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An exact algorithm for the resource constrained home health care vehicle routing problem

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Abstract

We consider a home health care routing problem with scarce resources which is a variation of the vehicle routing problem with resource constraints. In this problem, the services are provided by nurses and aids at patients’ homes. There are different types of patients that are categorized based on the services they demand and should be serviced by appropriate staff teams or single personnel where the teams are transported by rental vehicles. In this problem, the objective is to minimize the total transportation cost, by considering the patients’ requirements and the qualification of the staff, as well as the resource limitations. In this paper, we present a mathematical formulation of the problem that is modeled similar to a vehicle routing problem and propose a branch-and-price algorithm to solve it. Our proposed algorithm utilizes speed up techniques from the literature to enhance the effectiveness of the proposed solution method. A comprehensive computational study is conducted on Solomon based instances and the results illustrate the efficiency of the algorithm and the effective use of available resources.

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Acknowledgements

We would like to show our gratitude to Dr. Bülent Çatay and Dr. Gizem Özbaygın for comments that greatly improved the manuscript. We would also like to thank Nozir Shokirov for sharing his test instances and corresponding heuristic solutions.

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Authors and Affiliations

  1. Bahen Centre for Information Technology, University of Toronto, 40 St George St, Toronto, ON M5S 2E4, Canada

    Neda Tanoumand

  2. Sabanci University, 34956, Orhanli, Tuzla,Istanbul, Turkey

    Tonguç Ünlüyurt

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  1. Neda Tanoumand
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  2. Tonguç Ünlüyurt
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Correspondence to Tonguç Ünlüyurt.

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Appendices

Appendix A

The characteristics of the instances with 25 patients are demonstrated in Table 6. The first column shows the groups that each instance belongs to them. They are determined by using the information of Table 1. The second and the third columns are names and IDs of the instances. Following column is the best found objective function value of the instances (OFV). Next two columns refer to the available number of nurses and health aid providers for each instances. Based on the available resources, the service levels are determined in the next column; L_L, T_T, and medium mention to the loose, tight, and medium level resources. Finally, the last column is the completion time (CT) of the problem in which all vehicles visiting the patients should return to the health center.

Tables 7 and 8 present the similar information for instances with 50 and 100 patients, respectively.

Table 6 Characteristics of the instances with 25 patients
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Table 7 Characteristics of the instances with 50 patients
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Table 8 Characteristics of the instances with 100 patients
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Appendix B

The detailed results of both enhanced branch-and-price algorithm and CPLEX are presented in Tables 9–11 for the instances containing 25, 50, and 100 patients.

Table 9 Detailed results of solvers for Solomon based instances with 25 patients
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Table 10 Detailed results of solvers for Solomon based instances with 50 patients
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The labels and the results in the tables should be read as follows. The instances in these tables are labeled by their ID numbers. In the first row of the tables instance ID’s and enhanced branch-and-price algorithm are abbreviated as Inst. ID and BAP, respectively. In the tables, columns 2-4 report the outcomes of our enhanced branch-and-price algorithm; where columns 5-7 illustrate the results of CPLEX.

The numbers in the columns labeled as ’Status’ depict the final condition of the solvers within time limits; 0, 1, and 2 are representative of the conditions that a solver solves a specific instance optimally, feasibly, or the solver cannot find an incumbent solution, respectively.

The columns labeled as ’GAP(%)’ report the percent optimality gap of the solvers for each instance. For the instances that the solvers could not find an incumbent solution the optimality gap is depicted as a dash.

The columns with ’CPU(s)’ labels indicate the running time of the solvers in seconds for each instances. The characters ’M’ and ’C’ in front of some instances under the mentioned columns respectively state that the solvers face with memory error and/or CPU error during the run. Additionally, character ’T’ is written in front of the instances that the solvers hit time limit when tackling them.

Table 11 Detailed results of solvers for Solomon based instances with 100 patients
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Tanoumand, N., Ünlüyurt, T. An exact algorithm for the resource constrained home health care vehicle routing problem. Ann Oper Res 304, 397–425 (2021). https://doi.org/10.1007/s10479-021-04061-9

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