# A Bi-Objective Field-Visit Planning Problem for Rapid Needs Assessment under Travel-Time Uncertainty

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. Problem Definition

#### 3.1. Bi-Objective Lexicographic Maximin Approach to the SARP Model

**Definition 1**(Leximin ordering)

**.**

- Vector x leximin-dominates y (written as $x{\succ}_{leximin}y$), if and only if $\exists i\in \{1,\dots ,m\}$, such that $\forall j\in \{1,\dots ,i-1\}$, ${x}_{j}^{\uparrow}$ = ${y}_{j}^{\uparrow}$ and ${x}_{i}^{\uparrow}$>${y}_{i}^{\uparrow}$;
- x and y are indifferent (written as x ${\sim}_{leximin}$y), if and only if ${x}^{\uparrow}$ = ${y}^{\uparrow}$;
- $x{\u2ab0}_{leximin}y$ is the case where $x{\succ}_{leximin}y$ or x ${\sim}_{leximin}$y.

**Definition 2**(Dominance and Pareto optimality in the leximin–SARP)

**.**

- Let s and ${s}^{\prime}$ represent two solutions of the leximin–SARP. Solution s dominates solution ${s}^{\prime}$ iff ${c}_{s}\le {c}_{{s}^{\prime}}$ and ${l}^{s}{\u2ab0}_{leximin}{l}^{{s}^{\prime}}$, and either ${c}_{s}<{c}_{{s}^{\prime}}$ or ${l}^{s}{\succ}_{leximin}{l}^{{s}^{\prime}}$. Solution s is a Pareto-optimal solution iff no other solutions dominate s.

#### 3.2. Robust Optimization Approach to Deal with Uncertainty

## 4. Solution Method

#### 4.1. Multi-Directional Local Search

Algorithm 1 High-level overview of the MDLS procedure proposed by Tricoire [23]. |

1: pre-condition: F is a non-dominated set 2: repeat3: $x\leftarrow $ select a solution $\left(F\right)$ 4: $\mathbf{for}\phantom{\rule{2.84544pt}{0ex}}k\in \{1,2\}\phantom{\rule{2.84544pt}{0ex}}\mathbf{do}$ 5: $F\leftarrow F{\cup}_{\u2aaf}ALN{S}_{k}\left(x\right)$ 6: $\mathbf{end}\mathbf{for}$ 7: until timeLimit is reached8: return F |

#### 4.2. ALNS Operators

#### 4.2.1. Total-Route-Duration Objective Operators

#### 4.2.2. Leximin Objective Operators

- Random removal: q sites are selected randomly to be removed;
- Worst min removal: q sites with the lowest contribution to the solution’s minimum coverage ratio are selected to be removed.

- Highest max–min insertion: for this operator, we first insert the site with the highest contribution to the max–min value of the current solution;
- Highest Leximin insertion: this operator differs from the previous one in that we first insert the site with the highest contribution to the vector of coverage ratio (leximin objective) of the current solution.

## 5. Computational Results

#### 5.1. Instance Description

#### 5.2. Parameter Settings

#### 5.3. Pareto-Front Approximation at Different Levels of Uncertainty

#### 5.4. Trade-Off between Infeasibility and Solution Quality

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Sets/indices:**

- N = set of sites in the affected sites indexed by i, j∈${N}_{0}$
- ${N}_{0}$ = N∪$\left\{0\right\}$ where $\left\{0\right\}$ is the depot
- K = set of assessment teams indexed by k∈K
- C = set of characteristics indexed by c∈C

**Parameters:**

- ${\alpha}_{ic}$ = takes the value 1 if node $i\in N$ carries characteristic $c\in C$ and 0 otherwise
- ${\tau}_{c}$ = total number of sites that carry characteristic $c\in C$
- ${t}_{ij}$ = travel time between nodes i and j
- ${T}_{max}$ = total available time for each team

**Decision Variables:**

- ${x}_{ijk}$ = 1 if team k visits site j after site i and 0 otherwise
- ${y}_{ik}$ = 1 if team k visits site i and 0 otherwise
- ${u}_{i}$ = sequence in which site i is visited
- Z = minimum expected coverage ratio

**Mathematical formulation:**

## Appendix B

**Figure A1.**Two-dimensional visualization of non-dominated solutions at different levels of travel-time uncertainty for instances (

**a**) R8, (

**b**) RC11, (

**c**) R13 and (

**d**) RC14.

**Figure A2.**Two-dimensional visualization of non-dominated solutions at different levels of travel-time uncertainty for instances (

**a**) R20, (

**b**) RC23, (

**c**) R6 and (

**d**) RC5.

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**Figure 1.**Two-dimensional visualization of non-dominated solutions at different levels of travel-time uncertainty for instance R8.

**Figure 2.**Two-dimensional visualization of non-dominated solutions at different levels of travel-time uncertainty for instance RC12.

**Figure 3.**Histogram of violation amount of infeasible solutions in % of Tmax (average of 100 simulation runs)—instance RC5.

**Figure 4.**Histogram of violation amount of infeasible solutions in % of Tmax (average of 100 simulation runs)—instance R7.

**Table 1.**The SARP instances used in our computational study. $N-type/K/{T}_{max}$ represents the number of sites in each instance, the type of network (R or RC), number of teams and the maximum allowed duration for each route. The time limit column shows the run time for each cluster of instances in seconds.

Instance | N-Type/K/${\mathit{T}}_{\mathit{max}}$ | Time Limit (s) | Instance | N-Type/K/${\mathit{T}}_{\mathit{max}}$ | Time Limit (s) |
---|---|---|---|---|---|

R1 | 25_R/2/2 | 90 | RC1 | 25_RC/2/2 | 90 |

R2 | 25_R/2/3 | RC2 | 25_RC/2/3 | ||

R3 | 25_R/2/4 | RC3 | 25_RC/2/4 | ||

R4 | 25_R/3/2 | RC4 | 25_RC/3/2 | ||

R5 | 25_R/3/3 | RC5 | 25_RC/3/3 | ||

R6 | 25_R/3/4 | RC6 | 25_RC/3/4 | ||

R7 | 50_R/3/3 | 180 | RC7 | 50_RC/3/3 | 180 |

R8 | 50_R/3/4 | RC8 | 50_RC/3/4 | ||

R9 | 50_R/3/5 | RC9 | 50_RC/3/5 | ||

R10 | 50_R/4/3 | RC10 | 50_RC/4/3 | ||

R11 | 50_R/4/4 | RC11 | 50_RC/4/4 | ||

R12 | 50_R/4/5 | RC12 | 50_RC/4/5 | ||

R13 | 75_R/3/3 | 360 | RC13 | 75_RC/3/3 | 360 |

R14 | 75_R/3/4 | RC14 | 75_RC/3/4 | ||

R15 | 75_R/3/6 | RC15 | 75_RC/3/6 | ||

R16 | 75_R/5/3 | RC16 | 75_RC/5/3 | ||

R17 | 75_R/5/4 | RC17 | 75_RC/5/4 | ||

R18 | 75_R/5/6 | RC18 | 75_RC/5/6 | ||

R19 | 100_R/3/4 | 720 | RC19 | 100_RC/3/4 | 720 |

R20 | 100_R/3/6 | RC20 | 100_RC/3/6 | ||

R21 | 100_R/3/8 | RC21 | 100_RC/3/8 | ||

R22 | 100_R/6/4 | RC22 | 100_RC/6/4 | ||

R23 | 100_R/6/6 | RC23 | 100_RC/6/6 | ||

R24 | 100_R/6/8 | RC24 | 100_RC/6/8 |

Parameter | Name | Value |
---|---|---|

q | Ruin quantity used in the destroy operators | ∼ Random(1, 0.3*M) |

r | Reaction factor controlling the speed of weight-adjustment-algorithm changes | 0.1 |

${p}_{worst}$ | Degree of randomization for worst removal operator | 5 |

${p}_{related}$ | Degree of randomization for related removal operator | 3 |

**Table 3.**Percentage of infeasible solutions for deterministic and robust approaches (average of 100 simulation runs).

Instance | $\mathit{\rho}$ | Deterministic | Robust-Box |
---|---|---|---|

% of Infeasible Solutions | % of Infeasible Solutions | ||

RC5 | 0.1 | 25.3% | 0% |

0.2 | 36.3% | 0% | |

0.3 | 42.9% | 0% | |

0.6 | 52.1% | 0% | |

R7 | 0.1 | 8.6% | 0% |

0.2 | 15.8% | 0% | |

0.3 | 24.5% | 0% | |

0.6 | 40.4% | 0% |

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**MDPI and ACS Style**

Hakimifar, M.; Hemmelmayr, V.C.; Tricoire, F. A Bi-Objective Field-Visit Planning Problem for Rapid Needs Assessment under Travel-Time Uncertainty. *Sustainability* **2022**, *14*, 3024.
https://doi.org/10.3390/su14053024

**AMA Style**

Hakimifar M, Hemmelmayr VC, Tricoire F. A Bi-Objective Field-Visit Planning Problem for Rapid Needs Assessment under Travel-Time Uncertainty. *Sustainability*. 2022; 14(5):3024.
https://doi.org/10.3390/su14053024

**Chicago/Turabian Style**

Hakimifar, Mohammadmehdi, Vera C. Hemmelmayr, and Fabien Tricoire. 2022. "A Bi-Objective Field-Visit Planning Problem for Rapid Needs Assessment under Travel-Time Uncertainty" *Sustainability* 14, no. 5: 3024.
https://doi.org/10.3390/su14053024