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Vehicle routing problem for information collection inwireless networks

Luis Flores Luyo, Agostinho Agra, Rosa Figueiredo, Eitan Altman, EladioOcaña Anaya

To cite this version:Luis Flores Luyo, Agostinho Agra, Rosa Figueiredo, Eitan Altman, Eladio Ocaña Anaya. Vehiclerouting problem for information collection in wireless networks. ICORES 2019 - 8th InternationalConference on Operations Research and Enterprise Systems, Feb 2019, Prague, Czech Republic. �hal-02176511�

Vehicle routing problem for information collection in wireless networks

Luis Flores Luyo1,2, Agostinho Agra3, Rosa Figueiredo2, Eitan Altman2,4 and Eladio Ocana Anaya1

1Instituto de Matematicas y Ciencias Afines, Lima, Peru2Laboratoire Informatique d’Avignon, Avignon Universite, Avignon, France

3Departamento de Matematica, CIDMA, Universidade de Aveiro, Aveiro, Portugal4NEO - Inria Sophia Antipolis - Mediterranee, France

{lflores,eocana}@imca.edu.pe, rosa.figueiredo@univ-avignon.fr, aagra@ua.pt, eitan.altman@sophia.inria.fr

Keywords: Vehicle Routing Problem; Wireless Sensor Networks; Delay Tolerant Networks; Integer Programming

Abstract: Advances in computer network architecture add continuously new features to vehicle routing problems. Inthis work, the Wireless Transmission Vehicle Routing Problem (WT-VRP) is studied. It looks for a route tothe vehicle responsible for collecting information from stations as well as an efficient information collectionplanning. The new feature added here is the possibility of picking up information via wireless transmission,without visiting physically the stations of the network. The WT-VRP has applications in underwater surveil-lance and environmental monitoring. We discuss three criteria for measuring the efficiency of a solution andpropose a mixed integer linear programming formulation to solve the problem. Computational experimentswere done to access the numerical complexity of the problem and to compare solutions under the three criteriaproposed.

1 INTRODUCTION

The vehicle routing problem (VRP) is one of the moststudied problems in operations research. The inten-sive research on the topic is due not only to its com-putational complexity but also to the numerous appli-cations in fields such as logistic, maritime transporta-tion, telecommunications, production, among manyothers. Different variants of the VRP have beentreated in the past 50 years (see (Laporte, 2009)).Among the best-known variants we can cite the VRPwith Multiple Depots (Montoya-Torres et al., 2015),the Pickup-and-delivery VRP (Dethloff, 2001), theVRP with Time Windows (Agra et al., 2013), the VRPwith Backhauls (Toth and Vigo, 1997). Several hybridvariants of the problem are also described in the liter-ature, most of them inspired by real-life scenarios (see(Baldacci et al., 2007; Caceres-Cruz et al., 2014)).

As we could expect, different solution approachesfor the VRP have been explored in the literature.Sincethe VRP is an NP-hard problem, heuristic and meta-heuristic approaches are more suitable for practicalapplications, often defined on large scale graphs. De-spite the fact that they do not provide an approxi-mation guarantee on the obtained solution, they haveproved their efficiency empirically.

Exact methods for obtaining solutions with guar-

antee of optimality are often applied to small-sized in-stances and, sometimes, even to medium-sized ones(Fukasawa et al., 2006; Pecin et al., 2017). Amongthe exact approaches, we have solution methods basedon integer (ILP) and mixed integer linear program-ming (MILP) formulations (Feillet, 2010; Simchi-Levi et al., 2005) and other enumerative algorithms(Fischetti et al., 1994). Exact approaches providebenchmark instances to the VRP problem studied andallow us to evaluate the performance of heuristics.

In the last years, technological advances in com-puter network architecture have added new featuresand applications to vehicle routing problems. Delay-Tolerant Networks and Wireless Sensor Networks aretwo examples.

Delay-Tolerant Networks (DTN) (Fall, 2003; Jainet al., 2004) are wireless networks designed to toler-ate long delays or disruptions. One of the applica-tions of DTN’s that has gained popularity is to pro-vide web service to remote locations. For example,Daknet (Pentland et al., 2004) is a network for ru-ral connectivity that uses buses and local transport tocarry messages and web connection to small and re-mote villages. Daknet establishes small kiosks in ru-ral villages without web connectivity, allowing peopleto send email and information in an off-line manner.Another important application of DTN is to provide

connection for remote military stations which disposeof a set of vehicles to pick-up and deliver wireless in-formation (Malowidzki et al., 2016).

Wireless Sensor Networks (WSN) (Akyildiz et al.,2002) are often used in critical applications suchas habitat monitoring, war surveillance, submarinesurveillance and monitoring, detection of biological,chemical and nuclear attack (Mainwaring et al., 2002;Winkler et al., 2008; Basagni et al., 2014; Vieira et al.,2015). In these applications, sensors are deployed in aarea and are used to store information to be sent it thefuture to a control center via base stations. Mobile el-ements, such as unmanned aerial vehicles, have beenincorporated in the WSN design (Teh et al., 2008).

The applications mentioned here from both tech-nologies, DTN and WSN, need to provide vehiclerouting strategies with wireless information transmis-sion to the vehicles involved. Most of the workhas been done considering the routing protocol (Bhoiet al., 2017; Celik and Modiano, 2010; Moghadamet al., 2011; Velasquez-Villada et al., 2014) whilethere is still a gap in the development of vehicle rout-ing strategies. When developing efficient routing pro-tocols, many authors consider that the vehicle routeis already defined which is justified by applicationstaking advantage of an existing transportation infras-tructure (Velasquez-Villada et al., 2014). In (Celikand Modiano, 2010), the vehicle route is assumedto exist but authors suppose vehicles can adjust theirposition (in order to receive information from somestations) and study the delay performance in the net-work. Other works, like (Kavitha and Altman, 2009),suppose an architecture is defined for the vehicle rout-ing (cycle path or zig-zag path) and study the bestplacement for such architecture. To the best of ourknowledge, the authors in (Basagni et al., 2014) arethe only ones to investigate the vehicle routing prob-lem from scratch together with the wireless transmis-sion planning. They address an application of under-water wireless sensor networks for submarine moni-toring. The authors considered a scenario with a setof S surfacing nodes and |S| underwater nodes wherethey look for a routing to an autonomous underwa-ter vehicle (AUV) during a time period T . The AUVmust leave and return to a surface node while informa-tion generated by the set of underwater nodes is col-lected along a path that physically visits each stationwhere information is collected. The information gen-erated in a given underwater node i at a time point t1which arrives to a surface node at a time point t2 has agiven value vit1t2 . The strategy adopted by the authorsis the maximization of the value of the informationcollected. The authors in (Basagni et al., 2014) pro-posed an Integer Linear Programming (ILP) formula-

tion able to solve the problem with |S| ∈ {4,5,9,12}in a time that varies from a few hours to a few days.

In this work, we study a new version of the ve-hicle routing problem for information collection inwireless networks. The new characteristic added tothis well-known problem is the possibility of pickingup information via wireless transmission. In the con-text considered here, a vehicle is responsible for col-lecting data from a set of stations. The stations areequipped with technology capable of sending infor-mation via wireless connections to the vehicle whenit is located in another sufficiently close station. Si-multaneous transmissions are allowed. Time of trans-mission depends on the distance between stations, theamount of information transmitted, and other physicalfactors (e.g. obstacles along the way, installed equip-ment). Information to be sent outside of the networkis generated at each station at a constant rate. Given atime horizon, the vehicle must visit the set of stationsgathering information either when it is physically in astation or by wireless transmission. A unique basestation is connected with the outside. The vehiclestarts and ends its route at the base station. Any sta-tion can be visited several times but it can also neverbeing visited.

Unlike in (Basagni et al., 2014), the value of infor-mation collected does not decreases with time. How-ever, depending on the position of the base station,the vehicle can be conducted to return several timesto the base in order to have access to some stations.The VRP treated in this work looks for a route to thevehicle and for an efficient planning on how to collectinformation from stations. Since only one uncapac-itated vehicle is considered, the problem can also beseen as a version of the Traveling Salesman Problem(TSP) (Letchford et al., 2013). However, opposite tomost versions of TSP, no constraint is imposed nei-ther on the number nor on the frequency of visits tothe stations. The focus of this work is the investiga-tion of the strategies adopted in the vehicle routingproblem and its exact solution by an ILP formulation.

The paper is organized as follows. Section 2 for-mally describes the problem being solved while no-tations and assumptions are presented. A MILP for-mulation is presented in Section 3 with three differ-ent objective functions being discussed at this sec-tion. Computational experiments are presented onSection 4. Periodicity on the remaining informationat the stations after a sequence of vehicle routings isdiscussed in Section 4. Finally, some conclusions andresearch directions are presented at Section 5.

2 DESCRIPTION OF THEPROBLEM

The wireless network is modeled by a directedgraph D = (V,A). The node set V = {1, . . . ,n} rep-resents the n stations of the network and the arc setA represents m directed paths connecting pairs of sta-tions in V . A unique base station is considered anddenoted as node 1. Weights ti j and di j are associ-ated to each arc (path) (i, j) ∈ A representing, respec-tively, the time it takes to travel from station i tostation j and the distance among these two stations.Let T = {1,2, . . . , T} be the time horizon considered.At the beginning of the time horizon, each stationj ∈V \{1} contains an amount C j of information. Foreach station j ∈V \{1}, information is generated at arate of r j units per time point in T . Thus, the amountof information at station j at each time point k ∈ T ,denoted by q jk, is proportional to the elapsed timefrom the last extraction (either physically or througha wireless connection), i.e.,

q jk =

C j + kr j, if station j has not been visited

before time point k,(k− tlast)r j, otherwise, where tlast is the

time of the last extraction.

Only the base station is appropriately equipped forsending information outside the network. A uniquevehicle is in charge of collecting data from all the sta-tions in V \{1} and of transporting it to the base sta-tion. There is no capacity limit associated to the vehi-cle. At the beginning of the time horizon, the vehicleis located at the base station and at the end of the timehorizon, it must return to the base station. Multiplevisits are allowed to each node in V . Information canonly be transmitted when the vehicle is located in oneof the stations in V , i.e., no transmission is allowedwhile the vehicle is moving on an arc (i, j) ∈ A. Wealso assume that, once a station i starts a transmissionto the vehicle, all the current information located in iat that moment must be transmitted.

Wireless transmission can be used to transfer datafrom a station j ∈ V to the vehicle located in a sta-tion i ∈ V \ { j}. However, wireless transmission isonly possible for close enough stations. Let rcov be themaximum distance allowing wireless transmission. Astation j can wireless transfer its data to (the vehiclelocated in) station i whenever di j ≤ rcov. We define aset W = {(i, j) ∈V ×V | di j ≤ rcov}.

We assume that a transmission occurs with a fixedtransmission power of Pt . The received power Pr isgiven by

Pr = αPtD−η

where η is the pathloss parameter (which we shalltake in this paper to be equal to two but our resultscarry on to arbitrary positive value of η) and whereD is the distance between the receiver and transmit-ter. We shall assume that the vehicle has an antennawith an elevation of 1 unit. The coordinates of asensor are given by (xs,ys,0) and those of the vehi-cle are (xb,yb,1), which means, the antenna on thevehicle is elevated by one unit with respect to thesensors. Thus, if d =

√(xs− xb)2 +(ys− yb)2 then

D =√

1+d2. We use a linear approximation of theShannon capacity (as a function of the power) for thedata transmission rate (Tse and Viswanath, 2005) andwrite it as

T hp(d) = log(

1+αPr

σ

)∼ βPr

2σ=

βPt

2σ(1+d2)−1.

Here β is the antenna’s gain and σ is the noise at thereceiver (we assume independent channels and thusthere are no interferences of other transmissions onthe received signal from a sensor).

Thus, the time necessary for a wireless transmis-sion of q units of data between stations j and i is

α ji(1+d2i j)q, (1)

with α ji =βPt2σ

depending on physical factors in sta-tions i and j.

Simultaneous transmission is possible from a setof at most M stations to the vehicle located in a sta-tion i ∈V . In this case, the simultaneous data transferfinishes only when each individual wireless transmis-sion finishes. As a consequence, the time of a simul-taneous transmission corresponds to the highest max-imum individual wireless transmission.

The version of the VRP treated in this paper,denoted as Wireless Transmission VRP (WT-VRP),consists of finding a feasible routing for the vehicletogether with an efficient planning for collecting in-formation from stations V \{1}. The criteria for mea-suring the efficiency of a collection planning will bediscussed in the next section.

3 MATHEMATICALFORMULATION

In this section, we introduce a mixed integer linearprogramming (MILP) formulation to the WT-VRP.Let us define an artificial arc set A0 = {(i, i)∈V}. Foreach (i, j)∈ A∪A0 and k ∈ T , we define the followingdecision variables.

xi jk =

1, if the vehicle crosses path (i, j) and

arrives to node j at time point k,0, otherwise.

We observe that the action of crossing an artificialarc (i, i) ∈ A0 models the following action: the vehi-cle is located at node i without neither moving nortransferring data. Let t = (ti j) be the weight matrixassociated with A. We also define t = t + I the weightmatrix associated with A∪A0. For each ( j, i) ∈W ,k ∈ T and l ∈ Sk = {1, . . . , T − k}, we also define,

w jikl =

1, if node j is sending data to the vehiclewhile it is located in node i, withtransmission starting at time point kand lasting l time units,

0, otherwise.

Linear constraints defining the MILP formulationare presented next, divided in three sets according totheir modeling purposes.

3.1 Problem constraints

The first set, the Routing Constraints, characterizesthe way the vehicle moves around the set of stations.

∑s:(1,s)∈A∪A0

x1s(1+t1s) = 1, (2)

∑(s,1)∈A∪A0

xs1T = 1, (3)

xi jk ≤ ∑( j,p)∈A∪A0|k+t jp∈T

x jp(k+t jp)+ ∑(u, j)∈W

∑l∈Sk

wu jkl ,

∀(i, j) ∈ A∪A0,∀k ∈ T, (4)

xi j(k+ti j) ≤ ∑(u,i)∈A∪A0

xuik + ∑n|(n,i)∈W

k−1

∑l=1

wni(k−l)l ,

∀(i, j) ∈ A∪A0,∀k s.t. k+ ti j ∈ T, (5)

∑l∈Sk

w jikl ≤ ∑(i,n)∈A∪A0

∑s∈Sk|k+s+tin≤T

xin(k+s+tin),

∀i ∈V,∀ j s.t. ( j, i) ∈W,∀k ∈ T, (6)

∑(i, j)∈A

xi jk ≤ 1, ∀k ∈ T, (7)

k+ti j−1

∑s=k+1

∑(m,n)∈A∪A0

xmns ≤ ti j(1− xi j(k+ti j)),

∀(i, j) ∈ A, ∀k ∈ T s.t. k+ ti j ∈ T , (8)xi jk ∈ {0,1}, ∀(i, j) ∈ A,∀k ∈ T, (9)wuvkl ∈ {0,1},∀(u,v) ∈W,∀k ∈ T, ∀l ∈ Sk. (10)

Equations (2) and (3) ensure, respectively, that thevehicle starts and ends the routing at the base station.Inequalities (4) ensure that, if at a time point k ∈ T thevehicle arrives at station j ∈ V crossing arc (i, j) ∈A∪ A0, it either goes immediately to a neighboring

station (crossing an appropriate path ( j, p) ∈ A∪A0),or it stays at node j for a data transfer from anotherstation u ∈ V . Likewise, if at a time point k+ ti j ∈ Tthe vehicle arrives to station j ∈ V coming from sta-tion i, inequalities (5) impose that either the vehiclehas arrived to station i at time point k, or a data trans-fer from at least one another station was occurring andit has finished at time point k. Once the vehicle hasarrived to station i ∈ V , at a time point k ∈ T , if datatransfer happens, inequalities (6) force the vehicle toleave i, in a future time point k+ s, by crossing an arc(i,n) ∈ A∪A0. Inequalities (7) and (8) are responsi-ble for the elimination of simultaneous paths. On onehand, inequalities (7) force the vehicle to leave a givenstation along a unique path. On the other hand, if thevehicle cross the arc (i, j), leaving i at time point k, aninequality in (8) will prevent the vehicle from movingfrom time point k + 1 to k + ti j − 1. Constraints (9)and (10) impose binary conditions on the variablesdefined.

Inequalities defining the second set of constraintsare the Data Transfer Constraints.

1M ∑

( j,i)∈W∑

l∈Sk

w jikl ≤ 1,∀k ∈ T, (11)

w jikl ≤ ∑(p,i)∈A

xpik,

∀( j, i) ∈W, ∀k ∈ T, ∀l ∈ Sk, (12)

∑(i,n)∈A

l−1

∑s=0

xin(k+s+tin) ≤ l(1−w jikl),

∀( j, i) ∈W, ∀k ∈ T, ∀l ∈ Sk. (13)

Inequalities (11) define a bound of M simultane-ous data transfers. Inequalities (12) impose that, ateach time point k ∈ T , in order to start sending infor-mation to a station i ∈V , the vehicle must previouslyarrive to this station. Inequalities (13) prevent the ve-hicle to leave station i ∈V while a data transfer is oc-curring: if w jikl equals to 1, the vehicle cannot leavestation i from time point k to time point k + l − 1.When simultaneous data transfers occurs, this set ofinequalities will be in charge of defining the total du-ration of the simultaneous transfer.

Before presenting the last set of constraints, weneed to define the continuous variables of the MILPformulation. For each j ∈ V and k ∈ T , let q jk repre-sent the amount of data accumulated at station j (wait-ing for transfer out of the network) at time point k.The last set of constraints is presented next and they

are the Amount of Information Constraints.

q j1 =C j,∀ j ∈V \{1}, (14)

q jk+1 = q jk(1− ∑i|( j,i)∈W

∑l∈Sk

w jikl)+ r j,

∀k ∈ T \{T},∀ j ∈V, (15)

α ji(1+d2i j)q jkw jikl ≤ l,

∀( j, i) ∈W, ∀k ∈ T, ∀l ∈ Sk. (16)

Equations (14) set the initial load of each stationj ∈ V \ {1}. Equations (15) are in charge of updat-ing the load of stations along the time horizon. Theamount of data accumulated in node j, at time pointk + 1, q j(k+1), is set to r j in the case a data transferstarted at time point k, otherwise it equals to q jk + r j.Finally, inequalities (16) define the time necessary fortransferring q jk data units (as defined by (1)) when-ever w jikl = 1.

Constraints (15) and (16) are quadratic ones andcould be linearized by applying a classical change ofvariables. An alternative way of linearizing these con-straints is by replacing them with the following big-Minequalities:

q j(k+1) ≥ q jk + r j−M jk ∑j|( j,i)∈W

∑l∈Sk

w jikl ,

∀k ∈ T \{T},∀ j ∈V, (17)q jk ≥ r j, ∀ j ∈V, ∀k ∈ T (18)

α ji(1+d2i j)q jk−N jik(1−w jikl)≤ l,

∀( j, i) ∈W,∀k ∈ T, ∀l ∈ Sk, (19)

where

M jk = r j(k+1), j ∈V, k ∈ T,

N jik = α ji(1+d2ji)r jk, ( j, i) ∈W, ∀k ∈ T.

Consider a node j ∈ V and a time point k ∈ T .If a transfer occurs from station j at time point k,the associated inequality in (17) becomes redundantand the associated inequality in (18) defines the validlower bound q jk ≥ r j. On the other hand, if no trans-fer occurs, the associated inequality in (18) becomesredundant and a valid lower bound q j(k+1) ≥ q jk + r jis defined by (17). A minimization objective functionof the WT-VRP (discussed in the next subsection) to-gether with inequalities (17) and (18) will be in chargeof appropriately setting the value of variables q jk. Inthe same way, an inequality in (19) is either redun-dant (w jikl = 0) or it becomes an inequality in (16)(w jikl = 1).

3.2 Objective functions

As it has been described in Section 2, the WT-VRPlooks for an efficient way of collecting data located

in remote stations. We discuss in this section threedifferent criteria to measure the efficiency of a col-lection planning, each one giving birth to a differentlinear objective function. The first one maximizes thetotal amount of information extracted at the end ofthe time horizon T ; the second one maximizes the av-erage of the information in the vehicle at each timepoint; while the third one maximizes the satisfactionof each station at the end of the time horizon T .

Let us analyze the first criteria. Consider that ourgoal is the extraction of as much information as possi-ble from all stations, at the end of the finite time hori-zon T . Let η j be the quantity of information extractedfrom a given station j ∈V \{1}, i.e.,

η j =C j + r j(T −1)−q jT .

The amount of information extracted from all sta-tions in V \{1} is

∑j∈V\{1}

η j = ∑j∈V\{1}

C j+ ∑j∈V\{1}

r j(T−1)− ∑j∈V\{1}

q jT .

Since the only variables (from our MILP formula-tion) in this equation are the q jT variables, for j ∈V ,we have that

max

{∑

j∈V\{1}η j

}=

∑j∈V\{1}

C j + ∑j∈V\{1}

r j(T −1)−min

{∑

j∈V\{1}q jT

}.

The first objective function, denoted FO1, mini-mizes the remaining amount of information at the endof the time period, over the set of stations, i.e.,

Minimize ∑j∈V\{1}

q jT . (20)

For the first criterion, no assumption is madeabout data location security: only the amount of in-formation collected at the end of the time horizonmatters. The second efficiency criterion is motivatedby environments where information is safer once theyleave the station to be stored at the vehicle (Basagniet al., 2014). Among the factors behind this suppo-sition we have failures in station equipments, attackin case of military applications and energy resourceslimited. Let vk be the amount of information in thevehicle at a time point k ∈ T ,

vk = ∑j∈V\{1}

C j + ∑j∈V\{1}

r j(k−1)− ∑j∈V\{1}

q jk.

We look for a solution, i.e., a routing and a collec-tion planning, that maximizes the average over timeof the amount of information in the vehicle, i.e.,

max

{1T ∑

k∈Tvk

}= ∑

j∈V\{1}C j+

( ∑j∈V\{1}

r j)(T −1)

2− 1

Tmin

{∑k∈T

∑j∈V\{1}

q jk

}.

The second objective function, denoted FO2, min-imizes the total amount of remaining information onthe set of stations, over the whole time horizon, i.e.,

Minimize ∑k∈T

∑j∈V\{1}

q jk. (21)

In the third criterion, the satisfaction of each sta-tion will be taken into account. Let us define the sat-isfaction of a station j ∈ V as the total amount of in-formation extracted from j over the amount of infor-mation generated at j during the time horizon consid-ered, i.e.,

s j =C j + r j(T −1)−q jT

C j + r j(T −1)= 1−

q jT

C j + r j(T −1).

The maximization of the satisfaction over thewhole set of stations is

max

{1

n−1 ∑j∈V\{1}

s j

}=

= 1− 1n−1

min

{∑

j∈V\{1}

q jT

C j + r j(T −1)

}.

For the particular case where C j = r j,

= 1− 1(n−1)T

min

{∑

j∈V\{1}

q jT

r j

}.

Finally, the third objective function, denoted FO3,which maximizes the satisfaction of the stations, isdefined as,

Minimize ∑j∈V\{1}

q jT

r j. (22)

Consider the instance of the WT-VRP depicted inFigure 1 with five non-base stations. Figures 2–4 il-lustrate the solutions obtained by solving the MILPformulation, with the three different objective func-tions, assuming αi j = 0.03, for each i, j ∈V , rcov = 2,T = 20 and M = 3. The movement of the vehicle isillustrated in each one of these figures by a time ex-panded network. Each set of nodes aligned horizon-tally is associated to a same station i ∈ V in differenttime points. Each set of nodes aligned vertically rep-resents the whole set of stations in a unique time pointk ∈ T . Let us denote a node in the time expanded net-work by a pair ik, with i ∈ V and k ∈ T . The scalarvalue inside a node ik gives us the amount of informa-tion in station i at time point k. A filled arrow froma node ik to a node jl represents a movement of thevehicle, leaving station i at time point k and arriving

d =

0 4 5 4 7 74 0 2 4 3 65 2 0 2 2 14 4 2 0 5 17 3 2 5 0 27 6 1 1 2 0

t =

1 4 5 4 0 04 1 2 0 3 05 2 1 2 2 14 0 2 1 0 10 3 2 0 1 20 0 1 1 2 1

Figure 1: Directed (symmetric) graph describing an in-stance with 6 stations, data rates r = (0,4,1,5,3,2), timematrix t and distance matrix d.

Figure 2: Solution obtained for the instance depicted in Fig-ure 1 by the MILP formulation with objective function FO1defined in (20).

to station j at time point l. A dashed arrow from anode ik to a node jk indicates a data transfer occursfrom station i to station j, starting in a time point afterk. A weight associated with a dashed arc indicates theduration of the transfer.

The vehicle routing depicted in Figure 2 is 1→2→ 3→ 6→ 4→ 1. The vehicle leaves the base sta-tion 1 at time point 1 and arrives to station 2 at timepoint 5 with transmission from stations 2 and 3 start-ing immediately, each lasting 1 time unit. Then, thevehicle leaves station 2 at time point 6 arriving at sta-tion 6 at time point 9 for data transfer from stations4, 5 and 6; data transfer at station 6 lasts max{5,3,3}time units. The vehicle leaves station 6 at time point14 and continues its routing and data transfers untilreturning at time point 20 to the base station 1. The to-

Figure 3: Solution obtained for the instance depicted in Fig-ure 1 by the MILP formulation with objective function FO2defined in (21).

Figure 4: Solution obtained for the instance depicted in Fig-ure 1 by the MILP formulation with objective function FO3defined in (22).

tal amount of information generated during the wholetime horizon at each station, respectively from station2 to 6, is 80,20,100,60,40. For the solution depictedin Figure 2, the remaining information at each stationis, respectively, 60,15,25,33,10 which means, the to-tal amount of information collected from each stationis, respectively, 20,5,75,26,30. For this solution, thevalue of the three different objective functions are:143 for FO1, 1575 for FO2 and 51 for FO3.

The vehicle routing depicted in Figure 3 chosesfor not visiting stations 2 and 5; wireless transmis-sions are used to collected information from these sta-tions. For this solution, the value of the three differ-ent objective functions are: 162 for FO1, 1536 forFO2 and 48 for FO3. The vehicle routing depictedin Figure 4 choses for visiting each station in the net-work. This improves the satisfaction over the wholeset of stations though it increases the amount of re-maining information in the network. For this solution,the value of the three different objective functions are:145 for FO1, 1598 for FO2 and 45 for FO3.

4 COMPUTATIONALEXPERIMENTS

In this section, we report computational experi-ments carried out with the formulation presented inSection 3 comparing the solutions obtained with thedifferent objective functions proposed. The MILPproblems were solved by IBM CPLEX Optimizer12.6.1.0 (using 16 threads) on a server with a 16 pro-cessor IntelrXeonr Processor E5640 12M Cache,2.67 GHz and 32 GB of RAM memory. The CPUtime limit was set to 1h for all instances. Before pre-senting the obtained results, we briefly describe theset of instances used in our experiments.

Instances We evaluate the MILP formulation on aset of 80 instances. Each instance is defined by a ran-dom graph D = (V,A) and a given value T . A set of40 random graphs was generated as follows.

Let n and ∆ be, respectively, the total number ofvertices and a border on the density of the graph tobe generated. On a square of length B, the base sta-tion is located in the bottom-left vertex and the othern− 1 stations are placed randomly on the square oflength B− 2 in the upper-right (as shown in Fig-ure 5). We define A as the adjacency matrix as-sociated with the complete graph defined by the nstations. Let ∆(D) denote the density of the graphD = (V,A). We randomly select an arc (i, j) ∈ A suchthat D = (V,A\(i, j)) is a connected graph. We defineA = A \ (i, j). In case ∆−∆(D) > 0.01 we proceedwith the random elimination of arcs; otherwise westop and return the graph D = (V,A). Finally, a rater j is randomly generated for each j ∈ V \ {1}. Thedistance matrix d is defined by using the euclideandistance between each pair of vertices located in thesquare of side B. The time matrix t is defined ac-cording to the adjacent matrix considering a vehicleof speed equal to 1 which means, ti j = di j if (i, j)∈ A,ti j = 0 otherwise. We generate graphs with numberof vertices n ∈ {6,8,10,12} and upper bounds on thegraph density ∆ ∈ {0.5,0.7}. For each combinationof n and ∆, five random graphs are generated accord-ing to the procedure just described. In our experi-ments we considered the set of graphs just describedtogether with t ∈ {24,28}.

Objective functions comparison Tables 2 and 3exhibit the results obtained with the three objectivefunctions on the set of 80 random instances. Inboth tables, the first multicolumn exhibits informa-tion about the instances: |V | is the number of sta-tions, d is the graph density, T is the size of thetime horizon considered. The fourth column informs

Figure 5: Schema used for random graph generation: thebase station is located in the bottom-left vertex; the otherstations are randomly placed in the square located in theupper-right.

the total amount of information generated by all sta-tions in the network during the whole time hori-zon. The next five multicolumns display informa-tion about the solutions obtained with each differentobjective function: “CpuTime” is the time, in sec-onds, spent to solve the instance to optimality (“−”means the instance was not solved in the time limit);“Gap” is the MILP gap calculated between the bestinteger solution found and the final lower bound (aGap = 0 means the solution was solved to optimalityin the time limit); “In f .collected” display the totalamount of information collected by the vehicle dur-ing the whole time horizon; “Av.Vehicle” display theaverage amount of information in the vehicle overtime; “Av.Satis f action” display the average satisfac-tion over the set of stations.

Table 2 presents results obtained on instances with6 and 8 stations and show, as we expected, the sensi-bility of the ILP formulation to the total number ofstations in the network. In general, instances withmore than eight stations cannot be solved in one hourof computation. Looking the optimal solutions ob-tained, data transmissions took no more than threetime units. Thus, we solved a set of instances with8,10 and 12 stations by limiting data transmissions tofour time units which reduces the number of binaryvariables in the formulation: variables w jikl are de-fined for l ≤ 4. Table 3 presents the results obtainedon instances with 8, 10 and 12 stations.

From results on these two tables, we concludethat the WT-VRP problem becomes more difficult asthe number of stations and the time horizon increase.From Table 3, a total of 2 and 17 MILP problemswere not solved to optimality, respectively, for T = 24and T = 28 with gaps arriving to 30% when T = 28.From Table 2, a total of 7 and 29 MILP problems

were not solved to optimality, respectively, for T = 24and T = 28 with gaps arriving to 69% when T = 28.The graphics in Figure 6 display the results obtainedon the 47 instances solved to optimality by all threeobjective functions. In each graphic, the instancesare ordered in increasing order of the average of thetime spent to solve the three ILP problems. The threegraphics compare the optimal solutions obtained ac-cording to the value of “In f .Collected”, “Av.Vehicle”and “Av.Satis f action”. We can observe that, in fact,the WT-VRP problem became more difficult as the to-tal amount of information collected from the networkincreases.

From our results, we conclude the MILP formula-tion defined with objective function FO2 is computa-tionally easier: less instances not solved to optimal-ity and, in average, less time spending and smallergaps. From both tables, the total number of MILPnot solved to optimality in the time limit is equal to22, 11 and 22, respectively, for objective functionsFO1, FO2 and FO3. We can also observe from Fig-ure 6 that, on one hand, the solutions maximizingthe average satisfaction of the network impacts al-most equally the average amount of information inthe vehicle and the total amount of information col-lected. On the other hand, the solutions maximizingthe average amount of information in the vehicle overtime, impacts slightly more the average satisfactionof the network than the solutions maximizing the to-tal amount of information collected.

Periodicity We compare now the optimal solutionsobtained with each objective function with respect tothe information left in the network at the end of atime period T . Let C0 = [C0

0 ,C01 , . . . ,C

0n ] be a vec-

tor with the amount of information stored in each sta-tion i ∈ V . By considering the vector C0 as the ini-tial conditions in the MILP formulation, we obtain arouting for the vehicle, a collection planning and theamount of remaining information in each station, de-noted here as C1 = [C1

0 ,C11 , . . . ,C

1n ]. Thus, the MILP

formulation associated with an instance can be seenas a function F : Rn −→ Rn such that F(C0) = C1.In order to study the dynamics of function F , letus define Cm = Fm(C0) where Fk(C) = F(Fk−1(C)).We experimentally investigate the existence of valuesk,τ ∈ N such that Fτ(Ck) = Ck starting with the ini-tial conditions C0 = 0. When such values exist, wesay that the function F is periodic.

For this experiment, we use all the instancessolved in less than 300s by our MILP formulation(with all three different objective functions). Consideran instance of the problem and a MILP formulation,i.e. a function F : Rn −→ Rn. Starting with C0 = 0,

0

50

100

150

200

250

300

Instances

Inf

colle

cte

d

0

20

40

60

80

100

120

140

Instances

Av.

Ve

hic

le

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Instances

Av.

Sa

tisfa

ctio

n

FO1

FO2

FO3

Figure 6: Results obtained on the set of 47 instances solvedto optimality with all objective functions.

we solve this instance and obtain F(C0) =C1. Then,we continue to iterate until we discover that the func-tion is periodic or we have solved the MILP formula-tion 30 times.

Table 1 displays the results obtained. The firstmulticolumn exhibits information about the instancesused in this experiment as defined for the previoustables. The other three multicolumns inform us theresults obtained for each objective function: the val-ues obtained for k, τ and the number of isolated sta-tions #isol. An isolated station i is a station such that,in the successive applications of function F , we ar-rive to a given k, such that Ck′

i > Cki , for each k′ > k.

That means, the amount of information accumulatedat station i at iteration k arrives to a value that the opti-mal solution of the MILP formulation (due to the sizeof the time period T and the assumption that once atransmission starts all the information in the stationmust be transmitted) is not able to reduce it. Thus,station i became isolated from outside of the network.The entry “−” for k and τ means the solution becameperiodic for a subset of stations but with the presenceof isolated stations. Likewise, the entry “∗” for k andτ means after the limit of 30 iterations no periodic-ity was achieved. From the results in Table 1, we canconclude that the formulation maximizing the aver-age satisfaction of stations is more suitable to achieveperiodicity and with smaller values of k.

Table 1: Results obtained when the periodicity of the solu-tions obtained by the ILP formulations is studied.

|V| d TFO1 FO2 FO3

k τ #isol k τ #isol k τ #isol6 0.50 28 – – 1 – – 1 – – 1

– – 1 – – 1 – – 17 5 0 7 2 0 5 3 0– – 1 – – 1 – – 14 2 0 3 2 0 3 1 0

6 0.50 30 2 3 0 3 3 0 2 3 0– – 1 – – 1 – – 12 1 0 5 1 0 2 5 06 4 0 – – 1 6 3 04 1 0 3 2 0 5 1 0

8 0.53 32 – – 1 – – 1 – – 1* * 0 * * 0 5 3 1* * 0 * * 0 * * 03 2 0 7 4 0 5 2 0

12 4 0 * * 0 1 2 0

5 CONCLUSION

In this paper, we have introduced the WT-VRP,a version of the VRP defined on wireless networkswhere information can be delivered without physi-cally visiting a node in the network. A MILP for-mulation was provided to model the WT-VRP. Wediscussed three different criteria to measure the effi-ciency of a solution which results in three differentobjective functions for the MILP formulation. Com-putational experiments were conducted on randominstances. We conclude that the WT-VRP becomesmore difficult as the total amount of information col-lected increases and, as we could expect, as the sizeof the time period increases. Our MILP formulationis computationally easier to solve when the average(over time) of the amount of information collected ismaximized (FO2). However, we saw that the optimalsolutions obtained with this criteria impacts the aver-age satisfaction of the network (FO3). We have alsostudied the periodicity of the solutions obtained byeach different efficiency criteria. In average, the solu-tions obtained with FO3 are more suitable to achieveperiodicity. The results obtained in this work are im-portant to access the hardness of the WT-VRP. In thefuture, it is important to strength the MILP formula-tion proposed here as well as to study heuristic ap-proaches in order to be able to solve larger intances.Other characteristics could be considered in the futureas the existence of multiple vehicles and the possibil-ity of sending only part of the information located ina station. Another direction to continue this researchin to enrich the version of the problem by consideringa limit on the maximum traveling distance betweenconsecutive visits to the base station. This limitationis imposed by the autonomy of the vehicle used in

applications described in (Vieira et al., 2015; Mal-owidzki et al., 2016).

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Tabl

e2:

Res

ults

obta

ined

onth

ese

tofi

nsta

nces

with

n=

6,8

and

T=

24,2

8.

|V|

dT

∑r j

TC

puTi

me

Gap

Inf.

colle

cted

Av.V

ehic

leAv

.Sat

isfa

ctio

nFO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

36

0,5

2431

21,

641,

771,

840

00

6250

4428

,33

29,1

616

,41

0,24

0,16

0,25

264

3,21

6,8

3,26

00

012

912

912

965

,45

65,4

565

,45

0,5

0,5

0,5

408

2,83

4,14

2,89

00

011

111

111

157

,45

59,4

559

,45

0,32

0,32

0,32

432

26,7

59,

375,

870

00

9894

9447

,08

52,5

852

,58

0,28

0,29

0,29

312

3,39

3,91

3,9

00

014

013

096

68,3

371

,08

50,7

50,

350,

340,

428

364

4,01

3,6

3,5

00

013

413

413

276

,21

76,2

171

,50,

270,

270,

3930

834

67,4

849

,51

00

016

115

716

180

,14

83,6

482

,71

0,55

0,52

0,55

476

78,4

255

,88

76,9

50

00

174

166

170

91,5

793

,78

91,3

50,

40,

370,

4250

418

7,75

77,6

312

5,07

00

017

617

617

293

,14

93,1

480

,71

0,28

0,28

0,36

364

59,6

572

,37

80,2

90

00

193

193

184

100,

7810

0,78

92,7

10,

460,

460,

50,

7224

384

3,55

3,18

3,49

00

011

011

096

64,1

664

,16

540,

250,

250,

2740

82,

792,

452,

590

00

104

104

104

56,6

656

,66

56,6

60,

310,

310,

3131

22,

942,

332,

60

00

6756

4827

,79

3021

0,23

0,11

0,25

240

57,6

119

,91

68,7

70

00

141

141

136

6565

62,7

0,57

0,57

0,58

336

20,3

414

,48

40,2

90

00

178

176

178

90,1

695

,590

,16

0,48

0,4

0,48

2844

851

,46

10,9

211

,23

00

016

416

416

488

,28

88,2

888

,28

0,4

0,4

0,4

476

6,78

7,69

6,46

00

017

217

217

287

,57

87,5

787

,57

0,38

0,38

0,38

364

13,2

211

,14

11,8

00

010

510

598

52,1

753

,46

42,7

80,

20,

210,

3528

0—

898

—4,

250

16,9

918

618

617

489

,89

89,8

978

,53

0,67

0,67

0,64

392

2988

,72

332,

88—

00

22,9

524

523

623

912

3,89

127,

9612

0,89

0,56

0,48

0,56

369,

2018

6,79

80,3

027

,80

4,25

019

,97

142,

513

9,5

135,

172

,70

74,1

968

,31

0,39

0,36

0,41

80,

5324

672

7,97

7,48

10,2

60

00

144

144

142

85,1

685

,16

77,1

60,

190,

190,

2248

028

,63

2160

,62

00

015

715

714

282

,29

82,2

970

,79

0,3

0,3

0,3

528

296,

0183

,89

202,

910

00

184

174

158

8992

77,0

80,

350,

320,

3645

65,

685,

314,

920

00

120

120

117

7070

64,1

20,

170,

170,

2348

032

,85

20,9

827

,27

00

017

916

717

988

,54

90,7

988

,54

0,35

0,29

0,35

2878

417

7,97

177,

7918

1,17

00

022

021

219

611

912

1,85

104,

250,

280,

240,

2956

016

67,6

242

2,09

532,

580

00

244

221

244

118,

1412

0,1

118,

140,

410,

350,

4161

6—

——

23,4

215

,27

13,3

923

623

621

211

8,67

118,

6710

50,

450,

450,

4253

213

9,86

85,3

930

5,35

00

021

219

921

210

3,78

108,

2510

3,78

0,33

0,31

0,33

560

—11

60,5

522

25,2

530

,89

00

259

245

259

124,

3213

0,28

124,

320,

440,

380,

440,

7124

528

—11

02,7

2—

17,5

30

14,0

322

021

420

897

,511

4,41

105,

910,

350,

410,

4155

29,

417

,21

9,58

00

013

211

612

370

,25

71,6

651

,62

0,24

0,22

0,24

504

44,7

542

,79

50,3

60

00

169

152

169

83,0

483

,33

83,0

40,

360,

330,

3640

811

,51

11,5

424

,65

00

012

111

411

857

,91

63,5

52,5

0,3

0,29

0,3

480

137,

7891

,97

339,

780

00

169

169

168

81,0

485

,04

72,9

10,

350,

350,

3528

616

——

—17

,11

32,6

529

,77

300

294

254

123

146,

8912

3,85

0,43

0,4

0,43

644

—93

3—

16,3

60

29,5

422

422

422

411

4,85

114,

8511

4,85

0,32

0,32

0,32

588

—18

89,3

9—

14,1

30

16,1

213

199

213

105,

8910

8,5

105,

890,

390,

330,

3947

6—

602,

0514

58,9

26,2

50

016

616

116

672

,07

86,8

973

,57

0,37

0,36

0,37

560

—31

24,4

1—

17,4

70

5,83

217

205

215

98,9

610

7,32

93,1

70,

390,

360,

3855

1,20

213,

3454

4,42

388,

1120

,40

23,9

618

,11

194,

318

6,15

185,

9595

,17

100,

0990

,52

0,34

0,32

0,35

Tabl

e3:

Res

ults

obta

ined

onth

ese

tofi

nsta

nces

with

n=

8,10

,12

and

T=

24,2

8.

|V|

dT

∑r j

TC

puTi

me

Gap

Inf.

colle

cted

Av.V

ehic

leAv

.Sat

isfa

ctio

nFO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

3FO

1FO

2FO

38

0,53

2467

23,

153,

43,

080

00

126

117

9059

,16

61,1

246

,25

0,16

0,15

0,17

480

13,6

64,

4413

,96

00

015

715

715

782

,29

82,2

982

,29

0,3

0,3

0,3

528

59,3

13,4

941

,53

00

018

417

415

889

9277

,08

0,35

0,32

0,36

456

2,55

3,12

2,69

00

098

9890

50,4

552

,547

,25

0,16

0,16

0,17

480

9,02

11,0

118

,89

00

017

916

714

388

,54

90,7

966

,04

0,35

0,29

0,35

2878

426

,68

1374

00

017

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