Optimal film selection for multiplex theater.

ABSTRACT

The paper presents a film selection optimization model for a multiplex theater. In this model, the multiplex theater is treated as a single facility with multiple screens that can generate revenues though assignments of different films to those screens. The model identifies an optimal selection of the available films that maximizes revenue per multiplex theater subject to a variety of special constraints. These constraints include screen availability, limitation on the number of film copies, genre mix, time availability, and contractual obligations between theater owners and film distributors. The model was successfully used to develop optimal film selections and identify associated optimal revenues for a real multiplex theater for a period of 12 weeks. On the average, the optimal revenues in that period were almost 18% higher than the estimated revenues based on the actual film selections.

1. INTRODUCTION

The motion picture theater industry is dynamic business environment with the frequently changing tastes of cinema patrons and perishable nature of the product. The single screen theater of the past has given way to multi-screen theaters, and acquisitions have reduced the number of industry participants (Hinz, 2000). Theater customers are demanding more than just the movie; they are requesting stadium seating, gourmet food and beverages, and luxurious setting (Parry, 2000). This has caused dramatic expansion in theater capacity, resulting in the creation of multiplex (multi-screened) theaters.

Film patrons have consistently demanded a high-quality theater experience including updated sound systems, large screens, and comfortable seating. In recent years, theater owners (theater exhibitors) have built a number of multiplex theaters to replace older facilities and meet increasing demand (Armstrong 1999, Goldsmith 2002). However, several reports (The Economist 2000, Newman 2001) contended that theater profitability on a per-screen or a per-theater basis has not justified the capital investments required to fund new construction and new technology. The number of screens in the US is approximately 37,000 in an environment that may only support 25,000 (Goldsmith, 2002). At the same time, the yearly revenue generated on a per-film basis has continued to increase, which has been an incentive for theaters (theater owners) to add even more screen capacity. The decrease of revenues per screen and, at the same time, growth of revenues per film, may be indicative of the fact that the theater owners have expanded the theater capacities without properly identifying and scheduling the best films for their theaters.

The evidence suggests (Peers 1998, Farthing 2000, Rich 2001) that the traditional rules for film selection and distribution are changing significantly assuming the trend toward digital/electronic transmission of films to theaters. This makes the film distribution streamlined and enables theater owners to potentially better plan and schedule a cost-effective and quality-oriented selection of films that they demonstrate in the multiplex theaters. Usually, however, theaters' management does not have effective quantitative tools to ensure revenue maximization in their theaters in a rapidly changing environment of the film industry.

Since 1999 to date, the most important work in this area was done by Swami et al, who developed the SilverScreener model (Swati et al 1999). This model is a film selection and scheduling tool that will systematically select films for a multiplex theater based on maximizing the theater's revenues. The SilverScreener model (Swami et al 1999) develops optimal film schedules using the following assumptions:

* Movies availability is known in advance

* Weekly revenues can be estimated

* Replacement decisions are made weekly

* All screens in a theater are of equal capacity

* Movie replenishment is instantaneous.

The SilverScreener model is analogous to an assignment problem, where the theater screens represent "machines" and the individual films are "jobs" to be assigned to those "machines" to maximize total revenue of the assignments. The authors created a revenue function to assisting in the selection of films for a multiplex theater. The model contained only one constraint, according to which a film may not be shown more than a predetermined number of consecutive weeks.

However, the SilverScreener model has several major limitations. It assumes that the seating capacity of all theaters is the same. In reality, the capacity varies from theater to theater, which affects its performance and revenue. The SilverScreener model does not consider films' running times as a possible constraint in the film selection process. However, the running time of a film is an important limitation, because it directly affects the number of the film's shows per unit time. Besides, the SilverScreener model does not include some other important real-life constraints such as a number of identical genre films, amount of new and old films, a mix of distributors, and the number of identical films used on several screens. These necessitated the development of a new film selection model that would incorporate all real-life limitations of film selection in a multiplex theater.

2. MODEL CONSTRUCTION

We proposed a new integer linear programming model to optimize film selection in multiplexes. In this model, a multiplex theater is treated as a single facility with multiple screens that can generate revenues though assignments of different films to those screens. The model's objective is to identify a selection of films available for show in a multiplex theater that would maximize the theater's revenue for a certain time period. This model employs the following assumptions:

* The data on the revenue per film is available. This can be obtained via trade publications like the Variety magazine, marketing research, and forecasts of expected revenues based on available historical data.

* Each film must be shown a certain number of times to meet revenue estimates.

* New films are obtained instantaneously, without any delays.

* There is no time lag from the end of an old film to the start of a new film within a single theater screen. This can be approximated by adding to the film's length 15 minutes prior to the start and after finish of each film.

* The model requires the multiplex theater owners (managers) to update the input information on films, their revenues, and other parameters of the model on a weekly basis.

The model includes a variety of constraints to reflect real-life limitations of film selection. All available theater screens must be occupied with films during the operating time of the theater. However, the same films should not run concurrently on more than a few screens. All films must be shown at least a certain number of showings per day to hit revenue expectations. The running time of the films cannot exceed the available viewing time per day. A constraint should be also applied to ensure that the showings do not exceed a reasonable number of hours per day (for example, the number of hours that the multiplex is open for business).

Multiplex theaters require a mix of various films to meet the demand of the diverse customer population. Therefore, the theaters are constrained in terms of showing certain number of films within the same artistic genre (romance, comedy, horror, etc.). This number of required films of a specific genre can be set by theater owners (managers) or suggested by market research.

Based on the described assumptions and constraints, we defined the following parameters of the model: X = number of copies of a specific film shown in the multiplex theater

t = film's length

[alpha] = coefficient indicating the required percentage of new films

[beta] = number of showings per day for a specific film

R = revenue coefficient per film. This parameter indicates the amount of film revenue for a specific time period (week).

C = number of available screens (screen capacity).

N = maximum number of copies of the same film scheduled in the same multiplex.

T= total amount of time available per day for showing films for all screens.

[G.sub.i] = minimum number of films per genre (action, drama, comedy, etc.).

[D.sub.i] = maximum number of films per specific distributor.

i = film genre index

j = film distributors' index

m = binary index for a new film (m = 0) and a film which is more than one week old (m=l)

In the model, we consider [X.sup.m.sub.ij] as the number of copies associated with a specific film of genre i, distributor j, and "age" m shown in the multiplex theater per week. Based on the described definitions and assumptions, the model formulation is as following:

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The objective function reflects the total revenue of a multiplex theater per week. The revenue coefficient R represents the revenue for a specific film on a weekly basis. To identify R, we used the revenue per film as reported in the Variety magazine for the preceding week. For newly released films, we apply the revenue per film of a comparable genre and distributor.

Constraint (1) ensures that the theater screen capacity is fully utilized. Parameter C represents the total number of available screens in a multiplex theater, which in practice varies from 10 to 30 screens. Contraint (2) controls the maximum number of identical films that can be scheduled in the same theater. For most multiplexes, the maximum number of simultaneous showings of the same film rarely exceeds three screens. Constraint (3) represents an adequate genre mix. The parameter G in this constraint insures that the multiplex will have at least minimum required number of films in a specific genre. The value of G is defined by the theater owners or based on external requirements.

Constraint (4) ensures that the total running time of all films does not exceed the available showing time T per day. The [beta] coefficient reflects the number of times a film will be shown per day. These coefficients may be assigned by multiplex owners based on their subjective opinion or marketing data. Constraint (5) indicates the percentage of new versus old films in a multiplex. This percentage denoted as [alpha] is set by the theater owners (managers) depending on the number of new films available and marketing data. Finally, constraint (6) ensures that the films are selected from an appropriate mix of distributors.

3. IMPLEMENTATION OF OPTIMAL FILM SELECTION IN MULTIPLEX THEATER

Utilizing the described model, we were able to simulate optimal film selections and identify the revenue results for a real multiplex theater. We considered a large 20-screen multiplex theater in the San Francisco Bay Area for a period of 12 weeks (one quarter). The list of possible films to be included in the model for each Week was obtained from public sources such as Variety magazine, Reuters or some others. The revenue coefficient R in the objective function was obtained from the same sources. For each week, the model used the weekly revenue of a specific film from a preceding week. This data was refreshed on a weekly basis to reflect the changes in revenue for the upcoming week.

As discussed, constraint (1) represented the fact that all 20 screens in the multiplex theater must be utilized, i.e., C = 20. For constraint (2), we assumed that the distributor made available no more than 3 copies (N = 3) of the same film at any point of time. Based on the past films selections in the theater, we made another assumption with respect to the minimum numbers of film genres G used in the multiplex. We considered the following values: [G.sub.1] = 1 for horror films; [G.sub.2] = 2 for children's films; [G.sub.3] through [G.sub.5] equal to 3 for comedy, action, and drama genre, respectively; and [G.sub.6]= 2 for romance films.

For constraint (4), we applied the actual multiplex operating hours from 9 a.m. till 2 a.m. next morning, or 17 hours per day. With 20 screens at work, the total available hours of operations (parameter T in the constraint) is 340 hrs (17*20). A film's length t included the actual length of the film (in minutes) plus 30 minutes of cleanup, preparation, and advertisements per show. The value of [beta] parameters in constraint (4) was equal to 5 for all films, which means that each selected film will be shown five times a day. The value of [alpha] in constraint (5) was set as 0.20, indicating a requirement that 20% of the films shown in the multiplex should have been released in the previous week. Finally, for constraint (6) we assumed equal representation of different distributors (Buena Vista, Warner Brothers, Fox, Miramax, Paramount, and Columbia) in the multiplex. We considered that right-hand side value D in constraint (6) for all distributors was equal to 5, which means neither of the six distributors would have more than 25% (5 out of 20) of the total capacity in the multiplex theater.

The model was run every week for a period of 12 weeks. The actual model was run on the LINDO software. However, it may be also developed and run with Excel Solver or any common integer programming software. Each week we identified the optimal selection of films that might be shown in the multiplex theater and the maximum revenue that the multiplex would have obtained if the optimal selection of films was used. These data were compared with actual selection of films utilized in the multiplex in respective weeks. The results comparing actual and optimal selections of films are present in table 1.

For each week, we identified the number of films that were common for the generated optimal selection of films and the selection of films actually used in the multiplex theater (column 2 in Table 1). We also identified a number of films from the optimal selection that were not included in the actual film selection in a particular week (column 3 in Table 1). This number of films was used to measure the difference between the optimal and actual selection of films for each week as a ratio of that number to the maximum number of screens (20) in the multiplex. As can be seen from Table 1, the difference of films' selection varies from 15% to 40% with the average difference being more than 30%.

The comparison of estimated weekly revenues generated by showing the actual selection of films in the multiplex and the optimal revenues calculated by the model is given in Table 2.

The results in Table 2 clearly show the benefits of applying the optimal film selection model in the multiplex. On the average, the optimal revenues are almost 18% higher than those of actual film selection, with the revenue increase ranging from 11.4 to 22.0%. The comparison of the plots of revenue increase and differences in film mix as percentage of the total capacity (see Figure 1) show that they have similar patterns. A correlation coefficient between the two data sets is 0.84, which designates a statistically significant relationship between these data. The bigger the difference between the optimal and actual film selections, the greater the difference (increase) in the generated revenues.

CONCLUSION: MODEL BENEFITS AND AREAS FOR FUTURE INVESTIGATION

The optimal film selection model that we described in this paper has been proven to be an effective way to develop best film schedules in a multiplex theater. The model identifies a selection of films available for show in a multiplex theater that would maximize the theater's revenue in conjunction with fulfilling a variety of special constraints. These constraints include screen availability, limitation on the number of film copies, genre mix, time availability per day, and contractual obligations between theaters and distributors.

The model was used to develop the optimal film selections and associated optimal revenues for a real multiplex theater in a period of 12 weeks. The optimal revenues received in that period were consistently better than the projected revenues in the theater with the actual film selection. The average difference in the selections (optimal vs. actual) was slightly above 30%, and the average revenue of the optimal film selections was almost 18% higher than that estimated for the actual film selections.

Having been quite efficient, this model, however, assumes several areas that should be validated by future research. In particular, the method of calculating the revenue coefficients R in the objective function, specifically for new films, requires further investigation in terms of finding reliable data sources to better forecasting new films' revenues. Also, the model does not suggest how long the film should play. To incorporate this constraint, a cost utility function for changing from one film to another may be used. This function will enable to create decision rules for the exclusion of a film from the upcoming week selection. Another way to accommodate the films' length of play would be a simple rule of showing film within certain amount of weeks (for example, at lest 3 weeks and no more than 8 weeks).This can be enforced by adding a set of respective constraints to the original model.

Some films may be scheduled for showing even if the optimal selection result does not recommend them. This may be due to the theater's marketing strategies or subjective considerations of the theater owners. In this case, a constraint needs to be introduced in the model that would insure the inclusion of the specific films in the final selection. Similar constraints may be used to force the inclusion of films that have contractual requirements for showing for a period of time. However, as more of these constraints are introduced, the risk of infeasible solution increases. This can be alleviated be assigning priorities to the model constraints, and convert the integer model into a goal programming application.

TABLE 1. COMPARISON OF ACTUAL AND OPTIMAL SELECTION OF FILMS IN
MULTIPLEX

                Common films in     Difference in    Difference,
                both actual and     films (optimal   % of total
Weeks           optimal selection   vs. actual)      capacity

Week 1          12                  6                30.0
Week 2          14                  3                15.0
Week 3          10                  8                40.0
Week 4          12                  6                30.0
Week 5          10                  7                35.0
Week 6           9                  8                40.0
Week 7          12                  6                30.0
Week 8          11                  6                30.0
Week 9          13                  4                20.0
Week 10         13                  5                25.0
Week 11         10                  8                40.0
Week 12         12                  6                30.0
Average         11.5                6.1              30.4
St. Deviation    1.5                1.6               7. 8

TABLE 2. COMPARISON OF ESTIMATED AND OPTIMAL REVENUES IN MULTIPLEX

                Estimated   Optimal   Revenue Increase, %
Weeks           Revenue     Revenue   (optimal vs. estimated)

Week 1          182,354     214,365   17.6
Week 2          190,491     212,126   11.4
Week 3          192,672     235,002   22.0
Week 4          189,315     219,815   16.1
Week 5          205,689     240,359   16.9
Week 6          201,369     245,911   22.1
Week 7          198,201     235,345   18.7
Week 8          205,346     243,002   18.3
Week 9          188,567     219,310   16.3
Week 10         204,567     240,621   17.6
Week 11         210,567     250,220   18.8
Week 12         182,354     214,365   17.6
Average         195,958     230,870   17.8
St. Deviation     9,602      13,889    2.8

5. REFERENCES

Armstrong, David, "Multiplex Growth Spurt Seeks to Satiate Cinema-Savvy City Dwellers", Variety, 1999, 367.31.

"Apocalypse Now (Problems Affecting US Motion Picture Theaters)", The Economist, 2001, January 15, 88.

Farthing, Nicole, "The Changing Face of Cinema", Leisure and Hospitality Business, 2000, April 19, 22.

Goldsmith, Jill. "The A List: Wheeler-dealers keep biz oiled", Variety, 2002, April 1, 7.

Hinz, Greg, "Multiplex Momentum: Cinema's Final Bow; Curtain is Falling on Small Neighborhood Theaters", Crain's Chicago Business, 2000, 23, 15.

Newman, Morris, "Movie Screen Meltdown", Grid 3, 2001, No 4, 82.

Parry, Caroline, "Future Projections: Movie Theaters" Leisure Week, 2000, November, 23.

Peers, Martin, "Mega Building Punctures Profit: Multiplex Theater Building Boom in the US", Variety, 1998, 371, 8.

Rich, Laura, "And Now, Our Digital Presentation", The Industry Standard, 2001, April, 64.

Swami, Sanjeev, Eliashberg, Jehosua and Weinberg, Charles, "SilverScreener: A Modeling Approach to Movie Screens Management", Marketing Science, 1999, March, 352-372.

Zinovy Radovilsky, California State University, Hayward

Bradkley Vereen, California State University, Hayward

Author Profiles

Dr. Zinovy Radovilsky earned his Ph.D. at the Scientific Research Institute of Labor, Moscow in 1984. Currently he is a professor of management and finance at California State University, Hayward. He has published a number of papers in international and domestic referred journals. Also, Dr. Radovilsky has had an extensive consulting experience in quantitative business modeling and operations management.

Bradley Vereen earned his MBA at California State University, Hayward in 1999. Currently he is an Adjunct Professor at California State University, Hayward and also a Lead Commodity Manager at AMD.