Two-Stage Multi-Objective Stochastic Model on Patient Transfer and Relief Distribution in Lockdown Area of COVID-19
Abstract
:1. Introduction
2. Literature Review
- (1)
- Many studies have considered victim transfer and relief distribution separately, but few studies have considered the coordination between the two operations.
- (2)
- Some studies of emergency management have considered humanitarian issues, but few studies have been able to obtain a real Pareto optimal solution set.
- (3)
- No study has considered the characteristics of epidemic diseases and the control measures in a multi-objective victim transfer and relief distribution problem.
3. Problem Definition and Formulation
- (1)
- Each community has a fever clinic, in which the general physicians in the community diagnose illnesses and transfer confirmed cases to the appropriate hospitals;
- (2)
- The confirmed cases consisting of the infected people and critical patients can be estimated by the modified SEIR model;
- (3)
- We assume that untransferred confirmed cases do not affect the forecast results, because only a small number of confirmed cases are not transferred in time and will be preferentially transferred in the next period;
- (4)
- The infected people need to be transferred to the THs and the critical patients need to be transferred to the DHs;
- (5)
- The fever clinics provide a set of medical and ancillary supplies (such as an N95 mask and an additional protective suit) for each infected patient and critical patient;
- (6)
- Each established TRDC can distribute relief supplies to each fever clinic using a vehicle with a homogeneous capacity.
3.1. Time-Varying Epidemic Prediction Model
3.2. Notation
$S$: | Set of scenarios, $s\in S$; |
$T$: | Set of planning periods, $t\in T$; |
$I$: | Set of TRDCs, $i\in I$; |
$C$: | Set of fever clinics, $\mathrm{c}\in C$; |
$J$: | Set of THs, $j\in J$; |
$H$: | Set of DHs, $h\in H$. |
${P}_{s}$: | Probability of scenario s; |
${t}_{nm}^{p}$: | Transportation time from the node n to the node m; |
$F{c}_{i}$: | Fixed cost of the TRDC i; |
$F{t}_{j}$: | Fixed cost of the TH j; |
$F{d}_{h}$: | Fixed cost of the DH h; |
${c}_{1}$: | Holding cost of a relief supply; |
${c}_{2}$: | Penalty cost of a unit of relief supply; |
${C}_{cj}^{M}$: | Transfer cost for the infected people from the fever clinic c to the TH j; |
${C}_{ch}^{S}$: | Transfer cost for a critical patient from the fever clinic c to the DT h; |
$C{V}_{ic}^{R}$: | Transport fees for one vehicle from the TRDC i to the fever clinic c; |
$C{R}_{i}$: | Maximum capacity of the TRDC i; |
$Cav$: | Capacity of a vehicle to carry the relief supplies; |
$C{m}_{j}$: | Maximum capacity of the TH j; |
$C{s}_{h}$: | Maximum capacity of the DH h; |
$\tilde{I}{p}_{c}^{ts}$: | Number of infected people at the fever clinic c in scenario s; |
$\tilde{C}{p}_{c}^{ts}$: | Number of critical patients at the fever clinic c in scenario s; |
${\varphi}^{S}$: | Priority of satisfying the critical patient; |
$\alpha $: | Confidence levels of the chance-constrained model; |
${M}_{big}$: | A large positive number. |
${Y}_{i}$: | 1, if the TRDC i is established, 0 else; |
${Z}_{j}^{s}$: | 1, if the TH j is established under scenario s, 0 else; |
${K}_{h}^{s}$: | 1, if the designated hospital h be established under scenario s, 0 else; |
$V{N}_{i}$: | Amount of inventory at the TRDC i; |
${X}_{ic}^{ts}$: | Number of relief supplies transported from the TRDC i to the fever clinic c in scenario s. |
3.3. Model Formulation
3.4. A Chance-Constrained Model for Patient Transfer and Relief Distribution
4. Solution Algorithm
4.1. ε-Constraint Algorithm
- Step 1:
- Calculate a payoff table$$\left(\begin{array}{c}\begin{array}{cccc}{f}_{2}\left({x}_{1}^{\ast}\right)& {f}_{3}\left({x}_{1}^{\ast}\right)& ...& {f}_{m}\left({x}_{1}^{\ast}\right)\end{array}\\ ...\\ \begin{array}{cccc}{f}_{2}\left({x}_{2}^{\ast}\right)& {f}_{3}\left({x}_{3}^{\ast}\right)& ...& {f}_{m}\left({x}_{m}^{\ast}\right)\end{array}\end{array}\right)$$$${x}_{i}^{\ast}={\mathrm{arg}}_{x\in X}\mathrm{min}{f}_{i}\left(x\right),i=1,2,...,m$$
- Step 2:
- Set ${\overline{\epsilon}}_{i}={f}_{i}\left({x}_{i}^{\ast}\right),{\underset{\xaf}{\epsilon}}_{i}={f}_{i}\left({x}_{1}^{\ast}\right),i=1,2,...,m$ a
- Step 3:
- $P=\left\{{x}_{1}^{\ast},{x}_{2}^{\ast},...,{x}_{m}^{\ast}\right\},F=\left\{{f}_{1}\left({x}_{1}^{\ast}\right),{f}_{2}\left({x}_{2}^{\ast}\right),...,{f}_{m}\left({x}_{m}^{\ast}\right)\right\}$
- Step 4:
- Solve ${x}^{\ast}=opt\left({f}_{1},{\epsilon}_{2{j}_{2}},{\epsilon}_{3{j}_{3}},...,{\epsilon}_{m{j}_{m}}\right)$$${f}_{2}\left(X\right)\le {\epsilon}_{2{j}_{2}}\text{}\mathrm{for}\text{}{j}_{2}=0:1:{q}_{2}\phantom{\rule{0ex}{0ex}}{f}_{3}\left(X\right)\le {\epsilon}_{3{j}_{3}}\text{}\mathrm{for}\text{}{j}_{3}=0:1:{q}_{3}\phantom{\rule{0ex}{0ex}}\dots \phantom{\rule{0ex}{0ex}}{f}_{m}\left(X\right)\le {\epsilon}_{m{j}_{m}}\text{}\mathrm{for}\text{}{j}_{m}=0:1:{q}_{m}$$
- Step 5:
- $P=P\cup \left\{{x}^{\ast}\right\},F=F\cup \left\{{f}_{1}\left({x}^{\ast}\right),{f}_{2}\left({x}^{\ast}\right),...,{f}_{m}\left({x}^{\ast}\right)\right\}$
- Step 6:
- Return P and F
4.2. An Improved PICEA-g Algorithm
4.2.1. An Adaptive K-Nearest Neighborhood Method Based on a Novel Similarity Distance
- The similarity of the locations of emergency facilities
- 2
- The similarity between distribution planning and transfer planning
Algorithm 1: Framework of the K-nearest neighborhood method |
Input: solution S, neighborhood size K_{1} and K_{2} |
1. set the current neighborhood size K_{c} and K_{1}; |
2.for$gen=1:genMax$; |
3. Find non-dominated solutions _{NS} in _{S}; |
4. Generate offspring solutions _{OS} based on neighborhood size K_{c}; |
5. if$\mathrm{mod}(gen,5)==0$ |
6. Find non-dominated solutions _{NOS} in _{OS}; |
7. if${x}^{\prime}\prec x,\text{}x\in S,\text{}{x}^{\prime}\in OS$ |
8. ${K}_{c}={K}_{c}-1$; |
9. else |
10. ${K}_{c}={K}_{c}+1$; |
11. end if |
12. end if |
13. if${K}_{c}>{K}_{1}$ |
14. ${K}_{c}={K}_{1}$; |
15. else |
16. ${K}_{c}={K}_{2}$ |
17. end if |
18.end for |
4.2.2. The PICEA-g-AKNN Algorithm
- 1
- We define a dominant individual if the solution dominates all of the neighborhoods. It is probable that the dominant individual is close to the Pareto front. Therefore, an SBX local search strategy is used to modify this individual. The SBX local search strategy not only enables the dominant individual to move close to the Pareto fronts but also does not cause a large disturbance for the outstanding individual. Let $Pc$ is the probability of executing SBX and the SBX is described as Equation (46).$$c{x}_{il}=\{\begin{array}{c}0.5\cdot \left[\left(1+\beta \right){x}_{il}+\left(1-\beta \right){x}_{jl}\right],\text{}if\text{}rc\le Pc\\ 0.5\cdot \left[\left(1+\beta \right){x}_{il}-\left(1-\beta \right){x}_{jl}\right],\text{}otherwise\end{array}$$$$\beta =\{\begin{array}{c}\text{}{\left(2\cdot rand\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+\eta $}\right.},\text{}if\text{}rand\le 0.5\\ {\left[1/\left(2-2\cdot rand\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1+\eta $}\right.},\text{}otherwise\end{array}$$
- 2
- A solution is defined as an exploring individual if it is not dominated by any neighborhood and there are other non-dominant individuals in the neighborhood. The exploring individual is likely to obtain useful information from the non-dominant individual in the neighborhood. We use a classic and robust DE operator named “DE/rand/1” to generate offspring. Because more parents can be referenced in the DE, children individuals can exchange excellent information with individuals in the neighborhood [59]:$$c{x}_{il}={x}_{il}+F\cdot \left({x}_{r1,l}-{x}_{r2,l}\right)$$
- 3
- A solution is defined as a learning individual if it is dominated by individuals in the neighborhood. For the learning individual, it is necessary to try to learn from the outstanding individuals in the neighborhood. A directional search DE named “DE/current-to-dominance” has good global search ability and is used to generate offspring [60].$$c{x}_{il}={x}_{il}+K\cdot \left({x}_{dl}-{x}_{il}\right)+F\cdot \left({x}_{r1,l}-{x}_{r2,l}\right)$$
5. Computational Examinations
5.1. Simulation of Forecasting Phase
5.2. Simulation of the Second Stage
5.3. Computational Performance Analysis
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Parameters | r | β | q | α | δ_{I} | δ_{q} | γ_{I} | γ_{H} |
---|---|---|---|---|---|---|---|---|
2 | 0.045 | 1.0 × 10^{−6} | 2.7 × 10^{−4} | 0.13 | 0.13 | 0.007 | 0.014 |
Affected Areas | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
N | 150,005 | 113,553 | 175,424 | 226,502 | 200,740 | 164,807 | 187,551 | 159,922 | 210,337 | 209,542 |
E(0) | 125 | 131 | 320 | 225 | 477 | 205 | 320 | 103 | 98 | 155 |
I(0) | 5 | 3 | 9 | 8 | 5 | 7 | 9 | 4 | 7 | 3 |
R(0) | 37 | 41 | 33 | 36 | 40 | 23 | 35 | 20 | 15 | 32 |
H(0) | 1 | 3 | 5 | 4 | 2 | 5 | 2 | 3 | 0 | 1 |
S_{I}(0) | 219 | 178 | 409 | 291 | 611 | 321 | 400 | 184 | 129 | 233 |
E_{I}(0) | 139 | 171 | 230 | 180 | 320 | 189 | 306 | 99 | 102 | 174 |
dI(t)/dH(t) | Period | ||||||
---|---|---|---|---|---|---|---|
Scenario | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
1 | 15/4 | 28/8 | 38/11 | 46/13 | 51/15 | 55/16 | 57/17 |
2 | 15/5 | 31/10 | 45/14 | 56/18 | 64/21 | 70/23 | 77/25 |
3 | 15/4 | 30/9 | 43/13 | 54/16 | 61/18 | 67/19 | 70/20 |
4 | 25/7 | 47/14 | 65/18 | 78/22 | 88/25 | 94/27 | 98/29 |
${\mathit{c}}_{\mathit{n}\mathit{m}}/{\mathit{t}}_{\mathit{n}\mathit{m}}$ | Fever Clinic | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
TRDC | 1 | 5/1.2 | 5/1.1 | 6/2.0 | 7/1.6 | 3/1.1 | 6/1.5 | 4/2.2 | 7/3.0 | 9/2.5 | 4/1.3 |
2 | 7/2.0 | 4/2.2 | 5/2.3 | 3/1.5 | 2/1.0 | 5/1.3 | 5/1.3 | 6/1.5 | 4/1.8 | 6/2.4 | |
3 | 4/1.5 | 5/1.7 | 3/1.0 | 5/1.5 | 6/2.0 | 3/1.1 | 7/2.6 | 4/1.6 | 6/1.7 | 7/1.9 | |
TH | 1 | 6/1.6 | 7/1.9 | 5/1.5 | 4/1.7 | 6/2.4 | 5/2.2 | 8/2.5 | 3/1.0 | 5/1.1 | 7/1.4 |
2 | 7/2.3 | 6/2.2 | 6/3.0 | 3/1.2 | 4/1.2 | 6/1.8 | 5/2.0 | 7/2.0 | 5/1.5 | 6/1.4 | |
3 | 5/1.5 | 8/2.0 | 7/2.2 | 8/2.7 | 4/2.0 | 6/1.8 | 7/2.1 | 5/1.8 | 9/2.9 | 7/3.0 | |
DH | 1 | 5/1.2 | 6/2.0 | 3/1.4 | 6/1.8 | 4/1.1 | 7/1.5 | 5/1.1 | 8/2.4 | 3/1.1 | 8/2.0 |
2 | 7/1.9 | 5/1.5 | 7/2.6 | 6/2.5 | 5/1.4 | 8/2.0 | 8/3.2 | 6/1.5 | 3/2.0 | 7/2.3 | |
3 | 5/2.4 | 6/2.1 | 4/1.5 | 6/1.9 | 7/2.0 | 3/1.1 | 6/1.6 | 9/2.0 | 5/1.8 | 4/1.1 |
Capacity | Fixed Cost | ||
---|---|---|---|
TH | 1 | 2000 | 30,000 |
2 | 3000 | 40,000 | |
3 | 3000 | 44,000 | |
DH | 1 | 900 | 30,000 |
2 | 1000 | 50,000 | |
3 | 800 | 40,000 |
Fixed Cost | Variable Cost | ||
---|---|---|---|
TRDC | 1 | 30,000 | 1.2 |
2 | 25,000 | 1.5 | |
3 | 20,000 | 1.7 |
Scenario | TRDC | Fever Clinic | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
1 | 1 | 336 | 29 | 15 | 837 | 460 | |||||
2 | 35 | 456 | 652 | 521 | 931 | 513 | 279 | 274 | |||
3 | |||||||||||
2 | 1 | 263 | 51 | 46 | 18 | 69 | 41 | 713 | 57 | 26 | 393 |
2 | 160 | 380 | 630 | 527 | 820 | 495 | 67 | 252 | 259 | 71 | |
3 | |||||||||||
3 | 1 | 347 | 36 | 77 | 32 | 72 | 620 | 38 | 409 | ||
2 | 56 | 390 | 710 | 499 | 861 | 441 | 181 | 237 | 260 | 24 | |
3 | |||||||||||
4 | 1 | 403 | 17 | 21 | 114 | 666 | 94 | 19 | 393 | ||
2 | 102 | 389 | 766 | 559 | 828 | 417 | 58 | 192 | 258 | 83 | |
3 |
Scenario | TH | Fever Clinic | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
1 | 1 | 311 | 377 | 559 | 240 | 234 | 382 | ||||
2 | 433 | 760 | 441 | 469 | |||||||
3 | |||||||||||
2 | 1 | 382 | 398 | 594 | 288 | 263 | |||||
2 | 507 | 829 | 496 | 719 | 429 | ||||||
3 | |||||||||||
3 | 1 | 362 | 390 | 756 | 248 | 244 | |||||
2 | 470 | 791 | 478 | 714 | 394 | ||||||
3 | |||||||||||
4 | 1 | 418 | 781 | 295 | 274 | ||||||
2 | 541 | 866 | 266 | 752 | 435 | ||||||
3 | 523 | 283 |
Scenario | DH | Fever Clinic | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||
1 | 1 | 98 | 80 | 167 | 44 | 229 | 207 | 75 | |||
2 | |||||||||||
3 | 38 | 4 | 91 | 138 | 76 | 119 | |||||
2 | 1 | 134 | 212 | 251 | 221 | 82 | |||||
2 | |||||||||||
3 | 125 | 19 | 155 | 155 | 88 | 134 | |||||
3 | 1 | 115 | 12 | 182 | 54 | 240 | 218 | 79 | |||
2 | |||||||||||
3 | 112 | 90 | 147 | 79 | 124 | ||||||
4 | 1 | 161 | 165 | 262 | 227 | 85 | |||||
2 | 131 | 90 | |||||||||
3 | 73 | 195 | 168 | 135 |
Instance | |I| | |C| | |J| | |H| | |T| | |S| |
---|---|---|---|---|---|---|
T-4-20-4-4-7-4 | 4 | 20 | 4 | 4 | 7 | 4 |
T-4-20-4-4-7-8 | 4 | 20 | 4 | 4 | 7 | 8 |
T-4-20-4-4-7-12 | 4 | 20 | 4 | 4 | 7 | 12 |
T-4-20-4-4-14-4 | 4 | 20 | 4 | 4 | 14 | 4 |
T-5-20-4-4-14-8 | 4 | 20 | 4 | 4 | 14 | 8 |
T-5-20-4-4-14-12 | 4 | 20 | 4 | 4 | 14 | 12 |
T-5-30-5-5-7-4 | 5 | 30 | 5 | 5 | 7 | 4 |
T-5-30-5-5-7-8 | 5 | 30 | 5 | 5 | 7 | 8 |
T-5-30-5-5-7-12 | 5 | 30 | 5 | 5 | 7 | 12 |
T-5-30-5-5-14-4 | 5 | 30 | 5 | 5 | 14 | 4 |
T-5-30-5-5-14-8 | 5 | 30 | 5 | 5 | 14 | 8 |
T-5-30-5-5-14-12 | 5 | 30 | 5 | 5 | 14 | 12 |
T-8-50-8-8-7-4 | 8 | 50 | 8 | 8 | 7 | 4 |
T-8-50-8-8-7-8 | 8 | 50 | 8 | 8 | 7 | 8 |
T-8-50-8-8-7-12 | 8 | 50 | 8 | 8 | 7 | 12 |
T-8-50-8-8-14-4 | 8 | 50 | 8 | 8 | 14 | 4 |
T-8-50-8-8-14-8 | 8 | 50 | 8 | 8 | 14 | 8 |
T-8-50-8-8-14-12 | 8 | 50 | 8 | 8 | 14 | 12 |
Initial Values | N | E(0) | I(0) | R(0) | H(0) | Sq(0) | Eq(0) |
---|---|---|---|---|---|---|---|
[15,000~30,000] | [100~200] | [0~10] | [0~10] | [0~10] | [100~200] | [100~200] |
Parameters | PICEA-g-ANK | PICEA-g | MOEA/D | NSGA-II |
---|---|---|---|---|
Maximum generations maxGen Population size N Number of goal vectors Ng Probability pc Probability pm | 5000 100 100 0.8 0.2 | 5000 100 100 0.8 0.2 | 5000 100 - 0.8 0.2 | 5000 100 - 0.8 0.2 |
Neighborhood size T | N/20-N/5 | - | 20 | - |
Instance | $\mathit{C}\left(\mathit{A},\mathit{B}\right)$ | $\mathit{C}\left(\mathit{B},\mathit{A}\right)$ | $\mathit{C}\left(\mathit{A},\mathit{C}\right)$ | $\mathit{C}\left(\mathit{C},\mathit{A}\right)$ | $\mathit{C}\left(\mathit{A},\mathit{D}\right)$ | $\mathit{C}\left(\mathit{D},\mathit{A}\right)$ |
---|---|---|---|---|---|---|
T-4-20-4-4-7-4 | 0.213 | 0.253 | 0.202 | 0.183 | 0.212 | 0.013 |
T-4-20-4-4-7-8 | 0.266 | 0.135 | 0.214 | 0.288 | 0.302 | 0.044 |
T-4-20-4-4-7-12 | 0.184 | 0.117 | 0.25 | 0 | 0.360 | 0.161 |
T-4-20-4-4-14-4 | 0.20 | 0.304 | 0.202 | 0.104 | 0.105 | 0.255 |
T-5-20-4-4-14-8 | 0.496 | 0.121 | 0.24 | 0.166 | 0.406 | 0 |
T-5-20-4-4-14-12 | 0.211 | 0.058 | 0.057 | 0.204 | 0.237 | 0.187 |
T-5-30-5-5-7-4 | 0.419 | 0.013 | 0.358 | 0.072 | 0.023 | 0.303 |
T-5-30-5-5-7-8 | 0.168 | 0.263 | 0.072 | 0.227 | 0.042 | 0.164 |
T-5-30-5-5-7-12 | 0.283 | 0.105 | 0.299 | 0.056 | 0.183 | 0.133 |
T-5-30-5-5-14-4 | 0.218 | 0.152 | 0.195 | 0.012 | 0.153 | 0.389 |
T-5-30-5-5-14-8 | 0.285 | 0.151 | 0.158 | 0.221 | 0.248 | 0.242 |
T-5-30-5-5-14-12 | 0.480 | 0 | 0.184 | 0.118 | 0.223 | 0.294 |
T-8-50-8-8-7-4 | 0.515 | 0.124 | 0.231 | 0.132 | 0.171 | 0.40 |
T-8-50-8-8-7-8 | 0.317 | 0.15 | 0.147 | 0.294 | 0.297 | 0.145 |
T-8-50-8-8-7-12 | 0.275 | 0.164 | 0 | 0.412 | 0.194 | 0.290 |
T-8-50-8-8-14-4 | 0.360 | 0.218 | 0.159 | 0.113 | 0.315 | 0.047 |
T-8-50-8-8-14-8 | 0.581 | 0 | 0.125 | 0.177 | 0.307 | 0.031 |
T-8-50-8-8-14-12 | 0.338 | 0.081 | 0.149 | 0.033 | 0.238 | 0.155 |
Average | 0.3629 | 0.1506 | 0.2026 | 0.1758 | 0.2510 | 0.2031 |
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Long, S.; Zhang, D.; Li, S.; Li, S. Two-Stage Multi-Objective Stochastic Model on Patient Transfer and Relief Distribution in Lockdown Area of COVID-19. Int. J. Environ. Res. Public Health 2023, 20, 1765. https://doi.org/10.3390/ijerph20031765
Long S, Zhang D, Li S, Li S. Two-Stage Multi-Objective Stochastic Model on Patient Transfer and Relief Distribution in Lockdown Area of COVID-19. International Journal of Environmental Research and Public Health. 2023; 20(3):1765. https://doi.org/10.3390/ijerph20031765
Chicago/Turabian StyleLong, Shengjie, Dezhi Zhang, Shuangyan Li, and Shuanglin Li. 2023. "Two-Stage Multi-Objective Stochastic Model on Patient Transfer and Relief Distribution in Lockdown Area of COVID-19" International Journal of Environmental Research and Public Health 20, no. 3: 1765. https://doi.org/10.3390/ijerph20031765