Keywords: time-cost trade-off, project planning, linear programming, optimisation, project structure, project crashing.

One of the aims within project planning analysis is to develop the project time-cost curve and, further, to assess the minimum cost project duration. In particular, considering the structure of a project (the required activities and its interrelationships) and that each activity can generally be completed in a number of alternative ways (each of which is associated with particular duration and cost values), the basic objective of the analysis is to find the appropriate execution option for each activity so that the project is completed by a desired deadline and in an optimum way, i.e., with the minimum cost.

The time-cost trade-off problem has extensively been studied since 1960s and has been recognised as a particularly difficult combinatorial problem. Several solution schemes have been proposed, none of which is entirely satisfactory. They can be classified as exact methods based on linear and/or integer programming addressing the basic time-cost trade-off problem (e.g. [1]), approximate methods using decomposition or genetic algorithms to reduce the computational effort of exact methods (e.g. [2]), or methods directed to more realistic project cases (considering complex project structure and parameter uncertainty) (e.g. [3]).

Although a large number of methods exist, no one has been implemented in practical applications. This is due in part to the fact that existing methods have focused on the solution of the basic problem which results from simplifying assumptions regarding the precedence relations among project activities. In addition, other project characteristics, such as external time constraints for particular activities, bonuses or penalties for early or delayed project completion respectively have not been widely studied and incorporated in previous works. As a result, the application field of these methods is narrow and the methods are of little use for real life projects.

The present work aims to incorporate such parameters in the analysis and to develop a method for making optimal project time-cost decisions applicable to actual projects. The proposed method can model the following characteristics:

• Activity precedence relations: Besides the typical Finish-to-Start relation, other relations, such as Start-to-Start and Finish-to-Finish, are modelled. In addition, lead and lag times among activities can be handled together with any precedence relation type.
• Activity scheduling type: Activities that are to be executed either as soon as possible or as late as possible (within their slack range) can be modelled.
• External time constraints: In practical applications there are external constrains regarding the earliest or the latest time that an activity or set of activities can commence or finish. Such constraints are also modelled by the proposed method.
• Completion bonus or penalty: Any bonus for early project completion or penalty for delayed project delivery can be taken into account in this analysis.

The proposed method is based on linear/integer programming formulation and provides the optimal project time-cost curve and the minimum cost schedule. The variables of the problem are the start and finish times of each activity and additional zero-one variables for the various time-cost combinations of each activity. The objective function of the basic model includes the execution costs of the project activities and the project general expenses (indirect costs). As for the problem constraints, a first set is used to describe the relation of the start and finish times with the duration of each activity. Another set is used for the precedence relations among activities. A third set assure that all activity start and finish times are positive and fall within the duration of the project.

To force the program schedule each activity as soon as possible or as late as possible (within its slack time) according to the user preference, appropriate low- weighted variables are introduced to the objective function. External time constraints for particular activities, which cannot start earlier or finish later than a specific time point, are modelled by adding an appropriate constraint into the program. Finally, late penalty/early bonus situations are modelled by adding appropriate terms to the objective function and employing additional parameters to the program. The method has been applied with the use of a LP/IP computer program to a number of test cases with varying project size (number of activities), structure and constraints. Evaluation results indicate that the method can be reliably applied in actual engineering projects in terms of result accuracy and solution efficiency.

1

Shtub, A., Bard, J. and Globerson, S., "Project Management: Engineering, Technology and Implementations", Prentice Hall International Editions, 382- 392, 1994.

2

Feng, C.W., Liu, L. and Burns, S.A., "Using genetic algorithms to solve construction time-cost trade-off problems", ASCE Journal of Computing in Civil Engineering, 11(3), 184-189, 1997. doi:10.1061/(ASCE)0887-3801(1997)11:3(184)

3

Elmaghraby, S.E. and Kamburowski, J., "The analysis of activity networks under generalized precedence relations", Management Science, 38, 1245-1263, 1992. doi:10.1287/mnsc.38.9.1245

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