# Location Allocation Collaborative Optimization of Emergency Temporary Distribution Center under Uncertainties

Authors: Xiangyang Ren; Juan Tan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Emergencies cause uncertainty in supply chain environment and risks of disruption. Mitigating such risks in emergency supply chain relies on efficient relief material distribution, and in the distribution logistics system, emergency facility location interacts with material allocation clearly. This paper aims to provide a collaborative optimization for the location allocation of temporary emergency distribution centers, with objectives of minimizing rescue time and maximizing demand satisfaction rate. A location allocation model of emergency logistics is formulated by considering uncertain demand and supply information at the response stage of disaster relief. The model is solved by a plant growth simulation algorithm. At last, the feasibility and effectiveness of the model and algorithm in practical application are verified by evaluating a case of COVID-19 prevention and control in Handan city. This paper provides references for decision makers to accomplish the location allocation of emergency facilities and material distribution when dealing with actual situations.


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## Body

## 1. Introduction

Situations like emergency events such as a natural calamity or epidemic strike with no warning lead to a large degree of uncertainty in the supply chain environment such as a dramatic rise in the demand of emergency supplies. Such cases are often followed by supply shortage and risk of supply chain disruption. Therefore, implementing an efficient distribution plan of limited supplies to the affected areas by the local authority is critical in reducing morbidity and mortality [1]. The key decisions in setting up such a distribution plan include the number and locations of the distribution centers to be opened and the supplies and demand points assigned to each location [2].While planning for an emergency scenario, facility location combining material allocation simultaneously optimizes supply chain and guarantees timely material delivery [3]. There exists an amount of literature on facility location allocation problems dealing with a response to an emergency. Sherali et al. proposed a nonlinear mixed-integer programming model; the model selected a set of candidate shelters from given admissible alternatives with available resources and presented an evacuation plan with an objective of minimizing the total congestion-related evacuation time; a heuristic and an exact implicit enumeration algorithm were developed to solve the model [4]. Wei et al. designed an assignment system to support vehicles from candidate locations to deliver relief supplies to disaster-affected areas; the aim of this system was to find transfer depots to open, number of vehicles to use, and relief materials to supply; a hybrid ant colony optimization algorithm was used to identify the approximate Pareto frontier [5]. Panchalee et al. considered a shelter location-allocation problem responding to humanitarian relief logistics; objectives were set to minimize total costs, minimize total rescue time, and minimize the number of shelters required, in order to solve the problem. The epsilon constraint (EC) method and goal programming (GP) were employed [6]. Wang and Ma presented a dual objective mixed integer nonlinear programming (MINLP) model to analyze logistics center location and material allocation in urban emergency logistics systems, and a genetic algorithm was applied to obtain Pareto optimal solution [7].The above literature studies have made beneficial explorations in the characteristics, system, model design, and solution algorithm of emergency logistics location, respectively. But these studies locate the emergency problem as a deterministic problem and do not take into account the various uncertainties contained in the emergency problem. Disasters no matter natural or artificial are characterized by uncertainty and unpredictability [8]; therefore, it is more practical to consider uncertainties in the optimization of emergency location allocation problems. Methods often applied to address uncertainties in emergency response problems are stochastic programming, fuzzy theory, and robust optimization. Stochastic programming takes uncertain parameters as known probability distribution functions based on statistical results [9]. Sanci and Daskin proposed a two-stage stochastic programming model, which not only considered the facility location but also the location of restoration equipment; relief item distribution was carried out jointly, uncertainty was captured by incorporating repair times needed to restore the damaged roads, and a sample average approximation method was developed to solve this integrated model [10]. Based on managerial experience and judgment, the fuzzy theory uses a fuzzy set to incorporate imprecise information in a nonprobabilistic sense [11]. By assuming fuzzy demands of emergency supplies, Wang et al. established a fuzzy multiobjective optimization model to solve the vehicle-helicopter joint routing problem [12]. Robust optimization comes from robust control theory and can be regarded as a supplement of stochastic optimization and sensitivity analysis, which is not necessary to know the probability distribution of uncertain parameters [13]. Sun et al. proposed a biobjective robust optimization model to decide the facility location, emergency resource allocation, and casualty transportation plans in a three-level chain. The model considered various uncertainties in demand and was solved with theε-constraint method [14].While there exist lots of literature studies in the area of location allocation of emergency logistics, there have not been many papers considering the uncertainty of both supply and demand simultaneously. What is more, most models do not strictly distinguish emergency facilities as prepositioned or temporary. In this paper, we extend the model in work [7] to a model associated with fuzzy demand and stochastic scarce supply.The structural arrangement of this paper is as follows: Section2 introduces the location allocation model of emergency temporary distribution center, Section 3 discusses the solution approach, Section 4 presents a case study and the result of model solution, and Section 5 gives the research conclusions of this paper.


## 2. Location Allocation Model of Emergency Temporary Distribution Center under Uncertainties

### 2.1. Problem Description

This paper focuses on the emergency response phase. In this phase, the local authority deploys emergency services to protect people and reduce damages within disaster areas. Because the needed relief products are not available locally, those materials have to be dispatched from the production place to the demand points. As there are various relief products from different factories, distribution centers are required to serve as transit points. Although emergency demand is usually supplied through the allocation of resources at prepositioned facilities, prepositioned facilities with limited coverage may not be able to meet the requirements of affected areas by costing longer travel time and additional funds. In such a case, temporary distribution centers located close to demand points can transship relief resources and decrease response time [15]. Constraints in this kind of emergency setting are insufficient and face uncertain supply and difficulty in estimating the emergency demand.Based on this particular context, significant related decisions are to be made as follows:(1)
the number and the position of temporary distribution centers(2)
demand points assigned to each temporary distribution center and supply quality delivered to each demand pointIn order to solve the above problems, this paper extends the mixed integer nonlinear programming (MINLP) model in this work [7]. Two objectives of the MINLP model are minimum total rescue time and maximum satisfaction rate of relief demand.The following assumptions are put forward before constructing the model:(1)
Different transportation modes are not considered; single accessible road transportation is used in this paper.(2)
The centroid of each demand region, representing the demand points, that is demand zones are aggregated into demand points; various needed rescue resources are grouped into generic humanitarian functions.(3)
Working time limit and turnover time in distribution centers are not considered, and there is no vehicle capacity constraint.Based on the above problem description, parameter setting and model formulation are presented in the following study with improvements of stochastic supply and fuzzy demand based on work [7].


### 2.2. Parameter Setting

Parameter settings are shown in Table1.Table 1
Parameter settings.
ImplicationSetUSet of emergency material supply pointu,u=1,2,3,…,U.ISet of emergency distribution centeri,i=1,2,3,…,IJSet of resource demand pointj,j=1,2,3,…,JParameterwjWeight of material demand pointjAuSupply capacity of the emergency material supply pointu, which is a random variablexu,yuPosition coordinate of the emergency material supply pointuxi,yiPosition coordinate of the emergency distribution centerixj,yjPosition coordinate of the emergency material demand pointjrMaximum distribution radius of the emergency distribution centerssuiDistance between material supply pointu and distribution center isijDistance between distribution centeri and material demand point jvuiTransportation speed between material supply pointu and distribution center ivijTransportation speed between distribution centeri and material demand point jtuiTransportation time between material supply pointu and distribution center itijTransportation time between distribution centeri and material demand point jciRelative size of the emergency distribution centeriED˜jMaterial demand of emergency material demand pointj which is a triangular fuzzy numberhjMaterial demand satisfaction rate of material demand pointjZiMaterial reserve capacity of distribution centerimuiQuantity of emergency materials transported from material supply pointu to distribution center imijQuantity of emergency materials transported from distribution centeri to demand point jθjMaterial demand per person of material demand pointjDecision variableXi∈0,1Xi=1 means distribution center set at point i, and Xi=0 means distribution center not set at point iWui∈0,1Wui=1 means materials shipping from supply point u to distribution center iWui=0 means materials not shipping from supply point u to distribution center iWij∈0,1Wij=1 means materials shipping from distribution center i to material demand point j,and Wij=0 means materials not shipping from distribution center i to material demand point j


### 2.3. Multiobjective Model

Based on the above parameters, the location allocation model of temporary emergency distribution centers is established as follows:(1)minZ1=∑i=1I∑j=1JtuiWui+tijWij,(2)maxZ2=∑j=1Jwjhj,(3)s.t.hj=∑i=1IWijmijED˜j,(4)sui=xi−xu2+yi−yu2,(5)sij=xj−xi2+yj−yi2,(6)tui=suivui,(7)tij=sijvij,(8)ci=∑j=1Jmij∑i=1I∑j=1Jmij,∀i∈I,(9)∑i=1IWij=1,∀j∈J,(10)∑i=1IWijmij≤ED˜j,∀j∈J,(11)∑u=1UXimui=∑j=1JWij,∀i∈I,(12)∑i=1I∑j=1JWijmij=min∑u=1U∑j=1Jdj,Wij≤Xi,∀i∈I,j∈J,(13)Wij≤Xi,∀i∈I,j∈J,(14)∑i=1IXi≤I,(15)P∑i=1Imui−EAvarA≤Au−EAvarA≥α,∀u∈U,(16)sij≤r,∀i∈I,j∈J,(17)∑T=1UXimui≤Zi,∀i∈I,(18)Xi,Wui,Wij∈0,1,∀i∈I,j∈J,(19)mui,mij∈R+,∀u∈U,∀i∈I,j∈J.Objective function (1) is to minimize the emergency rescue time, and objective function (2) is to maximize the satisfaction rate of material demand. Constraint (3) calculates the demand satisfaction rate. Constraint (4) is the Euclidean distance from the material supply point u to the emergency distribution center i, and constraint (5) is the Euclidean distance from emergency distribution center i to material demand point j. Constraints (6) and (7) calculate the emergency rescue time. Constraint (8) calculates the relative scale of the emergency distribution center i. Constraint (9) replies that emergency supplies for each demand point are provided by one emergency distribution center only. Constraint (10) indicates that the quantity of emergency supplies provided to each material demand point does not exceed the demand at that demand point. Constraint (11) indicates that the emergency materials received by each emergency distribution center are equal to the number shipped by the emergency distribution center. Constraint (12) shows that if the supply capacity of the material supply point is less than the demand of the material demand point, all the emergency materials shall be transported to the material demand point; otherwise, the emergency rescue materials shall be distributed according to the demand of the emergency demand point. Constraint (13) says that only established emergency distribution centers can deliver emergency supplies to material demand points. Constraint (14) indicates that the number of emergency distribution centers cannot exceed the original upper limit. Constraint (15) indicates that the quantity of emergency supplies provided at each supply point does not exceed its supply capacity; P represents the probability, and α is the confidence level of supply. Constraint (16) indicates that the distances between the material demand point and the emergency distribution center do not exceed its maximum distribution radius. Constraint (17) indicates the storage capacity limits of the emergency distribution center. Constraints (18) and (19) represent the types of decision variables and several state variables.


### 2.4. Model Transformation

Firstly, asAu−EA/varA complies standard normal distribution, the above expression can be further converted as follows:(20)∑i=1Imui≤EA+1−αvarA,∀u∈U.Secondly, a triangular fuzzy number method [16, 17] is adopted in this paper to estimate the emergency material demand ED˜j, and it is converted to the determined value by the weighting method, which is(21)ED˜j=ω1EDjo+ω2EDjl+ω3EDjp.Then, the opposite number of objective functionZ2 is added to objective function Z1, making sure both objectives are the minimum value.(22)minZ1=∑i=1I∑j=1JtuiWui+tijWij−∑j=1Jwjhj.At last, the driving speed of the transport vehicle is assumed to be constant and unchanged; therefore, time can be presented by distance, and expression (22) is updated as follows:(23)minZ1=∑i=1I∑j=1JsuiWui+sijWij−∑j=1Jwjhj.


## 2.1. Problem Description

This paper focuses on the emergency response phase. In this phase, the local authority deploys emergency services to protect people and reduce damages within disaster areas. Because the needed relief products are not available locally, those materials have to be dispatched from the production place to the demand points. As there are various relief products from different factories, distribution centers are required to serve as transit points. Although emergency demand is usually supplied through the allocation of resources at prepositioned facilities, prepositioned facilities with limited coverage may not be able to meet the requirements of affected areas by costing longer travel time and additional funds. In such a case, temporary distribution centers located close to demand points can transship relief resources and decrease response time [15]. Constraints in this kind of emergency setting are insufficient and face uncertain supply and difficulty in estimating the emergency demand.Based on this particular context, significant related decisions are to be made as follows:(1)
the number and the position of temporary distribution centers(2)
demand points assigned to each temporary distribution center and supply quality delivered to each demand pointIn order to solve the above problems, this paper extends the mixed integer nonlinear programming (MINLP) model in this work [7]. Two objectives of the MINLP model are minimum total rescue time and maximum satisfaction rate of relief demand.The following assumptions are put forward before constructing the model:(1)
Different transportation modes are not considered; single accessible road transportation is used in this paper.(2)
The centroid of each demand region, representing the demand points, that is demand zones are aggregated into demand points; various needed rescue resources are grouped into generic humanitarian functions.(3)
Working time limit and turnover time in distribution centers are not considered, and there is no vehicle capacity constraint.Based on the above problem description, parameter setting and model formulation are presented in the following study with improvements of stochastic supply and fuzzy demand based on work [7].


## 2.2. Parameter Setting

Parameter settings are shown in Table1.Table 1
Parameter settings.
ImplicationSetUSet of emergency material supply pointu,u=1,2,3,…,U.ISet of emergency distribution centeri,i=1,2,3,…,IJSet of resource demand pointj,j=1,2,3,…,JParameterwjWeight of material demand pointjAuSupply capacity of the emergency material supply pointu, which is a random variablexu,yuPosition coordinate of the emergency material supply pointuxi,yiPosition coordinate of the emergency distribution centerixj,yjPosition coordinate of the emergency material demand pointjrMaximum distribution radius of the emergency distribution centerssuiDistance between material supply pointu and distribution center isijDistance between distribution centeri and material demand point jvuiTransportation speed between material supply pointu and distribution center ivijTransportation speed between distribution centeri and material demand point jtuiTransportation time between material supply pointu and distribution center itijTransportation time between distribution centeri and material demand point jciRelative size of the emergency distribution centeriED˜jMaterial demand of emergency material demand pointj which is a triangular fuzzy numberhjMaterial demand satisfaction rate of material demand pointjZiMaterial reserve capacity of distribution centerimuiQuantity of emergency materials transported from material supply pointu to distribution center imijQuantity of emergency materials transported from distribution centeri to demand point jθjMaterial demand per person of material demand pointjDecision variableXi∈0,1Xi=1 means distribution center set at point i, and Xi=0 means distribution center not set at point iWui∈0,1Wui=1 means materials shipping from supply point u to distribution center iWui=0 means materials not shipping from supply point u to distribution center iWij∈0,1Wij=1 means materials shipping from distribution center i to material demand point j,and Wij=0 means materials not shipping from distribution center i to material demand point j


## 2.3. Multiobjective Model

Based on the above parameters, the location allocation model of temporary emergency distribution centers is established as follows:(1)minZ1=∑i=1I∑j=1JtuiWui+tijWij,(2)maxZ2=∑j=1Jwjhj,(3)s.t.hj=∑i=1IWijmijED˜j,(4)sui=xi−xu2+yi−yu2,(5)sij=xj−xi2+yj−yi2,(6)tui=suivui,(7)tij=sijvij,(8)ci=∑j=1Jmij∑i=1I∑j=1Jmij,∀i∈I,(9)∑i=1IWij=1,∀j∈J,(10)∑i=1IWijmij≤ED˜j,∀j∈J,(11)∑u=1UXimui=∑j=1JWij,∀i∈I,(12)∑i=1I∑j=1JWijmij=min∑u=1U∑j=1Jdj,Wij≤Xi,∀i∈I,j∈J,(13)Wij≤Xi,∀i∈I,j∈J,(14)∑i=1IXi≤I,(15)P∑i=1Imui−EAvarA≤Au−EAvarA≥α,∀u∈U,(16)sij≤r,∀i∈I,j∈J,(17)∑T=1UXimui≤Zi,∀i∈I,(18)Xi,Wui,Wij∈0,1,∀i∈I,j∈J,(19)mui,mij∈R+,∀u∈U,∀i∈I,j∈J.Objective function (1) is to minimize the emergency rescue time, and objective function (2) is to maximize the satisfaction rate of material demand. Constraint (3) calculates the demand satisfaction rate. Constraint (4) is the Euclidean distance from the material supply point u to the emergency distribution center i, and constraint (5) is the Euclidean distance from emergency distribution center i to material demand point j. Constraints (6) and (7) calculate the emergency rescue time. Constraint (8) calculates the relative scale of the emergency distribution center i. Constraint (9) replies that emergency supplies for each demand point are provided by one emergency distribution center only. Constraint (10) indicates that the quantity of emergency supplies provided to each material demand point does not exceed the demand at that demand point. Constraint (11) indicates that the emergency materials received by each emergency distribution center are equal to the number shipped by the emergency distribution center. Constraint (12) shows that if the supply capacity of the material supply point is less than the demand of the material demand point, all the emergency materials shall be transported to the material demand point; otherwise, the emergency rescue materials shall be distributed according to the demand of the emergency demand point. Constraint (13) says that only established emergency distribution centers can deliver emergency supplies to material demand points. Constraint (14) indicates that the number of emergency distribution centers cannot exceed the original upper limit. Constraint (15) indicates that the quantity of emergency supplies provided at each supply point does not exceed its supply capacity; P represents the probability, and α is the confidence level of supply. Constraint (16) indicates that the distances between the material demand point and the emergency distribution center do not exceed its maximum distribution radius. Constraint (17) indicates the storage capacity limits of the emergency distribution center. Constraints (18) and (19) represent the types of decision variables and several state variables.


## 2.4. Model Transformation

Firstly, asAu−EA/varA complies standard normal distribution, the above expression can be further converted as follows:(20)∑i=1Imui≤EA+1−αvarA,∀u∈U.Secondly, a triangular fuzzy number method [16, 17] is adopted in this paper to estimate the emergency material demand ED˜j, and it is converted to the determined value by the weighting method, which is(21)ED˜j=ω1EDjo+ω2EDjl+ω3EDjp.Then, the opposite number of objective functionZ2 is added to objective function Z1, making sure both objectives are the minimum value.(22)minZ1=∑i=1I∑j=1JtuiWui+tijWij−∑j=1Jwjhj.At last, the driving speed of the transport vehicle is assumed to be constant and unchanged; therefore, time can be presented by distance, and expression (22) is updated as follows:(23)minZ1=∑i=1I∑j=1JsuiWui+sijWij−∑j=1Jwjhj.


## 3. Plant Growth Simulation Algorithm (PGSA)

### 3.1. Principle of PGSA

Literature [18] analyzed the growth mechanism of plants from the perspective of mathematics and established a probabilistic growth model to simulate plant tropism. The main idea can be summarized as follows: after completing the first growth of the initial growth point x0, plants are assumed to generate k growth points with a set of SM=SM1,SM2,…,SMK; the morphactin concentration at each node PM=PM1,PM2,…,PMK is calculated by following formulas:(24)PMi=fx0−fSMi∑i=1qfx0−fSMi+∑j=1lfx0−fSmj,Pmj=fx0−fSMi∑i=1qfx0−fSMi+∑j=1lfx0−fSmj,Where x0 is the initial feasible solution and f (·) is the objective function of each viable growth point. The growth point changes after each growth, which results in the dynamic value change of morphactin concentration; the value of whole plant morphactin concentration adds to 1.(25)∑i=1q∑j=1lPMi+Pmj=1.Once the morphactin concentrationPM=PM1,PM2,…,PMK is determined, the growth point for the next growth can be selected by certain principles. Depending on the tropism, the closer the growth point is to the light source, the greater the probability selected. The morphactin concentration of the (k + 1) growth points constitutes the growth space; the random numbers within [0,1] are systematically generated, and then the growth point corresponding to the random values is chosen as the next growth point. The higher the concentration of morphactin, the greater the probability of being selected. The previously selected growth points will be eliminated from the growth point set SM. Then, the plant continues to grow new branches, which become new growth points and are added to set PM. Repeating this process, the plant growth model can cover the entire growth space and obtain the optimal solution.


### 3.2. Algorithm Steps

The specific algorithm steps of the location allocation problem of emergency distribution center are as follows:Step 1.
Determine an initial solutionx0,Xmin=x0. To improve the quality of the initial solution, ensure its generality and make it close to the optimal solution as much as possible; it can be generated randomly by the greedy method. Set step sizeλ, calculate the optimal value function Fmin=fx0, and set the maximum number of iterations and the frequency of solution repetitions.Step 2.
Takex0 as the initial state, grow new growth points in 2n directions, respectively, according to the step size set in step 1 and calculate the function value of each growth point.Step 3.
Select the growth points which are better than the initial value, calculate their morphogen concentration, and retain the best growth point.Step 4.
Establish the probability space between [0,1] and use the random number generated by computer to select the growth point of the next growth.Step 5.
If no new branches are generated when the optimal value appears repeatedly to the set frequency or the number of loop iterations reaches the set maximum number of iterations, the growth process ends; otherwise, return to step 3.
The flowchart of the above steps is shown in Figure1.Figure 1
PGSA flowchart.


## 3.1. Principle of PGSA

Literature [18] analyzed the growth mechanism of plants from the perspective of mathematics and established a probabilistic growth model to simulate plant tropism. The main idea can be summarized as follows: after completing the first growth of the initial growth point x0, plants are assumed to generate k growth points with a set of SM=SM1,SM2,…,SMK; the morphactin concentration at each node PM=PM1,PM2,…,PMK is calculated by following formulas:(24)PMi=fx0−fSMi∑i=1qfx0−fSMi+∑j=1lfx0−fSmj,Pmj=fx0−fSMi∑i=1qfx0−fSMi+∑j=1lfx0−fSmj,Where x0 is the initial feasible solution and f (·) is the objective function of each viable growth point. The growth point changes after each growth, which results in the dynamic value change of morphactin concentration; the value of whole plant morphactin concentration adds to 1.(25)∑i=1q∑j=1lPMi+Pmj=1.Once the morphactin concentrationPM=PM1,PM2,…,PMK is determined, the growth point for the next growth can be selected by certain principles. Depending on the tropism, the closer the growth point is to the light source, the greater the probability selected. The morphactin concentration of the (k + 1) growth points constitutes the growth space; the random numbers within [0,1] are systematically generated, and then the growth point corresponding to the random values is chosen as the next growth point. The higher the concentration of morphactin, the greater the probability of being selected. The previously selected growth points will be eliminated from the growth point set SM. Then, the plant continues to grow new branches, which become new growth points and are added to set PM. Repeating this process, the plant growth model can cover the entire growth space and obtain the optimal solution.


## 3.2. Algorithm Steps

The specific algorithm steps of the location allocation problem of emergency distribution center are as follows:Step 1.
Determine an initial solutionx0,Xmin=x0. To improve the quality of the initial solution, ensure its generality and make it close to the optimal solution as much as possible; it can be generated randomly by the greedy method. Set step sizeλ, calculate the optimal value function Fmin=fx0, and set the maximum number of iterations and the frequency of solution repetitions.Step 2.
Takex0 as the initial state, grow new growth points in 2n directions, respectively, according to the step size set in step 1 and calculate the function value of each growth point.Step 3.
Select the growth points which are better than the initial value, calculate their morphogen concentration, and retain the best growth point.Step 4.
Establish the probability space between [0,1] and use the random number generated by computer to select the growth point of the next growth.Step 5.
If no new branches are generated when the optimal value appears repeatedly to the set frequency or the number of loop iterations reaches the set maximum number of iterations, the growth process ends; otherwise, return to step 3.
The flowchart of the above steps is shown in Figure1.Figure 1
PGSA flowchart.


## 4. Case Study

### 4.1. Case Description

This paper takes the COVID-19 prevention and control of Handan city which was affected by the epidemic in Shijiazhuang in early 2021 as an example to verify the effectiveness of the model and solution algorithm. Eighteen districts and counties in Handan are taken as 18 material demand pointsjj=1,2,3,…,18. By hypothetical conditions, material demand is aggregated in the seats of each district or county government. The weight of each material demand point is determined by the population proportion, and material demand is vaguely uncertain which is estimated by triangular fuzzy number by the population distribution and degree of disaster. According to the most likely value method proposed by Pishvaee et al. [19], ω1=ω3=1/6,ω2=4/6 are taken, the estimated demand is calculated by equation (21). The parameters of each demand point are shown in Table 2. The geographical coordinates of the two supply points uu=1,2 are (113.42, 36.32) and (115.76,36.65), respectively, and the available supply capacity at the first assignment of emergency response corresponds to normal distribution N (3900, 502) and N (4050, 702), respectively, and α is set as 0.01. According to the existing logistics service system in Handan city, decision makers need to select the appropriate number of sites from 6 candidate commercial logistics distribution centers ii=1,2,3,4,5,6 as temporary emergency distribution centers or temporary transit points. The parameters of 6 candidate sites are shown in Table 3.Table 2
Locations and demand of material demand points.
No.Material demand pointjGeographic coordinatexj,yjPopulationWeightwjFuzzy demand of emergency demand pointED˜j1Congtai(114.499307, 36.64270)5721750.065(500, 504, 510)2Hanshan(114.537471, 36.60024)6143030.069(660, 663, 670)3Fuxing(114.468586, 36.64537)3199800.036(310, 319, 320)4Fengfeng(114.239285, 36.42212)4292450.049(420, 422, 430)5Feixiang(114.806635, 36.55425)3722650.042(340, 345, 350)6Yongnian(114.543486, 36.74784)8512270.096(1000, 1090, 1100)7Wu’an(114.210535, 36.70264)8116310.092(900, 907, 910)8Linzhang(114.592178, 36.32938)5890030.067(630, 632, 640)9Cheng’an(114.676567, 36.44983)4010520.045(390, 394, 400)10Daming(115.154601, 36.29208)7263960.082(640, 648, 650)11Shexian(113.698024, 36.59137)3795590.043(360, 364, 370)12Cixian(114.380259, 36.38098)4407280.050(400, 402, 410)13Qiuxian(115.20645, 36.818644)2109980.024(130, 134, 140)14Jize(114.896634, 36.91699)2974250.034(260, 269, 270)15Guangping(114.955115, 36.49018)2640250.030(150, 154, 160)16Guantao(115.287453, 36.55409)3079120.035(320, 321, 330)17Weixian(114.945665, 36.36542)7968060.090(780, 782, 790)18Quzhou(114.963955, 36.77241)4637270.052(450, 455, 460)Table 3
Parameters of candidate distribution centers.
No.Candidate emergency distribution centersGeographic coordinateMaximum distribution radius rMaterial storage capacity1Wu’an(114.840707, 36.8750)0.630002Fengfeng(114.017121, 36.5449)0.740003Fuxing(114.366065, 36.6467)0.740004Feixiang(114.814008, 36.6072)0.640005Jinan xinqu(114.451021, 36.4778)0.535006Guangping(114.105367, 36.4694)0.73500Population data come from the Seventh National Census Bulletin of Hebei Province (No.2).


### 4.2. Test Results

Since the decision variables are 0-1 variable, step size is set asλ = 1, maximum number of iterations as 50, and the frequency of solution repetitions as 3. As there are 6 candidate locations and 18 demand points in this study case, the decision variables are designed as Boolean vectors in the iterative process of PGSA. X=x1,x2,…,x6 indicates whether to open a candidate distribution center; element value of 1 is open while 0 is not. Similarly, Wij decides which demand point the selected distribution center is to serve. Wij=1 indicates that distribution center i serves demand point j, while Wij=0 means not. Table 4 presents the iterative process of the optimal solution of opening 3 distribution centers calculated by MATLAB software.Table 4
The iterative process of PGSA.
No.XWijFunction value1(1 0 1 1 0 0)W1j. = (0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0)W3j. = (0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1)W4j. = (1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0)9.20262(0 1 1 1 0 0)W2j. = (0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)W3j. = (0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1)W4j. = (1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0)8.100250(0 1 1 0 0 1)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0)W6j. = (0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1)6.0748In order to illustrate the solution quality produced by PGSA, a comparison between PGSA and genetic algorithm (GA) is presented in Table5.Table 5
Computing result comparison between PGSA and GA.
AlgorithmXWijFunction valuePGSA(0 1 1 0 0 1)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0)W6j. = (0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1)6.0748GA(0 1 1 1 0 0)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0)W4j. = (0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1)6.2175According to the iterative process of PGSA described above, the optimal solutions of other distribution centers can be obtained in the same way. A comparison of test results with different number of opening distribution centers is presented in Table6 and Figure 2.Table 6
Comparison of test results of distribution centers in number from 2 to 6.
No. of distribution centers23456Distance from supply point to distribution center4.75447.14499.483611.878514.3502Distance from distribution center to demand point5.476710.21367.21147.67937.0549Total distance10.231117.358516.695019.557721.4051Optimal value of objective function6.14496.07486.28977.06437.5516Distribution center4, 23, 2, 63, 2, 5, 65, 2, 4, 3, 65, 1, 2, 3, 4, 6Demand satisfaction rate93.28%93.39%93.39%92.92%92.15%As shown in Table6 and Figure 2, when opening 3 temporary distribution centers, the objective function reaches the optimal value of 6.0748. The total shortest rescue distance (10.2311) is obtained when 2 distribution centers are selected. The highest demand satisfaction rate is achieved at both 3 and 4 locations. Therefore, opening 3 distribution centers is a fair solution. It is obvious that the difference of satisfaction rate among different distribution centers is quite small, which suggests that objective (2) is a weak determining factor when planning material allocation. Therefore, the key to solving unmet demand lies in sufficient supply while weighted allocation may not be very efficient. In addition, when opening 4 distribution centers, the solution has a highest demand satisfaction rate with a relatively short delivery distance. Consequently, the model proposed in this paper has flexibility responding to risk in emergency logistics. The appropriate decision of how many distribution centers to be open can be made when confronting changeable conditions.Figure 2
Comparison of test results of distribution centers in number from 2 to 6.Table7 and Figure 3 present the specific location result of the solution with 3 distribution centers. Figure 4 shows the location layout of other solutions.Table 7
Location result of 3 emergency distribution centers.
No.Geography coordinateDistribution centerRelative sizeNo. of material demand point1(114.366065, 36.646)Fuxing0.47291 2 3 6 7 8 122(114.017121, 36.544)Fengfeng0.09354 113(114.105367, 36.469)Guangping0.43365 9 10 13 14 15 16 17 18Figure 3
Locations of 3 emergency distribution centers.Figure 4
Locations of emergency distribution centers with Nos. 2, 4, 5, and 6.According to Table7, Fuxing distribution center undertakes the emergency material supply of Congtai, Hanshan, Fuxing, Yongnian, Wu’an, Linzhang, and Cixian, while Fengfeng distribution center distributes materials to Fengfeng and Shexian, and the Guangping distribution center is responsible for the material distribution of Feixiang, Cheng’ an, Daming, Qiuxian, Jize, Guangping, Guantao, Weixian, and Quzho. Distribution centers located in Fuxing and Guangping undertake the larger material turnover in the emergency logistics network, with 47.29% and 43.36%, respectively, while distribution center located in Fengfeng undertakes smaller material turnover of 9%.


## 4.1. Case Description

This paper takes the COVID-19 prevention and control of Handan city which was affected by the epidemic in Shijiazhuang in early 2021 as an example to verify the effectiveness of the model and solution algorithm. Eighteen districts and counties in Handan are taken as 18 material demand pointsjj=1,2,3,…,18. By hypothetical conditions, material demand is aggregated in the seats of each district or county government. The weight of each material demand point is determined by the population proportion, and material demand is vaguely uncertain which is estimated by triangular fuzzy number by the population distribution and degree of disaster. According to the most likely value method proposed by Pishvaee et al. [19], ω1=ω3=1/6,ω2=4/6 are taken, the estimated demand is calculated by equation (21). The parameters of each demand point are shown in Table 2. The geographical coordinates of the two supply points uu=1,2 are (113.42, 36.32) and (115.76,36.65), respectively, and the available supply capacity at the first assignment of emergency response corresponds to normal distribution N (3900, 502) and N (4050, 702), respectively, and α is set as 0.01. According to the existing logistics service system in Handan city, decision makers need to select the appropriate number of sites from 6 candidate commercial logistics distribution centers ii=1,2,3,4,5,6 as temporary emergency distribution centers or temporary transit points. The parameters of 6 candidate sites are shown in Table 3.Table 2
Locations and demand of material demand points.
No.Material demand pointjGeographic coordinatexj,yjPopulationWeightwjFuzzy demand of emergency demand pointED˜j1Congtai(114.499307, 36.64270)5721750.065(500, 504, 510)2Hanshan(114.537471, 36.60024)6143030.069(660, 663, 670)3Fuxing(114.468586, 36.64537)3199800.036(310, 319, 320)4Fengfeng(114.239285, 36.42212)4292450.049(420, 422, 430)5Feixiang(114.806635, 36.55425)3722650.042(340, 345, 350)6Yongnian(114.543486, 36.74784)8512270.096(1000, 1090, 1100)7Wu’an(114.210535, 36.70264)8116310.092(900, 907, 910)8Linzhang(114.592178, 36.32938)5890030.067(630, 632, 640)9Cheng’an(114.676567, 36.44983)4010520.045(390, 394, 400)10Daming(115.154601, 36.29208)7263960.082(640, 648, 650)11Shexian(113.698024, 36.59137)3795590.043(360, 364, 370)12Cixian(114.380259, 36.38098)4407280.050(400, 402, 410)13Qiuxian(115.20645, 36.818644)2109980.024(130, 134, 140)14Jize(114.896634, 36.91699)2974250.034(260, 269, 270)15Guangping(114.955115, 36.49018)2640250.030(150, 154, 160)16Guantao(115.287453, 36.55409)3079120.035(320, 321, 330)17Weixian(114.945665, 36.36542)7968060.090(780, 782, 790)18Quzhou(114.963955, 36.77241)4637270.052(450, 455, 460)Table 3
Parameters of candidate distribution centers.
No.Candidate emergency distribution centersGeographic coordinateMaximum distribution radius rMaterial storage capacity1Wu’an(114.840707, 36.8750)0.630002Fengfeng(114.017121, 36.5449)0.740003Fuxing(114.366065, 36.6467)0.740004Feixiang(114.814008, 36.6072)0.640005Jinan xinqu(114.451021, 36.4778)0.535006Guangping(114.105367, 36.4694)0.73500Population data come from the Seventh National Census Bulletin of Hebei Province (No.2).


## 4.2. Test Results

Since the decision variables are 0-1 variable, step size is set asλ = 1, maximum number of iterations as 50, and the frequency of solution repetitions as 3. As there are 6 candidate locations and 18 demand points in this study case, the decision variables are designed as Boolean vectors in the iterative process of PGSA. X=x1,x2,…,x6 indicates whether to open a candidate distribution center; element value of 1 is open while 0 is not. Similarly, Wij decides which demand point the selected distribution center is to serve. Wij=1 indicates that distribution center i serves demand point j, while Wij=0 means not. Table 4 presents the iterative process of the optimal solution of opening 3 distribution centers calculated by MATLAB software.Table 4
The iterative process of PGSA.
No.XWijFunction value1(1 0 1 1 0 0)W1j. = (0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0)W3j. = (0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1)W4j. = (1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 1 1 0)9.20262(0 1 1 1 0 0)W2j. = (0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0)W3j. = (0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1)W4j. = (1 0 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 0)8.100250(0 1 1 0 0 1)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0)W6j. = (0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1)6.0748In order to illustrate the solution quality produced by PGSA, a comparison between PGSA and genetic algorithm (GA) is presented in Table5.Table 5
Computing result comparison between PGSA and GA.
AlgorithmXWijFunction valuePGSA(0 1 1 0 0 1)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0)W6j. = (0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1)6.0748GA(0 1 1 1 0 0)W2j. = (0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0)W3j. = (1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0)W4j. = (0 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1)6.2175According to the iterative process of PGSA described above, the optimal solutions of other distribution centers can be obtained in the same way. A comparison of test results with different number of opening distribution centers is presented in Table6 and Figure 2.Table 6
Comparison of test results of distribution centers in number from 2 to 6.
No. of distribution centers23456Distance from supply point to distribution center4.75447.14499.483611.878514.3502Distance from distribution center to demand point5.476710.21367.21147.67937.0549Total distance10.231117.358516.695019.557721.4051Optimal value of objective function6.14496.07486.28977.06437.5516Distribution center4, 23, 2, 63, 2, 5, 65, 2, 4, 3, 65, 1, 2, 3, 4, 6Demand satisfaction rate93.28%93.39%93.39%92.92%92.15%As shown in Table6 and Figure 2, when opening 3 temporary distribution centers, the objective function reaches the optimal value of 6.0748. The total shortest rescue distance (10.2311) is obtained when 2 distribution centers are selected. The highest demand satisfaction rate is achieved at both 3 and 4 locations. Therefore, opening 3 distribution centers is a fair solution. It is obvious that the difference of satisfaction rate among different distribution centers is quite small, which suggests that objective (2) is a weak determining factor when planning material allocation. Therefore, the key to solving unmet demand lies in sufficient supply while weighted allocation may not be very efficient. In addition, when opening 4 distribution centers, the solution has a highest demand satisfaction rate with a relatively short delivery distance. Consequently, the model proposed in this paper has flexibility responding to risk in emergency logistics. The appropriate decision of how many distribution centers to be open can be made when confronting changeable conditions.Figure 2
Comparison of test results of distribution centers in number from 2 to 6.Table7 and Figure 3 present the specific location result of the solution with 3 distribution centers. Figure 4 shows the location layout of other solutions.Table 7
Location result of 3 emergency distribution centers.
No.Geography coordinateDistribution centerRelative sizeNo. of material demand point1(114.366065, 36.646)Fuxing0.47291 2 3 6 7 8 122(114.017121, 36.544)Fengfeng0.09354 113(114.105367, 36.469)Guangping0.43365 9 10 13 14 15 16 17 18Figure 3
Locations of 3 emergency distribution centers.Figure 4
Locations of emergency distribution centers with Nos. 2, 4, 5, and 6.According to Table7, Fuxing distribution center undertakes the emergency material supply of Congtai, Hanshan, Fuxing, Yongnian, Wu’an, Linzhang, and Cixian, while Fengfeng distribution center distributes materials to Fengfeng and Shexian, and the Guangping distribution center is responsible for the material distribution of Feixiang, Cheng’ an, Daming, Qiuxian, Jize, Guangping, Guantao, Weixian, and Quzho. Distribution centers located in Fuxing and Guangping undertake the larger material turnover in the emergency logistics network, with 47.29% and 43.36%, respectively, while distribution center located in Fengfeng undertakes smaller material turnover of 9%.


## 5. Conclusion

In this paper, we consider the problem of locating temporary distribution centers for relief products in response to disaster and risk emergency scenarios. The main contribution of this study is in designing a response strategy to address the location allocation problem with uncertain demand and supply. In addition, the model proposed in this work offers some managerial insights associated with the number of temporary distribution centers. Their locations and distribution strategy with limited supply in designing and deploying aid distribution networks. Specifically, a biobjective MINLP model with minimum emergency rescue time and maximum demand satisfaction rate is formulated. Then, a plant growth simulation algorithm is employed to solve the model. At last, the feasibility and effectiveness of the model and algorithm are verified in a COVID-19 prevention and control case in Handan city. In general, the model is suitable to the complex problem of multiobjective, multivariable, and nonlinear objective functions with great flexibility and adaptability. The inadequacy of this paper is that it only considers static demand of emergency supplies while the demand in emergency response is periodically changed which calls for the dynamic research of emergency logistics in the future.



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*Source: 6176756-2022-03-19.xml*