path with the longest expected duration = critical path)

Determine the amount required to reduce each path in the CPM network. The amount to be reduced on each path can be computed using Eqn.

Where, _{path–j}_{path–j}_{Com}

Calculate and tabulate the cost slope for each activity as well as defining the maximum crashing amount

Crash the critical activity with the least cost slope (LC-S) first based on the maximum amount to be crashed. If two or more critical paths occur, crash the activity appearing in both paths, or crash both activities with the minimum cost slope in each path.

Reschedule the CPM network using the crashed duration for that activity and calculate the project duration if the critical path does not change, continue to crash the activity until its maximum crash duration is exhausted.

Determine the total direct cost by multiplying the cost slope of the activity by the amount reduced, as well as the corresponding indirect cost by multiplying the estimated cost per day by the expected project duration.

Continue from step (g) until the deadline duration is reached.

Plot the total increase in direct cost against the resultant crash duration. Plot also the total project cost (direct + indirect costs) on the same graph and determine the optimum option with the least total cost.

The flowchart of the heuristic-based TCT optimization which describes the steps involved in analyzing the TCT problem is depicted in

Flowchart of the heuristic-based time-cost trade-off optimization

An example problem from a small construction project is considered to demonstrate the applicability and the impact of duration uncertainties on heuristic-based multiobjective TCT analysis. The scheduling project considered was previously applied to solve TCT problem by Biswas

Project data with deterministic durations

With reference to the data of the example problem, realistic durations through simulation were developed as shown in

Realistic duration of activities generated through simulation

Detailed project information of the example problem

To expedite the project, an optimization approach was used to reduced the project completion time of 144 days. In normal project crashing, reducing the duration increases the direct cost but decreases the indirect cost. Therefore, if the indirect cost is less than the direct cost, then an optimal or minimal cost can be achieved. Hence, the project can be crashed until the activity indirect cost becomes greater than its direct cost resulted from crashing the duration. However, in this study, a target completion time of 124 days was used to find the optimal cost of reducing the project duration of 144 days.

The network diagram in

Activities precedence network diagram

Cost slope of activities in the network

Using these realistic expected durations from simulation, the CPM network is drawn as depicted in

1^{st} cycle: This cycle uses the initial CPM data, where the total project completion duration = 144 days. Indirect cost = 144 days *100/day = 14400 and total direct cost = 49805. Total project cost = (49805+14400) = 64205

2^{nd} cycle: Identify critical path and activity with LC-S

Activity with the LC-S of this critical path = activity D (60) and this activity can be crashed by at least 8 days. Therefore, activity D becomes 22 days and the CPM is recalculated as shown in

Precedence network diagram after crashing activity D

3^{rd} cycle: Identify critical path and activity with LC-S.

No change in critical, Critical path = B – C – D – E but duration = 136 days.

Activity with LC-S of this critical path is still = activity D (60) which can further be crashed by 2 days. Therefore, activity D reaches its maximum crash duration of 20 days and the CPM is recalculated as shown in

Precedence network diagram after crashing activity D second time

4^{th} Cycle: Identify critical path and activity with LC-S. Here, two paths are critical: Path 1: B – C – D – E = 134 days and Path B – F – E = 134 days. The LC-S in Path 1 is activity D but cannot be crashed anymore since its maximum crash duration is exhausted. In Path 2, the LC-S is activity E which is also the least considering Path 1 when D is out. In this situation, both B and E can be crashed. Activity B and E both can crash by at least by 5 days. Hence, activity B becomes 17 days and activity E becomes 45 days (359; sum of 239 cost slope of B and 120 cost slope of E) and the network is recalculated as depicted in

Precedence network diagram after crashing activity B and E

In

Summary of associated costs of crashing the project duration

(a-d) Project time-cost relationship

Original time-cost plan against optimization time cost

This section discusses the results from the analysis of the TCT problem. Time and cost, which are independent of each other, are the two important attributes of any project. It is the objective of the project management team to complete the project within a minimum time and at the same time within a minimum cost. However, it is not easy to satisfy these two conflicting objectives at the same time. Optimization has been identified as the most suitable technique to model this kind of the problem of conflicting factors where an attempt to optimize (reduce) one objective increases the other objective factor. The results in

The goal of this study was to apply a heuristic approach to analyze a multiobjective time-cost optimization problem by considering the impact of risk and uncertainty of activity durations on achieving the optimum solution of the TCT problem under consideration. The study used a probabilistic approach through simulation to develop realistic durations and use these durations as input to reschedule the CPM network. A heuristic technique was used to crash the activity durations and the total project duration. A total of 10 days was reduced from the original CPM schedule of 144 days which has considered uncertainties in the activity durations. This reduction increased the total direct cost of the project from $49,805 to $50,405. This indicates that about 13.9% decrease in total project duration is achieved by increasing the total cost of the project by just 1.2%, which is quite satisfactory. Developing a heuristic-based time-cost optimization problem by considering the risk and uncertainty of activity duration will guide decision-makers in making efficient and effective decisions in TCT optimization problems.