Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5030619, 9 pages

http://dx.doi.org/10.1155/2016/5030619

## Optimization on Emergency Resources Transportation Network Based on Bayes Risk Function: A Case Study

^{1}School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China^{2}School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

Received 27 January 2016; Revised 14 May 2016; Accepted 5 June 2016

Academic Editor: Rita Gamberini

Copyright © 2016 Changfeng Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to coordinate the complex relationship between supplies distribution and path selection, some influential factors must be taken into account such as the insufficient remaining capacity of the road and uncertainty of travel time during supplies distribution and transportation. After the structure of emergency logistics network is analyzed, the travel time Bayes risk function of path and the total loss Bayes risk function of the disaster area are proposed. With the emergency supplies total transportation unit loss as the goal, an emergency logistics network optimization model under crowded conditions is established by the Bayes decision theory and solved by the improved ant colony algorithm. Then, a case of the model is validated to prove that the emergency logistics network optimization model is effective in congested conditions.

#### 1. Introduction

In the past ten years, nearly 9000 disasters have led to a great loss of property and casualties occurring in the world, on an average of about two new disasters every day [1]. Frequent natural disasters and other unexpected emergencies, especially like earthquakes, typhoons, floods, or other natural disasters, are not only a test of the government’s emergency response but also a challenge for emergency logistics. However, during the process of planning, transporting, and storing, humanitarian logistics, unlike normal logistics, have their own unique characteristics [2] such as the large amount of material transportation, the uncertainty of vehicle routing, the urgency of the rescue time, and the randomness of abnormal congestion events.

In order to make effective plans, humanitarian logistics network designers need to collect information and make a provisional decision based on the existing information [3]. And with the development of disasters, the decision will be continuously adjusted to the current situation. Lodree Jr. and Taskin [4] introduced a stochastic inventory control problem and formulated it as an optimal stopping problem using Bayesian updates based on hurricane predictions. Tofighi et al. [5] addressed a two-echelon humanitarian logistics network design problem and developed a novel two-stage scenario based possibility-stochastic programming (SBPSP) approach. However, there are many parameters that are difficult to quantify, making the humanitarian logistics optimization problem more complicated [6].

In the beginning of the disaster, as Özdamar and Balcik [7, 8] pointed out, traffic congestion will be heavier and it is easy to make traffic halted. Because of damaged roads and traffic congestion, common in disaster, section’s remaining capacity decreases while the impedance increases. The sensitivity of section travel time for the additional traffic becomes higher and the uncertainty of the section travel time is increasing. Thus, it is very hard to select the path of humanitarian logistics and construct humanitarian logistics network. Here, in order to overcome the problem, the Bayes theory is introduced to formulate the travel time Bayes risk function of section, which is used to rationally quantify the travel time of section.

The uncertainty [5] exists not only in the path selection but also in the relief demands [8]. Özdamar et al. [9, 10] pointed out that, because of the dynamics and uncertainties associated with various disaster characteristics, relief supplies are often available in limited quantities and precise relief demands are hard to rationally quantify during disaster. And there exists the game between affected areas [11, 12]. Each disaster area wants to get sufficient relief supplies, so the game will take place between affected areas. Furthermore, the different O-D distribution will lead to different forms of flow distribution in the transportation network [13]. So how to deal with the complex relationship between supplies distribution and path selection has become particularly important. In order to solve this problem, in consideration of the correlation between the total emergency loss and the delay of the emergency supplies, the total loss Bayes risk function of the disaster area is formulated. The total loss Bayes risk function consists of losses caused by the number of delayed relief supplies in the affected areas, which can reflect the effectiveness of emergency logistics network, and the losses of transportation time caused by real-time traffic condition, which can reflect the effectiveness of the travel time Bayes risk function.

Based on the total loss Bayes risk function, the emergency supplies total transportation unit loss is defined. With it as the goal, a multiperiod emergency logistics network optimization model under crowded conditions is established by employing multiperiod Bayesian decision theory, which can minimize the total loss Bayes risk function and the travel time Bayes risk function, and the dynamic user equilibrium (DUE) model [14], which makes full use of the remaining capacity of the road and minimizes transportation time [15]. Based on the model, an improved ant colony algorithm is proposed. And a case of the model is validated to prove that the emergency logistics network optimization model is effective in congested conditions.

#### 2. Literature Review

Optimization of emergency resources transportation network is a typical multipath selection problem under the influence of multifactors, such as the uncertainty of vehicle routing and the randomness of abnormal congestion events.

From the perspective of emergency logistics network, many scholars have made some researches on the optimization of emergency logistics network and some research achievements have been achieved. Aksen and Aras [16], Boloori Arabani and Farahani [17], and Farahani et al. [18] have studied the emergency facility location problem (FLP). Cosgrave [19] presented the three properties of emergency decision and used the Decision Participation Contingency Theory to construct a simple model of emergency decision-making process. Özdamar et al. [9] proposed a mixed multistage integer programming model which is solved by the Lagrangian Relaxation Technique. Pauwels et al. [20] explained the “irreversible effect” of decisions, which presented in economics research results, and applied the results to the withdrawal decision problems. Özdamar and Demir [7] described a hierarchical cluster and route procedure (HOGCR) for coordinating vehicle routing in large-scale postdisaster distribution and evacuation activities. Yong and Nan [21] had applied Bayesian analysis and optimization theory to establish emergency logistics system model which is solved by the genetic algorithms. Afshar and Haghani [22] had proposed a modeling integrated supply chain logistics in real-time large-scale disaster relief operations.

From the perspective of research methods, relevant scholars have studied this problem based on Bayesian theory; Zhan et al. [23] addressed multisupplier, multiaffected area, multirelief, and multivehicle relief allocation problem in disaster relief logistics. A multiobjective optimization model based on disaster scenario information updates is proposed in an attempt to coordinate efficiency and equity through timely and appropriate decisions regarding issues such as vehicle routing and relief allocation. Azoury [24] proposed a Bayesian solution to dynamic inventory models under unknown demand distribution conditions. Some relevant scholars also have studied this problem based on the network flow model [25], the game theory [26], scenario planning [27], and cooperative strategy [28, 29]. Existing research achievements have laid a certain foundation for this problem.

Nevertheless, optimization of emergency resources transportation network is complicated system engineering; there still are some facts that must be considered:(1)Section, as the basic unit of emergency logistics network, has special traffic attributes. During disaster, the road capacity will decline and the travel time of section is highly sensitive for the additional traffic and varies with time under the congested or disordered conditions.(2)The way of material distribution will affect the choices of vehicle routing. Because each disaster area wants to get sufficient relief supplies, the game will take place between affected areas. However, the different O-D distribution will lead to different forms of flow distribution in the transportation network [13]. So the way of material distribution will affect the choices of vehicle routing.

#### 3. The Analysis of Emergency Logistics Network Architecture

Being a special logistics network, emergency logistics network is a complex network composed of a large number of sides (sections or paths) and node (logistics node). From complex network perspective, the emergency logistics transportation network has the characteristics of multinetwork and multilevel. Here, in order to fully understand the complex relationships between subnetworks of emergency logistics network, the emergency logistics network will be divided into three network layers: the actual road network, the disaster distribution network, and the emergency materials transportation network.

The actual road network is composed of many sections and transportation hubs, which is the basis of all transportation activities. The intersections or hubs are considered as nodes of the actual road network, while the sections between nodes are the edges of the actual road network. The actual road network is defined as , whose edges have some characteristic value: the volume of traffic and length. So it can be seen as a weighted network. network can be expressed aswhere is a set of the actual road network nodes. is the set of edges, while the design capacity is the weight of edges between and . means that there is no link between and .

Assuming that the disaster information can be quantified and trusted, the disaster distribution network is composed of the damaged road and the demanded emergency supplies. The disaster distribution network is defined as , whose network nodes are the disaster areas. The edges of the network are the road linking the disaster areas and emergency supplies storage center. network can be expressed aswhere is a set of the demanded quantity of supplies of each disaster area at stage . is a set of the section’s remaining volume rate at stage (let the section’s remaining volume rate be equal to the actual capacity/the designed road capacity ). means that there is no link between and or the section has been interrupted.

The emergency materials transportation network is a set of emergency materials transportation path, which is decided after taking the actual road network and the disaster distribution network into account. As shown in Figure 1, the planner refers to the actual road network and collects information about the damaged road and disaster distribution and then make a provisional decision about the relief transportation network. Since the disaster distribution will evolve over time [3, 4], the relief transportation network should include a variety of transportation solutions to adjust to the complex and changeable situation.