Total cost for project scheduling, in general, consists of two independent costs, indirect costs and direct costs. Indirect costs are proportional to project duration, with the penalty rate factor. Direct costs are related to the activity durations. An important problem of cost-duration analysis is how to design the trade-off between indirect and direct costs. Academics and practitioners have been considering linear and nonlinear functions between them. In this paper, we introduce a logarithmic cost-duration function to design an optimal trade-off between indirect and direct costs. By employing this function, we show that the optimal project duration does not depend on the network configurations. However, total cost does depend on it. Moreover, we can modify the original total cost minimization problem to a maximization problem, aimed at finding optimal activity durations; for this maximization problem, the decision variables are certain flow variables, and duration costs to perform in one unit of time and learning curve factors are problem parameters, while the penalty rate factor is not. Thus, the optimal activity durations for the project can be derived by a single optimization covering all penalties. We also compared and contrasted the logarithmic activity cost-duration function approach with the well-known linear activity cost-duration function approach. A statistical analysis is presented, covering the case where some parameters are stochastic.
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