Combined overbooking and seat inventory control for two-class revenue management model

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1 Songklanakarn J. Sc. Technol. 38 (6), , Nov. - Dec Orgnal Artcle Combned overbookng and seat nventory control for two-class revenue management model Murat Somboon and Kannapha Amaruchkul Graduate School of Appled Statstcs, Natonal Insttute of Development Admnstraton (NIDA), Bang Kap, Bangkok, 040 Thaland. Receved: 0 November 05; Accepted: 6 February 06 Abstract We propose a two-class revenue management (RM) model, whch combnes two of the most mportant RM strateges, namely overbookng and seat nventory control for a passenger arlne. We derve a closed-form expresson for an optmal overbookng lmt that maxmzes the expected proft, and analytcally perform senstvty analyss by changng model parameters such as a revenue, a penalty cost assocated wth unsatsfed demand, a show-up probablty, a refund, a dened boardng cost, and a plane capacty. Keywords: overbookng, statc model, stochastc model, revenue management. Introducton In 04, the global economc crss damaged most economc sectors, but commercal arlne could stll grow.6% and were worth $704 bllon (Flght global, 05b). Accordng to 04 aerospace ndustry fnancal data, the commercal arlne ndustry would contnue to grow n the next ten years (Flght global, 05a). One of the keys to success s revenue management (RM) (Sabre Corporaton, 05). RM can be defned as sellng the rght product to the rght customer at the rght tme for the rght prce (Cross, 997). Major research areas n RM can be categorzed nto ) seat nventory control, ) overbookng, 3) prcng and 4) demand forecastng (McGll and van Ryzn, 999). In ths paper, we focus on two RM strateges, namely overbookng and seat nventory control, practced by a passenger arlne. Overbookng means that the arlne ntentonally sells more reservatons for a flght than physcal capacty on the arcraft to compensate for cancellatons and no-shows. The seat nventory control problem concerns wth mxng passengers n dfferent fare classes n the same arcraft Correspondng author. Emal address: muratsomboon@gmal.com compartment. In ths paper, we combne two strateges and propose a statc two-class overbookng model, n whch low fare (class-) customers arrve before hgh fare (class-) customers. The arlne ncurs a penalty cost for each rejected bookng request. The penaltes are dfferent for the two classes. The arlne may overbook class- customers. The two fare classes may have dfferent show-up rates. We want to fnd an optmal overbookng lmt that maxmzes the total expected proft. There are many multple-class bookng control models whch allow overbookng; see, e.g., Brumelle and McGll (989), Subramanan et al. (999), Gosav et al. (00), Lan et al. (0), Aydn et al. (0), and Lan et al. (05). These models are formulated as Markov decson processes, and most do not possess the closed-form solutons except Aydn et al. (0). Aydn et al. (0) assume that the random vector of bookng request follows a multnomal dstrbuton. Our model assumes general dstrbuton for the bookng request. In practce most commercal RM systems are based upon the two-class model, nstead of the mult-class model. In ths artcle, a closed-form soluton of the proposed twoclass model s obtaned. The two-class model, whch focuses on the bookng control problem, dates back to Lttlewood (97). All bookng

2 658 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , 06 requests show up at the tme of servce;.e., there are no cancellatons or no-shows: Lttlewood (97) does not allow overbookng. Shlfer and Vard (975) study the two-class overbookng model but does not nclude the bookng control problem. The two-class model, whch ncludes both overbookng and bookng control problem, can be found n Sawak (989) and Rngbom and Shy (00). Sawak (989) and Rngbom and Shy (00) extend Lttlewood (97) to allow no-show passengers. In Sawak (989), the bookng requests of the two classes are assumed to be contnuous, whereas those n Rngbom and Shy (00) follow bvarate normal. The show-up passenger n Sawak (989) and Rngbom and Shy (00) follows a bnomal dstrbuton. In ours, the bookng request needs not be normal, t can be any non-negatve nteger-valued random varable wth a general dstrbuton. In Rngbom and Shy (00), the refund s fully gven to class and the class receves no refund, whereas n ours, the refunds are gven to both classes, and the refund needs not be fully gven (.e., the refund can be expressed as a percentage of the fare). Smlar to other overbookng models, we accept the bookng requests up to the overbookng lmt, and addtonal requests are rejected. In Sawak (989), the arlne ncurs a penalty (loss-of-goodwll) cost for only class- rejected bookng request, whereas n ours, the penalty cost s gven to each rejected bookng request. Our refund and penalty scheme are more general and ft more cases n practce. Ths s the frst to nclude the refund cost and penalty cost smultaneously nto the twoclass RM model. The rest of the paper s organzed as follows. The model s formulated n Secton and analyzed n Secton 3. Secton 4 concludes our paper and provdes future research problems. All proofs are shown n Appendx.. Formulaton Let be the set of real numbers and + be the set of non-negatve ntegers. Let ( y) max(0, y) for y. The quantle functon of dstrbuton functon of random varable D s denoted as D F ( a) nf x : P( D x) a. Consder an arlne wth fxed capacty and two customer classes wth fares p p 0. We assume that all class- reservatons arrve before class- start reservaton. For each =,, the arlne earns revenue of p when a class- customer s accepted; on the other hand, f rejected the arlne ncurs a penalty cost g where g g 0. The penalty cost when a customer s rejected ncludes, e.g., the loss-of-goodwll cost, whch measures customer satsfacton, and the opportunty cost, whch measures future revenue loss. The loss-of-goodwll cost may be ntangble and can be dffcult to estmate n practce. The opportunty cost depend on what happens after the lost sales occur. If a customer s lkely to return to make a bookng request, then the opportunty cost s the expected revenue loss from ths event; however, f a customer never returns to make any bookngs wth the arlne, then the opportunty cost ncludes all future revenues the customer mght have brought to the arlne. Let x be an overbookng lmt of class : Class- bookng requests are accepted up to x. We allow overbookng;.e., can be greater than capacty. Let D be class- demand, the number of class- bookng requests. Assume that D and D are two ndependent -valued random varables. The number of class- reservatons s mn( x, D ), and the number of class- bookng requests rejected s ( D x). After class- reservatons all arrve, class- customers start ther bookng. The remanng capacty after class- arrves s mn( x, D ). We do not overbook class- because class- passengers are of hgh prorty or extremely hgh penalty cost. Class- customers are accepted up to the remanng capacty. For =,, let B ( x ) be the number of class- reservatons: B ( x) mn( x, D ), B ( x) mn(( B ( x)), D ). Some reservatons may cancel pror to or do not show up at the tme of servce. In ths model, we assume that cancellaton and no-show passengers are the same. Gven that the number of class- reservatons s B ( x) y, the number of class- show-ups, denoted by W ( y ), s assumed to follow a bnomal dstrbuton wth parameters y and where (0,] s the show-up probablty of class-. Note that when the show-up probablty of class- s equal to ( ) t means that all passengers of class- show up at the tme of servce. That the bnomal dstrbuton s an adequate model for the show-ups dstrbuton has been showed n Tasman Empre Arways (Thompson, 96). Each class- reservaton that does not show up receves a refund r, whch s a proporton of revenue cost where (0,) ; r p for =,. At the tme of servce, the number of show-up passengers may be over capacty. Recall that we overbook only class- passenger, so all dened boardng passengers are class-. The arlne pays a compensaton h to each dened boardng passenger where h p. Ths compensaton may nclude a fare of a hgher bookng class on a next flght, vouchers for cash or tckets for future travel, and/or hotel accommodaton. The arlne wants to choose an optmal overbookng lmt x that maxmzes ts expected proft: ( x) E p B ( x) r B ( x) W B ( x) E h W B ( x) E g ( D x) g D B ( x). () The frst term n () s the expected of revenue pb ( x ) mnus the expected refund cost pad to reservatons wth no shows r ( B ( x) W ( B ( x))). The second term s the expected dened boardng cost, the company pays to the dened boardng passengers when the number of show-ups s more than capacty. Recall that we do not overbook class- customer, so all dened boardng passengers are class-. The last term s the expected penalty cost, the expected revenue lost when

3 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , we reject the bookng requests of class and. Note that n () there are two sources of uncertanty, namely demand uncertanty and the number of show-ups. In practce, an optmal overbookng lmt s re solved perodcally to account for change n show-up probablty and proporton of refund cost over tme, resultng n overbookng lmts that vary over tme. The arlnes accept the reservatons at any tme up to the current overbookng lmts. Typcally, arlnes may only monthly update the optmal overbookng lmt for a flght before departure at least sx months (Phllps, 005). Most arlnes recalculate the optmal overbookng lmt everyday durng the last week before departure. 3. Analyses In ths secton, ncreasng (respectvely; decreasng) means non-decreasng (respectvely; non-ncreasng). For,, let p g r r and /. Let ( ;, ) t x j ( ) x j F t x j 0 j s the tal-sum probablty (complementary cumulatve dstrbuton functon) of bnomal dstrbuton wth parameters x and. Theorem. The expected proft functon ( x) s pecewse on x 0,,..., and x,,..., and t s unmodal n each pece.. For x 0,,...,, the expected proft ( x) has a local maxmum pont x gven by 0 ; 0 P( D ) ( ) ; ( ) ( 0) () ; P( D 0) x F P D P D D. For x,,..., f 0 / ( h ) F ( ;, ), then the expected proft ( x) has a local maxmum pont x gven by x arg mn{ x {,,...} : F ( ; x, ) }. (3) h Otherwse, the expected proft functon s ncreasng. From Theorem, we can fnd the optmal overbookng lmt x from three ponts; x, and x. Suppose that 0 / ( h ) F ( ;, ). Then arg max{ ( ), ( ), ( )}. x x x Suppose that / ( h ) F ( ;, ). If lm ( x) max{ ( x), ( )}, then x arg max{ ( x), ( )}. x Otherwse, ( x) s ncreasng, and the arlne should set the optmal overbookng lmt to be as large as possble. The optmal overbookng lmt n Theorem has a closed-form that s easy to calculate. Ths soluton can be extended to heurstc method n multple fare classes model whch s better than usng Markov decson process. Dfferent shapes of the expected proft functon are shown n Fgures and. In Fgures and, demand, D, s assumed to be Posson random varable wth mean for,, and capacty s 00. Although Posson random varable s assumed n numercal experments, the proof of Theorem does not need to assume Posson dstrbuton. Ths theorem holds for any non-negatve random varable. We set the penalty cost equal to the revenue;.e., g p and g p. We assume that the opportunty cost s the lost revenue from rejectng a reservaton from that partcular class. In Fgure, we do not overbook;.e., x. The maxmum expected proft n the frst pece (n Theorem ) s greater than that the second pece. In Fgure d, the optmal overbookng lmt s x 0 ; ths correspond to the frst case n (). It means that we do not accept class- reservatons when mean demand of class- s large. In Fgure a, b, c, the optmal overbookng lmt x s gven n the second case n (). Note that as the mean demand of class- ncreases, the optmal overbookng lmt decreases. In Fgure, overbookng occurs;.e., x. The maxmum expected proft n the second pece (n Theorem ) s greater than that n the frst pece;.e., ( x) ( x). Note that as the dened boardng cost h ncreases, the optmal overbookng lmt decreases. From Fgures and, we see that the optmal overbookng lmts change the locaton when some model parameters change. We formally perform senstvty analyss n the next Corollares. Corollary. For x 0,,...,, suppose that the rato / or capacty ncreases or D decreases wth respect to usual stochastc order. Then, the local maxmum pont x ncreases. Suppose that we do not overbook; for nstance, () mean demand of class- s larger than capacty, () mean demand of class- s much lower than capacty. There are many cases that ncrease the optmal overbookng lmt wth correspondng to / ncreases, e.g. () revenue cost of class- ncreases (or revenue cost of class- decreases), () penalty cost of class- ncreases (or penalty cost of class- decreases), (3) show-up probablty of class- ncreases (or show-up probablty of class- decreases), (4) refund cost of class- decreases (or refund cost of class- ncreases). In Corollary, we ndcate how the local maxmum pont x changes, when the model parameter of class- and capacty are vared. Corollary. For x,,..., suppose that the rato / ( h ) or capacty ncreases. Then, the local maxmum pont x ncreases. Suppose that we overbook; for nstance, () class- revenue s close to class- revenue, () mean demand of class- s much hgher than capacty. There are many cases that ncrease the optmal overbookng lmt wth correspond-

4 660 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , 06 Fgure. The expected proft when varyng 40, 60, 80 and 00. Other parameters are p g 00, p g 0, r 50, r 0, 0.9, 0.7 and h = 300. Fgure. The expected proft when varyng h = 00, 300 and 500. Other parameters are p g 00, p g 80, r 80, r 40, 0.9, 0.7 and 40. ng to / ( h ) ncreases, e.g. () revenue cost of class- ncreases, () penalty cost of class- ncreases, (3) show-up probablty of class- decreases, (4) refund cost of class- decreases, (5) dened boardng cost decreases. Corollary mples that the arlne may need to update the optmal overbookng lmt when there s an unusual stuaton such as dsaster, nsurgence or demonstraton whch affects some parameters n the model. For nstance, Bangkok bomb at Erawan shrne on 7 August 05 may decrease a show-up probablty from toursts who plan to travel to Bangkok. When the show-up probablty decreases, the arlne may need to set a hgher overbookng lmt. On the other

5 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , hand, durng long holday such as Chrstmas, new year festval, Songkran festval the show-up probablty may be hgher; consequently the arlne may decrease the overbookng lmt. 4. Conclusons In ths artcle, we propose a statc two-class overbookng model and derve an optmal overbookng lmt that maxmzes the expected proft. The parameters n the model are revenue cost, penalty cost, refund cost, show-up probablty, dened boardng cost and capacty. Senstvty analyses wth respect to changes n model parameters are performed: If t s optmal not to overbook, then the bookng lmt s affected by the demand of class and all of model parameters except for dened boardng cost. If t s optmal to overbook, then the overbookng lmt s affected by all of model parameters of class- ncludng dened boardng cost and capacty. It s possble to extend the study as follows: Estmatng the parameters n the model when demand s censored. The model, n whch demands are dependent, could be studed. Moreover, we can allow overbookng on class ; ths model would have two overbookng lmts. We hope to pursue some of these related problems n the future. Acknowledgements The authors would lke to express ther apprecaton to the Natonal Insttute of Development Admnstraton (NIDA), Thaland, for ther fnancal support. References Aydn, N., Brbl, S.L., Frenk, J., and Noyan, N. 0. Sngleleg arlne revenue management wth overbookng. Transportaton Scence. 47(4), Brumelle, S. and McGll, J A general model for arlne overbookng and two-class revenue management wth dependent demands. Techncal report, Workng Paper, Unversty of Brtsh Columba, Vancouver, BC., Canada. Cross, R.G Revenue Management: Hard-core Tactcs for Market Domnaton. Broadway Books, New York, U.S.A. Flght global. 05a. Sze matters: Ten bggest players n aerospace. Avalable from: com/news/artcles/sze-matters-ten-bggest-players-naerospace-46563/. [September, 05] Flght global. 05b. Top ten: How the world s bggest arlnes ranked n 04. Avalable from: flghtglobal.com/news/artcles/top-ten-how-the-worlds -bggest-arlnes-ranked-n-4574/. [September, 05] Gosav, A., Bandla, N., and Das, T.K. 00. A renforcement learnng approach to a sngle leg arlne revenue management problem wth multple fare classes and overbookng. IIE transactons. 34(9), Lan, Y., Ball, M.O., and Karaesmen, I.Z. 0. Regret n overbookng and fare-class allocaton for sngle leg. Manufacturng and Servce Operatons Management. 3(), Lan, Y., Ball, M.O., Karaesmen, I.Z., Zhang, J.X., and Lu, G.X. 05. Analyss of seat allocaton and overbookng decsons wth hybrd nformaton. European Journal of Operatonal Research. 40(), Lttlewood, K. 97. Forecastng and control of passenger bookngs, Proceedng of Agfors th Annual Symposum, October, 97, Nathanya, Israel, McGll, J.I. and van Ryzn, G.J Revenue management: Research overvew and prospects. Transportaton Scence. 33(), Muller, A. and Stoyan, D. 00. Comparson methods for stochastc models and rsks. Wley, U.S.A. Phllps, R Prcng and revenue optmzaton. Stanford Unversty Press, U.S.A. Rngbom, S. and Shy, O. 00. The adjustable-curtan strategy: Overbookng of multclass servce. Journal of Economcs. 77(), Sawak, K An analyss of arlne seat allocaton. Journal of Operatons Research Socety of Japan. 3(4), Sabre Corporaton. 05. Maxmum your revenue on each and every flght. Avalable from: sabrearlnesolutons.com/home/ndustry_challenges/ revenue_ growth/. [September, 05] Shlfer, E. and Vard, Y An arlne overbookng polcy. Transportaton Scence. 9(), 0-4. Subramanan, J., Stdham Jr, S., and Lautenbacher, C.J Arlne yeld management wth overbookng, cancellatons, and no-shows. Transportaton Scence. 33(), Thompson, H. 96. Statstcal problems n arlne reservaton control. Operaton Research. (3),

6 66 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , 06 Appendx Lemma. The expected proft ( x) can be wrtten as follows: For x 0, (4) ( x) E[ B ( x)] g E[ D ]. For x,,...,, ( x) [ E[ B ( x)] g E[ D ]]. For x,,..., (5) (6) ( x) ( h ) E[ B ( x)] E[ B ( x)] g E[ D ] h P( W ( B ( x)) t), t 0 where the expected number of class- reservatons s 0 ; x 0 E[ B ( x)] x P( D t) ; x,,... t 0 and the expected number of class- reservatons s P( D t) ; x 0 t 0 x E[ B ( x)] ( ) ( ) ( ) ;,,..., P D t P D t P D t x t 0 t x P( D t) P( D t) ; x,,... t 0 Proof. From (), we obtan ( x) ( p r ) E[ B ( x)] r E[ W ( B ( x))] he[( W ( B ( x)) ) ] g E[( D x) ] g E[( D ( B ( x)) ) ]. If x 0, we use a tal-sum formula for expectaton to fnd the expected number of class- reservatons (7) Smlarly, x E[ B ( x)] P(mn( x, D ) t) P( D t). t 0 t 0 t 0 E[ B ( x)] P(mn(( B ( x)), D ) t) P(( B ( x)) t) P( D t) t 0 t 0 [ P(( B ( x)) t)] P( D t). (8)

7 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , Clearly, f x 0, then E[ B ( x)] 0. If x 0, then E[ B ( x)] P( D t). t 0 If x,,...,, the probablty mass functon of B ( x) s gven as P( D x) ; k x P( B ( x) k) P( D k) ; k x,..., 0 ; otherwse (9) If x,,..., the probablty mass functon of ( B ( x)) s gven as P( D x) ; k 0 P(( B ( x)) k) P( D k) ; k,,..., 0 ; otherwse Substtuton (9) and (0) nto (8), we obtan (0) P( D t) ; x 0 t 0 x E[ B ( x)] ( ) ( ) ( ) ;,,..., P D t P D t P D t x t 0 t x P( D t) P( D t) ; x,,... t 0 The number of class- show-ups, W ( y ), has bnomal dstrbuton wth parameters y and (0,] where y s the number of class- reservatons and s the show-up probablty of class-. Then, E[ W ( B ( x)) B ( x) y ] y. So, E[ W ( B ( x))] E[ B ( x)] ;,. () We know that ( a b) a mn( a, b). Thus, Smlarly, E[( D x) ] E[ D mn( x, D )] E[ D ] E[ B ( x)]. () E[( D ( B ( x)) ) ] E[ D ] E[ B ( x)]. (3) The expected number of class- passenger who are dened boardng s E[( W ( B ( x)) ) ] E[ W ( B ( x))] E[mn( W ( B ( x)), )] E[ B ( x)] P( W ( B ( x)) t). t 0 Substtuton (), (), (3) and (4) nto (7), we obtan ( x) E[ B ( x)] g E[ D ] h E[ B ( x)] P( W ( B ( x)) t). t 0 (5) After substtute E[ B ( x )] and E[ B ( x )] nto (5), the expected proft becomes (4) - (6). (4)

8 664 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , 06 Proof of Theorem. For x 0,,,..., let ( x) ( x ) ( x) be the forward dfference of the expected proft. Denote G ( x) B ( x ) B ( x);,. Clearly, and E[ G ( x)] P( D x) x P( D x) P( D x ) ; x 0,,..., E[ G ( x)] 0 ; x,,... After some tedous algebra, we obtan the expresson for the dfference as follows. For x 0,,...,, ( x) P( D x)[ P( D x )]. For x, ( x) P( D x)[ P( D x )] he[( W ( B ( x)) ) ] For x,,..., ( x) P( D x)[ h F ( ; x, )], t x j x j where F ( t; x, ) ( ) s the tal-sum probablty of bnomal dstrbuton wth parameters x and j 0 j. We wll consder two pece: x 0,,..., and x,,.... Defne ( x) P( D x ). Consder the frst pece, ( x) P( D x)[ P( D x )] P( D x) ( x). We observe that ( x) has the same sgn as term ( x). Then. If ( x) 0 for x 0,,...,, then the expected proft ( x) s ncreasng and a local maxmum pont s. If ( x) 0 for x 0,,...,, then the expected proft ( x) s decreasng and a local maxmum pont s We wll show that ( x) s decreasng n x. If ( x) 0, x x and ( x) 0, x x, then there exsts a local maxmum pont s x such that ( x) s ncreasng for x x and decreasng for x x. A local maxmum pont s at x. If P( D 0) /, then ( x) 0 for all x 0,,...,, so ( x) 0 for x 0,,...,. The expected proft functon s ncreasng n x. A local maxmum pont s. If 0 / P( D ), then ( x) 0 for all x 0,,...,, so ( x) 0 for x 0,,...,. The expected proft functon s decreasng n x. A local maxmum pont s 0. Recall that P( D x ) s ncreasng n x so ( x) s decreasng n x. If P( D ) / P( D ), then (0) 0 and ( ) 0,.e., there exsts a local maxmum pont x such that ( x) 0, x x and ( x) 0, x x. So, ( x) 0, x x and ( x) 0, x x,.e., a local maxmum pont x gven by x arg mn{ x {0,,..., }: P( D x ) } (6) Let y x. Then, for x 0,,...,, we have that y,,...,. Also, y x. A local maxmum pont condton (6) becomes So, y y P D y F D arg max{ {,,..., }: ( ) } ( ) x F D ( ). Next, consder the second pece, x,,....

9 M. Somboon & K. Amaruchkul / Songklanakarn J. Sc. Technol. 38 (6), , ( x) h F( ; x, ). Let ( x) P( D x)[ h F( ; x, )] P( D x) ( x). We fnd that ( x) has the same sgn as term ( x). Then. If ( x) 0 for x,,..., then the expected proft ( x) s ncreasng and a local maxmum pont s set as large as possble.. We wll show that ( x) s decreasng n x. If ( x) 0, x x and ( x) 0, x x, then there exsts a local maxmum pont s x such that ( x) s ncreasng for x x and decreasng for x x. A local maxmum pont s at x. / ( h ), then ( x) 0 for all x,,..., so ( x) 0 for x,,.... The expected proft functon If s ncreasng n x. A local maxmum pont s set as large as possble. Recall that F ( ; x, ) s ncreasng n x, so ( x) s decreasng n x. If F ( ;, ) / ( h ) then (0) 0 and lm ( x) 0,.e., there exsts a local maxmum pont x such that ( x) 0, x x and ( x) 0, x x. x So, ( x) 0, x x and ( x) 0, x x,.e., a local maxmum pont s gven by x arg mn{ x {,,...}: F ( ; x, ) }. (7) h Proof of corollary. Note that a functon P( D x ) s ncreasng n x. The drectonal change of x wth respect to / s obvous n equaton (6). Let and ˆ be a functon that has a local maxmum pont x and ˆx respectvely. Snce, ˆ, then x xˆ. Note that P( D x ) s ncreasng n. Consder the drectonal change of x wth respect to. Let and ˆ be capacty such that ˆ, y x and yˆ ˆ x. P( D y) P( D y) ˆ F ( ; ) F ( ; ˆ ) D D ˆ F ( ; ˆ ) D Thus, x xˆ. Consder the drectonal change of x wth respect to D. Assume that D Dˆ st,.e., D smaller than ˆD wth respect to usual stochastc order (Muller and Stoyan, 00). Let F and G be dstrbuton functons of D and ˆD, and F ( t) P( D t) and G( t) P( Dˆ t). So, F( x ) G( x ) for all x {0,,..., }. Let x and x be local maxmum ponts of D and ˆD respectvely. Snce, F and G are ncreasng n x. Thus x xˆ. Proof of Corollary. Note that a functon F ( ; x, ) s ncreasng n x. The drectonal change of x wth respect to / ( h ) s obvous n (7). Let / ( h ) and let x be a local maxmum pont assocate wth. Smlarly, let ˆ ˆ / ( hˆ ˆ ) and let ˆx be a local maxmum pont assocate wth ˆ. Snce, ˆ, then x xˆ. Recall that F ( ; x, ) s decreasng n. Consder the drectonal change of x wth respect to. Let. and ˆ be capacty such that ˆ, F ( ; x, ) F ( ˆ ; x, ). Thus, x xˆ.