Demand-Driven Scheduling of Movies in a Multiplex

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1 Demand-Driven Scheduling of Movie in a Multilex Jehohua Eliahberg The Wharton School, Univerity of Pennylvania, Philadelhia, Pennylvania [email protected] Quintu Hegie School of Economic, Eramu Univerity Rotterdam, The Netherland, [email protected] Jaon Ho School of Buine Management, Ryeron Univerity, Toronto, ON, M5B 2K3 Canada, [email protected] Denni Huiman Econometric Intitute, Eramu Univerity Rotterdam, The Netherland, [email protected] Steven J. Miller Mathematic Deartment, Brown Univerity, Providence RI 02912,[email protected] Sanjeev Swami Deartment of Management, Faculty of Social Science, Dayalbagh Educational Intitute, Agra , UP, INDIA, [email protected] Charle B. Weinberg Sauder School of Buine, Univerity of Britih Columbia, Vancouver, BC V6T 1Z2 Canada [email protected] Berend Wierenga RSM Eramu Univerity Rotterdam, The Netherland, [email protected] May 1, 2007 NOT FOR REPRODUCTION OR QUOTATION WITHOUT WRITTEN ERMISSION P OF THE AUTHORS Abtract Thi aer decribe a model that generate weekly movie chedule in a multilex movie theater. A movie chedule ecifie within each day of the week, on which creen() different movie will be layed, and at which time(). The model conit of two art: (i) conditional forecat of the number of viitor er how for any oible tarting time; and (ii) an otimization rocedure that quickly find an almot otimal chedule (which can be demontrated to be cloe to the otimal chedule). To generate thi chedule we formulate the o-called movie cheduling roblem a a generalized et artitioning roblem. The latter i olved with an algorithm baed on column generation technique. We have alied thi combined demand forecating /chedule otimization rocedure to a multilex in Amterdam where we uorted the cheduling of fourteen movie week. The rooed model not 1

2 only make movie cheduling eaier and le time conuming, but alo generate chedule that would attract more viitor than the current intuition-baed chedule. Key word: otimization of movie chedule, integer rogramming, column generation, demand forecating. 1. Introduction The motion icture indutry i a rominent economic activity with total world wide box office revenue of $ billion in 2006, of which $ 9.49 billion i in the U.S.A 1. Movie forecating and rogramming in ractice tend to be aociated with intuition rather than analyi and thi alo characterize the tradition of deciion making in thi domain. However, many roblem in the film indutry are actually quite amenable to model building and otimization, and thi i increaingly recognized by movie executive. In thi aer we focu on one uch roblem: the detailed cheduling of a movie theater. The movie indutry run on a weekly cycle. A movie rogram or chedule in a theater i deigned for a week, and in the U.S.,for examle, each Friday a new movie week tart. Therefore, a movie theater ha to reare a new movie rogram every week. Thi i articularly comlex for multilex theater, the increaingly dominant movie theater format around the world. For examle in the Netherland, which i the etting of the emirical art of thi aer, multilexe with 8 or more creen rereent 24% of all the movie theater eat, and in 2005, 34% of total box office. It i clear that the rogramming of uch large cinema i not an eay matter. For each week movie rogram, management mut determine what movie will be hown, where they will be hown (on which creen), on which day, and at what time during the day. Tyically, on each creen, a theater can accommodate 3 to 5 howing er day, where a howing i defined a the creening of one movie, including trailer and advertiement. Thi mean that a 10-creen theater need to rogram around 280 howing er week. At reent, thi rogramming i motly done manually with encil and aer, by ecialit in the theater comany. Thee rogrammer combine an analytical mind with a broad knowledge about movie and about the audience of individual theater, often baed on many year of exerience. It i our belief that an analytical ytem can hel here. Thi would not only relieve 1 MPAA Annual Reort

3 theater from a labor-intenive tak recurring every week, but we believe that an analytically baed ytem will erform even better than the current manual rocedure. The rogramming roblem for an individual movie theater conit of two tage: (i) the election of the lit of movie, i.e., the movie to be hown in the articular week and (ii) the cheduling of thee movie over creen, day, and time of the day. Stage (i) include making agreement with movie ditributor and i comleted before tage (ii). In thi tudy we develo a olution for the econd tage, which involve contructing detailed chedule for where (which creen()) and when (day, time) the different movie will lay. For analytical rocedure that deal with the firt tage, i.e., deciding on the movie to be hown in a articular week, ee Swami, Eliahberg and Weinberg (1999), and Eliahberg, Swami, Weinberg and Wierenga (2001). Our cheduling roblem ha two ub-roblem. Firt, we need an anwer to the following quetion: if a articular movie would be hown on a articular day, at a articular time, how many viitor would it attract? Inut for making thee forecat are oberved viitor number in the reviou week (for exiting movie), characteritic of the movie (for newly releaed movie), and information about variable uch a ecific event (holiday) and the weather. Making uch conditional forecat i not an eay tak and it belong to the realm of the marketing diciline. Second, given thi demand aement, we have to find the chedule that maximize the number of viitor for the week, given contraint uch a theater caacity and runtime of the movie. Thi i a non-trivial roblem, for which we need the diciline of Oeration Reearch. The ecific olution aroach emloyed here i column generation, a method deigned to olve (integer) linear otimization roblem with many variable/column, but relatively few contraint. The remainder of thi aer decribe how by combining the two dicilinary aroache a olution to the movie cheduling roblem wa obtained. The aer firt take a cloer look at the movie cheduling roblem (Section 2). In Section 3 we decribe the column generation aroach to the otimization roblem. In Section 4, we dicu the method that we develoed to conditionally forecat the number of viitor of a how. Then, in Section 5, 3

4 we aly the comlete rocedure, through a model we call SilverScheduler, to fourteen movie week of the theater De Munt, a multilex with 13 creening room in central Amterdam. With over 1 million viitor a year, thi i the econd larget theater in the Netherland. The lat art of the aer (Section 6) dicue the reult, ut them in erective and dicue iue for future reearch. 2. Problem Decrition and the Reearch Project A movie theater have evolved from ingle creen theater to multilexe and even megalexe, the roblem of cheduling movie onto creen ha become increaingly comlex. Some of the key reaon for that comlexity are a follow. Firt, there are a large number of different movie that the theater want to how in a tyical week. Thi number i tyically larger than the number of creen. Moreover, thee movie have different running time. For examle, over the different movie running in the De Munt theater in Amterdam during our obervation eriod, the running time wa a hort a 71 minute for Plo en Kwiel (a kid movie) and a long a 240 minute for Ring Marathon. Second, the number of eat er creening room differ. In the De Munt theater the mallet room ha 90 eat and the larget room, 382. For ome movie there are contractual arrangement with ditributor about the ecific creening room in which a movie will be hown. Thi i tyically the cae for newly releaed movie, which ditributor like to have in large room. For other movie, the theater management i comletely free where to creen them. Third, there i variability in demand for movie. In our amle, for examle, in May 2005, the mot oular movie drew 8027 viitor in a week and the leat oular 133 viitor. In addition, movie demand varie by time of day and day of week. For examle, we found that Saturday evening at 8 m i the mot aealing time for movie goer. The exected demand for a movie influence what creening room it will be hown in; the demand for ome movie may be o large that it could be double or trile booked. 4

5 Fourth, there are different genre of movie, and thi alo ha imlication for their cheduling. For examle, children movie hould referably be hown at time when children are free from chool (weekend and Wedneday afternoon 2 ), and will not be hown during the evening. Fifth, there are many contraint oed by the logitic of a movie theater. An obviou limitation i the time that the theater i oen. (The cloing time i often determined in art by the ublic tranortation chedule: non-driver attending the lat how need to be able to get home). Other logitical contraint are the time needed for cleaning the room after a howing (deendent on the ize of the room), and the caacity limitation of the ticket office, the corridor, the taircae, and the conceion ale counter. To rovide high conumer atifaction with the theatrical exerience, crowding hould be avoided a much a oible, and major movie with many viitor hould not tart at the ame time, eecially if they are on the ame floor. Finally, management may imoe ecific contraint which are exected to contribute to the revenue of the theater. For examle, for the theater in our tudy, management ecified that the time between two movie tart hould not be more than 20 minute. In that cae an imule movie viitor, looking for entertainment without trong reference for a articular movie, never ha to wait more than 20 minute. We take the erective of management for thi roblem, which i to maximize the exected number of viitor to that theater for a given week. Another oible objective i rofit maximization. However, the cot and revenue tructure are uch that at thi level, attendance maximization and rofit maximization are likely to rovide quite imilar reult. We confirmed thi with emirical teting. Conequently, the roblem to be olved i how to generate a chedule of the different movie on the different creen during a given week, which obey all the requiite contraint and maximize the number of viitor. The Study 2 Primary chool children in the Netherland have free time on Wedneday afternoon. 5

6 Mot theater olve thi cheduling roblem by hand. Movie rogrammer work with hardcoy lanning heet to fill the total caacity of the theater, taking into account the variou contraint to the extent oible. In doing o, rogrammer follow certain rocedure (e.g., chooe the ize of the creening room for a movie, deendent on the exected attendance), and ue a mixture of hard fact (e.g., attendance figure of the at week) and intuition (e.g., the effect of an imortant occer match on TV on the number of theater viitor for a movie). It take a lot of exerience to become a killed movie rogrammer. And even then the olution i never erfect. For a human mind it i ractically imoible to find the bet oible olution while at the ame time dealing with all the contraint. Moreover, the chedule have to be made under time reure. A mentioned, in a movie theater every Thurday/Friday 3 a new movie rogram tart running, which ha to be finalized on the receding Monday. The critical time window i the Monday morning after the attendance figure of the recent weekend have become available and before the information about the new chedule ha to be ent to the newaer and oted online. The uroe of thi tudy i to develo a rocedure that make it oible to automate the movie rogramming roce a decribed above. We develoed a mathematical rocedure that roduce the (almot) otimal chedule, given the demand information and the variou contraint. Such an algorithm hould work fat and hould deliver the outut in a format that can directly be entered in the theater lanning rocedure. It hould alo be eay for a manager to make lat-minute change to the chedule a recommended by the rocedure. In deciion ituation like thi, there can alway be new information, (e.g., about a trike, a udden change in the weather), that i not included in the model and that may induce the manager to make lat minute change in the chedule. The generation of a recommended chedule can be automated, but for the imlementation of a chedule, human judgment till remain vital. We were aked to work on thi roblem by Pathé, the larget movie exhibitor in the Netherland. They view the current manual rogramming rocedure a being too cumberome and time-conuming. Our uroe wa not to jut ave the time of the rogrammer, but alo to do a better job in term of gener- 3 In the Netherland the new movie week tart on Thurday, in the USA and Canada thi i on Friday. 6

7 ating movie chedule that may attract larger number of viitor. It wa decided to take the etting of one articular Pathé location, the De Munt Theater, a our emirical environment. Management rovided comlete acce to the internal data they had, and cloely monitored the roject and it reult. A key inut to the cheduling algorithm i the demand information about the movie in the movie lit. For each movie in the movie lit we need to forecat the number of viitor that thi movie will generate in any given howing. Thi etimate ha to be available for each day of the week and for each different oible tarting time of the movie. For thi uroe, we develoed a forecating rocedure with two module. The firt module i for movie that have been running already. The oberved number of viitor are ued to etimate a forecating model. The econd module i for newly releaed movie, where the number of viitor are forecated in an indirect way, uing the characteritic of a movie a redictor variable. Thee forecating rocedure are decribed after the dicuion of the cheduling algorithm. We call the comlete model, the cheduling algorithm integrated with the demand aement rocedure, SilverScheduler. After the decrition of the cheduling art and the forecating art of Silver- Scheduler, we dicu it imlementation at the theater De Munt. 3. A Column Generation Aroach to Solve the Movie Scheduling Problem To roduce a movie rogram for a certain week, we need to find chedule for the different day in that week. We define the movie cheduling roblem (MSP) a the roblem of finding the otimal movie rogram for a ingle day given the lit of movie to be hown (the movie lit ), the running time of thee movie, the demand figure for thee movie, the caacitie of the different creening room, information about contractual agreement with ditributor about creening room for articular movie, and which take into account the different contraint a mentioned in Section 2. In Section 5, we decribe a rocedure to come u with a week chedule by olving a equence of MSP. The MSP ha many imilaritie with other well-known cheduling roblem uch a vehicle cheduling/routing, aircraft cheduling/routing, and crew cheduling (or combination of thee roblem). Thee kind of roblem are often formulated a et artitioning or covering roblem (eventually with 7

8 additional contraint). We alo ue a et artitioning tye of formulation. A drawback of uch a formulation i the large number of variable, which can be overcome by uing column generation technique. We refer to Barnhart et al. (1998), Lübbecke and Deroier (2005), and Deaulnier et al. (2005) for a general introduction on column generation. Some recent reference to alication of column generation to the aforementioned cheduling roblem are Löbel (1998) for vehicle cheduling, Deaulnier et al. (1997) for aircraft routing and cheduling, Huiman et al. (2003) for integrated vehicle and crew cheduling, and Sandhu and Klabjan (2006) for integrated airline lan. 3.1 Mathematical Formulation Before we reent the mathematical formulation of the MSP, we firt introduce ome notation. Analogou to the o-called time-ace network in tranortation roblem uch a vehicle and crew cheduling, we ue here time-movie network. Thee network are acyclic directed grah denoted by G =(N,A ), and are defined for each creen. Furthermore, let S and M be the et of creen and movie, reectively. Recall that due to caacity retriction and contractual agreement not all movie can be hown on each creen, therefore we define M a the ubet of movie that can be hown on creen. We denote for each movie m it duration and cleaning time a dr m and cl m, reectively. Define T a the et of oible time oint at which a movie can tart, where t i i the time correonding to time oint i. In each grah, a node (i,m) correond to tarting a movie m on time oint i on creen. Moreover, a ource and a ink are defined. There are arc from the ource to all intermediate node, and from them to the ink. If on a creen only one movie i allowed to lay throughout the day, an arc i defined between each air of node (i,m) and (j,n) if t j t i + dr m + cl m with m=n. In Figure 1, the time-movie network i deicted for thi articular cae. For intance, there i an arc between node (1,1) and (3,1), becaue the duration and cleaning time of movie 1 i not larger than the time between time oint 1 and 3. A ath in the network correond to a feaible chedule for the whole day on one creen. In thi cae, the dotted ath correond to howing movie 1 on time oint 1 and 3. 8

9 (1,1) (2,1) (3,1) (4,1) ( T,1) (1,2) (2,2) (3,2) (4,2) ( T,2) movie (1,3) (2,3) (3,3) (4,3) ( T,3) (1, M ) (2, M ) (3, M ) (4, M ) ( T, M ) time Figure 1: A time-movie network for one creen (with one movie er creen) If two movie can be hown on the ame creen, the network i extended to two layer. In each individual layer, there are only arc between node correonding to the ame movie, while the arc between the layer are between node with different movie. Since a witch of movie on a creen i not referred, a enalty Q i introduced when the arc between the different layer i choen. In thee network each ath from ource to ink i a feaible chedule for one creen. The cot of a ath i defined a c, which i equal to the um of the cot of the individual arc in the ath. Each arc (i,m,j,n) ha a cot -min {d im,ca } which i the minimum of the exected demand for movie m on time oint i and the caacity of creen if (i,m) and (j,n) are in the ame layer, and Q min{d im,ca } if the node are in a different layer. The baic movie cheduling, where every movie can only be hown on one creen, can now be een a finding a ath from ource to ink in each network uch that the total cot of the ath i minimal and each movie i in exactly one of the ath. In mathematical term, thi can be directly formulated a a et artitioning roblem with deciion variable x, which i 1 if ath i elected in network G and 0 otherwie. 9

10 .t. min S P S P a m x P c x x x = 1 = 1 { 01, } m M S S, P (1) (2) (3) (4) Herein, P i defined a the et of all ath in G, and arameter a m are 1 if movie m i in ath correonding to creen and 0 otherwie. Notice that the formulation can be extended to the ituation where a movie can be hown on different creen during the day. Since thi ituation i not the cae in our alication, we do not take it into conideration. Of coure, not all ractical aect have been taken into account in thi mathematical formulation yet. In the remainder of thi ection, we dicu three imortant aect that can be taken into account by adding extra et of contraint. The firt aect deal with the fact that only one movie can tart at the ame time on a floor during crowded eriod. Thi can be formulated with the following et of contraint: S P b e x f F,i T, (5) i f 1 1 where F i the et of floor and T 1 i the et of time oint over which thi condition hould hold. Parameter b i and e f are 1 if at time oint i a movie tart in ath belonging to creen i, and creen i on floor f, reectively, and 0 otherwie. A mentioned earlier, it i imortant for the management that in certain re-ecified interval (in our cae every 20 minute), there i at leat one movie tarting. However, thi extra requirement can reult in a too large reduction of the number of viitor. Therefore, we add a 0/1 deciion variable y l indicating whether thee contraint are atified or not (it i 1 if the contraint i not atified). The following et of contraint i added to the formulation: S P (b i b j )x + y l 1 l L, i,j T, l (6) 10

11 where L i the et of interval and T l are the time oint in interval L. Moreover, we add the term l L R y l in the objective function, where R i a enalty for violating one of thee retriction. The third and final aect that we exlicitly take into conideration ha to do with the cloure of the theater. Since it i undeirable for all movie to end at the ame time, in a certain fraction of r creen the lat movie ha to be finihed before a certain time: S P h x r, (7) where h i 1 if the lat arc in ath tart from a node (i,m) where t i + dr m i le than a certain reecified time. To ummarize, the MSP can be formulated a follow: (MSP) min S P (b S P.t. i c x S P S P b j + R a )x P S P m b i l L h x e x f + y x y = 1 x l l = r, m M, S, f F,i T, l L, i,j T, 1 l (8) (9) (10) (11) (12) (13) x { 01, } S, P. (14) 3. 2 The Prooed Algorithm A argued before, we will ue column generation technique to deal with the large number of variable. The general idea behind column generation, introduced by Dantzig and Wolfe (1960), i to olve a equence of reduced roblem, where each reduced roblem only contain a mall ortion of the et of variable (column). After a reduced roblem i olved, a new et of column i obtained by uing dual information of the olution. The column generation algorithm converge once it ha etablihed that 11

12 the otimal olution baed on the current et of column cannot be imroved uon by adding more column. Then the otimal olution of the reduced roblem i the otimal olution of the overall roblem. We will refer to the reduced roblem a the mater roblem and to the roblem of generating a new et of column a the ricing roblem. Traditionally, column generation for integer rogram i ued to olve their LP-relaxation. However, we will ue it in combination with Lagrangian relaxation (for the reader unfamiliar with Lagrangian relaxation, ee Fiher (1981)). Thi aroach ha been choen for the following reaon. For thi roblem, a et of column generated to comute the lower bound turn out to be a et from which we can elect a reaonably good feaible olution. Since we comute a lower bound on the otimal olution, we obtain an indication about the quality of the contructed feaible olution. Lagrangian relaxation ha been hown to rovide tight bound for et artitioning tye of roblem (ee e.g. Bealey (1995)). We refer to Huiman et al. (2005) for a more comrehenive dicuion about combining column generation with Lagrangian relaxation. In the remainder of thi ection, we reent the algorithm to olve the MSP. In Figure 2, we ummarize the algorithm main te. Ste 0: Initialization Contruct an initial feaible olution and take thoe column a tarting one. Ste 1: Mater roblem Obtain a good et of dual multilier by olving a Lagrangian dual roblem. Ste 2: Pricing roblem Generate for each creen new column with negative reduced cot. A long a we find new column, return to Ste 1. Otherwie, go to Ste 3. Ste 3: Contruct a feaible olution Solve the correonding integer rogram with the ubet of column generated in Ste 2 and the initial one from Ste 0 to otimality. Figure 2: Schematic icture of the algorithm In Ste 0, we contruct an initial feaible olution by howing each movie once and etting all y- variable equal to 1. Thi olution give an initial et of column. The firt time that we olve the mater roblem in Ste 1 we ue thi et column. 12

13 Mater roblem To olve the mater roblem (Ste 1), we ue a relaxation of (MSP) by relacing the = ign by ign and ubequently relaxing the contraint in a Lagrangian way. That i, we aociate Lagrange multilier λ m, µ ν fi, σ ijl and ω to Contraint (9-13), reectively. The remaining Lagrangian ubroblem can then be rewritten a: min S P c x + l L (R-σ ijl ) y l (15).t. x { 01, } y { 01, } l S, P l L,, (16) (17) where c = c m M f F i T 1 λ a µ ν b e σ ( b b ) ω. m m fi i f l L i T 1 j T 1 ijl i j The Lagrangian ubroblem can be olved by inection, i.e. x =1 if c < 0 and 0 otherwie, and y l = 1 if R < σ ijl and 0 otherwie. In thi way, we obtain a lower bound for the given et of column. We aly ubgradient otimization to find the bet lower bound. For the reader intereted in thi toic, leae ee, for examle, the urvey of Bealey (1995). We will not go to into further detail here. Pricing roblem In Ste 2 we generate new column with negative reduced cot, where the reduced cot of a column correonding to ath on creen i denoted by c. For each creen, we generate new ath by olving an all-air hortet ath roblem between each air of node (excluding the ource and ink node). From all thee oible ath, we add the k-mallet one to the mater roblem. By olving an allair hortet ath roblem, we get a large variety in column, which make it oible that only a few iteration in the column generation rocedure are neceary. Furthermore, we calculate a lower bound on the overall roblem by adding the reduced cot of all generated column by the lower bound obtained in Ste 1. If thi difference i mall enough, we quit the column generation rocedure and contruct a feaible olution. Otherwie, we return to Ste 1. 13

14 Feaible olution Finally, we contruct a feaible olution for the movie cheduling roblem. Thi i done by olving the roblem (MSP) with the initial column and the one generated during the column generation to otimality. For thi uroe, we ue the commercial MIP olver Clex Illutrative Comutational Reult To hel evaluate the algorithm erformance we conducted everal tet on Pathé data. We contructed daily chedule for a number of day, with varying number of movie. After dicuing the reult with Pathé, we et the value for the arameter Q and R (enaltie) to 100 and 10, reectively. Day # movie Ga (ab) Ga (in %) Cu* Ste 1 Cu Ste 2 Total cu** Average * The Cu time are meaured in econd on a Pentium III, 1 GHz PC (384MB RAM); ** Total cu time alo include Ste 0 and 3 Table 1: Comutational reult of the movie cheduling algorithm In Table 1, we reort for 14 intance (day) the number of movie, the abolute and relative ga between uer and lower bound on the otimal olution, and the comutation time (in econd) for the 14

15 mater roblem (Ste 1), ricing roblem (Ste 2) and the comlete algorithm. The number of creen i equal to 13 for all intance. A can be een in Table 1, the relative ga i between 0.60 and 3.27%. The average ga i 1.58%. The total comutation time i very reaonable: on average it take about 2.5 minute. The mot time, a exected, i ent in the column generation art of the algorithm, where the time to olve the mater and ricing roblem are almot equal. 4. The Conditional Forecating Method Develoing a week chedule require forecat for A jt, the attendance for a creening of movie j tarting at time t, for all movie available for creening and for all oible time (in ten minute interval) for every day of that week. Such forecat would be required each Sunday to be available for the otimization algorithm o that recommendation to management could be made on Monday. In order to ue the mot recent oible data, the forecating rocedure had to be comleted in le than 12 hour. Thi eentially ruled out ome imulation-baed forecating model uch a Hierarchical Baye model (e.g., Ainlie, Drèze and Zufryden 2005). A i well known, the age of a movie i an imortant factor in determining it aeal to the audience. A number of emirical tudie reort that moviegoer demand for mot movie tyically decreae acro week (e.g., Sawhney and Eliahberg 1996, Krider and Weinberg 1998, Lehmann and Weinberg 2000). We ue a two-arameter exonential model to cature thee effect. For mot movie, thee arameter can be etimated from the detailed attendance record maintained by Pathé. For movie for which we do not have attendance data for the De Munt Theater for at leat two week, we develoed a econdary rocedure to etimate the two arameter of the exonential model. Thi rocedure i decribed more fully below. Alo, there are other effect, uch a holiday, which may influence attendance and we include control variable for uch effect a well. In addition to thee weekly effect, there are more micro effect that need to be conidered. In articular, we introduce variable that account for day of the week and time within a day (in hourly inter- 15

16 val) at which movie are hown. We alo allow for other detailed factor uch a weather (temerature, reciitation) and whether the Dutch national football (occer) team i laying a major international match. Our choice of variable to include wa guided by reliminary analyi of the data and ublihed aer. A decribed below, we divided our rocedure into three te. 4.1 Method decrition Ste 1: Demand Model Etimation In the firt te, we ue at attendance figure to etimate a demand model, which earate out different time-varying attern. Formally, we model A jt, the attendance for a howing of movie j tarting at time t a: A jt = ex( θ j j I { j} + β h I { h} t + ω d I { d} t + h d +λ j AGE jt v γ v I { v} t +δ SATNIGHT SATNIGHT t +δ SUNPM SUNPM t +δ NG NG t +δ DTEMP DTEMP t +δ DPRECIP DPRECIP t +ε jt ) (18), where I {j} Indicator variable for the event that movie j i being hown AGE jt Age of movie j at time t (number of week ince movie j firt regular howing at the theater) I {h}t Indicator variable for the event that tarting time t i within hour h, h {10am 10m} I {d}t Indicator variable for the event that tarting time t i on day d, d {Monday, Tueday,, Sunday} ] 4 I {v}t Indicator variable for the event that tarting time t i on an Amterdam holiday or chool vacation v, v {Sring vacation, May vacation, Acenion Day, Whit Sunday, Whit Monday, Eater weekend, Summer vacation, Fall vacation, Chritma vacation} SATNIGHT t Indicator variable for the event that tarting time t i between 11m and 1am of a Saturday SUNPM t Indicator variable for the event that tarting time t i between 2m and 6m of a Sunday NG t Indicator variable for the event that tarting time t occur during the time when the Dutch national team i laying in a major tournament and i televied DTEMP t The maximum temerature of the day tarting time t i in DPRECIP t The indicator variable for the event that tarting time t i in a day with nonzero reciitation (e.g., rain or now). 4 For comletene, all oible value of the indicator variable are lited in the text. The bae cae for all the indicator variable are tated in the table reorting reult. 16

17 θ j and λ j are the two movie-ecific arameter caturing the time trend of an individual movie attractivene acro week. In articular, we aume the attractivene of each movie title follow an exonential decay, with θ j characterizing the cale of the movie attractivene (oening trength) and λ j caturing the weekly decay of the attractivene. A dicued earlier, the other tye of time trend inherent in the attendance i moviegoer time reference to watch a movie; thee effect are catured by three et of arameter, β h, ω d, and γ v. The γ v cature the time of year effect. In articular, a moviegoer tend to have more free time for leiure activitie uch a going to movie theater during holiday or chool vacation, a comared to normal work or chool day, we exect all of thee arameter to be oitive. Five chool vacation eriod, namely ring, May, ummer, fall and Chritma vacation are firt identified. Three ublic holiday outide thee vacation eriod, namely Acenion Day, Whit weekend and Eater weekend are then added to cature the other otential holiday effect. The day of week effect i catured by the ω d,. Seven arameter ω MON, ω TUE, ω WED, ω THU, ω FRI, ω SAT and ω SUN cature thi time trend. At the more micro level, the β h cature the time of day effect. We exect the β h correonding to the daytime to be maller than thoe for the evening. While the normal oerating hour of the DeMunt are bound by 11 m, occaionally ome movie are hown after 11 m on Saturday. We therefore ue the arameter δ SATNIGHT to cature the effect of thi extenion on Saturday. We alo include a arameter to rereent Sunday afternoon (δ SUNPM ) a our reliminary data analyi howed that attendance for thi day wa tyically higher in the afternoon than on other day. We want to control in our demand model three additional ytematic hift of the attendance: whether the movie i creened againt a trong cometing leiure activity (we focu on major tournament occer game layed by Dutch national team), the highet temerature of the day, and whether there i rain or now during the day. The arameter, δ NG, δ DTEMP and δ DPRECIP cature thee ytematic hift. After auming ε jt N(0, σ 2 ), we take the logarithm of (18) and etimate the tranformed model by OLS, uing all the available attendance figure u to the Sunday receding the Thurday the new 17

18 movie rogram tart. When additional attendance data are added each week, the demand model i reetimated with the extended data et. The current demand model aume that the attendance of movie j tarting at time t i not affected by which ecific movie are hown at the ame time. More comlex model that cature ubtitution effect are exlored in Ho (2005), but they rovide limited imrovement in the rediction ower, while greatly increaing the comlexity of the mathematical otimization. Conequently, we retained (18) a the demand model for the current roject. Ste 2: Determination of Movie-Secific Parameter In caturing individual movie attractivene, the two et of movie-ecific arameter, θ j and λ j are crucial to our demand forecating. We earate our forecating rocedure into two cae: (1) movie with attendance data for 2 or more week and (2) newly releaed movie with no data, a they are yet to be hown, or with one week of data, if they have been hown for jut one week at the De Munt Theatre. When there are two or more week of data for a movie title (cae 1), we have ufficient information to etimate both θ j and λ j from (18). To make forecat for uch a movie, we ue the etimate of θ j and λ j, obtained from the firt te with the mot recent data. On the other hand, for movie with either limited or no attendance data (cae 2), there are no etimate of θ j (and/or λ j ) from the firt te. To deal with thi iue, we firt built a regreion model that relate θ j (and/or λ j ) to movie attribute that might exlain θ j (and/or λ j ). Once we obtain the coefficient of thi regreion model, we ue thee number and the value of the attribute for a new movie to etimate that movie value of θ j (and/or λ j ). More ecifically, we regre (ay) the value ofθˆ j, the etimate of θ j from the firt te on variou movie attribute uing the following model: θˆ j = ρ 0 + ρ OU O-USA j +ρ OD O-DUTCH j +ρ OO O-OTHER j +ρ LE L-ENGLISH j +ρ LD L-DUTCH j +ρ LO L-OTHER j +ρ DD D-DUTCH j +ρ SQ SEQUEL j +ρ FC FRANCHISE j 18

19 +ρ BOW1 BOW1 j +ρ TNW1 TNW1 j +ρ BOW1W2 BOW1W2 j +ρ G MPAA-G j +ρ PG MPAA-PG j +ρ PG13 MPAA-PG13 j +ρ R MPAA-R j +ρ DRA DRAMA j +ρ ACT ACTION j +ρ COM COMEDY j +ρ RCOM RCOMEDY j +ρ MUS MUSICAL j +ρ SUS SUSPENSE j +ρ SF SCIFI j +ρ HOR HORROR j +ρ FAN FANTASY j +ρ WES WESTERN j +ρ ANI ANIMATED j +ρ ADV ADVENTURE j +ρ DOC DOCU j +ε j (19) See Aendix 1 for definition of thee characteritic. For Dutch and other movie for which there wa no U.S. releae, uch variable a US box office revenue are et to zero. The indicator variable i ufficient to cature the fact that no additional information from the US box office i available for etimation uroe. Uing the arameter etimate, we then redict θ j for the new movie yet to be oened (cae 2) by inerting the new movie characteritic into (19). (For the few movie that oened imultaneouly in the US and the Netherland, outide etimate of the US box office were ued.) For movie with one week of data, a imilar rocedure i ued for the decay rate, λ j. Ste 3: Attendance Forecat In thi te, we ue (18) to generate forecat for all movie available for creening (cae 1-2) at all oible time in the new movie rogram. Secifically, we take all the arameter etimate from the firt te and, when needed, the etimate of θ j and λ j from the econd te, to forecat the exected attendance for movie j tarting at future time t, E(A jt ): E(A jt ) = ˆ ex( θ j I { j} + λˆ j AGE jt j + h h ˆβ I { h} t + ˆω d I { d} t + ˆ γ d v v I { v} t + δˆ SATNIGHT SATNIGHT t + δˆ SUNPM SUNPM t + δˆ NG NG t + δˆ DTEMP DTEMP t + DRECIP δˆ DPRECIP t + ˆ σ 2 /2) (20) A our arameter etimate are obtained by taking the logarithm of (18) and then etimating the tranformed model with OLS, when we ue thee OLS etimate in the original non-linear model of (18), there 19

20 2 will be a downward bia in the forecat (Hanen et al 2003,.395). We ue the correction factor ˆ σ /2 2 ˆ in (20) to comenate for thi downward bia ( σ i the etimate for the variance of ε jt in (18)). Unlike other variable in (20), the future value of the two weather variable (DTEMP t and DPRECIP t ) are unknown at the time of forecating. We therefore ue the government weather forecat for Amterdam, where the De Munt i located, a the value for DTEMP t (=forecated maximum temerature) and DPRECIP t (=1 if robability of reciitation > 0.5). Data Management For our run of SilverScheduler in the 14 week in the eriod March-July 2005, we continuouly udated our etimate a new data became available. Secifically, we tarted with 57 week (January 29, 2004 February 27, 2005) of attendance data, giving u 23,148 obervation for etimation in the firt week. For each Sunday, the data et wa udated with a new week of data u to that day, adding around 400 new obervation (each week ha about 400 movie creening) 5. Some articular aect of our data collection roce are worth noting. While there were ome free admiion to movie, our etimation only included aid admiion. A only 3% of movie reached 90% of caacity and average utilization er howing wa about 26%, we do not conider caacity contraint in either our etimation or rogramming model. Alo, we do not conider rice effect. Although the De Munt theater charge lightly different rice for howing tarting in different time lot (e.g., matinee) and for moviegoer of different age (e.g., enior dicount), rice are the ame acro different movie title. A a reult, the rice variation i erfectly correlated with the time of day variation. Therefore, the rice effect will be comletely aborbed by the time of day arameter and there i no need to include any rice variable. 5 After comletion of the tet run, we erformed an audit of our data management roce and found ome minor data management roblem. For examle, the attendance at a movie that wa double booked in two creen wa combined into one data oint. Thee effect had a mall overall imact and led to le accurate forecat than otherwie would have occurred. (We verified thi by running our model for an additional week after dicovering thee roblem and found an increae in the R 2 between forecat ale and attendance comared to our mean reult.) In thi aer, we ue the forecat originally generated a they were the one ued in our interaction with management. 20

21 4.2 Evaluation of Forecating Performance Table 2 how the etimation reult for our main forecating model (18). For brevity, we reort only the etimation reult in the firt week. To identify the model (18), we et claic creened at 8m on Saturday a the bae cae. A can be een in Table 2, the reult demontrate high face validity and trong exlanatory ower. The R 2 for the firt week i 0.64; acro the 14 week in our tudy, the R 2 between redicted attendance and actual attendance averaged 0.65, with a high value of 0.81 in week 6 and a low value of 0.43 in week 8. A can be een in Table 2, all but one of the etimated coefficient are tatitically ignificant; thi i not urriing given the large amle ize. More imortantly, the value of the coefficient are conitent with exectation: 1) the etimate for β h increae a we go into the evening (8m i the mot referable time to watch a movie), 2) the etimate for ω d on Saturday and Sunday are the larget; 3) all the etimate for γ v are ignificantly oitive; 4) Sunday afternoon are better than thoe of other day (but other time on Sunday are equivalent to the bae cae); and 5) national occer game negatively affect movie attendance. In addition to the coefficient reorted in Table 2, movie ecific arameter value for θ j and λ j were obtained for each movie. For examle, baed on the data available for the week of March 3, the movie Paion of the Chrit had value of θ j =0.224 and λ j =-0.053, indicating a trong oening and a low decay rate. For movie in their firt or econd week of howing, the characteritic of the movie were ued to etimate value of θ j and λ j a hown in equation (19). A exected, the need to rely on etimate had an effect on the accuracy of our forecat. For examle, acro the 14 week, the average error for new movie wa admiion er howing; for movie with jut one week of data, the mean error wa 12.8 admiion er howing; and for movie with two or more week of data, the mean error wa -6.4 admiion ticket er howing 6. Future work hould examine method to obtain better etimate for new movie. Market reearch technique uch a thoe decribed in Eliahberg et al. (2000) or data from new data ource uch a the Hollywood Stock Exchange (Sann and Skiera 2003) could be uefully emloyed to obtain uch reult. 21

22 Movie Program: Mar 3-9, 2005 R-Square TUE <.0001 Adj R-Sq WED <.0001 N THU <.0001 Variable Parameter Etimate Pr > t FRI <.0001 Intercet <.0001 SUN am <.0001 SATNIGHT < am <.0001 SUNPM < m <.0001 National Soccer < m <.0001 Eater Weekend < m <.0001 Acenion Day < m <.0001 Whit Weekend < m <.0001 Sring Vacation < m <.0001 May Vacation < m <.0001 Summer Vacation < m <.0001 Fall Vacation < m <.0001 Xma Vacation < m <.0001 Daily Max. Tem <.0001 MON <.0001 Preciitation <.0001 Bae Cae: Claic creened at 8m on Saturday Table 2: Etimation Reult of Demand Model (18) 5 Alication of SilverScheduler to a Multilex in Amterdam 5.1. Emirical etting The comlete SilverScheduler rocedure (cheduling algorithm + conditional forecating method) wa alied to the De Munt Theater in downtown Amterdam. We had data for 14 different movie week for the De Munt, a well a data from the reviou year for etimating initial value of the model arameter. For each of the 14 week we received the following information from Pathé: The movie to be cheduled in that articular week (i.e., the lit of movie), with their running time. The average number of movie to be cheduled wa 18, ranging from 15 to 22. Contractual agreement with ditributor that certain movie will be hown in certain creening room. Tyically, uch agreement are made for newly releaed movie (1 to 3 er week). Sometime the creening room i ecified; ometime the agreement ay that the creening room hould be above a ecific eat caacity. 6 The Mean Abolute Error were alo calculated and were 37.2, 22.6, and 19.0 reectively for new movie, movie with one 22

23 Oening and cloing time of the theater for weekday and weekend day and information about the time required for cleaning the creening room between two how. Additional managerial requirement Firt, movie viitor hould never have to wait more than 20 minute until the next movie tart. In thi way, Pathé trie to aeal to eole who jut want to ee a movie, without a trong reference for a articular one. Therefore, the time between two ubequent movie tart hould not be more than 20 minute (imlemented with a enalty function). Second, for high otential movie there hould be the oibility of double of even trile booking. Thi mean that the ame movie i hown in different creening room at the ame time. We accommodated thi in our ytem, and to get the mot out of uch high otential movie, we decided that the tarting time of the ame movie in different creening room would be at leat one hour aart. Management informed u a to which movie hould be double or trile booked. Third, there i a roblem of crowding. The creening room of De Munt are on two different floor level, with room 1 to 8 on one floor and room 9 to 13 on the other. The corridor and hallway are quite narrow. To avoid overcrowding, a rule wa introduced that at the buy time (all evening and Saturday and Sunday afternoon) not more than one movie can tart in the ame time block on the creen 1 to 8, and the ame for the creen 9 to 13. Thi read in movie tart i alo favorable for the ale from the conceion counter, which have a more evenly ditributed demand in thi way. Fourth, management wanted to limit the number of creen change, i.e. the number of time that another movie ha to be mounted on the rojection ytem for a articular room. Movie reel are hyically quite big and difficult to handle, and. there i a (mall) robability that the movie reel will fall and be damaged. We limited the number of creen change by imoing a enalty for each creen change. A demand inut we ued the etimate roduced by the demand forecating rocedure, a decribed earlier. Thi mean that we have a forecat for the number of viitor for every movie for each oible tarting time/day combination in the week for which the chedule i roduced. Here we work with a grid of tarting time that are one hour aart. week of data, and for movie with 2 or more week of data. 23

24 From day chedule to a week chedule Emloying the rocedure decribed above, we ued SilverScheduler to make chedule for each of the 14 week in our dataet. A i clear from the decrition in ection 3, the cheduling algorithm, in rincile, make a chedule for a day (it olve the MSP). However, De Munt doe not have the ame movie chedule on each of the even day of the week. Firt of all, Saturday and Sunday are different from weekday. On both weekend day the theater oen at 10 am (comared to noon on weekday) and on thee day there are ome children movie hown (until 6 m). Furthermore, Saturday i different from Sunday, becaue Saturday i the only day that the theater i oen until 1.30 am. On all other day, it cloe at midnight. Finally Wedneday i different from the other weekday, ince children movie are hown on Wedneday afternoon. To take thee difference into account, we roceeded a follow. We firt alied SilverScheduler to generate a chedule for the four day (Thurday, Friday, Monday, and Tueday) that have the ame chedule. Subequently thi bae chedule erved a the tarting oint to make chedule for the other three day. Here the movie remain in the ame creening room a in the bae chedule, but the time are adated according to the requirement of the ecific day. Furthermore, on Saturday, Sunday, and Wedneday children movie were inerted during the day, relacing thoe movie of the bae chedule which had the lowet number of viitor. 5.2 Reult for one day To illutrate the SilverScheduler aroach to movie cheduling and to comare it to the correonding actual Pathé chedule, contructed manually, we ue the firt day (Thurday, March 3) in our dataet. There are 26 movie in total for thi week, including 2 coie of Contantine. (For a lit of movie and name abbreviation, ee Aendix 2.) The lat 3 movie lited are for dedicated howing in ecific creen. There are 4 children movie (not hown on Thurday, Friday, Monday or Tueday), o there remain 19 movie that need to be cheduled. 24

25 Figure 3 how (1) the actual chedule a roduced by Pathé management (left), and (2) the chedule generated by SilverScheduler (right). A can be een from Figure 3, we have divided the oening time of the theatre [from (noon) to (midnight)] into time interval of 10 minute each. The number behind the movie name are the forecated number of viitor for the articular how. For thi articular week there wa only one contractual agreement with a ditributor, requiring that the movie Hide and Seek (HS) hould be hown in one of the two larget creening room, i.e., either in Screening Room 3 (340 eat) or in Screening Room 11 (382 eat). From Figure 3 it can be een, that Pathé how HS in Screening Room 11, wherea SilverScheduler ut it into Screening Room 3. In Figure 3 the difference in running time between the different movie are clear. Manual Schedule Schedule contructed with SilverScheduler Screen Screen Caacity Caacity :00 12:00 CLO(5) SNL(8) VET(4) MTF(10) RYV(5) 12:10 12:10 12:20 TA(7) 12:20 BIR(3) 12:30 12:30 MM(5) 12:40 VET(4) 12:40 TA(7) 12:50 BAP(8) 12:50 13:00 AQUA(19) 13:00 WOO(8) BAP(18) 13:10 HS(17) WOO(8) 13:10 HS(17) 13:20 MTF(24) 13:20 SWD(10) 13:30 CON(25) 13:30 13:40 MM(11) AVI(11) RAY(20) 13:40 AQUA(19) 13:50 SWD(10) 13:50 14:00 14:00 SNL(25) 14:10 14:10 RAY(27) 14:20 14:20 CON(34) RYV(16) 14:30 MDB(55) 14:30 14:40 TA(23) 14:40 CON(34) BIR(10) 14:50 14:50 MM(15) 15:00 VET(13) 15:00 MDB(61) 15:10 15:10 15:20 15:20 WOO(12) 15:30 BAP(27) 15:30 BAP(27) 15:40 AQUA(29) WOO(12) 15:40 15:50 HS(25) 15:50 HS(25) SWD(16) 16:00 MTF(30) 16:00 SNL(22) 16:10 CON(31) MM(14) 16:10 16:20 SWD(13) 16:20 AQUA(24) 16:30 16:30 16:40 16:40 RYV(14) 16:50 RAY(24) 16:50 17:00 TA(25) 17:00 BIR(11) 17:10 VET(12) 17:10 MM(16) 17:20 AVI(15) 17:20 RAY(28) 17:30 MDB(58) 17:30 WOO(12) CON(36) 17:40 17:40 17:50 17:50 18:00 UNT(25) 18:00 BAP(31) MDB(72) 18:10 BAP(31) 18:10 CON(45) 18:20 AQUA(35) 18:20 SNL(32) 18:30 HS(30) 18:30 18:40 MTF(43) MM(20) 18:40 SWD(19) 18:50 SWD(19) 18:50 19:00 CON(85) 19:00 AQUA(65) RYV(39) 19:10 19:10 HS(56) 19:20 BIR(26) 19:20 BIR(26) 19:30 VET(29) 19:30 MM(37) 19:40 19:40 19:50 19:50 20:00 RAY(77) 20:00 WOO(32) 20:10 20:10 20:20 20:20 AVI(42) CON(99) 20:30 MDB(158) 20:30 UNT(54) RAY(77) 20:40 CON(99) 20:40 20:50 AVI(42) 20:50 CON(99) MDB(158) 21:00 AQUA(49) 21:00 21:10 MM(28) HS(42) 21:10 SWD(26) 21:20 SWD(26) 21:20 RYV(29) 21:30 MTF(60) 21:30 21:40 CON(63) WOO(21) 21:40 AQUA(49) BIR(19) 21:50 CLO(31) 21:50 HS(42) MM(28) 22:00 TA(52) 22:00 22:10 22:10 WOO(25) 22:20 22:20 22:30 22:30 22:40 22:40 22:50 22:50 23:00 23:00 23:10 23:10 23:20 23:20 23:30 23:30 23:40 23:40 23:50 23:50 00:00 00:00 00:10 00:10 00:20 00:20 00:30 00:30 Vitor Vitor Figure 3: Schedule for Thurday March 3, 2005 For examle, for Der Untergang (UNT) the running time i 165 minute, and for Vet Hard (VET) 105 minute. (Running time include advertiing and trailer.) 25

26 After each movie howing, there i a cleaning time before a new movie tart. The cleaning time i 20 minute for a mall room and 30 minute for a large room. Evaluation of the olution The number of viitor for Thurday March 3, 2005, i 1785 in the SilverScheduler chedule, againt 1661 in the manual chedule. For thi day, SilverScheduler generate 124 extra viitor. Silver- Scheduler manage to chedule more how on thi articular day: 57, againt 51 how in the manually contructed chedule. So in thi cae, notwithtanding the extenive et of contraint, SilverScheduler i able to accommodate more how. The SilverScheduler olution obey the requirement that at leat every 20 minute a new movie hould tart. It i quite difficult to reach thi goal when the chedule i made by hand. In the chedule for March 3 (Figure 3, left) thi requirement i violated everal time. For examle, there are no movie that tart between 13.50h and 14.30h, between 19.30h and 20.00h, and between 20,00h and 20.30h. Alo the other managerial requirement are violated reeatedly in the manual chedule. For examle, on the Saturday of the firt week of our dataet (March 5, 2005; not hown here) everal time the manual chedule allowed more than one movie to tart on the ame floor at the ame time. Thi will roduce undeirable crowding. Alo, it occurred that there were more than two different movie in the ame creening room on the ame day, which violate the contraint of no more than two different movie on one creen (i.e., too much creen witching). 5.3 Reult for the 14 week We evaluate our overall reult in everal way. Firt, we look at the cheduling efficiency, articularly the number of movie that are hown in each week. Second, we examine the revenue imlication. Table 3 how the number of movie ( movie howing ) that would be hown in the Silver Scheduler chedule a comared to the Pathé chedule. Although SilverScheduler ha to take into account a large number of contraint, it till manage to chedule lightly more movie than the rogrammer at 26

27 Pathé have done (+ 0.51%). But of coure, the main contribution of SilverScheduler hould be that it chedule better movie, which hould reult in higher ticket ale. Pathe chedule SilverScheduler Week Decrition Movie Showing Ticket ale Movie Showing Ticket ale Imrovement Mar 03 - Mar % Mar 10 - Mar % Mar 17 - Mar % Mar 24 - Mar % Ar 07 - Ar % Ar 14 - Ar % Ar 21 - Ar % Ar 28 - May % May 05 - May % May 19 - May % May 26 - Jun % Jun 02 - Jun % Jun 30 - Jul % Jul 14 - Jul % Total % Table 3: Comarion of the chedule of Pathé and SilverScheduler over all 14 week; Viitor rediction baed on redicted demand, uing Equation (18). It i clear from Table 3 that there i coniderable imrovement in attendance. If we comare both chedule, the Pathé chedule and thoe generated by SilverScheduler, and ue in both cae Equation (18) for redicting the number of viitor, we ee that SilverScheduler how an increae of 10.83%. Thi imrovement reflect the contribution of the cheduling algorithm, given the demand (i.e. the redicted number of viitor). In other word, thi i the contribution of SilverScheduler with erfect forecat. However, actual viitor number tend to deviate from forecated number. A we aw earlier, the correlation between actual and forecat i on average Of coure, with imerfect forecat, the imrovement obtained through SilverScheduler will be le. To invetigate thi, we have alo carried out a comarion of the chedule of Pathé and SilverScheduler baed on the actual demand, a manifet from the oberved viitor data (i.e. ex ot). For thi uroe, uing the actual viitor data for the week, a regreion model wa etimated exlaining the number of viitor er how, by the movie, the day of the week and the hour of the day. Thi model wa then ued to redict the number of viitor er how, for the chedule of Pathé, a well a the one from SilverScheduler. (For the Pathé chedule we could alo have ued the actual data directly, but we felt that uing the ex-ot regreion model rediction in both 27

28 cae, give a fairer comarion). Uing thi aroach, the etimated imrovement in number of viitor wa 2.53% over the 14 week. While lower than the level reorted in Table 3, thi i till a coniderable increae; it indicate there would be about 20,000 additional viitor er year, or about 150,000 in extra revenue ($ 187,000). In addition to generating more viitor, SilverScheduler alo rovide direct oerational efficiency by automating the rearation of the weekly movie chedule. SilverScheduler ave a coniderable amount of managerial time and effort. The current roce i cumberome, often generating a fair amount of management frutration. Furthermore, all the exreed management contraint are met. Taking thee managerial contraint into account (every 20 minute a new movie, le crowding) hould over time have a oitive effect on the number of viitor which i not taken into account in the current comutation of additional revenue. 6. Concluion and further develoment We have develoed an algorithm, SilverScheduler that chedule movie over the day of the week and the time of the day. Thi algorithm, which follow the column generation aroach, i able to roduce olution in a reaonable amount of time (on average 2.5 minute) and with very good accuracy (on average within 1.57% of the otimum). A forecating module wa develoed where the number of viitor are forecated uing a model etimated on data from reviou week. We carried out the cheduling for the De Munt Theater in Amterdam for 14 week in Comaring the SilverScheduler reult with the manual chedule, SilverScheduler generated imroved reult. Within the contraint et by logitical and managerial conideration, on average SilverScheduler cheduled about the ame number of how a management did. SilverScheduler, however, chedule better movie, and alo take better account of the managerial requirement comared to the manual olution. Auming accurate forecat, SilverScheduler generate nearly 11% more viitor than the manual chedule. When we make the comarion on the bai of actual demand in the given week, the imrove- 28

29 ment in viitor through SilverScheduler i 2.5%. The latter amount to $ 187,000 in extra revenue for the De Munt, on an annual bai. Additional reearch hould hel to further imrove SilverScheduler. The firt riority i to increae the accuracy of the forecat. A we have een, better forecat ignificantly imrove the erformance of SilverScheduler a a whole. One oibility i to combine the currently ued box office data with market reearch data f market. Thi will imrove the quality of the forecat, in articular for new movie. Furthermore, new methodologie have become available for the early rediction of the ucce of new movie (e.g. Eliahberg et al 2000). Another area to examine i the imact of movie cheduling on conceion ale. At reent Pathé management etimate a contant amount of conceion ale er viitor. However, conceion ale may vary by the time when movie are hown and may be deendent on the amount of queuing that occur. An extenion in thi regard would oe intereting technical challenge, but could lead to ubtantial rofit imrovement given the high margin on conceion ale. Alo, SilverScheduler ha to be develoed further in the direction of a deciion uort tool, o that it can eaily be ued by the theater manager. Thi include a uer-friendly interface, and intuitively-aealing creen that are eay to navigate. Pathé Nederland, the owner of the De Munt Theater, conider the imlementation of Silver- Scheduler a an intereting oortunity. In that cae, they will not jut ue it for De Munt, but for movie cheduling for all of their 12 movie theater in the Netherland. Pathé Nederland ha a lanning and ticketing oftware ytem in lace already, and SilverScheduler can be ued to deliver it chedule a the inut to thi ytem. While we decribe our develoment of the SilverScheduler algorithm in one theater, the roblem i a wideread one. There are 7000 theater in the US and 10,000 in Euroe, all faced with the ame roblem. Movie theater rogramming i a time-conuming activity, and a our examle illutrate, the required managerial contraint are not alway met. It i clear that the oibility to delegate at leat art of thi tak to a deciion uort ytem i an imortant te forward. Theater are increaingly moving from manual ticketing ytem to comuter baed ticketing ytem. Thi hould facilitate the adotion of 29

30 comuter baed deciion uort ytem. Probably even more imortant to the future imact of ytem uch a SilverScheduler in the movie indutry i the exected growth of digital cinema. When thi occur, movie will not be delivered to theater on huge reel anymore, but on imle dikette or electronically. Thi will make theater much more flexible in their cheduling. In uch a ituation, the cheduling oibilitie will multily, making it even more imortant to have an algorithm like SilverScheduler a art of a deciion uort ytem to hel management find the bet chedule for it theater. Reference Ainlie, A., X. Dreze, F. Zufryden Modeling Movie Lifecycle and Market Share. Marketing Sci Barnhart, C., E.L. Johnon, G.L. Nemhauer, M.W.P. Savelbergh, P.H. Vance Branch-and-Price: Column Generation for Solving Huge Integer Program. Oeration Reearch Bealey, J.E Lagrangean Relaxation. Page of: Reeve, C.R. (ed), Modern Heuritic Technique for Combinatorial Problem. McGraw-Hill, London. Dantzig, G.B., P. Wolfe Decomoition rincile for linear rogramming. Oeration Reearch Deaulnier, G., J. Deroier, Y. Duma, M.M. Solomon, F. Soumi Daily aircraft routing and cheduling. Management Sci Deaulnier, G., J. Deroier, M.M. Solomon (ed) Column Generation. Sringer, New York. Eliahberg, J., J.-J. Jonker, M.S. Sawhney, B. Wierenga MOVIEMOD: An Imlementable Deciion Suort Sytem for Prereleae Market Evaluation of Motion Picture Marketing Sci Eliahberg, J., S. Swami, C.B. Weinberg, B. Wierenga Imlementing and evaluating Silver- Screener: A marketing management uort ytem for movie exhibitor. Interface 31 S108- S127. Garey, M.R., D.S. Johnon Comuter and Intractability: a Guide to the Theory of NP- Comletene. Freeman, San Francico. Hanen, D.M., L.J.Paron, R.L. Schulz Market Reone Model: Econometric and Time Serie Analyi (ec. ed). Kluwer, Boton. Ho, J Marketing Model of Entertainment Product. Unublihed doctoral thei, Univerity of Britih Columbia. Huiman D., R. Freling, A.P.M. Wagelman Multile-Deot Integrated Vehicle and Crew Scheduling, Tranortation Sci Huiman, D., R. Jan, M. Peeter, A.P.M. Wagelman Combining Column Generation and Lagrangian Relaxation. Page of: Deaulnier, G., Deroier, J., Solomon, M.M. (ed), Column Generation. Sringer, New York. Krider, R.E., C.B. Weinberg Cometitive Dynamic and the Introduction of New Product: The Motion Picture Timing Game. Journal of Marketing Reearch Lehmann, D.R., C.B. Weinberg Sale through Sequential Ditribution Channel: An Alication to Movie and Video. Journal of Marketing Löbel, A Vehicle cheduling in ublic tranit and Lagrangean ricing. Management Sci Lübbecke, M.E., J. Deroier Selected Toic in Column Generation. Oeration Reearch

31 NVB Nederlande Vereniging van Biocooexloitanten, Annual Reort, Amtelveen, the Netherland. Sandhu, R., D. Klabjan Integrated Airline Fleeting and Crew Pairing Deciion. To aear in Oeration Reearch. Sawhney, M.S., J. Eliahberg A arimoniou model for forecating gro box-office revenue of motion icture. Marketing Sci Sann, M. and B. Skiera Internet-Baed Virtual Stock Market for Buine Forecating. Management Science Swami, S., J. Eliahberg, C.B. Weinberg SilverScreener: A modeling aroach to movie creen management. Marketing Sci

32 Aendix 1: Definition of Movie Attribute O-USA j Indicator variable for the event that movie j i made in the U.S. O-DUTCH j Indicator variable for the event that movie j i made in the Netherland O-OTHER j Indicator variable for the event that movie j i made in other countrie L-ENGLISH j Indicator variable for the event that movie j i in Englih L-DUTCH j Indicator variable for the event that movie j i in Dutch L-OTHER j Indicator variable for the event that movie j i in other language D-DUTCH j Indicator variable for the event that movie j i dubbed Dutch SEQUEL j Indicator variable for the event that movie j i a equel FRANCHISE j Average worldwide box office gro of all receding movie in the franchie of movie j, if movie j i a equel. BOW1 j Oening week box office ale in the U.S., if movie j i made in U.S. TNW1 j Oening week theater number in the U.S., if movie j i made in U.S. BOW1W2 j Box office ale ercentage change from oening week to the econd week in the U.S., if movie j i made in U.S. MPAA-G j Indicator variable for the event that movie j i rated G by MPAA and made in U.S. MPAA-PG j Indicator variable for the event that movie j i rated PG and made in U.S. MPAA-PG13 j Indicator variable for the event that movie j i rated PG13 and made in U.S. MPAA-R j Indicator variable for the event that movie j i rated R and made in U.S. DRAMA j Indicator variable for the event that movie j i claified by Variety.com to be in the drama genre and made in U.S. ACTION j Indicator variable for the event that movie j i in the action genre and made in U.S. COMEDY j Indicator variable for the event that movie j i in the comedy genre and made in U.S. RCOMEDY j Indicator variable for the event that movie j i in the romantic comedy genre and made in U.S. MUSICAL j Indicator variable for the event that movie j i in the muical genre and made in U.S. SUSPENSE j Indicator variable for the event that movie j i in the uene genre and made in U.S. SCIFI j Indicator variable for the event that movie j i in the ci-fi genre and made in U.S. HORROR j Indicator variable for the event that movie j i in the horror genre and made in U.S. FANTASY j Indicator variable for the event that movie j i in the fantay genre and made in U.S. WESTERN j Indicator variable for the event that movie j i in the wetern genre and made in U.S. ANIMATED j Indicator variable for the event that movie j i in the animated genre and i made in U.S. ADVENTURE j Indicator variable for the event that movie j i in the adventure genre and made in U.S. DOCU j Indicator variable for the event that movie j i in the documentary genre and made in U.S. 32

33 Aendix 2: Lit of movie for the movie week of March No Full Name Abbreviation Kid? Duration 3 1 Million Dollar Baby MDB No Meet The Focker MTF No Contantine CO1 No Contantine CO2 No Melinda And Melinda MM No The Aviator AVI No Birth BI No Team America TA No Ray RAY No Raie Your Voice RYV No Shall We Dance SWD No Songebob SNL No The Life Aquatic AQ No Bride And Prejudice BAP No Hide & Seek HS No Vet Hard VH No Cloer CLO No Der Untergang UNT No The Woodman WOO No Leel LPL Ye Plo & Kwiel PK Ye Incredible INC Ye Stree Wil Racen STR Ye Paion Of The Chrit 1 PAS No Goodbye Lenin 1 GL No Hitch 2 HIT No 133 1) One how, Sunday morning 2) One how, Saturday night 3) In minute. Advertiing and trailer included 33