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Dynamic Home Care Routing and Scheduling with Uncertain Number of Visits per Referral

Danial Khorasanian, Jonathan Patrick, Antoine Sauré

TL;DR

A Markov decision process (MDP) model is developed for the single-nurse problem to minimize the expected weighted sum of the rejection, diversion, overtime, and travel time costs and an intuitively explainable closed-form solution is derived for the optimal ALP parameters in a special case of the problem.

Abstract

Despite the rapid growth of the home care industry, research on the scheduling and routing of home care visits in the presence of uncertainty is still limited. This paper investigates a dynamic version of this problem in which the number of referrals and their required number of visits are uncertain. We develop a Markov decision process (MDP) model for the single-nurse problem to minimize the expected weighted sum of the rejection, diversion, overtime, and travel time costs. Since optimally solving the MDP is intractable, we employ an approximate linear program (ALP) to obtain a feasible policy. The typical ALP approach can only solve very small-scale instances of the problem. We derive an intuitively explainable closed-form solution for the optimal ALP parameters in a special case of the problem. Inspired by this form, we provide two heuristic reduction techniques for the ALP model in the general problem to solve large-scale instances in an acceptable time. Numerical results show that the ALP policy outperforms a myopic policy that reflects current practice, and is better than a scenario-based policy in most instances considered.

Dynamic Home Care Routing and Scheduling with Uncertain Number of Visits per Referral

TL;DR

A Markov decision process (MDP) model is developed for the single-nurse problem to minimize the expected weighted sum of the rejection, diversion, overtime, and travel time costs and an intuitively explainable closed-form solution is derived for the optimal ALP parameters in a special case of the problem.

Abstract

Despite the rapid growth of the home care industry, research on the scheduling and routing of home care visits in the presence of uncertainty is still limited. This paper investigates a dynamic version of this problem in which the number of referrals and their required number of visits are uncertain. We develop a Markov decision process (MDP) model for the single-nurse problem to minimize the expected weighted sum of the rejection, diversion, overtime, and travel time costs. Since optimally solving the MDP is intractable, we employ an approximate linear program (ALP) to obtain a feasible policy. The typical ALP approach can only solve very small-scale instances of the problem. We derive an intuitively explainable closed-form solution for the optimal ALP parameters in a special case of the problem. Inspired by this form, we provide two heuristic reduction techniques for the ALP model in the general problem to solve large-scale instances in an acceptable time. Numerical results show that the ALP policy outperforms a myopic policy that reflects current practice, and is better than a scenario-based policy in most instances considered.
Paper Structure (37 sections, 4 theorems, 45 equations, 5 figures, 6 tables)

This paper contains 37 sections, 4 theorems, 45 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Related literature
  4. Paper structure
  5. Problem description
  6. MDP formulation
  7. Decision epochs
  8. State space
  9. Action set
  10. State transition
  11. Cost function
  12. Bellman's equations
  13. ALP approach
  14. ALP formulation
  15. ALP policy
  16. ...and 22 more sections

Key Result

Proposition 1

Any referral of type $k$ coming from location $l$ is always rejected by the ALP policy if $\min \{\gamma \tau _{{{h}_{k}},kl,1}^{*}{{p}_{k,2}},\underset{t,t\le {{T}_{k}}-1}{\mathop{\min }}\,\{\gamma \tau _{tkl,0}^{*}\}\}>{{R}_{k}}$, and is always accepted if $\underset{t,t\le {{T}_{k}}-1}{\mathop{\m

Figures (5)

  • Figure 1: A circular grid area with Manhattan distances around the depot
  • Figure 2: A 2D area and its corresponding 1D array
  • Figure 3: Rejection rates for different policies, regions and service types
  • Figure 4: Referral assignment rates for different dates, policies and service types
  • Figure EC.1: Trade-off between the rejection and the diversion costs associated with the ALP policy as the value of $\varepsilon$ increases

Theorems & Definitions (4)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3