Emergency Logistics Management in Natural Disasters

WANG-Qiang School of Management

Harbin University of Commerce Harbin, P.R.China wqhlj@163.com

DAI-Bibo School of Management

Harbin University of Commerce Harbin, P.R.China

daibibo1399@163.com

Abstract—Logistics programming in natural involves dispatching commodities to distribution centres in disaster areas as soon as possible so that relief operations are accelerated. In this study, three planning models are to be integrated into a natural disaster logistics decision support system. The system readily decomposed into three problems the first one being fuzzy clustering disaster grades by historical data, the second one showing the grade distinguishment of new disaster areas based on fisher discrimination procedure, the last one giving plan incorporating relief commodities and equipment distributing in proper sequence in disaster areas. In emergency logistics context, supply is available in limited quantities at current time period but commodity demand is known with uncertainty at the time. So how to make grate use of the limited commodities under the uncertainty information, that is of great value. The mathematical models describe setting are considerably different than other conventional vehicle routing problem articles in the world. It suggests that macro adjustment is needed in emergency logistics programming in natural disasters, and it will drow more attention to this field in the near future.

Keywords- emergency logistics; natural disasters; fuzzy clustering analysis; fisher discrimination; multi-object programming

I. INTRODUCTION It has only been 5 years since a number of Asian countries

were devastated by the Asian tsunami, and the timeliness of our findings is again underscored by recent disasters in Sichuan, China and Myanmar. However, these three disasters are not isolated events but rather part of a rising trend in the number of reported natural disasters.While the reasons for this increase in the number of reported natural disasters are unclear, the need for more effective and better coordinated disaster relief is beyond doubt.

This study is concerned with planning logistics at a macro level in the presence of natural disasters. The logistics planning models proposed here is intended to be a component of a logistics decision support system linking all relevant databases (stocking units, aid distribution centres, national transportation networks, search and rescue teams, etc.) and the central aid coordination centre. Aid distribution centres decides on the quantities and destinations of relief commodities to be transported, and on the specific vehicles to be dis patched to carry these commodities. Natural disaster logistics decision support system is planed for the aid distribution centres. The

paper is organized as follows. First, the problem at hand is analysed how to clustering disaster grades by historical data. Next, grade distinguishment of new disaster areas based on fisher discrimination procedure is developed. And at last, plan incorporating relief commodities and equipment distributing in proper sequence in disaster areas.

II. ANALYSIS OF THE PROBLEM To the best of our knowledge, this hybrid problem has not

been dealt with in the literature.Four references that describe work relevant to emergency logistics are the studies made by Rathietal.(1993) ,Equietal.(1996), Linet Ozdamar (2004), and AllenYuhung Lai. (2009).

Rathietal.(1993) consider supply logistics in conflict or emergency situations. The authors develop LP models where routes and the amount of supply to be carried on each route are predetermined between each origin-destination pair. Their problem is to identify the optimal number of vehicles to be assigned to each route and the problem becomes an assignment problem. Real-valued optimal numbers of vehicles are rounded up to the next integer since the number of available vehicles in the system is assumed to be non-restrictive. This setting is far less complex than the emergency logistic planning problem considered here. Equietal.(1996) consider a combined transportation and scheduling problem in a supply chain where the transportation problem aims to identify the optimal number of trips to satisfy demand (customers) from a given number of supply nodes (plants). The scheduling problem, on the other hand, identifies the number of trucks (of homogeneous capacity ) that has to be allocated to make the trips. Again,the routes are pre-specified and a vehicle assignment problem is solved rather than a routing problem. Linet Ozdamar (2004) concerned with planning logistics at a macro level in the presence of natural disasters. The research is motivated by the re-organisation project of the Turkish Armed Forces Natural Disaster Coordination Centre. Their logistics planning model proposed is intended to be a component of a Logistics Decision Support System.

But this article gives a new vision to emergency logistics programming in natural disasters.

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.48

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2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.48

174

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.48

174

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.48

174

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.107

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2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.107

409

2010 International Conference on Computational and Information Sciences

978-0-7695-4270-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCIS.2010.107

409

III. FUZZY CLUSTERING DISASTER GRADES BY HISTORICAL DATA

A. Variable Explanation

ix —— object i to be gradeified, ;,,2,1 ni =

ijx —— score j of object i , ni ,,2,1= , mj ,,2,1= ; 'ijx —— standardized score, ni ,,2,1= , mj ,,2,1= ;

jx —— sample mean of all objects’ score j , mj ,,2,1= ;

js —— sample standard deviation of all objects’ score j , mj ,,2,1= ;

R —— fuzzy similarity matix ;

21 ii rr —— similarity coefficient between object 1i and

2i , nii ,,2,1, 21 = ; kR —— operate Boolean product on R k

times, ,2,,4,2 nk = ; T —— fuzzy equivalent matrix; λ —— threshold value;

λT —— cut-off matrix of T corresponding to threshold value λ ;

B. Disaster Grades Fuzzy Clustering 1) Discourse Domain Charaterization: Discourse domain

is consisted of disaster areas to be clustered. { }n21 x,,x,xX =

ix is characterized by a group of scores. ( )imijiii xxxxx ,,21 ,,,= ni ,,2,1= ， mj ,,2,1= .

2) Fuzzy matrix identification: Original score matrix ( )

mnijxO×

= is consisted of survey data from disaster areas by history data.

O is standardized as below.

∑=

=n

iijj x

nx

1

1 (1)

( )∑=

−=n

ijijj xx

ns

1

21 (2)

j

jijij s

xxx

−=' (3)

Standardized matrix is ( )mnijxN

×= ' .

Similarity coefficient 21 ii rr is calculated based on N .

∑=

−−=m

jjijiii xxcr

1

''2121

1 (4)

c is a undetermined constant. Fuzzy matrix is consisted of similarity coefficient.

( )nniirR

×=

21 (5)

3) Fuzzy Equivalent Matrix Identification: Original score matrix ( )

mnijxO×

= is consisted of survey data from disaster areas by history data.

Operate R by Boolean operator to find its transitive closure.

RR =1 , 112 RRR ⋅= , kkk RRR ⋅=2 , ( )( )

nnkii

k rR×

=21

, ( )( )nn

kii

k rR×

= 2221

(6)

( ) ( ) ( ){ }⎭⎬⎫

⎩⎨⎧=

≤≤

ksi

ksiiins

kii rrr

2121

21,minmax

,1

2 , ,2,,4,2 nk = (7)

If kk RR 2= , will be defined as equivalent matrix. ( ) k

nnij RtT ==×

, ,2,,4,2 nk = (8) 4) Threshold value identification: Find out cut-off

matrixes of T corresponding to different threshold value λ .

( )nniiT

×=

21λλ ，

⎪⎩

⎪⎨⎧

<

≥=

λλ

λ21

21

21 ,0

,1

ii

iiii r

r (9)

Generally,when threshold value reduced,the number of grades will become smaller. The best clustering scheme is chosen from all gradeifications according to realistic problem.

C. Method of Assignment of Disaster Areas Priority Average losses is regarded as scores of different

grades.Suppose disaster Grade p consists q objects.Score of Grade p is calculated as below.

∑ ∑= =

q

r

m

jjir

xmq 1 1

'11 (10)

Disaster area where the greater score it is,the higher priority grade it has.

IV. GRADE DISTINGUISHMENT OF NEW DISASTER AREAS BASED ON FISHER DISCRIMINATION PROCEDURE

Suppose that all disaster areas are divided into ( )nkk < grades in the clustering analysis above.Each grade includes

( )kini ,,1= areas. new disaster areas are gradeified by utilizing fisher discrimination analysis.

A. Variable Explanation

iG —— grade i , ki ,,2,1= ;

in —— number of areas in each grade, ki ,,2,1= ; ( )ixα —— observation vector α of

iG , ( ) ( ) ( )( )ip

ii xxx ααα ,,1= , ki ,,2,1= , in,,2,1=α ;

( )xy —— discriminant function； x —— observation vector of area to be discriminate,

( )'1 ,, pxxx = ;

c —— coefficient vector of the discriminant function，( )'

1 ,, pccc = ; ( )i

x —— area vector mean of iG ;

175175175175410410410

( )is —— area covariance matrix of iG ; ( )i

y —— area mean of ( )xy corresponding to iG ; 2iσ —— sample variance of ( )xy corresponding to iG ;

x —— the mean vector of all grades; y —— area mean of ( )xy corresponding to all grades; λ —— Fisher criterion statistics;

iq —— positive weight coefficient； E —— matrix of difference in a group; A —— matrix of difference between the groups;

lλ —— generalized eigenvalue of A and E , ml ,,2,1= ;

( )xyl —— discriminant function l , ml ,,2,1= ; ( )lc ——codfficient vector of ciscriminant function

( )xyl , ml ,,2,1= ;

lp —— indicator used to measure the discriminant ability of ( )xyl ;

0msp —— indicator used to measure the discriminant

ability of ( ) ( )xyxy m0,,1 ;

( )ily —— area mean of function iG corresponding to

l , 0,,2,1 ml = , ki ,,2,1= ;

B. The Establishment of Discriminant Function Suppose the discriminant function as follows:

( ) pp xcxcxcxcxy +++== 2211' (11)

So we have: ( ) ( )ii

xcy '= (12) ( )csc i

i'2 =σ (13)

xcy '= (14) Under conditions of multi-grade,Fisher discriminant

analysis is to select the coefficient vector c to maximize:

( )

∑

∑

=

=⎟⎠⎞⎜

⎝⎛ −

=k

iii

k

i

ii

q

yyn

1

2

1

σλ (15)

If 1−= ii nq :

EccAcc

'

'=λ (16)

( )∑=

=k

i

ii sqE

1

(17)

( ) ( ) '

1⎟⎠⎞⎜

⎝⎛ −⎟⎠⎞⎜

⎝⎛ −=∑

=

xxxxnAik

i

ii (18)

To get the maximum value of λ ， according to the necessary conditions for existence of extremum:

EcAcc

λλ =⇒=∂∂ 0 (19)

It indicates that λ is the generalized eigenvalue of A and E . c is the eigenvector of A and E corresponding to λ . A is un-negative definite ， non-zero eigenvalue must be positive,that is 021 >≥≥≥ mλλλ .m discriminate functions can be proposed：

( ) ( ) xcxy ll

'= ， ml ,,2,1= ； (20) Give indicators of discriminate ability to each discriminant

function:

∑=

=m

ii

llp

1

λ

λ (21)

Define the discriminant ability of 0m discriminant functions is as beblow：

∑

∑∑

=

=

=

==m

ii

m

llm

llm psp

1

1

1

0

0

0

λ

λ (22)

If 0msp can be up to a certain specified value（for example

85%），then 0m discriminant functions are sufficient.

C. Grade Distinguishment of New Disaster Areas Here gradeification method using non-weighted method：

If 10 =m

( ) ( ) ( ) ( )i

j

kj

iGxyxyyxy ∈⇒−=−

≤≤1min ; (23)

If 10 >m

( ) ( ) ( ) ( )r

m

l

illki

m

l

rll Gxyxyyxy ∈⇒⎥⎦

⎤⎢⎣⎡ −=⎥⎦

⎤⎢⎣⎡ − ∑∑

=≤≤

=

00

1

2

11

2min

.(24)

V. RELIEF COMMODITIES AND EQUIPMENT DISTRIBUTING IN PROPER SEQUENCE IN DISASTER AREAS

A. Model Assumption Without considering the interaction between relief

commodities and equipment， expected utility maximum is fitter for the reality of each material when the overall utility of commodities also reached the optimal,that is, the usefulness of the sum of each material.

In the extent above ,we made cluster analysis of the disaster areas, and then determined the priority of all grades.Now we distribute a certain material to the areas.Generally,three grades are sufficient.Suppose they are 1G , 2G and 3G .

176176176176411411411

{ })1(211 , ixxxG ⋅⋅⋅= ;

( ){ })2(1)1( 212 , iii xxxG ⋅⋅⋅= ++ ;

{ }nii xxxG ⋅⋅⋅= ++ 213 )2()2( , .

1G has the first priority, 2G has the second priority, 3G has the third priority.

B. Variable Explanation

ia —— maximum demand of certain relief material at area i ;

ic —— utility value of relief material applied to area i , ni ,,2,1= ;

is —— the amount of material distributed to area i , ni ,,2,1= ;

b —— the availavle amount of relief material; wZ —— objective valure in objective programming ,

5,,2,1=w ;

C. Model of Commodities Distribution 1) Restrictions on the total distribution of relief

commodities：

∑=

=n

ii bs

1, ni ,2,1= ; (25)

2) Target constraints on geographical distribution of relief commodities:

ii as <= , ni 2,1= ； (26)

3) Conditions for the distribution of relief commodities to achieve optimum:

−+⋅= 111min ndPZ (27)

∑∑=

−+

++

=

=+−⋅n

iinn

n

iii cddsc

111

1

(28)

{ }{ } { }∑

= ≤≤≤≤

≤≤

−

−=

m

j ijniijni

ijniiji xx

xx

mc

1 11

1

minmax

min1， ni ,,2,1= ； (29)

4) Distribution of relief commodities to the first priority area must meet the conditions as follows：

)(min )1(122−− +⋅= iddPZ (30)

⎪⎩

⎪⎨

⎧

=+−

=+−

−+

−+

)1()1()1()1(

1111

..

iiii adds

addsts , ni ≤≤ )1(2 ； (31)

5) Distribution of relief commodities to the second priority region must meet the conditions as below:

)(min )2()1( 133−−

+ +⋅= ii ddPZ (32)

⎪⎩

⎪⎨

⎧

=+−

=+−

−+

+−

++

++

)2()2()2()2(

)1()1()1()1( 1111..

iiii

iiii

adds

addsts , nii ≤≤+ )2()1( 1 ; (33)

6) Distribution of relief commodities to the third priority area must meet the conditions as follows：

)(min 144 )2(−−

+ +⋅= ni ddPZ (34)

⎪⎩

⎪⎨

⎧

=+−

=+−

−+

+−

++

++

nnnn

iiii

adds

addsts

1111 )2()2()2()2(

.. , ni ≤≤ )2(2 ； (35)

Use SPSS to solve the model.we can get the amount of relief material distributed to each disaster area easily.

Figure 1. Multi-modal transportation network

VI. CONCLUSION A natural disaster logistics decision support system is

developed in this study. The aim is to coordinate logistics support for relief operations. The system readily decomposed into three problems the first one being fuzzy clustering disaster grades by historical data, the second one showing the grade distinguishment of new disaster areas based on fisher discrimination procedure, the last one giving plan incorporating relief commodities and equipment distributing in proper sequence in disaster areas. It is obvious that the system can cope with natural disasters of similar size within a reasonable computation time.

REFERENCES [1] Aggarwal, C.C., M. Oblak, and R.R. Vemuganti, “A Heuristic Solution

Procedure for Multi-commodity Iinteger Flows,” Computers and Operations Research, 1995, 22, pp. 1075–1087.

[2] Awerbuch, B. and T. Leighton, “Multi-commodity Flows: A Survey of Recent Research,” Proceedings of ISAAC’93, 1993, pp. 297–302.

[3] Barnhart, C. and Y. Sheffi, “A Network Based Primal-dual Heuristic for Multi-commodity Network Flow Problems,” Transportation Science, 1993, 27, pp. 102–117.

[4] Barnhart, C, “Dual Ascent Methods for Large-scale Multi-commodity Network Flow Problems,” Naval Research Logistics, 1993, 40, pp. 305–324.

[5] Bodin, L.D, “Twenty Years of Routing and Scheduling,” Operations Research, 1990, 38, pp. 571–579.

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