A multi-objective model for resource-constrained multi-project scheduling with multiple modes subject to expected job duration.

The study considers a multi-modal project scheduling problem. A finite number of projects are to be scheduled. Each project is comprised of a set of jobs with known precedence relationships. To accomplish a project, all jobs comprising the project should be completed. However, to execute a job, a particular resource mode is needed. The available amount of resources is limited and the duration of a job is stochastic. There may exist more than one alternative mode for each job. Therefore, associated with each mode is the cost of using the mode, and job duration which is expressed as an expected value. Generally, jobs are completed in a shorter time if more costly resources are used. The objective of the study is to provide an optimal project schedule that indicates when should each job be started and what resources mode to use to execute the job while simultaneously minimizing the maximum expected project completion time and expected total project cost. The problem was formulated using Mixed Integer Non-Linear Program Model with multiple objectives under a continuous time horizon. This was then transformed into a Mixed Integer Linear Program Model. Sensitivity analysis showed that the objective pertaining to time and cost are conflicting when the ratio of the direct cost of the less costly mode to that of the more costly mode is less than the ratio of the reciprocal of their job duration. It was also found out that more costly modes with shorter job duration are opted by a system that gives more priority on time. Moreover, for this system, the optimal solution is not very sensitive to changes in direct and penalty cost. Since lesser weight is given to cost, only a fraction of their changes are realized by the weighted objective function. As such, the changes of these parameters have to be significant for them to affect the optimal solution. On the other hand, the optimal schedule of a system that gives more priority on cost is characterized by less costly modes with longer job duration. In addition, for this particular system, changes in direct and penalty cost does not necessarily result to a change in optimal solution. Rather, the optimal solution seeks a tradeoff between the expected total direct and expected total penalty cost