MATHEMATICAL MODELS FOR GROUP REVENUE MANAGEMENT
José Guadix, Juan La añe a, Luis Onie a
Indus ial O ganiza ion and Managemen Depa men
Indus ial Enginee ing School – Uni e si y o Se ille
Se ille, Spain
[email p o ec ed]
Abs ac – Re enue Managemen is a echnique ocus o
decision ules o maximizing p o i om sale o pe ish-
able in en o y uni s. This pape deals wi h he special case
o ho el e enue managemen , which can be sol ed using
de e minis ic and s ochas ic ma hema ical p og amming
echniques. We i s desc ibe he p oblem wi h a heo e i-
cal amewo k ha se s he e enue maximiza ion c i e ia
o a ho el. We conside he gene al case o he p oblem
ha accep independen and g oup gues s, wi h a gene al
mixed in ege linea p og amming model ha maximize
he o al o ecas ing. Finally, we made compa isons be-
ween di e en p oposed models and we e ound good-
quali y solu ions in sho unning imes.
Keywo ds: Yield Managemen ; G oup Re enue
Managemen ; Ho els; Ma hema ical Models.
I. INTRODUCTION
Recen yea s ha e seen an inc eased in e es in using
yield managemen echniques o maximize p o i abili y
in capaci y cons ained si ua ions. Mos o he cha ac-
e is ics unde lying his echnique ha e been used be-
o e in di e en indus ies. Pe ishable i ms, such as
bake s, g oce s, esh ui selle s o hea e manage s,
managed demand by a ying p ices in ime. A e US
Ai line De egula ion Ac in 1978, any ai line could
ope a e any ou e a any equency wi h wha e e a es
a e chose, Smi h e al. (1992). Companies adop ed di -
e en ia ed p icing in o de o be able o compe e o
p ice sensi i e a elle s, wi hou gi ing up he e enue
om hei exis ing, ull a e cus ome s.
Yield managemen , also e e ed o as e enue man-
agemen , is a sophis ica ed o m o supply and demand
managemen ha balances bo h p icing and in en o y o
maximize e enue o e e y a ailable uni o capaci y.
An inc easing numbe o se ice indus ies ha e ecog-
nized he apidly g owing impo ance o yield manage-
men in hei abili y o inc ease sales, especially p o i -
abili y.
Se ices indus ies (such as ai lines, ho els, en al ca
agencies, eigh anspo and b oadcas ad e ising),
ha e been able o ma ke hei se ices (sea on an ai -
c a , oom in a ho el, en a ca , spaces on coaches o
ad e ising ime pe iods) as a pe ishable p oduc .
In his way, Yield Managemen can be de ined as sell
he igh in en o y uni o he igh cus ome a he igh
ime. Fo Yield Managemen o be applicable he se -
ice indus y needs i e condi ions (Kimes 2000).
1. Limi ed capaci y. Yield Managemen is designed
o capaci y-cons ained se ices i ms. The uni s o
in en o y a e sold in a sho ime wi h a ixed capaci y,
measu ed by he numbe o ooms o he numbe o
sea s.
2. Ma ke segmen a ion. Se ices indus ies make use
o segmen a ion because hese can choose be ween
di e en ypes o cus ome s. A bi a y p ice is no al-
lowed, so he se ice should ha e some cha ac e is ic
ha dis inguish i . So he same uni o capaci y can be
used o deli e many di e en se ices.
3. Fu u e demand is unce ain. Yield Managemen
mus be able o o ecas he demand a iabili y so ha
manage s can inc ease he p ice du ing pe iods o high
demand and dec ease he p ice du ing pe iods o low
demand. Se ices i ms canno quickly a ailable capac-
i y o a ailable demand.
4. Pe ishable uni s o in en o y. The in en o y dis-
inguishes se ice i ms om manu ac u ing i ms.
Uni s o in en o y se ices indus ies unsold a e a
speci ic da e a e was ed, se ices canno be s o ed.
These cha ac e is ics decide o sell se ices in ad ance.
5. App op ia e cos and p icing s uc u e. Many se -
ices i ms p esen a ixed capaci y cos expensi e and
canno be apidly adjus ed demand. In he same way
addi ional cos associa ed o an addi ional isi o in
unused capaci y is e y low.
The e enue managemen models we s udy in his
pape include g oup accep ance in ho els. The e o e, in
his wo k we modelled he cus ome ypology as indi-
idual o g oup. We es ed a a ie y o di e en ooms
op imiza ion algo i hms, based on de e minis ic and
s ochas ic p og amming echniques.
The pape is o ganized as ollow. In Sec ion 2 we
s udy he pa icula case whe e he o ecas ing demand
is de e minis ic, in which g oups a e allowed o no . In
Sec ion 3 we o mula e he p oblem as a s ochas ic
model, wi hou and wi h g oups. Tha case is close o
eal-wo ld si ua ions, so he esul s we e be e . Compu-
a ional compa isons a e desc ibed in Sec ion 4, using
linea p og amming and mixed in ege linea p og am-
ming p oblem. Finally, conclusions a e d awn in Sec-
ion 5.
II. DETERMINISTIC MODEL
Yield managemen has ocused mainly on o ecas -
ing, ese a ion sys ems and op imiza ion models. Fo a
In e na ional Con e ence on Knowledge Enginee ing and Decision Suppo
375
comp essi e o e iew o he li e a u e we e e o
McGill and Van Ryzin (1999).
The model needs o o ecas he demand o cus ome
and ob ains he op imal alloca ion o ooms o e he
o ecas ed demand. Fo ecas is essen ially in he ho el
yield managemen sys em. Fo ho el o ecas ing we
need his o ical in o ma ion abou a i als by leng h o
s ay and a e ca ego y. We can use di e en me hods
(Lee, 1990): his o ical, ad anced and combined book-
ing models.
T adi ional o ecas ing echniques we e: mo ing a -
e age bookings, exponen ial smoo hing and ARIMA
imes se ies models. Ad anced booking models is used
o p edic cus ome pickup, i is he inc emen al book-
ings ecei ed du ing a ce ain ime in e al. The chosen
me hods we e addi i e and mul iplica i e models. Hy-
b id models can be eg ession me hods in which inde-
penden a iables we e numbe o ese a ions on hand
o a day pa icula day o economics pa ame e s om
cus ome coun ies and he dependen a iable was he
inal numbe o ooms sold.
I is no exi he bes me hod, e e y ho el own pa -
icula cha ac e is ics, and a ho el may use a o ecas ing
me hod depending on wha pe iod o calenda . In gen-
e al e ms eg ession model, linea o loglinea eg es-
sion, will be a good i ed da a. Unpublished s udies
used combina ion o ecas s,
A e using a good o ecas ing model, he sys em e-
lies on ills all a ailable capaci y and cha ges he high-
es uni p ice, his means ha ensu es ha hose cus om-
e s mos willing o pay o a oom can do so. Abou
op imiza ion models, Williamson (1992) s udy he p ob-
lem o maximizing e enue in he ai line indus y using
a de e minis ic ma hema ical p og amming. In he ho el
indus y, le k deno e depa u e day menus a i al day
(leng h o s ay), pj he a e class p ice, bi he capaci y o
he ho el on day i, dijk he demand o ecas ed o gues
a i ing on day i, s aying du ing k days a j a e class
and xijk he numbe o ooms ese ed o gues wi h ijk
cha ac e is ics. The ho el model is hen o mula ed as
ollows:
,,
()
Maximize
subjec o
0
in ege (DP)
jijk
ijk
ljk i
li j lk i
ijk ijk
ijk
kpx
xb i
xd
x
≤+>
⋅
≤∀
≤≤
∑
∑∑ ∑
The objec i e is o ind he oom alloca ion o maxi-
mize e enue om selling oom and sa is ies he capac-
i y cons ain s in a ho el.
This in ege p oblem sol es elaxing o a linea p o-
g am because he cons ain ma ix is unimodula . The
p oblem is able o sol e wi h an associa ed ne wo k
low, nodes ep esen days and a cs ep esen s ooms
will sell o cus ome s. In sec ion 4 we show di e en
examples compa ed he linea and ne wo k low solu-
ion.
We a e hus sol ing a subp oblem wi hin an o e all
app oach ha in eg a es o ecas ing and ese a ion
sys ems. In his way, i maximizes a e age p o i pe
a ailable uni , by an icipa ing he p ice sensi i i y o
di e en cus ome s and an icipa es selling o he highes
paying cus ome s. On he o he hand, his p ocess mi i-
ga es seasonali y o demand, by shi ing excess demand
away om peak pe iods and in o he o season.
In hese models we do no conside cancella ions
ooms, gues s which no show a e make a ese a ion.
In due o m, o e booking is no ake in o accoun he e.
O e booking occu s when a ho el accep s mo e ese a-
ions han i has ooms a ailable. I aises legal issues
when ho el manage use ai line o e booking as jus i i-
ca ion o he p ac ice. The di e ence is ha speci ic
ede al laws, which do no apply o ho els, go e n he
ai line indus y.
An ex ension o his model in oduced by S cek
(1991) includes g oup ese a ion. G oups a e special
clien s because usually bookings a e wi h ime, eques
blocks o ooms, need con e ence space and a e sensi-
i e abou p ice. We will deno e by i* he day o a i al,
λ
g numbe o days he g oup wan s o s ay,
µ
g g oup
size, cg g oup a e and bina y a iable xg ep esen i
accep ance o no his g oup. The g oup model is now
o mula ed as ollows:
{}
{}
,,
()
()
Maximize
subjec o *,..., *
*,..., *
jijk gg gg
ijk g
ljk i g
li j lk i
ljk g g i g
li j lk i
ijk ijk
kpx c x
xb iii
xxbiii
xd
λµ
λ
µ
λ
≤+>
≤+>
⋅
≤∀∉ +
+≤∀∈ +
≤≤
∑∑
∑∑ ∑
∑∑ ∑
+
0
{}
in ege
0,1 (DGP)
ijk
g
x
x∈
This p oblem will be a mixed-in ege p og amming
model.
In eal si ua ions, he g oup a e is usually nego ia ed
wi h ou ope a o s o a el agen s. Du ing nego ia ion
he ou ope a o s con ac s wi h he ese a ion supe i-
so and eques a speci ic numbe o ooms o a ime
pe iod. In addi ion, g oup needs ex a se ices such as
ood and be e age, con e ence ooms, e c. In his e-
ques he ho el equi ed he minimum p o i able oom
o accep o ejec decisions, and we will use i o con-
side o auc ion heo ies. A me hod o calcula e he
minimum g oup a e is sol ed his p oblem wice. The
i s ime we ind he e enue wi hou g oup, i chooses
he i s model. Then we calcula e he e enue wi h he
g oup, we sol e he second model. This di e ence e e-
In e na ional Con e ence on Knowledge Enginee ing and Decision Suppo
376
nue alue will be he minimum income ha ho el needs
o sell he oom blocks he leng h o s ay. G oup e-
ques s could displace indi idual cus ome s paying
highe a es. Some g oup cus ome s may occupy ooms
wi h highe expec ed ma ginal e enue han o he s
cus ome s. Bu he o al g oup e enue may be highe
han selling hese ooms o indi idual cus ome s.
III. STOCHASTIC MODEL
In his sec ion we assume ha demand is s ochas ic,
so he numbe o alloca es ooms could be di e en
om he o ecas ing ooms. He e we conside a s ochas-
ic p og amming wi h simple ecou se p oblem. This
case is equi alen o a sepa a e objec i e unc ion, a
linea pa o he le -hand side and unc ions o andom
a iables in he igh -hand side. These pa icula s o-
chas ic p oblems do no cause se e e compu a ional
di icul ies, Kall and Wallace (1994). De Boe e al.
(2002) in oduced a s ochas ic model o he ai line
indus y. Suppose ha he demand Dijk can ake on only
a limi ed numbe o disc e e alues
{
}
,1 ,2 ,
...
ijk ijk ijk
dd d<<< . This disc e e alues a e possi-
ble scena ios depends on cus ome demand.
,,
=1 , ,
,
1()
,1 ,1
,,,1
,
Max P ( )
s. . i
=2,...,
0 in ege
S
j ijk ijk ijk
ijk
S
ljk i
lijlki
ijk ijk
ijk ijk ijk
ijk
kp D d x
xb
xd
xdd S
x
=≤ +>
−
⋅⋅ ≥ ⋅
≤∀
≤
≤− ∀
≥
∑∑
∑∑∑ ∑
(SP)
In his model, he decision a iables xijk, is a in ege
a iable which ep esen he pa o demand Dijk ha
alls in he in e al . Numbe o ooms
ese ed x
,1 ,
(,
ijk ijk
dd
−]
) P ( )
ijk ijk ijk ijk
xd xd
−
===
. Howe e ,
ijk is di ide in possible scena ios, so we will
ind he decision a iables xijk, . The solu ion o his
model, xijk, will be di e en o ze o when xijk, -1 is equal
o dijk, -1, i.e. P (
he sum o x
,1 ,
ijk, ooms sold o cus ome s in S scena ios
will be ag ee wi h he daily capaci y cons ain .
We sol e a linea p oblem elaxa ion o he s ochas-
ic model because he cons ain ma ix is unimodula in
he same e ms, so he solu ion consis s o in ege al-
ues. The de e minis ic model is a pa icula model be-
cause i conside s only one demand scena io (o e S
possible).
Wi h h ee demand scena ios is enough o cap u e
mos o he ex a e enue gene a ed by ex a cus ome s.
These demands a e calcula ed om o ecas ing adding
up and ake away he s anda d de ia ion o e e y a e
class p ice.
Al hough we ha e used a s ochas ic model o indi-
idual cus ome , we can use i wi h g oup. In his
model, he objec i e unc ion has a s ochas ic e m wi h
he g oup e enue ha is modelled as a de e minis ic
demand.
{}
{}
S
,,
=1 , ,
S**
,
=1 ( )
S**
,
=1 ( )
Max P ( )
s. . ,...,
,...,
j ijk ijk ijk g g g g
ijk g
ljk i g
li j lk i
ljk g g i g
li j lk i
kp D d x c x
xb iii
xxb iii
λµ
λ
µ
λ
≤+>
≤+>
⋅⋅ ≥ ⋅ +
≤∀∉
+≤ ∀∈ +
∑∑ ∑
∑∑∑ ∑
∑∑∑ ∑
{}
,1 ,1
,,,1
,
- =2,...,
0 in ege
0,1
ijk ijk
ijk ijk ijk
ijk
g
xd
xdd S
x
x
−
≤
≤∀
≥
∈ (SGP)
+
We sol e he p oblem in simila way, needing he
minimum g oup a e o a new g oup. So wi h a o ecas -
ing dijk he model buil h ee di e en scena ios (wi h
he s anda d de ia ions as limi s) and could ind i sol -
ing he model wice, in he same way o he de e minis-
ic mixed linea in ege p oblem.
Ano he possibili y is wo king sol ing i he scena -
ios a e mo e impo an han p ices. Hence, he new
g oup displace he less possibili y scena io cus ome s a
i s ime, =3. When his scena io is emp y, he model
going on o displace he below scena io. This way o
ope a e was less eal han he i s one, so ha we will
no use i .
IV. EXPERIMENTAL RESULTS
The i s s age in ou compu a ional expe iences in-
ol ed he cons uc ion o a se o p oblems. To con-
s uc a se o ins ances we conside ed a ho el wi h 200
iden ical ooms and we ha e es ed he de e minis ic
and s ochas ic p og amming models o indi idual cus-
ome s and g oups. Indi idual gues s can be booked in
i e di e en a es, desc ibed in Table I.
Table I- Indi idual P ice Classes
Class P ice
P emie e / Luxu y a e 250 €
Business / Supe io a e 175 €
S anda d / No mal a e 125 €
Economy / Discoun a e 90 €
Supe economy / Supe discoun a e 75 €
O he s inpu s we e andomly gene a ed he ime ho-
izon o p oblems has been wi hin he in e al [0, 180]
o maximum leng h o s ay wi hin he in e al [0, 21].
Fo each SP p oblem a e used h ee scena ios; low,
a e age and high. Fo he p obabili ies, we a e checked
In e na ional Con e ence on Knowledge Enginee ing and Decision Suppo
377
h ee possibili ies: p1 0.8/0.6/0.4; p2 0.6/0.4/0.2 and p3
0.7/0.5/0.3.
To es DP model and SP model, we sol ed he same
se o p oblems using wice. Thus, ou ins ances we e
andomly gene a ed o each p oblem size, once o DP
model and h ee, once o scena io, in he SP model.
This means ha in o al we sol ed he SP model 24
imes.
To examine he impac o s ochas ic p og amming
we sol ed he same se o p oblems using DP, DGP, SP
and SGP models. In o de o measu e he e ec i eness
o he p oposed models, he a e age esul s show in he
igu e 1. Compu a ions we e done using CPLEX as a
sol e .
4,68E+06
4,69E+06
4,70E+06
4,71E+06
4,72E+06
4,73E+06
4,74E+06
4,75E+06
4,76E+06
12345678
Ins ances
Re enue (€)
DP
DGP
SP
SPG
Figu e 1- The a e age e enue o p oblems by he DP,
DGP, SP and SGP models
No e ha he e enue ob ained by DP is bigge han
SP models. The di e ence be ween de e minis ic and
s ochas ic models is called he expec ed alue o pe ec
in o ma ion, EVPI. I shows how much one could ex-
pec o win i one we e old wha would happen be o e
making one’s decision. I measu es he alue o an-
domness, bu i does no show ha he de e minis ic
models canno unc ion well. A small EVPI means ha
andomness plays a mino ole in he model han i
EVPI alue is bigge .
A he same ime, g oup models ob ained be e solu-
ions han indi idual models. Because i he ou ope a-
o o e is wo se han he expec ed e enue o indi id-
ual cus ome s, manage ho el e use he g oup.
The pe cen age e o s ha e been compu ed wi h e-
spec he maximum e enue. The di e ence be ween
hem is less han 8%, so we will use SPG model o
sol e cus ome p oblems.
Figu e 2 shows he summa y o ime ob ained by DP
and he a e age ime o SP and g oup models. The
compu ing ime equi ed by he p oposed models is e y
low. All unning imes a e gi en in CPU seconds on an
In el Pen ium III 850 MHz wi h 64 Mb o RAM.
0
0,5
1
1,5
2
2,5
3
3,5
4
12345678
Ins ances
Time
DP
DGP
SP
SGP
Figu e 2- The a e age ime o p oblems by he DP, DGP,
SP and SGP models
No e ha :
• DGP ound bes e enue solu ions. Al hough
wi h demands calcula ed om o ecas ing
models, e enue dec ease due o EVPI alue.
• Wi h ega d o compu a ion ime, he highes
model is below ou seconds.
The e o e, we could conclude ha he SGP model as-
su es qui e sa is ac o y esul s wi h low compu ing
equi emen s, and hence i could be easonably used o
sol e much g ea e p oblems.
V. CONCLUSION
In his pape , we ha e s udied an in en o y pe ish-
able p oblem unde limi ed capaci y, which is di e en-
ia ed wi h p ice policies. Fi s , we ha e conside ed a
special case o he p oblem, which is modelled as de-
e minis ic p og amming. Then, s ochas ic p og am-
ming has been used o sol e he same case. The quali y
o he solu ions imp o ed, i models a e compa ed.
Compu a ional esul s indica ed ha he SGP model
inds solu ions o e y good quali y in a easonable
compu a ion ime.
REFERENCES
[1] B. Smi h, J. Leimkuhle and R. Da ow, “Yield
Managemen a Ame ican Ai lines”, In e aces, Vol
22, No. 1 (Janua y 1992), pp. 8-31.
[2] S. Kimes, “A S a egic App oach o Yield Manage-
men ”, in Yield Managemen : S a egies o he se -
ice indus ies, A. Ingold, U. McMahon-Bea ie and
I. Yeoman, Ed. Con iuum: London, 2000, pp. 3-14.
[3] J. McGill and G. Van Ryzin, “Re enue Manage-
men : Resea ch O e iew and P ospec s”, T anspo -
a ion Science, Vol 33, No. 2 (May 1999), pp. 233-
256.
[4] A. Lee, “Ai line Rese a ions Fo ecas ing: P obabil-
is ic and S a is ical Models o he Booking P ocess”,
Massachuse s Ins i u e o Technology, 1990, Ed.
M.I.T. Lib a ies Theses Collec ion.
[5] E. Williamson, “Ai line Ne wo k sea in en o y
con ol: Me hodologies and e enue impac s”, Mas-
In e na ional Con e ence on Knowledge Enginee ing and Decision Suppo
378
sachuse s Ins i u e o Technology, 1992, Ed. M.I.T.
Lib a ies Theses Collec ion.
[6] T. S cek, “Modeling Ai line G oup passenge De-
mand o Re enue Op imiza ion”, Massachuse s In-
s i u e o Technology, 1991, Ed. M.I.T. Lib a ies
Theses Collec ion.
[7] P. Kall and S. Wallace, S ochas ic P og amming,
1994, John Wiley & Sons.
[8] S. Boe , R. F eling and N. Pie sma, “Ma hema ical
p og amming o ne wo k e enue managemen e-
isi ed”, Eu opean Jou nal o Ope a ional Re-
sea ch, Vol 137, (2002), pp. 72-92.
In e na ional Con e ence on Knowledge Enginee ing and Decision Suppo
379
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