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Column Generation based heuristic for the Column Generation based heuristic for the Column Generation based heuristic Vehicle Routing Problem withfor the Vehicle Routing with Vehicle Routing Problem Problem with Time-Dependent Demand Time-Dependent Demand Time-Dependent Demand Jorge F. Victoria, H. Murat Afsar, Christian Prins Jorge F. H. Afsar, Prins Jorge F. Victoria, Victoria, H. Murat Murat Afsar, Christian Christian ICD-LOSI, Universit´ e de Technologie de Troyes,Prins ICD-LOSI, Universit´ de de UMR-STMR CNRS ee6279, 12 Rue Marie Curie, ICD-LOSI, Universit´ de Technologie Technologie de Troyes, Troyes, UMR-STMR CNRS Rue Curie, CS 42060-10004 Troyes12 - France. UMR-STMR CNRS 6279, 6279, 12Cedex Rue Marie Marie Curie, CS 42060-10004 Troyes Cedex France. e-mail: {jorge.victoria, murat.afsar, christian.prins}@utt.fr) CS 42060-10004 Troyes Cedex - France. e-mail: christian.prins}@utt.fr) e-mail: {jorge.victoria, {jorge.victoria, murat.afsar, murat.afsar, christian.prins}@utt.fr) . .. Abstract: This paper presents a novel Capacitated Vehicle Routing Problem with TimeAbstract: This presents novel Vehicle Routing Problem with Dependent Demand (CVRP-TDD) to humanitarian logistics. This is a problem where Abstract: This paper paper presents aa applied novel Capacitated Capacitated Vehicle Routing Problem with TimeTimeDependent Demand (CVRP-TDD) applied to humanitarian logistics. This is a problem where the demandDemand is time (CVRP-TDD) dependent andapplied the objective is to maximize theThis totalissatisfied demand. Dependent to humanitarian logistics. a problem where the demand is dependent and the objective to the satisfied When a disaster strikes a territory, go is to shelters. If they do notdemand. receive the demand is time time dependent and the the people objective isdirectly to maximize maximize the total total satisfied demand. When a disaster strikes a territory, the people go directly to shelters. If they do not receive the first aid, water, food, etc. They tend to flee out of the shelters looking for the aid outside When a disaster strikes a territory, the people go directly to shelters. If they do not receive the first aid, They flee of looking the aid already outside of the area.food, Thisetc. mobilization generates an increase in for thethe chaos the firstaffected aid, water, water, food, etc. They tend tendofto topeople flee out out of the the shelters shelters looking for aid outside of the affected area. This mobilization of people generates an increase in the chaos already caused by the disaster. The aid must arrive at shelters as an quickly as possible to stop this of the affected area. This mobilization of people generates increase in the chaos already caused by disaster. The must arrive at as quickly as stop mobilization. developed mixed integer program a columnto caused by the theWe disaster. Thea aid aid must arrivelinear at shelters shelters as(MILP) quickly and as possible possible togeneration stop this this mobilization. We developed aa mixed integer program (MILP) and column (CG) algorithm the promising are generated dynamic (DP). mobilization. Wewhere developed mixed columns integer linear linear programusing (MILP) and aaprogramming column generation generation (CG) algorithm where the promising columns are generated using dynamic programming (DP). In CGalgorithm algorithm, two the dominance rules and one proposed toprogramming solve the problem. (CG) where promising columns are heuristic generatedare using dynamic (DP). In algorithm, two rules and heuristic are proposed to the problem. TheCG algorithm is tested on small and medium instances. good bounds more In CG algorithm, two dominance dominance rules and one one heuristicCG aregives proposed to solve solveand the find problem. The algorithm is on and instances. CG gives good bounds optimal solutions than those reported by MILP in less than we and showfind thatmore the The algorithm is tested tested on small small and medium medium instances. CG one giveshour. goodAlso, bounds and find more optimal solutions than those reported by MILP in less than one hour. Also, we show that the heuristic improves significantly the solution time. optimal solutions than those reported by MILP in less than one hour. Also, we show that the heuristic improves significantly the solution solution time. time. heuristic improves significantly the © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Vehicle routing, Time-Dependent Demand, Humanitarian logistics, Column Keywords: Vehicle routing, Time-Dependent Demand, Humanitarian Humanitarian logistics, logistics, Column Column Generation,Vehicle Elementary Shortest Path Keywords: routing, Time-Dependent Demand, Generation, Elementary Shortest Path Generation, Elementary Shortest Path 1. INTRODUCTION of optimization models focused in pre-disaster problems. 1. INTRODUCTION of optimization models focused in ¨ While Ozdamar and Ertem (2015) present a problems. survey of 1. INTRODUCTION of optimization models focused in pre-disaster pre-disaster problems. ¨ While Ozdamar and Ertem (2015) present a survey ¨ mathematical models, solution methods and information While Ozdamar and Ertem (2015) present a survey of of In last decade have occurred more than 4000 natural mathematical models, methods and information systems applications insolution post-disaster problems. In last decade have occurred more than 4000 natural mathematical models, solution methods and information disasters leaving approximately 1.7 billion affected people In last decade have occurred more than 4000 natural systems applications in post-disaster problems. disasters leaving 1.7 database billion affected affected people systems applications in to post-disaster problems. around the world,approximately according to the of the Center This work contributes the planning of routes in the disasters leaving approximately 1.7 billion people around the world, according to the database of the Center This work contributes to the planning of routes in the the for Research on the Epidemiology of Disasters (School of preparation phase to distribute aid in the aftermath of around the world, according to the database of the Center This work contributes to the planning of routes in for Research on the Epidemiology of Disasters (School of preparation phase to distribute aid in the aftermath of Public Health). Unfortunately, the number of available a disaster considering the displacement of people from for Research on the Epidemiology of Disasters (School of preparation phase to distribute aid in the aftermath of Public Health). Unfortunately, the number number of available available disaster considering the displacement of Immediately people from from resources to aid Unfortunately, the affected population is limited. The aashelters to places outsidethe of the affected area. Public Health). the of disaster considering displacement of people resources to aid the affected population is limited. The shelters to places outside of the affected area. Immediately distribution these cover the islargest number after the disaster strikes, the people in the affected area resources to ofaid theresources affected to population limited. The shelters to places outside of the affected area. Immediately distribution of to the disaster strikes, people in the affected of affected people hasresources been a challenge. go directly to shelters. Ifthe they do not receive the area first distribution of these these resources to cover cover the the largest largest number number after after the disaster strikes, the people in the affected area of affected people has been aa challenge. go directly to shelters. shelters. If tend they to doflee notout receive the first aid, water, food, etc. They of the shelters of affected people has been challenge. go directly to If they do not receive the first Humanitarian logistics is defined as the processes and aid, water, food, etc. They tend to flee out of the shelters looking for the aidetc. outside the to affected area. Theshelters flee do Humanitarian logistics is as the and aid, water, food, Theyoftend flee out of the systems involved in mobilizing people, skills and and looking Humanitarian logistics is defined defined as resources, the processes processes for the aid outside of the affected area. The flee not stop before the arrival of the aid. This displacement of systems involved in mobilizing people, resources, skills and looking for the aid outside of the affected area. The flee do do knowledge to help people affected byskills disaster systems involved in vulnerable mobilizing people, resources, and not stop before the arrival of the aid. This displacement of people generates an increase in the chaos already caused knowledge to help vulnerable people affected by disaster not stop before the arrival of the aid. This displacement of (Van Wassenhove andpeople its life-cycle composed knowledge to help (2006)) vulnerable affectedisby disaster people generates in chaos already caused by the disaster as an wellincrease as a possible post-disaster (Van Wassenhove (2006)) and life-cycle is people generates an increase in the thespread chaos of already caused by four phases. Mitigation to plans or mech- by (Van Wassenhove (2006)) phase and its itsrefers life-cycle is composed composed the disaster as well as a possible spread of post-disaster outbreaks. by four phases. refers to or by the disaster as well as a possible spread of post-disaster anisms as Mitigation training tophase reduce people vulnerability, by four such phases. Mitigation phase refers to plans plans or mechmech- outbreaks. anisms such as training to reduce people vulnerability, where the government and associations play an important Watson et al. (2007) and Kouadio et al. (2012) agree anisms such as training to reduce people vulnerability, outbreaks. where the associations play et (2007) et al. agree role. Preparation phaseand refers to operations or important strategies Watson that people displacement one of the risk factors where the government government and associations play an an important Watson et al. al. (2007) and andis Kouadio Kouadio et main al. (2012) (2012) agree role. Preparation phase refers to operations or strategies that people displacement is one of the main risk factors that must be planned before occurs.orThis phase that for outbreaks after a disaster. Inofaddition, Hopkins and role. Preparation phase refersa disaster to operations strategies people displacement is one the main risk factors that must be before occurs. This after aa disaster. In addition, Hopkins and is crucial, it incorporates past experiences in phase order for the outbreaks IFRC (2008) argue that depending on their nature, that must because be planned planned before aa disaster disaster occurs. This phase for outbreaks after disaster. In addition, Hopkins and is it past experiences in order order IFRC (2008) argue that on nature, to crucial, enhancebecause the response to disasters. two phases are the duration location, natural disasters results in is crucial, because it incorporates incorporates pastThese experiences in the IFRCand (2008) argue some that depending depending on their their nature, to enhance the response to disasters. These two phases are duration and location, some natural disasters results in known as pre-disaster phases. Response phase the third major disease outbreaks and deaths. Uribe-S´ a nchez et al. to enhance the response to disasters. These twoisphases are duration and location, some natural disasters results in known as phases. is third and Uribe-S´ nchez phase that begins immediately after aphase disaster strikes, (2011) disease presentoutbreaks the outbreak probability at aatime known as pre-disaster pre-disaster phases. Response Response phase is the the third major major disease outbreaks and deaths. deaths. Uribe-S´ nchezt et et>al. al.0 phase that begins immediately after a disaster strikes, (2011) present the outbreak probability at time t > 00 all the operations planned in previous phases must be (equation (1)) for each unaffected region. Each infectious phase that begins immediately after a disaster strikes, (2011) present the outbreak probability at time t > all the operations planned in previous phases must be (equation (1)) for each unaffected region. Each infectious carried on in order to reduce the casualties. Finally, the traveler is assumed to initiate a regional outbreak with an all the operations planned in previous phases must be (equation (1)) for each unaffected region. Each infectious carried on reduce the casualties. is to initiate initiate regional outbreak with an an Reconstruction phaseto to the processthe of traveler equal, time-homogeneous probability ω for the entirety of carried on in in order order torefers reduce therehabilitation casualties. Finally, Finally, the traveler is assumed assumed to aa regional outbreak with Reconstruction phase refers to the rehabilitation process of equal, time-homogeneous probability ω for the entirety infrastructure and the impact generated to the population his/hertime-homogeneous infection period and nt represents the number of of Reconstruction phase refers to the rehabilitation process of equal, probability ω for the entirety of infrastructure and the the infection period period and n the number of Cozzolino (2012). et generated al. (2012) to present a review his/her t represents infrastructure and Caunhye the impact impact generated to the population population his/her infection and n represents the number of t Cozzolino (2012). Caunhye et al. (2012) present a review Cozzolino (2012). Caunhye et al. (2012) present a review Copyright © 2016, 2016 IFAC 526Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2016 responsibility IFAC 526Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 526 10.1016/j.ifacol.2016.07.684

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infectious travelers in the region at time t. (1) Pt = 1 − (1 − ω)nt In this paper, we present a Capacitated Vehicle Routing Problem (CVRP) for humanitarian logistics with timedependent demand (CVRP-TDD). Commonly, in VRPs the demand of each node is known a-priori and constant in time. Here we consider that demand has a linear decreasing behavior and it is understood as the number of people to be attended at each shelter. The demand is represented by a linear function f (t) = a−bt, where a is the demand at time equal to zero and b is the number of people per time unit escaping from a shelter. This linear function represents the mobilization of people trying to get away to find aid. The objective is to maximize the total satisfied demand arriving as quickly as possible. This objective is quite similar to the Cumulative Capacitated Vehicle Routing Problem (CCVRP) introduced by Ngueveu et al. (2010). This paper is organized as follows: Section 2 shows the literature review. Section 3 presents the problem description and the MILP, then the CG algorithm is explained in Section 4. Computational results are presented in Section 5 and, followed by some conclusions in Section 6. 2. LITERATURE REVIEW As stated above, the studied problems and its solution methods are divided into: pre-disaster and post-disaster. Li et al. (2012) propose a stochastic bi-level approach with dynamic user equilibrium (DUE) for the shelter prepositioning and transportation planning under hurricane conditions. Evacuation planning problems can also be addressed as pre-disaster operations, Goerigk et al. (2014) use a multi-criteria mixed-integer programming model and genetic algorithm to solve the comprehensive evacuation problem (CEP). The objective is to minimize the evacuation time, the risk and the number of used shelters. In transportation problems, Sharif and Salari (2015) propose two mathematical models and develop a greedy randomized adaptive search procedure (GRASP) looking to minimize the routing cost plus the allocation cost using a ¨ limited number of vehicles. Ozdamar et al. (2004) minimize the unsatisfied demand using Lagrangean relaxation for a problem that integrates a multi-commodity network flow problem and the VRP. Afsar et al. (2014) shows the applicability of the generalized VRP with flexible fleet size to the humanitarian logistics. They propose an exact method based on CG and two metaheuristics to solve this problem. Rivera et al. (2015) propose two MILPs and an exact algorithm to the multi-trip cumulative capacitated singlevehicle routing problem which the objective is minimize the sum of arrival times at affected sites. Most of the time-dependent VRP (TDVRP) papers consider time-dependent travel times. Donati et al. (2008) solve this problem using ”Multi Ants Colony Systems” to minimize the number of tours and the total travel time. Kok et al. (2012) implement a modified Dijkstra algorithm and use restricted dynamic programming to minimize the total travel time, while Soler et al. (2009) transform the TDVRP with time windows (TD-VRPTW) into an asymmetric capacitated VRP (ACVRP) to solve it optimally. In VRP literature, we have identified two other issues that are time-dependent: the perishable food transportation, 527

527

where product quality depends on the distribution time, and the garbage collection where the inconvenience (smell) increases with the waiting time of the waste. In perishable food transportation, Song and Ko (2016) develop a nonlinear mathematical model to the multicommodity and multi-vehicle problem where the objective is to maximize the customer satisfaction which is a decreasing function according to the elapsed time from depot to a node and the number of storage door is open while a customer is visited. They propose a heuristic algorithm called priority-based heuristic to address the non-linearity of the model. Rong et al. (2011) integrate food quality degradation that depends on time and temperature environment in a multi-period production-distribution planning problem. A MILP is used to solve it. Coelho and Laporte (2014) propose a MILP for three different selling policies (Fresh first, Old first and optimized priority) and implement a branch & cut method to the multi-period perishable inventory-routing problem. Its objective is maximize the total profit, which includes sales revenue that depends on the perceived value given by the customer according to the age of the product, routing and inventory costs. Concerning waste collection, Amponsah and Salhi (2004) model the problem as a capacitated arc routing problem and implement a constructive heuristic with a look-ahead strategy, followed by a local search to solve the problem. Column Generation (CG) has been widely used to solve different problems. Desaulniers et al. (2006) give a complete overview to the CG and its applicability to optimization problems, including the VRPs. Feillet (2010) provides a tutorial on CG and branch and price to this specific problems. Authors as (Ceselli et al. (2009),Desrochers et al. (1992), Feillet et al. (2004), Righini and Salani (2006)) use the Elementary Shortest Path Problem with Resources Constraints (ESPPRC) as pricing problem in the CG. The ESPPRC is identified as NP-Hard (Dror (1994)) and has been first studied by Beasley and Christofides (1989). Desrochers et al. (1992) solve the VRP with time windows using the ESPPRC and the branch and bound is applied after not more columns are found by the ESPPRC. Feillet et al. (2004) propose a solution for the ESPPRC which extends the classical label correcting algorithm originally developed for non-elementary paths and an improvement of these DP algorithm is presented by Righini and Salani (2006) using bounded bi-directional search. Ceselli et al. (2009) solve a rich VRP that considers heterogeneous fleet, multiple depots, time windows, incompatibility constraints between goods, depots, vehicles and customers, and some extra constraints. 3. PROBLEM DESCRIPTION The CVRP-TDD is defined on a complete undirected graph G = (V, E), where V = {0, ...n, n + 1} is the node set and E is the edge set. Node 0 and node n+1 correspond to the depot and V = V \ {0, n + 1}. Each node has a demand ai at time zero and a demand variation βi . This variation corresponds to the number of people per time unit who flee from a node before the arrival of the humanitarian aids. Each arc (i, j) has a travel time cij . The objective is to identify a set of feasible routes so that every node is visited exactly once and the number of people

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attended is maximized. A feasible route begins at node 0 and ends at node n+1. A homogeneous fleet of K vehicles is available and the total travel time of each vehicle should be less than T max. di is a decision variable that represents the demand attended at node i, when a vehicle serves it. It has a decreasing behavior as di = ai − βi ti , where ti is the arrival time at node i. The total demand serviced by a route should not exceed the vehicle capacity Q. wi is the total amount of relief products distributed by the same vehicle between the depot and node i. It should be remark that T max is not large enough to avoid negative and zero values of the demand (di > 0 ∀i ∈ V ). The CVRP-TDD can be formulated as follows: max

s.t.

di

i∈V

j∈V

i∈V

Let P be the set of all feasible paths and P is a small sub-set of P (P ⊆ P ), dp is the satisfied demand of path p ∈ P and λp is a binary variable taking the value of 1 if the path p is in the optimal solution and 0 otherwise. The parameter γpu takes the value of 1 if path p ∈ P visits node u ∈ V . The model is formulated as follows:

s.t.

(3)

xji , ∀j ∈ V

(4)

xij = 1, ∀i ∈ V

(5)

tj ≥ ti + cij − M ∗ (1 − xij ), ∀i ∈ V, ∀j ∈ V (6) (7) wj ≥ wi + dj − M 1 ∗ (1 − xij ), ∀i ∈ V, ∀j ∈ V (8) di ≤ ai − βi ti , ∀i ∈ V (9) wi ≤ Q, ∀i ∈ V (10) ti + ci(n+1) ≤ T max, ∀i ∈ V t0 = 0 w0 = 0 xij ∈ {0, 1}, ∀i ∈ V, ∀j ∈ V ti ≥ 0, ∀i ∈ V di ≥ 0, ∀i ∈ V wi ≥ 0, ∀i ∈ V

(11) (12) (13) (14) (15) (16)

The objective function (2) is the maximization of the sum of satisfied demand at nodes. Constraints (3) limits the number of routes. Constraints (4) specify that a vehicle arriving at node i must leave it. Constraints (5) ensure that each node is served exactly once. Constraints (6) compute the arrival time at each node and prevent subtours. Constraints (7) calculate the amount of product transported until each node. Constraints (8) compute the demand that must be satisfied at each node, taking into account the mobilization of people and the arrival times to each them. Constraints (9) and (10) are capacity and time constraints, respectively. Constraint (11) sets the departing time from depot and Constraint (12) is the initial load at the depot. Finally, Constraints (13) - (16) define the decision variables. 4. COLUMN GENERATION ALGORITHM The MILP model is reformulated using Dantzig-Wolfe decomposition. Constraints (3) and constraints (5) are treated in the Master Problem (MP) and remaining constraints are encoded in the construction of feasible paths. These paths are constructed solving a longest path problem as pricing problem. 528

πu →

d p λp

(17)

p∈P

γpu λp = 1,

p∈P

πo →

i∈V

x0j = K

max

(2)

j∈V

xij =

4.1 Master Problem

λp ≥ 0,

p∈P

∀ u∈V

(18)

λp ≤ K

(19)

∀ p ∈ P

(20)

Constraints (18) impose that each node is visited once and constraint (19) guarantee that K paths should be selected. Dual variables πu ∀ u ∈ V and πo are associated to constraints (18) and constraint (19), respectively. 4.2 Pricing Problem The pricing problem should find new promising columns to add to the set P . When the MP is a minimization problem, these columns are those with negative reduced cost. As our MP is a maximization problem, these columns are those with positive reduced cost. The reduced cost of each column p ∈ P is: C p = dp γpu − πu γpu − πo , ∀ p ∈ P (21) u∈Vc

dp is a function of tu and it can be calculated as: dp = a u − β u tu .

(22)

u∈p

The result of replacing equation 22 in equation 21 is shown in equation 23. ∀ p ∈ P (23) Cp = (au − βu tu )γpu − πu γpu − πo , u∈V

πu is a dual variable associated with each node u ∈ V and πo is a non-negative cost associated with the node n+1. We use DP to solve the longest path problem as pricing problem. The DP has to find elementary and feasible paths from node 0 to node n+1 maximizing the reduced cost satisfying a set of resource constraints. An elementary path is a path in which all nodes are pairwise different (Irnich and Desaulniers (2005)). The DP finds these paths associating with each possible partial path a label indicating its reduced cost and resource’s consumption, and to eliminate labels with the help of dominance rules. We adapt the label and dominance definition given by Feillet et al. (2004) for the ESPPRC. With each partial path p = {0, ..., u} is associated a label Ru = (C p , qp , lenp , Vp1 , ..., Vpn ) where C p is the reduced cost, qp represents the accumulated quantity of vehicle capacity used, lenp is the length of p, and Vpk ∀ k ∈ V is a binary

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resource that takes the value of 1 if node k is visited in path p, or 0 otherwise. This latter helps to ensure that path is elementary. When a label Ru of p is extended at node v ∈ V , the reduced cost is computed as shown in equation 24. C p⊕v = C p + av − βv (lenp + tu,v ) − πv (24) In addition, if p is concatenate with another partial path p = {v, ..., n + 1}. The new reduced cost is: βj (lenp + tu,v ) (25) C p⊕p = C p + C p − j∈p

Then, the dominance is defined as follows: let p1 = {0, ..., u} and p2 = {0, ..., u} be two distinct paths with associated labels Ru1 and Ru2 , respectively. p1 dominates p2 if and only if, qp1 ≤ qp2 (26) lenp1 ≤ lenp2

(27)

Vpk1 ≤ Vpk2 , ∀ k ∈ V

(28)

C p1 ≥ C p2

(29)

Conditions (26-28) verify that any partial path p that is accessible from p2 must be accessible from p1 . These conditions ensure that only non-dominated labels will be extended.

529

Table 1. Results of the MILP Instance CVRP-TDD01

|V| 5

CVRP-TDD02

10

CVRP-TDD03

15

CVRP-TDD04

20

CVRP-TDD05

25

CVRP-TDD06

50

K 2 2 3 2 3 4 3 4 5 4 5 6 8 9 10

ai 2424

16024 38596

42979

58228

140853

Q 1515 10015 6677 24122 16082 12062 17907 13431 10745 18196 14557 12131 22145 19685 17717

Tmax 85 285 190 448 299 225 384 288 231 416 333 278 500 445 401

ZLB 2121.86 14691.31 14981.38 36220.05 36863.78 37179.58 40368.46 40828.75 41115.71 55133.09 55619.17 55884.41 131279.32 131745.30 132230.66

MILP ZU B 2121.86 14691.31 14981.38 37268.49 37246.65 37179.58 41708.09 41683.18 41614.00 56808.88 56811.31 56746.73 140510.66 140491.08 140450.86 Min. Max. Avg.

Time (s.) 0.46 40.20 6.88 3600.00 3600.00 1001.26 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 0.46 1001.26 262.20

experiments are conducted on an Intel(R) Core(TM) i74800MQ 2.70GHz with 16 GB of RAM running Windows 7 Professional. To the best of our knowledge, the CVRPTDD has not been addressed previously, therefore no instances are available in the literature. A set of instances is derived from R101 of the Solomon’s instances. In the adapted instances, |V | varies between 5 and 50 which are the first nodes on R101. Demand at time zero of nodes as well as people flow (demand variation) were generated in order to avoid negative values of demand at these nodes and given a number of fixed vehicles, the total capacity of vehicles correspond, approximately, to 20% more than total demand at time zero. Each instance has different T max values.

The experiments are done with both dominance rules and with the heuristic. The CG starts by calling the pricing Theorem 1. If qp1 ≤ qp2 , lenp1 ≤ lenp2 , Vpk1 ≤ Vpk2 ∀k ∈ V problem at each iteration. The longest path problem returns the paths with positive reduced cost. If the heuristic and C p1 − lenp1 (min βj ) > C p2 − lenp2 (min βj ) (30) is no longer capable of finding paths with positive reduced j ∈p / 2 j ∈p / 2 cost, the CG calls the corresponding DR1 or DR2, adding the condition (28). Finally, when no more paths with then p1 dominates p2 . positive reduced cost can be found, the MP is solved as an Proof. integer program with generated columns to have a lower C p1 − C p2 > (lenp1 − lenp2 ) min βj > (lenp1 − lenp2 ) βj bound. j ∈p / 2

j∈p

C p1 − lenp1 min βj > C p2 − lenp2 min βj j∈p

C p1 ⊕p > C p2 ⊕p

5.1 MILP results

j∈p

∀p accesible f rom p2

For every p accessible from p2 we can extend p1 by the same nodes and the later one has always a bigger reduced cost. 4.3 Pricing with heuristic dominance rules Two dominance rules and one heuristic are proposed. The first dominance rule (DR1 ) considers conditions (26-29) and the Theorem 1 gives the second one (DR2 ). In DR2, the condition (30) is evaluated, when condition (29) is violated. The proposed heuristic uses the same dominance rules, but the condition (28) is not checked (H+DR1 and H+DR2 ). 5. EXPERIMENTAL RESULTS The CG algorithm is implemented in JAVA using CPLEX 12.5 to solve the MP (17-20) and the MILP (2-16). The 529

The results of the MILP are presented in Table 1. The first six columns show the parameters of the problem. Instance name, number of nodes, number of vehicles, total demand at time zero, capacity of a vehicle and maximum trip duration, respectively. The next two columns are the results of the MILP. ZLB is the lower bound (LB) and ZU B is the upper bound (UB) reported by CPLEX. When an optimal solution was found by CPLEX in less than one hour (ZLB = ZU B ). The last column is the time in seconds reported by CPLEX. The results indicate that we can solve optimally instances up to 15 nodes in less than 17 minutes. The minimum, maximum and average execution time by CPLEX are 0.46, 1001.26 and 262.2 seconds, respectively. These are calculated considering only instances where optimal solution is found. In each instance, when the number of vehicles increases, the percentage of satisfied demand increase as expected. On average, 94% of the total demand is satisfied taking the LB as reference point when optimal solution was not found.

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differences between LB’s and UB’s found by the CG, but the difference in the average execution time is large.

5.2 Pricing without heuristic dominance Table 2 presents the results of the pricing without heuristic dominance. The first column is the instance name. The next five columns are the results using the DR1 and the last five are the results solving pricing problem with DR2. It is important to remember that it is a maximization problem, then the ZLB is the lower bound reported by the MP with integer variables (λp ∈ {0, 1} ∀ p ∈ P ) and the ZU B is the upper bound reported by the CG at the end. If the CG finds an optimal solution in less than one hour ZLB = ZU B , but if the CG does not stop within an hour ZU B = ” − ”. ”Iter.” is the number of times that the CG algorithm calls the pricing problem and ”# of Col.” are the number of paths generated by the pricing problem during the corresponding execution time. Although the CG is a method to find upper bounds to the problem. In 9 out of 15 cases, CG finds the optimal solution (shown in bold) without branching. The CG with the DR1 and with the DR2 can solve optimally instances up to 25 nodes in less than 34 minutes. In 8 out of 15 instances and 9 out of 15 instances, the number of iterations as well as the number of columns are less with the DR2 than with the DR1, respectively. In addition, in 9 out of 12 instances the execution time is less with the DR2. These instances are those where the CG stops in less than one hour. 5.3 Pricing with heuristic dominance Table 3 shows the results of the pricing with heuristic dominance. This table follows the same format of Table 2. In the fourth column, we report between parenthesis the number of iterations where we could not get positive reduced costs columns with the heuristic dominance rules and called the dominance rule. The last five columns are the results for H+DR2. Instances which the optimal solution was found are highlighted in bold. The H+DR1 and H+DR2 can solve instances up to 50 nodes in less than 9 minutes and 11 minutes, respectively. In most instances, the execution time as well as the number of iterations using the H+DR1 is smaller than using the H+DR2. 6. CONCLUSIONS A novel problem is proposed in humanitarian logistics where the demand is time-dependent due to the movement of people trying to flee of the nodes. A MILP is presented and solved on small and medium instances. Then a CG algorithm with a longest path problem as pricing problem is implemented. As a maximization problem, the columns with the positive reduced cost should be enter to the basis. Two dominance rules were deduced to improve the performance of the algorithm and a heuristic is proposed. This algorithm is tested on R101 Solomon’s instance which is adapted to the context of the problem. The MILP solved optimally 3 out of 15 instances, while the CG found 9 out of 15 optimal solutions and better LB’s and UB’s in less time that those reported by CPLEX. The results of the CG and the CG with heuristic shown the good behavior of this latter with the first dominance rule (H+DR1 ). This obtains the same optimal solutions and there is no big 530

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Table 2. Results of pricing without heuristic dominance Instance CVRP-TDD01 CVRP-TDD02 CVRP-TDD03

CVRP-TDD04

CVRP-TDD05

CVRP-TDD06

ZLB 2121.86 14691.31 14981.38 36220.05 36863.78 37179.58 40367.98 40828.75 41115.71 55230.84 55611.58 55884.41 133483.64 133630.86 134408.54

ZU B 2121.86 14691.31 14981.38 36220.05 36870.25 37179.58 40381.29 40828.75 41115.71 55230.84 55619.69 55884.41 -

DR1 Iter. 4 16 7 57 35 15 136 77 29 174 81 26 149 171 97

# of Col. 27 1092 376 4073 2405 904 9973 5513 1940 12558 5860 1611 11115 12784 7271 Min Max Avg.

Time (s.) 0.09 0.26 0.03 410.00 2.17 0.14 455.91 12.89 0.58 2011.09 10.43 0.65 3600.00 3600.00 3600.00 0.03 410.00 913.61

ZLB 2121.86 14691.31 14981.38 36220.05 36863.78 37179.58 40368.46 40828.75 41115.71 55230.84 55619.17 55884.41 133483.64 133976.10 134908.05

ZU B 2121.86 14691.31 14981.38 36220.05 36870.25 37179.58 40381.29 40828.75 41115.71 55230.84 55619.69 55884.41 -

DR2 Iter. 4 15 7 56 36 15 124 66 27 174 83 23 102 97 140

# of Col. 27 1002 376 3906 2431 876 9026 4677 1882 12792 5954 1535 7581 7203 10466 Min Max Avg.

Time (s.) 0.10 0.20 0.05 204.13 2.16 0.18 366.00 11.44 0.47 2002.45 9.51 0.44 3600.00 3600.00 3600.00 0.05 366.00 893.14

H+DR2 Iter. # of Col. 4(1) 26(0) 15(1) 911(0) 6(1) 355(0) 47(2) 3165(2) 32(3) 1905(3) 13(1) 789(0) 88(3) 6169(67) 60(3) 4090(32) 26(2) 1594(1) 130(4) 9014(12) 62(4) 4090(11) 22(1) 1398(0) 783(2) 58190(1) 610(1) 45412(0) 260(2) 19086(4) Min. Max. Avg.

Time (s.) 0.10 0.18 0.04 6.28 0.48 0.08 5.29 1.83 0.27 15.51 2.19 0.24 640.47 254.90 55.89 0.04 640.47 65.58

Table 3. Results of the pricing with heuristic dominance Instance CVRP-TDD01 CVRP-TDD02 CVRP-TDD03

CVRP-TDD04

CVRP-TDD05

CVRP-TDD06

ZLB 2121.86 14691.31 14981.38 36220.05 36863.78 37179.58 40368.46 40828.75 41115.71 55230.84 55612.13 55884.41 135686.44 135965.71 136166.69

ZU B 2121.86 14691.31 14981.38 36220.05 36870.25 37179.58 40381.29 40828.75 41115.71 55230.84 55619.69 55884.41 135688.73 135968.54 136174.59

H+DR1 Iter. # of Col. 4(1) 26(0) 14(1) 828(0) 7(1) 377(0) 49(3) 3366(11) 30(1) 1957(0) 13(1) 773(0) 85(1) 6082(0) 55(1) 3814(0) 27(1) 1769(0) 124(1) 9101(0) 63(2) 4288(4) 22(1) 1370(0) 748(3) 55675(5) 549(1) 41002(0) 295(1) 21934(0) Min. Max. Avg.

Time (s.) 0.09 0.11 0.03 18.02 0.59 0.10 4.70 1.49 0.30 12.29 2.31 0.23 512.70 209.13 84.34 0.03 512.70 56.42

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ZLB 2121.86 14691.31 14981.38 36220.05 36863.78 37179.58 40368.46 40828.75 41115.71 55230.84 55612.13 55884.41 135677.63 135965.71 136160.86

ZU B 2121.86 14691.31 14981.38 36220.05 36870.25 37179.58 40381.29 40828.75 41115.71 55230.84 55619.69 55884.41 135688.73 135968.54 136174.59

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