# Multi-Period Maximal Covering Location Problem with Capacitated Facilities and Modules for Natural Disaster Relief Services

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## Abstract

**:**

## 1. Introduction

- Location of the predefined number of facilities;
- Type and number of each module assigned to the located facilities in each time period during the planning horizon;
- Percentage of allocated demands of points to the assigned modules in each time period.

- We propose a new model called multi-period modular capacitated maximal covering location problem, which in addition to the facility location decisions, it determines module assignment and demand allocation decisions in different periods of the planning horizon.
- The configuration of the facilities using the modularity concept results in having three-level facilities. Different size of modules (first level) along with having a different kind of modules (second level) makes it possible to have different facilities (third level).
- An efficient Lagrangian decomposition-based method is developed to derive the upper bound. Furthermore, two different lower bounds are developed and compared to synthesize the proposed Lagrangian decomposition (LD) method.
- Different upper bounds and lower bounds from various approaches are compared to evaluate the efficiency and tightness of the bounds. Our findings approve the superiority of the bounds obtained from Lagrangian decomposition-based (upper bound) and heuristic method (lower bound) over large-scale instances.
- Sensitivity analysis for the problem is conducted to investigate the validity of the proposed model under different parameters.

## 2. Literature Review

#### 2.1. MCLP

#### 2.2. Modular Location Problem

#### 2.3. Lagrangian Relaxation

## 3. Problem Definition

#### 3.1. Multi-Period Modular Capacitated Maximal Covering Location Problem (MMCMCLP)

- Each module comes in different sizes. It can be chosen from different sizes to increase the service quality offered to demand points to overcome the service shortages or having idle units.
- The modules are portable and they can be transferred among the facilities when there is more request in another facility. The transferability is an important specification of modularity design that yields to flexibility in the system and reduces costs. The portability of most modules helps to provide a good level of service to demand points without having to provide more modules.
- As there are no constraints on the number of modules to be assigned to facilities, more of them can be established in the coming periods (if there is increasing demand) as expansion plans to have more coverage of demand points. The number and the location of the facilities after the decisions are made would be fixed for all time periods, but there is no such limitation for modules.
- Each size of the modules has a known lower and upper-level service capacity for which, the total number of allocated points should be between this lower and upper level capacity. On the other hand, the total number of demands allocated to all kinds of modules cannot exceed each facility’s capacity.

#### 3.2. Mathematical Formulation

$i\in I$ | The index and set of potential facilities. |

$j\in J$ | The index and set of demand points. |

$l\in L$ | The index and set of modules. |

${k}_{l}\in {K}_{l}$ | The index and set of size indices for module $l\text{}\left(l\in L\right)$. |

$t\in T$ | The index and set of time periods. |

$S$ | The maximum full coverage distance. |

${S}^{\prime}$ | The maximum partial coverage distance$\text{}({S}^{\prime}S)$. |

${d}_{ij}$ | Distance, traveling time or cost between facility $i\text{}$and demand point $j$. |

${g}_{ij}$ | The level of coverage provided by the facility $i$ to the demand point $j$. |

$${g}_{ij}=\{\begin{array}{c}\text{}1\hspace{1em}\hspace{1em}if\text{}{d}_{ij}\le S\\ f\left({d}_{ij}\right)\hspace{1em}ifS{d}_{ij}\le S\prime \\ \text{}0\hspace{1em}\hspace{1em}S\prime \text{}{d}_{ij}\end{array}$$
| |

$f\left({d}_{ij}\right)\text{}$ | Partial coverage function, where $0<f\left({d}_{ij}\right)<1$. |

${a}_{jlt}$ | The demand of point $j\text{}$from the service of module $l$ in period $t$. |

${b}_{l{k}_{l}}$ | The lower level of capacity for module $l\text{}$with the size indices of ${k}_{l}.$ |

${B}_{l{k}_{l}}$ | The upper level of capacity for module $l\text{}$with the size indices of ${k}_{l}$. |

${\alpha}_{j}$ | The profit gained by coverage of demand point $j$. |

${\beta}_{i}$ | The capacity of facility $i$. |

${h}_{il{k}_{l}t}$ | The cost of establishing module $l$ with size indices ${k}_{l}$ at facility $i$ at period $t$. |

$p$ | The number of facilities to be located. |

${v}_{i}\text{}$ | Binary variable which equals 1 if a facility is located at point$\text{}i.$ |

${y}_{il{k}_{l}t}$ | Binary variable which equals 1 if module $l$ with size indices of ${k}_{l}$ is sited at facility $i$ at period $t$. |

${x}_{ijlt}$ | Coverage amount of demand point $j\text{}$from module $l\text{}$of facility $i$ at period $t$. |

## 4. Solution Procedure

#### 4.1. Upper Bounds

#### 4.1.1. LR1

#### 4.1.2. LR2

#### 4.1.3. Lagrangian Decomposition (LD)

#### 4.2. Lower Bounds

#### 4.2.1. Lower Bound from Feasible Solutions

#### 4.2.2. Lower Bound Heuristic on MMCMCLP

Algorithm 1: The Heuristic Solution Method for MMCMCLP. |

Step 1. Choose the largest $p$ values of ${\delta}_{i}={w}_{1}{\beta}_{i}+{w}_{2}\sum _{j,l,t}{g}_{ij}\text{}{a}_{jlt}$. ${i}_{p}\text{}$is the set of $i$ that provides $p$ maximum values of ${\delta}_{i}$ and put the related ${v}_{i\in {i}_{p}}=1$, otherwise ${v}_{i}=0$. |

Step 2. Calculate ${\phi}_{{i}_{p}lt}={\mathrm{max}}_{k}\left({B}_{lk}/{h}_{{i}_{p}l{k}_{l}t}\right)$ for each ${i}_{p},l,k,t$.If ${\phi}_{{i}_{p}lt}\ge $ $\left[{\mathrm{max}}_{l,{k}_{l}}{B}_{l{k}_{l}}/mea{n}_{{i}_{p},l,{k}_{l},t}{h}_{{i}_{p}l{k}_{l}t}\right]$, assign the module $l$ to the located facility ${i}_{p}$ at time period $t$ in variable ${y}_{{i}_{p}l{k}_{l}t}=1$, otherwise ${y}_{{i}_{p}l{k}_{l}t}=0$. Calculate $\sum _{{i}_{p}l{k}_{l}t}\text{}{h}_{{i}_{p}l{k}_{l}t}{y}_{{i}_{p}l{k}_{l}t}$. |

Step 3. Sort ${\gamma}_{{i}_{p}jlt}={g}_{{i}_{p}j}\text{}{a}_{jlt}\text{}$with respect to $j$ for each ${i}_{p},l,t$. Set ${\pi}_{{i}_{p}t}=0\text{}$and ${\sigma}_{{i}_{p}lt}=0$. |

Step 4. For each ${i}_{p},t,\text{}$iterate ${\pi}_{{i}_{p}t}\leftarrow {\pi}_{{i}_{p}t}+{\gamma}_{{i}_{p}jlt}$ until ${\pi}_{{i}_{p}t}\le {\beta}_{{i}_{p}}{v}_{{i}_{p}}$. For each ${i}_{p},l,t$, iterate ${\sigma}_{{i}_{p}lt}\leftarrow {\sigma}_{{i}_{p}lt}+{\gamma}_{{i}_{p}jlt}\text{}$until ${\sigma}_{{i}_{p}lt}\le \sum _{{k}_{l}}{B}_{l{k}_{l}}\text{}{y}_{{i}_{p}l{k}_{l}t}$. For the demand points $j\text{}$involved in the iterations put ${x}_{{i}_{p}jlt}=1$, for others ${x}_{{i}_{p}jlt}=0$. |

Step 5. ${Z}_{HLB}$ = $\sum _{{i}_{p}jlt}\text{}{\alpha}_{j}\text{}{g}_{{i}_{p}j}\text{}{a}_{jlt}\text{}{x}_{{i}_{p}jlt}-\sum _{{i}_{p}l{k}_{l}t}\text{}{h}_{{i}_{p}l{k}_{l}t}{y}_{{i}_{p}l{k}_{l}t}.$ |

#### 4.3. Sub-Gradiant Method

Algorithm 2: LR1-LB |

Step0. Put $UB=+\infty \text{},\mathrm{LB}=-\infty ,\text{}\tau =0\text{}{\mu}_{jlt}^{0}=0\text{}\forall \text{}j,l,t,\text{}{\lambda}_{jlt}^{0}=0\text{}\forall \text{}i,l,t,\text{}{\gamma}_{jlt}^{0}=0\text{}\forall i,l,t,\text{}{\theta}^{\tau}=2$. |

Step1. Solve the problem LR1. Put $\tau =\tau +1$. Find the optimal values for variables and calculate ${Z}_{LR1}.$ |

Step2. Using a feasible solution of the problem MMCMCLP (Section 4.2.1), Calculate ${Z}_{LB}$. |

Step3. Update the upper and lower bounds. If ${Z}_{LB}>LB$, then put $LB={Z}_{LB}$. If ${Z}_{LR1}<UB$, then put $UB={Z}_{LR1}$. |

Step4 Update the step size parameter. If there was no improvement in upper bound for the past $b$ iterations then put ${\theta}^{\tau}=\rho {\theta}^{\tau \text{}}$ in which $0<\rho <1$. |

Step5. Check the termination conditions. Stop, if one of the following conditions holds.$\frac{UB-\mathrm{LB}}{UB}\le 0.01$ or,$\tau =80$. |

Step6. Update Lagrangian multipliers and go to Step 1. |

Algorithm 3: LD-HLB |

Step 0. Put $UB=+\infty ,\text{}\tau =0,\text{}{\zeta}_{jlt}^{0}=0\text{}\forall \text{}j,l,t,\text{}{\chi}_{it}^{0}=0\text{}\forall \text{}i,\text{}t,\text{}{\theta}^{\tau}=2$. |

Step 1. Solve the problem SubP1. Obtain ${I}_{o}$ and solve SubP2 for ${I}_{o}$. Put $\tau =\tau +1$ Find the optimal values for variables and calculate. ${Z}_{LD}$$={Z}_{LD1}+{Z}_{LD2}$. |

Step 2. Update the upper bound. If ${Z}_{LD}<UB$ then put $UB\text{}=\text{}{Z}_{LD}.$ |

Step 3. Update the step size parameter. If there was no improvement in upper bound for the past $b$ iterations then put
${\theta}^{\tau}=\rho {\theta}^{\tau \text{}}$ in which $0<\rho <1$. |

Step 4. Check the termination conditions. Stop, if one of the following conditions holds.$\frac{UB-{Z}_{HLB}}{UB}\le 0.01\text{}$or $\tau =80$. |

Step5. Update Lagrangian multipliers and go to Step 1. |

## 5. Numerical Experiments

#### 5.1. Model Validation

#### 5.1.1. Illustrative Example

#### 5.1.2. Sensitivity Analysis for the Capacity of Facilities

#### 5.1.3. Sensitivity Analysis for the Number of Facilities to be Located

#### 5.1.4. Sensitivity Analysis for the Cost of Module Assignment

#### 5.2. Comparison with the Conventional Methods

#### 5.2.1. Effectiveness of the Heuristic Method

#### 5.2.2. Bounds Provided by LR1, LR2, and LD for Small Size Problems

**#**3). From 19 test problems, HLB could obtain the best lower bounds for 14 problems while LR2-LB obtained the best lower bound for four problems (in case

**#**2, they obtain the same result). This fact approves the efficiency of the proposed heuristic method to compute feasible solutions as the lower bound of the problems. Regarding the upper bound, the competition is mainly between LD-HLB and LD-LB (except for two test problems 1 and 4). In particular, this fact approves that the best upper bounds are obtained from the LD problem. From 18 test problems, LD-HLB could produce best upper bounds for 10 problems, the upper bounds of two studied approaches are equal for 3 problems and LD-LB was superior only in 5 cases. According to the results, we suggest using HLB to compute lower bounds and LD-HLB to compute the upper bounds. The GAMS source codes of MMCMCLP are available at https://github.com/Alizadehroqayeh/MMCMCLP.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Authors | Problem | Approach | Decom. | Just LB | Both LB and UB |
---|---|---|---|---|---|

Ayala et al. [39] | Resource-constrained modular scheduling | LR | ✓ | ||

Nishi et al. [37] | Hybrid flowshop scheduling problem | LR with cuts | ✓ | ✓ | |

Nishi et al. [38] | Automated guided vehicles routing problem | LR with Petri nets | ✓ | ✓ | |

Yang, Chen and Chu [40] | Large scale new variant of capacitated clustering problem | LR approach with two phases of dual optimization | ✓ | ✓ | |

Litvinchev and Ozuna [41] | Two-stage capacitated facility location problem | Lagrangian heuristic producing feasible solutions | ✓ | ||

Diabat et al. [42] | Multi-echelon joint inventory-location problem | Improved LR-based heuristics | ✓ | ✓ | ✓ |

Gendron et al. [43] | Two-level uncapacitated facility location problem with single-assignment constraints | Lagrangian-Based Branch-and-Bound Algorithm | ✓ | ✓ | |

Marin et al. [23] | Multi-period stochastic covering location problems | LR based algorithm for high quality feasible solutions | ✓ | ✓ | |

Rafie-Majd et al. [44] | Three-echelon supply chain integrating inventory-location-routing problem under uncertainty | LR | ✓ | ||

Hamdan and Diabat [35] | Multi-objective and stochastic blood supply chain | Lagrangian heuristic | ✓ | ||

Fathollahi-Fard et al. [34] | Coordinated water supply and wastewater collection network design problem | Adaptive Lagrangian relaxation-based algorithm | ✓ | ||

Zhang et al. [36] | Train rescheduling and track emergency maintenance | Lagrangian-based decomposition | ✓ | ||

Current research | Dynamic modular capacitated maximal covering location problem | Lagrangian decomposition heuristic | ✓ | ✓ |

# | ${\mathit{\beta}}_{\mathit{i}}$ | Obj | Y | X |
---|---|---|---|---|

1 | (260,300) | 85.81 | 11 | 135.6 |

2 | (160,200) | 85.81 | 11 | 136.22 |

3 | (100,120) | 85.81 | 11 | 135.14 |

4 | (80,100) | 85.81 | 11 | 135.46 |

5 | (60,80) | 76.26 | 10 | 124.24 |

6 | (50,70) | 65.6 | 8 | 95.53 |

7 | (40,60) | 63.5 | 6 | 70.6 |

**Table 3.**Computational results to observe the effect of the increasing number of located facilities.

# | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{i}\mathit{l}{\mathit{k}}_{\mathit{l}}\mathit{t}}$ | P | Obj | Y | X |
---|---|---|---|---|---|---|

8 | (60,80) | 20,40 | 2 | 52.9 | 7 | 88.9 |

9 | (60,80) | 20,40 | 3 | 76.2 | 10 | 124.2 |

10 | (60,80) | 20,40 | 4 | 94.8 | 11 | 133.3 |

# | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{h}}_{\mathit{i}\mathit{l}{\mathit{k}}_{\mathit{l}}\mathit{t}}$ | P | Obj | Y | X |
---|---|---|---|---|---|---|

11 | (60,80) | 10,30 | 3 | 191.3 | 12 | 146.2 |

12 | (60,80) | 20,40 | 3 | 85.81 | 11 | 136.2 |

13 | (60,80) | 30,50 | 3 | 13.5 | 3 | 32.3 |

# | $\mathit{i}$ | $\mathit{j}$ | $\mathit{t}$ | $\mathit{l}$ | ${\mathit{\beta}}_{\mathit{i}}$ | $\mathit{p}$ |
---|---|---|---|---|---|---|

1–4 | 10 | 100 | 2,3 | 3,4 | (80,100) | 3 |

5–8 | 30 | 300 | 2,3 | 3,4 | (150,180) | 12 |

9–12 | 50 | 500 | 3,4 | 3,4 | (200,300) | 20 |

13–16 | 70 | 700 | 3,4 | 3,4 | (250,350) | 25 |

17–20 | 100 | 1000 | 3,4 | 3,4 | (300,400) | 35 |

# | CPLEX | Heuristic | ||
---|---|---|---|---|

Obj | Time | HLB | LO-Gap | |

1 | 56 | 1 | 55 | 0.018 |

2 | 71 | 2 | 69 | 0.029 |

3 | 85 | 2 | 80 | 0.063 |

4 | 114 | 2 | 114 | 0 |

5 | 1838 | 333 | 1617 | 0.137 |

6 | 2030 | 338 | 1819 | 0.116 |

7 | 2847 | 1002 | 2574 | 0.106 |

8 | 3056 | 503 | 2906 | 0.052 |

Average | 0.065 |

# | CPLEX | Heuristic | LR1-HLB | LR1-LB | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Obj | Time | HLB | UB | Time | UO-Gap | LB | UB | Time | LO-Gap | UO-Gap | |

1 | 56 | 1 | 55 | 80 | 37 | 0.429 | 47 | 75 | 67 | 0.191 | 0.339 |

2 | 71 | 2 | 69 | 97 | 50 | 0.366 | 68 | 94 | 89 | 0.044 | 0.324 |

3 | 85 | 2 | 80 | 108 | 51 | 0.271 | 80 | 110 | 85 | 0.063 | 0.294 |

4 | 114 | 2 | 114 | 177 | 54 | 0.553 | 101 | 168 | 97 | 0.129 | 0.474 |

5 | 1838 | 333 | 1617 | 2139 | 173 | 0.164 | 1649 | 2162 | 284 | 0.115 | 0.176 |

6 | 2030 | 338 | 1819 | 2312 | 242 | 0.139 | 1588 | 2465 | 504 | 0.278 | 0.214 |

7 | 2847 | 1002 | 2574 | 3289 | 262 | 0.155 | 2301 | 3402 | 560 | 0.237 | 0.195 |

8 | 3056 | 503 | 2906 | 3496 | 409 | 0.144 | 2367 | 3540 | 734 | 0.291 | 0.158 |

Average | 159 | 0.277 | 302 | 0.169 | 0.272 |

# | CPLEX | Heuristic | LR2-HLB | LR2-LB | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Obj | Time | HLB | UB | Time | UO-Gap | LB | UB | Time | LO-Gap | UO-Gap | |

1 | 56 | 1 | 55 | 72 | 30 | 0.286 | 36 | 72 | 40 | 0.56 | 0.286 |

2 | 71 | 2 | 69 | 98 | 33 | 0.380 | 69 | 98 | 49 | 0.03 | 0.380 |

3 | 85 | 2 | 80 | 109 | 40 | 0.282 | 47 | 109 | 52 | 0.81 | 0.282 |

4 | 114 | 2 | 114 | 151 | 59 | 0.325 | 68 | 117 | 93 | 0.676 | 0.026 |

5 | 1838 | 333 | 1617 | 2232 | 190 | 0.214 | 1764 | 2232 | 372 | 0.04 | 0.214 |

6 | 2030 | 338 | 1819 | 2388 | 267 | 0.176 | 1907 | 2388 | 513 | 0.06 | 0.176 |

7 | 2847 | 1002 | 2574 | 3394 | 280 | 0.192 | 2634 | 3393 | 578 | 0.08 | 0.192 |

8 | 3056 | 503 | 2906 | 3600 | 372 | 0.178 | 2894 | 3600 | 574 | 0.06 | 0.178 |

Average | 158 | 0.254 | 283 | 0.289 | 0.217 |

# | CPLEX | Heuristic | LD-HLB | LD-LB | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Obj | Time | HLB | UB | Time | UO-Gap | LB | UB | Time | LO-Gap | UO-Gap | |

1 | 56 | 1 | 55 | 52 | 1 | −0.071 | 24 | 51 | 77 | 1.333 | −0.089 |

2 | 71 | 2 | 69 | 80 | 51 | 0.127 | - | 80 | 51 | 0.127 | |

3 | 85 | 2 | 80 | 99 | 51 | 0.165 | 46 | 102 | 61 | 0.848 | 0.200 |

4 | 114 | 2 | 114 | 128 | 52 | 0.123 | 33 | 134 | 60 | 2.455 | 0.175 |

5 | 1838 | 333 | 1617 | 2120 | 148 | 0.153 | 1586 | 2122 | 333 | 0.159 | 0.155 |

6 | 2030 | 338 | 1819 | 2280 | 177 | 0.123 | 1755 | 2280 | 242 | 0.157 | 0.123 |

7 | 2847 | 1002 | 2574 | 3193 | 296 | 0.122 | 2317 | 3190 | 257 | 0.229 | 0.120 |

8 | 3056 | 503 | 2906 | 3456 | 275 | 0.131 | 2632 | 3456 | 453 | 0.161 | 0.131 |

Average | 131 | 0.135 | 191 | 0.763 | 0.147 |

# | LR1 | LR2 | LD | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

LB | UB | Time | Gap | LB | UB | Time | Gap | LB | UB | Time | Gap | |

9 | 2476 | 14,072 | 183 | 4.68 | 9832 | 11,630 | 672 | 0.18 | 9147 | 11,013 | 2356 | 0.20 |

10 | 2653 | 13,937 | 361 | 4.25 | 9095 | 11,806 | 2640 | 0.30 | - | 11,102 | 693 | - |

11 | 2122 | 18,532 | 205 | 7.73 | - | 15,282 | 930 | - | - | 14,429 | 713 | - |

12 | 3453 | 18,746 | 317 | 4.43 | 12,310 | 16,000 | 1301 | 0.30 | 11,134 | 15,141 | 790 | 0.36 |

13 | 7047 | 21,305 | 404 | 2.02 | - | 19,036 | 1941 | - | - | 17,748 | 1620 | - |

14 | 6088 | 21,228 | 766 | 2.49 | 12,534 | 19,117 | 1898 | 0.53 | 8007 | 20,443 | 287 | 1.55 |

15 | 7808 | 28,253 | 543 | 2.62 | - | 25,545 | 2157 | - | - | 23,953 | 1548 | - |

16 | 6616 | 29,053 | 844 | 3.39 | 18,372 | 26,282 | 2815 | 0.43 | 12,742 | 30,472 | 229 | 1.39 |

17 | 11,110 | 41,389 | 711 | 2.73 | - | 37,517 | 4466 | - | - | 35,123 | 4818 | - |

18 | 11,345 | 41,046 | 1784 | 2.62 | - | 37,768 | 8020 | - | - | 35,970 | 5243 | - |

19 | 14,002 | 54,636 | 1329 | 2.90 | - | 50,177 | 4759 | - | - | 46,720 | 4870 | - |

20 | 17,823 | 54,471 | 2101 | 2.06 | - | 50,043 | 21,466 | - | - | 47,399 | 6919 | - |

Average | 795 | 3.5 | 4422 | 0.34 | 2507 | 0.87 |

# | Heuristic | LR1 | LR2 | LD | ||||||
---|---|---|---|---|---|---|---|---|---|---|

HLB | UB | Time | Gap | UB | Time | Gap | UB | Time | Gap | |

9 | 7945 | 11,673 | 3703 | 0.47 | 11,617 | 585 | 0.46 | 11,118 | 451 | 0.40 |

10 | 10,237 | 11,870 | 4817 | 0.16 | 11,808 | 787 | 0.15 | 11,095 | 775 | 0.08 |

11 | 11,403 | 15,334 | 7016 | 0.34 | 15,283 | 617 | 0.34 | 14,368 | 582 | 0.26 |

12 | 13,817 | 16,029 | 8161 | 0.16 | 16,106 | 1093 | 0.17 | 15,056 | 765 | 0.09 |

13 | 12,358 | 18,935 | 7390 | 0.53 | 19,307 | 1151 | 0.56 | 17,699 | 969 | 0.43 |

14 | 16,096 | 19,191 | 8972 | 0.19 | 19,139 | 2555 | 0.19 | 18,060 | 1190 | 0.12 |

15 | 17,682 | 25,212 | 8859 | 0.43 | 25,100 | 1694 | 0.42 | 24,113 | 1294 | 0.36 |

16 | 23,590 | 26,289 | 21,429 | 0.11 | 26,295 | 2450 | 0.11 | 25,669 | 1847 | 0.09 |

17 | 20,740 | 41,389 | 1638 | 1.00 | 37,524 | 4354 | 0.81 | 35,143 | 2841 | 0.69 |

18 | 29,857 | 37,806 | 37,381 | 0.27 | 37,758 | 5418 | 0.26 | 35,937 | 4811 | 0.20 |

19 | 29,623 | 54,646 | 1295 | 0.84 | 49,827 | 5344 | 0.68 | 46,749 | 3890 | 0.58 |

20 | 37,469 | 54,471 | 5145 | 0.45 | 50,039 | 7414 | 0.34 | 47,354 | 5567 | 0.26 |

Average | 9650 | 0.41 | 2788 | 0.37 | 2081 | 0.3 |

# | Best LB | Best UB | ||
---|---|---|---|---|

LB | Method | UB | Method | |

1 | 55 | HLB | 72 | LR2-LB = LR2-HLB |

2 | 69 | HLB = LR2-LB | 80 | LD-HLB = LD-LB |

3 | 80 | HLB = LR1-LB | 99 | LD-HLB |

4 | 114 | HLB | 117 | LR2-LB |

5 | 1764 | LR2-LB | 2120 | LD-HLB |

6 | 1907 | LR2-LB | 2280 | LD-HLB = LD-LB |

7 | 2634 | LR2-LB | 3190 | LD-LB |

8 | 2906 | HLB | 3456 | LD-HLB = LD-LB |

9 | 9832 | LR2-LB | 11,013 | LD-LB |

10 | 10,237 | HLB | 11,095 | LD-HLB |

11 | 11,403 | HLB | 14,368 | LD-HLB |

12 | 13,817 | HLB | 15,056 | LD-HLB |

13 | 12,464 | HLB | 17,699 | LD-HLB |

14 | 16,096 | HLB | 18,060 | LD-HLB |

15 | 17,682 | HLB | 23,953 | LD-LB |

16 | 23,590 | HLB | 25,669 | LD-HLB |

17 | 20,740 | HLB | 35,123 | LD-LB |

18 | 29,857 | HLB | 35,937 | LD-HLB |

19 | 29,623 | HLB | 46,720 | LD-LB |

20 | 37,469 | HLB | 47,354 | LD-HLB |

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## Share and Cite

**MDPI and ACS Style**

Alizadeh, R.; Nishi, T.; Bagherinejad, J.; Bashiri, M.
Multi-Period Maximal Covering Location Problem with Capacitated Facilities and Modules for Natural Disaster Relief Services. *Appl. Sci.* **2021**, *11*, 397.
https://doi.org/10.3390/app11010397

**AMA Style**

Alizadeh R, Nishi T, Bagherinejad J, Bashiri M.
Multi-Period Maximal Covering Location Problem with Capacitated Facilities and Modules for Natural Disaster Relief Services. *Applied Sciences*. 2021; 11(1):397.
https://doi.org/10.3390/app11010397

**Chicago/Turabian Style**

Alizadeh, Roghayyeh, Tatsushi Nishi, Jafar Bagherinejad, and Mahdi Bashiri.
2021. "Multi-Period Maximal Covering Location Problem with Capacitated Facilities and Modules for Natural Disaster Relief Services" *Applied Sciences* 11, no. 1: 397.
https://doi.org/10.3390/app11010397