# Day and Night: Locating the General Practitioner’s Panel after Hours

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multiple Time Setting p-Median Problem

#### 2.2. Solution Approaches

#### 2.2.1. Independent Approach

#### 2.2.2. Sequential Approach

- Solve the standard p-Median problem for the regular setting (identical to the independent approach for the regular setting).
- Create a set (${I}^{\prime}$) of nodes from the solution containing nodes where regular time setting facilities are located.
- Solve a p-Median problem for the after-hours using ${I}^{\prime}$ instead of I as the set of possible facility locations.

- Solve the standard p-Median problem for the after-hours setting (identical to the independent approach for the after-hours setting).
- Solve a p-Median problem for the regular setting with additional constraint (15).

#### 2.2.3. Simultaneous Approach

#### 2.3. Model Formulation and Verification

## 3. Results

#### 3.1. Data

#### 3.2. Case Studies

#### 3.2.1. Halifax

#### 3.2.2. Halifax Approach Results

#### 3.2.3. Halifax Approach Comparison

#### 3.3. Cape Breton

#### 3.4. Cape Breton Modified

## 4. Discussion

#### 4.1. Limitations

#### 4.2. Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GPs | General Practitioners |

PCPs | Primary Care Practices |

OR | Operations Research |

RtA | Regular to After hours |

AtR | After hours to Regular |

LP | Linear Program |

CBC | COIN-OR Branch and Cut |

DA | Dissemination Area |

CMA | Census Metropolitan Area |

CA | Census Agglomeration |

LSCP | Location Set Covering Problem |

MCLP | Maximum Covering Location Problem |

## References

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**Figure 1.**Nova Scotia with Halifax (center) and Cape Breton (upper-right) Dissemination Areas Highlighted.

**Figure 4.**Minimum cost of p facilities for different ratios of ${p}^{\prime}$ (specified in legend). Independent approach in the Halifax setting is shown.

**Figure 8.**Minimum cost of p facilities for different ratios of ${p}^{\prime}$ (specified in legend). Independent approach in the Cape Breton Modified setting is shown.

**Figure 9.**Cape Breton modified percent cost increase for each approach compared with independent approach (${p}^{\prime}/p$ = 1).

Sets | |
---|---|

I | Set of all nodes where facilities may be placed, indexed by i. |

J | Set of all nodes where demand is located, indexed by j. |

Parameters | |

p | The number of primary facilities to locate. |

${p}^{\prime}$ | The number of secondary facilities to locate. |

${d}_{j}$ | The primary demand at node j. |

${d}_{j}^{\prime}$ | The secondary demand at node j. |

${c}_{ij}$ | The unit cost of supplying demand at node j from a facility at node i. |

W | The discount rate applied to the secondary setting. This can be thought of as |

relative cost; for example, $W=0.5$ suggests that cost incurred in the secondary | |

setting is considered half as important as cost in the primary setting. | |

Decision Variables | |

${x}_{ij}$ | The fraction of primary demand at node j satisfied by facility i. |

${x}_{ij}^{\prime}$ | The fraction of secondary demand at node j satisfied by facility i. |

${y}_{i}$ | A binary variable, 1 if a facility is located at site i and 0 otherwise. |

${y}_{i}^{\prime}$ | A binary variable, 1 if a secondary facility is located at site i and 0 otherwise. |

Approach | p = 60, ${\mathit{p}}^{\prime}$ = 12 | ||
---|---|---|---|

Day-Time | After-Hours | Total | |

Independent | 327,904 | 475,128 | 803,032 |

Sequential (RtA) | 327,904 | 482,107 | 810,011 |

Sequential (AtR) | 328,352 | 475,128 | 803,480 |

Simultaneous | 803,277 |

Approach | p = 60, ${\mathit{p}}^{\prime}$ = 12 | |||
---|---|---|---|---|

Regular | After-Hours | Overall | Std. Deviation | |

Independent | 0.8365 | 1.2121 | 2.0486 | 15.9845 |

Sequential (RtA) | 0.8365 | 1.2299 | 2.0664 | 16.0712 |

Sequential (AtR) | 0.8376 | 1.2121 | 2.0497 | 15.9852 |

Simultaneous | 2.0492 | 15.9855 |

Approach | p = 60, ${\mathit{p}}^{\prime}$ = 36 | ||
---|---|---|---|

Total | Difference | Percent Difference | |

Independent | 705,368 | 0 | 0% |

Sequential (RtA) | 706,736 | 1367 | 0.19% |

Sequential (AtR) | 705,633 | 265 | 0.04% |

Simultaneous | 705,578 | 210 | 0.03% |

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**MDPI and ACS Style**

Campbell, J.C.; Taghavi, M.; VanBerkel, P.T.
Day and Night: Locating the General Practitioner’s Panel after Hours. *Appl. Sci.* **2023**, *13*, 6273.
https://doi.org/10.3390/app13106273

**AMA Style**

Campbell JC, Taghavi M, VanBerkel PT.
Day and Night: Locating the General Practitioner’s Panel after Hours. *Applied Sciences*. 2023; 13(10):6273.
https://doi.org/10.3390/app13106273

**Chicago/Turabian Style**

Campbell, John Charles, Majid Taghavi, and Peter T. VanBerkel.
2023. "Day and Night: Locating the General Practitioner’s Panel after Hours" *Applied Sciences* 13, no. 10: 6273.
https://doi.org/10.3390/app13106273