A bi-level optimization model for aid distribution after the occurrence of a disaster

Journal of Cleaner Production xxx (2014) 1e12

Contents lists available at ScienceDirect

Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro

A bi-level optimization model for aid distribution after the occurrence of a disaster
-Fernando Camacho-Vallejo a, *, Edna Gonza
lez-Rodríguez a, F.-Javier Almaguer a, Jose
lez-Ramírez b Rosa G. Gonza a
ticas, Av. Universidad s/n, San Nicola
s de los Garza, Nuevo Leo
noma de Nuevo Leo
n, Facultad de Ciencias Físico-Matema
n CP 66450, Universidad Auto Mexico b
lica de Valparaíso, Av. Brasil 2241, 2362807 Valparaíso, Chile Escuela de Ingeniería Industrial, Pontificia Universidad Cato

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 February 2014 Received in revised form 26 August 2014 Accepted 18 September 2014 Available online xxx

In this paper the authors propose a bi-level mathematical programming model for humanitarian logistics to optimize decisions related to the distribution of international aid after a catastrophic disaster. In this situation, non-profit international organizations and foreign countries will offer to help by shipping certain necessary products, such as bottled water, food and medicine, but will want to minimize the shipping costs. At the same time, the affected country seeks to distribute the received aid to the affected areas as efficiently and quickly as possible. To deal with the considered problem, we reformulate the bilevel model and reduce it into a nonlinear single-level mathematical model; then we linearize it to obtain a mixed integer programming problem. To validate the proposed mathematical model, we consider as a case study the earthquake in Chile in 2010. We solve the reformulated model by using real data simulating the aid distribution scenario, and we interpret the obtained numerical results, giving some recommendations for aid distribution following future disasters. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Bi-level programming Humanitarian logistics Distribution problem Disasters

1. Introduction In recent years, several disasters have occurred with catastrophic consequences for the affected population e among them the earthquakes in Haiti and Chile in 2010 and the tsunami along the coast of Japan in 2011. These occurrences led many countries and international organizations to send money, volunteers and commodities as aid for these devastated regions. Some of the most commonly sent products e medications, canned food, bottled water and diapers e urgently need efficient distribution in the affected regions to avoid increasing deaths from starvation and disease. Caunhye et al. (2012), noted the necessity and importance of the efficient distribution of the sent products; also, they listed some relevant contributions done on aid distribution and casualty transportation. Furthermore, Wang and Rong (2007) and Ji and Zhu (2012) presented works emphasizing the importance of speed and efficiency in the supply chain in these situations.

* Corresponding author. Tel.: þ52 81 83294030x7125; fax: þ52 81 8352 5961. E-mail addresses: [email protected] (J.-F. Camacho-Vallejo), edna_
lez-Rodríguez), [email protected] (E. Gonza [email protected] (F.-J. Almaguer), [email protected] (R.G. Gonz
alez-Ramírez).

The branch of engineering devoted to studying these problems, called Humanitarian Logistics, focuses on analyzing problems associated with the storage and distribution of products required by the affected population due to the occurrence of a disaster. For example, Wamsler et al. (2013) and Dües et al. (2013), consider post-disaster response and recovery following climate-related disasters caused by the impact of growing urban areas. Gupta et al. (2002), pointed out the importance of efficient planning for building business centers because of the environmental risk triggered by a natural disaster. No matter the original cause, one can analyze any disaster situation in two forms: proactive and reactive. Proactive refers to the study of the problem of interest before a disaster happens, and reactive to the case when a disaster has already happened and some actions and decisions must occur. Altay and Green (2006) presented an extensive review of literature on the problems related to humanitarian logistics and divided the stages related to these disasters. Mitigation analyzes the actions needed to reduce the probability of occurrence of a disaster or to reduce the impact after one occurs. Preparation refers to the planning of activities to follow in case a disaster happens. Response involves the post-occurrence employment of resources and emergency procedures to preserve life, infrastructure, environment, and the social, economic and political structure of the affected

http://dx.doi.org/10.1016/j.jclepro.2014.09.069 0959-6526/© 2014 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

2

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

community. Finally, Recovery concerns the actions taken to return the affected area to normalcy, either in the short or medium term. This paper focuses specifically on analyzing a particular situation related to the Response stage; the considered problem addresses efficient aid distribution in the affected areas. In this stage, most of the situations studied involve distribution and transportation problems e generally difficult to model due to the many parameters or variables in real-life problems. Duman (2007) described the difficulties and the importance of this problem. If a disaster has occurred, the demands of the affected areas constantly change and even nodes and edges of the network disappear. Fiedrich et al. (2000) proposed an integrated dynamic model to minimize the number of expected deaths in each time period. To test their model, the authors implemented a tabu search algorithm and a simulated annealing method. Under this same objective function approach, Sheu and Chen (2005) proposed a three-stage algorithm: they first clustered the affected areas based on the demand characteristics and priority of each; after that, they identified the gravity center of each cluster for the delivery of aid; and, finally, they determined the vehicle routing for distribution. As a continuation of this work, Sheu (2007) added a case study of an earthquake in Taiwan, the proposed supply chain scheme appearing very similar to the one considered in our work. Vitoriano et al. (2011) proposed a model where they jointly consider structure of the transport network, vehicle routing and multi-commodity problems; they then applied the resulting multi-criteria optimization model to an illustrative case study based on the earthquake in Haiti in 2010. Rath and Gutjahr (2014) also considered the problem of storage centers responsible for receiving external help and developed a multi-objective function to minimize the cost of operating a storage center and maximize the covered demand. They reformulated the problem as a mixed integer linear programming model and then solved it with a stage approach by using a heuristic and comparing the results with those obtained directly after applying the well-known NSGA-II. Decision-makers also need systems to help make the decisions needed after an emergency. Rathi et al. (1993) developed a linear programming model in a supply chain for an emergency situation; it involved a deterministic demand for each affected area and the storage facility supplying the specific area. They reduced the problem to determine the number of trucks assigned to each route with no limitation on the number of available trucks e a simple € model in comparison with the one analyzed by Ozdamar et al. (2004). In the latter paper, they studied the case where the problem combined the multi-commodity flow problem in a network and the vehicle routing problem. They solved the problem in two stages; first, they generated the distribution plan and then the quantities of each product. Importantly, trucks did not have to return to the distribution center because they assumed easy relocation of the distribution centers based on demand. The selected resolution method used Lagrangian relaxation, and they experimented with the real case of an earthquake in Turkey. Also, Wex et al. (2011) provided two approaches to make efficient decisions related to the collection of donations in kind and their distribution. One of the proposed models, an allocation model, minimizes the time needed when helping in an affected area; meanwhile, the other model, a distributed collaboration one, could address simultaneously several affected areas. The previous models considered a known demand, whether obtained by a forecast, a fixed value or otherwise. Since we do not know the magnitude of a disaster or the time of its occurrence, however, one must consider the demand of the affected area as stochastic; thus, the existence of several models of two-stage stochastic programming to analyze humanitarian logistics problems. For example, Barbarosoglu and Arda (2004) proposed a model

where in the first stage they minimized transportation costs and resources and then in the second stage they minimized the costs related to flow and inventory. Using this stochastic problems' classification, but at the same time focused on the Preparation stage, research presented by Mete and Zabinsky (2010) sought to minimize either the duration of medical products' transportation or the unmet demand. In addition, the authors presented a case study of the city of Seattle. Finally, Rawls and Turnquist (2010) sought to have an inventory of products required for later distribution and proposed a heuristic algorithm based on the L-shaped method (capable of solving large scale problems); they validated it through a case study of a hurricane in the area of the Gulf Coast of the United States. Works also exist on humanitarian logistics problems analyzed by using predefined hierarchies involved in a disastrous situation. € Ozdamar and Demir (2012) considered a hierarchical model of clustering and routing. They conducted the clustering in a multilevel way, making smaller clusters of affected regions in each decision level. They considered the routing part, a capacitated network flow problem, independently from the clustering part and solved directly by an optimizer. Also, Liberatore et al. (2014) proposed a hierarchical model e first seeking to restore roads damaged by disaster and then distributing in a better way the assistance requested. They emphasized the importance of coordinating the distribution of aid through the restored roads. To show the validity of their model, they analyzed a case study based on an earthquake in Haiti. Although these models consider hierarchies, none of them belongs to the area of multi-level programming. Remarking on the differences between these two approaches, Gramani et al. (2011) concluded combining two problems with inter-related variables outperforms the solution obtained by considering the problems sequentially; they illustrated this fact considering a production planning problem combining lot sizing and cutting stock problems. The model we propose in this paper involves a bi-level programming problem, a particular case of multi-level programming considering only two decision levels. The problem considered focuses on the efficient delivery of aid in kind to storage centers. This aid must go to the affected areas quickly and effectively, but at the same time, the countries or international organizations providing help want to lower shipping costs. The following section of this paper describes in general the structure of a bi-level programming problem and presents a literature review of related works concerning bi-level problems in humanitarian logistics. In Section 3, we present a bi-level model of aid distribution. A reformulation of the bi-level model (reducing it to a single-level mathematical problem) appears in Section 4. Section 5 presents a case study based on an earthquake in Chile in 2010 and the computational and experimental analysis of the results. Finally, we conclude this paper with Section 6 providing some relevant observations and noting some areas of opportunity for future research. We present additional information about the solutions obtained during the computational experimentation in the Appendix. 2. Bi-level programming problems Many real situations involve decision-makers in two levels related by a preset hierarchy. The existence of this hierarchy precludes consideration of the problem in the category of multiobjective optimization; bi-level programming encompasses these problems. In them, one finds the decision-makers in the upper level called the leader and in the lower level called the follower. At both levels, constraints may exist and each of the involved decisionmakers will control a set of the variables of the problem and try to optimize his/her own objective function. The leader makes a

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

decision and, based on the action, the follower reacts, thus the follower optimizes his/her objective considering the decision made by the leader. Therefore, the leader should select the decision optimizing his/her objective function, taking into account the reaction of the follower; bi-level programming problems exist as mathematical programs, where a subset of the variables must contain the optimal solution of another mathematical program. A general model of a bi-level programming problem can look like the one proposed by Bracken and McGill (1973):

min Fðx; yÞ x2X

s:t:

Gðx; yÞ  0

min f ðx; yÞ y2Y

s:t:

gðx; yÞ  0

x; y  0 As seen in this model, the leader tries to minimize F(x,y), selecting a x 2 X and considering the best response of the follower, y*(x), minimizing f(x,y) in the lower level problem. One may also consider functional constraints in the form G(x,y)  0 and g(x,y)  0 in each level of the problem. These problems, generally nonconvex, are difficult to solve. Dempe (2002) discussed many specific complications arising in these problems and also provided some methodologies to analyze and reduce them into a single-level mathematical program. As already mentioned, a wide variety of applications exist, modeled with bi-level programming. Vicente and Calamai (1994), Bard (1998) and Colson et al. (2007) presented relevant literature reviews emphasizing this point. After an extensive literature review, we found in the area of humanitarian logistics very few works modeling real-life situations as bi-level programming problems. For example, from the point of view of analyzing disasters caused by man, Arroyo and Galiana (2005) considered the problem of a terrorist threat as a bi-level problem. At the upper level, the terrorist seeks to maximize the damage attacking the minimum number of lines in a power system while in the lower level a threatened company wants to minimize spilled loads caused by the attack. Those authors reduced the bi-level problem (proposed as a nonlinear mixed integer problem) to a single-level mixed integer linear problem by using the optimality conditions of KarusheKuhneTucker applying some restrictions to avoid nonlinearity. Recently, Aksen and Aras (2012) analyzed a leader-follower game with the aim of protecting the facilities to prevent reallocation of customers in case of a terrorist attack. The leader seeks to minimize the sum of the costs incurred to install, protect and use the facility. The follower seeks to destroy unprotected facilities to affect the supply capacity of the remaining plants. To solve the problem, the authors proposed an algorithm based on tabu search. Also, Losada et al. (2012) formulated the problem of protecting the facilities as a mixed integer linear bi-level program analyzing the recovery time of a system after an incident occurs considering the possibility of more incidents over time. None of these three papers analyzed a situation belonging to the Response stage. Regarding catastrophes caused by natural disasters, we only found a few articles. On the Preparation stage, Kongsomsaksakul and Yang (2005) proposed a bi-level model for locating shelters after a massive flood. The upper level consisted of locating shelters to minimize the total evacuation time and the lower level represented the evacuees, who chose the shelter to which they went and their evacuation route. To solve it, the authors proposed a genetic algorithm using real data obtained from a dam located in Utah. Related to this topic, Li et al. (2012) formulated a stochastic bi-level

3

programming problem where the upper level consists of a model to locate and allocate the shelters and the lower level contains the evacuees who seek the best current routes. Those authors analyzed a multiple scenario experimentation to measure the performance of their proposed heuristic algorithms. Finally, Barbarosoglu et al. (2002) analyzed a transportation problem of minimizing the cost of assigning pilots to helicopters and, then, minimizing the relief time by assigning helicopters to operation bases. Also, Feng and Wen (2005) considered the bi-level problem where an earthquake affected the area's transportation network, the leader tried to maximize the flow of vehicles entering the affected area to provide assistance, and the followers looked to travel through an unaffected route to minimize their total travel time. This situation generated traffic jams, negatively impacting relief efforts and recovery. Based on this, a government agency should regulate use of existing roads. Our problem of interest appears as the following bi-level problem: when devastating disaster strikes, many countries and international organizations send aid to the affected country. In the upper level, the affected country (leader) must choose the means of transportation and the speed of distribution of relief goods. Having many points (storage centers) where the aid can arrive, the affected sites will need aid requiring some products with different priority. Finally, on the lower level, countries or international organizations (followers) can choose the storage center where they ship the aid to minimize shipping cost. In a general way, both problems (upper and lower levels) seem similar to the supplier selection problem (see Sanayei et al., 2008), except for the inter-related decisions under a given hierarchy. For the first time, we consider those helping, not just the affected country. The ones sending help will seek to reduce the cost of their actions; this fact motivates us to consider the lower level of the proposed problem. Holguin-Veras et al. (2013) establish the importance of an adequate objective function in humanitarian logistics; they criticize consideration only of the distribution costs in the objective function in post-disaster cases. Also, Cruz (2013) mentions some benefits from taking into consideration the social responsibility of the involved parties in the supply chain. Accordingly, the bi-level objective function we propose reflects the total response time for delivering aid to the affected zones.

3. The bi-level programming mathematical model We can describe the model as follows: i 2 I are countries or international organizations helping the affected country; j 2 J are specific places where they can receive help in kind in the affected country (storage centers); k 2 K are the places in urgent need of help; l 2 L denotes a specific product (potable water, medicines, canned food, clothing, paper, etc.); and, finally, m 2 M stands for the means of transportation used to send or distribute products (land, air or sea). 1 as the time a shipment of the product l takes to get by Define tijlm means of transportation m from the helper country or international 2 organization i to the storage center j; similarly tjklm is the time it takes to distribute a shipment of the product l by means of transportation m from the storage center j to the affected area k. Also, we need to consider a volume vl occupying a cargo-hold of the product l and with a limited space capacity Vj in each storage center j. In each affected area k, we know demand Dkl for each article l, and every country or aid agency i will have a maximum quantity available Hil of each product l. Finally, let cijm equal the cost of sending a shipment by means of transportation m from the country or aid agency i to the storage center j. The decision variables for our problem are:

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

4

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

xijlm ¼ amount of shipments of the product l sent by means of transportation m from the country or aid agency i to the storage center j. yjklm ¼ amount of shipments of the product l sent by means of transportation m from the storage center j to the affected area k. Hence, we have the resulting model as follows:

XXX X

min

yjklm ;xijlm

s:t:

1 Tijlm þ

i2I j2J l2L m2M

X X X

vl yjklm  Vj ;

X X X X

2 Tjklm

(1)

j2J k2K l2L m2M

c j2J

(2)

k2K l2L m2M

X X

yjklm  Dkl ;

c k2K; l2L

(3)

c j2J; k2K; l2L; m2M

(4)

j2J m2M

yjklm 2Z þ ∪f0g xijlm 2arg min xijlm

s:t:

X X

XXX X

xijlm ¼

i2I m2M

X X

cijm xijlm

(5)

yjklm ;

(6)

i2I j2J l2L m2M

X X

c j2J; l2L

k2K m2M

xijlm  Hil ;

c i2I; l2L

(7)

j2J m2M

xijlm 2Z þ ∪f0g;

c i2I; j2J; l2L; m2M

(8)

1 and T 2 where, the auxiliary variables Tijlm are defined as follows: jklm

 1 Tijlm ¼

1 if xijlm  1 tijlm 2 and Tjklm ¼ 0 otherwise



2 if yjklm  1 tjklm 0 otherwise

The bi-level problem is defined by Constraints (1)e(8). In Constraint (1), the objective function of the upper level appears and it shows the leader wanting to minimize the total response time for delivering aid, i.e., the time needed to send from the aid agency i to the storage center j thence to the affected area k. Since the consideration that there are no restrictions about the number of available vehicles for sending aid is made; we also assume that those vehicles could leave their origins in a “simultaneous” way. In other words, we neglect the time between the departures. Thus, by considering (1) as the objective function, the total response time is 1 and interpreted as the time until help arrives. The definition of Tijlm 2 Tjklm implies that, for each means of transportation, if at least a single vehicle is sent for helping the needed country, the associated time will be considered in the objective function without regarding the amount of sent vehicles. Other objective functions may suit this model (e.g. minimization of penalties for late deliveries, unmet demand, human suffering and weight factors), but for this paper we decided to choose only equation (1). Constraint (2) reflects the available space in each storage center j; (3) dictates the problem must satisfy the demand for each product l needed in each affected area k; and Constraint (4) requires the number of shipments of each product be non-negative. Constraint (5) makes this problem a bi-level programming model; it implies variables xijlm must serve as the optimal solution of the problem defined by Constraints (5)e(8). Hence, Constraint (5) is called the objective function of the lower level and indicates the desire to minimize the cost of sending help from the aid agency i to the storage center j. Equation (6) says send only the

required amount for storage facility j of each product l. Expression (7) ensures a country or aid agency i cannot send more than the available quantity for each product l, and Constraint (8) indicates the non-negativity for the shipments of each product from the aid agency i to the storage center j. To consider a well-defined bi-level problem, if any leader's decision y has a non-unique optimal solution of the lower level problem, then the optimistic approach will be considered. In other words, when the follower has more than one optimal solution for a predefined leader's decision, the follower will make his/her decision as the more convenient also for the leader. In this case, if a country can send aid to two different storage centers with the same cost, it will make the decision associated with the shortest time. Throughout this paper, we assume knowledge of the location of the storage centers and agencies available to help. Also, supported by the relevant conclusions made in Schliephake et al. (2009), where they remark on the importance of good relations between the involved parties in the supply chain to find more efficient solutions with greater cost savings, we assume the existence of a central agency coordinating these countries and/or aid agencies to avoid sending unnecessary aid and to make resources work more efficiently. This assumption will help to show the importance of having a central agency fully coordinating the process, instead of considering each country sending aid without having communication with the others. As mentioned above, we also assume no restrictions on the number of vehicles available for the distribution of aid. In real situations, the number of transportation modes available to send aid is limited but large enough to fulfill the demand due to the small number of required vehicles, so this assumption also seems reasonable and common in humanitarian logistics. Without enough vehicles available, the demand may remain unsatisfied, causing undesired situations. And note, the same vehicles will have the same capacity; i.e., all planes will have the same capacity, different from the capacities of ships and trucks. The outline of the model is as follows: the leader (country concerned) decides how to distribute aid from the storage center to the places in need; this decision sets the variables yjklm. Because of this, a joint known demand will exist for each product l the affected country requires in each storage center j. Based on this demand, a humanitarian organization coordinates with the other countries or aid agencies wanting to participate to minimize the shipping cost and meet the demand conditions; this decision fixes the variables xijlm. With yjklm and xijlm, the leader evaluates his/her objective function seeking to minimize the total time of the supply chain required to distribute the aid to the affected areas k. The outline of the model considered in this paper looks similar to the classical multi-stage supply chain where one aims is to minimize both (1) transportation costs between plants and distribution centers and (2) the distribution cost incurred from the distribution centers to the customers, but the objective functions in humanitarian logistics are not commonly interested in minimizing costs. The diagram of the model appears in Fig. 1. 4. Model reformulation In this section, we reformulate the bi-level programming problem into a single-level nonlinear programming problem. If in the bi-level problem (1)e(8) we fix the variables yjklm, the lower level problem (5)e(8) will become a transportation problem, and, due to the uni-modularity of the matrix related to the constraints, a solution will be integer if the right hand sides are integer. Then we can relax the lower level variables' constraint (8) to xijlm  0. Hence, we can replace the lower level problem by the primal-dual optimality conditions. Define ajl, c j2J; l2L and bil, c i2I; l2L as the

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

Countries or agencies where the aid comes from

5

Catastrophically affected areas

Strategic storage centers in the affected country

Fig. 1. Representation of the problem.

dual variables corresponding to constraints (6) and (7), respectively. The dual problem associated with the lower level is formed by:

max ajl ;bil

s:t:

XX

X X

j2J l2L

k2K m2M

ajl þ bil  cijm ;

ajl urs;

! yjklm ajl þ

XX

0 bil @Hil 

Hil bil

yjklm 2Z þ ∪f0g;

c i2I; j2J; l2L; m2M

(10)

c j2J; l2L

s:t:

(12)

XXX X

1 Tijlm þ

i2I j2J l2L m2M

X X X

xijlm  0;

vl yjklm  Vj ;

yjklm  Dkl ;

X X X X

xijlm ¼

i2I m2M

X X

j2J

k2K; l2L

X X

(22)

yjklm ;

j2J; l2L

i2I; l2L

i2I; l2L

(23)

(14)

  xijlm  M 1  gijlm

(15)

cijm  ajl  bil  Mgijlm

(16)

k2K m2M

xijlm  Hil ;

i2I; j2J; l2L; m2M

(21)

(13)

2 Tjklm

j2J k2K l2L m2M

j2J m2M

X X

j2J; k2K; l2L; m2M

Constraint (13) is the objective function of the single-level nonlinear problem. Constraints (14)e(17), (21) and (22) provide the primal feasibility. Constraints (18) and (23) ensure dual feasibility. Finally, constraints (19) and (20) guarantee the achievement of the optimal value for the lower level problem; however, we will also lose the linearity in the reformulated model. To linearize constraints (19) and (20), first, for constraint (19) consider the auxiliary variables gijlm 2 {0,1} and let M be a sufficiently large positive constant. Since xijlm  0 and cijm  ajl  bil  0, we must add the following sets of constraints.

k2K l2L m2M

X X

(20)

(11)

c i2I; l2L

yjklm ;xijlm ;ajl ;bil

i2I; l2L

(9)

i2I l2L

Adhering to the existing theory on reductions to the bi-level programming, we consider the complementary slackness constraints. Hence, we obtain the following equivalent single-level nonlinear programming problem:

min

xijlm A ¼ 0;

j2J m2M

bil  0; bil  0;

1

X X

i2I; j2J; l2L; m2M

i2I; j2J; l2L; m2M

(24)

(25)

In a similar way, we linearize equation (20); let dil 2 {0,1} be auxiliary variables and, considering bil  0 and P P Hil  xijlm  0, the corresponding linearizing constraints are: j2J m2M

(17)

bil  Mdil

(18)

Hil 

(19)

Finally, the reformulated single-level mixed integer model is as follows:

i2I; l2L

(26)

j2J m2M

ajl þ bil  cijm ;

i2I; j2J; l2L; m2M

  xijlm cijm  ajl  bil ¼ 0;

i2I; j2J; l2L; m2M

X X

xijlm  Mð1  dil Þ

i2I; l2L

(27)

j2J m2M

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

6

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

XXX X

min

yjklm ;xijlm ;ajl ;bil ;gijlm ;dil

1 Tijlm þ

i2I j2J l2L m2M

X X X X

2 Tjklm

j2J k2K l2L m2M

The resulting problem is (13)e(18), (21)e(23) and (43). Equation (43) loses linearity in the model, and it cannot be straightforward to linearize.

(28) s:t:

X X X

5. Case study: earthquake, Chile, 2010

vl yjklm  Vj ;

j2J

(29)

k2K l2L m2M

X X

yjklm  Dkl ;

k2K; l2L

(30)

j2J m2M

X X i2I m2M

X X

X X

xijlm ¼

yjklm ;

j2J; l2L

(31)

k2K m2M

xijlm  Hil ;

i2I; l2L

(32)

j2J m2M

ajl þ bil  cijm ;

i2I; j2J; l2L; m2M

  xijlm  M 1  gijlm

i2I; j2J; l2L; m2M

cijm  ajl  bil  Mgijlm bil  Mdil Hil 

(33) (34)

i2I; j2J; l2L; m2M

(35)

i2I; l2L

X X

(36)

xijlm  Mð1  dil Þ

i2I; l2L

(37)

j2J m2M

yjklm 2Z þ ∪f0g;

j2J; k2K; l2L; m2M

(38)

xijlm 2Z þ ∪f0g;

i2I; j2J; l2L; m2M

(39)

bil  0;

i2I; l2L

gijlm 2f0; 1g; dil 2f0; 1g;

(40)

i2I; j2J; l2L; m2M

(41)

i2I; l2L

(42)

In the latter model, the objective function needs to be carefully 1 and treated when solving the problem due to the definition of Tijlm 2 . Also, it can be appreciated that two additional sets of variTjklm

ables are introduced. Let j$j denote the cardinality of a particular set. Hence, in total, jIjjJjjLjjMj þ jIjjLj binary variables are added. Also, the single-level mixed integer model has jIjjJjjLjjMj þ jIjjLj more constraints than the single-level nonlinear programming model. In any event, linearizing the model allows solving it with CPLEX optimizer. If this linearization were impossible, we would need a heuristic algorithm to solve the nonlinear model. Another way to reduce the bi-level problem into a single-level one is by removing equations (19) and (20) and considering the equality of the objective function for the lower level problem and the corresponding value for its dual problem:

XXX X i2I j2J l2L m2M

cijm xijlm ¼

XX

X X

j2J l2L

k2K m2M

þ

Hil bil

XX i2I l2L

! yjklm ajl (43)

To illustrate and apply our proposed model, we consider as a case study an earthquake in Chile in 2010. A group of countries helped Chile by sending basic products for distribution among the most affected zones. Countries needed to determine the most convenient transportation modes to ship the products. Products shipped to storage centers located at the most affected zones, then to the affected zones e possibly smaller storage centers located at suburbs or main cities. The sending countries intended to minimize the associated distribution costs of the products, while, on the other hand, Chile would have liked to have received the products as soon as possible to distribute them among the most affected people; this situation implies high distribution costs. Given the magnitude of the event, the demand of products corresponded to a planning horizon of 10 weeks e the time required in the first stage of recovering from the catastrophic event. Hence, the research question consists of determining the configuration of the international help shipments to the affected zone in a way to minimize distribution costs and to deliver products as soon as possible to the affected zones. To generate the instance of the problem, we analyzed several secondary sources of information, for example, press notes and information of different organizations, such as the National Emergency Office (ONEMI by its acronym in Spanish), Red Cross, etc. We consider 12 countries supporting Chile: Germany, Argentina, Bolivia, Brazil, Colombia, Cuba, Ecuador, United States, Spain, Mexico, Peru and Venezuela. Air, maritime and land transportation modes served as the alternatives for sending the products from the countries to Chile. Due to the proximity between the countries, land transportation mode occurred only for Peru, Bolivia and Argentina. Shipments went out from a single point from each country e an airport, maritime port or one of the main cities. On the other hand, for Chile all the products sent by air went to the Airport of the Army Forces (FACH by its acronym in Spanish) located at Pudahuel, Santiago. This location has an extremely large storage capacity. Products then went to storage centers by trucks. For those products sent by maritime transportation, they went to the Port of Valparaiso, and from there, by trucks to storage centers. Finally, products sent by land transportation mode went directly to storage centers by trucks. International and national organizations, such as Red Cross, Caritas and a Roof for Chile, as well student associations, operated the storage centers. We will consider only those storage centers used exclusively for products of international help. These storage centers satisfy the demand of among 2e4 communes. The affected zones could represent either a single commune or several small aggregated communes. The most affected zones with the earthquake and tsunami correspond to the South-Central Region of Chile, this area accounts for six Regions: Metropolitan Region of Santiago, the Region of Valparaiso, the Region Bernardo O'Higgins, the region of Maule, the Region of Bio-Bio and the Region of la Araucanía. Among these six regions, the Region of Maule and the Region of Bio-Bio suffered the most damage. We consider 17 storage centers and 44 affected zones, both single and multiple communes. According to the ONEMI, two million victims resulted as a total. Based on the Richter scale of the earthquake and the level of damage in each region, we distributed for the model victims among the 6 Regions. In general, the number of victims and the demand of products for a planning horizon of 10 weeks define the capacity of the storage centers.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

We consider four categories of products: liquids (water, milk, etc.), non-perishable food (canned food, rice, sugar, oil, etc.), personal products (hygiene, clothes, blankets, etc.) and medicine. Then, we consider two basic kits per victim. Kit-1 consists of food and liquids for a person for one week. Kit-2 consists of personal products and medicine, each for two and a half weeks. Hence, for the planning horizon of 10 weeks, a demand exists of ten Kits-1 and four Kits-2 per victim and a demand per kit based on the victims of each affected zone. To estimate the demand of each product, we consider the demand of the kits and the products and quantities required per kit, then translate demand into pallets, where one pallet equals 1000 litters and 1000 kg. We assume (1) countries have a limited capacity in terms of the products they can ship to the affected zones as a consequence of the economy of each nation and (2) in total, the 12 countries can send the total demand required for 10 weeks. The transportation mode and the distance to travel determine the traveling costs and time. One TEU equals 30 pallets. For land shipments, we consider as a base the cost of a fleet between Santiago and Valparaíso (120 km). For maritime shipments, we consider as a base the cost of a shipment between Barcelona and Valparaiso per TEU; and for air shipments, we consider the cost of sending one TEU from Barcelona to Santiago. The rest of the costs were estimated proportionally to the distance traveled. The transportation mode determines the traveling times and includes customs procedures, loading and unloading times, and times at the border point. For land transportation within Chile, the distance traveled determines costs and traveling times, including loading and unloading times. 6. Numerical experimentation In this section we describe the computational experimentation conducted to analyze the case study. To give some recommendations about the disastrous situation modeled in this paper, we propose a comparison between three different models from different points of view, but considering the same situation. We refer to the first model as the Leader's perspective, consisting of solving the problem of receiving and distributing the aid without considering the opinion of the countries helping; this means the objective function of the problem consists in minimizing both the time required for receiving the aid from the helping countries and the time incurred by distributing the products to the affected zones. The Leader's perspective model is defined by constraints (1)e(4) and (6)e(8). In this case, the leader controls both of the variables without considering the follower's objective function. The second model, named the Follower's perspective, has a description similar to the first one. In this model, we assumed the countries sending aid decide all the logistics in the affected area. In other words, the distribution time is not going to be the main objective of the problem; instead, the minimization of the shipping cost would be the aim. This model considers the objective function defined by constraint (5) subject to the constraints given by (2)e(4) and (6)e(8). Under this scheme, monetary costs alone determine all the decisions. Naturally, neither of the previously described models very accurately model the real situation happening after a natural disaster. The Leader's perspective and Follower's perspective models give us the extreme situations if we do not consider it as a bi-level programming problem. In the bi-level model defined by constraints (1)e(8), we take into account both decision-makers for selecting their best decisions, so we expect to find the equilibrium among them by considering a hierarchy in the decision process. The following describes the computational environment and the methodology used. The size of the instance obtained from the case study is defined by jIj ¼ 12, jJj ¼ 17, jKj ¼ 44, jLj ¼ 4 and jMj ¼ 3.

7

Table 1 Numerical results for the case study.

Leader's perspective Follower's perspective Bi-level model

F(x,y)

f(x,y)

t

3885.30 347,310.90 12,355.02

430,777,152 359,936,842 369,710,950

0.0936 0.1560 11.5129

The three different schemes of the case study were exactly solved with CPLEX 12.1 through a Cþþ code on a 3.00 GHz Pentium DualCore Processor with 2.00 GB RAM running under Windows 7 Professional operative system. For solving the bi-level model we used the equivalent reformulation presented at the end of Section 4 adding typical constraints that involve auxiliary binary variables 1 2 for dealing with Tijlm and Tjklm without affecting the analysis described in the mentioned section. The detail of all the values we used for the experimentation and the full description of this case study appear in http://www.fcfm.uanl.mx/jsp/posgrado/ JoseCamachoCOM.html. Table 1 presents the numerical results; we show the leader's objective function value F(x,y) representing the total time (in hours) for distributing the aid from all the countries to the storage centers and then to the affected zones, the follower's objective function value f(x,y) representing the costs (in dollars) associated with the distribution of the aid, and the time t consumed (in seconds) for solving the three models (Leader's perspective, Follower's perspective, the bi-level model) by using the computer described above. Due to the limitations of space and to improve the readability of this paper the distribution plan y and the decision x for sending the aid to Chile appear in the Appendix. As seen from Table 1, a hierarchical equilibrium results in the bilevel model for the countries sending aid (follower) and the affected country (leader), i.e., the FLeader < FBilevel < FFollower and fFollower < fBilevel < fLeader, where the sub-index represents the corresponding model. In other words, under the Leader's perspective we obtained the fastest aid distribution plan, but very expensively for the countries sending aid. Meanwhile, considering the Follower's perspective, the solution of the problem gave us the cheapest plan for sending aid, but the worst in relation to the required time for distributing all the products. On the other hand, the bi-level model allowed us to reach an intermediate point, keeping in mind the affected country has a justified hierarchy in the problem. In Fig. 2, a graphical representation of the optimal point for the bi-level solution appears.

Fig. 2. Graphical illustration of the bi-level equilibrium.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

8

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

7. Recommendations and conclusions

Table 2 Percentage of increase in the respective models against the best decision.

Leader's perspective Follower's perspective Bi-level model

Increase in time (%)

Increase in costs (%)

e 8839.101 217.994

19.681 e 2.716

The hierarchy considered in this paper appears in Table 2 where we make two comparisons. First, we compare the increase of the distribution time with respect to the value given by the Leader's perspective model; and second, we identify the cost associated with sending the distribution to the devastated country given by the Follower's perspective model and compare it with the increase in the costs given by the other models. The following formula provided the gap of increase:

% Increase ¼

ðCurrent valueÞ  ðBest valueÞ ðBest valueÞ

(44)

The equilibrium obtained after solving the bi-level model increases both objectives (time and costs); this result was expected because of the existing hierarchy among both levels. Due to the nature of the model, the consideration of the follower's problem (the minimization of the costs associated with sending aid to the affected country) clearly affects the total response time but allows to take into consideration the opinion of the helper countries. To show the advantages of the proposed bi-level model, we compute the gaps of the savings provided by this model against the Leader and Follower's perspectives. The gaps appear in Table 3 and derive from the formula (44) adjusting the values in the correct way. As Table 3 shows, by selecting the bi-level solution instead of the Leader's perspective solution, we reduce the total costs 16.517%. On the other hand, choosing the bi-level solution instead of the Follower's perspective gives us a huge time reduction of 2711.092%. In both cases, the reductions are more significant than the expected increase shown in Table 2. The negative values in Table 3 represent an increase in total time and costs, respectively. Referring to the solutions presented in the Appendix, under the bi-level scenario the 12 countries must send aid. The shipments occur by air and maritime transportation; only Argentina uses land transportation. United States sends the biggest amount of aid with 249,060 TEU, while Germany only sends 59,160 TEU, making it the country sending the least amount. It is worth mentioning that Bolivia, Ecuador, Cuba and Venezuela send the same amount of 84,220 TEU. On the other hand, with respect to the internal distribution in Chile, all the affected zones meet their demand using the 17 existing storage centers, i.e., all the storage centers are utilized and it is all made by land transportation mode. The total required time equals the sum of all the distribution and shipping times. Since these shipments may occur in a simultaneous way neglecting the time between departures, the real time needed for meeting the demand of the affected zones is increased. Also, all the countries will make their decisions in a simultaneous manner preventing us from estimating the real number of days for relief distribution. Nevertheless, the minimum time for sending all the shipments is achieved after solving the proposed bi-level model. Table 3 Percentage of decrease provided by the bi-level model.

Leader's perspective Follower's perspective Bi-level model

Decrease in time (%)

Decrease in costs (%)

68.553 2711.092 e

16.517 2.644 e

We propose a bi-level mathematical programming problem for distributing international aid after a disaster. Due to the lack of commercial software able to solve bi-level programming problems, we reformulated the bi-level problem into an equivalent singlelevel mixed integer problem. This reformulation solved a real case study from an earthquake occurring few years ago in Chile. To make some conclusions about the obtained numerical results, we compared the bi-level problem against two other models representing extreme situations related to this humanitarian logistics problem. Also, we clearly differentiated the bi-level optimal solutions from the optimal solutions for other related models (Leader's and Follower's perspective). As seen in Fig. 2, the optimal points for the extreme models are the solutions for the bi-objective model considering weighted objective function values; more precisely, the Leader's perspective point is the extreme when the weights are 1 and 0 for F(x,y) and f(x,y), respectively. On the other hand, the Follower's perspective is the other extreme when the weights are 0 and 1 for the corresponding objective functions.
(2010) analyzed the differences beIn fact, Calvete and Gale tween bi-level and bi-objective programming problems. The authors concluded, although some researchers have attempted to establish a formal relationship between both problems, a determinant conclusion about this relationship does not exist. Moreover, several counterexamples refuting the relationship between them appear in Candler (1988), Clarke and Westerberg (1988), Haurie et al. (1990), and Marcotte and Savard (1991). Those authors graphically showed optimal solutions of a bi-level programming problem do not occur in the Pareto optimal solution of a biobjective programming problem, as shown in Fig. 2. The obtained results show the importance of having a central agency coordinating the entire process, even without a perfect coordination between all the involved parties, this coordination would significantly help to make the societies more sustainable. Seuring and Müller (2008) established the coordination effort and the complexity as limitation factors for a sustainable supply chain. Hence, the central agency role would be very important during the Response stage after a disaster has occurred. Notably, the existence of a hierarchy in this problem allows us to analyze it more appropriately than the focus given by the multiobjective programming. In this case, the decision of the affected country has more importance than the decisions made in the countries or organizations sending aid. Therefore, considering the problem under this scheme is in the interest of the humanitarian logistics area. As a result of this research, we can conclude the bilevel model works well for this situation, leading us to recommend the necessity of having a central agency coordinating all the operations related to the reception and distribution of the international aid. For future research opportunities, the problem described here could be modeled as a bi-level problem with multiple nonindependent followers, i.e., each country and aid agency providing disaster relief has its own interest and separate budgets, hence they share a constraint associated with the demand at each storage center. Considering the fact one needs a rational reaction from the follower, an appropriate definition of the inducible region must occur. In the lower level, a Nash game takes place, while the bi-level problem can be viewed as a Stackelberg game. A lot of concepts would need introduction to solve this problem in a correct way. € Furthermore, we can analyze the approach considered in Ozkir and Basligil (2013) by incorporating stochasticity in the demands for aid and road conditions due to the uncertain environment.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

Based on the disaster's magnitude, the aid demand could vary, so we can create different scenarios and apply robust optimization techniques or simulation to solve the problem. Also, one may consider the disruption of the transportation networks because, when a disaster occurs, some of the transportation roads may close or have damage reducing good transportation flow. Finally, an interesting extension of this problem would consider balancing of received aid in the storage centers to avoid saturation of some, while others remain under-utilized. Acknowledgments This research received partial support from the Secretariat of Public Education (SEP by its acronym in Spanish) within the Academic Groups research project PROMEP/103.5/12/4953 and the
n (UANL) within the Support Autonomous University of Nuevo Leo Program for Scientific Research and Technology (PAICYT) with the Project CE960-11. The authors would like to thank Mr. Randy Smith for his editing review for improve the readability of this paper.

(continued ) Country

Storage center

Argentina Bolivia

Cauquenes Linares
n Concepcio Talca Linares
n Concepcio Lebu
n Concepcio
Los Angeles

Brazil

Cuba Spain United States Venezuela

Personal products Germany Argentina Bolivia

Brazil Cuba

Appendix Spain

In this Section appear the numerical results corresponding to the solutions xijlm and yjklm, i.e., the decision for sending aid to Chile from the different countries willing to help and the distribution planning for meeting the demand of the affected zones in Chile. Amount of TEU sent by air transportation:

Country Liquids Germany Argentina Bolivia Brazil

Cuba Spain United States Venezuela Medicines Germany Argentina Bolivia Brazil

Cuba

Spain

United States Venezuela

Non-perishable food Germany

Storage center

Amount


n Concepcio Chill
an Cauquenes
Los Angeles Chill
an Linares
Los Angeles

21,099 1 23,754 60,798 10,202 10,337 2085 128,578 55,221 15,779 45,224 60,776 211,000 71,000

Chill
an Lebu Chill
an
n Concepcio
Los Angeles
n Concepcio Linares


n Concepcio
Los Angeles Cauquenes
Los Angeles Chill
an Talca
Curico
n Concepcio
n Concepcio
Los Angeles Chill
an Lebu Temuco Provincia Malleco
n Concepcio Lebu Linares
Los Angeles


n Concepcio
Los Angeles Chill
an

United States Venezuela

17,517 2298 12,185

Amount 3562 10,999 1 1573 2436 9563 7928 11,000 16,250 32,000 1 10,999


n Concepcio Lebu Chill
an


n Concepcio Chill
an Cauquenes Talca Linares
n Concepcio
n Concepcio
n Concepcio Chill
an
n Concepcio Lebu
n Concepcio
Los Angeles

3267 1733 542 1237 246 317 3400 1 1799 1519 1081 2174 2826 1800

Linares

Amount of TEU sent by maritime transportation:

Country

1058 2 108 119 301 647 91 2 15 1 404 261 271 48 1059 1 407 13

9

Liquids Colombia Mexico

Peru

Ecuador

Medicines Colombia Peru

Ecuador

Non-perishable food Colombia

Mexico Peru

Storage center

Amount

Talca San Joaquín San Antonio Rancagua Pichilemu
Curico Linares ~ n ~ oa Nu Valparaíso San Fernando Talca ~ n ~ oa Nu Rancagua

106,000 8077 8500 12,981 5760 85,417 8242 9405 33,329 27,518 35,748 2398 68,602

Rancagua Valparaíso San Antonio Rancagua San Fernando San Joaquín Rancagua Pichilemu
Curico

96 151 38 267 124 90 9 24 297

San Joaquín Valparaíso
Curico Rancagua
Curico San Antonio San Fernando Pichilemu Talca
Curico

2982 4999 8269 10,939 4542 1274 4126 864 9985 1 (continued on next page)

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

10

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

(continued )

(continued )

Country

Storage center

Ecuador

Rancagua Talca

1296 9704

Rancagua San Fernando
Curico San Antonio Pichilemu Talca San Joaquín Valparaíso Talca
Curico
Curico

1864 628 108 193 129 658 454 761 1343 42 1800

Personal products Colombia

Mexico

Peru

Ecuador

Amount

Storage center

Affected zone

Amount

Linares

Linares Longavi Parral
n Concepcio
Tome Coronel Talcahuano Lebu Arauco
Los Angeles

41,079 24,250 24,250 197,725 45,331 93,716 130,451 22,050 35,649 82,950 18,681 22,028 77,280 38,640 38,640 29,249 15,209 15,209 10,154 5077 4870


n Concepcio

Lebu
Los Angeles


n Chilla

Temuco

Amount of TEU sent by land transportation from Argentina:

Country Liquids Argentina

Personal products Argentina

Non-perishable food Argentina

Medicines Argentina

Storage center

Amount

Lebu Temuco Provincia Mallerco

2478 59,667 20,101

Lebu Temuco Provincia Mallerco

237 1362 459

Lebu Temuco Provincia Mallerco

725 8949 3014


Los Angeles Provincia Mallerco

429 43

Provincia Mallerco

Personal products San Joaquín Valparaíso

San Antonio Rancagua

San Fernando Pichilemu

Talca

Cauquenes
Curico

Land distribution from storage centers to affected zones: Linares Storage center Liquids ~ n ~ ao Nu San Joaquín Valparaíso

San Antonio Rancagua

San Fernando Pichilemu

Talca

Cauquenes
Curico

Affected zone

Amount

Zona Metro. Santiago Zona Metro. Santiago
ndez Juan Ferna Valparaíso ~ a del Mar Vin
n Conco San Antonio Llolleo Rancagua Machalli San Vicente
pica Che San Fernando Pichilemu Litueche Paredones Cauquenes Pelluhue
Curico Cauquenes Pelluhue
Curico Molina Teno

11,803 7797 280 15,454 16,068 1807 7366 1134 34,745 23,419 23,419 13,759 13,759 1920 1920 1920 29,787 10,300 101,661 18,504 5250 49,123 18,147 18,147


n Concepcio

Lebu
Los Angeles


n Chilla

Temuco

Provincia Malleco

Non-perishable food San Joaquín Valparaíso

Yumbel Mulchen Chillan Yungay San Nicol
as Temuco Villarica Padre las Casas Angol Victoria Collipulli

Zona Metro. Santiago Juan Fern
andez Valparaíso ~ a del Mar Vin
n Conco San Antonio Llolleo Rancagua Machalli San Vicente
pica Che San Fernando Pichilemu Litueche Paredones
n Constitucio Pelarco Talca Cauquenes Pelluhue
Curico Molina Teno Linares Longavi Parral
n Concepcio
Tome Coronel Talcahuano Lebu Aracuco
Los Angeles Yumbel Mulchen Chill
an Yungay San Nicol
as Temuco Villarica Padre las Casas Angol Victoria Collipulli

448 6 353 367 41 168 25 794 535 535 314 314 43 43 43 680 235 2323 422 120 1122 414 414 938 554 554 4519 1036 2142 2981 504 814 1896 427 503 1766 883 883 668 347 347 232 116 111

Zona Metro. Santiago Juan Fern
andez Valparaíso ~ a del Mar Vin
n Conco

2940 42 2318 2410 271

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12 (continued )

(continued )

Storage center

Affected zone

San Antonio

San Antonio Llolleo Rancagua Machalli San Vicente
pica Che San Fernando Pichilemu Litueche Paredones
n Constitucio Pelarco Talca Cauquenes Pelluhue
Curico Molina Teno Linares Longavi Parral
n Concepcio
Tome Coronel Talcahuano Lebu Aracuco
Los Angeles Yumbel Mulchen
n Chilla Yungay
s San Nicola Temuco Villarica Padre las Casas Angol Victoria Collipulli

Rancagua

San Fernando Pichilemu

Talca

Cauquenes
Curico

Linares


n Concepcio

Lebu
Los Angeles

Chill
an

Temuco

Provincia Malleco

Medicines San Joaquín Valparaíso

San Antonio Rancagua

San Fernando Pichilemu

Talca

Cauquenes
Curico

Linares


n Concepcio

Lebu
Los Angeles

11

Zona Metro. Santiago
ndez Juan Ferna Valparaíso ~ a del Mar Vin
n Conco San Antonio Llolleo Rancagua Machalli San Vicente
pica Che San Fernando Pichilemu Litueche Paredones
n Constitucio Pelarco Talca Cauquenes Pelluhue
Curico Molina Teno Linares Longavi Parral
n Concepcio
Tome Coronel Talcahuano Lebu Aracuco
Los Angeles Yumbel Mulchen

Amount 1104 170 5211 3512 3512 2063 2063 288 288 288 4468 1545 15,249 2775 787 7368 2722 2722 6161 3637 3637 29,658 6799 14,057 19,567 3307 5347 12,442 2802 3304 11,592 5796 5796 4387 2281 2281 1523 761 730

89 1 70 73 8 33 5 158 107 107 62 62 8 8 8 136 47 464 84 24 224 82 82 187 110 110 903 207 428 596 100 162 379 85 100

Storage center

Affected zone


n Chilla

Chill
an Yungay San Nicol
as Temuco Villarica Padre las Casas Angol Victoria Collipulli

Temuco

Provincia Malleco

Amount 353 176 176 133 69 69 46 23 22

References Aksen, D., Aras, N., 2012. A bi-level fixed charge location model for facilities under imminent attack. Comput. Oper. Res. 39, 1364e1381. Altay, N., Green, W.G., 2006. Interfaces with other disciplines OR/MS research in disaster operations management. Eur. J. Oper. Res. 175, 475e493. Arroyo, J.M., Galiana, F.D., 2005. On the solution of the bi-level programming formulation of the terrorist threat problem. IEEE Trans. Power Syst. 20 (2), 789e797. Barbarosoglu, G., Arda, Y., 2004. A two-stage stochastic programming framework for transportation planning in disaster response. J. Oper. Res. Soc. 55, 43e53. € Barbarosoglu, G., Ozdamar, L., Cevik, A., 2002. An interactive approach for hierarchical analysis of helicopter logistics in disaster relief operations. Eur. J. Oper. Res. 140 (1), 118e133. Bard, J.F., 1998. Practical Bi-Level Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands. Bracken, J., McGill, J.T., 1973. Mathematical programs with optimization problems in the constraints. Oper. Res. 21 (1), 37e44. Caunhye, A.M., Nie, X., Pokharel, S., 2012. Optimization models in emergency logistics: a literature review. Socio-Econ. Plan. Sci. 46, 4e13.
, C., 2010. A multiobjective bi-level program for productionCalvete, H.I., Gale distribution planning in a supply chain. In: Ehrgot, M., et al. (Eds.), Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems, Lect. Notes Econ. Math, vol. 634. Springer-Verlag, Berlin, pp. 155e165. Candler, W., 1988. A linear bi-level programming algorithm: a comment. Comput. Oper. Res. 15, 297e298. Clarke, P., Westerberg, A., 1988. A note on the optimality conditions for the bi-level programming problem. Nav. Res. Logis. 35, 413e418. Colson, B., Marcotte, P., Savard, G., 2007. An overview of bi-level optimization. Ann. Oper. Res. 153, 235e256. Cruz, J.M., 2013. Modeling the relationship of globalized supply chains and corporate social responsibility. J. Clean. Prod. 56, 73e85. Dempe, S., 2002. Foundations of Bi-Level Programming. Kluwer Academic Publishers, Dordrecht, The Netherlands. Dües, C.M., Tan, K.H., Lim, M., 2013. Green as the new Lean: how to use Lean practices as a catalyst to greening your supply chain. J. Clean. Prod. 40, 93e100. Duman, E., 2007. Decision making by simulation in a parcel transportation company. J. Frankl. I 344, 672e683. Fiedrich, F., Gehbauer, F., Rickers, U., 2000. Optimized resource allocation for emergency response after earthquake disasters. Saf. Sci. 35, 41e57. Feng, C.M., Wen, C.C., 2005. A bi-level programming model for allocating private and emergency vehicle flows in seismic disaster areas. In: Proceedings of the Eastern Asia Society for Transportation Studies, vol. 5, pp. 1408e1423. Gramani, M.C.N., Franca, P.M., Arenales, M.N., 2011. A linear optimization approach to the combined production planning model. J. Frankl. I 348, 1523e1536. Gupta, A.K., Suresh, I.V., Misra, J., Yunus, M., 2002. Environmental risk mapping approach: risk minimization toll for development of industrial growth center in developing countries. J. Clean. Prod. 10, 271e281. Haurie, A., Savard, G., White, D., 1990. A note on: an efficient point algorithm for a linear two-stage optimization problem. Oper. Res. 38, 553e555.
rez, N., Jaller, M., Van Wassenhove, L.N., Aros-Vera, F., 2013. On Holguin-Veras, J., Pe the appropriate objective function for post-disaster humanitarian logistics models. J. Oper. Manag. 31, 262e280. Ji, G., Zhu, C., 2012. A study on emergency supply chain and risk based on urgent relief service in disasters. Syst. Eng. Procedia 5, 313e325. Kongsomsaksakul, S., Yang, Ch, 2005. Shelter location-allocation model for flood evacuation planning. J. East. Asia Soc. Transp. Stud. 6, 4237e4252. Li, A.C.Y., Nozick, L., Xu, N., Davidson, R., 2012. Shelter location and transportation planning under hurricane conditions. Transp. Res. E-Log 48, 715e729. ~ o, M.T., Tirado, G., Vitoriano, B., Scaparra, M.P., 2014. Liberatore, F., Ortun A hierarchical compromise model for the joint optimization of recovery operations and distribution of emergency goods in humanitarian logistics. Comput. Oper. Res. 42, 3e13. Losada, Ch, Scaparra, M.P., O'Hanley, J.R., 2012. Optimizing system resilience: a facility protection model with recovery time. Eur. J. Oper. Res. 217, 519e530. Marcotte, P., Savard, G., 1991. A note on the Pareto optimality of solutions to the linear bi-level programming problem. Comput. Oper. Res. 18, 355e359.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069

12

J.-F. Camacho-Vallejo et al. / Journal of Cleaner Production xxx (2014) 1e12

Mete, H.O., Zabinsky, Z.B., 2010. Stochastic optimization of medical supply location and distribution in disaster management. Int. J. Prod. Econ. 126, 76e84. € Ozdamar, L., Demir, O., 2012. A hierarchical clustering and routing procedure for large scale disaster relief logistics planning. Transp. Res. E-Log 48, 591e602. € Ozdamar, L., Ekinci, E., Kücükyazici, B., 2004. Emergency logistics planning in natural disasters. Ann. Oper. Res. 129, 217e245. € Ozkir, V., Basligil, H., 2013. Multi-objective optimization of closed-loop supply chains in uncertain environment. J. Clean. Prod. 41, 114e125. Rath, S., Gutjahr, W.J., 2014. A math-heuristic for the warehouse location-routing problem in disaster relief. Comput. Oper. Res. 42, 25e39. Rathi, A.K., Church, R.L., Solanski, R.S., 1993. Allocating resources to support a multicommodity flow with time windows. Logis. Transp. Rev. 28, 167e188. Rawls, C.G., Turnquist, M.A., 2010. Pre-positioning of emergency supplies for disaster response. Transp. Res. B-Methodol 44, 521e534. Sanayei, A., Mousavi, S.F., Abdi, M.R., Mohaghar, A., 2008. An integrated group decision-making process for supplier selection and order allocation using multi-attribute utility theory and linear programming. J. Frankl. I 345, 731e747. Schliephake, K., Stevens, G., Clay, S., 2009. Making resources work more efficiently e the importance of supply chains partnerships. J. Clean. Prod. 17, 1257e1263.

Seuring, S., Müller, M., 2008. From a literature review to a conceptual framework for sustainable supply chain management. J. Clean. Prod. 16, 1699e1710. Sheu, J.B., 2007. An emergency logistics distribution approach for quick response to urgent relief demand in disasters. Transp. Res. E-Log 43, 687e709. Sheu, J.B., Chen, Y.H., 2005. A novel model for quick response to disaster relief distribution. In: Proceedings of the Eastern Asia Society for Transportation Studies, vol. 5, pp. 2454e2462. Vicente, L., Calamai, H., 1994. Bi-level and multilevel programming: a bibliography review. J. Glob. Optim. 5 (3), 291e306. ~ o, M.T., Tirado, G., Montero, J., 2011. A multi-criteria optimization Vitoriano, B., Ortun model for humanitarian aid distribution. J. Glob. Optim. 51, 189e208. Wamsler, C., brink, E., Rivera, C., 2013. Planning for climate change in urban areas: from theory to practice. J. Clean. Prod. 50, 68e81. Wang, Q., Rong, L., 2007. Agile knowledge supply chain for emergency decisionmaking support. In: Proceedings of the 7th International Conference on Computational Science, Part IV, pp. 178e185. Wex, F., Schryen, G., Neumann, D., 2011. Intelligent decision support for centralized coordination during emergency response. In: Proceedings of the 8th International ISCRAM Conference.

Please cite this article in press as: Camacho-Vallejo, J.-F., et al., A bi-level optimization model for aid distribution after the occurrence of a disaster, Journal of Cleaner Production (2014), http://dx.doi.org/10.1016/j.jclepro.2014.09.069