Marginal Revenue-Based Capacity Management Models and Benchmark 1

Margnal Revenue-Based Capacty Management Models and Benchmark 1 Qwen Wang 2 Guanghua School of Management, Pekng Unversty Sherry Xaoyun Sun 3 Ctgroup ABSTRACT To effcently meet customer requrements, a manager must supply adequate quantty of products, or capacty, or servces at the rght tme wth the rght prce. Revenue management technque can help frms to use dfferental prcng strateges and capacty allocaton tactcs to maxmze revenue. In ths paper, we propose Margnal Revenue-Based Capacty Management (MRBCM) models based on revenue management prncple to manage stochastc demand at a mcro-level to create revenue opportuntes. In partcular, MRBCM models are created to generate order acceptance polcy, that s, to allocate avalable capacty for promsng to alternatve market segments. In our llustraton, products are classfed by revenue contrbuton n ther respectve capacty unt, low, mddle, and hgh. In three models (MRBCMa, MRBCMb, MRBCMc), the amounts of capacty are reserved for the hgh and mddle revenue classes. As an enhancement, MRBCMb and MRBCMc models consder opportunty revenue and cost n varous completed fashons. To evaluate these models, we desgn and conduct smulatons for 16 scenaros and compare three MRBCM models and two smple methods wth the Frst-Come-Frst-Served (FCFS) polcy n a sngle plannng horzon. The expermental results show that MRBCM models generate sgnfcant hgher profts over FCFS rule at each scenaro. 1. INTRODUCTION To meet dversfed customer needs, products and servces can be provded n assocate wth varant features, delvery tme, volume dscounts, and fnancal terms, and thus can be prced accordngly. Obvously hgher prced orders ought to be processed n hgher prorty. The entre capacty then can be segmented nto multple categores n respect to the prortes or prces (called market segmentaton). In stuatons when demand exceeds supply and expandng faclty s not avalable n the short term, a frm needs to develop a tactcal decson polcy to help best use of avalable resources. Therefore, to allocate lmted pre-exstng capacty for order promsng s very crtcal for both servce and manufacturng ndustres. Revenue management concepts shed great lghts on attackng ths type of problems. It nvolves two processes to balance demand and supply: regulatng demand by changng prces or dscount levels for products or servces and regulatng supply by adjustng producton levels or capacty 1 Ths research s supported by the Natonal Natural Scence Foundaton of Chna, No. 70471013. 2 Qwen Wang, wqw@gsm.pku.edu.cn 3 Sherry Xaoyun Sun, sherrysun@comcast.net 503

avalablty. An effectve revenue management means choosng the mx of supply and demand regulaton actvtes that maxmzes proft. Revenue management has been successfully appled to many servce ndustres, such as arlnes [Belobaba 1989; Smth et al. 1992; Weatherford and Bodly 1992; Weatherford 1998], car rentals [Carroll and Grmes 1995; Geraghty and Johnson 1997], hotel and resort [Kmes 1989; Badnell 1995], electrc utlty companes and telecommuncaton frms [Haas 1993]. In general, revenue management technque works partcularly well when short-run capacty s nflexble and orders are prce dscrmnable. Throughout the paper, revenue management s used for an order acceptance process that apples dfferental prcng strateges and stop-sales tactcs to manage demand, allocate capacty, and enhance delvery relablty and speed, and therefore to maxmze revenue from pre-exstng capacty. Well performed n servce ndustry, revenue management also has great potental n manufacturng envronment. In make-to-stock (MTS) producton, the revenue management tactcal rules nvolve stock ratonng. The nventory s selectvely ratoned to customers based on ther relatve mportance when avalable nventory s not suffcent to fulfll the orders [Haynsworth and Prce 1989; Gallego and Van Ryzn 1994]. In assemble-to-order (ATO) stuaton, on the other hand, revenue management consderatons are typcally related to allocate undfferentated unts of capacty to alternatve market segments of varyng proftablty. The capacty may be specal faclty or equpment that could be prcng dfferently accordng to customer order tme and delvery speed [Harrs and Pnder 1995]. A seres research by Balakrshnan et al. [1996, 1998] and Patterson et al. [1997] has developed capacty ratonng models for make-to-order (MTO) manufacturers and servce frms whch focus on how to allocate the avalable capacty to dfferent products or customers based on ther relatve proft or prorty. To ratonng scarce capacty between two or more product classes, a certan amount n total capacty needs to be reserved for the hgher-proft class, accordng to the proft contrbuton per capacty unt of the product orders. The smulaton results show that ratonng models produce sgnfcantly hgher profts than the model wth no capacty ratonng. Based on the revenue management prncple, ths paper proposes margnal probablstc models for Revenue-Based Capacty Management (MRBCM). MRBCM models are used to manage stochastc demand and offer stop-sales tactcal polces to maxmze expect proft. From the concept, MRBCM models can handle above stuatons we revewed n the manufacturng frms. More generally, the MRBCM-based tactcal decsons can help any types of frms that may nvolve 1) Allocaton of same type of products nto alternatve market segments assocated wth dfferent profts (e.g., n the servce ndustres, such as allocatng seats n arlnes, rooms n hotels, broadcast tme n advertsng frms, and cars n car-rentals, or an nsuffcent nventory n MTS frms); 2) Allocaton of capacty/servces nto dfferent products or customer channels as long as consdered capacty can be aggregated as undfferentated unts (e.g., ATO frms) or product order sze can be expressed n terms of capacty unts (e.g., MTO frms). For the sake of smplcty and wthout loss of generalty, ths paper assumes that a frm produces three classes of products and ther proft contrbutons per unt of capacty are low, mddle, and 504

hgh, respectvely. The product n the dfferent classes can be replaced by others. We refer to these three product classes as class 1 (for lowest proft), class 2 (for mddle proft), and class 3 (for hghest proft) and use R1, R2, and R3 to represent ther revenue per unt of capacty, respectvely. Three smple methods and three models, MRBCMa, MRBCMb, and MRBCMc, are developed to maxmze total expected revenue for avalable product capacty. The MRBCMb and MRBCMc models, n whch we consder opportunty revenue or/and costs of the hgher classes. The models and algorthms developed for three classes can be easly extended to multple classes. To benchmark our algorthms, we desgn and conduct smulatons and evaluate the models relatve to the base case wth no capacty reservaton,.e., Frst-Come-Frst-Served (FCFS) polcy. Ths paper s organzed as follows. The Secton 2 descrbes the man ssue and notatons. The Thrd Secton ntroduces three basc methods that buld a base to compare performance of dfferent models. The Forth Secton looks deeply nto the ssue and presents three MRBCM models wth formulas, optmalty condtons, and algorthms. Further expermental desgn s developed and the smulaton results are dscussed n Secton Fve. At last Secton, the Sxth, everythng stated above s clearly concluded. 2. BASIC VARIABLES AND ORDER ACCEPTANCE RULE In the real busness world, demand can never be precsely predctable and capacty s often lmted and nflexble. If all orders are confrmed on FCFS bass, the entre capacty wll soon be exhausted by lower-end orders. The busness may loss hgh margnal orders from emergent customers who are more lkely wllng to pay hgher premum for qucker delvery. To ensure the avalablty to those customer segments, certan porton of capacty must be held up front. Two decsve questons then arse: 1) What s the optmal amount of capacty should be reserved for hgh-end orders? 2) What knd of order acceptance rule should be set? We wll try to answer these questons n ths paper. 2.1 Parameters and Varables The followng notatons provde the defnton for parameters, varables, and functons used n ths paper. X -- the stochastc varable of demand for class, for =1,2,3. f (x) the probablty densty functon of varable X, for =1,2,3. F (x)= P (X x) s the dstrbuton functon of varable X, for =1,2,3. µ -- the mean of the dstrbuton of varable X, for =1,2,3. R -- the revenue or proft per unt from class, for =1,2,3. N -- the total avalable capacty for three classes of the product T -- the upper boundary of the arrval tme for demand orders Two decson varables are assocated wth ths decson problem: 1) Protecton Level (PL) the mnmum number of unts reserved for a partcular class,.e., to be protected from sale to all lower proft classes. 2) Order Quota (OQ) the maxmum number of unts allowed for sale to a partcular class. 505

A herarchcally nestng allocaton polcy s adopted for our order acceptance rule. Nestng means that, the upper class s allowed to take the capacty from the lower classes f need to, whle the lower class s prohbted to use the capacty reserved for upper classes. In the event of an unantcpated demand, nestng avods denyng order requests for hgher classes as long as there are capactes avalable n lower-proft classes. Besdes calculatng PL and OQ at the begnnng of the plannng horzon, we need track other three quanttes dynamcally n the order acceptance process: 1) Current Orders -- O(), the current total orders so far for class 2) Used capacty -- U(), the current number of used capacty unts that were reserved for class 2.2 Order Acceptance Rule Revenue Management Models manly deal wth demand uncertanty at a mcro-level to create revenue opportuntes. The model of allocaton polcy tres to generate an optmal protecton level for each class to avalable to promse. The capacty reservatons are set at the begnnng of the plannng horzon and reman unchanged. Followng assumptons for our models are hereby set forth. Frst, we assume random demands of three classes are Posson dstrbuted over the plannng horzon and the quantty of each order s ndependent. Second, a nestng structure s adopted regardng order acceptance rule. Thrd, the frm receves all orders over a sngle plannng horzon, ether accepts or rejects order as t arrves based on order acceptance rule. Fnally, we assume that demand or capacty cannot be carred over from one plannng horzon to another. The allocaton polcy needs to be set at each plannng horzon accordng to the forecasted demand dstrbutons and related profts. For smplcty, we consder each order has an average same sze and let the sze to be 1. For a gven plannng horzon T, the models are mplemented based on the followng order acceptance/rejecton rule. Algorthm Step 1: At t = 0, calculate three protecton levels PL(),for =1,2,3. Intalze O() = 0 and U() = 0 for = 1, 2, 3. Step 2: Receve an order accordng to the sequence of arrvals. If the order receved at tme t s for product class. Step 3: If t>t, go to Step 4. Otherwse, f U() <PL(), accept the order and let O()=O()+1, U()=U()+1; Otherwse, check class k=1 to 3 n turn; If R k < R and U(k) <PL(k), then set U(k)=U(k)+1 and O()=O()+1; Otherwse, reject the order. 3 If O ( ) < N, go to Step 2; otherwse go to Step 4. = 1 506

3 Step 4: Calculate the total revenue R O( ), then stop. = 1 3. BASIC METHODS In ths secton, we wll ntroduce three basc methods. These methods have very smple rules or formulas, and are very easy to understand. Usually, ther results are not qualfed compared to the complex models that are descrbed n secton 4. However, ther results can help us to measure the mprovement of the MRBCM models. 3.1 Frst-Come-Frst-Served Method (FCFS) The FCFS method does not set the protecton levels for hgher classes. It s equvalent to settng PL( )=N, for =1,2,3. 3.2 Mean-Weghted Capacty Management Method (MWCM) The MWCM method sets the protecton level for each class accordng to ts mean of the dstrbuton of orders,.e., 3 PL( )=N ( µ / µ j) for =1,2,3. j= 1 3.3 Mean-Revenue-Weghted Capacty Management Model (MRWCM) It s easy to know that the total revenue depends on the number of orders and the prce that s the revenue per unt of the order. The MRWCM method sets the protecton level for each class accordng to the product of the mean of the dstrbuton of orders and ts unt-revenue as follows: 3 PL( )=N ( R µ / R j µ j) for =1,2,3. j= 1 4. MARGINAL REVENUE-BASED CAPACITY MANAGEMENT MODELS (MRBCM) In ths secton, we wll provde three models, whch are non-lnear programmng models. Usng Kuhn-Tucker optmalty condtons to the frst model, we get margnal revenue equatons, whch s the base to buld our MRBCM models. 4.1 Model MRBCMa MRBCMa model, s a smple Revenue-Based Capacty Management (MRBCM) model. Its objectve s to maxmze total expected proft, not consderng usage of resources across classes. Formulaton Max E 1 (x 1 ) + E 2 (x 2 ) + E 3 (x 3 ) s.t. x 1 + x 2 + x 3 <= N 507

x 1, x 2, x 3 >= 0 (4.1) Where x = the sze of volume of capacty to be reserved for class, ( = 1,2,3), that s, the protecton level of class,.e., x =PL(), E (x ) = the expected proft for class. In order to calculate E (x ), we have x E ( x ) = R t f ( t) dt + R x f t dt o x ( ) (4.2) Note the second term of above equaton that when the demand s greater than x, only x unts of product for class s sold. In ths paper, we assume the demand for class obeys a Posson dstrbuton then the above equaton just needs change to a dscrete case,.e., from an ntegral to a sum as follows: j x E ( x ) = R jp( X = t) + R x P( X = j) (4.3) j = 1 j > x Optmalty Condtons Usng Kuhn-Tucker optmalty condtons [Luenberger 1989], maxmzaton for equaton (4.1) can be reached f followng condtons are met: R 1 P(X 1 x 1 ) = R 2 P(X 2 x 2 ) = R 3 P(X 3 x 3 ) x 1 + x 2 + x 3 = N x 1, x 2, x 3 0 (4.4) Where P(X t) s defned as the probablty that at least t number of unts can be sold to the class. In the contnuous dstrbuton, we have P(X t)= P(X >t). So we have R 1 P(X 1 >x 1 ) = R 2 P(X 2 >x 2 ) = R 3 P(X 3 >x 3 ) (4.5) Let G (x ) = P(X > x )=1-F (x ), R 1 G 1 (x 1 ) = R 2 G 2 (x 2 ) = R 3 G 3 (x 3 ) (4.6) Let EMR (x ) = R G (x ), the optmal allocaton condtons can be shortened as: EMR 1 (x 1 ) = EMR 2 (x 2 ) = EMR 3 (x 3 ) x 1 + x 2 + x 3 = N x 1, x 2, x 3 0 (4.7) EMR (x ) denotes Expected Margnal Revenue for product class when the number of quantty avalable to the class s x. EMR s just the revenue of the class multples by the probablty that the demand for class s greater than to x, or the probablty of the (x +1) th unt of products can be sold. The equaton (4.7) means that the expected margnal revenues of three classes are equal at the optmalty condton. In the dscrete dstrbuton, we cannot guarantee to fnd out x s to satsfy formula (4.7). The necessary condton becomes: EMR 1 (x 1 ) EMR 2 (x 2 ) EMR 3 (x 3 ) x 1 + x 2 + x 3 = N 508

x 1, x 2, x 3 0 (4.8) where means as closer as possble. In the followng algorthm, we can request the absolute dfference between any two of EMR(x) s less than a pre-specfed small postve number. As descrbed n early ths secton, a Posson dstrbuton s assumed for random demand. If the mean of the demand for product class n the plannng horzon s much larger than zero, t s reasonable to assume that demand approxmately obeys a Normal dstrbuton. 4.2 Model MRBCMb MRBCMb model s an extenson of model MRBCMa by takng nto account of lower classes opportunty revenue. The opportunty revenue means a capacty, whch s reserved for lower proft class, s sold for a hgher proft class n the plannng horzon because of nestng allocaton rules. Consderng opportunty revenue, EMR (=1, 2, 3) can be revsed as EMR 1 (x 1 ) =G 1 (x 1 )v 1 +(F 1 (x 1 )- F 1 (x 1-1))v 2 Where v 1 = F 2 (x 2 ) F 3 (x 3 ) R 1 +G 2 (x 2 ) F 3 (x 3 )[(R 1 G 1 (x 1 )+ R 2 G 2 (x 2 ))/(G 1 (x 1 )+ G 2 (x 2 ))] +F 2 (x 2 ) G 3 (x 3 )[(R 1 G 1 (x 1 )+ R 3 G 3 (x 3 ))/(G 1 (x 1 )+ G 3 (x 3 ))] +G 2 (x 2 ) G 3 (x 3 )[(R 1 G 1 (x 1 )+ R 2 G 2 (x 2 )+ R 3 G 3 (x 3 ))/(G 1 (x 1 )+ G 2 (x 2 ) + G 3 (x 3 ))], v 2 = G 2 (x 2 ) F 3 (x 3 )R 2 +F 2 (x 2 ) G 3 (x 3 )R 3 + G 2 (x 2 ) G 3 (x 3 )[(R 2 G 2 (x 2 )+ R 3 G 3 (x 3 ))/(G 2 (x 2 ) + G 3 (x 3 ))] EMR 2 (x 2 ) = G 2 (x 2 )F 3 (x 3 ) R 2 + G 2 (x 2 ) G 3 (x 3 )[(R 2 G 2 (x 2 )+ R 3 G 3 (x 3 ))/(G 2 (x 2 ) + G 3 (x 3 ))] +(F 2 (x 2 )- F 2 (x 2-1))G 3 (x 3 )R 3 EMR 3 (x 3 ) = G 3 (x 3 )R 3 (4.9) In whch, there s no opportunty revenue n class 3 snce t s the hghest class. 1) The frst term of EMR 1 (x 1 ),.e. G 1 (x 1 )v 1, s the approxmaton of the expected revenue when the order for class 1 s greater than ts protected level,.e. X 1 > x 1. Even though X 1 > x 1, the capacty of the x 1 unts can stll be sold to class 2 or 3 f the order for one of those hgher classes exceeds ts protected level, and the customer arrvals earler than the x 1 th customer of class 1. 2) The second term of v 1, G 2 (x 2 ) F 3 (x 3 )[(R 1 G 1 (x 1 )+ R 2 G 2 (x 2 ))/(G 1 (x 1 )+ G 2 (x 2 ))] s an estmate of the expected revenue when X 1 > x 1 and X 2 > x 2, but X 3 x 3. If we consder the margnal revenue for the x 1 th capacty unt, only the earlest one of the two orders of class 1 and 2 can be satsfed. What s the probablty that the x 1 th unt s sold to the customer of class 1? The answer depends on ther dstrbuton functon. We use G 1 (x 1 )/(G 1 (x 1 ) + G 2 (x 2 )) to estmate ths probablty, and use G 2 (x 2 )/(G 1 (x 1 ) + G 2 (x 2 )) to estmate the probablty that the x 1 th unt s sold to the customer of class 2. The same way s appled n term v 2, and EMR 2 (x 2 ). 509

3) It seems that the second term of EMR 1 (x 1 ) should be F 1 (x 1 )v 2, snce the sum of G 1 (x 1 ) and F 1 (x 1 ) equals one, represents the total probablty of the stochastc varable X 1. Here, the value of F 1 (x 1 )- F 1 (x 1-1) s the probablty that x 1-1 customers of class 1 have come, but the x 1 th customer has not appeared. So the x 1 th unt of capacty, whch s assgned to class 1, can be used by other classes. Term v 2 n EMR 1 (x 1 ) estmates the probabltes that the x 1 th unt for class 1 s used by class 2 or class 3. To make the formula complete, we should add terms related to F 1 (x 1-1)-F 1 (x 1-2), F 1 (x 1-2)-F 1 (x 1-3), and F 1 (1)-F 1 (0), whch represent 2,3,, and x 1 extra unts for class 1 are avalable for class 2 or class 3. We gnore these cases and only keep F 1 (x 1 )-F 1 (x 1-1) for two reasons. () Snce the mean of class 1 s much larger than that for class 2 and 3 n practce, the probablty that class 1 has more than 1 extra unt s very small. () The values of v 2 for cases related to F 1 (x 1-1)-F 1 (x 1-2), F 1 (x 1-2)-F 1 (x 1-3),, and F 1 (1)-F 1 (0) wll be dfferent and complete. We can calculate these value separately, but we can not use one value,.e. v 2, tmes F 1 (x 1 ). Recallng that we consder the margnal revenue only n order to obtan the optmal condton of the non-lnear programmng, the error of gnorng F 1 (x 1-1)- F 1 (x 1-2) to F 1 (1)- F 1 (0) s very small. 4.3 Model MRBCMc Model MRBCMc are margnal probablstc Revenue-Based Capacty Management models, whch maxmze total expected revenue by takng nto account of hgher classes opportunty cost. The opportunty cost here smple means a capacty waste n terms of capacty reservaton for hgh proft class that cannot be sold n the plannng horzon because of nestng allocaton rules. MRBCMc mproves MRBCMa model by not reservng excessve capacty for hgh and mddle proft classes. Countng n the opportunty cost, EMR (=1, 2, 3) n equaton (4.7) are revsed as EMR 1 (x 1 ) = G 1 (x 1 )R 1 EMR 2 (x 2 ) = G 2 (x 2 )R 2 - F 2 (x 2 )b 2 EMR 3 (x 3 ) = G 3 (x 3 )R 3 - F 3 (x 3 )b 3 (4.10) Where b 2 and b 3 are expressed as b 2 = G 1 (x 1 )R 1 b 3 = G 2 (x 2 )R 2 + F 2 (x 2 )G 1 (x 1 )R 1 The varable b 2 n EMR 2 represents the opportunty loss assocated wth unsold excessve capacty reservaton for class 2. It occurs when class 1 receves requests more than x 1 unts, and at meantme n class 2 and 3 there exst excessve capacty avalable but no more than x 2 nor x 3 unts are request. If the surplus s allocated to the lower class, the proft wll be R 1. Otherwse, t wll be wasted for nothng. Analogously, same stuaton may happen when class 3 has dffculty to sell. The varable b 3 n EMR 3 represents the opportunty loss assocated wth excessve capacty reservaton for class 3 but unused. It ncludes two parts: the one when recevng more than x 2 unt requests but rejected by class 2 and no more than x 3 request. And the other when recevng more than x 1 unt requests but not able to accept, also nether more than x 2 nor more than x 3 unts wll be request by the end of the plannng horzon for class 2 and 3, respectvely. 510

There s no opportunty cost n class 1 due to ts lowest rank, so b 1 =0. We have the followng general formula EMR (x ) = G (x )R F (x ) b for =1,2,3. (4.11) If G 3 (x3) and G 2 (x 2 ) are very small, and G 1 (x 1 ) s close to 1 (n the case that both mean 2 and mean 3 are sgnfcantly smaller than mean 1 ), the values of b 2 and b 3 are very close to R 1 G 1 (x 1 ). We denoted t as b. Replacng b 2 and b 3 wth b, we can get an approxmaton method. EMR 1 (x 1 ) = G 1 (x 1 )R 1 EMR 2 (x 2 ) = G 2 (x 2 )R 2 - F 2 (x 2 )G 1 (x 1 ) R 1 EMR 3 (x 3 ) = G 3 (x 3 )R 3 - F 3 (x 3 )G 1 (x 1 ) R 1 (4.12) Ths smplfed model has pretty good results based on our smulated cases n secton 5. However, we gnored n our result tables to reduce the length of ths paper. 5.1 Expermental Desgn 5 EXPERIMENTAL DESIGN AND RESULT DISCUSSION We compare sx models assocated wth dfferent capacty allocaton polces n our smulaton experments. They are: (1) FCFS: that s, for any classes, accepts any orders as they arrve untl all avalable capacty has been allocated. No capacty s reserved for the hgh and mddle proft classes; (2) MWCM; (3) MRWCM; (4) MRBCMa; (5) MRBCMb; and (6) MRBCMc. These models use same order acceptance rule as we descrbed n the Secton 3. We use the total revenue over the plannng horzon as model performance measure. If we assume order arrval for class s a Posson process wth mean µ, then the nter-arrval tmes of orders are IID exponental random varables wth common mean 1/ µ. Thus, we can randomly generate arrval tmes of orders for each class over the plannng horzon recursvely. After three streams of orders (for classes 1, 2, and 3) are generated, they are merged to be a sorted arrval tme to create a sngle combned sequence of orders for all three classes. For each scenaro, 10000 ndependent replcatons (usng dfferent random seeds) are performed. All models are coded n C and mplemented on Pentum personal computer. We assume total capacty for three classes of products s 300. Three major factors are examned when we make the expermental desgn for the smulaton. They are (1) the proft rato; (2) the rato of expected demand for three classes; and (3) the capacty tghtness or the rato of the total avalable capacty to expected total demand. The parameter settngs for the 16 scenaros are summarzed n Table 1. The second column shows the means of demand for three classes. The thrd column s ther profts assocated. 511

5.2 Results Dscusson Case ID Demand Mean Proft ($) µ 1 / µ 2 / µ 3 R 1 / R 2 / R 3 A1 450 / 75 / 25 600 / 800 / 1000 A2 450 / 75 / 25 500 / 800 / 1200 A3 450 / 75 / 25 400 / 800 / 1600 A4 450 / 75 / 25 300 / 800 / 2000 B1 400 / 100 / 50 600 / 800 / 1000 B2 400 / 100 / 50 500 / 800 / 1200 B3 400 / 100 / 50 400 / 800 / 1600 B4 400 / 100 / 50 300 / 800 / 2000 C1 350 / 125 / 75 600 / 800 / 1000 C2 350 / 125 / 75 500 / 800 / 1200 C3 350 / 125 / 75 400 / 800 / 1600 C4 350 / 125 / 75 300 / 800 / 2000 D1 300 / 150 / 100 600 / 800 / 1000 D2 300 / 150 / 100 500 / 800 / 1200 D3 300 / 150 / 100 400 / 800 / 1600 D4 300 / 150 / 100 300 / 800 / 2000 Table 1. Parameters settngs for 16 smulaton scenaros Table 2 presents the Protecton Levels for fve methods on 16 smulaton scenaros. We notce that, MWCM and MRWCM allocate more capacty to class 1. On contrast, MRBCM models reserve more capacty for class 2 and 3. The dfference of Protected Levels s the major drver to make sgnfcant dfferences among average revenues of these models, whch s shown n Table 3. In order to compare fve models wth FCFS from a statstcal perspectve, we run 10,000 replcatons for each smulaton scenaro and show the result n Table 3. If we look through Table 3 more carefully, from scenaro A1 to D4, we found that the mprovements of these models compared wth FCFS also depend on the parameters of scenaros. The larger of the dfference of revenues between class 1 and class 3 s, the more sgnfcantly better the performance of MRBCM models s. 6. CONCLUSION In ths paper, we develop three margnal probablstc optmzaton models for revenue-based capacty management. Specfcally, we assume that frms produce three classes of products, whch havng three dfferent unt proft contrbuton levels. Our MRBCM models generate an approxmate optmal protecton level for each of three classes for avalable to promse to relevant customer channels. The models are compared wth the base case of no capacty reservaton n 16 scenaros by a wde number of smulaton experments. The results ndcate these MRBCM models have sgnfcant ncreases n revenue compare to the FCFS polcy and other two smple 512

methods. Thus the models and algorthms developed wll have a great practcal value for any frms that need to reserve capacty for hgh proftable customer segments. It s clearly that the models we created n ths paper generate approxmate solutons for the complex non-lnear programmng. More detaled models could be developed and evaluated n further researches. MWCM MRWCM MRBCMa MRBCMb MRBCMc Method Case PL( PL( PL( PL(1PL(2PL(3PL(1PL(2PL(3PL(1PL(2PL(3 PL( PL(2PL(3 1) 2) 3) ) ) ) ) ) ) ) ) ) 1) ) ) A1 245 40 15 228 50 22 205 70 25 207 70 23 215 65 20 A2 245 40 15 214 57 29 200 73 27 203 72 25 210 68 22 A3 245 40 15 192 64 44 195 76 29 197 75 28 206 70 24 A4 245 40 15 165 73 62 190 79 31 193 77 30 201 73 26 B1 218 54 28 194 64 42 157 94 49 160 93 47 168 88 44 B2 218 54 28 176 70 54 150 98 52 154 96 50 162 92 46 B3 218 54 28 150 75 75 143 101 56 147 99 54 156 95 49 B4 218 54 28 120 80 100 138 104 58 142 102 56 150 98 52 C1 190 68 42 163 77 60 108 118 74 112 117 71 121 112 67 C2 190 68 42 143 82 75 100 122 78 105 120 75 113 116 71 C3 190 68 42 116 83 101 92 126 82 98 123 79 107 119 74 C4 190 68 42 88 84 128 86 129 85 92 126 82 100 123 77 D1 163 81 56 135 90 75 59 143 98 65 140 95 73 136 91 D2 163 81 56 115 92 93 50 147 103 58 143 99 65 140 95 D3 163 81 56 90 90 120 41 151 108 49 147 104 56 144 100 D4 163 81 56 65 87 148 34 155 111 43 150 107 50 147 103 Table 2. Protecton Levels for smulaton scenaros and methods 513

Case FCFS MWCM MRWCM MRBCMa MRBCMb MRBCMc Mean Mean Increase Increase Increase Increase Increase Mean Mean Mean Mean (%) (%) (%) (%) (%) A1 187369 193439 3.2 194254 3.7 193113 3.1 194242 3.7 196596 4.9 A2 161265 171836 6.6 173750 7.7 174479 8.2 175732 9.0 177799 10.3 A3 135587 153118 12.9 155181 14.5 160117 18.1 160780 18.6 163174 20.4 A4 109908 134400 22.3 138284 25.8 146547 33.3 147279 34.0 149058 35.6 B1 196548 202097 2.8 207337 5.5 207947 5.8 209127 6.4 210730 7.2 B2 176549 185973 5.3 195004 10.5 198602 12.5 199970 13.3 201933 14.4 B3 160003 175492 9.7 188138 17.6 198495 24.1 199714 24.8 201937 26.2 B4 143458 165010 15.0 184958 28.9 200123 39.5 201089 40.2 202659 41.3 C1 205851 210661 2.3 218650 6.2 222158 7.9 223645 8.6 225129 9.4 C2 192053 200254 4.3 217022 13.0 222459 15.8 224121 16.7 225775 17.6 C3 184807 198368 7.3 220673 19.4 237190 28.3 238890 29.3 240740 30.3 C4 177562 196483 10.7 228442 28.7 253593 42.8 254948 43.6 256475 44.4 D1 215178 219089 1.8 227902 5.9 236541 9.9 238166 10.7 239302 11.2 D2 207599 214332 3.2 237512 14.4 246365 18.7 248583 19.7 249811 20.3 D3 209686 220970 5.4 255301 21.8 275668 31.5 277787 32.5 279149 33.1 D4 211772 227609 7.5 273240 29.0 306897 44.9 308756 45.8 309852 46.3 Average 169129 180537 7.5 207228 15.8 217518 21.5 218927 22.3 220632 23.3 Table 3. Average revenue of fve methods compared wth FCFS over 10,000 replcatons REFERENCES Badnell, R. D., Background of an optmal, dynamc yeld management model, Techncal Paper, Vrgna Tech., 1995. Balachandran, B. V. and Ramakrshnan R.T.S., Jont Cost Allocaton for Multple Lots, Management Scence, 42(2), 247-258, 1996. Balakrshnan, N., V. Srdharan, and J.W. Patterson, Ratonng capacty between two product classes, Decson Scences, 27(2), 185-214, 1996 Balakrshnan, N., Dscrmnatng between multple product classes n make-to-order manufacturng envronment, European Jr. of Operatons Research, 1998. Belobaba, P.P., Applcaton of a probablstc decson model to arlne seat nventory control, Operatons Research, Vol. 37, No. 2, March-Aprl 1989. Carroll, W. and R. Grmes, Evolutonary change n product management: experences n the car rental ndustry, Interfaces 25(5), 84-104, 1995. 514

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