A Multicommodity Formulation for Routing in Healthcare Wireless

Body Area Networks

Pablo Adasme

, Abdel Lisser

and Chuan Xu

Departamento de Ingenier

ıa El

ectrica, Universidad de Santiago de Chile, Avenida Ecuador 3769 Santiago, Chile

Laboratoire de Recherche en Informatique, Universit´e Paris-Sud XI, Bˆatiment 650, 91405 Orsay Cedex, France

Keywords:

Healthcare wireless Body Area Networks, Multicommodity Netﬂow Formulation, Variable Neighborhood

Search Metaheuristic.

Abstract:

In this paper, we propose a minmax multicommodity netﬂow model for routing in healthcare wireless body

area networks (WBAN). The model is aimed at minimizing the worst power consumption of each bio-sensor

node placed in the body of a patient plus the total heating costs subject to ﬂow conservation and maximum

capacity energy constraints. The model is formulated as a mixed integer linear program (MILP). Thus, we

propose a variable neighborhood search (VNS) metaheuristic procedure to come up with tight near optimal

solutions. Our preliminary numerical results indicate the VNS approach obtains near optimal solutions with

integrality gaps no larger than 3.5 %. Finally, since the proposed model has two conﬂicting objectives, i.e,

heating costs and worst power consumption, we adopt a weighted sum criteria for each objective in order to

analyze the behavior of the model.

1 INTRODUCTION

Wireless sensor networks (WSN) have become one

of the most promising technologies to enhance qual-

ity of life around the globe. So far, there are appli-

cations for entertainment, testing weather conditions,

traveling location, retail industries, military logistics,

healthcare systems, and so on. Regarding healthcare

systems, a major concern is to address the problem of

pervasive preventive monitoring systems. Specially

for elderly population whose growth is directly pro-

portional to the development of countries (Kinsella

and Phillips, 2005). Moreover, this technology would

allow not only the elderly and chronically ill peo-

ple to be monitored, but also to provide high qual-

ity care services for little children in situations where

both parents have to work. It would also help peo-

ple living in rural areas where reaching hospitals and

medical centers is more difﬁcult. A major research

challenge in healthcare monitoring systems so far is

the development of new applications by integrating

efﬁcient wireless body area networks (WBAN) with

electronic devices such as mobile phones, laptop com-

puters, desk computers or tablets (Schmidt and Laer-

hoven, 2001). For example, being at home, a wire-

less personal area network (WPAN) may provide res-

idents and caregivers with continuous medical mon-

itoring, control of home appliances, and emergency

communication (Stanford, 2002; McFadden and In-

dulska, 2004). A WBAN is composed of tiny bio-

logical sensors (bio-sensors) which are placed in the

body of a person in order to remotely monitor health-

care status conditions such as fever, blood pressure,

body temperature, heart rate, blood glucose concen-

tration, among many others. Unlike typical WSN,

WBAN suffer from very limited energy resources and

hence preserving the energy of the nodes is of great

importance. Additionally, an extremely low transmit

power per node is required in order to minimize in-

terference and to cope with health concerns such as

avoiding tissue heating of skin on patients. One possi-

ble approach to minimize power consumption as well

as tissue heating of skin problems is by improving

the performance of routing protocols. So far, there

have been proposed some algorithmic approaches to

control bio-effects for WBAN. The authors in (Tang

et al., 2005) propose a Thermal Aware Routing Algo-

rithm (TARA) that balances the communication over

the sensor nodes in order to route data away from high

temperatures. The algorithm achieves better energy

efﬁciency levels and low temperatures, however it re-

quires that all nodes have complete knowledge about

the temperatures of all remaining nodes in the net-

work. Another attempt is a protocol known as Any-

409

Adasme P., Lisser A. and Xu C..

A Multicommodity Formulation for Routing in Healthcare Wireless Body Area Networks.

DOI: 10.5220/0004831704090416

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 409-416

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

body (Watteyne et al., 2007). The underlying idea

of Anybody is to form clusters and a backbone net-

work with selected cluster heads in order to reduce the

number of direct transmissions to the sink node. This

algorithm also achieves energy savings, but it does

not consider other aspects such as reliability of mes-

sages for example. In (Fang and Dutkiewicz, 2009),

the authors propose an energy efﬁcient medium ac-

cess control (MAC) protocol referred to as BodyMAC

which uses ﬂexible bandwidth allocation to improve

node energy efﬁciency. Besides, it includes a new ef-

ﬁcient sleep mode so as to reduce the idle listening

duration. In (Kwak et al., 2009), the authors com-

pare and analyze different protocols from WBAN re-

quirements to energy efﬁciency whereas in (Huang

et al., 2010) the authors propose a weighted random

value protocol for multiuser WBANs (WRAP). Other

efforts consider explicit mathematical programming

formulations in order to efﬁciently design optimal

routing protocols in WBANs (Abouzar et al., 2011;

Ababneh et al., 2012a; Ababneh et al., 2012b; Elias

and Mehaoua, 2012; Yan et al., 2012; Awad et al.,

2013). In a WBAN routing protocols must have self-

conﬁguration features and must be capable of ﬁnding

the best route for communication in order to increase

delivery insurance and decrease energy consumption

between nodes that comprise the network.

WBAN is an emerging research ﬁeld where new

routing protocols are mandatorily needed to minimize

power consumption at each node while simultane-

ously maximizing the lifetime of each node in the

network. In this paper, we present a minmax mul-

ticommodity netﬂow formulation to optimally route

sensed information by nodes in a WBAN. The model

minimizes the worst power consumption of each

bio-sensor plus the total heating costs produced by

the nodes subject to ﬂow conservation and maxi-

mum power available constraints for each node. The

model is formulated as a mixed integer linear program

(MILP) and thus, we propose a variable neighbor-

hood search (VNS) metaheuristic procedure to com-

pute tight near optimal solutions.

The paper is organized as follows. Section 2

presents the multicommodity netﬂow formulation un-

der study. In section 3, we present the VNS proce-

dure to compute near optimal solutions. In section 4,

we provide preliminary numerical results for the VNS

approach when compared to the optimal solution of

the problem. Since the proposed model has two con-

ﬂicting objectives, i.e, heating costs and worst power

consumption, here we also consider a weighted sum

criteria for each objective and numerically compare

the model behavior. Finally, in section 5, we conclude

the paper.

2 PROBLEM FORMULATION

We model a ﬁxed WBAN by the means of a graph

G = (N,A), where N denotes the set of sensor (bio-

sensor) nodes and A is a set of directed arcs. The as-

sumption of directed arcs is valid for WBANs since

before any message is transmitted, the route between

the source and the destination can be established

using Ad-hoc On-demand Distance-Vector routing

(AODV) protocols (Perkins and Royer, 1999). With-

out loss of generality, we assume that every node has a

ﬁxed initial power capacity E ∈ R

. The set of nodes

N is composed of a subset of source nodes N

which

sense and collect the data to be transmitted, a set of

intermediate transmitters N

and a set of sink nodes

where all data is received. For each node j ∈ N

we deﬁne the sets δ

−

( j ) = {i ∈ N : (i, j) ∈ A} and

( j ) = {i ∈ N : ( j,i) ∈ A}. We denote by C the set

of commodities to be transmitted where each com-

modity c ∈ C consists of routing D

packets from a

source node i ∈ N

to a destination node j ∈ N

. Let

i, j

denote the unitary energy needed for transmission

of packets on arc (i, j) ∈ A and deﬁne the total energy

consumption of node j ∈ N as

∑

c∈C

∑

i∈δ

−

( j)

i, j

where D

i, j

is the number of packets of commodity c

transmitted on arc (i, j). Note that this amount of en-

ergy is computed under the assumption that the trans-

mission energy requirement is negligible compared

to the energy required for receiving packets at each

node. This is valid assumption since an extremely

low transmit power per node is required in short range

ultra-wide band in WBANs and thus the effort is con-

siderably higher when the nodes are receiving pack-

ets (Shi et al., 2011). Moreover, this allows signiﬁ-

cant energy savings when using network coding tech-

niques with the objective of providing reliability un-

der low-energy constraints (Arrobo and Gitlin, 2011;

Shi et al., 2011). We consider the following multi-

commodity netﬂow formulation denoted hereafter by

min

f ,x

{

∑

i∈N

+ max

j∈N

∑

c∈C

∑

i∈δ

−

( j)

i, j

}

(1)

s.t.

∑

j∈δ

(i)

i, j

−

∑

j∈δ

−

(i)

j,i

= b

∀i ∈ N; c ∈ C (2)

∑

c∈C

∑

i∈δ

−

( j)

i, j

≤ Ex

,∀ j ∈ N (3)

−x

≤ b

≤ x

, ∀i ∈ N; c ∈ C (4)

i, j

∈ [0,1],∀(i, j) ∈ A; c ∈ C (5)

∈ {0,1},∀i ∈ N (6)

where the ﬂow variables f

i, j

represent the portion of

commodity c ∈ C to be transmitted on an arc (i, j) ∈ A.

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The binary variables {x

,i ∈ N} are used to decide

whether node i ∈ N will be active or not when trans-

mitting packets through the network. The objective

function in (1) is to minimize the total heating costs

,i ∈ N} produced by bio-sensors which are placed

in the body of a patient plus the worst case power con-

sumption of each active node in the network. The lat-

ter is a crucial aspect in a WSN since by deﬁnition, its

lifetime is equal to the minimum lifetime of all nodes

in the network (Deb, 2002; Jourdan and de Weck,

2004). In other words, the network lifetime ends as

soon as any node runs out of its battery. Let b

equal to 1 if node i ∈ N

, or be equal to -1 if node

i ∈ N

, and zero otherwise. Constraint (2) are ﬂow

conservation constraints for each node i ∈ N and for

each commodity

∈

while constraint (3) imposes

the condition that each node has a maximum avail-

able power to receive packets in the network. Note

that this constraint is forced to be equal to zero when

its respective node is set to an inactive state condition.

Constraint (4) imposes the condition that all source

and sink nodes should always be active, otherwise the

network can not sense or relay the data collected at

the sink node. Finally, constraints (5)-(6) are the do-

main constraints. Note that model P

can be easily

converted into a mixed integer linear programming

(MILP) problem by introducing an upper bounding

variable z instead of using the max term in its objec-

tive function as follows.

min

f ,x

z +

∑

i∈N

s.t. z ≥

∑

c∈C

∑

i∈δ

−

( j)

i, j

, ∀ j ∈ N

∑

j∈δ

(i)

i, j

−

∑

j∈δ

−

(i)

j,i

= b

, ∀i ∈ N; c ∈ C

∑

c∈C

∑

i∈δ

−

( j)

i, j

≤ Ex

, ∀ j ∈ N

−x

≤ b

≤ x

, ∀i ∈ N; c ∈ C

i, j

∈ [0,1],∀(i, j) ∈ A; c ∈ C

∈ {0,1},∀i ∈ N

We remark that model P

could be considered as part

of any existing WBAN MAC protocol (See references

in section 1) as it provides an optimal routing strat-

egy. However, it does not consider other technical as-

pects such as broadcasting control ﬂows and organi-

zation of the network. The routing strategy is manda-

tory in WBANs as it allows signiﬁcant power savings

when transmitting sensed data through the network

(Ababneh et al., 2012b; Elias and Mehaoua, 2012;

Yan et al., 2012; Awad et al., 2013). For a more gen-

eral, comprehensive and technological perspective re-

garding WBANs protocols, we refer the reader to the

works in (Ullah et al., 2010; Ullah et al., 2012).

In the next section, we propose a variable neigh-

borhood search metaheuristic approach to compute

near optimal solutions for P

3 THE VNS APPROACH

Metaheuristics are simple algorithmic procedures

commonly used to ﬁnd near optimal (suboptimal) so-

lutions for combinatorial optimization problems. In

practice, they have proven to be highly effective when

solving many of these hard problems (Glover and

Kochenberger, 2003). Especially when the dimen-

sions of the problem increase rapidly which is of-

ten the case in real world applications and where no

solver is available to solve these problems to opti-

mality. Perhaps, the most frequently utilized meta-

heuristics approaches are genetic algorithms, tabu

search, ant colony system, particle swarm optimiza-

tion, variable neighborhood search, simulated anneal-

ing, among others. For a more detailed explanation on

how these metaheuristics approaches work, we refer

the reader to the book in (Glover and Kochenberger,

2003). In principle, a genetic algorithm or a tabu

search approach would also serve to compute feasi-

ble solutions for our proposed multicommodity ﬂow

formulation in a straightforwardly manner. In this pa-

per, we choose VNS mainly due to its simplicity and

low memory requirements. In particular, we adopt a

reduced VNS strategy which drops the local search

phase of the basic VNS algorithm as it is the most

time consuming step (Hansen and Mladenovic, 2001;

Hansen et al., 2001). In order to compute feasible so-

lutions for problem P

using a VNS approach, we ob-

serve that for any ﬁxed assignment of vector x = ¯x in

, the problem reduces to solve the following linear

programming problem

min

s.t. z ≥

∑

c∈C

∑

i∈δ

−

( j)

i, j

, ∀ j ∈ N

∑

j∈δ

(i)

i, j

−

∑

j∈δ

−

(i)

j,i

= b

, ∀i ∈ N; c ∈ C

∑

c∈C

∑

i∈δ

−

( j)

i, j

≤ E ¯x

, ∀ j ∈ N

i, j

∈ [0,1],∀(i, j) ∈ A; c ∈ C

There are 2

|N|−|N

|−|N

feasible assignments for vector

x. It is obvious that some of them are not feasible as

they might turn problem

infeasible. We propose

a VNS approach to compute feasible solutions for P

by randomly generating these binary vectors.

VNS is a recently proposed metaheuristic ap-

proach (Hansen and Mladenovic, 2001) that uses the

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Input: A problem instance of P

Output: A feasible solution ( ˜x,

f , ˜v) for P

Step 0:

Time ← 0; H ← 1;

count ← 0; x

← 0,∀i ∈ N \ N

∪ N

← 1,∀i ∈ N

∪ N

Step 1:

For st = 2 to St

r ← min(a

,i ∈ N

)

← 1

end

Solve the linear problem

Let ( ¯x,

f , ¯v) be the optimal solution of

with

objective function value ¯v

¯v ← ¯v +

∑

i∈N

¯x

( ˜x,

f , ˜v) ← ( ¯x,

f , ¯v)

Step 2:

while (Time ≤ maxTime)

For j = 1 to H

choose randomly i

′

∈ N

if (x

′

= 0) x

′

← 1

else x

′

← 0

end if

end for

Solve the linear problem

Let ( ¯x,

f , ¯v) be the optimal solution of

with

objective function value ¯v

¯v ← ¯v +

∑

i∈N

¯x

Let ( ¯x,

f , ¯v) be the new solution found for P

with objective value function ¯v

if ( ¯v < ˜v)

H ← 1; ( ˜x,

f , ˜v) ← ( ¯x,

f , ¯v)

Time ← 0; count ← 0

else

Keep previous solution

count ← count + 1

if (H ≤ |N

|) and (count > η)

H ← H + 1; count ← 0

end if

end while

Figure 1: VNS Algorithm.

idea of neighborhood change during the descent to-

ward local optima and to avoid the valleys that contain

them. We deﬁne the neighborhood structure Ng(x) for

as the set of neighbor solutions x

′

in P

at a distance

“h” from x

where the distance “h” corresponds to the Ham-

ming distance between the binary vectors x and x

′

respectively. In our numerical tests, without loss of

generality, we adopt a grid layout conﬁguration as de-

picted in Figure 2 where a directed graph shows that

bio-sensor nodes are placed in each stage. In partic-

ular, in stage 1 we have the source nodes, from stage

2 to the ﬁnal stage we have the intermediate nodes.

Finally, we have only one node acting as a sink node

to receive all sensed and collected information sent by

the source nodes through the network. We denote by

the number of nodes placed at stage st ∈ {1,..,St}

where St refers to the ﬁnal stage in Figure 2.

Sink node

Source

nodes

Final Stage

Number of

nodes per

stage

Stage 2Stage 1

Intermediate nodes

Figure 2: WBAN with grid layout conﬁguration.

We further assume that all nodes at each stage are

consecutively connected with those of the next stage

(Arrobo and Gitlin, 2011). We also assume that the

number of nodes at each stage is the same, i.e., N

··· = N

. The grid layout conﬁguration is a valid

assumption in WBANs as it provides more reliable

communications when using cooperative network and

diversity coding transmission schemes with enhanced

throughput (Arrobo and Gitlin, 2011). There are other

network conﬁgurations such as star, tree, mesh (Ullah

et al., 2012) and grid topologies in the literature (Ar-

robo and Gitlin, 2011). The most common topology is

a star one where the nodes are connected to a central

coordinator in star manner (Ullah et al., 2012). How-

ever, the star conﬁguration follows a single hop strat-

egy which is not always the best choice. In (Reusens

et al., 2009), the authors discuss about energy efﬁ-

cient topology designs for WBANs. They consider a

tree network topology and discuss on the energy sav-

ings when using single hop and multi hop strategies.

They conclude that the distance between nodes plays

an important role and that both single hop or multi

hop strategies achieve energy savings under different

conditions (Reusens et al., 2009).

The VNS approach we propose is presented in

Figure 1. It receives an instance of problem P

as in-

put and provides a tight feasible solution for it. We

denote by ( ¯x,

f , ¯v) the ﬁnal solution obtained with the

algorithm where ¯v represents the objective function

value. The algorithm is simple and works as follows.

In Step 0, we initialize all the required variables.

Then, in Step 1 we obtain an initial feasible assign-

ment for vector x = ¯x by simply setting ¯x

= 1 such

that r = min(a

,i ∈ N

) is the minimum value for that

particular stage. This allows solving

and obtaining

an initial feasible or infeasible solution ( ˜x,

f , ˜v) for P

that we keep. During the execution of the while loop

in the VNS algorithm, if for any x = ¯x model

is in-

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412

feasible, then the solution is discarded and not consid-

ered as a valid solution. Next, the algorithm performs

a variable neighborhood search by randomly assign-

ing binary values in H ≤ |N

| positions of vector x

where these positions belong to the set N

. Initially,

H ← 1 while it is increased in one unit when there

is no improvement after new “η” solutions have been

evaluated. On the other hand, if a new current solution

is better than the best found so far, then H ← 1, the

new solution is recorded and the process goes on. The

whole process is repeated until the cpu time variable

“Time” is less than or equal to the maximum available

“maxTime”. Note we reset “Time ← 0” when a new

better solution is found. This gives the possibility to

search other “maxTime” units of time with the hope

of ﬁnding better solutions.

4 NUMERICAL RESULTS

In this section, we ﬁrst present preliminary numeri-

cal results for the proposed VNS approach using only

one sample for the input data of the instances. Sub-

sequently, we provide preliminary numerical compar-

isons for P

while adopting a weighted sum criteria

for the objective function of P

in order to analyze the

behavior of the model. Finally, we compute average

numerical results.

4.1 Numerical Results for the VNS

Approach

In order to present preliminary numerical results for

problem P

using the proposed VNS algorithm, the in-

put data is randomly generated as follows. The entries

in matrix (e

i, j

) are uniformly drawn from [0, 1] while

the heating costs {a

,i ∈ N} and packets {D

,c ∈ C}

are uniformly distributed in [0,10]. The maximum

available energy for each node is set equal to E =

0.4 ∗

∑

i∈N

i,1

∗

∑

c∈C

|C|

. The value of η in the VNS

algorithm is calibrated to η = 20. We set the maxi-

mum number of commodities be equal to |C| = |N

i.e. we assume that each source node can only sense

one type of commodity. This is a valid assumption as

bio-sensors are usually designed for sensing special-

ized information in a WBAN. Finally, we set the pa-

rameter maxTime = 100. A Matlab program is imple-

mented using CPLEX 12 to solve problem P

, its lin-

ear programming (LP) relaxation, and each LP relax-

ation

within each iteration of the VNS algorithm.

The numerical experiments have been carried out on

a Pentium IV, 1 GHz with 2 GoBytes of RAM under

windows XP. In Table 1, column 1 shows the number

of nodes considered in each instance. Commonly, the

maximum number of nodes in WBANs can be up to

256 nodes (Fang and Dutkiewicz, 2009). Columns 2

and 3 show the number of stages and number of nodes

per stage for each of the instances. Columns 4 and 5

provide the optimal solution of P

and the cpu time in

seconds CPLEX needs to get that solution. Columns

6 and 7 provide the optimal solutions for the LP re-

laxation of P

and the CPLEX cpu time in seconds as

well. Similarly, columns 8 and 9 show the best solu-

tion found with the VNS approach and its cpu time

in seconds. Finally, columns 10, 11 and 12 give the

gaps for the LP relaxation, the gaps obtained using

the initial solution of the VNS approach and the gaps

for the best solution found with the VNS approach,

respectively.

The gaps are computed as Gap

(

−LP

)

∗ 100

for the LP case, Gap

Ini

V NS

(

V NS

ini

−P

)

∗ 100 for the

initial solution obtained with the VNS algorithm and

Gap

V NS

(

V NS−P

)

∗ 100 for the best solution found

with VNS, respectively. The numerical results pre-

sented in Table 1 are computed using only one sam-

ple for the input data of each instance. From Table

1, we mainly observe that the gaps obtained with the

VNS algorithm are near optimal for all the instances

we test, e.g. not larger than 3 % from the optimal so-

lution of the problem. Regarding the cpu times, we

observe that the VNS approach requires more time

when the number of stages is less than the number

of nodes per stage. Furthermore, this cpu time is even

larger than the time required by CPLEX. This is easy

to check for instances with 61 nodes in Table 1, for

example. From our preliminary numerical results, we

observe that this is mainly caused by the fact that the

VNS approach needs to solve many linear programs

in this case. Ultimately, we observe that the gaps

obtained when using the initial solutions found with

VNS are not very tight which shows somehow the ef-

fectiveness of the VNS approach. On the other hand,

when the number of stages is larger than the number

of nodes per stage, we observe that the VNS approach

is signiﬁcantly faster than CPLEX. Moreover, in this

case we see that the initial solution found with the

proposed algorithm is very tight and in some cases

optimal, e.g. this is the case for instances with 49 and

61 nodes. In particular, we see that the cpu time re-

quired by CPLEX becomes prohibitive for some of

these instances. Finally, we observe that the gaps ob-

tained with the LP relaxation of P

are far from the

optimal solution of the problem.

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413

Table 1: Numerical results for the VNS approach.

N # stages nodes/stage P

cpu(s) LP

cpu(s) VNS cpu(s) Gap

% Gap

Ini

V NS

% Gap

V NS

13 3 4 31.5214 1.8120 24.6043 0.7340 31.5214 0.3910 21.9442 13.9145 0

46 3 15 89.0458 2.5000 81.0329 1.9060 89.0458 67.4370 8.9986 26.4712 0

61 3 20 141.9789 5.7810 125.7967 4.3280 142.3221 1044.1710 11.3976 17.6738 0.2418

17 4 4 53.2150 0.8910 32.0876 0.7500 53.2150 0.0320 39.7019 2.1233 0

61 4 15 99.6237 10.9060 91.0988 3.2190 99.6858 222.8290 8.5571 22.8606 0.0623

21 5 4 51.7760 0.7660 46.6627 0.7340 51.7760 0.0010 9.8757 0 0

41 5 8 73.5483 16.3120 50.3013 1.2030 73.5483 135.1730 31.6078 17.6137 0

51 5 10 63.5951 12.7030 50.8429 1.7970 65.2285 208.0150 20.0522 38.6343 2.5685

33 8 4 56.3391 3.7350 27.9300 0.8280 56.7766 2.1240 50.4251 1.4147 0.7767

49 8 6 60.9010 24.8910 38.4090 1.3440 62.5080 0.1570 36.9320 2.6387 2.6387

41 10 4 55.9875 12.3280 26.8474 0.9220 55.9875 416.3600 52.0474 9.8009 0

61 10 6 83.3481 792.2180 44.5579 1.7500 84.4382 25.8290 46.5400 9.8488 1.3078

49 12 4 76.0640 18.1090 31.3246 1.6250 76.0640 0.2354 58.8180 0 0

73 12 6 83.1798 17204.7970 40.4708 2.2970 84.9211 0.6880 51.3455 4.4978 2.0934

61 15 4 70.8630 31.1720 33.4660 1.0630 70.8630 13.9380 52.7736 0 0

91 15 6 75.5382 2096.2040 42.1599 2.5940 77.6941 0.3265 44.1873 4.6672 2.8540

4.2 Weighted Objective Function

In order to explore the behavior of model P

when

the two conﬂicting objectives have different degree of

importance, we adopt a weighted objective function

criteria and write the objective function in P

min

{ f ,x}

αz + (1 − α)

∑

i∈N

(7)

where α ∈ [0,1]. From a practical point view, the

weighted objective function in (7) provides an alterna-

tive way to handle the trade-off between the total heat

generated on patients with more delicate skin versus

power energy savings in order to maximize the net-

work lifetime. This would allow to avoid possible

hazardous damages on patients.

Table 2: Numerical results for weighted objectives.

α P

cpu(s) LP

cpu(s) Gap

Inst.1: N = 13, #stages = 3 and nodes/stage = 4

0 30.8022 0.5780 24.6294 0.4220 20.0400

0.25 24.0651 0.3900 20.1701 0.3600 16.1853

0.5 18.1543 0.4530 14.7102 0.3440 18.9714

0.75 10.5910 0.3900 9.1875 0.3600 13.2519

1 3.5855 0.3750 3.5855 0.5470 0

Inst.2: N = 41, #stages = 5 and nodes/stage = 8

0 37.9941 1.1410 35.4446 0.8910 6.7103

0.25 34.7115 2.3280 27.6292 0.9530 20.4034

0.5 28.3399 10.2190 18.9793 0.9530 33.0298

0.75 17.7986 21.2970 10.2668 0.9370 42.3172

1 1.5070 1.1570 1.5070 0.8900 0

Inst.3: N = 49, #stages = 8 and nodes/stage = 6

0 49.8155 1.5630 37.1658 0.9370 25.3931

0.25 41.4989 3.8130 29.4254 0.9690 29.0936

0.5 32.2754 17.5310 21.2694 0.9690 34.1004

0.75 20.4248 15.7180 13.1134 0.9540 35.7968

1 4.9574 1.1560 4.9574 1.1400 0

In Table 2, we present preliminary numerical re-

sults for different values of parameter α and for three

instances having different number of nodes, stages

and nodes per stage. More precisely, in column 1 we

give the value of α. In columns 2, 3, and 4, 5 we

present the optimal function value of P

(resp. LP

)

and their cpu time in seconds that CPLEX requires

to obtain that solution, respectively. Finally, column

6 shows the gaps for the LP relaxation we compute

exactly as in Table 1. Without loss of generality, the

input data is randomly generated exactly as for Table1

as well.

From Table 2, we mainly observe that the gaps

of the LP relaxation tends to zero when the value of

α → 1. This means that solving the LP relaxation of

, in this case, sufﬁces to obtain the optimal solution

of the problem. On the opposite, when 0 ≤ α < 1,

the gaps of its LP relaxation deteriorates considerably

which turns the problem more difﬁcult to solve.

In order to provide more insight regarding our

VNS approach, we further present average numerical

results for the instances presented in Table 1. These

results are presented in Table 3 and the column infor-

mation is exactly the same as for Table 1. We generate

50 samples for the input data of the instances in rows

1-13 while for the instances in rows 14-16 we use only

10 samples to compute the averages as their cpu times

become highly prohibitive. In particular, we arbitrar-

ily set the maximum time for CPLEX to solve these

instances be at most 3600 seconds.

From Table 3, we observe similar trends as in Ta-

ble 1 concerning the gaps obtained with VNS. They

are not larger than 3.5 % for all the instances we test

when compared to the optimal solution of the prob-

lem. We also see that the cpu times required by

the VNS approach are larger than those required by

CPLEX for instances in rows 1-9 while for instances

in rows 10-16, CPLEX requires more cpu time. In

particular, the instances in rows 14-16 require a huge

amount of cpu time using CPLEX while the VNS ap-

proach ﬁnds very tight near optimal solutions with

gaps no larger than 2% in less than 25 seconds ap-

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Table 3: Average numerical results for the VNS approach.

N # stages nodes/stage P

cpu(s) LP

cpu(s) VNS cpu(s) Gap

% Gap

Ini

V NS

% Gap

V NS

13 3 4 39.8464 0.4312 32.0752 0.4062 39.8464 0.9370 19.6396 - 0

46 3 15 95.0189 3.4718 80.4871 1.7472 95.7131 110.3312 15.2916 - 0.5546

61 3 20 152.7756 7.9654 129.0012 3.8741 153.9987 477.3697 16.2044 - 1.1142

17 4 4 42.2965 0.4716 32.9802 0.3938 42.2965 9.4488 22.1685 - 0

61 4 15 96.7648 17.4686 80.2268 2.8404 98.8593 252.7788 17.0192 - 2.2219

21 5 4 45.2427 0.6064 32.5676 0.4186 45.2427 26.4560 26.9386 - 0

41 5 8 72.1552 10.3562 52.7956 0.9094 74.8638 36.1282 27.0548 - 3.1426

51 5 10 85.3136 38.3716 65.2315 1.5314 87.1930 42.5282 23.5664 - 2.9688

33 8 4 58.1523 2.5686 31.7155 0.5250 58.7456 10.6092 46.3102 4.0809 1.3087

49 8 6 69.7592 82.4720 46.7561 1.0534 71.0608 17.0594 33.0253 9.5012 1.8658

41 10 4 63.9525 10.1404 37.6166 0.6188 65.2207 7.5086 40.8058 3.7043 2.0128

61 10 6 68.0209 100.3878 40.9167 1.3532 70.3692 47.8566 40.1894 8.8179 3.2827

49 12 4 66.3268 30.0688 29.9523 0.7094 67.0870 0.1336 54.8053 4.1534 1.7767

73 12 6 75.0030 3456.5126 44.3070 1.8002 76.2619 16.2030 40.9237 5.2012 1.9912

61 15 4 72.5088 1441.3000 35.3752 0.9186 72.7524 12.2218 50.7315 1.0480 0.2773

91 15 6 80.5427 2740.1874 44.6374 2.6750 81.0830 20.6280 44.8002 3.2151 0.8021

-: No feasible initial solution found.

proximately. Another observation is that the initial

solutions obtained with VNS approach are not feasi-

ble for instances in rows 1-8. Which means at least in

one of the samples, the initial solution was infeasible.

Conversely, ﬁnding initials solutions for instances in

rows 9-16 is easier. In general, the gaps of the initial

solutions are not larger than 10%. Finally, the gaps

obtained with the LP relaxation of P

are not tight

when compared to the optimal solution of the prob-

lem. In general, we note that the LP gaps deteriorate

signiﬁcantly when the number of stages is larger than

the number of nodes per stage which is the case for

instances in rows 9-16.

5 CONCLUSIONS

In this paper, we proposed a minmax multicommod-

ity netﬂow formulation to optimally route data pack-

ets in a healthcare wireless body area network. The

aim of the model is to minimize the worst power con-

sumption of each bio-sensor node over the body of a

patient plus the total heating costs subject to ﬂow con-

servation and maximum capacity energy constraints.

The model is formulated as a mixed integer linear pro-

gram. Thus, we proposed a variable neighborhood

search metaheuristic procedure to obtain near optimal

solutions. Preliminary numerical results indicate that

the VNS approach obtains near optimal solutions with

integrality gaps no larger than 3.5%. As future work,

we plan to consider other layout conﬁgurations while

using more realistic data, and new algorithmic pro-

cedures to solve efﬁciently the LP relaxations within

each iteration of the VNS approach.

ACKNOWLEDGEMENTS

The author Pablo Adasme is grateful for the ﬁnan-

cial support given by Conicyt Chilean government

through the Insertion project number: 79100020.

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