Kris Coolen

Sequential testing policies for complex systems under precedence constraints

We study the problem of sequentially testing the components of a multi-component system to learn the state of the system, when the tests are subject to precedence constraints and with the objective of minimizing the expected cost of the inspections. Our focus is on k-out-of-n systems, which function if at least k of the n components are functional. A solution is a testing policy, which is a set of decision rules that describe in which order to perform the tests. We distinguish two different classes of policies and describe exact algorithms (one branch-and-bound algorithm and one dynamic program) to find an optimal member of each class. We report on extensive computational experiments with the algorithms for a representative data set.

Scheduling modular projects on a bottleneck resource

In this paper, we model a research-and-development project as consisting of several modules, with each module containing one or more activities. We examine how to schedule the activities of such a project in order to maximize the expected profit when the activities have a probability of failure and when an activity’s failure can cause its module and thereby the overall project to fail. A module succeeds when at least one of its constituent activities is successfully executed. All activities are scheduled on a scarce resource that is modeled as a single machine. We describe various policy classes, establish the relations among them, develop exact algorithms to optimize over two different classes (one dynamic program and one branch-and-bound algorithm), and examine the computational performance of the algorithms on two randomly generated instance sets.

Project Scheduling with Modular Project Completion on a Bottleneck Resource

In this paper, we model a research-and-development project as consisting of several modules, with each module containing one or more activities. We examine how to schedule the activities of such a project in order to maximize the expected profit when the activities have a probability of failure and when an activity’s failure can cause its module and thereby the overall project to fail. A module succeeds when at least one of its constituent activities is successfully executed. All activities are scheduled on a scarce resource that is modeled as a single machine. We describe various policy classes, establish the relationship between the classes, develop exact algorithms to optimize over two different classes (one dynamic program and one branch-and-bound algorithm), and examine the computational performance of the algorithms on two randomly generated instance sets.

Minimum-cost diagnostic strategies for k-out-of-n systems with imperfect tests

A k-out-of-n system configuration requires that, for the overall system to be functional, at least k out of the total of n components be working. We consider the problem of sequentially testing the components of a k-out-of-n system in order to learn the state of the system, when the tests are costly and when the individual component tests are imperfect, which means that a test can identify a component as working when in reality it is down, and vice versa. Each component is tested at most once. The stopping criterion for the inspection is the attainment of a lower bound on the confidence level regarding the system state. We define different classes of inspection policies and we examine global optimality of each of the classes. We show that a globally optimal policy can be found in polynomial time when the predictive error probabilities are the same for all the components.

A Note on “Discrete Sequential Search with Group Activities”: Discrete Sequential Search with Group Activities

This note comments on a paper published by Wagner and Davis ( Decision Sciences (2001), 32(4), 557–573 ) . These authors present an integer-programming model for the single-item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer-programming model, hence contradicting their conjecture for the specific linear program that they specify. To the best of our knowledge, the conjecture settled in this note was still an open question. [Submitted: April 18, 2012. Revised: June 11, 2012. Accepted: June 13, 2012.] Subject Areas: Discrete Sequential Search, Integer Programming, and Linear Programming.

A Note on “Discrete Sequential Search with Group Activities”

This note comments on a paper published by Wagner and Davis (Decision Sciences (2001), 32(4), 557–573). These authors present an integer-programming model for the single-item discrete sequential search problem with group activities. Based on their experiments, they conjecture that the problem can be solved as a linear program. In this note, we provide a counterexample for which the optimal value of the linear program they propose is different from the optimal value of the integer-programming model, hence contradicting their conjecture for the specific linear program that they specify. Furthermore, we show that the discrete sequential search problem is equivalent to scheduling a set of jobs on a single machine to minimize the sum of weighted completion times with a special bipartite graph representing the precedence constraints amongst jobs. The latter type of problems is well-studied in the field of operations research and operations management. Finally, we prove that the scheduling problem equivalent to the discrete sequential search problem studied by Wagner and Davis is strongly NP-hard. This complexity result implies that, unless P = NP, it is impossible that there exists any (compact-size) linear program for solving the discrete sequential search problem studied. To the best of our knowledge, the conjecture settled in this note was still an open question.