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Cited by 139Quantum Computing Algorithms and ArchitectureScheduling and Timetabling SolutionsConstraint Satisfaction and Optimization

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Application of Quantum Annealing to Nurse Scheduling Problem

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Ikeda, Kazuki, Nakamura, Yuma, Humble, Travis S.·2019·Scientific Reports
DOI 10.1038/s41598-019-49172-3PMC PMC6731278
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Abstract

Quantum annealing is a promising heuristic method to solve combinatorial optimization problems, and efforts to quantify performance on real-world problems provide insights into how this approach may be best used in practice. We investigate the empirical performance of quantum annealing to solve the Nurse Scheduling Problem (NSP) with hard constraints using the D-Wave 2000Q quantum annealing device. NSP seeks the optimal assignment for a set of nurses to shifts under an accompanying set of constraints on schedule and personnel. After reducing NSP to a novel Ising-type Hamiltonian, we evaluate the solution quality obtained from the D-Wave 2000Q against the constraint requirements as well as the diversity of solutions. For the test problems explored here, our results indicate that quantum annealing recovers satisfying solutions for NSP and suggests the heuristic method is potentially achievable for practical use. Moreover, we observe that solution quality can be greatly improved through the use of reverse annealing, in which it is possible to refine returned results by using the annealing process a second time. We compare the performance of NSP using both forward and reverse annealing methods and describe how this approach might be used in practice.

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Crossref · 10.1038/s41598-019-49172-3

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Figure 1 Graphical examples of the computed shift schedules for the case of N=3 nurses for D=4 days. A black square indicates that a nurse is scheduled to work. The horizontal axes labels the days and the vertical axes labels the nurses. The left figure represents a schedule that satisfies all the constraints, while the right figure represents a schedule that fails to satisfy the soft nurse constraint of every nurse being assigned some days of work.
Figure 1 Graphical examples of the computed shift schedules for the case of N=3 nurses for D=4 days. A black square indicates that a nurse is scheduled to work. The horizontal axes labels the days and the vertical axes labels the nurses. The left figure represents a schedule that satisfies all the constraints, while the right figure represents a schedule that fails to satisfy the soft nurse constraint of every nurse being assigned some days of work.CC BY · PubMed Central
Figure 2 The probability of finding a ground state of the Hamiltonian by using D-Wave 2000Q (left) and simulate annealing (right) for (blue) N=3 and (red) N=4 with respect to the number of days D in the schedule. Each error bar is generated by the standard deviation (1000 samples for 10 shots for D-Wave 2000Q and 500 annealing times 10 shots for SA).
Figure 2 The probability of finding a ground state of the Hamiltonian by using D-Wave 2000Q (left) and simulate annealing (right) for (blue) N=3 and (red) N=4 with respect to the number of days D in the schedule. Each error bar is generated by the standard deviation (1000 samples for 10 shots for D-Wave 2000Q and 500 annealing times 10 shots for SA).CC BY · PubMed Central
Figure 3 The Hamming distance between consecutive forward annealing solutions with respect to number of schedule days for (blue) N=3 nurses and (red) N=4 nurses. Error bars are based on the standard deviation.
Figure 3 The Hamming distance between consecutive forward annealing solutions with respect to number of schedule days for (blue) N=3 nurses and (red) N=4 nurses. Error bars are based on the standard deviation.CC BY · PubMed Central
Figure 4 A reverse annealing schedule used in our study plotted with respect to the evolution of the dimensionless parameter s.
Figure 4 A reverse annealing schedule used in our study plotted with respect to the evolution of the dimensionless parameter s.CC BY · PubMed Central
Figure 5 Probability of finding the ground states of the NSP Hamiltonian by reverse annealing. Each error bar is generated by the standard deviation (1000 samples times 10 shots for each problem).
Figure 5 Probability of finding the ground states of the NSP Hamiltonian by reverse annealing. Each error bar is generated by the standard deviation (1000 samples times 10 shots for each problem).CC BY · PubMed Central
Figure 6 Hamming distance between consecutive reverse annealing solutions (1000 samples for each problem). Error bars are based on the standard deviation.
Figure 6 Hamming distance between consecutive reverse annealing solutions (1000 samples for each problem). Error bars are based on the standard deviation.CC BY · PubMed Central
Figure 7 Probability of finding the ground states of the NSP Hamiltonian by reverse annealing. Error bars are based on the standard deviation (1000 samples times 10 shots for each problem).
Figure 7 Probability of finding the ground states of the NSP Hamiltonian by reverse annealing. Error bars are based on the standard deviation (1000 samples times 10 shots for each problem).CC BY · PubMed Central
Figure 8 The Hamming distance of consecutive reverse annealing solutions (1000 samples for each problem). Error bars are based on the standard deviation.
Figure 8 The Hamming distance of consecutive reverse annealing solutions (1000 samples for each problem). Error bars are based on the standard deviation.CC BY · PubMed Central
Figure 9 Solutions of the proposed three-shift NSP with additional constraints. DT, EN and LN refer to daytime, early night and late night work shift. Among (a–c), the day-off requests are tuned differently as g(n,d)=0 for (a), g(1,4)=1, g(2,6)=1.5 and g(3,5)=2 in (b) and g(1,6)=1, g(2,6)=1.5 and g(3,4)=2 in (c).
Figure 9 Solutions of the proposed three-shift NSP with additional constraints. DT, EN and LN refer to daytime, early night and late night work shift. Among (a–c), the day-off requests are tuned differently as g(n,d)=0 for (a), g(1,4)=1, g(2,6)=1.5 and g(3,5)=2 in (b) and g(1,6)=1, g(2,6)=1.5 and g(3,4)=2 in (c).CC BY · PubMed Central

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Research works printed but not linked: 2

  1. Stollenwerk, T. et al. Quantum Annealing Applied to De-Conflicting Optimal Trajectories for Air Traffic Management, arXiv e-prints, arXiv:1711.04889, arXiv:1711.04889 [quant-ph] (2017). · arXiv
  2. Neukart, F. et al. Traffic Flow Optimization Using a Quantum Annealer, Frontiers in ICT 4, 29 (2017). · DOI

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Metadata

Authors
Ikeda, K., Nakamura, Y., & Humble, T. S.
Affiliations
  • The University of Osaka (JP)
  • University of Tennessee at Knoxville (US)
  • Oak Ridge National Laboratory (US)
Corresponding author
Ikeda, Kazuki
ORCID
Venue
Scientific Reports
Publisher
Springer Science and Business Media LLC
Published
2019-09-06
Volume
9
Issue
1
Pages
12837-12837
ISSN
2045-2322 (ISSN-L)
DOI
10.1038/s41598-019-49172-3
PMC
PMC6731278
OpenAlex
W2972088227
Cited by
139
References
26
Citations by year
2026: 9 · 2025: 21 · 2024: 25 · 2023: 28 · 2022: 29 · 2021: 13 · 2020: 13 · 2019: 1
FWCI
6.39
Type
journal article
Language
en
Open access
Licence
Locations
  • Scientific Reports (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Funders
  • U.S. Department of Energy (AC05-00OR22725, DE-AC05?, 00OR22725)
  • Battelle (DE-AC05- 00OR22725, AC05-00OR22725, DE-AC05)
  • UT-Battelle (DE-AC05, DEAC05-00OR22725, AC05-00OR22725)
  • Ministry of Education, Culture, Sports, Science and Technology
  • Ministry of Science, ICT and Future Planning
  • National Research Foundation of Korea
  • Office of Science (DEAC05- 00OR22725, AC05-00OR22725)
  • Japan Society for the Promotion of Science (19J11073)
  • Oak Ridge National Laboratory (AC05-00OR22725)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Social Sciences › Decision Sciences › Management Science and Operations Research › Scheduling and Timetabling Solutions
  • Physical Sciences › Computer Science › Artificial Intelligence › Constraint Satisfaction and Optimization
Keywords
quantum annealing, nurse scheduling, D-Wave 2000Q, Ising Hamiltonian, reverse annealing, combinatorial optimization, hard constraints, solution quality, solution diversity, shift assignment

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