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Int. J. Mathematics in Operational Research, Vol. 1, Nos. 1/2, 2009
Optimal material distribution decisions based on epidemic diffusion rule and stochastic latent period for emergency rescue Haiyan Wang and Xinping Wang Institute of Systems Engineering, Southeast University, Nanjing, Jiangsu 210096, P.R. China Fax: + 86 25 52090759 E-mails:
[email protected];
[email protected]
Amy Z. Zeng* Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA Fax: + 1 508 831 5720 E-mail:
[email protected] *Corresponding author Abstract: Demand of emergency materials is usually uncertain and varies quickly as the latent period changes. With the consideration of the delay caused by the latent period of an epidemic, we construct a multi-objective stochastic programming model with time-varying demand for the emergency logistics network based on the epidemic diffusion rule. The genetic algorithm coupled with Monte Carlo simulation is adopted to solve the optimisation model, and the application of the model as well as a sensitivity analysis of the latent period is given by a numerical example. Keywords: emergency logistics; epidemic diffusion rule; genetic algorithm; latent period; optimisation; SEIR model; stochastic programming. Reference to this paper should be made as follows: Wang, H., Wang, X. and Zeng, A.Z. (2009) ‘Optimal material distribution decisions based on epidemic diffusion rule and stochastic latent period for emergency rescue’, Int. J. Mathematics in Operational Research, Vol. 1, Nos. 1/2, pp.76–96. Biographical notes: Haiyan Wang currently is a Full Professor at the Institute of Systems Engineering in Southeast University in Nanjing, China. He obtained his bachelors degree in Applied Mathematics in 1986, masters degree in the same speciality in 1989 and a doctoral degree in System Engineering in 2001, all from the Southeast University. His research interests lie in the areas of supply chain and logistics systems, complexity of social and economic system, and analysis of chaotic time series. Xinping Wang is currently a graduate student pursuing her masters degree in the Institute of Systems Engineering at the Southeast University in Nanjing, China. She obtained her bachelors degree in Information and Computing Science in 2007 from the Southeast University. Her research effort focuses on investigating the dynamic complexity of supply chain systems. Copyright © 2009 Inderscience Enterprises Ltd.
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Amy Z. Zeng has been on the Faculty of the Department of Management at Worcester Polytechnic Institute in Massachusetts, USA since 1999. She earned her doctoral degree from the Smeal College of Business Administration at Pennsylvania State University and an MS in Engineering from the University of Washington. Her expertise lies in the areas of the efficient operations and tactics of supply chain processes. She has published over 60 articles in professional journals, books and conference proceedings. She is also a frequent speaker at various occasions, and a recipient of numerous awards and recognitions for her teaching, research and service.
1
Introduction
The World Health Organisation defines a disaster as any event that causes damage, ecological disruption, loss of human life, deterioration of health and health services on a scale sufficient to warrant an extraordinary response from outside the affected community or area (Haghina and Oh, 1996). Natural disasters such as earthquakes, hurricanes, floods, drought, volcanic eruption, famine, epidemics and others can occur at any time and anywhere. They have catastrophic effects on the society and everyone’s daily life in many aspects such as injuries, property damage and even loss of life. Epidemics have always been one of the strongest enemies threatening human health and life, and the well-being of a national economy. For example, SARS, a serious respiratory infectious disease that started in Guangdong Province of China in September, 2002, swept more than 30 countries and districts, and is called the first plague that humanitarian society has ever experienced in the 21st century. Moreover, due to inefficient implementation of prevention and treatment, some infectious diseases such as cholera, swamp fever, white plague and others have again appeared. Therefore, it remains imperative for every country to be prepared for emergency rescue in the event that an infectious disease breaks out. After an infectious disease occurs, the public officials face with many critical issues, the most important of which is how to ensure the availability and supply of emergency medicines so that the loss of life can be minimised and the efficiency of each rescue can be maximised. As such, emergency logistics is more complex and difficult in meeting the requirements for material supply and distribution, and differs from business logistics in the following aspects. First of all, a disaster usually happens suddenly and causes a surge of demand for a particular medicine during a very short period of time. Hence, the emergency materials must be transported to affected areas as quickly as possible. Second, the demand information is quite limited and varies rapidly with time. It is often very difficult to predict the actual demand based on historical data (and for many disasters, the historical data may not even exist), especially for the epidemics with a stochastic latent period. Third, time is a critical factor in emergency rescue and any delay in transportation can cause more deaths and greater losses. Finally, unlike logistics management in which all the activities are triggered based on customer orders, emergency logistics must deal with sudden and random demand for a particular type of material after a disastrous event occurs. The particularity of various decisions pertinent to emergency rescue opens a wide range of applications of Operations Research/Management Science techniques. For example, Larson (2004) points out a number of topics related to homeland security that
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can be solved by Operations Research models; Green and Kolesar (2004) specifically address how Operations Research/Management Science tools can help improve emergency response operations; Recently, an article by Altay and Green (2006) offers a summary of literature survey that identifies potential research directions in disaster operations by using Operations Research/Management Science techniques. This article presents an application of two Operations Research related techniques, namely, stochastic programming and genetic algorithm, to study the distribution decision in emergency rescue which can be quite challenging due to the reasons listed above. As mentioned earlier, speed and availability of a specific medicine in the event of the breakout of a contagious disease significantly affect the efficiency of a rescue. Unfortunately, the demand for a medicine often changes fast and is influenced by the latent period. Existing studies have not addressed the impact of latent period explicitly on the distribution of medical supplies; therefore, this article studies emergency logistics distribution problem in the context that an epidemic has a stochastic latent period, for example, influenza. Specifically, a stochastic demand for an emergency medical supply caused by the latent period is considered, and a responsive emergency distribution procedure for the entire rescue period until the epidemic is under control is presented. We treat the entire rescue period such that it can be divided into several parts, and we develop a time-varying rescue demand forecast model, a multi-objective stochastic programming model and a consumption time calculation model for each part. Considering both the uncertain and dynamic features of the rescue demand, we propose a unique forecast mechanism to predict the demand in the epidemic area. In addition, we solve the distribution problem based on two objectives: minimising the total transportation costs and minimising the rescue time in each part of the rescue period. The remainder of the article is organised as follows. Section 2 contains a review of the relevant literature from the following two perspectives: one is focused on the prevention and control as well as the modelling methods of epidemics, and the other looks at the analysis of emergency logistics planning and distribution design. Section 3 introduces our framework for the proposed approach, including the time-varying demand forecast model based on the epidemic diffusion rule, the emergency distribution mechanism and a consumption time calculation model. The solution procedure for the models is explained in Section 4. A numerical example and a sample sensitivity analysis are given in Section 5. Finally, the limitations of the proposed models and future research directions are reported in Section 6.
2
Literature review
Considering the relationship between an unexpected event and the associated emergency logistics decisions, we review two streams of recent research efforts here: one is focused on the modelling methods for prevention and control of epidemics, and the other is related to emergency logistics planning and distribution design.
2.1 Previous studies of epidemics Many recent research efforts have been devoted to understanding the prevention and control of epidemics. Numerous mathematical models have been created to analyse and study the general characteristics of every epidemic. According to the propagation
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mechanism of epidemics, examples of epidemic models include SI, SIR, susceptible– infected–susceptible (SIS), SIRS, SEI, SEIR and others. A study by Xu et al. (2006) presents a modified SIS model on complex networks, small-world and scale-free, to study the spread of an epidemic by considering the effect of time delay that is introduced in the infected phase. Based on the topologies of the networks, both uniform and degree-dependent delays are studied during the contagion process. The study has found that the existence of delay enhances both outbreaks and prevalence of infectious diseases in the networks. Meng and Chen (2007) propose a new SIR epidemic model with vertical and horizontal transmissions, and the dynamics of this disease model under constant and pulse vaccination are analysed. Yuan and Yang (2008) develop an SEI model with acute and chronic stages in a population with exponentially varying size. A system with the following two equilibriums is obtained: a disease-free equilibrium and an endemic equilibrium, and the stability of these two equilibriums are studied. Surveys and discussions of the existing SEI models and their extensions can be readily found in articles by Gao, Mena-Lorca and Hethcote (1995) and Li and Zhen (2005). Pulse vaccination is an efficient way applied to prevent and control the epidemics. Using the SIR epidemic model, Shulgin, Stone and Agur (1998) show that under a planned pulse vaccination regime, the system converges to a stable point where the number of infected individuals equals zero. They also show that the pulse vaccination leads to epidemics eradication if certain conditions regarding the magnitude of vaccination proportion and the period of the pulse are met. An effort by Stone, Shulgin and Agur (2000) analyses the rationale of the pulse vaccination strategy in the simple SIR epidemic model. The research has proven that repeatedly vaccinating the susceptible population in a series of ‘pulse’ is possible to eradicate the measles infection from the entire model population. In terms of the structure of the models, most of the existing studies rely on different kinds of differential equations; for example, the first-order partial differential equations are used to integrate the age structures; the second-order partial differential equations are suitable when a diffusion term exists; and the integral differential equations or differential equations are often used when time delay or delay factors are considered. Two articles by Hethcote (1994, 2000) provide a systematic review and summary of the applications of the differential equations in modelling epidemics.
2.2 Emergency logistics planning and distribution design There exists a great deal of research on logistics management, but only a limited amount of literature is dedicated to emergency logistics management. We herein give a brief review of the existing studies. Kemball-Cook and Stephenson (1984) are among the first group of scholars to point out that logistics management is needed to improve the transportation efficiency when rescue materials are transported. By incorporating knowledge-based rules into a linear programming model, Knott (1988) addresses the issue of vehicle scheduling for supplying bulks of food to a disaster area. Fiedich, Gehbauer and Rickers (2000) propose a dynamic combinatorial optimisation model and a heuristic to determine an optimised resource schedule for assigning resources to the affected areas after strong earthquakes occur. A study by Sheu (2006) presents a hybrid fuzzy clustering-optimisation approach to the operation of an emergency logistics system in response to the urgent demand
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during the crucial rescue period. Based on a proposed three-layer conceptual framework, the proposed methodology involves three recursive mechanisms: 1
the forecasting of time-varying relief demand
2
the grouping of the disaster-affected areas
3
the co-distribution of emergency relief.
To deal with the complexity and difficulty in solving the emergency logistics distribution problems, we see a trend in solution methodologies, that is, decomposing the original problem, which can be a multi-commodity, -modal, or -period model, into several mutually correlated sub-problems, and then solve them systematically in the same decision scheme. For instance, Guelat et al. (1990) present a multi-commodity, multi-modal network assignment model for strategic planning with the aim to predict multi-commodity flows over a multi-modal network in the minimal total routing and transfer costs. Another study by Barbarosoglu, Ozdamar and Ahmet (2002) proposes a bi-level hierarchical decomposition approach for helicopter mission planning during a disaster relief operation. The top-level model is formulated to deal with the tactical decisions, covering the issues of helicopter fleet management, crew assignment and the number of tours undertaken by each helicopter. The base-level model aims to address the corresponding operational decisions, including routing, loading/unloading, rescue and re-fueling scheduling. In the third example, Ozdamar, Ekinci and Kucuk yazaci (2004) propose a planning model that is integrated with a natural disaster logistics decision support system. The model is decomposed into two multi-commodity network flow problems with the first being linear (for conventional commodities) and the second being integer (for vehicle flows). In a recent study, Yi and Ozdamar (2005) present a meta-heuristic based on ant colony optimisation for solving the logistics problem arising in disaster relief activities. The logistics planning involves dispatching commodities to distribution centres in the affected areas and evacuating the wounded people to medical centres. The proposed method decomposes the original emergency logistics problem into a two-phase decision-making process: the vehicle route construction, and the multi-commodity dispatching. Yi and Kumar (2006) study the similar problem, but have taken the facility location issue into consideration. A very recent research effort by Yan and Shih (2008) considers how to minimise the length of time required for emergency roadway repair and relief distribution, as well as the related operating constraints. The weighting method is adopted and a heuristic is developed to solve this problem in practice. It is worth mentioning that the majority of the previous work has been carried out under the assumption that the relief demand is deterministic. In practice, the demand for emergency materials is usually uncertain, and the existence of a latent period can make the demand even more unpredictable. In this article, we consider stochastic demand and latent period simultaneously, apply the epidemic diffusion rule and the genetic algorithm, and study how to ensure sufficient supply of a medicine after a dangerous disease strikes an area. As such, the article distinguishes from existing efforts in that it takes a number of issues and factors concurrently into consideration.
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The modelling framework and components
In this article, we study how to choose the storage points to distribute the relevant medicines to an affected area during the entire rescue cycle in highest cure rate and lowest total costs. As illustrated in Figure 1, there exist several storage places, both close to and far from the epidemic area, surrounding the epidemic area. We want to determine which storage points should be chosen to satisfy the demand for the associated medicines at the lowest costs and the shortest time. Since most epidemics divide people into four classes: the susceptible people (S), the people during the latent period (E), the infected people (I), and the recovered people (R), we will use an SEIR model to describe the developing epidemic process. Based on the specified epidemic diffusion rule, we consider the following situation for the emergency operations. Given the occurrence of an epidemic, the emergency operation system is triggered immediately and the initial materials are collected at the storages. A series of operational phases is then executed afterwards, as shown in Figure 2. Such a four-phase sequential operational routine is continued until the epidemic is under control and the number of infected people finally reduces to zero. As Figure 2 shows, the total rescue cycle time is first calculated in Phase 1. Then the next three phases, which contain demand forecast, medicine delivery and the computation of the consumption time, are executed iteratively until the total consumption time exceeds the total rescue cycle. The number of the infected people and the amount of inventory at each storage place are updated at each cycle during the entire rescue process. Figure 1
A conceptual framework of the emergency logistics network
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Figure 2
The operational procedure of the proposed emergency logistics system
To facilitate the model formulation in the following sections, we make the following three assumptions. 1
Once the epidemic breaks out in an area, this area will be isolated from other areas to avoid the spread of the disease. As such, we consider only one epidemic area in this article.
2
The locations of the storage places are known. Practically, the number of storage places to be used can be preset by a national disaster management programme.
3
The time-varying relief demand in the epidemic area is highly correlated with the number of infected people predicted based on the epidemic diffusion model.
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3.1 SEIR epidemic diffusion model with uncertain latent period Since epidemic models with time delay match actual situation much better, a great deal of attention has been paid to studying these models. Time delay is a basic but critical factor of epidemic models, playing an important role in the propagation process of epidemics and in reflecting the practical phenomena of latent period, infection period and immune period. Therefore, models with time delay and stochastic latent periods are considered in this article; in particular, the following SEIR model (e.g. Gao, Chen and Teng, 2006) is adopted:
dS ° dt A dS (t ) E S (t ) I (t ), ° ° dE E S (t ) I (t ) E S (t W ) I (t W ) dE (t ), ° dt ® ° dI E S (t W ) I (t W ) (d D J ) I (t ) ° dt ° dR ° J I (t ) dR (t ). ¯ dt
(1)
In this epidemic diffusion model, the time-based parameters, S (t ), E (t ), I (t ), R(t ) , denote the number of susceptible people, the number of people during the latent period, the number of infected people, and the number of recovered people, respectively. Other parameters include: A is the constant input rate of the population; d is the natural mortality of the population;̓ D is the mortality because of disease; J is the recovery rate of the infected people; E is the propagation coefficient of disease; W is the latent period, and is assumed to be a random variable following a particular probability distribution. Furthermore, A, d , E , J ! 0 , D t 0 and everyone moving into the region under consideration is assumed susceptible to the disease. As the above statement shows, I is the number of infected people and can be calculated by solving the differential model in Equation (1) when the initial values of S, E, I and R are given. It is desired that I stays at a value as low as possible, which implies that the situation is stable and that the spread of the epidemic is under control. According to the SEIR model, the change of I mainly depends on the population of the recovered people and the onset people at the end of the latent period. Intuitively, the number of recovered people reduces the amount of I, while the quantity of the onset people increases it. Therefore, the objective is to increase the number of recovered people and reduce the number of onset people during the latent period, which is equivalent to improving the recovery rate (J) and reducing the propagation coefficient (Eҏ). In the context of emergency rescue, there should be enough medicines to cure the patients so that the recovery rate can be improved, thereby decreasing the value of I effectively. Thus, the focus of this article is placed on improving the recovery rate and maintaining it at a high level through the proposed emergency rescue methods.
3.2 The forecasting model for the time-varying demand As introduced in Figure 2, when an epidemic with the diffusion rule expressed by Equation (1) breaks out, the quantity of the medicine needed to rescue the epidemic area
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in each cycle should be obtained before delivery. In this section, we are going to discuss how to forecast the rescue demand, which is the first important step in efficient emergency rescue. In this study, we consider only one kind of medicine that is most needed when the epidemic strikes suddenly. Based on the diffusion rule expressed by the SEIR model, the time-varying forecast model (Sheu, 2006) is formulated as
D (t )
a u I (t ) u L z1D u STD(t ) u L ,
(2)
where D(t ) is the demand forecast for the rescue medicine at time t. The first term in Equation (2) is the average demand during lead time, which is determined by the average daily demand of the medicine per infected person (a) and the upper bound of the lead time (L) to guarantee two consecutive emergency distributions to the given epidemic area. Note that we use the upper bound of the lead time L to ensure enough time available for the trucks to return to the storage places and prepare for the next distribution. The second term in Equation (2) represents the safety stock of the medicine when the tolerable demand shortage is set to be D, i.e. Prob{demand during L d a u I (t ) u L z1D u STD(t ) u L } 1 D ,
(3)
where STD(t ) represents the standard deviation of the number of the infected people, which can be calculated by k
STD(t )
¦
[ I l (t ) I (t )]2
l 1
k 1
(4)
where Il (t ) are the samples of I (t ) , and k is the number of samples. In this section, we have introduced the time-varying forecasting model, which can predict the demand of rescue medicine in each rescue cycle. In what follows, we will propose an emergency distribution model to optimise the total costs of each rescue cycle based on the forecasted demand.
3.3 The optimisation model for emergency rescue medicine distribution Usually, the first issue under consideration in emergency rescue is to determine the allocation and transportation schemes to meet the demand of the epidemic area. Hence, a time limit for transporting the emergency material is often set, that is, the allocation and transportation of emergency medicine is effective only if they are completed within the time limit. If the time limit is exceeded, then a kind of penalty applies. In this article, we use a penalty coefficient, which is determined by the increase of the objective function value per unit unsatisfied material. On the other hand, cost savings should also be considered when the time constraint is satisfied. According to the above explanation and assumptions, the parameters and variables of the model are defined as follows. ai
Storage place i (i 1, 2, ! , m)
qi (t )
Available capacity of storage i at time t (i 1, 2, ! , m)
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ci
Unit transportation cost of each storage i (i 1, 2, !, m)
wi
Transportation capacity of storage i (i 1, 2, ! , m) . It is assumed that each storage point uses one kind of truck
ni (t )
The number of available trucks at each storage i at time t (i 1, 2, ! , m)
tni
Time needed from storage i to the epidemic area (i 1, 2, ! , m)
D
Emergency material demand of the epidemic area
Tw
Transportation time threshold (penalty should be given if exceeding Tw )
Ei
Penalty coefficient, the amount of the increased objective function value per unit unsatisfied material, i 1, 2, ! , m
Z
Confidence level of meeting the emergency materials demand
xi (t )
Whether or not the storage is chosen to deliver the emergency material at time t, with 0 indicating it is not to be chosen, and 1 to be chosen.
Let TC be the total cost of emergency material transportation. Based on the description and assumptions of the model, we construct a multi-objective stochastic programming model for this emergency logistics network as follows. The objective functions are:
min TC1 (t )
min tni u xi (t ),
(5)
and m
min TC2 (t )
¦ c u min{q (t ), w u n (t )}u x (t ). i
i
i
i
i
(6)
i 1
The first objective expressed in Equation (5) is to minimise the longest transportation time from each storage place to the epidemic area, and the second objective given in Equation (6) is to minimise the total transportation cost. There are four sets of constraints, as listed below. ° m ½° Pr ® xi (t ) u min{qi (t ), wi u ni (t )} t D ¾ t Z °¯ i 1 °¿
¦
(7)
D
D(t )
(8)
xi (t )
1,if storage a i joins in emergency rescue i 1, 2, ! , m. ® ¯0,if storage a i does not join in emergency rescue
tni t 0, i 1, 2, ! , m.
(9) (10)
The first set of constraints in Equation (7) is to guarantee that the amount of the medicine delivered from the chosen storage places can satisfy the demand of the epidemic area with a probably of Z. The second constraint in Equation (8) relies on the result from the
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forecasting model discussed in the preceding section. The remaining two sets of constraints are straightforward based on the definitions of the parameters. For this model, a critical value of the emergency rescue time limit is set and the time is also treated as a part of the total cost, which we refer to as ‘time cost’ and define it as the product of the penalty coefficient and the time delay. If the rescue materials cannot be transported to the epidemic area within the time limit, the time cost will increase; otherwise it is zero. As a result, the total cost consists of the time cost and the transportation cost, and the two objective functions in Equations (5) and (6) can then be combined into a single objective which is to minimise the sum of the two cost elements. The new optimisation model is given below (For simplicity, TC(t) is still used to denote the total cost). m
¦
min TC(t )
m
ci u min{qi (t ), wi u ni (t )} u xi (t )
i 1
¦ E u{max{0, tn T }}u x (t ) i
i
i
(11)
i 1
° m ½° °Pr ® xi (t ) u min{qi (t ), wi u ni (t )} t D ¾ t Z ° ¯° i 1 ¿° ° ° D D(t ) s.t. ® 1,if storage a i joins in emergency rescue ° ° xi (t ) ®0,if storage a does not join in emergency rescue i 1, 2, ! , m. i ¯ ° °¯tni t 0, i 1, 2, ! , m.
¦
(12)
3.4 Calculating the consumption time after one distribution Since the demand of the epidemic area is stochastic and varies as time goes by, the quantity of the medicine that has been delivered to the epidemic area is usually not exactly equal to the actual demand. Thus, we introduce a new parameter called consumption time that is denoted by T (t ) . The consumption rate depends on the number of infected people I (t ) and the amount of medicine delivered to the area yi (t ) . The number of infected people at time t L / 2 is used to represent the average during the consumption time. Thus, the consumption time can be computed as m
¦ y (t ) i
T (t )
i 1
,
(13)
min{qi (t ), wi u ni (t )} u xi (t )
(14)
a u I (t L / 2)
where
yi (t )
It should be mentioned that the consumption time is calculated to determine the length of the on-going rescue cycle and the starting time for the next rescue cycle. If the starting time for the next rescue cycle exceeds the entire rescue cycle, then the medicine transportation will become a daily event.
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Solution methodology
We use the ‘dde23 function’ in Matlab to solve the SEIR model, and calculate the forecasted demand and the consumption time using Equations (2) and (13), respectively. Thus, in this section, we focus on the solution methodology for the optimisation model of emergency distribution. Since the available techniques for solving the mixed integer stochastic programming problems are still limited, and the genetic algorithm based on Monte Carlo simulation technology is commonly used, we adopt the genetic algorithm to solve the models in Equations (11) and (12). Genetic algorithm is a search method based on biological natural selection and principle of genetics. This algorithm introduces the biological evolution principle to the coding serials of the parameters to be optimised and selects among the individuals according to specific fitness function and a series of genetic operations so that it can keep the individuals whose fitness are high for forming new groups. This process repeats until the fitness of every individual in the group becomes higher and satisfies pre-specified conditions. The individual with the highest fitness is the optimal solution of parameters to be optimised. In what follows, we explain how to use the genetic algorithm to solve our emergency distribution problem.
4.1 Operating instruction of genetic algorithm 4.1.1 Chromosome coding The first step of genetic algorithm is to define the coding method of chromosomes. Each chromosome contains m bit gene, where m is the number of storage points. Every bit has two values, with 0 indicating that a storage place is not chosen for distribution, and 1 being chosen. Using a binary system can tell easily which storages are to join the emergency rescue; for example, if there are eight storages, a chromosome coding of 10010101 indicates that storages a1 , a4 , a6 and a8 are chosen.
4.1.2 Fitness definition The fitness of each individual (h) is obtained by computing the objective function, m
TC(t )
¦
m
ci u min{qi (t ), wi u ni (t )} u xi (t )
i 1
¦ E u{max{0, tn T }}u x (t ) i
i
i
i 1
which contains two parts. The first part is the total transportation costs from the chosen storage places to the epidemic area, and the second part is the time cost from the storages to the epidemic area. The value of the fitness function is the reverse of the objective function value; hence, the fitness can be calculated by f h 1/ TC(t ) , and the higher f h is, the better h is, and much closer to the optimal solution.
4.1.3 Selection operator Selection process is based on keeping down the best individual and rotating a roulette n times to select new chromosomes for a new group. A five-step selection process is given below.
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Step 1. Reorder the chromosomes according to the values of fitness functions. The higher f h , the better h, and has a smaller serial number. Keep down the chromosome whose serial number is one to the next generation, and select other chromosomes as follows. Given the parameter, a (0,1) , the evaluation function based on the orders is defined as follows. a(1 a)i 1 , i 1, 2, ! , n
eval (hi , f hi )
Note that when i 1 , the chromosome is the best, and is the worst when i
n.
Step 2. Compute the cumulative probability pi for each chromosome (hi). i
p0
0, pi
¦ eval (h , f j
hj
), i 1, 2, ! , n. .
j 1
Step 3. Produce a random number such that r (0, pn ) . Step 4. If pi 1 r d pi , then select the chromosome hi. Step 5. Repeat Step 3 and Step 4 n 1 times and obtain n 1 chromosomes. The last chromosome has been obtained in Step 1.
4.1.4 Crossover operator This step is to choose the same positions of two individuals to propagate the next generation randomly and exchange the code on them. This process reflects a stochastic information exchange, aiming at producing new gene combinations, that is, new individuals. The single-point crossover is adopted in this article. The general range of the crossover probability pc is 0.25 d pc d 0.75 .
4.1.5 Mutation operator During biological inheritance, mutation is operated on some bits of individuals at a probability of pm . This probability is generally small and is set such that 0.001 d pm d 0.2 in this article. When mutating, we reverse the corresponding bits of individuals to be mutated, that is, change 1 to 0, and change 1 to 0.
4.2 The solution procedure We present the solution procedure in Table 1. The flowchart of the procedure is also given in Appendix A. It is worth mentioning that Monte Carlo simulation technology is used to check whether the stochastic constraint condition of the distribution model is satisfied.
Optimal material distribution decisions based on epidemic diffusion rule Table 1
89
The solution procedure
Step 1
Initialise parameters, including the number of storages (m), population size (n), crossover probability ( pc ) , mutation probability ( pm ) , and iteration number (G)
Step 2
Initialise generation gen : 0 and determine length of chromosome coding according to the number of storages. Step 2.1. i : 1 Step 2.2. if i d n , generate a chromosome pi (0) of initial population p (0) randomly by computer. Then pick up numbers of storages which join in emergency rescue according to the coding of chromosome pi (0)
Step 3
Check whether constraint condition of opportunity is satisfied by Monte Carlo simulation technology Step 3.1. set N '
0
Step 3.2. generate a group of samples of W randomly, solve the epidemic model with DDE23 and get a group of samples I1 , I 2 , ! , I N of infective people I Step 3.3. if
¦
m
x (t ) u min{qi (t ), wi i 1 i
u ni (t )} t D , N '
Step 3.4. repeat Step 3.2 and Step 3.3 N times Step 3.5. if N ' / N t Z , set J Step 4
If J
1, i
1 ; else set J
0
i 1 , go to Step 2.2; else go to Step 2.2
Step 5
Calculate fitness values of all the chromosomes
Step 6
Copy chromosomes for the next generation according to selection operator
Step 7
Execute crossover operation according to the crossover operator and go to Step 3
Step 8
If J
Step 9
Execute mutation operation according to the mutation operator and go to Step 3
Step 10 If J
1 , go to Step 9; else go to Step 7 1 , go to Step 11; else go to Step 9
Step 11 Set gen : gen 1 , if termination conditions for genetic algorithm are satisfied, end the programme and output globally optimised fitness value and corresponding chromosome, else go to Step 5
5
A numerical example and implications
5.1 A numerical example In this section, we rely on a numerical analysis to demonstrate the efficiency of the proposed method for emergency distribution when an epidemic strikes. We assume that a special flu breaks out in a certain place, and there are eight storages around the area (m = 8). Given the related information, the proposed method is employed to determine the time-varying emergency deliveries during the entire rescue period. We follow the
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operational sequence for emergency rescue shown in Figure 2 to solve this example problem. The values of the parameter in the epidemic diffusion model are given as follows: A
0; d
0.0001; D
0.0005; J
0.9995; E
0.0045.
The latent period W is assumed to follow a Poisson distribution with mean being 1. We obtain 50 (k = 50) samples of W from the Poisson function of MATLAB7.1 and the numerical solutions of the above epidemic model with time delay under different samples of W, which are to be used for the Monte Carlo test (MCT). Figure 3 is a numerical simulation of this epidemic model. The four curves respectively represent the number of four groups of people (S, E, I, R) as time goes by. It is seen from Figure 3 that the quantity of the infected people (I) becomes zero on about the 16th day, that is, the epidemic will be under control in 16 days if the recovery rate is maintained by emergency rescue distribution. Therefore, the entire rescue cycle is 16 days. At the first stage, we assume that the existing medicine at the epidemic area is enough to be consumed for at least 2 days. As such, the first distribution is started on the second day after the epidemic breaks out. In the time-varying forecast model, we set a 3, L 4 (days), and D 0.05 . The demand is forecasted as D (t1 )
a u I (t1 ) u L z1D u STD(t1 ) u L
3 u I (t1 ) u 4 1.6 u STD(t1 ) u 4,
where k
l 1
STD(t1 )
2
¦[I (t ) I (t )] l 1
k 1
1
.
Table 2 shows the values of parameters in this emergency rescue material distribution model. Figure 3
Solution of the SEIR epidemic diffusion model (see online version for colours)
Optimal material distribution decisions based on epidemic diffusion rule Table 2
91
Primary parameter set in the distribution model at the 1st stage
i
1
2
3
4
5
6
7
8
qi (t1 )
350
300
280
200
1,000
800
650
400
tni
0.5
1
2
1.5
3
1
3
2
ci
0.8
1.3
1.5
1
1.7
0.8
1
1.3
wi
25
15
18
20
25
20
20
25
ni (t1 )
20
25
15
15
20
12
20
20
Ei
100
T
24
Z
0.95
Table 3
Storage selection decisions at the first stage
Number of runs
1
2
3
4
5
6
Storages being chosen
a1 , a6 , a7
a1 , a6 , a7
a1 , a6 , a7
a1 , a4 , a7
a1 , a6 , a7
a1 , a6 , a7
872
872
872
880
872
872
TC(t1 )
Let n 50, pc 0.6, pm 0.01 . We iterate 200 generations according to the above genetic algorithm procedure (in Section 4.2) for six times. The optimal decisions are reported in Table 3. It can be seen that the genetic algorithm is stable for running more than once, and the error of the results is 0.9%. Based on the results of the six runs, the scheme, ( a1 , a6 , a7 ), has the minimum total cost, TC(t1 ) 872 . The consumption time is m
¦ y (t ) i 1
T (t1 )
i 1
a u I (t1 L / 2)
4 (days).
Therefore, the first stage stops on the 6th day (four consumption days + 2 initial days), and the second delivery starts on the 6th day after the epidemic breaks out. The demand of the second stage is found as D (t2 )
a u I (t2 ) u L z1D u STD(t2 ) u L
where k
l
STD(t2 )
2
¦[I (t ) I (t )] 2
l 1
k 1
2
.
3 u I (t2 ) u 4 1.6 u STD(t2 ) u 4
92
H. Wang, X. Wang and A.Z. Zeng
During the second stage, the amount of the medicine and the trucks are replenished. It is easy to see that the amount of medicine replenished at storage places, ( a1 , a6 , a7 ), should be much more than the others because the inventory at these three storage points has dropped to zero after the first stage. In this article, we assume that there is no limit in available supplies and that each storage place can always be replenished completely. Table 4 shows the values of parameters of the emergency material distribution model at the second stage. Note that the parameters, n 50, pc 0.6, pm 0.01 , stay unchanged. We again iterate the problem 200 generations according to above genetic algorithm for six times, and the results are reported in Table 5. It can be found that the genetic algorithm is stable for running more than once, and the error of the results is 6.25%. This time, the optimal decision is to choose the scheme, ( a6 , a7 ), with the minimum total cost, TC(t2 ) 800 . The consumption time at this stage is computed as m
¦ y (t ) i
2
i 1
T (t2 )
a u I (t2 L / 2)
13 (days)
Since the time to consume the total amount of medicine distributed for the second stage (19 = 6 + 13) has exceeded the entire rescue cycle (16 days), the solution process can be stopped. Table 4
Primary parameter set in the distribution model at the 2nd stage
i
1
2
3
4
5
6
7
8
qi (t2 )
250
350
330
250
1,050
760
700
450
tni
0.5
1
2
1.5
3
1
3
2
ci
0.8
1.3
1.5
1
1.7
0.8
1
1.3
wi
25
15
18
20
25
20
20
25
ni (t2 )
30
25
15
20
20
25
20
25
Ei
100
T
24
Z
0.95
Table 5
Storage selection decisions at the second stage
Number of run
1
2
3
4
5
6
Storages being chosen
a6 , a7
a1 , a4 , a7
a6 , a7
a1 , a4 , a7
a6 , a7
a6 , a7
800
850
800
850
800
800
TC(t2 )
Optimal material distribution decisions based on epidemic diffusion rule
93
5.2 An example of sensitivity analysis In this section, a sensitivity analysis of the latent period W on the distribution scheme and the total rescue cost is conducted. Specifically, we perform the proposed operational procedure of the emergency logistics system five times under five different latent periods. Holding all the other parameters fixed as in the numerical example given in Section 5.1 except that W takes on five values ranging from 0.25 to 1.5 with an increment of 0.25, we have obtained the distribution decisions and reported the results in Table 6. As indicated in Table 6, the length of the emergency rescue increases with the mean of the latent period, because the time at which the number of the infected people achieves the highest is pushed off as the latent period grows longer. For example, when the mean of the latent period equals 1.5, the entire rescue requires three stages and the demand reaches the largest at the second stage. However, when W equals 0.25, 0.5, 0.75 and 1.0, respectively, the entire rescue needs only two stages and the highest demand occurs at the first stage. In addition, the number of storage places selected at each stage also differs under different latent periods. The above analysis confirms that the latent period plays an important role in emergency distribution decisions. For a small change of the latent period at 0.25, the final distribution decisions, including the number of rescue stages and of storage places being chosen, and the costs at each stage, can change significantly. Unfortunately, predicting the length of the latent period for an epidemic is difficult and is usually done empirically. As the accuracy of the latent period is vital to the success of emergency rescue, a great deal of effort needs to be devoted to scientifically estimating the latent periods of different epidemics.
6
Discussions and conclusions
In this article, the optimal decision of emergency medicine distribution problem with uncertain demand caused by the latent periods of epidemics is investigated. Taking the effect of the latent period into consideration, we have studied the optimal distribution of the emergency rescue materials based on the epidemic diffusion rule and developed a solution procedure that involves three parts. Table 6 Stages
W
Distribution decisions on different mean values of the latent period The first stage Scheme
The second stage
TC( t1 ) T( t1 )
The third stage
Scheme
TC( t2 )
T( t2 )
Scheme
TC( t3 )
T( t3 )
0.25
a1 , a2, a6 , a7
1,262
5
a7
400
30
–
–
–
0.5
a1 , a4, a6 , a7
1,072
4
a1 , a6
600
19
–
–
–
0.75
a1 , a6 , a7
872
4
a4 , a6
650
14
–
–
–
1.0
a1 , a6 , a7
872
4
a6 , a7
800
13
–
–
–
1.5
a1 , a4, a6
672
3
a4, a6 , a7
950
7
a4
150
10
94
H. Wang, X. Wang and A.Z. Zeng
1
forecast of the time-varying relief demand
2
distribution of the emergency rescue materials
3
calculation of the consumption time.
We have developed a multi-objective stochastic programming model which is solved by the genetic algorithm based on Monte Carlo simulation. A relatively good result is obtained in a short amount of time. To obtain more accurate results, we can increase the number of generations and the pop-size so as to expand the coding scale and search the optimisation solution in a larger space. It is necessary to point out some limitations of this research. First of all, only one epidemic area is considered. When multiple epidemic areas need to be studied, a collaborative allocation of materials among these areas can be effective, but must be justified by considering the increased transportation costs. Second, only one kind of emergency material is considered. In reality, many types of medical materials may be needed simultaneously. As the number of emergency material type increases, the decisions on the storage places will be more complex. Third, the latent period is assumed to follow a Poisson distribution, which may not be true for all the epidemics. Other distributions such as exponential and normal distributions can be used. Finally, the assumption that the amount of inventory at each storage place and at each stage is independent upon the replenishment time and the consumption time can be relaxed. All these areas represent our future research directions.
Acknowledgements This work is supported by the project, titled ‘Operations and simulation of emergency response logistics network in the anti-bioterrorism systems’ and supported by the National Natural Science Foundation of China (70671021) and the National Key Technology R&D Programme of China (2006BAH02A06). The authors are grateful for the valuable comments and suggestions of the anonymous referees that have improved the quality of the article.
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Appendix A
The flowchart of genetic algorithm based on Monte Carlo test
