EpidemicLogistics Network Considering Time Windows and Service Level
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Abstract
In this chapter, we present two optimization models for optimizing the epidemiclogistics network. In the first one, we formulate the problem of emergency materials distribution with time windows to be a multiple traveling salesman problem. Knowledge of graph theory is used to transform the MTSP to be a TSP, then such TSP route is analyzed and proved to be the optimal Hamilton route theoretically. Besides, a new hybrid genetic algorithm is designed for solving the problem. In the second one, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level. We formulate the problem to be a mixedinteger nonlinear programming model and develop an effective algorithm to solve the model. In this chapter, we present two optimization models for optimizing the epidemiclogistics network. In the first one, we formulate the problem of emergency materials distribution with time windows to be a multiple traveling salesman problem. Knowledge of graph theory is used to transform the MTSP to be a TSP, then such TSP route is analyzed and proved to be the optimal Hamilton route theoretically. Besides, a new hybrid genetic algorithm is designed for solving the problem. In the second one, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level. We formulate the problem to be a mixedinteger nonlinear programming model and develop an effective algorithm to solve the model.
In this chapter, we present two optimization models for optimizing the epidemiclogistics network. In the first one, we formulate the problem of emergency materials distribution with time windows to be a multiple traveling salesman problem. Knowledge of graph theory is used to transform the MTSP to be a TSP, then such TSP route is analyzed and proved to be the optimal Hamilton route theoretically. Besides, a new hybrid genetic algorithm is designed for solving the problem. In the second one, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level. We formulate the problem to be a mixedinteger nonlinear programming model and develop an effective algorithm to solve the model. In this chapter, we present two optimization models for optimizing the epidemiclogistics network. In the first one, we formulate the problem of emergency materials distribution with time windows to be a multiple traveling salesman problem. Knowledge of graph theory is used to transform the MTSP to be a TSP, then such TSP route is analyzed and proved to be the optimal Hamilton route theoretically. Besides, a new hybrid genetic algorithm is designed for solving the problem. In the second one, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level. We formulate the problem to be a mixedinteger nonlinear programming model and develop an effective algorithm to solve the model.
13.1 Emergency Materials Distribution with Time Windows
13.1.1 Introduction
With rapid development of the global economy, a new biological virus can get anywhere around the world in 24 h. Virus which lurked in the forest or other biological environment before, have been forced to face human ecology when its nature ecology environment destroyed, and this would cause some new type diseases such as Marburg hemorrhagic fevers in Angola, SARS in China, Anthrax mail in USA, Ebola in Congo,smallpox and so on. Bioterrorism threats are realistic and it has a huge influence on social stability, economic development and human health. Without question, nowadays the world has become a risk world, filling with all kinds of threaten from both nature and man made.
Economy would always be the most important factor in normal materials distribution network. However, timeliness is much more important in emergency materials distribution network. To form a timeliness emergency logistics network, a scientific and rational emergency materials distribution system should be constructed to cut down the length of emergency rescue route and reduce economic loss.
In 1990s, America had invested lots of money to build and perfect the emergency warning defense system of public health, aiming to defense the potential terrorism attacks of biology, chemistry and radioactivity material. Metropolitan Medical Response System (MMRS) is one of the important parts, which played a crucial role in the “9.11” event and delivered 50 tons medicine materials to New York in 7 h [1]. In October 2001, suffered from the bioterrorism attack of anthrax, the federal medicine reserve storage delivered a great deal of medicine materials to local health departments [2].
Khan et al. [3] considered that the key challenge of antibioterrorism is that people don’t know when, where and in which way they would suffer bioterrorism attack, and what they can do is just using vaccine, antibiotics and medicine to treat themselves after disaster happened. Because of this, Venkatesh and Memish [4] mentioned that what a country most needed to do is to check its preparation for bioterrorism attacks, especially the perfect extent of the emergency logistics network which includes the reserve and distribution of emergency rescue materials, and the emergency response ability to bioterrorism attacks. Other antibioterrorism response relative researches can be found in Kaplan et al. [5].
Emergency materials distribution is one of the major activities in antibioterrorism response. Emergency materials distribution network is driven by the biological virus diffusion network, which has different structures from the general logistics network. Quick response to the emergency demand after bioterrorism attack through efficient emergency logistics distribution is vital to the alleviation of disaster impact on the affected areas, which remains challenges in the field of logistics and related study areas [6].
In the work of Cook and Stephenson [7], importance of logistics management in the transportation of rescue materials was firstly proposed. References Ray [8] and Rathi et al. [9] introduced emergency rescue materials transportation with the aim of minimizing transportation cost under the different constraint conditions. A relaxed VRP problem was formulated as an integer programming model and proved that’s a NPHard problem in Dror er al. [10] Other scholars have also carried out much research on the emergency materials distribution models such as Fiedrich et al. [11], Ozdamar et al. [12] and Tzeng et al. [13].
During the actual process of emergency material distribution, the Emergency Command Center(ECC) would always supply the emergency materials demand points(EMDP) in groups based on the vehicles they have. Besides, each route wouldn’t repeat, which made any demand point get the emergency materials as fast as possible. To the best of our knowledge, this is a very common experience in China. Under the assumption that any demand point would be satisfied after once replenishment, the question researched would be turn into a multiple traveling salesman problem (MTSP) with an immovable origin. In the work of Bektas [14], the author gave a detailed literature review on MTSP from both sides of model and algorithm. Malik et al. [15], Carter and Ragsdale [16] illustrate some more results on how to solve the MTSP.
To summarize, our model differs from past research in at least three aspects. First, nature disaster such as earthquake, typhoons, flood and so on was always used as the background or numerical simulation in the past research, such kind of disaster can disrupt the traffic and lifeline systems, obstructing the operation of rescue machines, rescue vehicles and ambulances. But situation in antibioterrorism system is different, traffic would be normal and the epidemic situation could be controlled with vaccination or contact isolation. Second, to the best of our knowledge, this is the first time to combine research on the biological epidemic model and the emergency materials distribution model, and we assume that emergency logistics network is driven by the biological virus diffusion network. Therefore, it has a different structure from the general logistics network. Third, the new hybrid genetic algorithm we designed and applied in this study is different from all the traditional ways, we improved the twopart chromosome which proposed by Carter and Ragsdale [16], and custom special set order function, crossover function and mutation function, which can find the optimal result effectively.
13.1.2 SIR Epidemic Model
Note that number of the susceptible and the infective persons would be gotten via computer simulation with value of the other parameters preset. Then, the emergency materials each point demanded can be also calculated based on the number of sick person.
13.1.3 Emergency Materials Distribution Network with Time Windows

Step 1: Using SIR epidemic model in Sect. 13.2 to forecast number of the susceptible and infective people, and then, confirm the emergency distribution time in each EMDP;

Step 2: Generate the original population based on the code rule;

Step 3: Using the custom set order function to optimize the original population and make the new population have finer sequence information;

Step 4: Estimate that whether the results satisfy the constraints (4) to (10) in the model, if yes, turn to the next step, if not, delete the chromosome;

Step 5: Using the fitness function to evaluate fitness value of the new population;

Step 6: End one fall and the best one doubled policy are used to copy the population;

Step 7: Crossover the population using the custom crossover function;

Step 8: Mutate the population using the custom mutation function;

Step 9: Repeat the operating procedures (3)–(8) until the terminal condition is satisfied;

Step 10: 10 approximate optimal routes would be found by the new hybrid genetic algorithm and then the best equilibrium solution would be selected by the local search algorithm.
13.1.4 Numerical Tests
From Figs. 13.6 and 13.7, though length of the route in group 9 is the shortest one, it isn’t the best equilibrium solution. In other words, some demand points can be supplied immediately but others should wait for a long time. This is not the objective we pursue. From Fig. 13.7, inside deviation of group 7 is the minimum one, which means route in group 7 is the best equilibrium solution, though it isn’t the shortest route. In other words, all the demand points can be supplied in the minimum time difference at widest possibility. Another problem worthy to be pointed out is that group 10 is the suboptimal to group 7, and this can be used as a candidate choice for commander under the emergency environment.
13.1.5 Discussion
In fact, results in the prior section are too idealized, for we just considered emergency materials distribution at the beginning of the virus diffusion (\( Day = 5 \)) and we assume that each EMDP has the same situation. In fact, it is impossible. Each parameter preset would affect the result at last immensely. Some of them are discussed as follows.
 (1)
Time consumed with different initial size of \( S \)
 (2)
Time consumed with different initial size of \( I \)
 (3)
Time consumed with different initial size of β
Based on the analysis above, we can see that time consumed in the first 30 days always stay in a lower level. It is important information for emergency relief in the antibioterrorism system, which means the earlier the emergency materials distributed, the less affect would be brought by parameters varied. This also answers the actual question that why emergency relief activities always get the best effectiveness at the beginning.
13.1.6 Conclusions
Emergency materials distribution problem with a MTSPTW characteristic in the antibioterrorism system is researched in this study, and the best equilibrium solution is obtained by the new hybrid GA. Modeling the MTSP using the new twopart chromosome proposed has clear advantages over using either of the existing one chromosome or the two chromosome methods. Besides, combined with the SIR epidemic model, relationship between the parameters and the result are discussed at last, which makes methods proposed in this study more practical.
A problem worthy to be pointed out is that the shortest route between any two EMDPs in the new hybrid GA is calculated by Dijkstra algorithm, so, the optimal result would be gotten even if some sections of the roadway are disrupted, which makes applicability range of the method projected in this study expanded. Research on the emergency materials distribution is a very complex work, only some idealized situations are analyzed and discussed in this study, and some other constraints such as loading capacity of the vehicles, death coefficient for disease, distribution mode and so on, which could be directions of further research.
13.2 An Improved LocationAllocation Model for Emergency Logistics Network Design
Emergency logistics network design is extremely important when responding to an unexpected epidemic pandemic. In this study, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level (ESL). We formulate the problem to be a mixedinteger nonlinear programming model (MINLP) and develop an effective algorithm to solve the model. The numerical test shows that our model can provide tangible recommendations for controlling an unexpected epidemic.
13.2.1 Introduction
Over the past decade, various types of diseases have erupted throughout the world, i.e., SARS (2003), human avian influenza (2004), H1N1 (2009), and Ebola (2014–2015). These unconventional diseases not only seriously endanger humanity’s life, but also have significant impacts on economic development. A recent example is the 2014–2015 Ebola pandemic occurring in West Africa, which infected 28,610 individuals, causing 11,300 fatalities and $32.6 billion in economic losses.
To satisfy the emergency demand of epidemic diffusion, an efficient emergency service network, which considers how to locate the regional distribution center (RDC) and how to allocate all affected areas to these RDCs, should be urgently designed. This problem opens a wide range for applying the OR/MS techniques and it has attracted many attentions in recent years.
For example, Ekici et al. [17] proposed a hybrid model, which estimated the spread of influenza and integrated it with a locationallocation model for food distribution in Georgia. Chen et al. [18] proposed a model which linked the disease progression, the related medical intervention actions and the logistics deployment together to help crisis managers extract crucial insights on emergency logistics management from a strategic standpoint. Ren et al. [19] presented a multicity resource allocation model to distribute a limited amount of vaccine to minimize the total number of fatalities due to a smallpox outbreak. He and Liu [20] proposed a timevarying forecasting model based on a modified SEIR model and used a linear programming model to facilitate distribution decisionmaking for quick responses to public health emergencies. Liu and Zhang [21] proposed a timespace network model for studying the dynamic impact of medical resource allocation in controlling the spread of an epidemic. Further, they presented a dynamic decisionmaking framework, which coupled with a forecasting mechanism based on the SEIR model and a logistics planning system to satisfy the forecasted demand and minimize the total operation costs [22]. Anparasan and Lejeune [23] proposed an integer linear programming model, which determined the number, size, and location of treatment facilities, deployed medical staff, located ambulances to triage points, and organized the transportation of severely ill patients to treatment facilities. Büyüktahtakın et al. [24] proposed a mixedinteger programming (MIP) model to determine the optimal amount, timing and location of resources that are allocated for controlling Ebola in WestAfrica. Moreover, literature reviews on OR/MS contributions to epidemic control were conducted in Dasaklis et al. [25], Rachaniotis et al. [26] and Dasaklis et al. [27].
In this study, we propose an improved locationallocation model for emergency resources distribution. We define a new concept of emergency service level (ESL) and then formulate the problem to be a mixedinteger nonlinear programming (MINLP) model. More precisely, our model (1) identifies the optimal number of RDCs, (2) determines RDCs’ locations, (3) decides on the relative scale of each RDC, (4) allocates each affected area to an appropriate RDC, and (5) obtains ESL for the best scenario, as well as other scenarios.
13.2.2 Model Formulation
 (1)
Definition of ESL
 (2)
Mathematic Model
Parameters:
\( I \): Set of SNSs, \( i \in I \).
\( J \): Set of RDCs, \( j \in J \).
\( K \): Set of affected areas, \( k \in K \).
\( \alpha \): Weight coefficient for the two parts of ESL.
\( d_{k} \): Demand for emergency supplies in affected area \( k \).
\( (x_{k} ,y_{k} ) \): Coordinates of affected area \( k \).
\( (x_{i} ,y_{i} ) \): Coordinates of SNS \( i \).
\( C_{TL} \): Unit transportation cost from SNS to RDC.
\( C_{LTL} \): Unit transportation cost from RDC to affected area.
\( C_{j}^{RDC} \): Cost for operating a RDC. It is decided by the relative size of the RDC \( j \).
\( U_{i} \): Supply capacity of SNS \( i \).

Variables:

\( D_{ij} \): Distance from SNS \( i \) to RDC \( j \). For simplify, the Euclidean distance is adopted.

\( D_{jk} \): Distance from RDC \( j \) to affected area \( k \).

\( \varepsilon_{jk} \): Binary variable. If RDC \( j \) provides emergency supplies to affected area \( k \), \( \varepsilon_{jk} = 1 \); otherwise, \( \varepsilon_{jk} = 0 \).

\( z_{j} \): Binary variable. If RDC \( j \) is opened, \( z_{j} = 1 \); otherwise, \( z_{j} = 0 \).

\( x_{jk} \): Amount of emergency supplies from RDC \( j \) to affected area \( k \).

\( y_{ij} \): Amount of emergency supplies from SNS \( i \) to RDC j.

\( (x_{j} ,y_{j} ) \): Coordinates of RDC \( j \).
Constraint (13.21) indicates that each affected area is serviced by a single RDC. Constraint (13.22) specifies that the supplies to each affected area should not be more than its demand. Constraint (13.23) is a flow conservation constraint. Constraint (13.24) shows that only the opened RDC can provide distribution service. Constraint (13.25) specifies the upper bound of RDC number. Constraint (13.26) is the supply capacity constraint of each SNS. Finally, constraints (13.27)–(13.29) are variables constraints.
13.2.3 Solution Procedure
The proposed model for emergency services network design is a MINLP model as it involves multiplication of two variables (i.e., \( \varepsilon_{jk} x_{jk} \)). More importantly, the optimization model includes a continuous facility locationallocation model with unknown number of RDCs. To avoid the complexity of such MINLP model, we modify it by adding two auxiliary variables. The detail of the modification was introduced in McCormick [28]. Our solution procedure integrates an enumeration search rule and a genetic algorithm (GA), which are applied iteratively. As GA is a mature algorithm [29], details of the GA process are omitted here. We summarize the proposed solution methodology as below.

Step 1: Data input and parameters setting, which includes I, J, K, α, d_{K}, \( (x_{k} ,y_{k} ),(x_{i} ,y_{i} ),C_{TL} ,C_{LTL} ,\,{\text{and}}\,C_{j}^{RDC} \) and the related parameters for GA.

Step 2: Initialization. Generate the original population according to the constraints.

Step 3: Evaluation. Fitness function is defined as the reciprocal of ESL.

Step 4: Selection. Use roulette as the select rule.

Step 5: Crossover. Singlepoint rule is used.

Step 6: Mutation. A random mutation is applied.

Step 7: If termination condition is met, go to the next Step; else, return to Step 4.

Step 8: Output the results.
13.2.4 Numerical Test
 (1)
Data Setting
 (2)
Test Results
Location and relative scale of each RDC
RDC  Location  Relative scale (%)  Affected area 

1  (18.1651, 33.8696)  33.23  2, 7, 9 
2  (38.2318, 39.6607)  10.24  10 
3  (75.9550, 37.1063)  21.24  4, 6 
4  (61.6731, 93.3449)  13.41  1, 5 
5  (48.1101, 84.0045)  21.88  3, 8 
The proportion of demand satisfaction in each affected area
Number  Affected areas  Demand  Supply  Proportion (%) 

1  (81.5, 15.7)  141  98  69.5 
2  (90.6, 89)  149  149  100 
3  (31.7, 85.7)  158  158  100 
4  (48.5, 31.3)  170  170  100 
5  (3.2, 70)  188  130  69.15 
6  (8.7, 4.2)  191  191  100 
7  (27.8, 42.1)  208  208  100 
8  (54.7, 91.6)  214  214  100 
9  (55.8, 79.2)  208  208  100 
10  (36.4, 26)  233  174  74.68 
 (3)
Sensitivity Analysis
 (1)
Impact of α on the ESL
Sensitivity analysis on weight of ESL
α  \( {\text{ESL}}_{1} \)  Proportion (%)  \( {\text{ESL}}_{ 2} \)  Proportion (%)  ESL 

0.4  0.3537  88.425  0.5466  91.1  0.9003 
0.5  0.4381  87.62  0.4839  96.78  0.9220 
0.6  0.5398  89.96  0.386  96.5  0.9258 
0.7  0.6378  89.99  0.2822  94.07  0.9200 
0.8  0.7269  90.86  0.1876  93.8  0.9145 
0.9  0.8182  90.91  0.0914  91.4  0.9096 
 (2)
Sensitivity analysis on different demand in each affected area
13.2.5 Conclusions
In this study, we propose an improved locationallocation model with an emphasis on maximizing the emergency service level (ESL). We formulate the problem to be a mixedinteger nonlinear programming model and develop an effective algorithm to solve the model. Moreover, we test our model through a case study and sensitivity analysis. The main contribution of this research is the function of ESL, which considers demand satisfaction and emergency operation cost simultaneously. Our definition of ESL is different from the existing literature and has a significant meaning for guiding the actual operations in emergency response. Future studies could address the limitations of our work in both the disease forecasting and logistics management. For example, the dynamics of epidemic diffusion could be considered and thus our optimization problem can be extended to a dynamic programming model.
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