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Transportation Research Part E 69 (2014) 160–179

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Multi-objective open location-routing model with split delivery for optimized relief distribution in post-earthquake Haijun Wang, Lijing Du ⇑, Shihua Ma School of Management, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

a r t i c l e

i n f o

Article history: Received 16 August 2013 Received in revised form 23 May 2014 Accepted 7 June 2014

Keywords: Emergency logistics Relief distribution Location routing problem Multi-objective optimization

a b s t r a c t The effective distribution of critical relief in post disaster plays a crucial role in postearthquake rescue operations. The location of distribution centers and vehicle routing in the available transportation network are two of the most challenging issues in emergency logistics. This paper constructs a nonlinear integer open location-routing model for relief distribution problem considering travel time, the total cost, and reliability with split delivery. It proposes the non-dominated sorting genetic algorithm and non-dominated sorting differential evolution algorithm to solve the proposed model. A case study on the Great Sichuan Earthquake in China expounds the application of the proposed models and algorithms in practice. 2014 Elsevier Ltd. All rights reserved.

1. Introduction Earthquakes have been among major fatal natural disasters, causing a massive death toll. Examples in recent years include the Great Sichuan Earthquake killing nearly 70 thousand people in May 2008(Fawu et al., 2009), the Haiti Earthquake in January 2010 claiming approximately 230,000 lives (Patrick and Anna, 2012), the Chile Earthquake with at least 708 victims in February 2010 (BBC, February 28, 2010), and the Great East Japan Earthquake in March 2011 with 15,883 confirmed deaths (Christine, November 22, 2013). When an earthquake happens, immediate distribution of emergency supplies is pivotal in minimizing the damage and the fatality and therefore the main focus of post-earthquake relief operations. As the time and the resources are limited, emergency logistics decision-makers have to make optimal decisions in the allocation of limited time, funds and other resources. In comparison with the traditional logistics, disaster relief distribution is more complicated and challenging in emergency logistics (Sheu, 2007). The features of relief distribution in post-earthquake are critical for decision-making. First, With ‘‘burstiness’’ of earthquake events, earthquakes cannot be predicted in advance with reasonable accuracy, which is different from hurricanes (Horner and Downs, 2007). Thus, decision makers need to establish temporary DCs where people can more effectively gain access to goods in post-earthquake stage, while for hurricane disaster the strategic location decision is made in preparedness planning stage (Horner and Downs, 2007, 2010; Rawls and Turnquist, 2010; Widener and Horner, 2011). Second, vehicles wait at their last node without returning to DCs until the next order is specified in post-earthquake, for that any given disaster areas receiving aids can be the new DC or the former DC may no longer provide service in the next mission. The open vehicle routing problem has been extensively discussed in traditional logistics (Schrange, 1981; Sariklis and Powell, 2000); however, few works have focused on OLRP in emergency logistics. Third, given the large demand of relief at the

⇑ Corresponding author. Tel.: +86 13638697612. E-mail address: [email protected] (L. Du). http://dx.doi.org/10.1016/j.tre.2014.06.006 1366-5545/ 2014 Elsevier Ltd. All rights reserved.

H. Wang et al. / Transportation Research Part E 69 (2014) 160–179

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affected areas in post-earthquake, an affected area can be served more than one time once the demand of the disaster area is greater than the capacity of the serving vehicle. This is called split delivery in the relevant literature (Dror and Trudeau, 1989, 1990; Dror et al., 1994; Özdamar et al., 2004; Yi and Kumar, 2007; Yi and Özdamar, 2007). Allowing split deliveries can lead to substantial cost savings, which has been introduced and empirically shown by Dror and Trudeau (1989, 1990). The fourth is potential secondary disaster, such as after-shocks and debris flow following the main earthquake, posing threat to relief personnel. This potentially differentiates planning for earthquakes from other types of disasters, for example, terrorism attacks and hurricanes. The connections between disaster areas become uncertainty with the influenced highway, bridge and railway etc. Reliability can be defined as the probability for vehicles travelling through the connections between disaster areas in time in post-earthquake. Disaster relief distribution with enhanced reliability not only protects the rescue personnel but also ensures timely delivery of emergency supplies to those in needs. High reliability in distribution can be achieved through routing vehicles such that supplies on the vehicles could reach their intended destination for sure in a timely manner. Based on above features, we study a relief distribution problem in post-earthquake with multiple conflicting objectives by considering travel time, the total cost, and reliability. The relief distribution involves the location of distribution centers (DCs) and vehicle routing and scheduling in post earthquake. The problem can be classified as multi-objectives open location-routing problem (OLRP) with split delivery. In general, location of DCs and vehicle routing are addressed individually for emergency logistics (Haghani and Oh, 1996; Özdamar et al., 2004; Barbaroso and Arda, 2004). However they are highly correlated (Ballou and James, 1993). Studies on the design of mathematical models and solution algorithms for the integration of location and route problem are much fewer in a post-disaster situation. In Fiedrich et al. (2000), DCs are definite and fixed in the distribution network, and resources are directly sent from DCs to demand points. Single objective model is designed to minimize the total number of casualties. Tabu Search and Simulated Annealing are used with fictitious data to test the models. Yi and Özdamar (2007) describe an integrated location-distribution model for coordinating resource supply and wounded people evacuation operations in response activities. Vehicle routing and scheduling is not considered in the above. Sheu (2007) investigate a hybrid fuzzy clusteringoptimization approach for multi-objective dynamic programming model. Weighting method is used to convert the two objectives of minimizing distribution cost and maximizing demand fill rate into one objective; however, the reliability of the infrastructure is neglected. Vitoriano et al. (2011) present an original multi-criteria optimization model based upon cost, time, priority etc. for humanitarian aid distribution. It helps to select vehicles and design routes; however, the location of DCs is not considered. For multi-objective optimization models, classical optimization methods (Chanta et al., 2011; Dror and Trudeau, 1990; Sariklis and Powell, 2000; Tzeng et al., 2007) take multiple runs to find multiple optimal solutions by converting the multi-objectives into a single-objective. Over the past few years, some researchers have been engaged in the development of multi-objective evolutionary algorithms which can find multiple solutions in a single run due to their population-based approach. Strength Pareto evolution algorithm (SPEA) (Zitzler and Thiele, 1999), multi-objective scatter search (MOSS) algorithm (Beausoleil, 2006), non-dominating sorting genetic algorithm-II (NSGA-II) (Deb et al., 2002), etc., constitute the pioneering multi-objective methods. Non-dominated sorting differential algorithm (NSDE) is proposed by Rakesh and Babu in 2005 as an extension of differential algorithm (Storn, 1996; Storn and Price, 1997) to solve multi-objective problems. In this paper, NSGA-II and NSDE are used separately for the specific multi-objective OLRP, and some comparisons are made to verify the efficiencies of the two algorithms. Within the scope of the study defined above, the proposed relief distribution models and approach in emergency logistics is unique with the following distinctive features: (1) Multi-objective open location and routing scheduling problem with split delivery are considered to coordinate supplies and demands of relief. Heterogeneous vehicles with different capacities and velocities are used. Vehicles may wait at their last node without returning to starts until the next order is specified in OLRP. With split delivery, each disaster area can be served by more than one time once the volume of relief needed is bigger than vehicle’s capacity. These distinct features make the problem closer to the real emergency situation. (2) Two model approaches: the three-index method and flow variables and constraints method are used to model the proposed multi-objective OLRP with split delivery. The relationship among multi-objective OLRP with split delivery, location problem (P-median) and typical VRP are described. Research on multi-objective OLRP with split delivery is more limited compared with the extensive literature on pure location problems, VRPs and their variants. (3) Reliability, as well as time and cost, is considered as an objective for relief distribution in the operation process. Reliability is one way to model high uncertainty in emergency logistics. The higher the reliability is, the safer the rescue workers are. (4) Both NSGA-II and NSDE are used to solve the Multi-objective OLRP with split delivery. The natural number permutation encoding is used to represent solutions. Fast-non-dominated sorting and Crowding-distance Sorting operators are both used for the two algorithms in selection operation. The difference lies to the mutation operator and the crossover operator. We make comparisons using instances generated randomly. The remainder of this paper is organized as follows: we describe the multi-objective OLRP in emergency logistics and the mixed integer programming mathematic models for the OLRP in Sections 2 and 3 respectively. In Section 4 we propose NSGA-II and NSDE to solve the multi-objective models. Finally, the models and algorithms are applied in Great Sichuan

162


Earthquake and sets of instances generated randomly are depicted to test the performance of NSGA-II and NSDE in Section 5 and conclusions are drawn in Section 6.

2. The problem description A general overview of post-earthquake relief distribution network can be described as graph G = (V, E) provided in Fig. 1, where V denotes the vertex set and E = {(i,j):i,j e V, i – j} is the set of available traffic links. Vertex set V contains two subsets: M and N, M = {n + 1, . . ., n + m} represents the set of candidate DCs and N = {1, . . ., n} is the set of disaster areas. Each vertex in M features none of demands. The distance matrix D = (dij) is assumed to satisfy the triangle inequality, i.e., dij < dik + dkj for all i,j,k e V. To ensure that vehicles move on proper links, the allowable velocity vij of link (i,j) is defined. Reliability rij is defined as basic data the probability for traversing the link (i,j) successfully in the network. It is difficult to evaluate, especially in this higher dynamic traffic network after earthquake; however, the values of these data applied are based on subjective perceptions rather than on observed data (Vitoriano et al., 2011). Thus, we assume the reliability rij is known a priori. Furthermore, the three attributes are assumed to be symmetric: dij = dji, vij = vji, and rij = rji. In order to select appropriate vehicles, some nonnegative attributes related to vehicles are defined: vk means the normal velocity of k type vehicles, ck denotes the cost per unit length of available links for k type vehicles, and Lk describes the loading capacity. The problem proposed aims at coordinating the transportation of relief from distribution centers to disaster areas. Sleeping bags and water are critical supplies, which are urgently demanded in case of an earthquake. The amount of relief is considered for one day needed by helpless people. First, suppliers originate from DCs established and are loaded on scheduled vehicles. Second, suppliers will be given to the disaster areas allocated on routes of dispatched vehicles. Split deliveries are required once the demands of any disaster area break the capacity of serving vehicles. And vehicles stay at the end point without returning to the depot until the next mission is received. The goals of our problem are: (1) to determine the subset of DCs to open; (2) to allocate disaster areas and vehicles to DCs; (3) to plan the routes from DCs to disaster areas considering the capacities of vehicles with split delivery. The following three objectives are considered for the OLRP in emergency logistics: (1) minimization of the maximum route travelling time; (2) minimization of the total cost, including the fixed establishing costs of DCs and the vehicle travelling cost; (3) maximization of the minimum route reliability for all the serving vehicles. Objective 1 is for the goals of effectiveness and fairness. The maximum vehicle route travelling time means the latest service completion time among all the disaster areas. Objective 2 is designed for the pursuit of the economic value. Owing to the limited funds, the distribution cost should not be neglected. Further, using donations economically increases the public trust in government. Objective 3 is designed in consideration of reliability. In this paper, the reliability of route k means the possibility for drivers to deliver relief to all the demand points on route k successfully. We need to avoid the secondary casualties to the largest extent. At the initial stage of earthquakes, objective 1 and 3 are more essential than objective 2. Victims are more likely to survive with earlier rescue work. With the recovery of damaged infrastructures at a later stage, it is easier for rescue workers to arrive at the disaster areas. Thus, economic value is relevantly more critical for the rescue work in later phases.

3. Model formulations Two model approaches are used in this section. Firstly we extend the three-index formulation of LRP proposed by Perl and Daskin (1985). Secondly inspired by models proposed by Hansen et al. (1994), we use flow variables and constraints to model the OLRP.

i

1

d ij , vij , rij

DC

DC

j

2

3 DC

4

Candidate DC

Disaster area

Available link

Fig. 1. A sample network for OLRP in emergency logistics.


163

3.1. Assumptions (1) The number of disaster areas and candidate DCs is known. The available links, including distance, maximum velocity can be obtained with advanced disaster detection technology in real time and transportation research community’s expertise. The possibility for successfully travelling through the available links is known a priori. (2) Each vehicle is allowed to stow multiple types of relief for any given transportation assignment. At same time, different types of relief are permitted to be loaded in the same vehicle. (3) Only the disaster areas that are still reachable by vehicles through the current traffic network are considered, and those need helicopter or other extraordinary means of transportation are ignored. Based on the aforementioned assumptions, the multi-objective OLRP under consideration can be described as in the following subsections. 3.2. Notations and definitions Let us first propose the notations and definitions in three-index formulation for the OLRP, and then introduce additional symbols in flow variables and constraints method. Sets and indices N ¼ f1; . . . ; ng M = {n + 1, . . ., n + m} V = {1, . . ., n + m} K = {1, . . ., j} L = {1, . . ., s} E = {(i,j),i,j e V, i – j} i,j l k Parameters m n

j fj dij

vij rij Dil UVl Ql ck

vk Lk Variables xj yijk zik qilk devil VFik

Set of disaster areas; Set of candidate DCs; Set of nodes, V = M [ N; Set of vehicles; Set of relief; Set of available traffic links; Indices to nodes i,j e V; Indices to relief, l e L; Indices to vehicles, k e V; Number of candidate DCs; Number of disaster areas; Number of vehicles; Fixed cost of establishing the DC j, "j e M; Distance of link (i,j), "(i,j) e E; Maximum allowable travelling velocity link (i,j), "(i,j) e E; Probability of crossing arc (i,j) successfully, "(i,j) e E; Quantity of relief l demanded by disaster area i, "l e L, "i e N; Unit volume of relief l, "l e L; Amount of relief l available in traffic network, "l e L; Transportation cost per kilometer of vehicle k, "k e K; Normal velocity of vehicle k, "k e K; Loading capacity of vehicle k, "k e K; 1, if candidate DC j is opened, 0, else, "j e M; 1, if i precedes j in route of vehicle k, 0, else, "k e K, "(i,j) e E; 1, if i is on route of vehicle k, 0, else, "k e K, "(i,j) e E; Quantity of relief l distributed by k to demand point i, "i e N, "l e L, "k e K; Amount of unsatisfied demand f relief type l at node i at the end of the operation, "i e N, "l e L; 1, if the last demand point serviced by vehicle k is node i e N; 0, else, "i e N, "k e K;

The OLRP under investigation can also be alternatively formulated through using flow variables and constraints. We use qijlk as the flow of relief along the distribution network: P P qijlk: Quantity of relief l shipped from i to j with vehicle k, "k e K, (i,j) e E and qilk = j/(j,i)eEqjilk j/(i,j)eEqijlk. 3.3. Multi-objective Relief Distribution Model The descriptions of objectives are the same in the models through using the two methods.

164


3.3.1. Objective 1: Minimization of the maximum vehicle route travelling time The time tijk for vehicle k traversing arc (i,j) e E is dependent on the length of the dij, the maximum allowable velocity vij, and the normal velocity vk associated with vehicle k. It is calculated as t ijk ¼ dij = minðv ij ; v k Þ. Without serving time (pickup and delivery time) considered, assuming the leaving time for vehicles from DCs is 0, the travelling time tkis equal to the total P P d y travelling time through all the links on route of vehicle k, tk ¼ ði;jÞ2E t ijk ¼ ði;jÞ2E minðijv ijk;v Þ. Let ftime denote the headway for ij

k

relief distribution in emergency network, the objective 1 can be formulated as:

(

ftime ¼ Minimax

) dij yijk ;k2K : minðv ij ; v k Þ ði;jÞ2E X

ð1Þ

3.3.2. Objective 2: Minimization of relief distribution cost The OLRP is to simultaneously determine the number, locations of DCs, assignment of disaster areas to DCs, and the corresponding vehicle delivery routes, such that the total costs consisting of two components are minimized: the fixed cost fj for P establishing DC j (j e M), and the vehicle travelling cost ði;jÞ2E ck dij yii’k (k e K). The objective function is as follows:

fcost

( ) X XX ¼ Min fj xj þ ck dij yijk j2M

ð2Þ

k2K ði;jÞ2E

3.3.3. Objective 3: Maximization of the minimum route reliability We describe the reliability of route k as the possibility for rescue workers to deliver relief to all the demand points on route k completely. The first link in route k might have a reliability of 0.97 of working properly for a delivery; the second one might have a reliability of 0.98 of work without major disruption, and so forth. We can use the reliability of each link to calculate the reliability of the corresponding route as a whole. For example, Fig. 2 presents nodes and links in route of vehicle k. The probability of reaching node j from i is rij. Let Pk denote the reliability for vehicle k to accomplish the corresponding distribution mission successfully in Fig. 2. Assuming the links on route rk are independent of each other, Q Pk ¼ r 01 r12 ::: rn1 rn ¼ ði;jÞ2routek r ij . In this way, we could compute the reliability for each route in a solution. The objective 3 is to maximize the minimum route reliability for urgent relief distribution. Let freliability denote the reliability for the whole relief distribution work in emergency network. The formulation for objective 3 is given by

freliability

8 < Y ¼ Maximin :ði;jÞ2E;y

r ij ; k 2 K

ijk ¼1

9 =

ð3Þ

;

The objective 1 and 2 are suitable for other disasters with limited time and funds, such as hurricanes and terrorists. These two objectives in post-earthquake could put the disaster relief needs associated with earthquakes in a broader context. The objective 3 is the specific concern of relief distribution in earthquake disasters, which considers the safety of rescue personnel with aftershocks and debris following earthquake. 3.3.4. Constraints by the three-index method

xi P yijk ; 8i 2 M; ði; jÞ 2 E; k 2 K

ð4Þ

xi P zik ; 8i 2 M; k 2 K zik P yijk ; 8i 2 V; ði; jÞ 2 E; k 2 K

ð5Þ ð6Þ

zik P VF ik ; 8 2 V; k 2 K X VF ik ¼ 1; 8k 2 K

ð8Þ

ð7Þ

i2V

X

yjik

j=ðj;iÞ2E

X j=ði;jÞ2E

yijk

8 VF ik ¼ 1; i 2 N > < 1; ¼ 1; zik ¼ 1 i 2 M; 8k 2 K > : 0; else

r12

r01 0

ð9Þ

r ij i

2

1 DC

Disaster area

route k Fig. 2. Reliability example.

j

n


XXX qjlk zik 6 Q l ; 8l 2 L

165

ð10Þ

i2M k2K j2N

dev jl ¼ Djl

X qjlk P 0; 8j 2 N; l 2 L

ð11Þ

k2K

XX UV l qilk 6 Lk ; 8k 2 K

ð12Þ

i2N l2L

qilk P 0; 8ði; jÞ 2 E; l 2 L; k 2 K

ð13Þ

xj 2 ð0; 1Þ; 8j 2 M

ð14Þ

yijk 2 ð0; 1Þ; 8ði; jÞ 2 Ek 2 K

ð15Þ

zik 2 ð0; 1Þ; 8i 2 Vk 2 K

ð16Þ

VF ik 2 ð0; 1Þ; 8i 2 N; k 2 K

ð17Þ

Constraints (4) and (5) ensure that only the opened DCs can provide service. Constraints (6) ensure that vehicle k can travel through link (i,j) only and if only node i is on route of vehicle k. Constraints (7) characterize that the node at the end of the route k should be serviced by vehicle k. Constraints (8) mean that each vehicle should stay at only one node finally. Constraints (9) ensure that vehicle k arrives at and departs from the same point, and stays at the last node on its route without going back to the leaving DC. Constraints (10) stipulate that the amount of relief distributed to the disaster areas from all the DCs does not exceed the corresponding relief amount available. Constraints (11) guarantee that the amount of relief l distributed to node i does not exceed the amount of demands of node i. Constraints (12) guarantee the volume of all the relief distributed to disaster areas by vehicle k without exceeding its capacity. Binary integer and nonnegative values constraints for the decision variables are given in the remaining constraints (13)–(17). Here we provide valid sub-tour elimination and connectivity constraints for the OLRP with split delivery. Let di denote the P P outgoing degree of node i: di = keK jeVyijk (i e N). Following the model proposed by Dror et al. (1994), constraints (18) allow eliminating sub-tours disconnected from DCs.

XX

yijk 6

k2K i;j2N

X di j

ð18Þ

i2N

Here j denotes the number of vehicles.

XX yjik 6 1; 8ði; jÞ 2 E; k 2 K

ð19Þ

j2M i2N

XX yjik P 1; 8i 2 N

ð20Þ

k2K j2V

X yjik 6 1; 8i 2 N; k 2 K

ð21Þ

j2V

Constraints (19)–(21) are connectivity constraints. Constraints (19) ensure that each vehicle k departs from no more than one DC. Constraints (20) imply that any disaster area i will be visited at least once. Constraints (21) mean that any disaster area i can be served no more than one time by the same vehicle, also eliminate sub-tour for route of vehicle k. 3.3.5. Constraints in flow variables and constraints formulation

X X

qjilk

k2K j=ðj;iÞ2E

XX X

X X

qi jlk þ dev il ¼ Dil ; 8i 2 N; l 2 L

ð22Þ

k2K j=ði;jÞ2E

qijlk 6 Q l ; 8i 2 M; l 2 L

ð23Þ

i2M k2K j=ði;jÞ2E

X X X X X yjik ¼ yijk þ VF ik ; 8i 2 N

j=ðj;iÞ2E k2K

j=ði;jÞ2E k2K

X X yijk ¼ j; 8i 2 M

ð24Þ

k2K

ð25Þ

j=ði;jÞ2E k2K

X qijlk UV l 6 Lk ; 8ði; jÞ 2 E; k 2 K

ð26Þ

l2L

qijlk P 0; 8ði; jÞ 2 E; l 2 L; k 2 K

ð27Þ

166


Constraints (22) are equations for load flow balance at disaster areas, and guarantee that the total quantity of relief l distributed to node i does not exceed the amount of demands in node i. Constraints (23) are flow conversation constraints at DCs and mean that the amount of relief l e L distributed by all the vehicles from all the DCs does not exceed the corresponding amount of relief available. Constraints (24) and (25) are the flow balance of the dispatched vehicles equations at disaster areas and DCs respectively. Constraints (26) define the volume of all the relief loaded by vehicle k without exceeding its capacity. Constraints (27) mean that the quantity of relief distributed by vehicles is nonnegative. The flow conservation constraints guarantee that sub-tours formed by disaster areas are not feasible; since each disaster area requires a positive amount of flow. Hence, elimination constraints (18) and connective constraints (19)–(21) are redundant in flow formulation. Constraints (4), (14), (15), (17) are also necessary for flow variables and constraints model. The key difference between the flow variables and constraints model and the three-index model is the number of the constraints. The number of constraints in flow formulation is polynomial with respect to the problem size; that makes it tractable in practice, when solution approaches are investigated. As far as the number of the variables is concerned, flow formulation uses the flow variables qijlk, which are not present in the three-index model; on the other hand, it does not use the variable zik and qilk. Constraints (5), (6), (7), and (16) with zik are redundant; Constraints (13) concerning qilk in model by the three-index method are replaced by (23), (26), and (27). Relief flow conservation constraints (22) substitute constraints (11), and constraints (9) is replaced by constraints (24) in the flow variables and constraints model. With the presence of flow conservation constraints, constraints (8) and (13) are redundant in flow variables and constraints model as well as the sub-tour elimination constraints (18) and connective constraints (19)–(21). Therefore, depending on the number of links, disaster areas and candidate DCs in the input distribution network, and the size of the fleet of vehicles, the flow variables and constraints model may also be more compact than the three-index model in the number of the variables. The above OLRP models integrate the facility location problem and vehicle routing problem. The classical facility location problem selects the best p locations among a range of potential DCs, which is called p-median problem (ReVelle and Swain, 1970; Jackson et al., 2007). The objective of p-median model is minimizing the average distance from open DCs to disaster areas. However, according to objective (1), (2), and (3), in OLRP, what needs to be established are temporary locations where relief goods can be distributed to disaster areas effectively, efficiently and safely. The p-median structure imposes that each customer should be served by exactly one distribution center. According to constraints (11) in three-index formulation and constraints (22) in flow variables and constraints formulation, each disaster area can be served by multiple vehicles, which depart from different DCs. Compared with typical VRP in business logistics with urgent demands, the OLRP with split delivery in formulations is distinct in the following: 1) Since each dispatched vehicle k stays at the last nodes on the route of vehicle k, variable VFik is defined in our OLRP model, and used in constraints (9) in the three-index model, and (24) in the flow variables and constraints model. This P P is critically different from typical VRP in which jeVyjik jeVyijk = 0, i e N. 2) With split delivery strategy, constraints (20), (22), (23) imply that each disaster area can be served more than once by different vehicles. This is different from typical VRP without considering split delivery, in which the total quantity of relief distributed to each node is provided only once. 3) Sub-tour elimination constraints for typical VRP are not valid for the OLRP with split delivery. The sub-tour elimination constraints in three-index formulation are shown in constraints (18). Refer to literature by Dror et al. in 1994 for more information about the elimination constraints developed for the typical VRP. Furthermore, the flexibility of allowing the vehicles to be different types with different capacities and velocities in the problems is added instead of the homogeneous type. From above on, OLRP with split delivery is an extension of typical VRP with location decisions considered.

4. NSGA-II and NSDE Algorithms OLRP with split delivery results in nonlinear mixed integer programming models. Since they are generalized LRP and NPhard problem (Balakrishnan et al., 1987), the use of heuristics should be justified. NSGA-II and NSDE are applied to solve multi-objective OLRP models in this paper. The NSDE integrates the DE operations (Rakesh and Babu, 2005) with the selection operation in NSGA-II (Deb et al., 2002). The NSGA-II differs from NSDE in the mutation and crossover operators. The more details are shown in the following parts.

4.1. Principle of Multi-objective Optimization Optimizing different objectives simultaneously is required for most practical problems, and these objectives often conflict and compete with each other.


167

(1) Pareto dominance: Considering all objectives, if solution x1 is at least as equal as x2, and better than x2 with at least one objective value, solution x1 dominates solution x2 (denoted as x1 x2). In formal terms, for minimizing (f1, . . ., fW), x1 x2 iff:

ð8w 2 f1; 2; . . . ; Wg : fw ðx1 Þ 6 fw ðx2 ÞÞ ^ ð8w0 2 f1; 2; . . . ; Wg : fw ðx1 Þ < fw ðx2 ÞÞ

ð28Þ

(2) Pareto optimum: A solution x1 is called Pareto optimal or non-dominated solution if and only if there is no any solution x2 that satisfies x2 x1. (3) Pareto front: Furthermore, if x1 is Pareto optimal (non-dominated), then f(x1) = {f1(x1), . . ., fW(x1)} is said to be non-dominated vector. The set of all non-dominated vectors is called Pareto front (or non-dominated frontier). 4.2. NSGA-II and NSDE Operations Series of sets of solutions called generations are computed by both of the two algorithms. Each generation consists of several individuals called chromosomes. And each chromosome denotes one strategy for relief distribution. The initial operation, mutation and crossover operations are applied to generate new solutions. The selection operation is used to choose the elites into the offspring. 4.2.1. Initial population In this work, we use the natural number permutation encoding to represent solution in both NSGA-II and NSDE to solve the particular multi-objective OLRP. In our approaches, each chromosome consists of three sub-strings, as shown in Eq. (29), where g = 1, 2, . . ., NP, NP is the number of chromosomes in population, and t is the number of generation, t = 0 for initial population.

X tg ¼ fðxtg11 ; xtg12 ; . . . ; xtg1j Þ; ðxtg21 ; xtg22 ; . . . ; xtg2j Þ; ðxtg31 ; xtg32 ; . . . ; xtg3n Þg |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} xtg1

xtg2

ð29Þ

xtg3

Sub-string xtg1 is a permutation of j vehicles. Sub-string xtg2 is a j-dimensional vector with each gene randomly generated from 1 to m, where m is the number of candidate DCs. Meanwhile, sub-string xtg3 is a permutation of n demand points. From above, each chromosome is a j + j + n dimensional vector. If all the numbers from 1 to m appear in sub-string xtg2 ; it indicates all the candidate DCs s are open. Thus, the DCs location decision is decided in this way in the solution denoted by chromosome X tg . Sub-string xtg1 and xtg2 together make the vehicle allocation decision. The vehicle represented by the first gene in substring xtg1 is located at the DC denoted by the first gene in sub-string xtg2 ; the vehicle denoted by the second gene in sub-string xtg1 is located at the DC denoted by the second gene in sub-string xtg2 ; and so forth. The set of disaster areas served by each vehicle, the servicing sequences and the quantity of suppliers distributed to disaster areas in each route of vehicle can be identified by Sub-string xtg1 and xtg3 together. The vehicle in the first gene of sub-string xtg1 firstly distributes suppliers to the disaster area denoted by the first gene of sub-string xtg3 . We first check if the vehicle available loading capacity is enough for the demands in the disaster area. If not, and the vehicle capacity is bigger than the volume of the total suppliers remained in the emergency system, the suppliers are distributed to the disaster area according to the demand quantity by the vehicle, and the whole relief distribution work is finished. Else if the available remaining quantity of suppliers in the emergency system is larger than the vehicle loading capacity, the vehicle is full loaded and stays at the disaster area when service is finished, the demand of the disaster area will be updated. And the vehicle denoted by the next gene in sub-string xtg1 will be used from the next gene in sub-string xtg3 . Else, the disaster area in the first gene of sub-string xtg3 will be served by the vehicle in the first gene of sub-string xtg1 ; and the quantity of each supplier distributed is equal to the demand. And the remaining available capacity of the vehicle and the quantity of suppliers in the emergency system will be updated. Then the demand point in the next gene of sub-string xtg3 will be considered for the vehicle. We repeat the above processes till all disaster areas are served or all the vehicles are used. If all the vehicles are used, then a feasible scheme is obtained. Else if the vehicles are not used up, and there are still available relief remained, we repeat the procedure from the first gene of sub-string xtg3 to satisfy the remaining demands of disaster areas from the current gene in sub-string xtg1 : In this way, split delivery occurs and some disaster areas could be serviced more than once by different vehicles. 4.2.2. Mutation operations Mutation is used to generate mutant vector V tg for each generation t, where V tg is represented as (30):

V tg ¼ fðv tg11 ; v tg12 ; . . . ; v tg1j Þ; ðv tg21 ; v tg22 ; . . . ; v tg2j Þ; ðv tg31 ; v tg32 ; . . . ; v tg3n Þg |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v tg1

v tg2

v tg3

ð30Þ

168


Where

v tg1 ; v tg2 and v tg3 are mutant vectors for the three sub-strings in X tg

respectively.

(1) In NSGA-II, reverse sequence mutation (Abdoun et al., 2012) is applied to avoid the local optimization for the three sub-strings in X tg separately. The inverse scheme randomly selects two points within a chromosome and then inverse their contents. The example is shown in the following: Step1: Randomly select the inverse points. Parent = [1 2/3 4 5 6 7 8/9 10]; Step 2: Inverse the genes between the two points, then get the child. Child = [1 2/8 7 6 5 4 3 8/9 10]; In this way, the perturbation in NSGA-II occurs in accordance with a random quantity. (2) In NSDE, the mutation strategy used is known as DE/rand/1 (Storn, 1996), in which the population is perturbed by adding the weighted difference between two individuals to the third one at each generation t (in Eq. (31)).

V tg ¼ X tc þ FðX ta X tb Þ

ð31Þ

Where g = 1, 2, . . ., NP, and a, b, c e [1, NP] are chosen randomly a – b – c – g. The scaling factor F is responsible for scaling the step size resulting from ðV ta V tb Þ; and lies in the range of [0, 1.2]. If any gene lies beyond valid ranges after implementing formulation (31), an auxiliary operator must be considered to modify it. For sub-string v tg1 and sub-string v tg3 ; the largest-order-value (LOV) rule proposed by Qian et al. (2009) is adopted here. In detail, for v tg1 the largest value of genes in mutant vector gives the maximum vehicle number j, the second is j 1, etc. Taking v tg1 = [7.1, 1.3, 5.6, 2.5, 3.7, 0] (the sum of vehicles is 6) as an example, the modified vector by using LOV is [1, 5, 2, 6, 3, 4]. The operation of v tg3 is similar to v tg1 . For sub-string v tg2 ; the LOV rules do not make sense on account that the values of some genes may equal to each other. Some modifications are required to ensure that value of each gene is valid. First, to ensure each gene is nonnegative and integer, we make v tg2 ¼ djv tg2 je; where dxe means the ceiling function, such as d3:2e ¼ 4; second, we replace the genes that violate boundary with values generated randomly within the allowable range. Then, the discrete values can be treated as an integer feasible variable. 4.2.3. Crossover operation Crossover operation is employed to produce the trial vectors U tg after mutation operator in generation t. U tg can be described as Eq. (32):

U tg ¼ fðutg11 ; utg12 ; . . . ; utg1j Þ; ðutg21 ; utg22 ; . . . ; utg2j Þ; ðutg31 ; utg32 ; . . . ; utg3n Þg |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} utg1

utg2

ð32Þ

utg3

(1) In NSGA-II, the two- points crossover based on De Jong (1975) is used to produce trial vector utg1 and utg3 ; and singlepoint crossover operator is used to produce utg2 : Take utg1 as an example, we first randomly select the crossover points in mutant vectors

v tg1 and v tðgþ1Þ1 :

v tg1 ¼ ½1 2 3 j 4 5 6 7 j 8 9 10; v tðgþ1Þ1 ¼ ½4 5 2 j 1 8 7 6 3 j 10 9; Secondly, retain the digitals before the first crossover point which do not appear between the crossover points in each mutant vector, and then crossover the digitals between the points in the two mutant vectors into trial vector utg1 and utðgþ1Þ1 :

utg1 ¼ ½2 1 8 7 6 3 ; utðgþ1Þ1 ¼ ½2 4 5 6 7 ; Thirdly, copy the digitals behind the first crossover point in the parent chromosome, which do not appear in the identified part into trial vector.

utg1 ¼ ½2 1 8 7 6 3 4 5 9 10; utðgþ1Þ1 ¼ ½2 4 5 6 7 1 8 3 10 9; (2) In NSDE, the crossover operation (Storn, 1996) is carried out through generating a random number, and if the random number is bigger than CR (crossover probability), copy the corresponding genes of the target vector X tg into the trial vector U tg ; otherwise copy genes in the mutant vector V tg : CR ranging from 0 to 1 controls the diversity of the population. A self-adapting CR is proposed in (33) to avoid the local optimum.

CR ¼ 1 t=Maxgen

ð33Þ

Maxgen is the total number of iterations. CR changes from large to small with iteration number t. In the early stages, CR is large and can improve the global searching capability; in the later stage of iteration, CR is small and can accelerate the convergence.


169

4.2.4. Selection operation The selection operation used in NSGA-II and NSDE is proposed by Deb et al. in 2002, and simulation results on difficult test problems have shown that the proposed selection operation is able to find much better spread of solutions and better convergence near the true Pareto-optimal front. A mating pool by combining the parent and offspring populations is created through using the selection operation. With fast-non-dominated-sorting and Crowding-Distance-Sorting scheme, the NP best solutions with respect to fitness and spread are selected as the next population. 4.2.4.1. Fast-non-dominated-sorting. Non-Dominated-Sorting is a procedure ranking all the individuals of the current population to form non-dominated fronts, hence, classifying the chromosomes into several non-dominated solutions fronts. Front 1 (F1) is created as the set of all the non-dominated solutions in the current population; Front 2 (F2) consists of all the nondominated solutions in the set of solutions obtained by removing front 1; recursively, Fi (r > 1) includes all non-dominated solutions in the set of solutions got by removing fronts 1 to r 1. In this paper, we use the fast non-dominated sorting approach proposed by Deb et al. (2002). The pseudo code is described as follows: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Fast-non-dominated-sort (P) //{Sort the current population according the Pareto-optimal front each solution p belongs to} for each p e P Sp = £ ; Np = 0. // {Sp: The set of solutions dominated by p, Np: the number of solutions which dominate the solution p} for each q e P if (p q) then // {(p q: p dominates q} Sp = Sp [ {q} else if (q p) then Np = Np + 1 end end if Np = 0 then prank = 1 //{Solutions in the first front have Np = 0, prank : the front of p} F1 = F1 [ {P} i=1 end while Fi – £ Q = £// {Q the solution set of the next front} for each p e Fi for each q e Sp Nq = Nq 1 //{visit each solution (q) in of set Sp, for each p e Fi , and reduce its domination count by one} if Nq = 0 then qrank = i + 1 Q = Q [ {q} end end end end i=i+1 Fi = Q end //{This process continues until all fronts are identified}

4.2.4.2. Crowding-Distance-Sorting. To preserve diversity, crowding distance, the normalized distance to closest neighbors in objective space, is computed to estimate the solution density surrounding a particular solution in the population. In this study, Harmonic-Average-Distance-Assignment procedure proposed by Huang et al. (2005) is used to as the Crowding-Distance-Sorting method. Assuming the a -nearest neighbor distances around one solution is d1, d2, d3, . . ., da, the harmonic distance is: d = a/(1/d1 + 1/d2 +, . . ., + 1/da). Here a = NP, it means all the individuals in current population are considered in Harmonic-Average-Distance. In this way the potential abnormal influence on the computation of density degree can be eliminated. For solutions in the same Pareto front, the one with a larger Harmonic-Average-Distance is much better for preserving the diversity of solutions. 4.2.4.3. Selection procedure. First we combine the current population Xt and corresponding trial population Ut to form Rt with 2NP size, and evaluate each individual in Rt; Second, Fast-Non-Dominated-Sorting approach is used in Rt to get Pareto-optimal fronts F1, F2, F3. . .;

170


Third, Let Si denotes the number of individuals in Fi, find non-dominated front rank r, which satisfies:

8 r1 X > > > Si 6 NP > < i¼1

ð34Þ

r > X > > > Si P NP : i¼1

Then we compute the crowding distances for individuals in front Fr using Harmonic-Average-Distance-Assignment operation, and sort them in descending order. Pr1 Fourth, individuals in ranks from F1 to Fr-1, and first NP i¼1 Si individuals in Fr are selected into next generation Xt+1. The conceptual schema of Selection is illustrated in Fig. 3.

4.3. NSGA-II and NSDE Procedure Based on the above introduction, the procedures of NSGA-II and NSDE in the present study are summarized as follows: Step 1: Generate parent population Xt of size NP. Initial population X0 (t = 0) is generated randomly. Step 2: Perform mutation, crossover operators over each individual in the population Xt. In this way trial vectors Ut of size NP are generated. Step 3: Combine the parent population of Xt and trial population Ut together to form population Rt. Compute the objective values for each chromosome in Rt. Since all parent and current population are included in Rt, elitism is guaranteed. Step 4: Classify all the individuals in Rt into several ranks based on non-domination obtained by applying fast-non-dominated-sorting operation. Step 5: Calculate the crowding distance for each individual in any front F of Rt. Then sort all the individuals in front F in descending order of magnitude according to crowding distance. Step 6: Select the best NP individuals based on their ranking and crowding distances. In the next generation, these NP individuals act as the parent population. Step 7: Stop the procedure if the generation t is bigger than maxgen (maximum of iteration times), else turn to Step 2. The flowchart is illustrated in Fig. 4.

5. Illustration of the approaches on Great Sichuan Eearthquake 5.1. Test Instances The Great Sichuan Earthquake measured 8.0 on the Richter scale occurred in China is studied in this part. With the disaster-related information including the candidate DC locations, relief demand from disaster areas, and the status of routes, the proposed model and method are employed to determine the relief distribution flows in post disaster period. The study is aimed at the 11 most severely and reachable disaster areas in Sichuan with 3 candidate DCs to supply relief, whose distribution includes tents and water. The relief is gathered from other cities in China, which is beyond the domain of the research.

Non-DominatedSorting F1 F2 .. . Fr

Xt

Crowding Distance-Sorting

Xt+1 . . .

}

Ut

reject

Rt Fig. 3. Selection Procedure based on Deb et al. (2002).


171

In addition, the parameters of the multi-objective OLRP model must be determined preceding the integrated DCs location and route scheduling for relief distribution. On account that some data are not yet released for public use by the government, we insert man-made data to test the model and algorithm, which will not lead to essentially different results. The corresponding parameters are set as follows: (1) Candidate DCs information. Once the candidate locations for DCs are selected, the cost for establishment can then be estimated. The candidate DCs in this case are Chengdu, Mianyang, Guangyuan, which are shown in Table 1. We assume the capacity of each candidate DC is large enough. (2) Relief. The size of a tent is used as the criterion for measuring volume equivalents. The parameters of relief can be found in Table 2. (3) Vehicle. The vehicles used for transportation are of 3 types (10 large military vehicles, 6 medium military vehicles and 9 small Civilian trucks). For more detail see Table 3. (4) Demand information. We assume the relief demands for each disaster area on the first day of the earthquake are as in Table 4. (5) Traffic network status. For each of disaster areas, the relevant information is obtained from the government reports. Beside, in advance to vehicle dispatches, the state of the traffic network needs to be determined; it can be obtained through a bird’s-eye view from helicopter immediately after the earthquake. In addition, on the basis of the manpower and relative equipment layout, the length and reliability of links, as well as the maximum permissible road speed, can be estimated.

2

4

Fig. 4. The flowchart of NSGA-II and NSDE.

172


The national and provincial highway, county roads are taken into consideration in this test instance study. More details about the emergency logistics network are exhibited in Table 5, and each of the three values in the form denotes length, the maximum allowable velocity and reliability of the links between two nodes, if ‘–’ means there is no direct path. The first three rows j1 j2 j3 denote the parameters of links between 3 candidate DCs (Chengdu, Mianyang, Guangyuan) and 11 disaster areas. And other rows denote status of links among the 11 disaster areas. The topology of the transportation network is illustrated in Fig. 5, which is superimposed on the map of Sichuan. It is constructed based on the road condition of Google Map. The demanding nodes labeled from 1 to 11 and 3 candidate DCs labeled from 12 to 14 are shown in the network. The node labeled 15 is working as transship node. For simplicity’s sake, straight lines denote links between disaster areas. All the 34 available links are shown in Fig. 5 in different color based on reliability and different thickness depending on their maximum speed of the vehicles travelling through them (take the black color as an example for illustration in the label). It is safer for drivers to travel through more reliable links. 5.2. Computational results The algorithm described in Section 4 is programmed at Matlab6.5, and all the calculation work is finished with the same computer. Table 6 is the configure parameters of the computer. With considerable research data, the best values of control variables are determined separately by trial & error: NP = 66, for NSDE F = 0.5, CR = 1-t/maxgen; for NSGA-II, the mutation rate pm = 0.7, and crossover rate pc = 0.7.

Table 1 Candidate DCs parameters. Name of candidate DCs

Fixed cost fj (Yuan)

Chengdu j1 Mianyang j2 Guangyuan j3

10,000 15,000 20,000

Table 2 Parameters of relief commodity. Item

Available quantity Ql

Calculation unit

Unit volume UVl (cm3)

Tent l1 Mineral water l2

2000 6000

Each Box (1410 ml,12 bottles)

45 25 11 36 26 30

Table 3 Parameters of the vehicles. Vehicle name

Loading capacity Lk (cm3)

Normal velocity

Military vehicle k1 Military vehicle k2 Civilian truck k3

600 250 175 280 200 145 231 150 130

50 30 20

vk (km/h)

Cost per unit of length ck (Yuan/km) 10.0 3.1 1.7

Table 4 Parameters of the disaster areas. Demand points

Demands (nylon sleeping bags, mineral water) Dil ðDi1 ; Di2 Þ

Wenchuan i1 Jinzhu i2 Beichuan i3 Qingchuan i4 Maoxian i5 Dujiangyan i6 Anxian i7 Pingwu i8 Pengzhou i9 Jiangyou i10 Deyang i11

(3458,1153) (3647,1216) (969,323) (1545,515) (818,273) (439,146) (1348,449) (3215,1072) (577,192) (1002,334) (3199,1066)

173

H. Wang et al. / Transportation Research Part E 69 (2014) 160–179 Table 5 Parameters of the traffic network (distance (km), maximum velocity (km/h), reliability.

j1 j2 j3 i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 i11

i1

i2

i3

i4

i5

i6

i7

i8

i9

i10

i11

– – – 0,0,1

– 88,46,1 – – 0,0,1

– – – – 66,44,1 0,0,1

– 185,52,1 127,52,1 – – – 0,0,1

– – – 39,24,0.5 – 89,22,0.5 251,27,0.5 0,0,1

59,47,1 – – 78,29,0.8 85,32,0.9 – – – 0,0,1

– 17,43,1 – – 48,48,0.9 37,45,0.8 – – – 0,0,1

– – 235,42,0.9 – – 131,26,0.8 116,30,0.5 211,24,0.5 – – 0,0,1

41,44,1 – – – 55,27,0.9 – – 41,44,0.9 35,37,1 – – 0,0,1

– 46,64,1 145,64,1 – – 48,39,0.8 155,47,1 131,22,0.8

74,53,1 53,60,1 – – 36,38,1 85,35,0.8 – – – – – – – 0,0,1

48,48,1 120,29,0.9 – 0,0,1

Fig. 5. Transportation network in Sichuan.

Table 6 Configuration parameters of computer. Type CPU Memory Operation System

Intel Celeron D Processor Master Frequency Cache Memory

2.67 GHz 256 KB 512 MB Microsoft Windows XP Professional

174


1 NSDE NSGA-II

Minimum route reliability

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 5

8 6

10 20 Maximum route travelling time(h)

4

4

15 2

x 10 Total cost(Yuan)

Fig. 6. Approximate Pareto optimal solutions set.

Fig. 7. Solution minimizing the maximum route time.


175

It is advisable to get the approximate solutions due to the fact that the problem considered is NP-hard. And for realistic instance, we cannot expect to obtain the set of all Pareto optimal solutions exactly. Fifteen runs were performed for the test instance with NSGA-II and NSDE. The values given in the following tables and figures refer to the average of the fifteen runs. The sets of approximate Pareto optimal solutions after 500 generations for this specific accident scenario are shown in Fig. 6. It can be observed that the NSDE algorithm got solutions with more diversity than NSGA-II. However, the approximate Pareto optimal solutions obtained by NSGA-II dominate almost all the solutions got by NSDE. Decision makers can choose the specific solution from the obtained approximate Pareto optimal solutions sets to be applied in this scenario according to their preferences. Fig. 7 illustrates the Pareto-optimal solution based on Objective 1 (minimizing the maximum route time), and Fig. 8 describes the best solution of Objective 2 (minimizing the total cost), the Pareto-optimal solution depending on Objective 3 (maximizing the minimum route reliability)is displayed in Fig. 9. Associated labels (a, b, c) are made for all the disaster areas where a, b and c represent the number of three types (large, medium, small, respectively) of vehicles distributing tents and water for them. Vehicles crossing the nodes without relief distribution are not included. Here only the actually used links are displayed in the following figures. Table 7 summarizes the features of solutions with the three objectives. In Fig. 7 the DCs labeled 12, 13 are open. As described in Table 7, the longest route travelling time is 7.6 h. We can observe that only the Military vehicle K1 and K2 with high velocity and large capacity are used for nodes far from DCs, such as disaster areas 1, 4, 5, 8, which can reduce the travelling time and the distribution frequency to minimize the maximum route time. The node close to DC served by small capacity vehicles get lower demand fill rate, node 2, for example, with demand fill rate of only 24%. Compared to Fig. 7, more links with high travel speed are chosen in Fig. 8. Also the unreliable link 4–8 is used to minimize maximum route time solution, which is not safe for drivers. In the solution given by Fig. 8 where total cost is taken into account, only the candidate DC labeled 12 is chosen to reduce high fixed cost. The DC labeled 13 is working as transship node without relief distributed. Compared to solutions in Figs. 7 and 9, the decrement in fixed cost is 15,000 (about 53% of the total cost), which can be observed from Table 7, and the longest route travelling time is 19.9 h. Using the slow vehicle with low transportation cost per kilometer to serve disaster areas far from DCs could reduce the travelling cost, such as disaster area 4 and 8. However, the unreliable link 1–5 is used in Fig. 8. The solution illustrated in Fig. 9, maximizing minimum route reliability, does not include unreliable links, which are in black color. We can see from Table 7 that the minimum route reliability in Fig. 9 is 0.8. It means that the relief distribution work is much safer than work scheduled by Figs. 7 and 8 with minimum reliability 0.45, 0.36 respectively.

Fig. 8. Solution minimizing total cost.

176


6. Additional experiments to show performance of the two algorithms To further verify the performance of the NSGA-II and NSDE we randomly generated five sets of problem instances (SET 1 to SET 5). The number of disaster areas (n) ranges from 20 to 120, with candidate DCs (m) ranging from 3 to 15 in the distribution networks. All DCs and disaster areas are uniformly distributed within a square of 100 distance units of width. The fixed costs of DCs are generated randomly from 0 to 3000 Yuan. Relief demands of water and tent in each disaster area are generated in [0, 3000] and [0, 1000] randomly. And the maximum permissible speed of each link is less than 80 km per hour. Each problem instance can be denoted by In_m. The size of population is 5 (n + m), and other parameters are same as in 5.2. Table 8 summarizes the performance results of the two algorithms considered. For each algorithm, we report on the values of the three objectives: T (Min–max route time, hour), C (Min total cost, Yuan) and R (Max–min route reliability, measured as probability), as well as NNS (the average number of non-dominated solutions) and RT (the run time, second) respectively. From Table 8, NSGA-II got better results in T and C for all the instances in SET1, 2, 3, 4 and 5(from small to larger), and most instances in R. For SET1, 2(small) and 3(medium), NSDE got non-dominated solutions with more diversity; while for larger instances in SET4 and 5, NSGA-II almost provides more Pareto solutions and maintains better diversity properties than NSDE. In terms of execution time, NSGA-II runs faster than NSDE. This efficiency is due to the fact that NSDE do the mutation operation for each chromosome and crossover for each gene in the population. We also conclude from Table 8 that, the runtime of NSDE and NSGA-II increases with the number of DCs increasing in each SET, as well as the number of disaster areas increasing with the same quantity of DCs, on account that the number of genes in each individual and the size of population increase with larger number of DCs and disaster areas according to the initial operator. NSGA-II outperforms NSDE greatly in obtaining better quality of the approximate Pareto frontiers for larger instances in SET 4 and 5. For example, Fig. 10 illustrates the approximate Pareto frontiers obtained by the two methods for the problem instance I80_12. Visually, the trade-off surfaces of these two approaches are very similar, and NSGAII results in the solutions dominate all of the approximate Pareto solutions obtained by NSDE.

Fig. 9. Solution maximizing minimum route reliability.

177

H. Wang et al. / Transportation Research Part E 69 (2014) 160–179 Table 7 Objective values for Pareto-optimal solutions with different objectives. Min–max route time (h)

Solution1: Min–max time 7.6

Solution2: Minimize total cost 19.9

Solution1: Max–min reliability 9.3

Max–min route reliability

0.45

0.36

0.80

Total cost (Yuan)

Location cost Travel cost

25,000 18,683

10,000 18,242

25,000 15,215

Demand fill rate (%)

1 2 3 4 5 6 7 8 9 10 11

92 54 100 100 100 100 91 76 100 83 67

81 95 81 92 100 100 100 24 100 95 87

57 100 80 75 100 100 100 56 100 100 82

Table 8 Computation results for test instances with NSDE and NSGA-II. Instances

TNSDE

CNSDE

RNSDE

NNSNSDE

TNSGA-II

CNSGA-II

RNSGA-II

SET 1

I20_3 I20_6 I20_9 I20_12 I20_15

2.04 1.74 1.63 1.81 1.55

8954 6202 6179 5882 6256

0.16 0.15 0.16 0.18 0.19

38 37 27 46 57

RTNSDE 434 412 491 558 616

1.58 1.49 1.41 1.06 0.95

5556 5889 4686 4109 4313

0.23 0.35 0.27 0.35 0.19

NNSNSGA-II 15 20 50 25 39

RTNSGA-II 277 275 307 372 418

SET 2

I30_3 I30_6 I30_9 I30_12 I30_15

1.96 1.52 1.58 1.53 1.54

5819 6317 6248 6377 6025

0.27 0.23 0.23 0.29 0.28

58 52 68 66 44

437 665 756 854 900

1.61 1.23 1.16 0.91 0.93

4727 4947 3819 4385 4386

0.23 0.34 0.30 0.48 0.39

17 21 23 68 54

328 545 562 657 715

SET3

I50_3 I50_6 I50_9 I50_12 I50_15

1.42 1.67 1.66 1.76 1.59

6324 5664 5508 4973 5451

0.33 0.30 0.25 0.28 0.29

23 41 23 41 34

1044 1117 1232 1341 1428

1.01 0.92 0.81 0.81 0.59

4164 3526 3533 4241 4543

0.29 0.28 0.46 0.55 0.50

17 22 33 19 19

756 764 803 931 995

SET4

I80_3 I80_6 I80_9 I80_12 I80_15

1.68 1.35 1.46 1.46 1.41

7007 5391 5089 4822 4649

0.29 0.33 0.28 0.33 0.29

57 37 41 29 46

2311 2529 2680 2900 3096

1.37 1.24 0.88 0.78 0.66

3540 3347 3358 2752 3415

0.46 0.58 0.65 0.53 0.49

62 26 53 68 76

1661 1830 1821 1881 2004

SET 5

I120_3 I120_6 I120_9 I120_12 I120_15

1.83 1.50 1.63 1.38 1.38

7329 5116 5125 4896 5023

0.31 0.30 0.30 0.34 0.31

52 49 47 48 22

5185 5361 5571 6139 6398

0.79 0.83 0.66 0.66 0.73

3713 2341 2562 2182 2110

0.58 0.49 0.36 0.53 0.67

108 33 97 182 253

3593 3420 3675 3779 3989

7. Conclusion and future research In this paper, we present multi-objective OLRP models with split delivery, which are motivated by real-world disaster relief problem. We propose NSGA-II and NSDE to solve the models for the delivery of relief in a post-earthquake situation. With these approaches, Pareto-optimal solution sets are presented to facilitate the decision-making according to the preference. A numerical study with a real-large scale earthquake disaster occurred in Sichuan is conducted to illustrate the applicability of the proposed methods. To further evaluate the performance of the two methods, five sets of instances with different size are generated randomly. And the comparison results exhibit that NSGA-II outperforms NSDE in most cases. It will be useful for investigating the application of NSDE and NSGA-II to other multi-objective combinatorial optimization problems with discrete variables. In the future research, the multi-periods OLRP will be considered. Firstly, the post disaster situation under consideration is characterized by a high grade of uncertainty; therefore a method should be developed to consider the stochastic components of the problem. Secondly, the coordination of relief supply from other cities and demand in disaster areas is critical, which can reduce wastes of fresh relief such as blood.

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NSDE NSGA-II

0.7

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0.6 0.5 0.4 0.3 0.2 0.1 0 0.5

15000 10000 1

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Fig. 10. The approximate Pareto frontiers obtained by the NSGA-II and NSDE for I80_12.

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