# Column Generation based heuristic for the Vehicle Routing Problem with Time-Dependent Demand

## Column Generation based heuristic for the Vehicle Routing Problem with Time-Dependent Demand

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IFAC Conference on Manufacturing Modelling, IFAC Conference Manufacturing Modelling, Management and on Control IFAC Conference on Manufacturing Modelling, Management and Control June 28-30, 2016. Troyes, Available online at www.sciencedirect.com Management and Control France June 28-30, 2016. Troyes, France June 28-30, 2016. Troyes, France

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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

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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.

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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

REFERENCES Afsar, H.M., Prins, C., and Santos, A.C. (2014). Exact and heuristic algorithms for solving the generalized vehicle routing problem with flexible fleet size. International Transactions in Operational Research, 21(1), 153–175. Amponsah, S. and Salhi, S. (2004). The investigation of a class of capacitated arc routing problems: the collection of garbage in developing countries. Waste Management, 24(7), 711 – 721. Beasley, J.E. and Christofides, N. (1989). An algorithm for the resource constrained shortest path problem. Networks, 19(4), 379–394. Caunhye, A.M., Nie, X., and Pokharel, S. (2012). Optimization models in emergency logistics: A literature review. Socio-Economic Planning Sciences, 46(1), 4 – 13. Ceselli, A., Righini, G., and Salani, M. (2009). A column generation algorithm for a rich vehicle-routing problem. Transportation Science, 43(1), 56–69. Coelho, L.C. and Laporte, G. (2014). Optimal joint replenishment, delivery and inventory management policies for perishable products. Computers & Operations Research, 47, 42 – 52. Cozzolino, A. (2012). Humanitarian logistics and supply chain management. In Humanitarian Logistics, Springer Briefs in Business, 5–16. Springer Berlin Heidelberg. Desaulniers, G., Desrosiers, J., and Solomon, M.M. (2006). Column generation, volume 5. Springer Science & Business Media. Desrochers, M., Desrosiers, J., and Solomon, M. (1992). A new optimization algorithm for the vehicle routing problem with time windows. Operations Research, 40(2), 342–354. Donati, A.V., Montemanni, R., Casagrande, N., Rizzoli, A.E., and Gambardella, L.M. (2008). Time dependent vehicle routing problem with a multi ant colony system. European Journal of Operational Research, 185(3), 1174 – 1191. Dror, M. (1994). Note on the complexity of the shortest path models for column generation in vrptw. Operations Research, 42(5), 977–978. Feillet, D. (2010). A tutorial on column generation and branch-and-price for vehicle routing problems. 4OR, 8(4), 407–424. Feillet, D., Dejax, P., Gendreau, M., and Gueguen, C. (2004). An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems. Networks, 44(3), 216– 229. for Research on the Epidemiology of Disasters(School of Public Health), C. (2009). Em-dat the international disaster database. http://www.emdat.be. Goerigk, M., Deghdak, K., and Heler, P. (2014). A comprehensive evacuation planning model and genetic solution algorithm. Transportation Research Part E: Logistics and Transportation Review, 71, 82 – 97. Hopkins, T.J. and the IFRC (2008). The Johns Hopkins and International Federation of Red Cross and Red Crescent Societies Public Health Guide for Emergencies.

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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

Irnich, S. and Desaulniers, G. (2005). Shortest path problems with resource constraints. In G. Desaulniers, J. Desrosiers, and M. Solomon (eds.), Column Generation, 33–65. Springer US. Kok, A., Hans, E., and Schutten, J. (2012). Vehicle routing under time-dependent travel times: The impact of congestion avoidance. Computers & Operations Research, 39(5), 910 – 918. Kouadio, I.K., Aljunid, S., Kamigaki, T., Hammad, K., and Oshitani, H. (2012). Infectious diseases following natural disasters: prevention and control measures. Expert Review of Anti-infective Therapy, 10(1), 95–104. Li, A.C., Nozick, L., Xu, N., and Davidson, R. (2012). Shelter location and transportation planning under hurricane conditions. Transportation Research Part E: Logistics and Transportation Review, 48(4), 715 – 729. Ngueveu, S.U., Prins, C., and Calvo, R.W. (2010). An effective memetic algorithm for the cumulative capacitated vehicle routing problem. Computers & Operations Research, 37(11), 1877 – 1885. ¨ Ozdamar, L., Ekinci, E., and K¨ uc¸u ¨kyazici, B. (2004). Emergency logistics planning in natural disasters. Annals of operations research, 129(1-4), 217–245. ¨ Ozdamar, L. and Ertem, M.A. (2015). Models, solutions and enabling technologies in humanitarian logistics. European Journal of Operational Research, 244(1), 55 – 65. Righini, G. and Salani, M. (2006). Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optimization, 3(3), 255 – 273. 531

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

Rivera, J.C., Afsar, H.M., and Prins, C. (2015). Mathematical formulations and exact algorithm for the multitrip cumulative capacitated single-vehicle routing problem. European Journal of Operational Research. Rong, A., Akkerman, R., and Grunow, M. (2011). An optimization approach for managing fresh food quality throughout the supply chain. International Journal of Production Economics, 131(1), 421 – 429. Sharif, M.T. and Salari, M. (2015). A GRASP algorithm for a humanitarian relief transportation problem. Engineering Applications of Artificial Intelligence, 41, 259 – 269. Soler, D., Albiach, J., and Mart´ınez, E. (2009). A way to optimally solve a time-dependent vehicle routing problem with time windows. Operations Research Letters, 37(1), 37 – 42. Song, B.D. and Ko, Y.D. (2016). A vehicle routing problem of both refrigerated- and general-type vehicles for perishable food products delivery. Journal of Food Engineering, 169, 61 – 71. Uribe-S´anchez, A., Savachkin, A., Santana, A., PrietoSanta, D., and Das, T.K. (2011). A predictive decisionaid methodology for dynamic mitigation of influenza pandemics. OR Spectrum, 33(3), 751–786. Van Wassenhove, L.N. (2006). Humanitarian aid logistics: supply chain management in high gear. Journal of the Operational Research Society, 57(5), 475–489. Watson, J., Gayer, M., and Connolly, M. (2007). Epidemics after natural disasters. Emerging Infectious Diseases, 13(1).