Project Crashing Optimization Strategy with Risk Consideration

Schedule Network Analysis

Work activities and a network logic diagram with activity durations on the critical path can be obtained from schedule network analysis. As shown in the schedule sensitivity index (Figure 5), construction plays an important role in the entire plan, especially in the activity "CK0035 central building arch. 4 to a shelter and outside," which has 89% schedule sensitivity. Figure 9 provides a bar chart illustrating the critical path of the case project. The activities located in the longest bar of Figure 9 are defined as the most critical items of the case project. These critical items and correlative activities are used to make a critical path network diagram shown in Figure 10. The completion time of the case project under different paths can be solved based on the precedence relationships of each node and its activity time in the network diagram. Nine nodes and ten activities exist.

Figure 9 Schedule network bar chart.

Figure 10 Critical path network diagram.

4.3.2. Model Numerical Application

Table 2 shows the input data of the case analysis. The duration of activity G is longer than that of activity F, though both can be conducted simultaneously. G is defined as one activity located in the critical path for subsequent analysis. Similarly, activity A and activity J can be conducted at the same period. The completed duration of activity A is longer than that of activity J. A is defined as one activity located in the critical path for subsequent analysis. Hence, the critical path is A+B+C+D+E+G+H+I after analysis. Table 3 shows the different crash cost for different crash duration of each activity. Three segments for crash durations (1-10 days as segment 1, 11-20 days as segment 2, and $\geq$ 21days as segment 3) are proposed with different crash costs for each activity. For example, the crash cost (10 days x 5,000USD/day + 6 days x 5,500USD/day) shall be paid when B is crashed by 16 days.

Table 2 Input data for case analysis.

Note: segment 1: crash duration 1-10 days.
Segment 2: crash duration 11-20 days.
Segment 3: crash duration ≥ 21 days.

Table 4 shows the calculation results of project duration under different scenarios by applying the integer linear programming Model 1. Under optimistic time, the project can be completed in advance within the contractual deadline. Hence, we will not discuss this case. But under pessimistic time, the project will be overdue for 146 days (1,122 days–976 days). As mentioned in Section 4.1, the ceiling penalty is 10% of the contract amount, so the maximum delay penalty cost accepted by the contract is converted into 100 days. Hence, the next step is to focus on finding the optimum solution for crash cost and crash schedule in the cases of average time, most likely time, and pessimistic time.

Table 4 Project duration in case of different expected times.

Tables 5–7 show the calculation results of project crash plans by applying integer linear programming Model 2. With an integer linear programming technique, the overall project cost can be reduced using less expensive resources, and project planners can adjust the resource selection to shorten the project duration. Table 5 indicates that when the project is completed under average time, the optimum project duration meets the requested date, resulting in no delay penalty with a crash cost of 133,500 USD. Analysis shows that the delay penalty cost is 53,000 USD/day (0.1% of the contract price per day of delay), and the crash cost of several activities, such as J, B, C, and I, is much lower than the delay penalty cost per day. Hence, after the calculation of the integer optimization model via CPLEX R12.6.2 software, the crash plan for the average time case is to crash activity B for 10 days, activity C for 13 days, and activity I for 6 days to meet the contracted deadline with the lowest total cost. The required crash days of each activity for the average time case after analysis fall in the reasonable range as indicated in the crash time constraint of Table 2.

Table 5 Optimum project duration and crash plan in case of average time.

Table 6 Optimum project duration and crash plan in case of most likely time.

Table 7 Optimum project duration and crash plan in case of pessimistic time.

Table 6 shows that when the project is completed by considering the case of the most likely time, the optimum project duration meets the requested date, resulting in no delay penalty with a crash cost of 306,000 USD. After the calculation of the integer optimization model via CPLEX software, the crash plan for the most likely time case is to crash activity B for 22 days, activity C for 16 days, and activity I for 18 days to meet the contracted deadline with the lowest total cost. The required crash days of each activity for the most likely time case after analysis fall in the reasonable range as indicated in the crash time constraint of Table 2.

Table 7 shows that when the project is completed by considering the case of pessimistic time, the optimum project duration meets the requested date, resulting in no delay penalty with a crash cost of 1,149,000 USD. After the calculation of the integer optimization model via CPLEX software, the crash plan for the pessimistic time case is to crash activity A for 10 days, activity B for 32 days, activity C for 24 days, activity D for 10 days, activity E for 10 days, and activity I for 60 days to meet the contracted deadline with the lowest total cost. The required crash days of each activity for the pessimistic time case after analysis fall in the reasonable range as indicated in the crash time constraint of Table 2.

4.3.3. Crash Plan Feasibility

In the case of average time, we set parameter L (project duration after crashing) to 976-981 and analyze the differences in the crash cost and delay penalty cost of the project. As shown in Table 8 and Figure 11, the slope of the decline of the crash cost with many overdue days is less steep than the slope of the rise of the total extra cost. When this project has longer duration, the gap between crash cost and total extra cost is getting bigger and delay penalty cost is increased accordingly. Given that the delay penalty cost of this case is high, it is suggested that the crash plan be conducted to make the project schedule completion meet the contracted deadline. Any extension of the project duration increases the total extra cost for this case project. However, from the viewpoint of total project cost minimized, if the cost of crash activities is higher than the delay penalty, then conducting the crash plan becomes an unsuitable option. Hence, optimizing the relationship between crash cost and crash time to achieve the best solution is worthy of further evaluation.

Table 8 Analysis of crash cost and delay penalty cost in average time case.

Figure 11 Illustration for crash cost and delay penalty cost in average time case.

After the several analyses in this study, the company can find the appropriate execution option for each activity so that the project can be completed by a desired deadline with the minimum cost. The company can make a final decision with support data to approach this case project with confidence.