A multi-objective resource allocation problem in dynamic PERT networks

Abstract

We develop a multi-objective model for the resource allocation problem in a dynamic PERT network, where the activity durations are exponentially distributed random variables and the new projects are generated according to a Poisson process. This dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the generation rate of new projects. It is assumed that the mean time spent in each service station is a non-increasing function and the direct cost of each activity is a non-decreasing function of the amount of resource allocated to it. The decision variables of the model are the allocated resource quantities. To evaluate the distribution function of total duration for any particular project, we apply a longest path technique in networks of queues. Then, the problem is formulated as a multi-objective optimal control problem that involves three conflicting objective functions. The objective functions are the project direct cost (to be minimized), the mean of the project completion time (min) and the variance of the project completion time (min). Finally, the goal attainment method is applied to solve a discrete-time approximation of the original optimal control problem. We also computationally investigate the trade-off between accuracy and the computational time of the discrete-time approximation technique.

Introduction

Since the late 1950s, Critical Path Method (CPM) techniques have become widely recognized as valuable tools for the planning and scheduling of large projects. In a traditional CPM analysis, the major objective is to schedule a project assuming deterministic durations. However, project activities must be scheduled under available resources, such as crew sizes, equipment and materials. The activity duration can be looked upon as a function of resource availability. Moreover, different resource combinations have their own costs. Ultimately, the schedule needs to take account of the trade-off between project direct cost and project completion time. For example, using more productive equipment or hiring more workers may save time, but the project direct cost will increase.

In CPM networks, activity duration is viewed either as a function of cost or as a function of resources committed to it. The well-known time–cost trade-off problem (TCTP) in CPM networks takes the former view. In the 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.

Studies on TCTP have been done using various kinds of cost functions such as linear [10], [12], discrete [7], convex [14], [5], and concave [9]. When the cost functions are arbitrary (still non-increasing), the dynamic programming (DP) approach was suggested by Robinson [15] and Elmaghraby [8]. Tavares [17] has presented a general model based on the decomposition of the project into a sequence of stages and the optimal solution can be easily computed for each practical problem as it is shown for a real case study.

Weglarz [18] studied this problem using optimal control theory and assumed that the processing speed of each activity at time t is a continuous, non-decreasing function of the amount of resource allocated to the activity at that instant of time. This means that time is considered as a continuous variable. Azaron et al. [1] proposed an approximation technique to deal with time–cost trade-off in classical PERT networks.

Recently, some researchers have adopted computational optimization techniques such as genetic algorithms to solve TCTP. Chau et al. [6] and Azaron et al. [2] proposed models using genetic algorithms and the Pareto front approach to solve construction time–cost trade-off problems.

Although project scheduling and management has been investigated by many researchers, one cannot find many models regarding dynamic project scheduling in the literature. Actually, as the classical definition of project indicates, it is a one-time job which consists of several activities. Therefore, the models representing the project scheduling, including the above models, are all static. In reality, during the implementation of a project some new projects are generated, in which the activities associated with successive projects contend for resources.

Dynamic PERT does not take into account the time–cost trade-off. Therefore, combining the aforementioned concepts to develop a time–cost trade-off model under uncertainty and dynamic situations would be beneficial to scheduling engineers in forecasting 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 a dynamic PERT network. In fact, in real world, there are many jobs with similar structure of activities sharing the same facilities. We consider a service center serving various projects with the same structure. Thus, although each one acts individually as a project represented as a classical PERT network, they cannot be analyzed independently since they share the same facilities. Like every other PERT project, the completion time is stochastic since the processing time of each activity is random.

Each dynamic PERT network is represented as a network of queues, where the service times represent the durations of the corresponding activities and the arrival stream to each node follows a Poisson process with the generation rate of new projects. All projects have the same activities and the same sequences.

In our proposed method, first we transform each network of queues into a proper stochastic network. Then, the distribution function of the longest path in this stochastic network, which would be equal to the project completion time distribution in the original dynamic PERT network, is determined through solving a system of linear differential equations. By applying a continuous-time Markov process technique, this system of differential equations is constructed.

Then, we develop a multi-objective model for the time–cost trade-off problem in dynamic PERT networks. It is assumed that the activity durations are independent random variables with exponential distributions. It is also assumed that the amount of resource allocated to each activity is controllable, where the time spent in each service station (activity duration plus waiting time in queue) is a non-increasing function of this control variable. The direct cost of each activity is also assumed to be a non-decreasing function of the amount of resource allocated to it.

The problem is formulated as a multi-objective optimal control problem, where the objective functions are the project direct cost (to be minimized), the mean of the project completion time (min) and its variance (min). Then, we apply the goal attainment technique, which is a variation of the goal programming technique, to solve this multi-objective problem.

It is proved that solving the resulting multi-objective optimal control problem using the standard optimal control tools is impossible. Therefore, we use a discrete-time approximation technique to solve it. We also computationally investigate the trade-off between accuracy and the computational time of the discrete-time approximation technique.

In Section 2, we compute the project completion time distribution in dynamic PERT networks with exponentially distributed activity durations, analytically. Section 3 presents the multi-objective resource allocation formulation. Section 4 presents the computational experiments, and finally we draw the conclusion of the paper in Section 5.