5 research outputs found
FPTAS for half-products minimization with scheduling applications
Cataloged from PDF version of article.A special class of quadratic pseudo-boolean functions called “half-products” (HP) has recently been introduced. It has been
shown that HP minimization, while NP-hard, admits a fully polynomial time approximation scheme (FPTAS). In this note, we
provide a more efficient FPTAS. We further show how an FPTAS can also be derived for the general case where the HP function is
augmented by a problem-dependent constant and can justifiably be assumed to be nonnegative. This leads to an FPTAS for certain
partitioning type problems, including many from the field of scheduling.
c 2008 Elsevier B.V. All rights reserved
Minimizing weighted mean absolute deviation of job completion times from their weighted mean
Cataloged from PDF version of article.We address a single-machine scheduling problem where the objective is to minimize the
weighted mean absolute deviation of job completion times from their weighted mean. This
problem and its precursors aim to achieve the maximum admissible level of service equity.
It has been shown earlier that the unweighted version of this problem is NP-hard in the
ordinary sense. For that version, a pseudo-polynomial time dynamic program and a 2-
approximate algorithm are available. However, not much (except for an important solution
property) exists for the weighted version. In this paper, we establish the relationship
between the optimal solution to the weighted problem and a related one in which the deviations
are measured from the weighted median (rather than the mean) of the job completion
times; this generalizes the 2-approximation result mentioned above. We proceed to
give a pseudo-polynomial time dynamic program, establishing the ordinary NP-hardness
of the problem in general. We then present a fully-polynomial time approximation scheme
as well. Finally, we report the findings from a limited computational study on the heuristic
solution of the general problem. Our results specialize easily to the unweighted case; they
also lead to an approximation of the set of schedules that are efficient with respect to both
the weighted mean absolute deviation and the weighted mean completion time.
2011 Elsevier Inc. All rights reserved
Routing and scheduling optimisation under uncertainty for engineering applications
The thesis aims to develop a viable computational approach suitable for solving large vehicle routing and scheduling optimisation problems affected by uncertainty. The modelling framework is built upon recent advances in Stochastic Optimisation, Robust Optimisation and Distributionally Robust Optimization. The utility of the methodology is presented on two classes of discrete optimisation problems: scheduling satellite communication, which is a variant of Machine Scheduling, and the Vehicle Routing Problem with Time Windows and Synchronised Visits. For each problem class, a practical engineering application is formulated using data coming from the real world. The significant size of the problem instances reinforced the need to apply a different computational approach for each problem class. Satellite communication is scheduled using a Mixed-Integer Programming solver. In contrast, the vehicle routing problem with synchronised visits is solved using a hybrid method that combines Iterated Local Search, Constraint Programming and the Guided Local Search metaheuristic.
The featured application of scheduling satellite communication is the Satellite Quantum Key Distribution for a system that consists of one spacecraft placed in the Lower Earth Orbit and a network of optical ground stations located in the United Kingdom. The satellite generates cryptographic keys and transmits them to individual ground stations. Each ground station should receive the number of keys in proportion to the importance of the ground station in the network. As clouds containing water attenuate the signal, reliable scheduling needs to account for cloud cover predictions, which are naturally affected by uncertainty. A new uncertainty sets tailored for modelling uncertainty in predictions of atmospheric phenomena is the main contribution to the methodology. The uncertainty set models the evolution of uncertain parameters using a Multivariate Vector Auto-Regressive Time Series, which preserves correlations over time and space. The problem formulation employing the new uncertainty set compares favourably to a suite of alternative models adapted from the literature considering both the computational time and the cost-effectiveness of the schedule evaluated in the cloud cover conditions observed in the real world. The other contribution of the thesis in the satellite scheduling domain is the formulation of the Satellite Quantum Key Distribution problem. The proof of computational complexity and thorough performance analysis of an example Satellite Quantum Key Distribution system accompany the formulation.
The Home Care Scheduling and Routing Problem, which instances are solved for the largest provider of such services in Scotland, is the application of the Vehicle Routing Problem with Time Windows and Synchronised Visits. The problem instances contain over 500 visits. Around 20% of them require two carers simultaneously. Such problem instances are well beyond the scalability limitations of the exact method and considerably larger than instances of similar problems considered in the literature. The optimisation approach proposed in the thesis found effective solutions in attractive computational time (i.e., less than 30 minutes) and the solutions reduced the total travel time threefold compared to alternative schedules computed by human planners. The Essential Riskiness Index Optimisation was incorporated into the Constraint Programming model to address uncertainty in visits' duration. Besides solving large problem instances from the real world, the solution method reproduced the majority of the best results reported in the literature and strictly improved the solutions for several instances of a well-known benchmark for the Vehicle Routing Problem with Time Windows and Synchronised Visits.The thesis aims to develop a viable computational approach suitable for solving large vehicle routing and scheduling optimisation problems affected by uncertainty. The modelling framework is built upon recent advances in Stochastic Optimisation, Robust Optimisation and Distributionally Robust Optimization. The utility of the methodology is presented on two classes of discrete optimisation problems: scheduling satellite communication, which is a variant of Machine Scheduling, and the Vehicle Routing Problem with Time Windows and Synchronised Visits. For each problem class, a practical engineering application is formulated using data coming from the real world. The significant size of the problem instances reinforced the need to apply a different computational approach for each problem class. Satellite communication is scheduled using a Mixed-Integer Programming solver. In contrast, the vehicle routing problem with synchronised visits is solved using a hybrid method that combines Iterated Local Search, Constraint Programming and the Guided Local Search metaheuristic.
The featured application of scheduling satellite communication is the Satellite Quantum Key Distribution for a system that consists of one spacecraft placed in the Lower Earth Orbit and a network of optical ground stations located in the United Kingdom. The satellite generates cryptographic keys and transmits them to individual ground stations. Each ground station should receive the number of keys in proportion to the importance of the ground station in the network. As clouds containing water attenuate the signal, reliable scheduling needs to account for cloud cover predictions, which are naturally affected by uncertainty. A new uncertainty sets tailored for modelling uncertainty in predictions of atmospheric phenomena is the main contribution to the methodology. The uncertainty set models the evolution of uncertain parameters using a Multivariate Vector Auto-Regressive Time Series, which preserves correlations over time and space. The problem formulation employing the new uncertainty set compares favourably to a suite of alternative models adapted from the literature considering both the computational time and the cost-effectiveness of the schedule evaluated in the cloud cover conditions observed in the real world. The other contribution of the thesis in the satellite scheduling domain is the formulation of the Satellite Quantum Key Distribution problem. The proof of computational complexity and thorough performance analysis of an example Satellite Quantum Key Distribution system accompany the formulation.
The Home Care Scheduling and Routing Problem, which instances are solved for the largest provider of such services in Scotland, is the application of the Vehicle Routing Problem with Time Windows and Synchronised Visits. The problem instances contain over 500 visits. Around 20% of them require two carers simultaneously. Such problem instances are well beyond the scalability limitations of the exact method and considerably larger than instances of similar problems considered in the literature. The optimisation approach proposed in the thesis found effective solutions in attractive computational time (i.e., less than 30 minutes) and the solutions reduced the total travel time threefold compared to alternative schedules computed by human planners. The Essential Riskiness Index Optimisation was incorporated into the Constraint Programming model to address uncertainty in visits' duration. Besides solving large problem instances from the real world, the solution method reproduced the majority of the best results reported in the literature and strictly improved the solutions for several instances of a well-known benchmark for the Vehicle Routing Problem with Time Windows and Synchronised Visits
An FPTAS for agreeably weighted variance on a single machine (Extended abstract)
We investigate the following scheduling problem: There is a single machine and a set of jobs. Every job is specified by its processing time and by its weight. The goal is to find a schedule that minimizes the sum of squared deviations from the weighted average job completion time. Jobs with small processing times have large weights, and hence the weights are agreeable.
This problem is NP-hard. In 1995, Cai derived a fully polynomial time approximation scheme for the special case where the weights of the jobs are polynomially bounded in the number n of jobs. In this paper we completely settle the approximability status of this scheduling problem: We construct a fully polynomial time approximation scheme for the general case, without putting any restrictions on the weights of the jobs
An FPTAS for agreeably weighted variance on a single machine (Extended abstract)
We investigate the following scheduling problem: There is a single machine and a set of jobs. Every job is specified by its processing time and by its weight. The goal is to find a schedule that minimizes the sum of squared deviations from the weighted average job completion time. Jobs with small processing times have large weights, and hence the weights are agreeable. This problem is NP-hard. In 1995, Cai derived a fully polynomial time approximation scheme for the special case where the weights of the jobs are polynomially bounded in the number n of jobs. In this paper we completely settle the approximability status of this scheduling problem: We construct a fully polynomial time approximation scheme for the general case, without putting any restrictions on the weights of the jobs