Time-cost trade-off in PERT networks using a genetic algorithm Amir Azaron1, Cahit Perkgoz2, Kosuke Kato3, Hideki Katagiri4, Masatoshi Sakawa5 Department of Artificial Complex Systems Engineering, Graduate School of Engineering, Hiroshima University, Kagamiyama 1-4-1, Higashi-Hiroshima, Hiroshima, 739-8527 Japan Abstract-We develop a multi-objective model for the time- cost trade-off problem in PERT networks with generalized Erlang distributions of activity durations, using a genetic algorithm. The mean duration of each activity is assumed to be a non-increasing function and the direct cost of each activity is assumed to be a non- decreasing function of the amount of resource allocated to it. The decision variables of the model are the allocated resource quantities. The problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. The objective functions are the project direct cost (to be minimized), the mean of project completion time (min), the variance of project completion time (min), and the probability that the project completion time does not exceed a certain threshold (max). It is impossible to solve this problem, optimally. Therefore, we apply a genetic algorithm for numerical optimizations of constrained problems (GENOCOP) to solve this multi-objective problem, using goal attainment technique. 1. INTRODUCTION In the time-cost trade-off problem (TCTP), the objective is to determine the duration of each activity in order to achieve the minimum total direct and indirect costs of the project. The well-known TCTP in CPM networks takes the former view. Studies on TCTP have been done using various kinds of cost functions such as linear [4], discrete [2], convex [7], and concave [3]. Nevertheless, PERT takes account of time-cost trade-off. Therefore, developing a time-cost trade-off model under uncertainty would be beneficial to scheduling engineers in the forecast of a more realistic project completion time and cost. In this paper, we develop a multi-objective model for the time-cost trade-off problem in PERT networks, using a genetic algorithm. The problem is formulated as a multi-objective optimal control problem that involves four conflicting objective functions. For the problem concerned in this paper, as a general-purpose solution method for nonlinear programming problems, in order to consider the nonlinearity of problems and to cope with large-scale problems, we apply the revised GENOCOP V [8], which is a direct extension of the genetic algorithm for numerical optimizations of constrained problems (GENOCOP V) [5]. Furthermore, the feasibility of the proposed method is shown through illustrative numerical examples. 2. PROJECT COMPLETION TIME DISTRIBUTION IN PERT NETWORKS In this section, we present an analytical method to compute the distribution function of project completion time in PERT networks. To do that, we extend [6], because this method is an analytical one, simple, easy to implement on a computer and computationally stable. Let G=(V,A) be a PERT network. Duration of activity A a (T a ) exhibits a generalized Erlang distribution of order and the infinitesimal generator matrix G as: a n a 0 0 . . 0 0 0 0 ... ... ... ... ... 0 0 . 0 0 0 . 0 0 . 0 2 2 1 1 a a n a n a a a a a a G . First, we transform the original PERT network into a new one, in which all activity durations have exponential distributions. For constructing this network, we use the idea that if the duration of activity is distributed according to a generalized Erlang distribution of order and the infinitesimal generator matrix , it can be decomposed to n exponential series of arcs with the parameters a a n a G a a n a a a ,..., 2 , 1 . Then, we substitute each generalized Erlang activity with series of exponential activities with the parameters a n a n a a a ,..., , 2 1 . Now, Let G be the transformed network, in which V and represent the sets of nodes and arcs of this transformed network, respectively, where the duration of each activity ' ) , ( ' ' ' A V ' ' A A a is exponential with parameter a . The source and sink nodes are denoted by s and t, respectively. For , let ' A a ) (a be the starting node of arc a, and ) a ( be the ending node of arc a. 1