Continuous Linear, Discrete Time Cost Trade-Off Problems?

1.1 General In project scheduling, a set of activities with specified precedence relations has to be performed so as to create a service or product. An activity of the project can be performed in several different ways with different times and/or costs. Usually, it may be possible to accelerate the process by reducing the activity durations. The activity durations can be reduced by dedicating extra resources. These resources can be human, tool, machine, alternative processing options, like the subcontracting. An increase in resource consumption naturally increases the cost and decreases the time. According to the resource consumption levels, the alternatives can be defined. Each alternative defines its own cost and
The opinion can be either discrete or continuous depending on the available resources/methods or simply a contractor’s preference. Figure 1.1(B) shows an example the contractor can choose one of the three (A, B or C) methods to complete the activity. Each method has its associated time and cost. This type of time cost relation in literature is referred to as Discrete Time Cost Trade-off Problem. Figure 1.1(B): Discrete TCT relation

The classification used is based on the relation of time and cost figures. Problems with continuous time cost relations are studied in two subsections: Linear time/cost relations and nonlinear time/cost relations.

1.3 Objective Function and constraints in DTCTP
The objective function and the constraints of the problem give rise to three different Discrete Time Cost Trade-off Problem categories:
Deadline Problem
Budget Problem Time/Cost Curve
The objective is to minimize the project duration. A special case of the problem where all activities have a single mode is easily solvable by the Critical Path Method. The Critical Path Method simply finds the minimum project duration with given activity durations. The problem is shown to be NP-hard in the strong sense by Dunne et al. (1997).
The Time/Cost Curve Problem generates all efficient solutions relative to the total cost, and project duration objectives. A non-dominated, i.e., an efficient, set is defined using the successive solutions of the Deadline or Budget Problems. The Time/Cost Curve Problem is strongly NP-hard, as the Deadline Problem and Budget Problem is strongly NP-hard.
In this study, we consider the Budget Problem for the discrete time/cost alternatives. We have used a Branch and Bound Algorithm and a computer code for this algorithm.
In the literature, the optimization procedures designed for the Deadline and Time/Cost Curve Problems ranges from simple enumeration, heuristic to genetic algorithm to solve only moderate-sized instances of small number of activities with few modes. However the need for even a moderate size construction project is the capabilities of handling hundreds of activities with at least 2-3 different modes for few activities. The present study thus aims to handle such type of project network