A robust optimization model for logistics planning in the earthquake response phase

Article history: Received February 25, 2016 Received in revised format: March 28, 2016 Accepted May 2, 2016 Available online May 3 2016 In this paper, a robust bi-level model is proposed to optimize decisions related to distribution and evacuation aid after earthquake. Usually in disastrous situation foreign countries help the affected country by sending relief commodities. In this problem, the foreign countries try to minimize their shipping costs and the affected country seeks to minimize its total costs which include inventory, operation, and transportation expenses. This situation is a game between different decision makers after a catastrophic disaster. To deal with this situation, a bi-level model is proposed in which the affected country is the leader and suppliers are the followers. To validate the proposed robust model, we consider Tehran probable earthquake in region 1 as a case study. Then the advantages of using bi-level modeling against considering just one player's point of view is provided. The sensitivity analysis of the experiments are presented to explore the effects of various parameters to show managerial insights that can guide DMs under a variety of conditions. Growing Science Ltd. All rights reserved. 6 © 201


Introduction
Humanitarian relief operations is not new idea and, in fact, the modern roots of international humanitarianism can be traced back to the formation of the Red Cross in 1863.Humanitarian logistics have gained academic attention after the 2004 Indian Ocean Tsunami and hurricane Katrina in 2005.The impact of the Indian Ocean Tsunami and hurricane Katrina was so devastating, thousands of dead and millions homeless left, these statistics reveal the problems in humanitarian logistics and disaster management.Several definitions for disaster management has been expressed in the literature.The most comprehensive and most widely accepted definition of the crisis has been presented by the World Health Organization.According to Barbarosoğlu and Arda (2004) "any event that causes destruction, environmental damage, loss of human life or even suffering, threaten the health and health services, so that affected areas need help and response from other areas is called disaster".The earthquake is one of the disasters that usually leads to widespread destruction and human injuries.Earthquake's damages have been increased in recent years due to excessive condensation of population in accident-prone areas.This fact illustrates the need of an effective plan to respond to such possible crisis.Response phase is called actions which is proceeded after the incident and is considered as initial actions.
In this paper a linear bi-level model by considering evacuation and transportation injured people from affected areas to hospital for response stage is developed.In upper level the affected country seeks to minimize its total costs while international suppliers in lower level tries to minimize their transportation expenses.Afterward, a robust optimization model is developed to deal with uncertainty in demanded relief commodities.
The remainder of this paper is organized as follows.Section 2 introduces optimization related studies in disaster management.Third section defines the considered problem and its basic assumptions.In Section 4, the proposed model is defined.Section 5 explains the robust modeling technique used in this study.In section 6 the numerical results are presented.In this section the performance of the bi-level and the single-level models are compared.Finally, in section 7 conclusion and future researches are presented.

Literature Review
Logistics of essential commodities and rescue operation are the most important activities after any earthquake.Aid distribution is necessary to deliver essential commodities such as food, medicine, sanitation, shelter and water to save human life and to prevent the expansion of diseases.In addition, rescue operation is vital to carry injuries from affected areas to medical centers.In fact, the efficient plan for these operations after earthquake can significantly reduce the death rate.
At the end of 1980, researchers began their studies on the crisis management.Most of the researches have studied the supply and logistics of a particular type of goods and services to the injured people.Knott (1987) proposed a linear model considering food transportation problems to minimize the transportation costs and maximized the amount of food delivered to the affected areas.Knott (1988), next, developed a linear model to schedule the rout of vehicles by maximizing the amount of food delivered.Barbarosoğlu et al. (2002) used helicopter for rescuing actions during a natural disaster by routing and transportation problem during the response phase.Özdamar et al. (2004) proposed a network model for planning logistics of commodity.They presented an algorithm that determines the origin and destination of each amount of commodities.Tzeng et al. (2007) presented a multi objective model for relief commodities that can be used for a real life relief delivery systems.Their model included three objective functions, minimizing total cost, minimizing total travel time and maximizing the minimal satisfaction in the given period.Sheu (2007) proposed an approach based on fuzzy clustering for optimization to coordinate the relief materials flow in a relief supply network during the crucial rescue period.He also presented a three-step dynamic model for emergency logistics actions under imperfect data.These steps are data fusion to estimate relief demand in affected area, classifying affected areas using fuzzy clustering and determining the priority of each group which generated in last step.Nolz et al. (2011) provided a model for post disaster with three objective functions to minimize the risk, to maximize the coverage gained by the model and to minimize the total traveling time.Lin et al. (2011) proposed a multi objective model by considering different types of vehicles and commodities to minimize the total unsatisfied demands and total travel time between affected areas.Moreover, Zhan and Liu (2011) proposed a stochastic model for a disaster network including uncertain data about demand, supply and availability of transportation paths by minimizing the expected travel time and unsatisfied demands using chance constraint and scenario planning.Afshar and Haghani (2012) proposed a disaster network model to schedule the flow of commodities and determined the optimal position for facilities.Their model considered routing problem as well.
Others focused on transportation in different ways, using different vehicles and approaches.As some example, Fiedrich et al. (2000) presented a model by considering allocation of resources to transport injuries in the best way.Also Jotshi et al. (2009) used robust optimization for scheduling routes for vehicles after the disaster.Yi and Özdamar (2007) presented a dynamic model including both evacuation of injuries and satisfaction of demands.Yi and Kumar (2007) considered medical centers to serve injuries and distribution centers for commodities.Ozdamar (2011) established a mathematical model for transporting of injuries from affected areas to hospitals and deliver medical commodities to victims using helicopter to minimize the medical mission time.Also, Özdamar and Demir (2012) used a hierarchical clustering algorithm to organize the rout of vehicles in large scale disasters in the response phase.
Many researchers considered the uncertainty in disaster management in their scientific papers.Some of them used stochastic programming to consider uncertainty.Barbarosoğlu and Arda (2007) proposed a two stage stochastic model for the response phase of an earthquake and also used scenario planning to account data uncertainty.Ma et al. (2010) presented a min-max robust model for a disaster network with uncertainty in distances between nodes.Furthermore, Zhan and Liu (2011) considered a locationallocation problem using chance constraint programming with uncertainty in demand and the availability of paths and supply of commodities.Finally Najafi et al. (2013) developed a multiobjective, multi-mode, multi-commodity and multi-period stochastic model to manage the logistics of both commodities and injured people in the earthquake response stage.
After earthquake, international countries and local suppliers offer to help the affected country by sending aid commodities.Regularly, local and international suppliers and affected country have different goals.This situation is a multi-decision maker problem which should consider comprehensively.Multi-decision maker models in critical situations investigated by a few studies and most of them focused on unnatural crisis such as terrorist attacks.For instance, Arroyo and Galiana (2005) considered the problem of a terrorist threat as a nonlinear mixed integer bi-level problem where the terrorist seeks to maximize the attacking damages at the upper level, while in the lower level a threatened company seeks to minimize spilled loads caused by the attack.The authors used the equivalent single-level mixed integer linear problem obtained by Karushe-Kuhne-Tucker instead of the bi-level problem.They also applied some restrictions to avoid nonlinearity.In another study, Aksen et al. (2013) formulated a static Stackelberg game between a defender as the leader and an attacker as the follower in order to investigate facility location problem.They considered facilities in both protected and unprotected modes.The leader tries to minimize the sum of install costs, protection and usage of the facility while the follower seeks to destroy unprotected facilities to affect the supply capacity of the remaining plans.In another facilities protection problem, Losada et al. (2012) proposed a mixed integer linear bi-level programming .
There are a few researches concern about bi-level programming in natural disasters.with focusing on preparation stage, Barbarosoğlu et al. (2002) investigated the helicopter relief mission using bi-level programming to minimize the total cost of assigning pilots to helicopters and then, minimizing the relief operation time.Feng and Wen (2005) proposed a bi-level model to manage roads traffic after earthquake.In upper level, the leader attempts to send more vehicles for relief operations while the evacuees in lower level seek to travel through unaffected roads in order to minimize their total travel time.This process causes heavy traffic in roads and interrupts sending aids and evacuation operations.Kongsomsaksakul et al. (2005) used bi-level programming to model a Stackelberg game in order to describe a shelter location model.They considered authority as the leader and evacuees as follower while leader determines the shelter locations to minimize the total evacuation time and the followers choose the destinations and routes for evacuation.To solve the model, authors developed a genetic algorithm using real data from a dam located in Utah.Li et al. (2012) also considered the same problem but for hurricane situation.The upper level decides about location and allocation the shelters and evacuees in lower level seek to find better routs.Finally, Camacho-Vallejo et al. (2014) proposed a bi-level model to optimize decisions related to aid distribution after earthquake.They considered international organization and foreign countries as suppliers.In upper level the affected county tries to minimize total response time while international organization and foreign countries in lower level seek to minimize their total transportation cost.The authors also considered an earthquake in Chile in 2010 to illustrate the application of their proposed model.
It appears there are some scientific gaps in multi-level models especially for natural disasters.In every catastrophe there are numerous decision makers and victims, so taking them into account is necessary to model the disaster situation properly.In this paper, a bi-level model by considering supply and transportation of relief commodities for response stage is developed.In upper level the affected country seeks to minimize the total cost of evacuation and distribution while international suppliers in the lower level try to minimize their transportation costs.We assume the existence of a coordinator agency between affected country and local and international suppliers to avoid sending unnecessary relief commodities.To deal with the uncertainty in demands of essential commodities, a robust model based on Mulvey et al.'s (1995) research is developed.Finally Tehran probable earthquake is investigated and sensitivity analysis and numerical experiments are proposed.

Statement of the Problem
When an earthquake occurs, foreign countries help the affected country by sending relief goods such as water, food, medicine and shelter.At the upper level, the affected country is responsible for transporting aids from warehouses to affected sites and also carrying injuries from affected areas to medical centers.At the lower level, international suppliers should deliver relief commodities to the located warehouses and try to minimize their transportation expenses.Since suppliers usually have communication with affected country to determine needed commodities, we assume there is a central agency coordinating affected country with local and international suppliers to avoid sending unwanted commodities and make resources more efficient.Fig. 1 shows the general schema of this humanitarian logistics, which contains four nodes: foreign suppliers, warehouses, affected areas, and hospitals.

Fig. 1. General schema of humanitarian logistics network
We proposed a bi-level multi-commodity and multi-modal model to optimize decisions related to aid distribution and evacuation after earthquake by considering international suppliers.At the upper level the affected country decides how to distribute aids from warehouses to demand nodes, and how to save injuries and transport them to the medical centers in order to minimize the total costs.At the lower level, international suppliers send their aid commodities to the located warehouses and try to minimize their shipping costs.
The main assumptions of the proposed model is as follows:  The locations of warehouses and hospitals are known. Warehouses have limited storages capacity.
 Warehouses are equipped with an initial inventory of aid commodities  Rescue operation should be done in the specific times.
Demand points can be satisfied by more than one warehouse, and all of them should be satisfied, at least partially. The position of demands, intermediates, and suppliers and warehouses nodes and distance between them are known. There are different type of vehicles for evacuation with known capacities.

Mathematical model
Based on previous section, the proposed bi-level model is developed.The sets used in model are as follows: Also the parameters used in model are as follows: Definition Parameter Demand of aid commodity type q at affected area i at period t Binary variable represents 1 if demand for aid commodity type q at affected are n is satisfied by warehouse j in period t, otherwise 0 t ijq x Amount of transported aid commodity type q from warehouse j to affected area i at time t.

t ijq x
Amount of transported aid commodity type q in warehouse j at the end of time t.

t jq I
Binary variable represents 1 if rescue vehicle l from warehouse j at time t transports injuries from affected area i to hospital k, otherwise 0 t ijlk  Amount of transported aid commodity type q from supplier e to warehouse j at time t.(1) min ( ) , , , , , , , , , , , , At the upper level, objective function ( 1) is the leader's goal which minimizes the total costs (transportation, operation, and inventory costs of distribution and evacuation).Constraint (2) shows for distribution, the warehouse should be assigned to the affected area.Constraint (3) assures each demand point can be satisfied by more than one warehouses.Constraint (4) ensures all warehouses attend in distribution operation.Constraint (5) shows demands for rescue operation must be satisfied.Constraint ( 6) and ( 7) demonstrate raduis covegare for distribution evacuation.Constraint (8) assures the inventory level of aid commodities at warehouses.Constraint (9) shows rescue operation for each demand point must be done in the specific time.Constraint ( 10) and ( 11) show how demands for aid commodities and resuce operation satisfy.Constraint (12) ensures injuries who are carried to each hospital must be less than the hosptial's available beds.constraint ( 13) is related to upper level decision variables.At the lower level, objective function ( 14) shows the followers' goal which minimizes transportation costs.Constraint (15) assures maximum capacity of foreign suppliers to transport aid commodities.Constraint ( 16) asserts the relation between transported aid commodities from foreign suppliers and distributed commodities at the affected areas.Finally, constraint (17) definess the followers' decision variables.

Robust Modeling
Mulvey et al. (1995) introduced a robust optimization due to optimal design of supply chain in the real world and uncertain environments.By expressing the value of vital input data in a set of scenarios, robust optimization tries to approach the preferred risk aversion.This approach results in a series of solutions that are less sensitive to the model data from a scenario set.Two sets of variable act in this approach control and design variables.The first ones are subject to adjustment once a specific realization of the data is obtained, while design variables are determined before realization of the uncertain parameters and cannot be adjusted once random parameters are observed.Constraints can be divided into two types as well too: structural and control constraints.Structural constraints are typical linear programming constraints which are free of uncertain parameters, while the coefficients of control constraints are subject to uncertaint.Since the uncertainty is an integral part of demand after earthquake, the demand considered as an uncertain parameter which represents by defined scenarios.
The symbol S represents different scenarios.Now to determine the robust model the following additional parameters and decision variables are defined.

Definition Parameter
Demand of aid commodity type q at affected area i at period t under scenario s

  
Unsatisfied demand of aid commodity type q at affected area i at period t under scenario s ts iq


Unsatisfied demand of aid commodity type q at affected area i at period t under scenario s ts i

 
Unsatisfied demand of aid commodity type q at warehouse j at period t under scenario s ts jq

 
, , , , , , , , , , , , , , ) 2 ] ts s q ej ejq s S e E j J q Q t T ts ts s q ej ejq s q ej ejq s s S e E j J q Q t T s S e E j J q Q t T ts s jq j J q Q t T s S p Tr d y p T rd y pT rd y p , , , ts ts q ej ejq s q ej ejq s e E j J q Q t T s S e E j J q Q t T

Tr d y p Tr d y s
Where constraints (30) and ( 35) are used to linearize the robust model and rest of constraint are the same as previous model.

Numerical Results
To illustrate and apply our proposed model, we consider the probable earthquake in Tehran region one as a case study.After occurrence of earthquake, groups of foreign will help the affected zone by sending necessary commodities.Suppliers choose the cheapest transportation plan in order to distribute commodities among warehouses.Commodities send to pre-considered storage centers by suppliers and affected country will distribute them from storage centers to demanded points.The suppliers try to minimize their transportation costs.The affected country, Iran, also must satisfy rescue demand at affected areas and transport injuries from these points to medical centers.Iran, in this paper, would have like to minimize total costs.Given the magnitude of the event, the demand of products and services are corresponded to a planning horizon of 2 days_ the maximum time in which human can survive without water.Shahr-e Rey, Mosha and the south and north Tehran faults are four active faults which can lead to an earthquake in Tehran.The model's scenarios is defined according to which one of this faults will be activated.Each fault causes earthquake with different magnitude in Region 1.We assume ten countries including Iraq, Turkey, Azerbaijan, Syria, UAE, Qatar, Germany, Turkmenistan, Kuwait, and Afghanistan will help Tehran after earthquake and send their products.
The products are including food, water, medicine, and shelter.Rescue operation can be done by air and land transportation for transporting injuries to medical centers.Trucks, ambulances and helicopters considered as rescue vehicles.We also assume Red Cross and Helal-e Ahmar agency coordinating Tehran with foreign to avoid sending unnecessary commodities to warehouses.Region 1 consists of ten area.We assume the center of each area as the demand points.Fig. 2 displays the location of warehouses and hospitals in this region.The used parameters for Tehran case study are shown in Table 1 to Table 4.

Table 1
The latitude and longitude of the affeced areas, located hospitals and located warehouses and foreign suppliers

Fig. 2. The location of hospitals and warehouses in Region one
Table 1 shows the latitude and longitude of the affeced areas, located hospitals and located warehouses and foreign suppliers.according to the following equation and Table 1 disances between nodes can be calculated.Warehouses' capacity of aid commodities are seen in Table 2. Package of aid commodities including water, food and shelter and medicine consist 1000, 1000, 1 and 1000 units, respectively.Table 3 consists of information about rescue vehicles and their characteristics.Warehouses' radius coverage for distributing aid commodities and evacuating is about 4 km.In addition, based on: (1) earthquake intensity, (2) population of the affected area we defined five scenarios, so Table 4 and Table 5 respectively show demand for aid commodity and rescue vehicle and maximum time related to rescue operation under each scenario and for two time periods.We assume that operation costs of rescue vehicles are $100, $200, and $1000.Unit of transportation cost for these vehicles are $0.01,$ 0.01 and $0.15.Finally Table 6 shows foreign suppliers' available package of aid commodities for each periods.Each package of these commodities consist of 1000, 1000, 1, 1000, 1000 units of water, food, shelter, and medicine type 1 and type 2, respectively.In this section we propose the numerical results conducted to the proposed case study.To show the benefits of bi-level modeling, a comparison between three points of view proposed.First, the model is investigated from the standpoint of Iran, the leader of bi-level model.Second, the standpoint of foreign countries and finnaly the leader-follower model in a bi-level structure is investigated.In order to solve the bi-level model, we used Fuzzy Goal Programming (FGP) approach and reduced the bilevel model to an equivalent single level model.Obviously, considering just one player's perspective will improve his/her objective function but cause disregarding others point of view which may be important and vital.

Table 7
The objective functions from different points of view for suppliers.Fig. 3 is a graphical representation of the three points of view performance.In Table 8 a comparison between each perspective against the best decision are made to show the advantages of using bi-level model for Tehran earthquake.First the amount of leader objective function in bi-level and follower based models compared with the leader's perspective objective function.Then the transportation costs of suppliers compared with the follower's perspective objective functions.The bi-level solution increases both upper and lower level objective functions but takes all decision makers points of view into account.On the other words, in leader's perspective the amount of its total costs will decrease but suppliers costs ignore.This can affect the performance of model in real situation.Also followers perspective model presents the cheapest transportation plan but larger total cost for Iran will accur.To show the benefits 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 8 and derive from the formula (37) adjusting the values in the correct way.As Table 9 shows, by choosing the bi-level solution instead of the Leader's perspective solution, we hugely reduce the total transportation costs 226.35% for international s.Then again, choosing the bilevel solution instead of the Follower's perspective gives us total cost reduction about 63.30%.In both cases, the reductions are more significant than the expected increase shown in

Conclusion and future research
In this paper, a robust bi-level optimization model for supplying and distributing of aid products after an earthquake has been developed.Supplires consist of foreign countries which play as the followers in the game and the affected country plays as the leader.To illustrate and apply the proposed model, Tehran probable earthquake in Region 1 was investigated.To show the advantages of using bi-level model, a comparison between leader, follower and bi-level model points of views was proposed.The results have been indicated that considering just leader or follower perspective generates solutions while bi-level model offers reseonable solutions concerning both the affected country and suppliers points of view.The amount of savings obtained by bi-level model also provided.In addition the experiments of sensitivity analysis for the capacity of warehouses and their radius coverage were proposed.To put in a nutshell, our contributions can be summarized as follow: 1. We developed a multi-period robust model for humanitarian logistics which captured uncertainty in the value of some input data.2. According to the absolute scientific gap in multi-level models we presented a bi-level model to capture different DMs points of view.3. Our model was developed not only to distribute aid commodities but also to evacuate injured people from affected areas.
Future studies can be aimed at new game conditions such as defining priority for suppliers to access to the warehouses vacant spaces.In addition, the proposed bi-level model could be modeled as a multi non-indipendent followers for example researchers can consider foreign and local supplier as non-independent followers.Finally, another intresting extension of this problem is considering roads condition stochasticity and model it by fuzzy programming to show the partial availability of roads after catasrophic earthquakes.

DU
Demand of rescue operation at affected area i at period t t i D Unit of transportation cost of warehouses for each unit of commodity type q q Tr Unit of transportation cost for rescue vehicle type l l Tr Unit of transportation cost of supplier e for each unit of commodity type q eq Tr Operation cost of warehouses for each unit of commodity type q q Op Operation cost for rescue vehicle type l l Op Inventory cost of warehouses for each unit of commodity type q q Inv Average velocity of rescue vehicle l l V Radius coverage of warehouse j j R Maximum time that rescue operations should be done at affected area i in period t Capacity of warehouse j for aid commodity type q jq C Available aid commodity type q in foreign supplier e in period t t eq b A very big number M Distance between affected area i and warehouse j ji d Distance between affected area i and hospital k ik d Distance between supplier e and warehouse j ej d Finally decision variables are as follows: Definition Parameter

D
Demand of rescue operation at affected area i at period t under scenario s t i D Maximum time that rescue operations should be done at affected area i in period t under scenario s Weight used in objective functions to show the importance of expected value against the standard deviation and penalty at the upper level , Weight used in objective functions to show the importance of expected value against the standard deviation and penalty at the lower level ,

Fig. 3 .
Fig. 3.The objective functions from different points of view

Fig. 4 .
Fig. 4. Impact of warehouse capacity on the amount of transportation costs

Fig. 5 .
Fig. 5. Impact of warehouses radius coverage on optimal bi-level model solutions The objective function of the leader under scenario s ,

Table 2
Warehouses' capacity of aid commodities and rescue vehicles

Table 4
Demand for aid commodities and rescue operation under each scenario and for two time periods

Table 5
Maximum time related to rescue operation under each scenario and for two time periods Considering just Iran's point of view will decrease the first level objective function but hugely increases second level objective functions while considering second level point of view will reduce suppliers transportation costs and increases the amount of Iran's total cost.Proposed bi-level model stablished a trade off between leader anf follower goals.As shown in Table7, the bilevel model performance is more reseonable since both leader and followers points of view are considered.In bi-level model, affected country's objective function value is between leader's perspective and follower's perspective value.In fact, considering just one perspective generates extreme solutions while bi-level model presents resoanable and moderate solutions.In other words, bilevel model results an equilibirium as leader

Table 8
Percentage of increase in the respective models against the best decision

Table 9
Percentage of decrease provided by the bi-level model

Table 8 .
The negative values in Table9represent an increase in Iran's total costs and suppliers' transportation costs.To explore the effects of various problem parameters, the example problem is accompanied by sensitivity

Table 10
Impact of warehouses capacity on optimal bi-level model solutions Table10represent the sensitivity analysis of warehouses' capacities for relief commodities including water, food, and shelter.For capacity values equal or lower than 10 for each relief commodity the solution will be infeasible.By increasing the capacity of warehouses for each relief commodity at beginning the objective functions start decreasing exponentially, because of decreasing in unsatisfied demand.But after turning point the objective function starts growing because of increasing in operational and inventory costs.Increasing in warehouses capacity for more than 225 does not create changes in first level objective function.In fact, capacities for more than 225 applies as big M since this parameter located in right hand side of constraints.

Table 11
Impact of warehouses radius coverage on optimal bi-level model solutions

Table 11
represents the sensivity analysis of proposed model to warehouses' radius coverage.The first level objective function reveals sensivity to changes in radius coverage values.But second level objective function does not show noticeable sensitivity to changes in raduis coverage values.Fig 5 is the graphical representation of these objective functions sensitivity.For radius coverage values equal or lower than 2.5 for each warehouse the solution will be infeasible.