Mathematical models for group revenue management

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. 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