A mixed (continuous + discrete) time-cost trade-off model considering four different relationships with lag time

Numerous Time-Cost Trade-Off (TCTO) models have been developed to identify the best combination of time and cost in a critical path network. Although there are four relationships in the critical path network, most of the models developed so far have considered only the Finish-Start (F-S) relationship. Thus, an advanced TCTO model that considers all four relationships between activities was developed in this study to accurately present a project, and is presented in this paper. The model also takes into account the lag time between activities. Previous TCTO models minimize the total project cost based on the given crash scenario. Moreover, to enhance the practicality, the model was developed to work with various TCTO scenarios such as continuous, discrete, and even mixed (continuous + discrete). The combined scenario reflects the most realistic situation. Two independent scenarios cannot be combined without mathematical modifications since a rudimentary mixing of two scenarios may provide an incorrect solution. A new formulation technique is introduced to merge the two independent scenarios mathematically and it guarantees the optimal solution.