Algebraic Approaches to Stochastic Optimization

The dissertation presents algebraic approaches to the shortest path and maximum flow problems in stochastic networks. The goal of the stochastic shortest path problem is to find the distribution of the shortest path length, while the goal of the stochastic maximum flow problem is to find the distribution of the maximum flow value. In stochastic networks it is common to model arc values (lengths, capacities) as random variables. In this dissertation, we model arc values with discrete non-negative random variables and shows how each arc value can be represented as a polynomial. We then define two algebraic operations and use these operations to develop both exact and approximating algorithms for each problem in acyclic networks. Using majorization concepts, we show that the approximating algorithms produce bounds on the distribution of interest; we obtain both lower and upper bounding distributions. We also obtain bounds on the expected shortest path length and expected maximum flow value. In addition, we used fixed-point iteration techniques to extend these approaches to general networks. Finally, we present a modified version of the Quine-McCluskey method for simplification of Boolean expressions in order to simplify polynomials used in our work

A Study of Approximating the Moments of the Job completion Time in PERT Networks

The importance of proper management of projects has not gone unrecognized in industry and academia. Consequently tools like Critical Path Method (CPM) and Program Evaluation Review Technique (PERT) for project planning have been the focus of attention of both practitioners and researchers. Determination of the Time to Complete the Job (TCJ) in PERT networks is important for planning and bidding purposes. The complexity involved in accurately determining the TCJ has led to the development of many approximating procedures. Most of them ignore the dependence between paths in the network. We propose an approximation to determine the TCJ which explicitly recognizes this dependency. Experimental results which demonstrate the accuracy of our approximation for a wide variety of networks are presented

Resource-constrained project scheduling for timely project completion with stochastic activity durations.

We investigate resource-constrained project scheduling with stochastic activity durations. Various objective functions related to timely project completion are examined, as well as the correlation between these objectives. We develop a GRASP-heuristic to produce high-quality solutions, using so-called descriptive sampling. The algorithm outperforms other existing algorithms for expected-makespan minimization. The distribution of the possible makespan realizations for a given scheduling policy is studied, and problem difficulty is explored as a function of problem parameters.GRASP; Project scheduling; Uncertainty;

Calculating Project Completion in Polynomial Processing Time

Technology-based organizations and knowledge organizations rely on large activity networks to manage Research & Development (R&D) projects. Avoiding optimistic completion times due to the characteristic Program Evaluation and Review Technique (PERT) assumptions is a problem that can grow exponentially in complexity with the number of activities. A recursive technique that solves the problem in a polynomial number of steps has been developed, assuming that all duration times follow beta distributions. It is important to notice that the only two 100% valid approaches to calculate the project completion time are simulation and the stochastic sum for each and every path in the network. Nevertheless, both require finding the shape parameters, and that is precisely the main contribution of this paper: a system of equations to calculate the shape parameters of each activity and the overall project

Applications of Semidefinite Optimization in Stochastic Project Scheduling

We propose a new method, based on semidefinite optimization, to find tight upper bounds on the expected project completion time and expected project tardiness in a stochastic project scheduling environment, when only limited information in the form of first and second (joint) moments of the durations of individual activities in the project is available. Our computational experiments suggest that the bounds provided by the new method are stronger and often significant compared to the bounds found by alternative methods.Singapore-MIT Alliance (SMA