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  DYNAMIC OVERBOOKING LIMITS FOR GUARANTEED AND NONGUARANTEED RESERVATIONS Stanislav Ivanov, Ph. D.International University College3, Bulgaria Str., 9300 Dobrich, Bulgariatelephone: 3!9 !" #!!#$%e&'ail: stanislav(h(ivanov)' Abstract :*he research note a++resses the proble' o +yna'ic overboo-ing li'its in the contet o guarantee+ an+ nonguarantee+ boo-ings. *he note sho/s ho/ the opti'al nu'ber o overboo-ings /ill change /hen the hotel receives a reuest or a speciic +ate given the current guarantee+1nonguarantee+ reservations ratio or the sa'e +ate. Key words  hotel 'anage'ent, overboo-ing, yiel+ an+ revenue 'anage'ent$  I!trod"ct#o! 2verboo-ing is a /ell&-no/n an+ /i+ely use+ practice in the hotel in+ustry. Its goal is to 'ai'ise roo' revenue by conir'ing 'ore roo's than the available capacity o the property. ro' a theoretical perspective it has been usually analyse+ in the contet o yiel+ an+ revenue 'anage'ent 4e.g. 5ain an+ Bo/'an, %00!6 7ee&8oss an+ 5ohns, $996 etessine an+ Shu's-y, %00%6 Sanche; an+ Satir, %00!6 Ivanov, %00#<. =ost o the research has ocuse+ so ar on calculating a  static  opti'al overboo-ing level 4or econo'etric 'o+elling o static overboo-ing li'its see etessine an+ Shu's-y, %00%6 Ivanov, %00#<. It has been argue+ that a hotel can aor+ to overboo- until the 'arginal revenues ro' the overboo-ings ecee+ the 'arginal costs associate+ /ith the'. >lbeit their 'erits, overboo-ings are not /ithout their shortalls. ?irt; et al 4%00%:$"< e'phasi;e that overboo-ings an+ other inventory control revenue 'anage'ent practices coul+ cause perceive+ unairness or perceive+ lac- o custo'er appreciation by clients. In or+er to avoi+ the large inancial costs, 'isse+ revenues an+ negative /or+&o&'outh associate+ /ith overboo-ings, hotel co'panies shoul+ collect +etaile+ inor'ation on no&sho/s, cancellations, an+ last 'inute a'en+'ents 4@i'es an+ =cAuire, %00$:$$<.>lthough 'uch research has ocuse+ on overboo-ings, the proble' o dynamic  overboo-ing li'its has not receive+ 'uch attention. In our research note /e a++ress the proble' by posing the uestion:  How will the optimal number of %  overbooked rooms change when we receive a booking request for a specific date  given the current guaranteed/nonguaranteed reservations ratio for the same date  *he opti'al level o overboo-ings is inversely relate+ to the a'ount o the cancellation charge  the closer the cancellation charge to the roo' rate, the lo/er the 'isse+ beneit ro' the unoccupie+ roo', an+ the less the sti'uli to overboo- 4Ivanov, %00#:%9<. I a hotel accepts both guarantee+ an+ non&guarantee+ boo-ings then the opti'al nu'ber o overboo-ings 'ust be set separately or each boo-ing type. Met$odo%o&y 7et  N   be the nu'ber o nonguarantee+ boo-ings at the hotel or a speciic +ate, η    the average percentage o unuse+ nonguarantee+ boo-ings, A  the nu'ber o guarantee+ boo-ings, γ    the average percentage o unuse+ guarantee+  boo-ings, an+ C    the total available capacity o the hotel 4or si'plicity /e shall assu'e that all roo's are i+entical an+ co'plete substitutes<. *he epecte+ use+ capacity o the hotel ep C   is +eine+ as:4$< ( ) ( ) γ η   −+−=  $.$. ep G N C  ?e +ierentiate 4$< to  N   an+ G an+ arrive to:4%< η  −=∂∂ $ ep  N C   , an+43< γ  −=∂∂ $ ep GC   .3  or'ulas 4%< an+ 43< sho/ the increase in the epecte+ use+ roo' inventory ep C   as a result o one a++itional nonguarantee+ or guarantee+ boo-ing, respectively. *hey also i'ply that the epecte+ use+ capacity /ill increase less than the nu'ber o a++itional roo's boo-e+.o/ever, reservation 'anagers are 'ore intereste+ in ho/ the overboo-ing li'its /ill change i they conir' particular reuest. *o a++ress this proble', /e epress  N   as a unction o G :4E< η γ η   −−−−= $$.$ GC  N  .Dierentiating 4E< lea+s to:4!<       −−−=∂∂ η γ  $$ G N  Fpression 4!< sho/s the 'arginal rate o substitution o nonguarantee+  boo-ings by guarantee+ ones. or ea'ple, i γ  G!H an+ η  G%!H, then:4#< %.$%!.0$ 0!.0$ $$ −=     −−−=      −−−=∂∂ η γ  G N  *he result sho/s that a guarantee+ boo-ing o E roo's /ill have the sa'e i'pact on the epecte+ use+ capacity as a nonguarantee+ boo-ing o ! roo's 4E$.%G!.0"<.7et us consi+er that a hotel receives only nonguarantee+ boo-ings. I it has $00 si'ilar roo's an+ η  G%!H, then the hotel 'ust conir':4< $33%!.0$ $00$ 'a  =−=−= η  C  N   roo's.E  >ccor+ing to 4< the hotel /ill overboo- 33$00$33 'a  =−=−  C  N   roo's.In the other case, i all boo-ings are guarantee+ an+ γ  G!H, the hotel shoul+ conir':4"< $0!0!.0$ $00$ 'a  =−=−= γ  C G  roo's,/hich correspon+s to only !$00$0! 'a  =−=−  C G  overboo-e+ roo's.*he t/o analyse+ situations +eter'ine the upper an+ lo/er overboo-ing li'its 4see igure $.<igure $ presents a population o points /hich are a linear co'bination o the nu'ber o guarantee+ G  an+ nonguarantee+  N   reservations an+ have an ! G N  0 G maxmax G ii  E  igure !"  8elationship bet/een guarantee+ an+ nonguarantee+ reservations
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