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Path selection model and algorithm for emergencylogistics management

Yuan Yuan a,*, Dingwei Wang b

a Department of Management Science and Engineering, School of Business Administration, Northeastern University, Shenyang, Chinab Institute of Systems Engineering, School of Information Science and Engineering, Northeastern University, Shenyang, China

Available online 26 September 2008

Abstract

Path selection is one of the fundamental problems in emergency logistics management. Two mathematical models forpath selection in emergency logistics management are presented considering more actual factors in time of disaster. First asingle-objective path selection model is presented taking into account that the travel speed on each arc will be affected bydisaster extension. The objective of the model is to minimize total travel time along a path. The travel speed on each arc ismodeled as a continuous decrease function with respect to time. A modified Dijkstra algorithm is designed to solve themodel. Based on the first model, we further consider the chaos, panic and congestions in time of disaster. A multi-objectivepath selection model is presented to minimize the total travel time along a path and to minimize the path complexity. Thecomplexity of the path is modeled as the total number of arcs included in the path. An ant colony optimization algorithm isproposed to solve the model. Simulation results show the effectiveness and feasibility of the models and algorithms pre-

sented in this paper. 2008 Elsevier Ltd. All rights reserved.

Keywords: Emergency logistics management; Path selection; Disaster extension; Path complexity; Modified Dijkstra algorithm; Antcolony optimization algorithm

1. Introduction

In recent years, frequent natural disasters and man-made catastrophic events have brought great loss tohuman beings (Ergonul, 2005; Chandre, Baskett, & Gallagher, 2007). As an emerging research area, emer-gency management is attracting more and more attention of researchers (Kuwata & Takada, 2004; Mathieu,

2006; Nezih & Walter, 2006; Tufekci & Wallace, 1998). Logistics support is one of the major activities in disas-ter response (Ozdamar, Ekinci, & KUCUKYAZICI, 2004). Commodities such as food, shelter and medicinemust be sent from the supply center to the affected area as quick as possible to support rescue operation andhelp wounded people. Furthermore, important or hazardous materials must be transferred from the affectedareas to safety areas.

0360-8352/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2008.09.033

* Corresponding author. Tel.: +86 24 83671468.E-mail address:neuyy@126.com(Y. Yuan).

Available online at www.sciencedirect.com

Computers & Industrial Engineering 56 (2009) 10811094

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Path selection is one of the fundamental problems in logistics management. Among the existing researcheson emergency logistics management (Dai, Wang, Yang, & Lv, 1995; Chang, Tseng, & Chen, 2007; Sheu, 2007;Yi & Ozdamar, 2007), several complicated models have been built considering the disaster conditions. Butmost of them consider the travel time on each arc of the logistics network as a constant. In fact, the travel

conditions on the arcs will be greatly affected by disaster extension especially under some disasters like hur-ricane and flood that will extend gradually in time and space (Farahmand, 1997; Tufekci, 1995). The travelspeed on each arc will decrease continuously under disaster extension, and the decrease extent will differ withthe positions of the arcs and the severity of the disaster.

On the other hand, most of the existing research works on emergency logistics management took timeas the most important parameter to be considered. The objective of the model was to minimize the timeneeded to complete the logistics transmission process. But disasters may cause great chaos and conges-tions. People will be in great panic under dangerous conditions and the sense of panic will affect theirability to follow a suitable path. Therefore the logistics transmission path should be as simple as possibleso that it can be followed more easily under disaster conditions. The complexity of the logistics transmis-sion path should also be taken into account as an objective of the path selection model for emergencylogistics management.

In this paper, we focus on the path selection problem in emergency logistics management and buildmathematical models to select the best path. The motivation of this research is to consider more actualfactors in time of disaster when building models. The factors we take into account include the real-time effect of disaster extension on the travel conditions of the arcs and the path complexity of thepath. Two mathematical models are built in our study and algorithms are developed to solve themodel. Our method for studying path selection problem in emergency logistics management is pre-sented in Fig. 1.

As described inFig. 1, first a mathematical model is built for path selection in emergency logistics manage-ment considering the real-time effect of disaster extension. The objective of the model is to minimize total tra-vel time along the path. The travel speed on each arc is modeled as a continuous decrease function with respectto time. The proposed problem is different from classical shortest path problem since the travel speed on eacharc will decrease with time continuously. A modified Dijkstra algorithm is designed to solve the time-variedshortest path problem presented in the model. Based on this single-objective model, we take path complexity

Consider real-time effect of

disaster extension

Single-objective time-varied

shortest path model

Consider the complexity ofthe path

Multi-objective time-variedshortest path model

The travel speed on the arc: a

continuous decrease function

with respect to time

Modified Dijkstra algorithm

The complexity of the path:

total number of arcs included

in a selected path

Ant colony optimization

algorithm

Fig. 1. Method for studying path selection problem in emergency logistics management.

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into consideration and a multi-objective path selection model for emergency logistics management is built. Thetwo objectives of the model are to minimize total travel time along the path and to minimize the complexity ofthe path. The complexity of the path is modeled as the total number of arcs included in a selected path. Thismulti-objective time-varied shortest path problem cant be solved through classic algorithms and the modified

Dijkstra algorithm presented above. An ant colony optimization algorithm is proposed to solve the multi-objective optimization model. Simulation results show the effectiveness and feasibility of the model andalgorithm.

This paper is organized as follows. A single-objective model to minimize the travel time along a path and amulti-objective model to minimize the travel time and path complexity are presented in Section 2. In Section3,a modified Dijkstra algorithm is designed to solve the single-objective path selection model. An Ant ColonyOptimization algorithm is designed to solve the multi-objective path selection model in Section4. In Section5,numerical experiments are carried out to test the models and algorithms presented in this paper. Finally, wepresent our conclusions in Section6.

2. Path selection model for emergency logistics management

2.1. Definition of variables and parameters

(1) An emergency logistics network is defined by a directed graph G(V, A), whereV= {t1,t2, , tn} is theset of nodes andA # V Vis the set of arcs. Let t1,t2, ,tndenote the nodes in the network, wheret1is the source node and tn is the destination node.

(2) lijdenotes the length of the arcs between nodes tiand tj, where (ti, tj) e A.(3) s0ijis the travel speed on arc (ti,tj) under normal conditions. Define sij(t) as the travel speed on arc (ti,tj)

at time t under disaster conditions.

Observing the extension processes of some disasters such as flood and hurricane, we can find that the travelspeed on each arc of the network will decrease with the extension of disasters in time and space. The decreaseextent of the travel speed is affected by the position of the arc and the type of the disaster, etc. Hence, without

loss of generality, we assume the decrease function of the travel speed as follows:

sijt s0ij aij e

bijt

Here, aij, bijare the decrease parameters that determine the decrease extent of the travel speed function s ij(t).aij,bijcan be estimated according to the distance from arc (ti,tj) to the disaster center, the vulnerability of thearc and the type of the disaster etc.

(4) Lettijbe the time needed to travel through arc (ti,tj).tidenotes the time when the logistics reach node ti,tjdenotes the time when the logistics reach node tjalong arc (ti, tj). It is obvious that tij= tjti .

(5) xijis the decision variable in the model. x ij= 1 when arc (ti, tj) is included in the fixed path and xij= 0when arc (ti, tj) is not included in the fixed path.

(6) Pdenotes the transmission path of logistics which is a sequence of nodes in the network. Let pkbe the

sequence number of node vpkin the network, then path Pcan be represented as vp1 ; vp2 ; ; vpk; ; vpK,where 1 6 pk6 n,kis the travel sequence of nodevpkalong pathP. PathPstarts at the source node andterminates at the destination node, that is, p1= 1 and pk=n. PathPshould not have circles consideringthe time pressures in emergency logistics management.

(7) Let ETP; vpk denote the travel time from node vp1 to node vpk along path P vp1 ; vp2 ; ; vpk, where1 6 pk6 n. We can easily obtain:

ETP; vpk Xk1m1

tpmpm1 tp2 tp1 tp3 tp2 tpk tpk1 tpk

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2.2. Recursive method for computing the travel time of a path

Based on the definitions above, for an already known path P vp1 ; vp2 ; ; vpk, we can obtain:

tp1 t1 0 1Z tpktpk1

spk1pktdt lpk1pk; 2 6 k6 K 2

In Eq.(2), if we have already known the lower limit of the integraltpk1 , the integrandspk1pkt and the integralresult lpk1pk, the upper limit of the integral tpkcan be obtained through solving Eq.(2).

The recurrence relations presented by Eqs. (1) and (2) show that the total travel time of path Pcan beobtained through calculating the arriving time tpkat node vpk1 6 pk6 n recursively.

2.3. Path selection model for emergency logistics management under real-time effect of disaster extension

The formulation of path selection model under real-time effect of disaster extension is described as follows:Model I:

minXni1

Xnj1

tijxij 3

s.t. Z tjti

sijtdt lij 4

tij tj ti 5

t1 0 6

sijt s0ij aij e

bijt 7

Xn

j 1

ji

xijXn

j 1

ji

xji

1 i 1

1 i n

0 otherwise

8>: 8

Xnj 1

ji

xij6 1 in

0 i n

9

xij 0; 1; i 1; 2; ; n;j 1; 2; ; n 10

The objective of model I is to minimize the total travel time along a path. Eqs. (4), (5) and (6)are the recursionformula of total travel time along a path, which means that arc (ti,tj) is traveled through with the speed s ij(t)

during time period t ij. Eq.(7) is the decrease function of the travel speed on arc (ti, tj) under real-time effectdisaster extension. Constraint(8)restricts the value ofxijto constitute a feasible path from the source node t1to the destination nodetn. Constraint(9)ensures that there are no circles in the path. Constraint (10)is the 01integer constraint of the decision variable xij.

2.4. Multi-objective path selection model for emergency logistics management

Related researches have shown that most congestion and panic happen at the intersections of two arcs inthe emergency network. For example, most traffic delays in regional evacuations occur at intersections ( Tho-mas & Justin, 2003). When traveling along a path, the fewer arcs included in the path, the more easily the pathcan be followed. The complexity of a path can be simply represented as the total number of arcs included in a

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fixed path. Based on the single-objective path selection model presented in 2.3, a multi-objective path selectionmodel can be built taking into account both time factor and path complexity factor. The objectives of themodel are to minimize travel time along a path and to minimize the path complexity respectively. The modelcan be formulated as follows:

Model II:

minf1Xni1

Xnj1

tijxij 3

minf2Xni1

Xnj1

xij 11

s.t.(4)(10)The constraints in model II are the same with those of model I.

3. Modified Dijkstra algorithm to solve model I

Dijkstra algorithm is one of the classical algorithms to solve shortest path problem effectively (Ahuja Rav-indra, Magnanti Thomas, & Orlin James, 1993). The basic idea of the algorithm is to find shortest path fromthe source node t1 step by step. Dijkstra algorithm maintains a label Twith each node t i, which is an upperbound of the total weight on the shortest path from node t1to node t i. The algorithm also sets a label Pforeach node that denotes the total weight of the shortest path from source nodet1to each nodeti. At any inter-mediate step, the algorithm modifies the Tlabels of nodes and setPlabel for a certain node, then it adds thenode to the set of nodes with Plabels. Thus the number of nodes with Plabels will increase by one after eachstep and the shortest paths from the source node t1to all the other nodes in the network will be found after atmost (n1) steps.

As described in Section 2.2, in path selection problem for emergency logistics management under real-time effect of disaster extension, the travel time tij on arc (ti, tj) is determined not only by the length of

the arc lijand the travel speed function sij(t), but it is also determined by the time when the logistics reachthe origin node t iof arc (ti, tj) since the travel speed on each arc is a decrease function with respect to time.A modified Dijkstra algorithm is designed to solve model I as the classical Dijkstra algorithm cant beapplied to solve the time-varied shortest path problem presented in this paper directly. The method toachieve the travel time of a path in classical Dijkstra algorithm is modified in terms of the recursive methoddescribed in Section 2.2, and the correctness of the algorithm is proved according to the characteristics ofthe model and the algorithm.

3.1. The modified Dijkstra algorithm

LetP(tj) be thePlabel of nodetjand T(tj) theTlabel of node tj.Siis the set of nodes with Plabels afterith

step of the algorithm. Letk(tj) be the predecessor of node tjalong a pathPfrom the source nodet1to nodetj.M is a large enough positive number. Based upon the above discussion, the proposed modified Dijkstra algo-rithm can be summarized as follows:

Step1: Initialization (iterative step i= 0). Let S0= {t1}, P(t1) = 0, for "tkt1, let T(tk) = +1, k(tk) = Mand m = 1 .Step2: Iftn eSi, we can obtain that P(tn) is the shortest travel time needed from the source nodet1to nodetn, then the corresponding path is the best path selected, the algorithm terminates. Otherwise go to step3.Step3: For each nodetjwhere (tm,tj) eA and tjR Si, lettm= P(tm), solve the equation

Rtjtmsmjtdt lmjto

achieve tj. Iftj

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3.2. Proof of the modified Dijkstra algorithm

To prove the correctness of the proposed algorithm, first we have Lemma 1as follows.

Lemma 1. tjis a monotone increasing function of tiin the equationRtjtisijtdt lij with respect to tjif sij(t) is a

monotone decreasing and integrable function with respect to t when te[0,+1).

Let tHij be the travel time needed on arc (ti, tj) e A along path H. Define tHj;...;kas the time needed to travel

from nodetjto nodetkalong path H. LetPbestvkbe the shortest travel time needed from node t1to nodetk.Based upon above definitions and the conclusion ofLemma 1, the correctness of the algorithm can be provedthrough mathematical induction.Proof. To prove the correctness of the algorithm, we only need to prove that in each step of the algorithm, forevery tk eSi, thePlabel of nodetki.e.P(tk) is the shortest travel time needed from source node t1to node tk.

(1) When i= 0, the conclusion is right obviously.(2) Suppose the conclusion is right when i=n, i.e. for "tj e Sn, ETPbestvj ; vj Pvj .

When i= n+ 1, from step3 and step4 of the algorithm, we can obtain that Tvjn minvjRSn

fTvjg;

Sn1 Sn[ fvjng. Suppose H is an arbitrary feasible path from node t1 to node vjn . Since t1eSn andvjn RSn, along path Hthere must exist an arc whose origin node is in set Sn while destination node is not

in set Sn. Suppose (tr,tl) is the first one among those arcs along path H, i.e. tr e Sn ,vl R Sn, thenETH; vjn ETH; vr t

Hrl t

Hl;...;jn

. From the inductive assumption, P(tr) is the shortest travel time needed

from nodet1to nodetr, i.e.ETH; vrP ETPbestvr; vr. Then fromLemma 1and Eqs.(1) and (2)which standfor the recursive method for computing the travel time along a path, we can get:

ETH; vr tHrl ETH; vlP ETPbestvr; vr t

Pbestvrrl ETPbestvr; vl:

According to the rule for modifying the T labels of nodes in the algorithm, since tr e Sn vl R Sn, we can get

ETPbestvr; vlP Tnvl. Furthermore, from the rule for selecting node vjn to add in set S, we can getTnvlP Tnvjn. Then

ETH; vjn ETH; vl tHl;...;jnP Tnvl t

Hl;...;jnP Tnvjn Pvjn;

i.e. ETPbestvjn ; vjn Pvjn. The conclusion is obtained. h

4. Ant colony optimization algorithm to solve model II

4.1. Convert model II to a single-objective model

First we use the ideal point method to deal with the model and convert the multi-objective path selectionmodel into a single-objective model as follows:

MinF r1f1 f1f1

r2f2 f2f2

12

s.t.(4)(10)Here f

*

f1;f

2 T is the ideal point. r

*2R f r

*jri P 0;

Pri 1; i 1; 2g is the weight vector.

According to the theorem about ideal point method, the optimum solution of the model(12), (4)(10)is anoninferior solution of model II.

After converting to a single-objective model, model II can be solved through solving three single-objectivemodels. The method for solving model II can be described as follows:

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Step1: Use modified Dijkstra algorithm presented in Section 3.1to get the optimal solution of the single-objective model(3)(10)and obtain the corresponding optimal value f1*.Step2: Use classical Dijkstra algorithm to get the optimal solution of the single-objective model (11), (4)

(10)and obtain the corresponding optimal value f2*.Step3: Put the ideal point into(12)and solve the single-objective model(12), (4)(10)to get the noninferiorsolution of the original multi-objective model (model II).

4.2. Algorithm to solve the multi-objective path selection model

Model II presented in this paper is a multi-objective model and the traveling speed on each arc is not aconstant but a continuous decrease function with respect to time, so model II is a multi-objective time-variedshortest path problem. Therefore although the ideal pointf1* can be obtained through the modified Dijkstraalgorithm presented in Section3and the ideal pointf2* can be obtained through classical Dijkstra algorithms,the single-objective model (12), (4)(10)cant be solved through either of the above two algorithms.

As a novel metaheuristic algorithm, ant colony optimization (ACO) algorithm was proposed firstly to solveTSP problem (Dorigo, Maniezzo, & Colomi, 1996). In recent years, ACO has been applied to solve many dif-ferent combinational problems successfully. ACO should also be suitable to solve the path selection problempresented in this paper since it was proposed based on the simulation of the way real ant find the shortest routebetween a food source and their nest. Furthermore, every ant can construct a feasible path in every cycle of theACO algorithm. This theorem can effectively avoid the problem of unfeasible solutions caused by crossoverand mutation process in genetic algorithms. So we choose ACO to solve model II.

After converting the multi-objective path selection model into a single-objective model by ideal pointmethod, an ACO algorithm is designed to achieve the noninferior solution of the multi-objective path selec-tion model for emergency logistics management.

The ACO algorithm to solve the multi-objective path selection model is as follows:

Step1: Initialize (cycles counter NC = 0). Set the value of fundamental parameters of the ACO algorithmincluding the maximum number of cycles NCmax, the number of ants M. For every edge (ti,tj) set an initialpheromone value sij(0) =c and set the initial pheromone updating value Dsij= 0.Step2: Every ant constructs a path from the source node to the destination node.

First place all the Mants on the source node. At node i, the mth ant chooses node jto move to with theprobability:

pmij

saij

gb

ijPnj1

saij

gb

ij

ifvi; vj 2A

0 otherwise

8