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Systems Engineering Procedia 5 (2012) 248 – 253

International Symposium on Engineering Emergency Management 2011

A Model on Emergency Resource Dispatch under Random Demand and Unreliable Transportation Xiang Lia*, Yongjian Lib a

College of Economic and Social Development, Nankai University, Tianjin 300071, P.R. China b Business School, Nankai University, Tianjin 300071, P.R. China

Abstract This paper develops an emergency resource dispatching model in which the demand for resource is random and the transportation channel can be unreliable. There is also a reliable but more costly transportation channel. A basic model is set up to optimize the expected total costs while ignoring the capacity constraint of reliable channel and the demand fill rate. We derive analytical solutions for the basic model and obtain managerial insights from them. The model is extended to incorporate the capacity constraint of reliable channel and the prescription of demand fill rate. A multi-period model framwork is also established on a more realistic multi-period situation. ©2012 2011Published Published Elsevier Selection and peer-review under responsibility Desheng © by by Elsevier Ltd.Ltd. Selection and peer-review under responsibility of DeshengofDash Wu. Dash Wu Open access under CC BY-NC-ND license. Keywords: Emergency resource dispatch; Unreliable transportation;Transportation engineering

1. Introduction Unconventional Emergency usually refers to some super natural and sudden disasters, creating a devastating threat to the national security and economy. On May 12, 2008, an 8.0 magnitude earthquake hit Sichuan Province in China, and the neighboring 10 provinces also suffered damages of different levels. The total earthquake damaged area was over 100,000 km2; as many as 69,226 people were killed and 17,923 people missing in this giant disaster. In such unconventional emergency, emergency resource dispatch includes the resource allocation and route assignment which is an important problem in transportation engineering. A good dispatching scheme could effectively help to protect the population and infrastructure, to reduce both human and property loss, and to rapidly recover. There have been extensive researches on this field. For example, Fiedrich et al. [1] studies the dispatching and transportation problem for emergency resources after an earthquake, with the objective to minimize the casualty. Fiorucci and Gaetani [2] studies the configuration and scheduling of emergency resources under a fire disaster by establishing and analyzing a dynamic model. Linet et al. [3] explores the support decision system for emergency resource transportation under consideration of the case that the transportation paths could be destroyed. Liu et al. [4-5] establishes emergency resource scheduling models under the constraints of the emergency rescue starting time and the rescue team number, respectively. Wang et al. [6] proposes a solution for emergency resource allocation among multiple disaster places to solve the conflict in allocation, but it does not consider the constraints in the allocation or any disruption risk.

2211-3819 © 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. doi:10.1016/j.sepro.2012.04.039

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Xiang Li and Yongjian Li / Systems Engineering Procedia 5 (2012) 248 – 253

This paper establishes a framework to model emergency dispatching problem under the situation that the demand for resource is random and there is also disruption risk in the transportation path of the dispatching. Besides the unreliable transportation path, we assume there is also a reliable transportation channel. We establish a basic model to determine the optimal dispatching quantities for multiple resources from each unreliable and reliable channel, with analytical solutions obtained. We then extend the model into the cases of capacitated reliable channel and prescribed demand fill rate. Finally, the multi-period model is developed. 2. Model Description Consider an emergency event, say, an earthquake, occurs at a location A and requires resources such as rescue materials, recovery equipments, and personality to be transported to A. Suppose there are n resources to be transported, but due to the sudden and high variation in the occurrence of the emergency, the demand of resource i is only known as a random variable Di , with distribution and density functions Fi () and fi () . If the demand is not satisfied, it may cost more severe property damage and casualties, and we transfer such loss into cost bi , for each unsatisfied demand of resource i . The nearest station to transport needed resource i is called station i , and the transportation cost is ci for each unit of resource i . However, potential disruption in the traffic due to the emergency event can cause the transportation failed from station i to location A. For example, an earthquake can cause great damage to the traffic network around disaster area. We denote the state of transportation from station i as T i , while we use T i ! 0 and

Ti

0 to denote the transportation ‘‘path’’ from station i to location A is up and down, respectively, with is drawn from a Discrete Time Markov Chain (DTMC) with state space * i {0,1,..., ni } . In other words, we use a DTMC to describe the disruption risk of transportation as time flows, for each resource i . This modeling approach has been utilized in the supply chain management literature, such as Ozekici and Parlar [7], and is adopted here to depict

[ wilm ] denote the transition probability matrix of DTMC of path state for resource i . We also assume for each i , Wi is aperiodic and irreducible, resulting that

the disruption caused by emergency event and disaster. Let Wi

DTMC is ergodic as it has a finite state space. Therefore, it has a steady distribution which we denote by

S i {S i0 , S i1 ,..., S in } . Hence, S i0 i

0

is the long-run disruption probability for resource i , and 1  S i is the long-

run probability that it is not disrupted. There is also a common resource center R in which all types of resources can be stored. Also, from R they can be transported to location A with no disruption. For example, after the earthquake the rescue goods can transported by air which is not influenced to the road destruction, which is usually called ‘‘green channel’’ in China. In contrast, we call the unreliable path from station i to A as ‘‘regular channel’’. However, the transport through this reliable green channel is more costly than through the regular channel as it requires extra investment in facility and personnel. Let the unit transportation cost be

ciG for resource i . We have ciG ! ci .

The problem is how many units of each resource to dispatch through each channel to satisfy the sudden occurred and stochastic demand Di , to minimize the expected total costs while satisfying constraints such as demand fill rate or green channel capacity constraint, dependent on the real scenario. The next section studies the basic, single period model ignoring these constraints, while the ones considering these constraints and the multi-period model are formulated in Section 4. 3. The Basic Model In this section we study the single period model while disregarding the constraints such as demand fill rate, etc. In this sense, the damage or loss of needed resource are totally reflected by shortage penalty bi . The advantage of this basic model is that we can derive analytical solution for the problem and generate explicit managerial insights.

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Xiang Li and Yongjian Li / Systems Engineering Procedia 5 (2012) 248 – 253

Si ( x) bEDi [ Di  x] , when [ x] max[0, x] . We also use event function 1() , in which 1( A) 1 if A is true and 1( A) 0 otherwise. For the single period model, the long-run probabilities of the state 0 0 of regular channel S i and 1  S i are used. The decision variables include the dispatch quantities of resource ))& & i through regular and green channels, qi and qiG . Denote q and q G be the decision vector consisting of qi and Denote

qiG , respectively. The problem thus can be formulated by: n n & ))& C (q, q G ) ET i ,i 1,...,n {¦1(Ti ! 0)[ci qi  Si (qi  qiG )]  ¦1(Ti i 1

i 1

n

0)[ Si (qiG )]  ¦ ciG qiG } i 1

n

¦{(1  S

0 i

)[ci qi  Si (qi  qiG )]  S i0 Si (qiG )  ciG qiG }

(1)

i 1

qi t 0, qiG t 0,

s.t.

i 1,..., n

According to the second equation in (1), we find that the optimization problem of total costs of dispatch n resources can be separated into n independent sub-problems which optimize the problem of dispatching one

& ))& G

resource. Note that C ( q, q ) is a jointly concave function. Thus, the optimal solutions satisfy the following KKT conditions of problem (1):

­(1  S i0 )[bi Fi (qi  qiG )  (bi  ci )] Pi ° ° Pi qi 0 °° Pi t 0 ® 0 0 G G G °(1  S i )[bi Fi (qi  qi )  bi ]  S i [bi Fi (qi )  bi ]  ci Oi °O q G 0 ° i i °¯Oi t 0 In (2), Oi , Pi are the Lagrangian multipliers.We have the following result by solving (2).

(2)

Proposition 1. The optimal dispatching quantities for basic problem can be characterized by: 0 0 G ­ *G 1 bi  (ci  (1  S i )ci ) / S i [ ( )] q F i ° i b ° i , i 1,..., n. ® b c  * 1 G  °q F ( i i )  q i i °¯ i bi

Proof. It can be easily verified that the solutions

qi* qi*G

Fi 1 (

bi  ci )  qiG , Pi bi

0 , qi*

Fi 1 (

Oi

bi  ci ) , Pi bi

(3)

qi*G

[ Fi 1 (

bi  (ciG  (1  S i0 )ci ) / S i0 )] , bi

0 satisfy the KKT conditions when S i0 !

ciG  ci , and the solutions b  ci

0, Oi ! 0 satisfy the KKT conditions when S i0 d

ciG  ci . The proposition b  ci

has been proved. ƶ Proposition 1 provides analytical solutions for the optimal dispatching quantities for n sources. It can be seen that unreliable regular channel is always adopted, but the reliable green channel is not. It is also interesting to note

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Xiang Li and Yongjian Li / Systems Engineering Procedia 5 (2012) 248 – 253

*G

that the total dispatching quantity, qi

 qi*

Fi 1 (

bi  ci ) is a constant only related to the transportation cost bi

through regular channel, the shortage penalty and the demand distribution, while it is irrelevant to the disruption risk and the transportation cost through green channel. Also, from Proposition 1 we can immediately obtain the following result. Corollary 1.

qi*G is increasing in S i0 . Also, the green channel is adopted if and only if S i0 !

ciG  ci . b  ci

Corollary 1 shows that the reliable green channel will be open to transport resource i if and only if the disruption risk of using the regular channel is sufficiently high, i.e., the probability that the regular channel is down is higher than a threshold. Moreover, the dispatching quantity through green channel is larger when the disruption risk increases. This result links the tendency of adopting reliable green channel with the disruption risk caused by the emergency event. Corollary 2. The total dispatching quantity

qi*G  qi* and dispatching quantity through green channel qi*G are

increasing in b . Corollary 2 shows that more units of resource should be transported totally and also by green channel, when the disaster area is more in need of the emergency resource, i.e., with larger shortage penalty b . This result links the tendency of adopting reliable green channel with the extent of emergency event need for resource. 4. Model Extension

4.1. Green channel capacity In some cases, the green channel has capacity, i.e., the maximum units of all resources transported is infinite. For example, the transportation flight has limited space, or the common resource center has limited storage capacity, etc. We denote this capacity as K . The problem with the green channel capacity is formulated as follows:

& ))& C ( q, q G )

ET i ,i

n

n

i 1

i 1

G 1,..., n {¦ 1(T i ! 0)[ci qi  Si ( qi  qi )]  ¦ 1(T i

n

0)[ Si (qiG )]  ¦ ciG qiG } i 1

n

¦{(1  S

0 i

)[ci qi  Si (qi  qiG )]  S i0 Si (qiG )  ciG qiG }

i 1

s.t.

G i

qi t 0, q t 0,

(4)

i 1,..., n

n

¦q

G i

d C.

i 1

It is easy to see that when C is large the optimal dispatching plan should be the same as in the basic model. However, for a generalized capacity, the problem cannot be optimized independently for each resource i like in the basic model because there is a common resource constraint, and thus there is no analytical solution for the optimal decisions. However, we can work on the problem by solving the following KKT conditions.

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Xiang Li and Yongjian Li / Systems Engineering Procedia 5 (2012) 248 – 253

­(1  S i0 )[bi Fi (qi  qiG )  (bi  ci )] Pi ° ° Pi qi 0 °(1  S 0 )[b F (q  q G )  b ]  S 0 [b F (q G )  b ]  cG  J i i i i i i i i i i i i °° G ®Oi qi 0 ° n °J ( q G  C ) 0 i ° ¦ i 1 ° °¯ Pi t 0, Oi t 0, J t 0

Oi (5)

4.2. Demand fill rate As the demand for resources are stochastic, it is difficult to ensure that all demands can be satisfied. However, it is possible to prescribe a demand fill rate, which is defined by the possibility that the demand is fully satisfied. We denote the demand fill rate for resource i as ri . According to the above analysis, we have

ri (1  S i0 ) Fi (qi  qiG )  S i0 Fi (qiG ) . If we prescribe that the demand fill rate should be above ai for resource i , then the problem can be formulated as: n n n & ))& C (q, q G ) ET i ,i 1,...,n {¦1(Ti ! 0)[ci qi  Si (qi  qiG )]  ¦1(Ti 0)[ Si (qiG )]  ¦ ciG qiG } i 1

i 1

i 1

n

¦{(1  S

0 i

)[ci qi  Si (qi  qiG )]  S i0 Si (qiG )  ciG qiG }

(6)

i 1

s.t.

qi t 0, qiG t 0,

i 1,..., n

(1  S i0 ) Fi (qi  qiG )  S i0 Fi (qiG ) t ai The problem has the separation property, and can be solved according to KKT conditions as follows:

­(1  S i0 )[bi Fi (qi  qiG )  (bi  ci )  Ei f i (qi  qiG )] Pi ° ° Pi qi 0 °(1  S 0 )[b F (q  q G )  b  E f (q  q G )]  S 0 [b F (q G )  b  E f (q G )]  c G ° i i i i i i i i i i i i i i i i i i i ® G °Oi qi 0 ° E (a  (1  S 0 ) F (q  q G )  S 0 F (q G )) 0 i i i i i i i ° i i °¯ Pi t 0, Oi t 0, E i t 0

Oi

4.3. Multi-period Model We can also extend the model into a multi-period setting. Consider the aftermath of emergency event is evolving as time goes by. We divide the total time horizon into T periods. Thus, the demand for resources can be different during each period, and we suppose the next period demand can be forecasted by the current one in an updating way, with probability distribution FDit 1|Dit () for resource i . We also assume the disruption risk state as a DTMC with transition probability matrix Wi

[ wilm ] for resource i . Finally, we assume the unmet demand of the current

period can be further satisfied by the resources transported in next period. The problem can be formulated by the following stochastic dynamic programming:

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Xiang Li and Yongjian Li / Systems Engineering Procedia 5 (2012) 248 – 253 n

Cit (dit 1 , Tit 1 , xit 1 ) n

¦1(T

max q

G it , qit

ETit , Dit {¦1(Tit ! 0)[ci qi  Si (qi  qiG )]  i 1

n

it

i 1

0)[ Si (qiG )]  ¦ ciG qitG Cit ( Dit , Tit , xit )} i 1

G it

qit t 0, q t 0,

s.t.

Tit 1 o Tit Dit 1 o Dit xt

with with

Wi

[ wilm ]

(7)

FDit 1|Dit ()

Dt  qtG  1(Tit ! 0) qi

t 1,..., T

 i 1,..., n

The problem can be solved by the standard backwards recursive approach, to first solve the sub-problem in period T , and then T  1 , until back to the beginning of time horizon. However, this procedure can be very complicated as there are 3 states, so a good heuristic or artificial optimization approach could be used. 5. Conclusion This paper develops an emergency resources dispatching model for the case of random demand and unreliable transportation channel. It is an attempt to use the transportation engineering theory into emergency management. We also incorporate a reliable but more costly transportation channel, which is usually called green channel. A basic optimization model is set up without considering the capacity constraint of reliable channel and the demand fill rate, with analytical solutions derived and managerial insights generated. The model is extended to the cases with constraints of reliable channel capacity and demand fill rate. Finally, a multi-period model framework is also established. Future research directions include to more developing good algorithms to solve the extended problems which have no analytical solutions. Furthermore, the determination of reliable channel capacity is also an interesting problem. Finally, we can also incorporating vehicle routing and facility location into this problem.

Acknowledgments This work was partly supported by National Natural Science Foundation of China (NSFC) Nos. 70971069, 71002077, and 91024002.

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