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.Scheduling; Uncertainty; Research and development; Activity failures; Modular precedence network;

Models and algorithms for search and sequencing problems.

1. Introduction 2. Exact algorithms for MP1 3. Heuristic algorithms for MP1 4. Sequential searchnrpages: 133status: publishe

Sequential testing of 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. Since tests are imperfect, even when all components are tested the state of the system is not necessarily known with certainty, and so reaching a lower bound on the probability of correctness of the system state is used as a stopping criterion for the inspection. We define different classes of inspection policies and we examine global optimality of each of the classes. We find that a globally optimal policy for diagnosing k-out-of-n systems with imperfect tests can be found in polynomial time when the predictive error probabilities are the same for all the components. Of the three policy classes studied, the dominant policies always contain a global optimum, while elementary policies are compact in representation. The newly introduced class of so-called `interrupted block-walking' policies combines these merits of global optimality and of compactness.COME

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

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 representative datasets.status: publishe

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.status: publishe

Complexity analysis of the discrete sequential search problem with group activities

This paper studies the computational complexity of the discrete sequential search problem with group activities, in which a set of boxes is given and a single object is hidden in one of these boxes. Each box is characterized by a probability that it contains that object and the cost of searching that box. Furthermore, each box may be related to one or more ‘group activities’. For ‘conjunctive’ group activities, a box can be searched only when all the associated group activities have been performed whereas for ‘disjunctive’ group activities, a box can be searched as soon as at least one of the associated group activities has been executed. A cost is also incurred when performing a group activity. The goal is to find a sequence in which the boxes are to be searched and the group activities will be executed to minimize the expected total cost while satisfying the precedence constraints imposed by the group activities. In this paper, we prove that this problem is strongly NP-hard both for conjunctive group activities and for disjunctive group activities, and we discuss some special cases that can be solved in polynomial time.status: publishe

Complexity analysis of the discrete sequential search problem with group activities

We study the computational complexity of the discrete sequential search problem with group activities, in which a set of boxes and group activities is given, and a single object is hidden in one of these boxes. The goal is to find a sequence in which the boxes are to be searched and the group activities will be executed to minimize the expected total cost of finding the object while satisfying the precedence constraints imposed by the group activities. We prove that this problem is strongly NP-hard both for conjunctive group activities and for disjunctive group activities, and we discuss some special cases that can be solved in polynomial time.nrpages: 11status: publishe

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.

A fast greedy heuristic for scheduling modular projects

We propose a greedy heuristic for scheduling modular projects. Exact solutions for this NP-hard problem can be obtained with existing branch-and-bound and dynamic-programming algorithms, but only for simple instances. The proposed heuristic, by contrast, can also be used for large or otherwise difficult instances. We leverage results from the sequential testing literature, which will be briefly reviewed. The heuristic performs very well: for a dataset of 360 instances, the average relative optimality gap is below 0.5%.status: publishe