An application of membrane computing to humanitarian relief via generalized Nash equilibrium

Vol.:(0123456789)
Jou nal o Memb ane Compu ing
h ps://doi.o g/10.1007/s41965-025-00187-y
RESEARCH PAPER
An applica ion o memb ane compu ing ohumani a ian elie
iagene alized Nash equilib ium
Alejand oLuque‑Ce pa1· Da idO ellana‑Ma ín2,3· MiguelÁ.Gu ié ez‑Na anjo3
Recei ed: 1 Oc obe 2024 / Accep ed: 13 Ma ch 2025
© The Au ho (s) 2025
Abs ac
Na u al and poli ical disas e s, including ea hquakes, hu icanes, and sunamis, bu also mig a ion and e ugees c isis, need
quick and coo dina ed esponses in o de o suppo ulne able popula ions. In such disas e s, nongo e nmen al o ganiza ions
compe e wi h each o he o inancial dona ions, while people who need assis ance su e a lack o coo dina ion, conges ion
in e ms o logis ics, and duplica ion o se ices. F om a heo e ical poin o iew, his p oblem can be o malized as a
gene alized Nash equilib ium (GNE) p oblem. This is a gene aliza ion o he Nash equilib ium p oblem, whe e he agen s’
s a egies a e no ixed bu depend on he o he agen s’ s a egies. In his pape , we show ha memb ane compu ing can
model humani a ian elie as a GNE p oblem. We p opose a amily o P sys ems ha compu e GNE in his con ex , and we
illus a e hei capabili ies wi h Hu icane Ka ina in 2005 as a case s udy.
Keywo ds Memb ane compu ing· Game heo y· Nash equilib ium
1 In oduc ion
Acco ding o [1], he economic cos o damages as a esul o
global na u al disas e s in 2023 was 203.35 billion o USA
dolla s. They include d ough s, loods, ex eme wea he ,
ex eme empe a u es, landslides, d y mass mo emen s,
wild i es, olcanic ac i i y, and ea hquakes. No only na u-
al bu also poli ical disas e s, such as mig a ion and e u-
gee c ises, need quick and coo dina ed esponses o suppo
ulne able popula ions. In such disas e s, nongo e nmen al
o ganiza ions (NGOs) a e he main ac o s o educe su e ing
and mo ali y and suppo quali y o li e. Howe e , se e al
s udies (e.g., [2, 3]) poin ou ha humani a ian aid has no
been success ul due o a lack o coo dina ion, conges ion in
e ms o logis ics, and duplica ion o se ices.
Al hough he e a e huge di e ences be ween humani a -
ian logis ics and comme cial logis ics [4], bo h p oblems can
be o malized in a simila way. In [5], he au ho s compa e
und aising wi h and wi hou an ea ma king op ion using
op imiza ion models. I seems o be he i s s udy whe e
game heo y is conside ed in o de o model he in e ac ion
be ween dono s and humani a ian o ganiza ions.
In [3], he au ho s de elop a gene alized Nash equilib-
ium ne wo k model o pos -disas e humani a ian elie by
nongo e nmen al o ganiza ions which is he s a ing poin
o ou s udy by using Memb ane Compu ing echniques. In
such a pape , he au ho s conside Hu icane Ka ina as a
case s udy, he cos lies disas e in he his o y o he Uni ed
S a es. This hu icane caused huge damage o p ope y and
in as uc u e, le 450,000 people homeless, and ook 1833
li es in Flo ida, Texas, Mississippi, Alabama, and Louisi-
ana. Fo his wo k, we ake he da a o he disas e p o ided
by Nagu ney e al. [3].
F om a heo e ical poin o iew, his p oblem can be o -
malized as a gene alized Nash equilib ium (GNE) p oblem.
This is a gene aliza ion o he Nash equilib ium p oblem,
whe e he agen s’ s a egies a e no ixed bu hey depend
on he o he agen s’ s a egies and i can be conside ed as
a p oblem in he a ea o E olu iona y Game Theo y (EGT)
* Alejand o Luque-Ce pa
luque@chalme s.se
Da id O ellana-Ma ín
[email p o ec ed]
Miguel Á. Gu ié ez-Na anjo
[email p o ec ed]
1 Depa men o Compu e Science andEnginee ing,
Chalme s Uni e si y o Technology, Go henbu g, Sweden
2 Resea ch G oup o Na u al Compu ing, Uni e sidad de
Se illa, Se ille, Spain
3 Depa men o Compu e Science andA i icial In elligence,
Uni e sidad de Se illa, Se ille, Spain
A.Luque-Ce pa e al.
[6]. Beyond he in insic in e es o GNE as a heo e ical
p oblem, i can be applied o a wide ange o applica ions,
such as he con ol o in e ac ing ehicles [7], he in e sec-
ion managemen p oblem [8] o he ene gy ma ke [9]. GNE
(o i s simpli ied e sion, Nash equilib ium) has also p o-
ided a heo e ical model o o he helping ac ions such as
blood dona ions [3, 10], compe i ion o medical supplies
[11], o logis ics o humani a ian se ices [12, 13] among
many o he s.
Memb ane compu ing is a well-known bio-inspi ed
compu ing pa adigm [14, 15], whose de ices, called P
sys ems, ha e been used o success ully model many eal-
li e p oblems, such as he dynamics o he popula ion o
gian panda in cap i i y [16], aul p opaga ion pa hs in
powe sys ems [17] o he ecosys em o some sca enge
bi ds [18] among many o he s. To he bes o ou knowledge,
he i s ime ha a GNE p oblem was simula ed by
memb ane compu ing de ices was in Luque-Ce pa e al.
[19]. This pape ollows he esea ch line s a ed in ha
pape by showing ha he Memb ane Compu ing pa adigm
can be use ul in he simula ion o humani a ian elie . In
opposi ion o ha pape , we p opose a amily o P sys ems
ha compu es GNE on a game ou o he amewo k o
E olu iona y Game Theo y.
The pape is o ganized as ollows: Sec .2 ecalls some
heo e ical aspec s ela ed o how GNE can be used in he
amewo k o humani a ian elie a e a disas e . Sec ion3
gi es some basic in o ma ion abou he P sys em model used
o he simula ion. In Sec .4, he design o ou Memb ane
Compu ing de ice o dealing wi h humani a ian elie
based on he GNE p oblem is p esen ed. Nex , Sec .5
shows, as a case s udy, he use o ou de ice o simula ion
o humani a ian elie wi h he da a o Hu icane Ka ina,
and inally, he pape ends wi h some conclusions and some
u u e esea ch lines.
2 GNE model o pos ‑disas e humani a ian
elie
The gene alized Nash equilib ium p oblem (GNEP), i s
p esen ed by G. Deb eu [20] in 1952, is a gene aliza ion
o he Nash Equilib ium p oblem in which he playe s’
s a egies depend on hei i als’ s a egies. Cons ain s
in he game de ine hese dependencies, and he payo s
ob ained by each playe depend on he s a egies selec ed.
Each playe can choose only one s a egy a a ime,
al hough his choice can be changed o subsequen
ac ions. The guiding idea he e is ha playe s ypically
selec cou ses o ac ion ha p o ide hem wi h g ea e
ewa ds, bu he selec ed s a egy may a y o e ime
o adap o he beha io o he es o he playe s. When
no playe in his si ua ion may change i s s a egy and
whe eas o he agen s keep using hei cu en ones, a Nash
equilib ium is eached [21].
Gene alized Nash equilib ium p oblem can be
conc e ized in o p oblems om many di e en a eas.
Among o he applica ions, in [3] au ho s model he
humani a ian elie pos -disas e p oblem as a GNEP. This
p oblem is hen ans o med in o an op imiza ion p oblem.
The o mula ion is as ollows.
Le us ha e m NGOs ha p o ide disas e elie o n
di e en loca ions. NGOs compe e o p o ide elie i ems
o demand poin s. Le
qij ≥0
be he amoun o i ems ha
he NGO i p o ides o he demand poin j. Le
si
be he
maximum amoun o elie i ems ha he NGO i can
p o ide. Then, he ollowing equa ion mus hold o each
NGO i:
n
∑
j=1
qij ≤s
i
; ha is, he o al o he i ems p o ided
by each NGO canno be highe han he maximum elie
i ems ha such an NGO can p o ide.
Fu he mo e, o each NGO i and each loca ion j, we
can conside he mapping
cij ∶ℝ+
→
ℝ+
, which maps
qij
o
cij(qij)
; ha is, he mapping
cij
ep esen s he cos o
sending he elie o he loca ion j by he NGO i.
Simila ly, each NGO i ge s u ili y associa ed wi h he
elie i ems i sen o he loca ion j. The u ili y o e all
demand poin s is measu ed by
n
∑
j=1
𝛾ijq
ij
, whe e
𝛾ij
is a posi i e
ac o . Each NGO i has a posi i e weigh
𝜔i
associa ed wi h
such a u ili y measu e ha ep esen s he mone iza ion o
his u ili y, and he mone iza ion o an NGO i is measu ed
by
𝜔
i
n
∑
j=1
𝛾ijqij . Finally, each NGO i ecei es unding om
dono s based on media a en ion and he isibili y o NGOs
a loca ion j. These unds a e ep esen ed by
𝛽
i
n
∑
j=1
Pj(q
)
,
whe e q is he ec o o all he i em lows o all he NGOs o
all he demand poin s,
Pj(q)
ep esen s he unds in dona ion
dolla s due o he isibili y o all NGOs a loca ion j, and
𝛽i
ep esen s he p opo ion o o al dona ions collec ed ha is
ecei ed by NGO i.
O he cons ain s a e usually imposed by an au ho i y.
Fo example, he e a e lowe (
dj
) and highe bounds (
dj
)
o he numbe o elie i ems needed a a loca ion j. These
cons ain s a e exp essed as
m
∑
i=1
qij ≥d
j
and
m
∑
i=1
qij ≤d
j
,
which limi he amoun o objec s dis ibu ed by all NGOs
o a speci ic loca ion j. Ano he impo an assump ion is
ha NGOs ha e enough esou ces o comply wi h he
An applica ion o memb ane compu ing ohumani a ian elie iagene alized Nash equilib ium
lowe bounds o elie i ems equi ed by all loca ions, ha
is,
m
∑
i=1
si≥
n
∑
j=1
d
j
. Finally, he op imiza ion p oblem aced
by he NGO i can be exp essed as
which, subjec o he cons ain s indica ed, is equi alen
(Theo em1, [3]) o he ollowing op imiza ion p oblem:
No e (Exis ence and uniqueness) A solu ion
q∗
o he
op imiza ion p oblem (1) is gua an eed o exis when he
objec i e unc ion consis s o con inuous unc ions and he
easible se de ined by he cons ain s is compac . In [3], he
unc ions employed sa is y hese condi ions, and because he
objec i e unc ion is s ic ly con ex, he uniqueness o he
solu ion is also gua an eed.
2.1 Eule me hod applied o hegame heo y
model
To sol e he minimiza ion p oblem (1), we can use a
a ia ional inequali y o mula ion o he p oblem1 and hen
apply he Eule me hod [22, 23] o ob ain closed- o m
exp essions in he
qij
and Lag ange mul iplie s
𝜆
i,𝜆
1
j
,𝜆
2
j
associa ed wi h he cons ain s. Such Eule me hod
app oxima es he solu ion i e a i ely. We ha e he ollowing
exp essions o
q +1
kl
a each i e a ion o
k=1, ..., m
;
l=1, ..., n
:
Minimize
−𝛽i
n
∑
j=1
Pj(q)−𝜔i
n
∑
j=1
𝛾ijqij +
n
∑
j=1
cij(qij
)
(1)
Minimize
−
n
∑
j=1
Pj(q)−
m
∑
i=1
n
∑
j=1
𝜔
i
𝛾
ij
𝛽i
qij +
m
∑
i=1
n
∑
j=1
1
𝛽i
cij(qij
)
(2)
q
+1
kl =max {0, {q
kl +a (
n
∑
l=1(𝜕Pj(q
)
𝜕qkl )+𝜔k𝛾
kl
𝛽k
−1
𝛽
k
𝜕ckl(q
kl)
𝜕q
kl
−𝜆
k+𝜆1
l
−𝜆2
l
)}}
(3)
𝜆
k=max
{
0, 𝜆
k+a
(
−sk+
n
∑
l=1
q
kl
)}
(4)
𝜆
1
l
+1=max
{
0, 𝜆1
l
+a
(
−
m
∑
k=1
q
kl +dl
)}
(5)
𝜆
2
l
+1=max
{
0, 𝜆2
l
+a
(
−dl+
m
∑
k=1
q
kl
)}
whe e he sequence
{a }
mus sa is y
a >0, a
→
0
y
∞
∑
=0
a
=∞
o gua an ee he con e gence o he i e a i e
scheme [22].
When designing he P sys em ha will model hese
equa ions, he main p oblem o sol e is dealing wi h he
wo pa ial de i a i es in Eq.2. The disc e e encoding o he
in o ma ion o P sys ems as mul ise s o objec s is an added
di icul y in o de o deal wi h de i a i es, he e o e, we ha e
made he ollowing decisions in he design o he P sys em:
• The unc ions
Pj(q)
, which ep esen s he unds in
dona ion dolla s due o he isibili y o all NGOs a
loca ion j, in many cases can be exp essed as
wi h
kj>0
, hus
and
This usually gi es he e m
a ious o de s o magni ude lowe han he o he e ms
on Eq.2. Because o his, we ha e decided o igno e i . In
ac , in Sec .5, we expe imen ally show ha he impac
de i ed om his decision is low.
• Simila ly, he cos unc ions a e o he o m
wi h
akl,bkl >0
, so
𝜕
ckl(q
kl)
𝜕
q
kl
=2a2
klqkl +2aklb
kl
. Finally, we
can ew i e hen Eq.2 as:
We no e ha he condi ions ha gua an ee he exis ence and
uniqueness o he solu ion ha e no been comp omised. In
Sec .4 a P sys em ha simula es he algo i hm de ined by
Eqs.3, 4, 5 and 6 is designed.
P
j(q)=kj
√
√
√
√
m
∑
i=1
qij
𝜕
Pj
(
q
)
𝜕qkl
=0 i j≠
l
𝜕
Pj(q )
𝜕qkl
=kj
2
(
m
∑
i=1
qij
)−
1
∕
2
when j=
l
𝜕P
j
(q )
𝜕qkl
cij
(q
ij
)=(a
kl
q
ij
+b
kl
)
2
(6)
q
+1
kl =max {0, {q
kl +a (
𝜔
k
𝛾
kl
𝛽k
−
1
𝛽k
(2a2
klqkl +2aklbkl
)
−𝜆
k+𝜆1
l
−𝜆2
l
)}}
1 An in e es ed eade can ind a de ailed desc ip ion in [3].
A.Luque-Ce pa e al.
3 T ansi ion P sys ems wi hmemb ane
pola iza ion
In his sec ion, we in oduce he chosen P sys em model
o designing ou solu ion o he GNE p oblems. We ha e
chosen he model o ansi ion P sys ems [14] endowed wi h
memb ane pola iza ion, i.e., he memb anes ha e a cha ge
ha ac s as a s a e o he memb ane. Tha s a e con ols he
e olu ion o he objec s wi hin he memb ane. Fo some
de ini ions abou memb ane sys ems and o mal languages,
we e e he eade o [15, 24]. A ansi ion P sys em wi h
memb ane pola iza ion o deg ee
q≥1
is a uple
whe e:
1.
Γ
is an alphabe whose elemen s a e called objec s;
2.
𝜇
is a hie a chical ee-like s uc u e;
3.
M1,…,Mq
a e mul ise s o objec s o e
Γ
;
4.
R1,…,Rq
a e se s o ules o he ollowing o m:
•
[u
→
]𝛼
h,u, ∈M(Γ),h∈{1, …,q},𝛼∈{0, +,−}
is an objec e olu ion ule;
•
[
u]𝛼
h
→ []
𝛽
h
,u, ∈M(Γ),h∈{1, …,q},𝛼,𝛽∈{0, +,
−}
is
an send-ou communica ion ule;
•
u
[]
𝛼
h
→[ ]
𝛽
h
,u, ∈M(Γ),h∈{1, …,q},
h
is no
he label o he skin memb ane
,𝛼,𝛽∈{0, +,−}
is an
send-in communica ion ule;
5.
𝜌1,…,𝜌q
a e pa ial weak ela ions be ween ules om
R1,…,Rq
, espec i ely;
6.
iou ∈{0, 1, …,q}
is he ou pu egion (i
iou =0
, he
ou pu egion is he en i onmen o he sys em).
A ansi ion P sys em wi h memb ane pola iza ion
can be seen as a se o q memb anes o ganized in a oo ed
ee-like s uc u e whose oo node is called he skin
memb ane, each o hem ha ing a pola iza ion among 0,
+
,
o −. A con igu a ion o
Π
in an ins an can be desc ibed
by he mul ise s o objec s in each memb ane in such a
momen and he pola iza ion o each memb ane and he
mul ise o objec s in he en i onmen , deno ed by
M0,
;
ha is,
C = ((M1, ,𝛼1, ),…,(Mq, ,𝛼q, ),M0, )
. I can
also be desc ibed in a mo e g aphical way such ha
[u]𝛼
h
ep esen s ha memb ane h con ains he mul ise o objec s
u and i s cha ge is
𝛼
. I he g aphical desc ip ion con ains a
memb ane
h′
such ha i is in he memb ane h, like
[[ ]
h
�
]h
,
hen
h′
is a child memb ane o h. The ini ial con igu a ion
o
Π
is
C0= ((M1
,0
)
,
…
,
(Mq
,0
)
,
�)
. A con igu a ion is
called a hal ing con igu a ion i no mo e ules a e applicable
o i . Fo he sake o simplici y, we will use he no a ion
Π=(Γ
,𝜇,
M1
,
…
,
Mq
,
(R1
,𝜌
1)
,
…
,
(Rq
,𝜌
q)
,i
ou ),
Π=(Γ
,𝜇,
M1
,
…
,
Mq
,
(R1
,𝜌
1)
,
…
,
(Rq
,𝜌
q)
,i
ou )
ℂ
o deno e speci ic pa s o he P sys em in such a way
ha
ℂ =𝜇�
, whe e
𝜇′
is a sub ee o
𝜇
, and he mul ise s
o objec s associa ed o each memb ane h o
𝜇′
a an ins an
is deno ed by
[u]h
, whe e u is a s ing ha ep esen s he
mul ise
Mh,
.
An objec e olu ion ule
[u
→
]𝛼
h
,
u
,
∈M(Γ),
h∈{1, …,q},
𝛼
∈{0, +,−}
is applicable o a con igu a ion
C
i he e exis s a memb ane labeled by h in
Π
such ha
i con ains he mul ise o objec s u and i s pola iza ion is
𝛼
. Applying such a ule leads o he emo al o u om he
memb ane h and he gene a ion o he mul ise o objec s
in he memb ane h.
An send-ou communica ion ule
[
u]𝛼
h
→ []
𝛽
h
,u, ∈M(Γ)
,
h∈{1, …,q},𝛼,𝛽∈{0, +,−}
is
applicable o a con igu a ion
C
i he e exis s a memb ane
labeled by h in
Π
such ha i con ains he mul ise o objec s
u and i s pola iza ion is
𝛼
. The applica ion o such a ule
leads o he emo al o u om such a memb ane h, he
gene a ion o he mul ise o objec s in he pa en egion
o h and he change o pola iza ion o such a memb ane h
om
𝛼
o
𝛽
.
An send-in communica ion ule
u
[]
𝛼
h
→[ ]
𝛽
h
,u
,
∈M(Γ),h∈{1, …,q},h
is no he label o he skin
memb ane
,𝛼,𝛽∈{0, +,−}
is applicable o a con igu a ion
C
i he e exis s a memb ane labeled by h in
Π
such ha
i s pa en memb ane con ains he mul ise o objec s u and
he pola iza ion o such memb ane h is
𝛼
. The applica ion
o such a ule leads o he emo al o u om he pa en
memb ane, he gene a ion o he mul ise o objec s in
such a memb ane h, and he change o pola iza ion o such
a memb ane h om
𝛼
o
𝛽
.
In addi ion, i
( 1, 2)∈𝜌h
, i means ha
1
has p io i y
o e
2
in he ollowing sense: Rule
2
is applicable only i
he emaining objec s om memb ane h canno i e ule
1
any mo e imes. I one objec can i e mo e han one ule,
hen i will be selec ed in a non-de e minis ic way.
Apa om ha , objec e olu ion ules a e applied in a
maximal pa allel way; ha is, any objec ha can i e an
applicable ule will i e i . A mul ise o ules U is maximal
i he e is no o he mul ise o ules
U′
in he se o mul ise s
o applicable ules such ha
U⊂U′
.
Send-in and send-ou ules a e applied in a maximal
pa allel way as ollows: Le
1
and
2
wo ules whe e ei he
1
and
2
can be a send-in o a send-ou ule. Le
h1
(
h2
,
espec i ely) be he label o he memb ane a ec ed by he
ule
1
(
2
, esp.). Le
𝛼1
(
𝛼2
, espec i ely) be he pola iza ion
o he le -hand side o he ule ( ha is, he pa o he ule
ha is o he le o he a ow) o ule
1
(
2
, esp.), and le
𝛽1
(
𝛽2
, esp.) be he pola iza ion o he igh -hand side o he
ule
1
(
2
, esp.). We say ha
1
and
2
a e compa ible i he
ollowing holds:
h1=h2∧𝛼1=𝛼2
→
𝛽1=𝛽2
; ha is, wo
ules ha a ec he same memb ane and can be applied in
he same momen a e compa ible i and only i he esul ing
An applica ion o memb ane compu ing ohumani a ian elie iagene alized Nash equilib ium
pola iza ion by he applica ion o each ule is he same. Each
mul ise o applicable ules can con ain only compa ible
ules.
A ansi ion o a compu a ional s ep o
Π
is made by
applying he ules in he a o emen ioned way in all memb anes
a he same ime, and i is deno ed by
C
⇒
ΠC +1
.
A compu a ion o
Π
is a sequence o con igu a ions
C=(C0,C1,…,Cn)
such ha
C0
is he ini ial con igu a ion
o
Π
and o each ,
C
⇒
C +1
. A compu a ion is hal ing i
n∈ℕ
, and
Cn
is a hal ing con igu a ion.
3.1 An example o aP sys em
Fo he sake o simplici y, we gi e a simple example o
explain he beha io o ansi ion P sys ems wi h memb ane
pola iza ion. Le
be a ansi ion P sys em wi h memb ane pola iza ion o
deg ee 1 whe e:
1.
Γ={a,b,c,d,e}
;
2.
𝜇=[ ]
1
;
3.
M1={a3,d}
4.
R1={ 1
≡
[a2
→
b]1, 2
≡
[a
→
c]1, 3
≡
[d
→
e]1}
;
5.
𝜌1= {( 1, 2)}
;
6.
iou =0
.
The ini ial con igu a ion can be ep esen ed as
ℂ0
=[a
3
,d]
0
1
.
Then, he ollowing mul ise s o ules a e applicable:
•
U1={
1,
2}
;
•
U2={ 1}
;
•
U3={
1,
3}
;
•
U4={ 1, 2, 3}
;
Since we look o maximal mul ise s o applicable ules, we
can obse e ha
U1,U2,U3⊂U4
. The e o e, he mul ise
U4
will be applied in he i s compu a ional s ep, leading o he
ollowing con igu a ion:
ℂ1=[b,c,e]1
.
4 Design and unc ioning o  heP sys em
Le us conside he Eqs.3, 4, 5 and 6 de ined in Sec .2.1. In
his sec ion, a P sys em ha compu es he solu ions o hese
equa ions is analyzed.2
We i s need o de ine a sequence
{a }
ha sa is ies
Π=(Γ,𝜇,M1,(R1,𝜌1),iou )
as indica ed in Sec .2.1. The sequence
a =0.1
⋅
b
is de ined
by
{b }=
aking copies o he elemen
0.1
⋅
1∕2w
wi h
=max{1024, 2w}
and
w=0, 1, …
. This sequence
gua an ees con e gence and can be easily upda ed and
employed using he e olu ion ules o a P sys em. In
opposi ion, he implemen a ion o he sequence used in
[3] can aise he complexi y o he model. The e ms
a
a e
ep esen ed h ough objec s s in he P sys em.
The compu a ion o he P sys em can be summa ized in a
loop o h ee s ages, ep esen ed in Algo i hm1.
Algo i hm 1 Gene al o e iew o he P sys em
compu a ion
Nex , an analysis o he compu a ion o each s age is
p o ided. The esul o he analysis is ha , o each ime
s ep, he numbe o ansi ion s eps in he Ini ializa ion and
Compa ison s ages is bounded by a cons an . Meanwhile,
he numbe o ansi ion s eps in he Upda e s age is cons an
o he i s 10,240 i e a ions (1024 iden ical elemen s o
each o he i s 10 alues in he sequence
a
) and hen
g ows loga i hmically wi h ime. This means ha he ime
complexi y o he global compu a ion only depends on he
numbe o i e a ions equi ed by he o iginal algo i hm.
In he compu a ion analysis,
ℂ𝜏
ep esen s he
𝜏
- h
con igu a ion o each i e a ion; ha is,
ℂ0
is he ini ial
con igu a ion in each i e a ion. The main di e ence be ween
i e a ions is gi en by he exis ence o objec s s, and his ac
is conside ed along he compu a ion.
S age 1: ini ializa ion
In his s age, all he necessa y objec s o upda ing he
alues
q
+
1
kl
,𝜆
+
1
k
,𝜆1
+1
l
and
𝜆
2
+1
l
a e c ea ed and dis ibu ed o
he co esponding memb anes. We assume ha his s age s a s
wi h some objec s
xk,l,lak,la1,l,la2,l
inside he memb ane INIT
which ep esen he alues
q
kl
,𝜆
k
,𝜆1
l
and
𝜆
2
l
a a ime s ep
. Fo simplici y, we conside only he objec s
xk,l
, bu he
beha io o he o he objec s is analogous. I we only conside
a
>0, a →0 and
∞
∑
=0
a
=∞
{
1, 1,
…
,1
⏟⏞ ⏞⏟⏞ ⏞⏟
1024 imes
,1
∕
2, 1
∕
2,
…
,1
∕
2
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
1024 imes
,
…
,1
∕
1024, 1
∕
1024,
…
,1
∕
1024
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
1024 imes
,
1∕2048, 1∕2048, …,1∕2048
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
2048 imes
,…}
2 A de ailed desc ip ion o he P sys em can be ound in he Appen-
dix A.

A.Luque-Ce pa e al.
he memb anes in ol ed in each s ep, we ha e he ollowing
con igu a ion:
In he nex ansi ion s ep, di e en copies o each objec in
he memb ane INIT a e ejec ed in o he skin memb ane. The
objec s y a e applied o coo dina e he compu a ion:
In one ansi ion s ep, objec s ep esen ing he cons an pa
o equa ions (3) o (6) (be o e mul iplying by
a
) a e c ea ed
in he co esponding memb anes. Fo simplici y, only he
memb anes
Qk,l
, which ep esen he Eq.6, a e conside ed.
The objec s
p0
ep esen o iginally he e m
𝜔
k
𝛾kl
𝛽k
, while
objec s
c 0
ep esen he e m
−2a
kl
bkl
𝛽k
(see ule
RS1,6
in he
Appendix A). In one ansi ion s ep:
Now ha he cha ge o he memb anes is nega i e,
objec s ep esen ing
q
kl
,𝜆
k
,𝜆
1
l
and
𝜆2
l
a e inse ed in o he
co esponding memb anes. The e ms
q
kl
a e ep esen ed by
he objec s p and
c0
, while he
𝜆
- e ms a e ep esen ed by
objec s
p0
o
n0
. These objec s will be used o upda e he
a iables using Equa ions (3) o (6) (see ules
RS1,10
o
RS1,19
in he Appendix A). In one ansi ion s ep:
In his s ep, all memb anes
Qk,l
,
LAMBk
,
LAMB1,l
and
LAMB2,l
ha e he necessa y objec s o s a he Upda e s age,
and all o hem a e coo dina ed h ough he usage o he
objec
y1
. The Ini ializa ion s age is hen comple ed in h ee
ansi ion s eps.
S age 2: upda e
In his s age, he alues
q
+1
kl
,𝜆 +1
k
,𝜆1
+1
l
and
𝜆
2
+1
l
a e
compu ed ollowing Equa ions (3) o (6). The compu a ion has
wo phases: one o compu ing he pa en heses in each equa ion
and mul iplying i by
a
, and one o compu ing he
max
unc ion. Fo simplici y, only memb anes
Qk,l
a e conside ed,
bu he beha io o memb anes
LAMBk
,
LAMB1,l
and
LAMB2,l
is analogous. This s age s a s wi h he con igu a ion
ℂ3
. In wo ansi ion s eps, all e ms in he pa en heses o
Equa ions (3) o (6) will be ep esen ed by objec s
p0
o
n0
depending on hei sign:
In one ansi ion s ep, objec s
y3
will be in oduced in he
memb anes REDUCE, changing hei cha ge o nega i e. In
one mo e ansi ion s ep, objec s
p0
and
n0
will be in oduced
ℂ
0= [[x
q
k,l
k,l
]
0
INIT
y0]
0
1
ℂ
1=[x
q
k,l
k,l
,x
q
k,l
k,l
,xl
q
k,l
0,k,l
,xl
q
k,l
1,k,l
,xl
q
k,l
2,k,l
,y0,k,l,ylk,yl1,l,yl2,l]
0
1
ℂ
2=[[y0,p
𝜅
0
0,c
𝜅
1
0]
−
Q
k,l
x
q
k,l
k,l,x
q
k,l
k,l,xl
q
k,l
0,k,l,xl
q
k,l
1,k,l,xl
q
k,l
2,k,l]
0
1
ℂ
3=[[y1,p
𝜅
0
0,c
𝜅
1
1,p
q
k,l,c
q
k,l
0]
−
Q
k,l
]
0
1
ℂ
5=[[y3,p𝜅0
0,n𝜅1
+⌊
2⋅a
2
kl
∕
𝛽k
⌋
⋅qk,l
0,pqk,l]−
Q
k,l
]
0
1
in he memb anes REDUCE. Fo he i s 1024 i e a ions,
in he nex s ep objec s
p0
and
n0
will be added, gene a ing
objec s p o n, and he objec
y6
will be gene a ed, changing
he cha ge o memb anes REDUCE o posi i e. In he es ,
he e will be objec s s p esen ha will cause he educ ion
in he numbe o objec s p and n by mul iplying hem by
10
⋅
a
. The e is one mo e ansi ion s ep o each objec s
p esen , and he e a e n objec s s p esen o he sequence
e m
a =0.1
⋅
1∕2w
. Assuming we end wi h objec s p
(analogous o he wise), in a leas h ee ansi ion s eps we
ha e he con igu a ion:
In wo mo e ansi ion s eps, he objec s
y6
will e ol e and
change he cha ge o memb anes REDUCE back o neu al
and he cha ge o memb anes
Qk,l,LAMBk
,
LAMB1,l
and
LAMB2,l
o posi i e. Du ing hese s eps, he numbe o
objec s p o n ha we e p e iously in memb anes REDUCE
will be mul iplied by 0.1 and added o he es in memb anes
Qk,l,LAMBk
,
LAMB1,l
and
LAMB2,l
, upda ing he e ms
q
+1
kl
,𝜆 +1
k
,𝜆1
+1
l
and
𝜆
2
+1
l
:
In he wo nex ansi ion s eps, i objec s p and n a e p esen
hey cancel each o he . I objec s p a e le , hey a e ejec ed
o he skin as objec s
okl
,
ikl
and
x 1,k,l
, and i objec s n a e
le hey a e dele ed.
The Upda e s age is hen comple ed in a minimum o
nine ansi ion s eps, and he numbe o ansi ion s eps
g ows linea ly o nine een o he 10240 i s i e a ions and
loga i hmically wi h o he es . In p ac ice, a solu ion
wi h con e gence ole ance
10−10
is ound in he i s 2048
i e a ions, so we ha e an e ec i e uppe bound o en
ansi ion s eps.
S age 3: compa ison
In his s age, he new alues
q +1
kl
a e compa ed wi h he
p e ious alues
q
kl
. I a di e ence is ound, he execu ion
con inues. O he wise, he new alues a e sen o he memb ane
OUTPUT, and he compu a ion inishes. This s age s a s wi h
he ollowing con igu a ion:
In wo ansi ion s eps, objec s
x k,l
and
x 1,k,l
a e in
memb ane COMP o be compa ed:
ℂ
≥8=[[y6[p10⋅a ⋅
(
𝜅0
−
𝜅1
−⌊
2⋅a
2
kl
∕
𝛽k
⌋
⋅qk,l
)
]
+
REDUCE
0,k,l
pqk,l]
−
Q
k,l
]
0
1
ℂ
≥10 =[y8,k,l[[]
0
REDUCE
0,k,l
pq
+1
kl ]+
Q
k,l
]
0
1
ℂ
≥12 =[ym⋅n
10 ,o(q
+1
k,l)
k,l
,i(q +1
kl )
k,l
,x (q +1
kl )
1,k,l
[[]
0
REDUCE
0,k,l
]0
Q
k,l
]
0
1
ℂ
≥12 =[ym⋅n
10
,o(q
+1
k,l)
k,l
,i(q
+1
kl )
k,l
,x (q
+1
kl )
1,k,l
,x (q
kl)
k,l
[]
0
COMP
[]
0
INIT
[]
0
OUTPUT
]
0
1
ℂ
≥
14
=[i(q
+1
kl )
k,l
[y
12
,o(q
+1
k,l)
1,k,l
,x (q
+1
kl )
1,k,l
,x (q
kl)
k,l
]0
COMP
[]
0
INIT
[]
0
OUTPUT
]
0
1
An applica ion o memb ane compu ing ohumani a ian elie iagene alized Nash equilib ium
In he nex wo ansi ion s eps, objec s
x k,l
and
x 1,k,l
cancel
each o he . I a leas one o hem is p esen , hen
y16
e ol es
o
y13
and is ejec ed as
y14
. O he wise, an objec s op is
c ea ed and ejec ed. The es o he objec s
x k,l
and
x 1,k,l
a e dele ed.
A his momen , wo hings can happen. The i s one is ha
in he nex h ee ansi ion s eps, he s op objec changes he
memb ane OUTPUT cha ge o nega i e, he objec s
o3,k,l
go
in o i , and he compu a ion inishes a e changing again he
cha ge o COMP o neu al:
The o he possibili y is ha , in he ollowing wo ansi ion
s eps, he
y14
objec changes he cha ge o memb ane INIT,
he objec s
ik,l
ge in o memb ane INIT, and an objec
y0
is
c ea ed, lea ing he P sys em in a s a e whe e a new i e a ion
can s a again:
In his las s ep, a
coun 0
objec is also c ea ed. This objec
will con ibu e o he upda e o
a
. We can see hen ha his
s age is comple ed in six ansi ion s eps excep o he las
i e a ion, which is comple ed in se en s eps.
The esul o his analysis is ha he whole compu a ion
o he sys em only depends on he numbe o i e a ions
equi ed by he o iginal algo i hm. The algo i hmic ime
complexi y o an i e a ion o he algo i hm is cons an o
S age 1, loga i hmic wi h o S age 2, and cons an o
S age 3. The algo i hmic ime complexi y o he whole
compu a ion is hen
O( log( ))
. In p ac ice, i is
O( )
, as
explained in S age 2. In con as , he o iginal algo i hm
has o compu e Eqs.3, 4, 5, and 6 a e e y i e a ion, so
he algo i hmic ime complexi y o he o iginal algo i hm is
O((m
⋅
n+m+2n)
⋅
)
. Consequen ly, we p esen a educ ion
o a ac o o
m
⋅
n+m+2n
.
5 Expe imen s
Di e en P sys em simula o s a e a ailable, such as P-lingua
(MeCoSim) [25–27] o UPSimula o [28, 29]. To pe o m
expe imen s, MeCoSim has been upda ed and chosen o
implemen ou P sys em. The las simula o upda e allows
o he design o ansi ion P sys ems wi h memb ane
pola iza ion.
To es he co ec beha io o ou P sys em, we ha e con-
side ed se e al examples aken om [3], whe e he nume i-
cal da a o some case s udies a e p o ided. Namely, he
ℂ
≥
16
=[o(q
+1
k,l)
3,k,l
,i(q
+1
kl )
k,l
,y
14||
s op []
0
COMP
[]
0
INIT
[]
0
OUTPUT
]
0
1
ℂ
≥19 =[i(q
+1
kl )
k,l
[]
0
COMP
[]
0
INIT
[o(q
+1
k,l)
k,l
]0
OUTPUT
]
0
1
ℂ
≥18 =[y0[]
0
COMP
[x(q
+1
kl )
k,l
,coun 0]0
INIT
[]
0
OUTPUT
]
0
1
au ho s p o ide ou oy examples and he eal-wo ld case
s udy o Hu icane Ka ina. All o hese examples ha e been
simula ed. Fo he i e nume ical examples, he con e gence
ole ance chosen is
10−5
, while o he Hu icane Ka ina
case s udy, he ole ance is
10−10
. A solu ion was ound o
all he nume ical examples in less han 1024 i e a ions. The
Upda e s age ook hen nine ansi ion s eps o e e y i e a-
ion. Fo he Hu icane Ka ina case s udy, a solu ion was
ound in less han 1100 i e a ions. The Upda e s age ook
hen nine ansi ion s eps o he i s 1024 i e a ions, and
en ansi ion s eps o he es (see Sec .4).
The esul s o he simula ions o he ou oy nume ical
examples can be ound in Tables1, 2, 3 and 4, whe e he
ob ained alues o
qij
(i.e., he numbe o i ems ha each
NGO i p o ides o a demand poin j) a e compa ed. On he
igh , he equilib ium alues ob ained in [3] a e shown,
and, on he le , he alues ob ained wi h ou P sys em.
Conside ing ha he ue alues a e ounded o one decimal,
i can be obse ed ha he esul s e u ned by ou P sys em
Table 1 Resul s o he
simula ion o Example 1 o [3]
qij
P sys em Solu ion
q11
352.50012 352.5
q21
247.50012 247.5
Table 2 Resul s o he
simula ion o Example 2 o [3]
qij
P sys em Solu ion
q11
352.50012 352.5
q12
452.50004 452.5
q21
247.50012 247.5
q22
347.50004 347.5
Table 3 Resul s o he
simula ion o Example 3 o [3]
qij
P sys em Solu ion
q11
423.75003 423.8
q12
471.25002 471.3
q13
436.87498 436.9
q21
176.25011 176.3
q22
328.75007 328.8
q23
563.12492 563.1
Table 4 Resul s o he
simula ion o Example 4 o [3]
qij
P sys em Solu ion
q11
411.25004 411.3
q12
458.75002 458.8
q13
499.37496 499.4
q21
138.75012 138.8
q22
291.25007 291.3
q23
750.62488 750.6
A.Luque-Ce pa e al.
coincide pe ec ly wi h he ue alues, e en a e eplacing
Eq.2 wi h Eq.6 as discussed in Sec .2.1.
Rega ding he Hu icane Ka ina case s udy, he
eplacemen o Eq.2 wi h Eq.6 has impac ed he solu ions
e u ned, as can be obse ed in Table5. Howe e , he
solu ion e u ned by he P sys em is a good app oxima ion:
he a e age e o be ween he co ec solu ion and he
solu ion e u ned by ou P sys em is 1.98%, he median is
0.82%, he maximum e o is 7.89%, and only in ou cases
ou o hi y is he e o highe han 5%.
These expe imen s suppo ou claim ha he con ibu ion
o he unc ions
Pj(q)
can be igno ed wi hou incu ing
signi ican e o s. This implies ha he inancial unds due o
he isibili y o all NGOs a each loca ion ha e li le impac
on he inal solu ion o he p oblem o how o dis ibu e
humani a ian elie subjec o he cons ain s s a ed i hey
ollow he assump ions in [3].
6 Technical conclusions
The simula ion o con inuous p ocesses by in insically
disc e e models is always a complex ask, and each eal-
wo ld p oblem needs o be deeply s udied o adop he bes
possible solu ion. In his pape , we ha e conside ed he
minimiza ion ques ion exp essed in Eq.2. Such an equa ion
in ol es wo e ms wi h de i a i es. A e a deep s udy o
he p oblem, we can conclude ha he e m wi h he i s
de i a i e is se e al o de s o magni ude lowe han he
emaining e ms. By emo ing his e m, we lose accu acy;
howe e , om a p ac ical poin o iew, his loss is no
signi ican , as he expe imen s show. Conce ning he second
e m wi h a de i a i e, a solu ion o a gene al unc ion
cij
is no possible, bu i can be eached in his case bea ing in
mind ha i can be exp essed as a second-g ade polynomial
equa ion.
F om a memb ane compu ing pe spec i e, we would
like o emphasize ha ou design shows ha he numbe o
ansi ion s eps in he Ini ializa ion and Compa ison s ages
is bounded by a cons an and he numbe o ansi ion s eps
in he Upda e s age g ows linea ly o he i s 10240 i e a-
ions and hen g ows loga i hmically wi h ime. This can be
conside ed as he main con ibu ion o his pape om he
heo e ical side. Due o he in insic massi e pa allelism o
he Memb ane Compu ing de ices, he ime complexi y o
he global compu a ion, conside ed on he basis o he P
sys em s eps, only depends now on he numbe o i e a ions
equi ed by he o iginal algo i hm. Besides, he modula i y
o he design allows us o ex end he p ocess wi h new agen s
by in oducing new modules and only minimal changes in
he o he modules. Addi ionally, he objec -based app oach
is simila o agen -based models in he sense ha he beha -
io s o hese indi iduals can be acked, ha ing a one- o-
one ela ionship be ween objec s and eal-li e esou ces, and
leading o a ine-g ained model ha can be calib a ed o
di e en scena ios.
As a inal ema k, i is wo h s essing ha , al hough he
use o ansi ion P sys ems wi h pola iza ions o gene alized
Nash equilib ia was in oduced in [19], a i s de ailed eal-
li e use case, speci ically applied o model humani a ian
elie , is p esen ed he e.
7 Final ema ks
In his pape , we ha e shown ha Memb ane Compu ing
is a use ul compu a ional pa adigm o model humani a ian
elie . The main con ibu ion o his pape is wo old. On
he one hand, we con ibu e o highligh ing ha i is unac-
cep able ha e o s o dis ibu e humani a ian aid a e no
enough success ul due o a lack o coo dina ion, conges ion
in e ms o logis ics, and duplica ion o se ices. This si u-
a ion demands an answe om he scien i ic communi y,
and many heo e ical e o s mus be made o op imize he
esou ces. One possible solu ion is o model he p oblem
in e ms o GNE, bu o he op imiza ion me hods can be
Table 5 Resul s o he
simula ion o he use case o
Hu icane Ka ina in [3]
O he s Red c oss Sal a ion a my
Loca ion P sys em Solu ion E o P sys em Solu ion E o P sys em Solu ion E o
S .Cha les 16.10 17.48 7.89% 29.47 28.89 2.01% 4.35 4.192 3.77%
Te ebonne 267.93 267.02 0.34% 410.31 411.67 0.33% 73.38 73.57 0.26%
Assump ion 47.92 49.02 2.24% 77.59 77.26 0.43% 13.09 12.97 0.93%
Je e son 264.57 263.69 0.33% 405.34 406.68 0.33% 72.30 72.45 0.21%
La ou che 186.55 186.39 0.09% 287.22 287.96 0.26% 51.11 51.18 0.14%
O leans 466.04 463.33 0.58% 710.66 713.56 0.41% 126.63 127.1 0.37%
Plaquemines 20.53 21.89 6.21% 37.02 36.54 1.31% 4.37 4.23 3.31%
S .Ba na d 74.52 72.31 3.06% 120.12 115.39 4.10% 17.13 16.22 5.61%
S .James 57.66 58.67 1.72% 92.31 92.06 0.27% 15.77 15.66 0.70%
S .John Bap is 16.83 18.2 7.52% 30.60 29.99 2.03% 4.51 4.40 2.50%
An applica ion o memb ane compu ing ohumani a ian elie iagene alized Nash equilib ium
explo ed. On he o he hand, o he bes o ou knowledge,
his is he i s pape b idging Memb ane Compu ing wi h
he dis ibu ion o humani a ian elie and we ha e shown
ha P sys ems can be a use ul ool o model complex si ua-
ions in his a ea.
As u u e esea ch, many applica ions o Memb ane
Compu ing o logis ics in humani a ian aid can be explo ed,
no only wi h ansi ion P sys ems bu wi h o he P sys ems
models. In pa allel, o he op imiza ion me hods can be
conside ed o op imize humani a ian aid dis ibu ion. As a
consequence o he good beha io o he model, i seems
easonable o s udy speci ic si ua ions whe e he gene al case
is no enough o model eal-li e e en s, such as abno mal
dona ion pa e ns, changes in he poli ical landscape, o any
o he ac ha could impac he dis ibu ion o humani a ian
elie .
Appendix AAppendix: De ini ion o  heP
sys em
Le
N={1, ..., m}
and
D={1, ..., n}
wi h m and n he
numbe o NGOs and demand loca ions espec i ely. Le
P−1=10−p
wi h
p∈ℕ
be he ole ance o con e gence o
he algo i hm. Le us conside he ollowing ansi ion P
sys em wi h memb ane pola iza ion
whe e he alphabe o objec s is gi en by:
The se o memb ane labels is gi en by:
Π=⟨Γ,H,EC,𝜇,{wh}h∈H,(R,𝜌)⟩
Γ={y
0
,y
1
,y
2
,y
3
,y
4
,y
5
,y
6
,y
7
,y
10
,y
11
,y
12
,y
13
,y
14
,y
15
}∪
∪{y0,k,l,yl0,k,yl1,l,yl2,l|k∈N,l∈D}∪
∪{y8,k,l,y9,k,l,yla8,k,yla9,k,yla1,8,l,yla1,9,l,yla2,8,l,yla2,9,l|k∈N,l∈D}∪
∪{xk,l,x k,l,x 1,k,l,xl0,k,l,xl1,k,l,xl2,k,l|k∈N,l∈D}∪
∪{lak,laq0,k,l,la0,k,la1,l,laq1,k,l,la1,l,la2,l,laq2,k,l,la2,l|k∈N,l∈D}∪
∪{p0,n0,c 0,c 1,c 2,c0,c1,p,n,o}∪
∪{ik,l,lao0,k,lao1,l,lao2,l,ok,l,o0,k,l,o1,k,l,o2,k,l,o3,k,l,o4,k,l,|k∈N,l∈D
}∪
∪{ em,s op,s,s0,coun n,u0,un,maxn
|
n≥1}∪
∪{sq
k,l
,sl
k
,sl
1,l
,sl
2,l|
k∈N,l∈D}
H
={1, INIT,COMP,OUTPUT}∪{Qk,l|k∈
N
,l∈
D
}∪{LAMBk|
k∈N}∪{LAMB1l|l∈D}∪{LAMB2l|l∈D}∪{REDUCE0,k,l
|
k∈N,l∈D}∪{REDUCE1,k|k∈N}
∪{REDUCE
2,l
|l∈D}∪{REDUCE
3,l
|l∈D}.
The memb ane s uc u e
𝜇
can be de ined as ollows:
• The skin memb ane wi h label 1, inside o which he e
exis :
1. One memb ane wi h label INIT.
2. One memb ane wi h label OUTPUT.
3. One memb ane wi h label COMP.
4. One memb ane wi h label
Qk,l∀
k
∈N
;
∀
l
∈D
. Inside
o each memb ane
Qk,l
:
– One memb ane wi h label
REDUCE0,
k
,
l∀k∈
;
∀l∈
5. One memb ane wi h label
LAMBk
∀k∈
N
. Inside o
each memb ane
LAMBk
:
– One memb ane wi h label
REDUCE1,k
∀k∈N
6. One memb ane wi h label
LAMB1l∀l∈D
. Inside o
each memb ane
LAMB1l
:
– One memb ane wi h label
REDUCE2,l∀l∈D
7. One memb ane wi h label
LAMB2l∀l∈D
. Inside o
each memb ane
LAMB2l
:
– One memb ane wi h label
REDUCE3,l∀l∈D