Demand-driven scheduling of movies in a multiplex
Introduction
The motion picture industry is a prominent economic activity with total worldwide box office revenue of $ 26.7 billion in 2007, of which $ 9.6 billion is in the U.S.A.2 Movie forecasting and programming in practice tend to be associated with intuition rather than formal analysis, and this also characterizes the tradition of decision-making in the film industry. However, many problems in the film industry are actually quite amenable to model building and optimization, as movie executives have increasingly recognized. In this paper, we focus on one such problem: the detailed scheduling of a movie theater in Amsterdam. Movie marketing and modeling is a new area for application of marketing decision support systems, originally developed for the fast-moving consumer goods industry (Wierenga & Oude Ophuis, 1997). Developing models that deal with real problems of decision-makers in practice has been an issue of continuing concern in marketing (Eliashberg and Lilien, 1993, Leeflang and Wittink, 2000). In this project, we have followed a “market-driven” approach, where we have designed a model with capabilities that are custom-designed to managerial needs (Roberts, 2000).
A movie program or schedule in a theater is designed for a week. In the Netherlands, for example, a new movie week starts each Thursday. Therefore, a movie theater has to prepare a new movie schedule at the beginning of every week. This is particularly complex for the multiplex theaters, the increasingly dominant movie theater format around the world. In the Netherlands, multiplexes with eight or more screens represent 24% of all movie theater seats and 34% of the total box office. It is clear that programming such large cinema facilities is not an easy matter.
For each week's movie program, management must determine what movies will be shown, on which screens, on which days, and at what times. Typically, on each screen, a theater can accommodate three to five showings per day, where a “showing” is defined as the screening of one movie, including trailers and advertisements. This means that a 10-screen theater needs to program around 280 showings per week. Presently, this programming is mostly done manually with pencil and paper by specialists in the theater company. Despite the combination of a programmer's analytical mind and a broad knowledge about movies and about the audience of individual theaters, often based on many years of experience, it is our belief that an analytical system can help in movie programming. This would not only relieve theaters from a repetitive labor-intensive task, but also achieve a better performance in scheduling than the current mental, manual procedure.
The programming problem for an individual movie theater consists of two stages: (i) the selection of the list of movies, i.e., the movies to be shown over the course of the particular week and (ii) the scheduling of these movies over screens, days, and times. Stage (i) (macro-scheduling) includes making agreements with movie distributors and is completed before stage (ii). In this study, we develop a solution for the second stage (micro-scheduling), which involves constructing detailed schedules for where (which screen(s)) and when (days, hours) the different movies will play. For analytical procedures that deal with the first stage, (i.e., deciding on the movies to be shown in a particular week), see Swami, Eliashberg, and Weinberg (1999), and Eliashberg, Swami, Weinberg, and Wierenga (2001).
Our scheduling problem has two sub-problems. First, we need to answer the question: if a particular movie were shown on a particular day at a particular time, how many visitors would attend? Inputs for making these forecasts are based on numbers of observed visitors in previous weeks (for existing movies), various characteristics of the movie (for newly released movies), coupled with information about variables such as specific events (holidays) and the weather. Making conditional forecasts is not an easy task, and it belongs to the realm of marketing. Second, given this demand assessment, we have to find the schedule that maximizes the number of visitors for the week, given constraints such as theater capacity and run-times of the movies. This is also a non-trivial problem, for which the discipline of operations research is useful. The specific solution approach employed here is column generation, a method designed to solve (integer) linear optimization problems with many variables/columns with relatively few constraints. The remainder of this paper describes how, by combining the two disciplinary approaches, a solution to the movie micro-scheduling problem was obtained. The structure of the movie scheduling problem of this paper is similar to the problem of optimal scheduling of TV programs, which has been addressed in the marketing/OR literature (Danaher and Mawhinney, 2001, Horen, 1980, Reddy et al., 1998). In the TV program scheduling problem, there is also a demand forecasting module that predicts the size of the television audience, and an optimization procedure that finds the best schedule. Several new and unique features arise in our problem, however, as the TV scheduling problem has no constraint for audience size and also the same program cannot be consecutively scheduled many times.
In this paper, we first take a closer look at the movie scheduling problem (Section 2). In Section 3, we describe the column generation approach to the optimization problem. In Section 4, we discuss the method that we developed to conditionally forecast the number of visitors of a show. In Section 5, we apply the complete procedure through a model we call SilverScheduler to fourteen movie weeks of the De Munt theater, a multiplex with 13 screening rooms located in central Amsterdam. The De Munt theater is owned by Pathé Nederland, which has the largest share of the movie exhibition business in Holland. With over 1 million visitors a year, the De Munt theater is the second largest theater in the Netherlands. The last part of the paper (Section 6) discusses the results and places them in perspective, and discusses issues for future research.
Section snippets
Problem description and the research project
As movie theaters have evolved from single screen theaters to multiplexes and even megaplexes, the problems of scheduling movies on screens has become increasingly complex. Here, we will now discuss some of the key reasons for that complexity.
First, there are a large number of different movies that the theater wants to show in a typical week. This number is typically larger than the number of screens. Moreover, these movies have different run-times. For example, among the different movies
A column generation approach to solve the movie scheduling problem
To produce a movie program for a certain week, we need to find schedules for the different days in that week. We define the movie scheduling problem (MSP) as the problem of finding the optimal movie program for a single day given the list of movies to be shown (the “movie list”), the run-times of these movies, forecasted demand, capacities of different screening rooms, and information about contractual agreements with distributors about screening rooms for particular movies, accounting for the
Conditional demand forecasting
Developing a week's schedule requires forecasts for Ajt, the attendance for a screening of movie j starting at time t, for all movies available for showing at all possible starting times (broken down into hourly intervals) for every day of that week. These forecasts are required to be available each Sunday for the scheduling algorithm so that recommendations to management become available on Monday. To use the most recent data possible, the forecasting procedure had to be completed in less than
Empirical setting
The complete SilverScheduler procedure (scheduling algorithm plus conditional forecasting method) was tested and evaluated in the De Munt theater in downtown Amsterdam. The goal was to demonstrate to Pathé's management the efficiency of SilverScheduler and the difference between SilverScheduler and manually generated schedules for 14 weeks, employing data from the previous year for estimating initial values of the model parameters.
For each of the 14 weeks we received the following information
Conclusions and further development
We have developed a model, SilverScheduler, which schedules movies over the days of the week and the times of the day. The model's algorithm, which follows the column generation approach, is able to produce solutions in a reasonable amount of time (on average 2.5 min) with very good performance (on average within 1.57% of the optimum). A forecasting module was also developed where the numbers of visitors are forecasted using a model estimated on data from previous weeks.
We generated and
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Partially supported by NSF Grant DMS0600848.