The Time Window Assignment Vehicle Routing Problem

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The Time Window Assignment Vehicle Routing ProblemRemy Spliet, Adriana F. Gabor

To cite this article:Remy Spliet, Adriana F. Gabor (2014) The Time Window Assignment Vehicle Routing Problem. Transportation Science

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© 2014 INFORMS

The Time Window AssignmentVehicle Routing Problem

Remy Spliet, Adriana F. GaborEconometric Institute, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands

{[email protected], [email protected]}

In this paper we introduce the time window assignment vehicle routing problem (TWAVRP). In this problem,time windows have to be assigned before demand is known. Next, a realization of demand is revealed, and

a vehicle routing schedule is made that satisfies the assigned time windows. The objective is to minimize theexpected traveling costs. We propose a branch-price-and-cut algorithm to solve the TWAVRP to optimality.We provide results of computational experiments performed using this algorithm. Finally, we offer insight onthe value of an exact approach for the TWAVRP by comparing the optimal solution to the solution found byassigning time windows based on solving a vehicle routing problem with time windows with average demand.

Keywords : vehicle routing problem; time window assignment; pricing problem with linear node costsHistory : Received: April 2012; revisions received: January 2013, June 2013, September 2013; accepted: October

2013. Published online in Articles in Advance.

1. IntroductionIn many distribution networks, deliveries are madeat regular intervals and take place within a sched-uled time window. Typically, these time windows areendogenously imposed. The supplier and customermight, for instance, agree on a specific time windowfor delivery. These endogenous time windows arelong-term decisions in certain industries. In retail itis very common that deliveries at a store are alwaysmade within a specific time interval on the same dayof the week for an entire year. This is crucial formany operational processes such as inventory man-agement and the scheduling of personnel. Distribu-tion networks are often also faced with exogenoustime windows. For example, an exogenous time win-dow might be imposed by a local government thatforces trucks to make deliveries in a populated areaduring daytime hours only. Hence an endogenoustime window can only be chosen within the exoge-nous time window.

Demand is usually unknown at the moment thatendogenous time windows are assigned and mostoften fluctuates per delivery. When the demand ofcustomers becomes known, a vehicle routing schedulehas to be determined for making the deliveries withinthe endogenous time windows. This problem, knownas the vehicle routing problem with time windows(VRPTW), is well studied in the scientific literature;see, for instance, the surveys by Baldacci, Mingozzi,and Roberti (2012) and Kallehauge et al. (2005).

The time windows assigned to customers greatlyaffect the transportation costs because they affect the

choice of delivery routes. Because customer demandis not known at the moment of assigning time win-dows, or it fluctuates per delivery, it is not straightfor-ward to assign time windows that allow for deliveryroutes with low transportation costs. Despite the factthat many organizations work with endogenous timewindows, and much scientific literature is devoted tosolving the VRPTW, only limited research has beendone on actually assigning time windows in thissetting.

In this paper we present a model to assign timewindows before demand is known. The problem ofassigning time windows is referred to as the time win-dow assignment vehicle routing problem (TWAVRP).A finite number of scenarios is given, with each sce-nario describing a realization of demand for eachcustomer. Furthermore, the probability with whicheach scenario occurs is known at the moment ofscheduling. The TWAVRP consists of assigning a timewindow to each customer and constructing a vehi-cle routing schedule for each scenario satisfying thesetime windows such that the expected costs are mini-mized. The TWAVRP is NP-hard because for one sce-nario it is the VRPTW.

The TWAVRP is similar to the consistent vehi-cle routing problem (ConVRP) introduced by Groër,Golden, and Wasil (2009), but the problems differin the following ways: In the ConVRP, a customerdoes not necessarily require service in each scenario.Also, each customer needs to be visited by the samedriver in all scenarios in which it requires service.Finally, there are restrictions on the total driving

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Spliet and Gabor: The Time Window Assignment Vehicle Routing Problem2 Transportation Science, Articles in Advance, pp. 1–11, © 2014 INFORMS

time of each driver. The ConVRP has applicationsin small-package shipping. Our experience with spe-cific Dutch retail chains suggests that the ConVRPis too stringent for application in their case. Here,every customer requires goods on each day of deliv-ery. Furthermore, personalization of service is not anissue, as both the supplier and customers are partof the same retail chain; hence, the same-driver con-dition is not necessary. Finally, delivery routes will(almost) never exceed the maximum allowed driv-ing time because capacity constraints in many retailsettings prohibit long routes, unlike in the small-package shipping industry to which Groër, Golden,and Wasil applied the ConVRP. In their paper, com-putational experiments are provided with ConVRPinstances with up to 12 customers and three scenarios.They solve these instances to optimality using a com-mercial mixed integer program solver. They reportcomputation times of up to several days. Further-more, they develop a local search heuristic to findsolutions to instances of up to 3,715 customers and25 scenarios.

Another closely related problem to the TWAVRP isthe vehicle routing problem with stochastic demand(SVRP). In this problem, vehicle routes are deter-mined before demand is known. When demandbecomes known, routes may be infeasible because ofthe limited capacity of each truck, in which case arecourse action is required. Among the most popu-lar recourse actions is the strategy suggested by Dror,Laporte, and Trudeau (1989), in which the originalvehicle route is followed until the load of the truckis depleted and is resumed after a visit to the depotto restock. An advantage of this recourse action isthat the expected costs of a vehicle route can be com-puted efficiently. Studies on this model include thework by Laporte, Louveaux, and van Hamme (2002),who solve the SVRP to optimality using an inte-ger L-shaped method; and Novoa and Storer (2009),who develop an approximate dynamic programmingapproach for the single vehicle variant. In this model,however, the time of service at the customer is nottaken into account.

In the paper by Jabali et al. (2013), another relatedproblem is considered. In their paper, demand isassumed to be known at the moment of assign-ing the time windows and travel time is stochastic.They develop a tabu search algorithm to find goodsolutions for this problem. Furthermore, Agatz et al.(2011) consider the problem of deciding on a set oftime windows to offer to potential customers in differ-ent zip code areas for a Web store offering home deliv-eries. A customer may then select one time windowfrom the ones available in its area. Neither the cus-tomers nor their demand are known at the momentof deciding on the sets of time windows. Agatz et al.

propose a local search heuristic to find solutions tothis problem.

In this paper, we propose a relevant new problemwhich we have encountered in practice, the TWAVRP.We develop a column generation algorithm to findlower bounds on the optimal solution value of theTWAVRP. We apply route relaxation to allow cyclicroutes and apply the algorithm by Ioachim et al.(1998) to solve the pricing problem. Furthermore, tostrengthen the lower bound, we eliminate routes con-taining 2-cycles and modify the algorithm by Ioachimet al. (1998) accordingly. We incorporate this col-umn generation algorithm into a branch-price-and-cut algorithm to find optimal integer solutions tothe TWAVRP. We show by means of computationalexperiments that this algorithm is capable of solvinginstances with up to 25 customers and three scenariosto optimality within one hour of computation time.Finally, we compare the solutions to the TWAVRPwith solutions found by solving a VRPTW using aver-age demand.

The rest of this paper is organized as follows:A formal definition of the TWAVRP is given in §2.In §3, the branch-price-and-cut algorithm is pre-sented. The results of our computational experimentsare provided in §4. Section 5 presents conclusions.

2. Problem DefinitionConsider a complete graph G = 4V 1A5, where V =

801 0 0 0 1n + 19 is a set of locations such that 0 repre-sents the starting depot, n+1 is the ending depot, andV ′ = 811 0 0 0 1n9 are the customers. Let cij ≥ 0 be the costto travel along arc (i1 j), and let tij ≥ 0 be the corre-sponding travel time. Both the travel costs and traveltimes satisfy the triangle inequality. Furthermore, anunlimited number of vehicles of equal capacity Q isavailable.

Let ì be a set of scenarios, where each scenariorepresents a realization of demand. The probabilitythat scenario � occurs is p�. Let demand at location vin scenario � ∈ ì be given by d�

v , where 0 <d�v ≤Q.

For ease of notation, let d�0 = d�

n+1 = 0. Associatedwith each location v ∈ V is the exogenous time win-dow 6sv1 ev7, which should not be confused with theendogenous time window.

In this paper we use the term “route” to refer to apair (P1 t) where P is a path in G starting at 0 andending at n + 1 and t is a vector containing arrivaltimes at each location on the path. Let avr be the num-ber of times customer v ∈ V ′ is visited by route r .Furthermore, let tvr be the cumulative time of serviceof customer v ∈ V ′—that is, if location v is not visited,tvr = 0; if it is visited once, tvr is the time of service;and if customer v is visited multiple times, tvr is thesum of the times of service. To each route r with arcs8r11 0 0 0 1 rk9 we assign costs cr =

∑ki=1 cri .

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Spliet and Gabor: The Time Window Assignment Vehicle Routing ProblemTransportation Science, Articles in Advance, pp. 1–11, © 2014 INFORMS 3

A route is considered feasible for scenario � if(i) the capacity constraint in scenario � is satisfied,(ii) the exogenous time window constraints are satis-fied, and (iii) the service time at location j is not beforethe service time at location i plus the travel time tij iflocation j is visited directly after i. Note that waitingat a customer is allowed. Let R4�5 be the set of allfeasible routes for scenario �.

An endogenous time window of width wv has tobe assigned to each customer v ∈ V ′ within whichit will receive its delivery. The assignment is madebefore the realization of demand is learned. Prior tothe dispatching of vehicles, demand becomes known,and an optimal routing schedule will be designed tomake the deliveries within the assigned time win-dows. The TWAVRP is to assign time windows beforedemand is known and to select feasible routes in eachscenario � ∈ ì that satisfy these time windows. Theobjective is to minimize the expected traveling costs.

Next, we provide a mixed integer linear program-ming formulation for the TWAVRP. Let the time win-dow variable yv be the start time of the endogenoustime window at each location v ∈ V ′. Note that yv ∈

6sv1 ev − wv7, and we assume sv ≤ ev − wv. Let thebinary route variable x�

r indicate whether route r isused for scenario �. The TWAVRP can be formulatedusing the following mixed integer linear program:

min∑

�∈ì

p�∑

r∈R4�5

crx�r 1 (1)

∑

r∈R4�5

avr x�r = 1 ∀v ∈ V ′1 ∀� ∈ì1 (2)

∑

r∈R4�5

tvr x�r ≥ yv ∀v ∈ V ′1 ∀� ∈ì1 (3)

∑

r∈R4�5

tvr x�r ≤ yv +wv ∀v ∈ V ′1 ∀� ∈ì1 (4)

x�r ∈ 80119 ∀� ∈ì1 ∀ r ∈R4�51 (5)

yv ∈ 6sv1 ev −wv7 ∀v ∈ V ′0 (6)

Here, (1) are the expected total costs of a time win-dow assignment. Constraints (2) ensure that everylocation is visited exactly once; constraints (3) and(4) ensure that all locations are visited within theassigned time windows. Finally, note that becausetime is continuous, the number of routes in R4�5for all � ∈ ì—and therefore also the number ofvariables—is infinite, unless wv = 0 and sv = ev for allv ∈ V ′.

3. Solution ApproachIn this section we propose a branch-price-and-cutalgorithm to solve the TWAVRP. First, we present acolumn generation algorithm to find lower boundsby solving the linear programming (LP) relaxation of

the TWAVRP formulated in (1)–(6). We consider tworoute relaxations, allowing the path of a route to benonelementary. Moreover, we discuss the algorithmto solve the pricing problem, an acceleration strategy,and the addition of valid inequalities. Finally, we dis-cuss the branch-price-and-cut algorithm.

3.1. Column Generation AlgorithmWe propose using a column generation algorithmto solve the LP relaxation of (1)–(6), referred to asthe master problem. We consider the master prob-lem where only a subset of routes is included, alsoknown as the restricted master problem. At each iter-ation of the column generation algorithm, a restrictedmaster problem is solved, followed by solving a pric-ing problem to identify feasible routes with negativereduced costs. Routes with negative reduced costs areadded to the restricted master problem. If no suchroute exists, the current solution to the restricted mas-ter problem is optimal for the master problem.

We decompose the pricing problem into severalproblems, one for each scenario. For scenario �, thepricing problem is to find a feasible route (P1 t) suchthat P is elementary, with minimum reduced costs.Let us denote the dual variables corresponding to(2), (3), and (4) by �, �, and �, respectively. For easeof notation, let � = � − �. Observe that both � and� are unrestricted. The reduced costs correspondingto route variable x�

r are given by

p�cr −∑

v∈V ′

��v a

vr −

∑

v∈V ′

��v t

vr 0 (7)

We model the pricing problem for scenario � usinggraph G. With each node v ∈ V ′, we associate demandd�v , time window 6sv1 ev7, and the cost coefficient −��

v .Furthermore, with each arc 4i1 j5 ∈ E, we associate thetravel time tij , and costs p�cij − ��

j if j ∈ V ′ and p�cijotherwise. For each route (P1 t), we calculate the cor-responding reduced costs in scenario � as the sum ofthe costs of the arcs on path P and the costs at eachnode v. These costs are linear in the arrival time tvwith coefficient −��

v . The pricing problem is solvedby finding an elementary shortest path in G with acapacity constraint, time window constraints, and lin-ear node costs.

Because of the complexity of our pricing prob-lem, we suggest using route relaxations, i.e., allowingnonelementary routes, yielding a less complex pricingproblem. Observe that the optimal integer solutionof the TWAVRP does not change when cyclic routesare used in the formulation. However, the LP valuewill decrease. Route relaxation has been successfullyused by Desrochers, Desrosiers, and Solomon (1992),for instance, to solve the VRPTW, which is a closelyrelated problem to the TWAVRP. They solve a pricingproblem in which they allow cyclic routes. Moreover,

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they eliminate 2-cycles, i.e., cycles of the form i–j–i,to strengthen the LP bound with respect to allow-ing all cycles, at the cost of increased computationalcomplexity. Other examples of route relaxations thatprovide stronger LP bounds rather than allowing allcycles at the cost of increased computational com-plexity are k-cycle elimination for k ≥ 3, as describedby Irnich and Villeneuve (2006), and the ng-routerelaxation, as introduced by Baldacci, Mingozzi, andRoberti (2011). In this paper, we consider allowing allcyclic routes and allowing all cyclic routes not includ-ing 2-cycles. Next, we present the algorithms used tosolve the corresponding pricing problems.

3.2. Pricing Problem with All Cyclic RoutesWhen all cyclic routes are allowed, the pricing prob-lem is a shortest-path problem on G with a capacityconstraint, time window constraints, and linear nodecosts. To solve this problem, we introduce an auxiliaryacyclic graph G� for each scenario �, in which onlythe paths in G with a total load less than or equal toQ are represented. This allows us to model the pric-ing problem of scenario � as a shortest-path prob-lem with time window constraints and linear nodecosts on the acyclic graph G�, but without a capac-ity constraint. Moreover, the labeling algorithm byIoachim et al. (1998) can be directly applied to G� tosolve the pricing problem of scenario �. We empha-size that applying this algorithm to G� means thatthe capacity constraint is taken into account indirectly.In particular, dominance is never evaluated duringthe execution of the labeling algorithm for any twopaths with different total loads. Furthermore, observethat the pricing problem cannot be solved by apply-ing the algorithm of Ioachim et al. (1998) directly toG because this algorithm is designed to solve theshortest-path problem with time window constraintsonly and linear node costs on an acyclic graph. In thissection, we formally define the auxiliary graphs anddiscuss the labeling algorithm by Ioachim et al. (1998).

Let G�= 4V �1A�5 be the auxiliary graph in sce-nario �. The set V � includes a node for (i) the startingdepot 0; (ii) each triple (v1m1q) such that for v ∈ V ′,there exists a (01v)-path in G visiting exactly m loca-tions and with a total load of q ≤ Q in scenario �;and (iii) each pair (n + 11 q) such that there exists a(01n+1)-path in G with a total load of q ≤Q. By con-struction, to each node u ∈ V � there is a correspond-ing node in V denoted by o4u5. We refer to o4u5 as theoriginal node corresponding to u. We also associatewith each node u such that o4u5= i, demand d�

u = d�i ,

time window 6su1 eu7= 6si1 ei7, and the linear node costcoefficient cu, which is −��

i if i ∈ V ′ and 0 otherwise.The set A� includes (a) the arcs (01v) for every node

v ∈ V � representing a triple (o4v5111d�o4v5); (b) the arcs

(v1u), where v ∈ V � represents the triple 4o4v51m1q5

and u ∈ V � represents the triple 4o4u51m+11 q+ d�o4u55,

such that 4o4v51 o4u55 ∈ A; and (c) the arcs 4v1u5,where v ∈ V � represents the triple 4v1m1q5 and u ∈

V � represents the pair 4n+11 q5. With each arc 4v1u5 ∈

A� where o4v5= i and o4u5= j , we associate the traveltime tvu = tij and costs cvu = cij − ��

j if j 6= n + 1 andcosts cvu = cij if j = n+ 1.

With each pair (P1 t), where P is a path in G� andt are service times at each node visited on P , we asso-ciate costs. These costs are computed as the sum of thecosts of each arc traversed by P and the costs at eachnode v ∈ V � visited by P equal to the time of servicetv multiplied with the linear node cost coefficient cv.

Observe that there exists a bijection between thepaths in G and G�. Moreover, the costs of correspond-ing paths in G and G� are equal when the times ofservice coincide. Hence, when allowing all cycles, thepricing problem can be solved by finding a shortestpath in G� with a capacity constraint, time windowconstraints, and linear node costs.

To solve the pricing problem, we apply the algo-rithm by Ioachim et al. (1998) to G�. We describe thisalgorithm next. First note that even when a path P inG� is given, because the linear node costs cv of eachnode v ∈ V � on the path can be positive or negative,determining the optimal times of service is an opti-mization problem in itself. To address this, Ioachimet al. (1998) introduced a node cost function gP 4T 5that provides the minimum costs of using path Pwhere service at the last node in P is performed beforetime T . They show how to construct this function andprove that it is piecewise linear, is convex, and con-tains at most �P � line pieces.

Consider the set of partial paths çv starting at thedepot and ending at node v ∈ V �. Define the dom-inance function Dv4T 5 = min8gP 4T 5 � P ∈ çv9, whichprovides the minimum costs of servicing node vbefore time T . Ioachim et al. (1998) prove that Dv

is piecewise linear, nonincreasing, but not necessarilyconvex or continuous. Next, we describe the dynamicprogramming algorithm they propose to constructthis function.

Let fv be the number of line pieces of Dv restrictedto the interval 6sv1 ev7. We refer to the start andend points of these line pieces as the breakpointsb1v1 0 0 0 1 b

fv+1v . Line piece lkv, 1 ≤ k ≤ fv, of the domi-

nance function can be represented by

lkv = 4bkv1Dv4bkv51h

kv51 (8)

where bkv is the start of the line piece, Dv4bkv5 is the

value of the dominance function at the start of the linepiece, and hk

v is the slope. The dominance functioncan be described using the set of line pieces 8lkv � 1 ≤

k ≤ fv9. Note that Dv4T 5 is not defined for T < b1v, and

Dv4T 5=Dv4bfvv 5+hfv

v 4bfv+1v − bfvv 5 if T > bfvv 0 (9)

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Every line piece corresponds to a label in the label-ing algorithm. The label extension operator that isused to extend a label from node v to u is defined asfollows:

EXTENDvu4lkv5

= 4max8su1 bkv + tvu91Dv4b

kv5+ cvu

+ cu max8su1 bkv + tvu91min801hk

v + cu950 (10)

Note that extended labels with max8su1 bkv + tvu9 >min8eu1 bk+1

v + tvu9 are removed, as the path corre-sponding to such a line piece does not satisfy thetime window constraint. In this case, either the earli-est arrival is later than the end of the time windowor the latest arrival is earlier than the start of the timewindow and the next extended label dominates theoption of waiting.

We denote by EXTENDvu4Dv5 the extension opera-tor that provides the set of extended labels for eachlabel lkv for 1 ≤ k ≤ fv, and an additional label ifbfv+1v + tvu < eu. This additional label represents a line

piece that provides the minimum costs for servicecommencing before T , for T ∈ 6b

fv+1v + tvu1 eu7, and is

defined by

l′u=

4bfv+1v + tvu1Dv4b

fv+1v 5

+ cvu+ cu4bfv+1v + tvu5105 if bfv+1

v +tvu<eu1

� otherwise.(11)

The extension operator on a dominance function isdefined as

EXTENDvu4Dv5=8EXTENDvu4lkv5 �1≤k≤fv9∪8l′u90 (12)

The set of labels EXTENDvu4Dv5 describes a piece-wise linear function. Let F denote the operator thatfinds the minimum of a set of piecewise linear func-tions represented by a set of labels, which we use toconstruct the dominance functions. Furthermore, letV � be ordered as follows: first 0, next the nodes rep-resenting triples (v1m1q) in increasing order of m andordered lexicographically in v and q, and finally thenodes representing the pairs 4n + 11 q5 ordered withrespect to q. The labeling algorithm is summarized inAlgorithm 1.

Algorithm 1 (Labeling algorithm to solve the pricingproblem)

Initialize Lv = � for all v ∈ V �.Initialize l10 = 4s010105, and f0 = 1.Initialize L0 = 8l109.for all v ∈ V � do

Dv = F 4Lv5.for all 4v1u5 ∈ A� do

Add EXTENDvu4Dv5 to Lu.end for

end for

This labeling procedure yields the dominance func-tions Dv for all v ∈ V �. Backtracking allows us tofind the shortest paths corresponding to the labels ofthe dominance functions Dv for o4v5 = n + 1. In ourexperiments, we add all found routes with negativereduced costs. Hence, at each iteration of the columngeneration algorithm, multiple routes might be addedto the restricted master problem for one scenario.

A similar pricing problem is encountered byLiberatore, Righini, and Salani (2011) for the vehi-cle routing problem (VRP) with soft time windows.In our algorithm we evaluate dominance of labelswith the same load and the same number of visits,whereas Liberatore, Righini, and Salani developed alabeling algorithm in which dominance is evaluatedfor labels with a different load and number of visits.To be precise, for two labels with loads q and q′ andnumber of visits m and m′, respectively, in our case,a label can only be dominated if q = q′ and m = m′;by contrast, in the algorithm of Liberatore, Righini,and Salani, a label can be dominated if q ≥ q′ and m≥

m′. Because they use a stronger dominance criterion,potentially fewer labels would need to be comparedand stored during the execution of the labeling algo-rithm. We were not aware of this approach until afterour paper had been submitted for publication. Notethat our computational experiments show that, usingthe weaker dominance criterion, we can provide solu-tions that are superior to those based on solving aVRPTW with average demand.

3.3. Pricing Problem with 2-Cycle EliminationTo improve the LP-bound obtained when allowingall cyclic routes, we propose to eliminate 2-cycles.A 2-cycle i–j–i in G is represented in G� by a partialpath v–v′–v′′, where o4v5= o4v′′5. Next we discuss themodifications to the labeling algorithm to eliminate2-cycles.

Let Pred4lkv5 be the predecessor of v on the pathin G� corresponding to line piece lkv. Similarly, letPred4P5 be the node preceding the last node on pathP in G�. For T ∈ 6bkv1 b

k+1v 7, define D′

v4T 5= min8gP 4T 5 �

P ∈ çv1 o4Pred4P55 6= o4Pred4lkv559, providing the mini-mum costs of servicing node v before time T , consid-ering only the paths with a different original previouscustomer than o4Pred4lkv55. The main idea of the mod-ified algorithm is to extend the path correspondingto D′

v4T 5 at time T instead of the path correspond-ing to Dv4T 5, when extending the latter would yielda 2-cycle.

For the labeling algorithm with 2-cycle elimina-tion, we redefine the extension operator. We asso-ciate with every line piece lkv of Dv the line piece lk

′

v ,k′ < k, as the last line piece such that bk

′

v ≤ bkvwith a different original predecessor customer; i.e.,o4Pred4lkv55 6= o4Pred4lk′

v 55. When extension of the path

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corresponding to lkv yields a 2-cycle, lk′

v is used instead.In this case, the extended line piece represents a pathwhere v is serviced at time bk

′+1v and the vehicle waits

at least until time bkv before traveling to node u. Eventhough the resulting label will never be part of Du, itmight be part of D′

u. The new extension operator isdefined as follows, using the same notation for linepieces of D′

v:

EXTEND′

vu4lkv5

=

EXTENDvu4lkv5 if o4Pred4lkv55 6=o4u51

4max8su1bkv+ tvu91Dv4bk′+1v 5

+ cvu+ cumax8su1bkv+ tvu91

min801cu95 otherwise0

(13)

Let F ′ denote the operator that finds D′v. The label-

ing algorithm is summarized in Algorithm 2.

Algorithm 2 (Labeling algorithm to solve the pricingproblem with 2-cycle elimination)Initialize Lv = � for all v ∈ V �.Initialize l10 = 4s010105, and f0 = 1.Initialize L0 = 8l109.for all v ∈ V � do

Dv = F 4Lv5.D′

v = F ′4Lv5.for all 4v1u5 ∈ A� do

Add EXTEND′

vu4Dv5 to Lu.Add EXTEND′

vu4D′v5 to Lu.

end forend for

3.4. Acceleration StrategyThe column generation algorithm requires solving apricing problem for each scenario � ∈ ì at everyiteration. These pricing problems differ only in thevalues of the dual variables, the demand of each cus-tomer, and the scenario probabilities that are part ofthe reduced costs. Therefore, we propose the follow-ing acceleration strategy: whenever a route is foundas a solution to the pricing problem of some sce-nario �, it is also added to the restricted master prob-lem in scenario �′ if it is feasible and has negativereduced costs in that scenario as well. In this case, thepricing problem of scenario �′ is not solved duringthis iteration. This procedure potentially reduces thenumber of pricing problems that have to be solved.The column generation algorithm is summarized inAlgorithm 3.

Algorithm 3 (Column generation algorithm, reusingroutes)

Initialize R4�5 for all �.repeat

Solve the restricted master problem using theroutes R4�5 for scenario �.

Set ì=ì.while ì 6= � do

Choose � ∈ ì and remove it from ì.Solve the pricing problem for scenario �, to

find a set of routes R.Add all routes in R that have negative reduced

costs for scenario � to R4�5.for All � ∈ ì do

Let R⊆R be all routes that are feasible andhave negative reduced costs for scenario �.

if R 6= � thenAdd all routes in R to R� and remove �

from ì.end if

end forend while

until No new routes are added to the master problem.

3.5. Valid InequalitiesTo improve the LP bound of the TWAVRP we addvalid inequalities. In particular, we consider inequali-ties that are valid for the vehicle routing problem, asthey are also valid for each scenario in the TWAVRP.These inequalities include capacity, comb, hypotour,and multistar inequalities (Lysgaard, Letchford, andEglese 2004). We have experimented with addingthese inequalities using the separation routines ofLysgaard (2003). Preliminary experiments showedthat adding only capacity inequalities yields the low-est computation time.

Next, we briefly discuss the capacity inequalities.Let z�ij be the arc flow in G on arc 4i1 j5 in scenario �.Let b4S5 be the minimum number of vehicles neededto visit all customers in S ⊆ V ′. The capacity inequal-ities are

∑

i∈S1 jyS

z�ij ≥ b4S5 ∀S ⊆ V ′1 ∀� ∈ì0 (14)

As is common, we replace b4S5 by the lowerbound �

∑

i∈S d�i /Q�. These constraints can be refor-

mulated using the route variables x�r . When capacity

inequalities are added, the pricing problem remainsa shortest-path problem with a capacity constraint,time window constraints, and linear node costs. How-ever, the costs on each arc are modified as follows. Let��S be the dual variable associated with the capacity

inequalities for subset S in scenario �. We subtract ��S

from the initial costs of each arc 4i1 j5 ∈ A such thati ∈ S and j 6∈ S.

Other valid inequalities for the VRP and VRPTWthat might also be applied here are the following:the k-path inequalities introduced by Kohl et al. (1999)and extended to generalized k-path inequalities byDesaulniers, Lessard, and Hadjar (2008) have beenused to solve the VRPTW successfully. These inequal-ities are strongest when capacity and time window

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constraints are tight. Because we focus on instanceswith wide exogenous time windows, we have cho-sen not to include these inequalities. Also, the sub-set row inequalities introduced by Jepsen et al. (2008)have been used to solve the VRPTW using a branch-price-and-cut algorithm. However, the pricing prob-lem changes substantially when adding these inequal-ities, making it more difficult to solve. Therefore wehave chosen not to include these inequalities.

3.6. Branch-Price-and-CutNext we describe the branch-price-and-cut algorithmwe propose to solve the TWAVRP. Lower bounds areobtained by using the column generation algorithm tosolve the LP relaxation of (1)–(6) and adding capacityinequalities. In our implementation, capacity inequal-ities are only separated during the iterations of thecolumn generation algorithm where no new routeswith negative reduced costs are found.

With each feasible solution to the LP relaxation of(1)–(6) we associate an arc flow in G for each sce-nario �. Observe that an integer arc flow in eachscenario corresponds to an integer solution of theTWAVRP, even when the route variables are frac-tional. In our branch-price-and-cut algorithm, we per-form special ordered subset branching on the arcs asfollows.

For scenario � and customer v, let �−�4v5 and �+

�4v5be the sets of in and out arcs, respectively. Next, acustomer v′, a scenario �′, and an arc type o′ ∈ 8−1+9are selected with the highest number of arcs a in�o′

�′4v′5 for which z�′

a > 0. Let �o′

�′4v′5 = 8a11 0 0 0 1 ak9 beordered such that z�′

ai≥ z�

′

ajif i < j . The arcs are divided

into two groups, S and its complement S, where S =

8a11 0 0 0 1 ai9 is such that∑

a∈S z�′

a ≥ 005 and∑

a∈S\8ai9z�

′

a <005. In one branch we disallow the use of the arcsin S, and in the other we disallow the use of thearcs in S. Observe that the pricing problem remains ashortest-path problem with a capacity constraint, timewindow constraints, and linear node costs. However,fewer arcs are included in the graph.

Upper bounds are obtained when a solution withinteger arc flow in each scenario is found to the LPrelaxation. At each iteration of the branch-price-and-cut algorithm, the node with the lowest lower boundis selected.

4. Computational ResultsIn this section we present the results of numericalexperiments using our algorithms. First, we discussthe test instances that we have generated. Next, weshow results of solving the LP relaxation for theseinstances, obtained by using the column generationalgorithm. This is followed by the results of usingthe branch-price-and-cut algorithm. Finally, we com-pare the optimal solution value of the TWAVRP to the

value of the solution found by solving a VRPTW withaverage demand. In all experiments, a one-hour timelimit is used.

All algorithms are coded in C++, and ILOG CPLEX12.4 is used to solve the restricted master problemat each iteration of the column generation algorithm.The experiments were performed on an Intel® CoreTM

i5-2450M CPU 2.5 GHz processor.

4.1. Test InstancesWe have generated a total of 40 test instances consist-ing of 10 instances with 10, 15, 20, or 25 customers.1

These instances are inspired by Dutch retail chains.Customer locations are generated using a uniform

distribution over a square with sides of length 5.The depot is located in the center of the square. Boththe travel costs and times are equal to the Euclideandistance between locations, rounded to two digits.The depot has the exogenous time window 661227.Each customer is given one of three exogenous timewindows, and each is assigned a fixed frequency.The exogenous time window 6101167 is given to 10%of the customers, 681187 to 60%, and 671217 to 30%.The endogenous time window width is set to 2 for allcustomers. The vehicle capacity is 30.

For every instance, three scenarios are gener-ated, each occurring with equal probability. To varydemand throughout the scenarios for each customer,we generate it by computing d�

v = �u�vdv�. Here, dv is

drawn from a normal distribution with an expecta-tion of 5 and a variance of 1.5. Furthermore, for each� ∈ ì, the multiplier u�

v is drawn from a uniformdistribution on the interval 600710087, 60095110057, or610211037 to generate scenarios with low, medium, orhigh demand, respectively.

4.2. Computational Results of Using theColumn Generation Algorithm

Next, we provide the results of solving the LP relax-ation of (1)–(6) using the proposed column generationalgorithm. We compare the two route relaxations con-sidered in this paper, allowing all cyclic routes and2-cycle elimination.

Preliminary experiments suggest that the col-umn generation algorithm employing the accelerationstrategy of reusing routes, as summarized in Algo-rithm 3, is faster than without this acceleration strat-egy. Therefore, we only present results obtained usingthis algorithm.

The column generation algorithm is initialized byincluding single customer routes, i.e., routes of theform 4401v1n+ 151 4t − t0v1 t1 t + tv1n+155, for each cus-tomer v ∈ V ′ in each scenario, for different values

1 Details on all instances are available at http://people.few.eur.nl/spliet.

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of t in the exogenous time window. More precisely,we use the values t = s′

v1 s′v +wv1 0 0 0 1 s

′v + kwv, where

s′v = max8sv1 s0 + t0v9 is the earliest possible arrival

times at customer v, e′v = min8ev1 en+1 − tv1n+19 is the

latest possible arrival times at customer v, and k =

�4e′v − s′

v5/wv�. This way, for every endogenous timewindow assignment, feasible routes are included inthe restricted master problem in each scenario satis-fying the assigned time windows.

Table 1 shows the results of the experiments usingthe column generation algorithm. In the first twocolumns, the instance and the number of customers inthat instance, �V ′�, are indicated. For each instance, wereport the results obtained when allowing all cyclesand when 2-cycles are eliminated. In the columns“T.Time,” the total computation time in seconds isreported. The columns “P.Time” report the total timein seconds spent on solving the pricing problems.The columns “Iter.” indicate the total number of iter-ations before termination. Finally, the columns “LP”contain the value of the LP relaxation per instance foreach route relaxation.

First observe that the computation time increaseswith the number of customers. Furthermore, almostall the computation time is spent on solving the pric-ing problems. When comparing the two route relax-ations, Table 1 indicates that the column generationalgorithm is significantly faster when all cycles areallowed. However, the values of the LP relaxations arehigher when 2-cycles are eliminated. In the next sec-tion, we discuss how the branch-price-and-cut algo-rithm is affected by the increase of both the LP valueand computation time in the case of 2-cycle elimina-tion, as opposed to allowing all cycles.

4.3. Computational Results of Using theBranch-Price-and-Cut Algorithm

In this section we present the results of the computa-tional experiments performed with the branch-price-and-cut algorithm. In this algorithm, lower boundsare obtained by using Algorithm 3 and by addingcapacity inequalities in iterations where no newroutes are found. We compare the branch-price-and-cut algorithms using the two route relaxations, allow-ing all cycles and 2-cycle elimination.

The results of applying the branch-price-and-cutalgorithm to the test instances when allowing allcycles and when eliminating 2-cycles are shown inTable 2. The columns “Opt.Gap” provide the per-centage difference between the best obtained upperand lower bounds after termination of the algorithm.The columns “LP gap” and “Root gap” show thepercentage difference between the value of the LPrelaxation and the best found upper bound, beforeand after adding capacity inequalities, respectively.The columns “Nodes” provide the number of nodes

Table 1 Column Generation Results

All cycles allowed 2-cycle elimination

Instance �V ′� T.Time P.Time Iter. LP T.Time P.Time Iter. LP

1 10 0044 0020 13 14022 0069 0067 15 170642 10 0039 0037 15 13048 1005 1003 12 140273 10 0092 0086 17 14066 2072 2070 18 160634 10 0016 0016 13 16051 0070 0066 17 180485 10 0030 0030 15 14015 0075 0075 11 140846 10 0037 0034 19 17027 0089 0089 12 180007 10 0048 0048 14 13034 1053 1051 15 160488 10 0016 0014 12 19061 0058 0058 12 220659 10 0047 0047 15 17038 1006 1003 13 1907410 10 0061 0061 17 14069 2000 2000 15 1505811 15 3040 3038 29 14096 13067 13062 28 1702812 15 1023 1017 20 22073 3073 3068 21 2403313 15 0080 0078 25 25083 2053 2041 27 2706514 15 1011 1011 25 18083 4049 4045 22 2203415 15 1025 1023 24 21030 3084 3079 24 2209116 15 1058 1057 30 19071 3068 3062 27 2002017 15 1009 1005 24 19044 3045 3040 21 2103018 15 1044 1039 26 19098 4032 4029 20 2009719 15 1054 1050 24 24040 5029 5027 33 2505920 15 1089 1086 28 20032 5045 5040 23 2104221 20 2032 2029 37 26044 8061 8047 40 2708022 20 3003 2095 34 27029 9098 9089 37 2905623 20 3010 3004 30 26039 10023 10016 34 2802024 20 4062 4053 36 22069 15082 15069 33 2304425 20 2017 2003 35 28000 7041 7027 35 2808326 20 4026 4013 35 26094 13099 13093 37 2803827 20 3028 3018 36 23050 9022 9011 36 2505828 20 3024 3016 35 24030 9031 9025 29 2506029 20 3020 3011 35 23069 13023 13009 36 2508330 20 5077 5068 40 24049 17071 17056 40 2502631 25 6047 6040 49 28004 23024 23003 42 3005932 25 9008 8092 48 28031 40090 40062 57 2903733 25 8056 8039 47 30091 37005 36074 51 3204234 25 5021 5001 47 31061 16088 16074 51 3205735 25 6080 6063 49 26097 26036 26012 51 2707136 25 4051 4038 36 28047 20066 20050 46 2909137 25 15058 15038 45 25048 73032 72094 68 2704038 25 4090 4078 42 31085 19070 19048 48 3405639 25 4077 4068 37 28088 17002 16081 42 3103640 25 4006 3087 49 29008 14049 14026 52 30080

processed in the search tree, and the columns “CI”give the number of added capacity inequalities.

Table 2 shows that when allowing all cycles, two 15-customer instances, two 20-customer instances, andnine 25-customer instances remain unsolved withinthe one-hour time limit. For the other instances, theLP gap ranges from 4.04% to 21.65%. After addingcapacity inequalities, these gaps are all tightened toless than 0.85%, and the gap even closes completelyfor 13 instances.

When eliminating 2-cycles, four 25-customer in-stances are solved of the above-mentioned unsolvedinstances when allowing all cycles. Moreover, for thepreviously unsolved instances 13 and 25, an integersolution is found and the optimality gap is closed to

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Table 2 Branch-Price-and-Cut Results

Allowing all cycles 2-cycle elimination

Instance �V ′� T.time Opt.gap LP gap Root gap Nodes CI T.time Opt.gap LP gap Root gap Nodes CI

1 10 2028 0 19046 0 6 41 0067 0 0005 0 1 22 10 86025 0 13039 0.19 543 54 127053 0 8029 0.17 483 303 10 1028 0 15086 0 1 43 4015 0 4053 0 1 134 10 21092 0 10083 0.14 241 65 31017 0 0014 0.14 193 175 10 5019 0 11093 0.29 19 53 2064 0 7066 0 2 116 10 1044 0 4004 0 4 41 1072 0 0 0 2 07 10 3087 0 21065 0 5 34 5076 0 3016 0 4 48 10 402 0 1709 0.85 62 50 3093 0 5019 0.65 29 119 10 4003 0 14042 0 17 41 3045 0 2082 0 7 610 10 9022 0 9094 0 21 45 6088 0 4051 0 5 3211 15 35001 0 15085 0 12 76 102048 0 2079 0 22 3412 15 3,600 — — — 31932 212 3,600 — — — 11070 8313 15 3,600 — — — 31634 304 3,600 0.25 5087 1.11 41391 17214 15 47046 0 18074 0.03 53 123 68011 0 306 0 45 5115 15 22082 0 11081 0.08 42 112 34059 0 501 0 36 5616 15 64041 0 6029 0.13 151 86 108058 0 3096 0.1 98 2817 15 2907 0 1108 0 42 97 26094 0 3035 0 15 2418 15 709016 0 10043 0.42 11107 142 123086 0 5096 0.2 98 5319 15 232058 0 7099 0.71 318 206 157045 0 3049 0.56 133 6520 15 24052 0 8011 0 31 101 46044 0 3011 0 25 5921 20 3,600 — — — 11824 382 3,600 — — — 864 10322 20 237068 0 804 0.12 150 239 196017 0 008 0.03 62 7423 20 104068 0 12091 0.11 78 171 159071 0 6092 0 65 10124 20 161057 0 601 0.3 111 199 142087 0 2098 0.03 27 6825 20 3,600 — — — 21040 361 3,600 0.25 3059 0.83 21130 18426 20 100045 0 9036 0 80 177 80079 0 4051 0 16 10627 20 92035 0 11027 0 53 209 10102 0 3039 0 22 9828 20 120096 0 7005 0.04 85 240 146066 0 2009 0 47 10829 20 9405 0 10096 0 37 261 59052 0 2092 0 10 8730 20 11009 0 701 0 54 176 5701 0 4017 0 4 11231 25 3,600 — — — 11161 433 312075 0 2067 0.13 39 19932 25 704047 0 7082 0.22 192 293 11371048 0 4037 0.07 232 14433 25 3,600 — — — 22 353 3,600 — — — 880 28634 25 3,600 — — — 778 479 138033 0 2031 0 18 28535 25 3,600 — — — 827 460 389074 0 4059 0 58 25336 25 3,600 — — — 11422 504 3,600 — — — 11333 26637 25 3,600 — — — 288 388 3,600 — — — 232 17938 25 3,600 — — — 765 537 3,600 — — — 978 29439 25 3,600 — — — 11343 506 3,600 — — — 21083 27940 25 3,600 — — — 11030 461 11749058 0 4017 0.3 539 286

0.25% within one hour. The other seven previouslyunsolved instances remain unsolved with 2-cycleelimination and no integer solutions are found.

As can be seen from Table 2, the LP gaps are sig-nificantly smaller when 2-cycles are eliminated. Afteradding capacity inequalities, the gap is completelyclosed for the 13 previously closed instances, as wellas for five other instances. For the remaining instancesfor which the optimum is found, the root gap issmaller when 2-cycles are eliminated. Note that atighter root gap is not guaranteed, as a heuristic pro-cedure is used to separate capacity inequalities.

Of the 27 instances that are solved when all cyclesare allowed, 13 instances are solved faster when 2-cycles are eliminated and 14 instances are solved moreslowly.

4.4. Solutions Based on VRPTW withAverage Demand

In this section, we compare the optimal solutions to theTWAVRP with the solutions found by a heuristic proce-dure, based on solving a VRPTW with average demandto optimality. Consider the instance of the VRPTW thatcorresponds to an instance of the TWAVRP in the fol-lowing way. The average demand over all scenariosin the TWAVRP is used as demand in the VRPTW,and the exogenous time windows of the TWAVRP areused as time windows of the VRPTW. We solve thisVRPTW to optimality and use the resulting arrivaltime at each customer as a point of reference for theassigned endogenous time window. More precisely,for tv the arrival time at customer v, the endogenous

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time window 6yv1yv +wv7 is computed as

6yv1yv +wv7

=

6sv1 sv +wv7 if tv −wv/2 ≤ sv1

6ev −wv1 ev7 if tv +wv/2 ≥ ev1

6tv −wv/21 t +wv/27 otherwise.(15)

Note that this resembles a procedure that is oftenused in practice to assign time windows. Theseendogenous time windows are used to construct asolution to the TWAVRP. Routes for each scenarioare found by solving another VRPTW to optimality,using the endogenous time windows as time win-dows. These routes, together with the endogenoustime windows as defined by (15), form a feasible solu-tion to the TWAVRP. We have implemented this pro-cedure and used it to solve the test instances.

Table 3 shows the results of using this procedure.The column “Value” shows the value of the solution

Table 3 VRPTW with Average-Demand-Based Solutions

Instance �V ′� Value Opt. Gap

1 10 17065 17065 02 10 16017 15056 30923 10 17042 17042 04 10 18051 18051 05 10 16015 16007 00526 10 18000 18000 07 10 17002 17002 08 10 23097 23089 00339 10 21041 20031 504210 10 16054 16031 1041

Average gap 1016

11 15 18005 17078 105414 15 24005 23018 307715 15 24087 24015 300016 15 21011 21003 003617 15 23022 22004 503518 15 23003 22030 302719 15 26066 26052 005420 15 22073 22011 2080

Average gap 2058

22 20 30047 29080 202623 20 30092 30030 200524 20 24030 24016 005726 20 29072 29072 027 20 27048 26048 307828 20 27005 26014 304729 20 27016 26061 200830 20 26036 26036 0

Average gap 1077

31 25 31082 31043 102532 25 31086 30071 307434 25 34054 33034 305935 25 29066 29005 201140 25 32022 32014 0026

Average gap 2019

based on solving a VRPTW with average demand,and the column “Opt.” gives the optimal value ofeach instance. The column “Gap” provides the per-centage difference between the solution value andthe optimum. Only the test instances that have beensolved to optimality are included in Table 3.

As can be seen from Table 3, the procedureprovides the optimal solution for five 10-customerinstances and two 20-customer instances. For theother instances, the differences are up to 5.42%. Theaverage difference over all instances is 1.85%. Afterone hour of computation time, the branch-price-and-cut algorithm with 2-cycle elimination found a solu-tion for instance 13 with a value 29.37 and forinstance 25 with a value 29.07. The VRPTW withaverage-demand-based solution values of instances 13and 25 is 29.55 and 30.04, respectively. The differencesbetween the solutions obtained by these proceduresare 0.57% and 0.45%, respectively.

5. ConclusionIn this paper we introduce the time window assign-ment vehicle routing problem, the TWAVRP, whichmodels the problem of assigning time windows tocustomers before demand is known. In this model,demand realizations occur according to a predefinedset of scenarios with known probability distribution.After demand becomes known, an optimal vehiclerouting schedule is made adhering to the assignedtime windows. The problem is to assign time win-dows such that the expected total traveling costs areminimized.

We propose a branch-price-and-cut algorithm tosolve the TWAVRP. We have considered two routerelaxations, one allowing all cycles and the othereliminating 2-cycles. Moreover, we strengthen theLP bound by adding capacity inequalities. Compu-tational experiments show that the proposed branch-price-and-cut algorithm can solve instances of theTWAVRP of up to 25 customers and three scenarios.Using 2-cycle elimination in the branch-price-and-cutalgorithm increased the number of instances that weresolved to optimality. However, neither route relax-ation yields a branch-price-and-cut algorithm that issuperior with respect to running times.

We compared the optimal solution to a solutionobtained by solving a VRPTW with average demand.In our experiments, the solutions based on solving aVRPTW with average demand have costs that are upto 5.42% higher than the optimum and are on average1.85% higher.

The branch-price-and-cut algorithm that is pre-sented in this paper is a first step toward solvingthe TWAVRP. In future research, this algorithm couldbe improved so that it can solve instances with a

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larger number of customers and scenarios. One pos-sible research direction to this end is the develop-ment of valid inequalities that are specific to theTWAVRP. Another possible research direction that canbe explored is the development of a method in whichtime windows are selected from a limited set of can-didate time windows, instead of choosing time win-dows from exogenous time windows.

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Documents

The Time Window Assignment Vehicle Routing Problem