Optimal television schedules in alternative competitive environments

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

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European Journal of Operational Research 104 (1998) 451-473 Theory and Methodology

Optimal television schedules in alternative competitive environments Christina M.L. Kelton

a,*,

Linda G. Schneider Stone b

a Department of Economics, McMicken College of Arts and Sciences, University of Cincinnati, Cincinnati, OH 45221-0371, USA b ACNielsen Analytical Services, 8401 WayzataBlvd., Minneapolis, MN 55426, USA

Received 26 June 1995; accepted3 September1996

Abstract

We formulate network-television-scheduling problems as integer programs for three different competitive environments-myopic, Nash competitive, and cooperative. To provide input data for the scheduling models, we develop and estimate a regression model in which show-part ratings are regressed on variables that influence television viewership, including day, time slot, show attribute, and competitive effects, as well as lead-in from the previous show part. We apply our models by solving for optimal myopic (noncompetitive) and Nash competitive week-long prime-time schedules for the three major networks for two specific weeks. We find that there are substantial gains to optimization and that those gains are not diminished much by competition. We illustrate the cooperative problem for six time slots and show how the solution differs from the Nash solution. We discuss the use of simple programming heuristics such as counterprogramming. (~) 1998 Elsevier Science B.V. Keywords: Integer programming;Televisionscheduling; Competition;Game theory; Ratings forecasts

1. Introduction

According to A d $ Summary, advertisers spent $23 billion on television advertising in the United States in 1993. The top 100 advertisers accounted for $15 billion of this amount. Procter & Gamble alone spent over $1 billion and General Motors spent over $700 million on television advertising. In comparison to the overall $8 billion spent on magazine advertising in 1993, the $12 billion spent on newspaper advertising, and the $1.7 billion spent on radio advertising, television has clearly emerged as the marketing manager's favorite advertising medium. Furthermore, the big three

broadcast TV networks reported advertising revenues for 1993 as follows: $3.5 billion for ABC, $3.2 billion for CBS, and $3.0 billion for N B C - - i n total, close to $10 billion (Broadcasting & Cable, May 16, 1994). The ratings of television shows drive the revenues that networks receive and the relative desirability of the networks from the marketing manager's point of view. With so many dollars at stake on both sides (the cost of advertising for the product advertisers and the advertising revenue for the networks), it becomes important to understand the underlying influences on show ratings that drive these fortunes. Assuming a fixed pool of shows of given quality, the two main influences on show ratings are viewership behaviors and network schedules. For example, when CBS

* Corresponding author. 0377-2217/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PII S0377-2217 (96) 00369-4

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C.M.L. Kelton, L.G. Schneider Stone/European Journal of Operational Research 104 (1998) 451-473

moved 48 Hours from one time slot to another, its ratings improved tremendously. On the other hand, when FOX moved 21 Jump Street and The Tracy Ullman Show from their Sunday-night slots, they floundered. When David Letterman refused to let CBS air his show later in the evening to accommodate the 1994 Winter Olympics, CBS was forced to reduce its late-night Olympic ratings estimates from 5.5 to 3.5 and provide rebates to some advertisers (Advertising Age, November 15, 1993). Our emphasis in this paper is on the scheduling effect on show ratings, although the work also sheds light on aggregate viewership behavior. The literature dealing specifically with the network's scheduling problem is sparse. Horen (1980) formulated an integer-programming optimization model to determine the optimal program schedule for any television network. Each network was assumed to maximize total network reach. As inputs to his model, he used forecasted show-part ratings that he obtained using a simple regression model. (See Gensch and Shaman 1980, Rust and Alpert 1984, and Reddy, Aronson, and Stam 1992 for other regressiontype ratings forecasting models.) Horen's scheduling model did not explicitly incorporate competition between networks, nor was his forecasting model flexible enough to accommodate the scheduling of new shows, movies, and specials. Rust and Eechambadi (1989) used an extended version of the Rust and Alpert (1984) audience-flow model to schedule shows for a network, using a heuristic technique for solving the general week-long prime-time problem. They found a global optimum for a smaller threenight scheduling problem. Henry and Rinne (1984) discussed various general programming strategies without a formal modeling framework. In this paper, we formulate and solve integerprogramming problems to determine the optimal schedule of television programs for each network under several different sets of assumptions about the competitive conditions in the industry. We study three different competitive settings. First we consider a myopic environment in which each network maximizes weekly reach assuming known and fixed schedules of the other networks. Next, we solve for Nash equilibrium schedules, as the solution to a simultaneous scheduling game, from which no network has an incentive to alter its schedule unilaterally. We suggest and illustrate an algorithm to solve for the

Nash schedules. Finally, we formulate a cooperative or "quasi-monopoly" problem in which a single "entity" solves for optimal schedules across all networks (although we do not let shows migrate across networks). The cooperative problem provides a bound for the maximum ratings in the system. In two sets of applications, we compare optimal schedules across competitive environments as well as contrast the optimal schedules to the actual schedules. We discuss the implications of our optimization models for programming heuristics in the industry such as protecting newcomers, starting fast, homogeneity, counterprogramming, and bridging. Our model is practical and could be adopted as a prototype by a television network. The objective function could be modified or constraints added, as required by the network using the model. Moreover, data are available for periodically updating and adjusting the model's parameters. The scheduling model is useful not only for scheduling regular mature shows for another season, but also new shows as well as movies and specials--since we describe all shows as attribute bundles. The model is meant to be used as a planning tool prior to the beginning of the season to set the schedule. Clearly, schedule changes are costly, and would generally not be made mid-season. Our model for forecasting ratings is based on consumer preferences for show attributes. Each show part in our sample is assigned one or several of some basic attributes. The relative strengths of consumer preferences for the attributes are assessed by estimating a regression model with the basic attributes as independent variables; the main effects of the attributes as well as their interactions with days of the week and time slots are included. The regression results are interesting in their own right. We find, for example, that the lead-in audience, either from an earlier part of the same show or from a different show, is very important in affecting show-part ratings (a somewhat surprising result given the popular notion of "channel surfing" and the large number of households with remote controls--70% of homes according to Zufryden, Pedrick, and Sankaralingam 1993). We find as well that competition between comedy shows can adversely affect show ratings on a network (the ratings for the Cosby show, for example, dropped considerably after the FOX network competed head to head with its flagship comedy show, the Simpsons). There

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

are also many interesting time-slot and day effects on ratings. We emphasize throughout the paper that, in practice, the regression model should be updated and reestimated as the need arises. It would be naive to suggest that, in such a dynamic industry influenced by both network strategic moves and viewership behavioral changes, the coefficient estimates we obtain for 1980s data will hold up during the 1990s, let alone through the turn of the century. The rest of our paper is organized as follows. In the next section, we formulate and discuss the network scheduling problems. We discuss the myopic noncompetitive problem first, then the Nash competitive problem, and finally the cooperative problem. Each is formulated as either a binary integer-programming problem, or a series of them, with the objectives and constraints expressed explicitly. In Section 3 we develop, explain, and estimate a regression model for forecasting ratings for all show parts in all time slots. Our Arbitron ratings data are for twelve different weeks taken from the 1981-1982 through the 1988-1989 television seasons. Then, in Section 4, we present two sets of applications of our models--one for the week of February 1, 1989 (a week in our regression sample), and one for the week of February 6, 1991, for out-ofsample model validation. We solve for optimal myopic and Nash-competitive schedules for both weeks, comparing them to the networks' actual schedules in both cases. In that section, we also have an illustration of the cooperative solution--comparing it to the Nash-competitive schedules. A brief discussion concludes the paper.

2. The networks' scheduling problems In this section we formulate the networks' scheduling problems for three different competitive scenarios: myopic, Nash competitive, and cooperative. In each case the problem is cast as a binary integer program, the solution to which is an optimal assignment of show parts to the same number of time slots; typically a time slot is a half hour, and a show part is a half-hourlong segment that might be part of a longer show. The networks seek to maximize, in the context of the particular competitive scenario, their total weekly ratings subject to the following scheduling constraints:

453

• Each time slot is filled by exactly one show part. • Each show part is assigned to exactly one time slot. • Show parts of multi-part shows are shown consecutively, in the correct order (part 2 follows part 1, part 3 follows part 2, etc.), and on the same day. We discuss the ratings metric and measurement in Section 3. Before formulating the problems, we define notation that will be common to all competitive scenarios, and briefly discuss some general relationships: N = the number of networks included in the problem. In our examples we consider only the major broadcast networks, so take N = 3. D = the number of days in a viewing week, assumed to be the same for each network. For example, if weekends are regarded as being outside the scheduling problem, D = 5 (our model, formulation, and notation are completely general, however). T = the number of time slots per day, which are assumed to be of equal length and the same for each network. For instance, if we restrict attention to conventionally defined evening prime time, T = 6 half-hour slots. We assume that each day has the same number of time slots, although this could be relaxed. The number of time slots to be filled during the week for each network is thus TD. S(n) = the number of shows to be scheduled during a week for network n. Note that a show could be longer than a single time slot; for instance, a two-hour movie would occupy four halfhour slots but is still regarded as a single show. In any case, S(n) < TD. Ps(n) = the number of parts in show s on network n. If time slots are half hours and show s on network n is a two-hour movie, then Ps (n) = x-'s(n) Ps(n) = TD, and that T 4. Note that z_.~=x must be at least maxs=l.....S(n) Ps(n) to allow network n to schedule all its shows without breaking any of them up from one day to the next, as we require. nT(n) is a vector representing a weekly schedule of shows for network n in the form of assignment of specific show parts to specific time slots through the week.

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454

2.1. The myopic problem

Xspdt(no) = 0

We first consider, as a reasonably tractable starting point, an environment without competitive reactions to the scheduling decisions of a particular network no; it maximizes its ratings given fixed and known schedules of its competitors, # ( n ) for n E {1 . . . . . N} \ {no} ( \ denotes set differencing). Additional notation for this problem is as follows: Rspdt(n) = the ratings for network n in time slot t on day d if part p o f its show s is shown then. Although not explicit in the notation, Rspdt (no) for a particular network no generally depends on the schedules o f the other networks, ~ ( n ) for n C

f o r d = 1 . . . . . D; s = 1 . . . . . S(no); p=l

. . . . . Ps(n0);

t
t > T - Ps(no) + p + l, (5) Xwdt(no) E {0, 1} for s = 1 . . . . . S(no); p = 1 . . . . . Ps(no); d=l

. . . . . D;

t = 1 . . . . . T.

(6)

{1 . . . . . N} \ {no}. xwdt(n) is the binary choice variable, determined as 1 if network n decides to schedule part p o f its show s in time slot t on day d, and 0 otherwise, for n = 1 . . . . . N; s = 1 . . . . . S ( n ) ; p = 1 . . . . . Ps(n); d = 1 . . . . . D ; a n d t = 1 . . . . . T. The myopic problem may then be formulated as S(no)

max x ,~, ( no )

s=l

D

Ps(no)

T

~

ZZRspdt(no)Xspdt(no),

p=l

d=l

(1)

t=l

subject to S(~)~(~)

Y~ Z s=l

XWd'(no) = 1

p=l

ford=

1 . . . . . D;

t = 1 . . . . . T, (2) D

T

Z Xspd,(no)= 1 d=l t=l

f o r s = 1 . . . . . S(no); p = 1 . . . . . Ps(no), (3)

Xspdt(no)

=

Xs,p+l,d,t+l(nO)

f o r d = 1 . . . . . D; t = 1 .....

T-

1;

s = 1 . . . . . S(no);

p = I. . . . . Ps(no) - 1, (4)

Each element of this formulation has an intuitive interpretation. The objective function in (1) simply adds up applicable ratings for the show parts that are to be scheduled in particular time slots on particular days. The left-hand side o f constraint (2) is the number of show parts assigned to time slot t on day d, and forcing this to one for all t and d ensures that each time slot on each day will be filled by exactly one show part. Similarly, the left-hand side of constraint (3) counts the number o f time slots during the week to which part p o f show s is assigned, and holding this to one for all show parts guarantees that each will be used for exactly one time slot on one day during the week. Constraints (4) and (5) apply only to multipart shows (they are vacuous for shows s for which Ps(no) = 1). Constraint (4) requires that successive parts of a multipart show are either both shown or both not shown in successive time slots, ensuring proper sequencing of show parts. Constraint (5) prohibits us from scheduling a show part so early in the day that the beginning of the show would have to have been put in an earlier day, or so late in the day that the end o f the show would have to go over to a later day. For instance, for a three-part show (Ps(no) = 3), constraint (5) expands to forcing Xs2dl (no) , Xs3dl (no), and Xs3d2 ( no ) to zero (preventing the scheduling of a part too early in a day for the first part of the show to have fit in earlier on the same day), as well as forcing Xs2ar (no), Xsl~(no), and Xs, l,d,r-l(no) to zero (preventing us from scheduling a part too late in a day to have time for the end of the show later on the same day).

C.M.L. Kelton, L.G. Schneider Stone/European Journal of Operational Research 104 (1998) 451-473

Finally, constraint (6) formally states that the choice variables are to be binary, and also facilitates counting up the dimensionality (number of choice variables) for the problem: ~--]~) Ps (n) DT = (DT) 2, which is 900 for a six-slot prime-time problem over five days, and 1,764 if weekends are included. In Section 4 we give specific solutions to the myopic problem using real data, generating optimal weekly schedules that differ from actual network practice. The myopic solution should give some initial indication of the gains to optimization versus the use of programming heuristics like counterprogramming.

2.2. The Nash competitive problem A more realistic competitive scenario is not to treat competing networks' schedules as given, but rather allow all networks' schedules to respond to the scheduling decisions of all other networks. We treat optimization in this competitive environment as a simultaneous game and search for a Nash equilibrium, which we denote as a vector of schedule vectors [ ~* (1) . . . . . ff~*(N) ] that • is feasible, that is, for each network separately, satisfies constraints ( 2 ) - ( 6 ) above; and • for each network no = 1. . . . . N, ff~*(no) is the solution to ( 1 ) - ( 6 ) above with ~ ( n ) = ~ * ( n ) for n E {1 . . . . . N} \ {no}. The first condition says that any Nash-equilibrium schedule vector must satisfy the constraints for the myopic problem for each network. The second condition says that any Nash-equilibrium schedule vector satisfies the condition of each network's doing the best it can (achieves the highest weekly ratings) given that each of the other networks is doing the best that it can. At a Nash equilibrium there is no incentive for any of the networks to alter its weekly schedule unilaterally. Although we cannot prove existence or uniqueness of a Nash equilibrium for this problem, it is clear that if the following algorithm converges, then the schedules generated by the final round of the algorithm are a Nash equilibrium: ( 1 ) To begin round 1 of the algorithm, network 1 solves ( 1 ) - ( 6 ) given initial arbitrary schedules for networks 2 . . . . . N. (2) Network 2 solves ( 1 ) - ( 6 ) given the myopically optimal schedule for network 1 from Step 1 and the initial arbitrary schedules for

455

networks 3 . . . . . N.

(3) Network 3 solves ( 1) - ( 6 ) given the schedules for networks 1 and 2 from Steps 1 and 2, and the initial arbitrary schedules for networks 4 . . . . . N. (N) Network N solves ( 1) - (6) given the schedules for networks 1 . . . . . N - 1 from Steps 1. . . . . N - 1, completing round 1. (N+I) To begin round 2, network 1 solves ( 1 ) - ( 6 ) given the round-1 schedules for networks 2 . . . . . N determined in Steps 2 . . . . . N. ( N + 2 ) Network 2 solves ( 1 ) - ( 6 ) given the round1 schedules for networks 3 . . . . . N from Steps 3 . . . . . N and the round-2 schedule for network 1 from Step N + 1. (2N) Network N solves ( 1 ) - ( 6 ) given the round2 schedules for networks 1 . . . . . N 1 from Steps N + 1. . . . . 2N - 1, completing round 2. The algorithm stops when each network's schedule is the same as it was in the previous round. Note that in each step, a binary integer programming problem of dimension (DT)2 must be solved. In our applications in Section 4 below, the algorithm converged in every case, and fairly rapidly; the optimal Nash schedule vectors also differ in interesting ways from the collection of N myopically optimal schedules.

2.3. The cooperative problem Finally, we model network scheduling behavior as a cooperative maximization problem. That is, we consider a "quasi-monopoly" problem (a monopoly problem with shows constrained not to migrate between networks). We do so not because we think that the networks behave collusively, but because the solution should provide an upper bound on (that is, a "benchmark" for) available weekly ratings (revenues). We should then be able to decompose a gain in total ratings into a gain due to optimal scheduling and a gain due to cooperative effort. In an application we will see that, while a Nash schedule vector has comedies showing head to head on two competing networks, an

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

456

optimal cooperative schedule does not since the cooperating entity suffers "doubly" when the two comedies pull audiences away from each other. Since all N networks are being scheduled simultaneously, we need to enrich the earlier notation: g' denotes a vector [sl . . . . . SN] of shows from networks 1 . . . . . N, respectively. ff denotes a v e c t o r [ p l . . . . . PN] of parts of shows in the respective entries o f ~, which is assumed to have been already set. 1'~ is a row vector o f length N with the nth entry = 1 and with 0s elsewhere, for n = 1 . . . . . N. R~'ffd t = the total ratings for all N networks combined in time slot t on day d if part pn of network n's show Sn, for all n = 1. . . . . N, are shown then. X~fdt is the binary choice variable, determined as 1 if the cooperating entity decides to schedule part Pn of network n's show Sn, for all n = 1 . . . . . N, in time slot t on day d, and 0 otherwise. )--]~,f formally denotes the multiple summation opx'--~S(1) ~-'~Ps I (1)

erator 2_.,s~=l Z-~p¿=l

~-.,S(N) ~-~Psu(N) " " " ~..~SN=I Z-~pN=I " v~S(n) ~-~P,.(n)

~--]g,:,n denotes )--]~,f except skipping z_.,s.=l z-.~t,.=l , f o r n = 1 . . . . . N. The cooperative problem may then be formulated

ford=

1 . . . . . D;

t = 1. . . . . T -

1;

n = 1 . . . . . N;

sn = 1 . . . . . S ( n ) ; Pn =1 . . . . . P , . ( n ) - l ,

(10)

X~:dt = 0 f o r n = 1 . . . . . N; d = 1 . . . . . D;

Sn = 1 . . . . . S ( n ) ; p. = 1 .....

Ps.(n);

t < Pn - 1 or t>_T-Ps.(n)+pn+l,

(11)

x~:dt E {0, 1} f o r n = 1 . . . . . N; Sn = 1 . . . . .

S(n);

Pn = 1. . . . . Ps.(n); d = l . . . . . D; t = 1 . . . . . T.

(12)

as:

D

max Z X~fdt

T

Y~- Y ~ R ~.:a,x~:at,

g',ff d=l

(7)

t=l

subject to

ZXgFdt = 1

ford= 1

s,p

D;

t = 1 . . . . . T, D

(8)

T

~-'] ~-'~X~':d,= l g',ff,n d=l

t=l

for n = 1 . . . . . N; s. = 1 . . . . . S ( n ) ;

p.

=

I .....

Z X~'ffdt = E X~,,ff+f,,,d,t+l ~,,ff,n s~,p.n

Ps.(n),

(9)

The objective function in (7) adds up the total ratings across all networks for the joint scheduling vectors representing which combinations of show parts are to be scheduled in which time slots, and corresponds closely to the myopic objective in (1). The constraint (8) ensures that each time slot on each day, now viewed across all networks, is filled by exactly one N-vector of show parts. The left-hand side of constraint (9) counts, for part Pn of show Sn on network n, the number of time slots for that network to which this part will be assigned, which must be forced to 1; the outer summation operator ~--~,f,~ is required to force this result regardless o f what is being scheduled for the other networks. Constraints (10) and ( 11 ) apply only to multipart shows, forcing them to be shown in sequence and within a single day; they correspond to constraints (4) and (5), respectively, for the myopic problem. Finally, constraint (12) requires the choice variables to be binary, and indicates that the dimen-

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

sionality of this problem is (DT) N+I. The largest cooperative-scheduling problem that we were able to solve exactly in the hardware/software environment available to us was for two networks over two days of six prime-time slots per day, which is of dimension [ (2) (6) ]2+1 = l, 728. Even for small problems, however, we obtain some interesting results--the cooperative solution differs from the competitive Nash solution.

2.4. Model extensions We view our models as prototypes or "starting points" for scheduling practitioners. We are providing models that capture the essential aspects of the scheduling problem. To add sophistication to the models, either more constraints could be added or the objective function could be modified. In this section, we present three examples. First, suppose that a network n were obliged not to air programs with high adult content in the 7:00 p.m. and 7:30 p.m. time slots on any day. To accommodate this requirement, we would simply add constraints disallowing a show part with adult content to be scheduled so early in the evening. In the myopic problem, this would involve "zeroing out" the appropriate Xspdt(n)'s. Second, suppose a network wanted to devote the first two hours of Thursday prime time to comedy--which would make sense if viewers (over the years) have come to expect to see comedy on Thursday night. Again, as far as the model is concerned, one could simply disallow shows that do not have a comedy component to be aired during those Thursday slots. Third, our models can be extended to accommodate differences among viewing segments. Rust and Alpert (1984) suggest age, education, and gender as important variables in segmentation; Rust, Kamakura, and Alpert (1992) suggest geographic segmentation; and Lehmann (1971) suggests psychographic segmentation. We show here one way to modify (1) if there are G segments:

max ~

,AispdtRispdt(n )

~

x'~t'dt(n) S=I p=l d=l t=l "Xspdt(n).

i=1

(13)

457

Here, I~ispdt is the weight assigned to segment i for part p of show s in time slot t on day d, and Rispdt is the ratings for segment i for that show-part-day-slot combination. The ,~ispdt'S could be estimated empirically or set judgmentally. Setting them in a particular way could induce schedules for particular time slots targeted to specific segments. If ,~ispdt = the proportion of households in segment i (independent of s, p, d, and t), then (13) is equivalent to ( 1) above. Another realistic problem that the prototype models as formulated do not address would be movie serials that are aired over the course of several days. This consideration could be addressed by adding constraints to the models.

3. Forecasted ratings To make the myopic, Nash, and cooperative models for television-program scheduling operational, we obtain predicted ratings i~spdt(n) in the case of myopic and Nash schedules, a n d [~ffdt for cooperative schedules. Although there are many other ways to predict ratings (such as managerial-judgment approaches or pure time-series models), we chose to develop a regression model that accounts for many of the major influences on television viewership. In this section, we discuss our data, our regression model, and our predicted ratings. In the next section we incorporate these predictions into the television-scheduling models formulated above to generate optimal schedules in various competitive environments. Just as our scheduling models are to be viewed as prototypes for practitioners, so is our forecasting model. As required by a network, additional sophistication can be modeled.

3.1. The dam For our dependent variable--ratings of a show part in a particular time slot for a network in a given week--we used Arbitron ratings (percents or shares of households) data for Minneapolis-St. Paul for twelve separate weeks over the 1981-1982 through 19881989 television seasons. (In a pilot study we noted differences in viewing patterns for various areas in the country--in particular, Indianapolis versus San Francisco. Hence, we decided to base our study on one restricted geographic area. Of course, our study could

458

C.M.L. Kelton, L.G. Schneider Stone/European Journal of Operational Research 104 (1998) 451-473

easily be generalized to other areas of the country or to a national study.) We did not want to consider earlier years due to concern that either viewership behavior or network strategies differed considerably in the 1970s from those in the 1980s. We reserved some data from the 1990s to generate some out-of-sample schedules for purposes of model validation. The particular weeks in our sample are the weeks of 2/3/82, 10/20/82, 2/2/83, 10/19/83, 10/17/84, 1/30/85, 10/2/85, 1/29/86, 10/1/86, 11/4/87, 2/3/88, and 2/1/89. (These dates are Wednesdays since Arbitron weeks begin on Wednesday and continue through the following Tuesday.) We selected these particular weeks due to data availability and degree of "ordinariness" of the weekly schedule (for example, we attempted to avoid the World Series weeks in October). We considered only prime-time programming (7:0010:00 p.m. in Minneapolis-St. Paul) for week nights, Monday through Friday, on the affiliates of the three major networks--KARE (NBC), WCCO (CBS), and KSTP (ABC). We recognize the existence of nonnetwork viewing alternatives. However, the low variability in the ratings data for the other networks for the forecasting model, plus the need for tractability for the scheduling models, argued for focusing on the three major networks for purposes of this paper. We thus collected twelve weeks of data across 30 time slots for three networks, resulting in a total of 1080 individual observations on show-part ratings. To classify the shows (including movies and specials) by type, we relied on the TV Guide classification. This generated 24 different show types: adventure, children's, comedy, comedy adventure, comedy drama, crime drama, documentary, drama, family drama, fantasy, game, magazine, musical, mystery, news, newsmagazine, religion, science, science fiction, serial drama, sports, thriller, variety, and western. Finally, we used McNeil (1991) to determine the first season of each show and the number of seasons a show had been on the air.

3.2. Theoretical background and empirical methodology In developing a model general enough to schedule new shows as well as movies and specials, we regard a show as a bundle of fundamental attributes or characteristics (this approach is based on Lehmann

1971 as well as on the general attribute literature, e.g. Lancaster 1971 ). We chose these attributes somewhat arbitrarily but with considerable appreciation for the particular shows in our data set--that were described in TV Guide and McNeil (1991). We chose the maximum number of attributes with which we could work tractably. (More attributes would likely have strained the degrees of freedom in the regression model.) Specifically, we chose the following nine show attributes: news, comedy, drama, action, family, fantasy, suspense, serial, and sports. We then determined systematically which attributes the show possessed. For example, we assigned adventure shows like MacGyver the action and drama attributes. We assigned documentaries and newsmagazines like 20/20 the news attribute only. And we assigned crime dramas like Miami Vice the suspense, drama, and action attributes. Lighthearted crime dramas like Moonlighting were assigned the comedy attribute as well. A family drama like Highway to Heaven would be characterized by the drama and family attributes. Tables 1 and 2 specify the shows in our two applications, and their assigned attributes, one of which we designate the dominant attribute to form our competition variables only. (The other classifications for our sample of shows are available from the authors on request. Note that the movies and specials are also assigned show attributes--in the same way as regular shows.) We include the attributes as binary variables in our regression model. We then consider day and time-slot effects by two sets of binary variables--four day-of-the-week dummy variables (Monday omitted) and five timeslot dummies (the 7:00 time slot omitted). Also included are 32 (8 x 4) attribute-day interaction variables and 40 (8 x 5) attribute-slot interaction terms. The interaction variables capture the relative desirabilities of scheduling particular show characteristics on specific days and in particular time slots. These 72 variables are critical to the network's scheduling problem. Without them, there would be little variation across feasible schedules (only some variation introduced through the competitive and lead-in variables discussed below). To account for the effects of competitive programming on viewership choice in the model, we did the following. For each show part we counted the number of other networks that were showing a show part with

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473 Table 1 Shows in 1989 application and their assigned attributes

Show

Dominant attribute Other attributes

20/20 News 48 Hours News Alf Comedy Beauty and the Beast Fantasy Billy Graham Special News Cheers Comedy China Beach Drama Cosby Comedy Dallas Serial Different World Comedy Dynasty Serial Empty Nest Comedy Equalizer Suspense Falcon Crest Serial Fine Romance Drama Full House Comedy Golden Girls Comedy Growing Pains Comedy Head of the Class Comedy Heartbeat Drama Hooperman Comedy In the Heat of the Night Suspense Just the Ten of Us Comedy Kate & Allie Comedy Knots Landing Serial Letterman Anniversary Special Comedy MacGyver Action Main Event Sports Matlock Suspense Miami Vice Suspense Drama Midnight Caller Moonlighting Suspense Movie: Columbo Suspense Movie: Lonesome Dove Action Mr. Belvedere Comedy My Two Dads Comedy Newhart Comedy Night Court Comedy Nightingales Drama Paradise Action Perfect Strangers Comedy Roseanne Comedy thirtysomething Drama Tour of Duty Drama UNSUB Suspense Unsolved Mysteries News Who's the Boss Comedy Wiseguy Suspense Wonder Years Drama

Drama

Drama Drama Drama, Action Drama Comedy

Drama, Action

Drama Drama Comedy, Drama Drama, Action Comedy, Drama, Action Drama, Action Drama

Drama

Drama, Action Drama Drama, Action Comedy

459

Table 2 Shows in 1991 application and their assigned attributes

Show 20/20 48 Hours Anything but Love Barbara Waiters Special Blossom Cheers Coach Cosby Dallas Dark Shadows Davis Rules Designing Women Different World Doogie Howser, M.D. Evening Shade Face to Face Special Family Matters Fantasies Father Dowling Mysteries Flash Fresh Prince of Bel-Air Full House Going Places Good Sports Growing Pains In the Heat of the Night Jake and the Fatman Knots Landing L.A. Law Law & Order MacGyver Major Dad Matlock Midnight Caller Movie: Deadly Intentions Movie: Not of this World Movie: Perry Mason Murphy Brown Night Court Perfect Strangers Prime Time Live Rescue 911 Roseanne Seinfeld Sons and Daughters Super Bloopers thirtysomething Top Cops Unsolved Mysteries War Special Who's the Boss Wings Wonder Years

Dominant attribute Other attributes News News Comedy News Comedy Comedy Comedy Comedy Serial Drama Fantasy Drama, Suspense Comedy Comedy Comedy Drama Comedy Comedy News Comedy Fantasy Drama Suspense Comedy, Drama Action Drama Comedy Comedy Comedy Comedy Comedy Suspense Drama, Action Suspense Drama, Action Serial Drama Drama Suspense Drama, Action Action Drama Comedy Suspense Comedy, Drama Drama Drama Suspense Fantasy Drama, Action, Suspense Suspense Drama Comedy Comedy Comedy News News Drama Comedy Comedy Drama Comedy Drama News Drama News Drama News Comedy Comedy Drama Comedy

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C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

the same dominant attribute during the same day and time slot--0, 1, or 2 other networks. Then we created eighteen binary variables (two for each a t t r i b u t e ) one corresponding to one other network's airing a show with the same dominant attribute and the other corresponding to two other networks' doing so. We expect these 18 competitive variables to have a negative impact on a show part's ratings, with stronger effects for competition by two other networks rather than one. The greater the competition in the same dominant characteristic as the show being evaluated, the lower the likelihood of viewing the show in question. Put differently, we expect that, for any day and time slot, if a competing network goes head to head in its scheduling with the same type o f show, it should draw away some o f the audience from the given network. (In practice this may be violated for some types of shows. In 1988 NBC ran boxing against A B C ' s Winter Olympics. Based on the success of this strategy, ABC repeated the tactic in an effort to ambush CBS's Olympic coverage. See Advertising Age, December 23, 1991.) We also considered a lead-in effect by including, as an independent variable in the regression, ratings of the show part in the previous time slot on the same network. Although "channel surfing" with remote controls allows individuals to choose their preferred program every half hour (or even more often), anecdotal evidence suggests that there may be a countervailing inertial effect that reduces channel-changing. We try to capture this effect through several lead-in variables. We postulate that these variables will positively affect a show's ratings. Lead-in is also a key element in the Rust and Alpert (1984) audience-flow model. Finally, we included two network binary variables (one for CBS and one for A B C ) , a binary variable for a Winter (January or February) observation, a timetrend variable (1 for the 1981 season, 2 for the 1982 season, etc.), a dummy for the show part's being a movie, a specials dummy, and show-season variables (both linear and quadratic terms) indicating the number of seasons the show had been on the air at the time of the observation. These variables are included to account for the following phenomena. First, we want to account for the tendency to watch more television in the winter months than in the other seasons of the year, particularly in Minneapolis-St. Paul. Moreover, ratings for the major networks have trended downward

over the last fifteen years with competition from cable television, other networks and independents, and videos. Finally, a program's season is meant to capture a show's maturity and, in some sense, quality. We postulate that, over the course of a show's "life," ratings first rise with the number of seasons that it is on the air (shows rarely start out as hits, having first to develop an audience), peak after some number of seasons, and then decline. The importance of modeling show "wearout" is discussed in Gensch and Shaman (1980). 3.3. The forecasting model We now formally express our regression model. Let Dx = 1 if x = d and 0 otherwise for x = 2 . . . . . 5 (that is, Tuesday through Friday); Ty = 1 if y = t and 0 otherwise for y = 2 . . . . . 6 (that is, the 7:30 time slot through the 9:30 time slot); Nz = 1 if z = n and 0 otherwise for z = 2,3 (that is, CBS and ABC, respectively); Aq = 1 if show s has characteristic q and 0 otherwise, for q = 2 . . . . . 9 (that is, comedy, drama, action, family, fantasy, suspense, serial, and sports); W = 1 if the observation is for a week in Winter (January or February) and 0 otherwise; es,p,d,t_ 1(n) is the rating of part pt of show s t on network n in slot t - 1 on day d; X, the same-show lead-in-enhancement variable, is 1 if s = s t and 0 otherwise; ADqx = Aq × Dx for q = 2 . . . . . 9 and x = 2 . . . . . 5 (the attribute-day interaction variables); ATqy = Aq × T v for q = 2 . . . . . 9 and y = 2 . . . . . 6 (the attribute-slot interaction variables); Ctq = 1 i f p a r t p on show shas dominant characteristic q and l other networks have scheduled shows on day d in time slot t with dominant characteristic q and 0, otherwise, for l = 1,2 and q = 1 . . . . . 9 (these are the competitive variables); M = 1 if show s is a movie and 0 otherwise; Q = 1 if show s is a special and 0 otherwise;

CM.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473 L = 1+ the program season at the time of the observation minus the program's first season (we code the 1981-1982 season as 1981, the 1982-1983 season as 1982, and so forth); H is a time-trend variable equal to the television season at the time of the observation minus 1980 (with 1981 representing the 1981-1982 television season, etc.) ; is a random error term; and F-sprit(n) i = 1 . . . . . 117 are coefficients to be escr and fli timated. Then

R spdt ( n ) = a + f l l D 2 + f12D3 -}- f13D4 q- f14D5

+fisT2 + fl6T3 + f17T4 d- f18T5 -q- fl9T6 +fll0Nz q- flllN3 -~-fl12A2 + "'" "~ ~19A9

+fl2oW -~-/~21C1,1 -~- . . . -~-/~29C1,9 -~-/~30C2,1 -+- . . . -~-/~38C2,9

-~-f139Rs,p,d,t-I ( n) -~- 1~40(XRs,p,d,t-1 ( n) ) +fl41AD2,2 + " " + f172AD9,5 + f173AT2,2 q- " " d- fll I2 AT9,6 + f l l l 3 M + flll4Q + fill5 H + flll6L

+ill 17L 2

-q-ewdt(n ) .

(14)

Note that the "base" for the binary variables is a regular (not movie or special) news show part on the NBC network at 7:00 on Monday night during a week in the Fall when there is no competition (no other network showing news). 3.4. Estimated coefficients

We estimated our model (14) using ordinary least squares. We omitted several of the competitive dummies and interaction dummies for which we had no observations during our time period (C1,9, 6"2,4, C2,5, C2,6, C2,7, C2,8, C2,9, AD6,2, AD6,3, AD9,4, AD4,5,

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AT8,2, AT4,5, and AT4,6). We also omitted AD8,5 and AT8,6 as they turned out to be, for our sample, perfect linear combinations of some of the other binary variables, so introduced perfect multicollinearity into the model. Hence, we were left with 1 1 8 - 1 6 = 102, rather than 118, parameters to estimate. With a larger data set, these omissions could presumably have been avoided. For the most part, unless unrealistic, we assigned a zero value to their effects in the forecasting model. We present many of our estimated coefficients and accompanying t statistics in Table 3. The estimated coefficients for the attribute-day and attribute-slot interaction effects are found in Tables 4 and 5. We note that, with an R 2 of 0.698, we are able to explain approximately 70% of the variation in ratings with our regression model. Note that our data are essentially cross-sectional in nature, rather than a time series. Our R 2 value is very close to those reported in Horen (1980). Our F statistic of 22.4 allows us to reject (at the 1% level) the null hypothesis of no significant independent-variable effects. We ran a collinearity diagnostic procedure consistent with the recommendations of Belsley (1991). Results indicate very little potential degradation of the estimated coefficients. According to the procedure, only eight coefficients might be unreliable to some degree: those on D2, D3, D4, D5, A2, A3, L, and L 2. None of the coefficients on the interaction variables (the "schedule drivers") is suspect according to the procedure. In addition, a 101 x 101 simple-correlation matrix indicates very few (61 out of 5050 pairs) correlation estimates either above 0.5 or below - 0 . 5 . All evidence suggests that multicollinearity is not a major problem in our data set. See Cohen and Cohen (1975) for a statistical-scaling justification to include both main effects and interaction effects in a regression model. Although we intend our regression model primarily as a forecasting tool, the coefficient estimates in Table 3 give some interesting insights into aggregate television viewership behavior. Both of the estimated coefficients on our "inert-couch-potato" variables (gs'p'd,t-I (n) and XRs,p,da-1 ( n ) ) are positive and statistically significant. The higher the ratings of the show part shown in the previous time slot on the same network, the higher the ratings of the show in the current time slot. Moreover, there is an enhanced lead-in effect if the earlier show part is the previous show part for the same show, as seems perfectly reasonable. All

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C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

Table 3 Least-squares results for selected variables (1080 observations); dependent variable R spdt( n ) Independent variable Description Notation Intercept Tuesday D2 Wednesday D3 Thursday D4 Friday D5 7:30 T2 8:00 T3 8:30 T4 9:00 T5 9:30 T6 CBS N2 ABC N3 Comedy A2 Drama A3 Action A4 Family A5 Fantasy A6 Suspense A7 Serial A8 Sports A9 One News Competitor Cl,l One Comedy Competitor Ct,2 One Drama Competitor Ci.3 One Action Competitor C1.4 One Family Competitor C1,5 One Fantasy Competitor Ci,6 One Suspense Competitor Ci,7 One Serial Competitor C1,8 Two News Competitors C2A Two Comedy Competitors C2.2 Two Drama Competitors C2,3 Movie M Special Q Lead-In Rs,,p,d.t_l(n) Same-Show Lead-In XRs, p,,d.t_ l (n) Winter W Trend H Linear Show Season L Quadratic Show Season L2

Estimated coefficient Value t Statistic 6.38a -2.17 -1.05 -0.07 -1.53 -0.92 -0.68 -1.61 2.07 --2.63 c -0.16 -0.48 3.37 b --2.84c 2.49b 2.40 11.24a -5.06 a 3.16b 6.10b 1.24 -1.13 b -0.11 -1.58 c -1.11 0.61 0.17 - 1.55 1.38 -3.19 0.33 1.41a - 1.96b 0.54a 0.17a 0.03 -0.27 a 1.42a -0.12 a

3.85 -1.42 -0.66 -0.05 -0.96 -0.74 --0.43 --1.03 1.54 -i.91 -0.43 -1.20 2.28 - 1.86 1.99 1.16 5.44 -3.27 2.29 2.26 0.99 -2.33 -0.20 -1.78 -0.54 0.43 0.26 - 1.27 0.78 -1.34 0.35 2.92 -2.10 21.33 6.27 0.11 -3.99 6.33 -5.05

aSignificant at the 1% level (two-tailed test). bSignificant at the 5% level (two-tailed test). cSignificant at the 10% level (two-tailed test).

o f t h e d a y d u m m i e s h a v e e s t i m a t e d n e g a t i v e effects on ratings (all days have lower ratings than Monday), t h o u g h n o n e h a s a statistically s i g n i f i c a n t coefficient e s t i m a t e . O f all o f t h e slot d u m m i e s , o n l y t h e o n e for t h e 9 : 3 0 slot h a s a statistically s i g n i f i c a n t effect

Table 4 Estimated coefficients on attribute-day interaction variables (1080 observations); dependent variable Rspdt(n) Independent variable Description Notation Comedy x Drama x Action x Family x Suspense x Serial x Sports x Comedy x Drama x Action x Family x Suspense × Serial x Sports x Comedy × Drama x Action x Family × Fantasy x Suspense x Serial x Comedy x Drama x Action x Fantasy x Suspense x Sports x

Tuesday Tuesday Tuesday Tuesday Tuesday Tuesday Tuesday Wednesday Wednesday Wednesday Wednesday Wednesday Wednesday Wednesday Thursday Thursday Thursday Thursday Thursday Thursday Thursday Friday Friday Friday Friday Friday Friday

AD2,2 AD3,2 AD4,2 ADs,2 AD7,2 ADs,2

ADg,2 AD2,3 AD3,3 AD4,3 ADs,3 AD7.3

ADs,3 AD9,3 AD2,4

AD3,4 AD4,4 ADs,4

AD6,4 ADT,4 ADs,4

AD2.5 AD3,5 AD4,5 ADr.5 ADT,5

AD9.5

Estimated coefficient Value t Statistic 1.76 2.28 -0.96 0.39 0.09 --7.54a 1.34 -1.25 0.65 0.35 1.27 0.20 -3.13 b 7.56a 2.15 -2.32 -1.45 4.56 -10.38 a 3.63 b - 1.84 -1.39 -1.11 0.26 -7.12 a 2.06 0.03

1.20 1.44 -0.81 0.12 0.06 -3.22 0.57 -0.83 0.41 0.25 0.58 0.12 -2.07 2.81 1.57 -1.51 -0.96 0.88 -3.84 2.22 - 1.26 -0.94 -0.63 0.18 --3.48 1.40 0.01

aSignificant at the 1% level (two-tailed test). bSignificant at the 5% level (two-tailed test). cSignificant at the 10% level (two-tailed test).

o n ratings, a n d its effect is negative. T h e o t h e r t i m e slots, w i t h t h e e x c e p t i o n o f 9:00, also h a v e e s t i m a t e d

negativeeffects--though nonsignificant. Interestingly, our Winter dummy has a positive, but n o t a statistically significant, effect o n r a t i n g s . O u r resuits i n d i c a t e that, at least f o r o u r t w e l v e - w e e k s a m p l e , p e o p l e d o n o t t e n d to w a t c h m o r e t e l e v i s i o n in J a n u a r y o r F e b r u a r y t h a n they d o in O c t o b e r o r N o v e m ber. O n t h e o t h e r h a n d , t h e r e is a s t a t i s t i c a l l y s i g n i f i c a n t effect o f t r e n d o n ratings. T h e e s t i m a t e d t r e n d coefficient is - 0 . 2 6 5 w i t h a t statistic o f - 3 . 9 9 2 . T h i s m i g h t s u g g e s t t h a t v i e w e r s are s w i t c h i n g to cab l e or o t h e r e n t e r t a i n m e n t o p t i o n s . It m a y b e p o s s i ble, t h r o u g h b e t t e r s c h e d u l i n g c h o i c e s , to w i n b a c k a portion of these switchers. The movie binary variable

CM.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473 Table 5 Estimated coefficients on attribute-slot interaction variables ( 1080 observations); dependent variable Rspar(n) Independent variable Description Notation Comedy Drama Action Family Fantasy Suspense Sports Comedy Drama Action Family Fantasy Suspense Serial Sports Comedy Drama Action Family Fantasy Suspense Serial Sports Comedy Drama Action Fantasy Suspense Serial Sports Comedy Drama Action Fantasy Suspense Sports

x x x x x x x x x × x x x x x x × x x × × × x × x x × x × × × x x x x ×

7:30 7:30 7:30 7:30 7:30 7:30 7:30 8:00 8:00 8:00 8:00 8:00 8:00 8:00 8:00 8:30 8:30 8:30 8:30 8:30 8:30 8:30 8:30 9:00 9:00 9:00 9:00 9:00 9:00 9:00 9:30 9:30 9:30 9:30 9:30 9:30

AT2,2

AT3,2 AT4.2 ATs,2 AT6.2 ATT,2 AT9.2 AT2.3 AT3,3 AT4,3 ATs,3 AT6.3 ATT,3 ATs.3 AT9.3 AT2,4 AT3.4 AT4,4 ATs,4 AT6,4 ATT,4 ATs,4

AT9,4 AT2.5 AT3,5 AT4,5 AT6.5 ATT,5 AT8,5 AT9.5 AT2.6 AT3,6 AT4,6 AT6,6 AT7,6 AT9.6

Estimated coefficient Value t Statistic -2.59 ~ 0.29 -1.19 -3.14 c -2.25 2.42 -5.25 c -2.99 b 3.56b -1.58 --8.82 b -5.80 b 1.92 5.47a -3.02 -2.66 c 2.09 -1.59 -6.00 --3.26 2.06 0.94 -4.53 c --3.64 a -0.53 -2.74 c -7.01 6.60a 1.31 -8.42 a -2.52 c 2.91 b --1.23 -4.94 2.87 --4.30

-2.16 0.21 -0.95 -1.70 -1.20 1.63 -1.95 -2.07 2.14 -1.13 -1.98 -2.43 1.16 3.87 -1.10 -1.84 1.25 -1.12 -1.35 -1.37 1.23 0.67 -1.65 -2.63 --0.36 -1.69 -1.62 3.64 1.01 -3.01 -1.82 1.98 --0.76 -1.14 1.59 -1.55

aSigniflcant at the 1% level (two-tailed test). bSignificant at the 5% level (two-tailed test). cSignificant at the 10% level (two-tailed test). has a positive and statistically significant coefficient, while our results for specials are the opposite; a negative, statistically significant coefficient was obtained for the specials dummy. The show-season variables (approximate indicators o f "show quality") work in the direction we hypothesized and are seen to have statistically significant estimated coefficients. The linear term has a positive and statistically significant effect

463

at the 1% level; the quadratic term has a negative and statistically significant effect at the 1% level. W h i l e both of the network dummies (one for CBS and one for A B C ) have negative coefficient estimates, they are not significant. The main attribute effects are estimated as follows: positive statistically significant effects for comedy, action, fantasy, serial, and sports, and negative statistically significant effects for drama (at the 10% level) and suspense (at the 1% level). Put differently, comedy, action, fantasy, serial, and sports do better than news; drama and suspense do worse. The effect of family is positive but not statistically significant. Some of the estimated competitive effects are as expected. Both C1,2 and C2,2 have estimated negative effects on ratings. That is, when either one or two other networks show comedy at the same time as the network in question, the ratings o f that network's comedy are adversely affected. However, whereas C1,2 has a statistically significant (at the 1% level) estimated coefficient, C2,2 has a nonsignificant coefficient estimate. The only other statistically significant competitive effect is that for action shows. When one other network is showing an action show, the action show's ratings on the network in question are hurt. From the attribute-day and attribute-slot interaction effects estimates in Tables 4 and 5, we find statistically significant (at at least the 5% level) negative effects o f serial × Tuesday and serial x Wednesday. Moreover, we find a statistically significant positive effect o f sports x Wednesday. For Thursday, the fantasy × Thursday interaction effect is a statistically significant negative one, and likewise for Friday. We find a different result for s u s p e n s e - - t h e suspense × Thursday estimated effect is positive and statistically significant (at the 5% level). The attribute-slot results are suggestive as well. For each of the time slots, the comedyslot interaction effects are estimated to be negative and statistically significant (at at least the 10% level), suggesting that all slots are worse, to varying extents, for comedy than the earliest s l o t - - t h e 7:00 ("base") slot. Family and fantasy do worse at 8:00 than at 7:00; both have statistically significant negative interactions with the 8:00 slot. On the other hand, drama and serial do significantly better (at the 5% level) at 8:00 than at 7:00. The suspense x 9:00 interaction variable has a positive and statistically significant (at the 1% level) effect on ratings.

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

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3•5. Using the forecasted ratings in the scheduling problems We now discuss in detail how the regression model's estimated coefficients are incorporated into the network program-scheduling problems. We can schedule new shows, movies, and specials, along with older, mature shows, for any week for which we feel that the regression results are reasonable• In practice the regression model should be updated and estimated periodically with current ratings data.

Myopic ratings In solving the myopic problem for each network, given in ( 1 ) - ( 6 ) above, we obtain estimates for the R~pdt(n)'s directly from the estimated regression coefficients: [~spdt( n ) = ( ~/ q- 7"l'ls 'J¢- 7"I'2s q- q'l'3s ~- q'F4s q- 71"5d -~-71"6t q- "71"7spdt -1- 77"8spdt + 7r9sd -~- 7/'10st) • ( 1 -- 97"l l s p d t ) - l ,

(15)

where y = the coefficient estimate on the regression intercept, plus the coefficient estimate on the relevant network binary variable, plus the coefficient estimate on the winter binary variable if relevant, plus the coefficient estimate on the trend variable multiplied by the appropriate number of years since the 1981-1982 season, the sum of the coefficient estimates on the ,n-ls relevant attribute variables that characterize show ,,/7-2s =

7F3s

7r4s

7/'5d =

s,

the coefficient estimate on the movie dummy (1•406) if show s is a movie, and 0 otherwise, the coefficient estimate on the specials dummy ( - 1 • 9 5 6 ) if show s is a special, and 0 otherwise, the coefficient estimate on L multiplied by the number of seasons a show has been on the air (1 for new shows) plus the coefficient estimate on L 2 multiplied by the square of the number of seasons a show has been on the air, the coefficient estimate for day d (0 if d corresponds to Monday),

7/'6t ----- the coefficient estimate for slot t (0 if t corresponds to the 7:00 time slot), ¢rTspat = the relevant coefficient estimate on the competitive variable corresponding to one other network's scheduling a show with the same dominant attribute as show s in slot t on day d, or 0 if no other networks or two other networks have shows scheduled with the same dominant characteristic, 71"8spat = the relevant coefficient estimate on the competitive variable corresponding to two other networks' scheduling a show with the same dominant attribute as show s in slot t on day d, or 0 if no other networks or one other network has a show scheduled with the same dominant characteristic, 7r9sd = the sum of the coefficient estimates on all of the interaction variables involving show s's attributes and day d, 7rlost = the sum of the coefficient estimates on all of the interaction variables involving show s's attributes and slot t, 7rllspdt = the lead-in coefficient, equal to 0.54 (from Table 3) for a single-part show or the last part of a multi-part show, equal to 0.71 (0.54 + 0.17) for all but the final part of a multi-part show, and equal to 0 for any show part in the final (9:30) time slot for any day. Although our 7rll~pdt lead-in component of ratings may at first blush seem mysterious, it is not. In the network's scheduling problem we cannot capture lead-in directly since a priori we do not know which show precedes or follows any other show on a network before the integer-programming scheduling problem is worked. It turns out, however, as Horen (1980) demonstrated, that consistently treating leadin as lead-out will not affect the optimal schedule resulting from solving ( 1 ) - ( 6 ) • So we augment each show's rating by dividing the other components of the ratings forecast by 1 - 7rl Ispdt, that is, by 0.465 or by 0.292 depending on the situation. In the case of the sixth slot, we divide by 1; there is no ratings enhancement in this case. The zero lead-in effect that we hypothesize for 7:00, the first prime-time slot, is simply "reassigned" to a zero lead-out effect for the sixth slot of the evening. For a more formal argument, see Horen (1980).

C.M.L. Kelton, L.G. SchneiderStone~European Journal of Operational Research 104 (1998) 451-473

Note that, generally, there will be several optimal myopic schedules (as well as Nash-competitive and optimal cooperative schedules). Our model should be indifferent between regularly-scheduled show parts with the same assigned attributes and belonging to a show with the same maturity, that is, on the air for the same number of seasons. Hence, if an optimal schedule, such as those we generate below, has a secondseason comedy scheduled at 7:00 on Thursday and a second-season comedy scheduled at 7:30 as well, these two shows can be interchanged without affecting the value of the objective function. Moreover, since our model in (15) does not enhance the ratings in the last slot of the evening, all of our optimal schedules should, ceteris paribus, have the worst-performing shows in the last slots--an intuitive result, since they should have the weakest lead-in effect for other shows. (This assumes that late-night viewing is of secondary importance to the network. Certainly, the model could be extended to later time slots if the link between primetime and late-night viewing is considered important to the network.) Since we include a measure of show maturity (including linear and quadratic effects), a show is more likely to be scheduled in the last slot, ceteris paribus, when it is either immature or very mature. Put differently, our model supports, at least to some degree, the heuristics to protect newcomers (place new shows after strong old shows) and to start fast (schedule the best programs early in the evening). Nash ratings To obtain ratings estimates for the Nash competitive problem, we used (15) for each step in the Nash algorithm. In each step, however, we updated the schedules of the other two networks. For example, Step 2 would start with the optimal NBC schedule from Step 1. Step 3 would start with optimal schedules of both NBC and CBS. In this way, the competitive components ¢rTspdt and rrsspdt change for each step as different numbers of networks have shows scheduled on a given day and in a given time slot with the same dominant attribute as show s. Cooperative ratings To obtain estimated ratings for the cooperative problem in ( 7 ) - ( 1 2 ) - - t h a t is, the /~rffd/'s--we first estimated ratings for each network using (15) and then added those estimated ratings together. In the

465

two-network case (the largest problem of dimension ( D T ) N+I that we were able to solve below), each network was assigned the relevant competitive coefficient from the regression model if both networks had shows scheduled of the same dominant type during a particular time slot, and 0 otherwise. With three networks the competitive effect is somewhat more complicated, but could be handled as follows. If the three networks schedule shows having different dominant characteristics, then each show is assigned a zero competitive effect; if two networks have shows with the same dominant characteristic, then those two shows would be assigned the estimated coefficient modifying the relevant Clq in the regression model; and, if all three networks have shows with the same dominant characteristic, then all three shows would be assigned the estimated coefficient modifying the relevant C2q from the regression model. Computation All of our integer-programming computations were done using the LINGO optimization modeling language (Lindo Systems, Inc. 1991). Runs were made on a Sun SPARCStation IPX with 32 megabytes of RAM and running at 28.5 MIPS. Our version of the software allowed for a maximum of 4,000 integer variables and 2,000 constraints--sufficient to solve realistic week-long scheduling problems for either the myopic or Nash competitive environment. Even the largest version of the software, however, with a 32,000-variable maximum, could not accommodate a thirty-slot, three-network cooperative problem of dimension 810,000.

4. Applications In this section we present and discuss five sets of empirical results: optimal myopic and Nash schedules for the week of February 1, 1989 (the twelfth week in our sample); optimal myopic and Nash schedules for the week of February 6, 1991 (an out-of-sample week); and an optimal cooperative schedule for a smaller two-network, one-day, six-slot problem, along with an appropriate comparison to the Nash schedules for the same problem. In each case the networks' actual schedules differ significantly from the optimal

466

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

schedules; a substantial ratings gain is suggested by scheduling optimally.

Table 6 Total weekly ratings for the week of February 1, 1989

4.I. The week of February I, I989 Network

Fig. 1 presents the actual network prime-time schedules for the week of February 1, 1989, while Fig. 2 presents the optimal myopic schedules, obtained by solving ( 1 ) - ( 6 ) with the forecasted ratings from (15), for each of the three networks--NBC, CBS, and ABC, respectively. We took, for the "fixed and given" schedules of the other networks, their actual schedules for that week. For each network, the constrained integer-programming problem is of dimension (DT) 2 = 900. (Note that Monday and Tuesday in the schedules refer to the Monday and Tuesday from the next calendar week--that is, to Monday, February 6, and Tuesday, February 7.) In describing differences between actual and optimal schedules, we reiterate that the optimal schedule is not unique. Besides the sixth slot for any evening, which has no lead-out component (equivalent to the first slot's having no lead-in component), any two show parts with the same attributes assigned to it as well as the same maturity can be exchanged between time slots without affecting total ratings, that is, t h e value of the objective function for the maximization problem. So the fact that My Two Dads appears in the optimal schedule in Fig. 2 at 7:30 on Monday, whereas Alf, another situation comedy, was actually scheduled in that slot, is not a real difference. Our model would allow those two comedies to be interchanged without affecting the forecasted weekly ratings. There are many real differences between actual and optimal schedules for all three networks, however. For example, for NBC, the hour-long show Main Event, classified as a sports (wrestling) show, is assigned to Wednesday evening in the optimal assignment rather than Friday evening in the actual schedule. Furthermore, as opposed to the actual schedule, several comedies in the optimal schedule are assigned to the 7:00 and 7:30 time slots rather than during the 8:00-9:00 hour. For ABC we see that its "Monday Night Movie" Columbo would have contributed more to total ratings as a "Friday Night Movie" Table 6 shows, for each network, the actual "total weekly ratings" (individual show-part ratings added up for the week) for the actual schedules; the fore-

NBC CBS ABC

Actual total

Forecasted total

for

for

optimal

actual schedule

actual schedule

myopic schedule

401 459 365

395.7 429.9 364.7

Total for

500.4 554.4 452.3

casted total weekly ratings for the actual schedules; and, finally, the forecasted total weekly ratings for the optimal myopic schedules. (Summing over all networks and prime-time slots, the theoretical maximum for weekly ratings is 3000, that is, 100% x 30 slots. Moreover, for the week of February 1, 1989, Arbitron reported a percentage of households with at least one television turned on during each time slot. Based on these percentages, the weekly maximum was 1800. Our predicted total for the myopic solutions of 1507 is below even this more conservative ceiling, because of the large number of non-network viewing options.) We are encouraged by how closely the model predicts actual total ratings, especially for the NBC and ABC networks (in the case of NBC, a forecasted total of 395.7 versus the actual total of 401; in the case of ABC, a forecasted total of 364.7 versus the actual total of 365). In the case of CBS, our regression model underpredicted by about 30 "total percentage points" the actual total weekly ratings of 459. (In that particular week CBS scheduled an excellent movie, Lonesome Dove, with unusually high ratings for a western.) We next compare the forecasted weekly totals for the actual network schedules with the forecasted weekly totals for the optimal myopic schedules. We see that ratings rise significantly if an optimal scheduling strategy is used. In particular, there is a 26% improvement for NBC; a 29% improvement for CBS; and a 24% improvement for ABC. These ratings improvements would translate into close to $1 billion in additional advertiser revenues for each network, assuming that each network currently receives about $3 billion in revenue. Moreover, as we will see in the next section, these improvements hold up even in a competitive Nash environment. That is, we will be able to conclude that large ratings gains are available from using an optimization model as opposed to heuristics--regardless of the competitive environment

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

t O,y

I v:oo

Mort. 2/6/89

"Puts. 2/7/89

Wed. 2/1/89

Thurs. 2/2/89

Fri. 2/3/89

I v:3o

Cosby Special Newhart

Alf Kate & Allie

18:oo Golden Girls Movie

I 8:30

19:oo

I Empty Nest

I Cheers Special

I 9:3o Night Court Special

I Net I NBC

CBS

MacGyver

Movie

Matlock

In the Heat of the Night Movie

Midnight Caller

Moonlighting

thirtysomething

ABC

Nightingales

NBC

Wiseguy

CBS

Tour of Duty Who's the

] Roseanne

Boss

I

Unsolved Mysteries Billy Graham Special Growing l Head of the Pains I Class Cosby J Different [ World 48 Hours Fine Romance Main Event Beauty and the Beast Perfect Full Strangers House

ABC NBC CBS

Night Court Equalizer

My Two Dads

Wonder Years Cheers

Hooperman China Beach Letterman Anniversary Special Knots Landing Heartbeat

Paradise Dynasty Miami Vice Dallas Mr. Belvedere

467

I Just the I Ten of Us

ABC

NBC CBS ABC

UNSUB

NBC

Falcon Crest 20/20

CBS ABC

i

Fig. 1. A c t u a l n e t w o r k schedules f o r the w e e k o f February !, 1989.

I Day Mort. 2/6/89

Tues. 2/7/89

Wed. 2/1/89

I 7:00

I v:30

Cosby My Two Special Dads Beauty and the Beast MacGyver Night Court Special Movie

Different World

Mr, Belvedere Main Event Paradise

Who's the Boss

20/20

Thurs, 2/2/89

Cheers Special Newha~ Hooperman

Pri. 2/3/89

Cosby

Golden Girls Kate & Allie Full House Night Court Just the Ten of Us

18:30

Nightingales

19:00 AIf

19:30 I Empty I Nest

Movie

I Net. ] NBC CBS

thirt~omething

Heartbeat

ABC

Midnight Caller

Matlock

NBC

Tour of Duty Perfect Strangers UNSUB

CBS

Fine Romance Unsolved Mysteries Dallas Dynasty

48 Hou~

Growing Pains

I 8:00

Cheers Knots Landing Roseanne Miami Vice Falcon Crest Movie

Equalizer China Beach I Letterman Anniversary I Special Billy Graham Special Head of the Moonlighting Class In the Heat of the Night Wiseguy

Fig. 2. O p t i m a l m y o p i c s c h e d u l e s for the w e e k o f F e b r u a r y 1, 1989.

Wonder Years

ABC NBC CBS ABC NBC CBS ABC NBC CBS ABC

468

C.M.L. Kelton, L.G. SchneiderStone~European Journal of Operational Research 104 (1998) 451-473

Table 7 Total weekly ratings under compefifionforthe week of February 1, 1989 Network

Total for Nash schedule

NBC CBS ABC

491.1 560.2 440.7

in which the networks make scheduling decisions. The Nash schedules for the week of February 1, 1989, are found in Fig. 3. Interestingly, for NBC, there are no significant differences between the Nash and the optimal myopic schedules. For CBS there are several minor differences, while greater differences are observed for ABC. For instance, ABC's serial drama Dynasty is scheduled on Monday evening under Nash rather than on Wednesday. Table 7 has the total weekly ratings under competition for the three networks. Relative to the forecasted totals for the actual schedules, there is still an improvement of 24% for NBC for scheduling optimally; a 30% improvement for CBS for optimizing; and a 21% increase for ABC. Whereas competition seems to diminish the gains from optimizing for ABC, it serves to enhance (slightly) the gains from optimizing for CBS. Our intuitive algorithm for finding Nash equilibrium schedules converged after only two rounds, that is, after solving six integer-programming problems. 4.2. The week of February 6, 1991

We then repeated our analyses for the week of February 6, 1991, for the purposes of out-of-sample model validation as well as to be able to consider other show types that happened not to occur during our week in 1989. Figs. 4 and 5 are the counterparts for 1991 to Figs. 1 and 2 above for 1989; that is, they contain, for each of the three networks, the actual prime-time schedules and the optimal myopic schedules, respectively. (Note that this is a full two years outside our estimation sample, and, hence, provides a rather harsh validation test. Moreover, this is the only study in this area of which we are aware that provides any out-of-sample check.) As in our discussion for the week in 1989, not all of the differences between the actual and optimal myopic schedules are significant (since our optimal myopic

Table 8 Total weekly ratings for the week of February 6, 1991 Actual total Forecastedtotal Total for for for optimal Network actualschedule actualschedule myopicschedule NBC CBS ABC

323 281 405

308.8 334.7 274. l

456.4 470.2 462.2

schedule is not unique in the particular way explained above). Real differences include a schedule shift for the movie Perry Mason and the starting out with comedy on Tuesday and Friday under the optimal myopic schedule. Table 8 is the counterpart to Table 6; it shows the actual total weekly ratings resulting from the actual schedule; the forecasted total weekly ratings for the actual schedule; and the forecasted total weekly ratings for the optimal myopic schedule. We first note that the model does not predict total ratings as well for a week in 1991 as it does for a week in 1989. Although forecasted ratings totaling 308.8 are reasonably close to actual ratings of 323 for NBC, the model overpredicts CBS's ratings (334.7 versus 281 only), and considerably underpredicts ABC's total weekly ratings (274.1 as opposed to 405). Between 1989 and 1991, ABC significantly altered its mix of shows, scheduling a total of twelve half-hour comedies and comedy dramas during prime time. Although our model estimates a "premium" for comedy (a premium of 3.370, to be exact, from the regression model), these particular comedies were, generally, much more successful than average (for the 1980s). Furthermore, our model gives ABC a "penalty" ( - 0 . 4 7 9 ) relative to NBC that may not have been warranted in 1991. Nevertheless, this underestimate argues again for periodic updating and reestimation of the forecasting regression model if it were adopted in practice. As for the week in 1989, our myopic results suggest substantial gains to optimization versus the use of programming heuristics. Compared to the forecasted ratings for the actual schedule for the week of February 6, 1991, the optimal myopic schedule leads to a 48% improvement for NBC; a 40% improvement for CBS; and a noteworthy 69% increase for ABC. Measuring improvement relative to the actual totals would result in smaller estimated gains for NBC and ABC,

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

I Day Mon. 2/6/89

2/7/89

Wed. 2/1/89

Thurs. 2/2/89

Fri. 2/3/89

17:oo

17:30

Golden Girls Movie

AIf

18:00

l s:30

19:oo

19:3o

I

Net. [ NBC

Cosby I Empty Special [ Nest Tour of Duty thirtysomething

CBS

Unsolved Mysteries Movie

Matlock

NBC

Nightingales

ABC

Head of the I Roseanne Class I My Two I Night Court Dads [ Special Beauty and the Beast Fine Romance

Dynasty

China Beach

Growing Pains

Main Event

Midnight Caller

UNSUB

NBC

Paradise

Falcon Crest MacGyver

Wiseguy

CBS

Heartbeat

ABC

Mr. Belvedere Cheers Special Kate & Allie Just the Ten of Us Night Court 48 Hours

I Perfect Strangers Cosby Newhart Who's the Boss I Cheers

I

20/20

Different World Dallas Hooperman Miami Vice Knots Landing Movie

CBS [ Wonder [ Years

ABC

NBC

LettermanAnniversary Special Billy Graham Special Full Moonlighting House In the Heat of the Night Equalizer

CBS ABC NBC CBS ABC

Fig. 3. Nash schedules for the week of February 1, 1989.

[ Day Mon. 2/11/91

Tues.

I 7:00 Fresh Prince of BeI-Air Evening Shade MacGyver

I 7:30 Blossom Major Dad

Matlock Rescue 911

Wed. 2/6/91

Thurs. 2/7/91

]~'i.

2/8/91

Wonder Years Cosby

Murphy Brown Movie

Is:30 I Designing I Women

Davis Rules

] Growing I Pains Different World Flash

Top Cops Father Dowling Mysteries Super Bloopers War Special Full Family House Matters

] 9:00

I 9:30

Face to Face with Connie ChunK Special

I Net. I NBC I CBS ABC

In the Heat of the Night Movie

2/12/91 Who's the Boss Unsolved Mysteries 48 Hours

I g:oo Movie

Law& Order

NBC CBS-

Roseanne

Coach

thirtysomething

ABC

Night Court Jake and the Fatman Doogle Howser MD Cheers

Seinfeld

L.A. Law Special Top Cops Barbara Waiters Special L A Law

NBC

Anything but Love Wings Good Sports

Fantasies Dark Shadows Dallas Perfect Strangers

J Going

Knots Landing Prime Time Live Midnight Caller Sons and Daughters 20/20

I Places

Fig. 4. Actual network schedules for the week of February 6, 1991.

CBS ABC NBC CBS ABC NBC CBS ABC

469

470

C.M.L. Kelton, L.G. SchneiderStone~European Journal of Operational Research 104 (1998) 451-473

I Day

Mon. 2/11/91

"I~es. 2/12/91

Wed. 2/6/91

Thurs.

2/7/91

I 7:00

I 7:30

Blossom

Cheers

Murphy Brown Fantasies

Evening Shade

Fresh Prince of BeI-Air Major Dad Family Matters Dark Shadows Movie

Different World Top Cops Coach

Pedect Strangers Super Bloope~ Designing Women Rosoanne

Fri. 2/8/91

Cosby

J 8:30

Good Sports Who's the Boss Seinfeld

I 9:30

I Net. I NBC

In the Heat of the Night Jake and the Fatman

thirtysomething

20/20

ABC

L.A. Law

Matlock

NBC

Sons and Daughte~ Father Dowling Mysteries Law & Order Top Cops Barbara Waiters Special

CBS

Rescue 911 Wonder

] Dcogie I HowserMD

Special t Anything but Love

19:00

Unsolved Mysteries Flash

Yea~ LA. Law

48 Hours Full House

I S:00

} MacGyver Midnight Caller Knots Landing Movie

CourtNight

CBS

] Wings

Face to Face with Connie Chung Special

NBC CBS ABC NBC CBS ABC

Movie

NBC

Dallas Prime Time Live

ABC

Growing Pains

War Special Going Places

CBS

I Davis I Rules

ABC

Fig. 5. Optimal myopic schedules for the week of February 6, 1991. but there would still be a gain. (Again, our optimal network total of 1389 is below both the theoretical maximum of 3000 and the more conservative Arbitron ceiling of 1623.) The optimal Nash schedules for the week of February 6, 1991, are found in Fig. 6 (the counterpart to Fig. 3 above), for NBC, CBS, and ABC, respectively. Again, convergence was achieved after just two rounds. Whereas there are many important differences between actual and optimal myopic schedules for this week, there are relatively few differences between optimal myopic and optimal competitive schedules. Note that both Figs. 5 and 6 contain the single occurrence of the bridging programming heuristic (that is, scheduling shows of one hour or longer in duration in time slots so that the competing programs end and begin in the middle of these shows). ABC's hour-long Prime Time Live is scheduled from 7:30-8:30 under both the optimal myopic and Nash schedules. Table 9, the 1991 counterpart to Table 7 above, shows the forecasted weekly ratings totals under Nash. Now, relative to the actual weekly totals, we see that under competition, NBC's estimated gains from optimizing are essentially the same as for the myopic problem, as are CBS's. Again, as for 1989, it is ABC

Table 9 Total weekly ratings undercompetitionfor the week of February 6, 1991 Network NBC CBS ABC

Total for Nash schedule 457.7 467.8 451.8

that "loses the most" under competition--although the estimated 65% gain from optimization is still quite impressive.

4.3. An illustration of the cooperative scheduling problem To illustrate the cooperative scheduling problem, we solved a one-day, six-slot problem for two networks (a (DT) N+I = 63 = 216-dimensional integerprogramming problem). We selected four shows from NBC and another four from ABC that were actually scheduled during the week of February 6, 1991. For NBC, we selected two comedies--Night Court and Cheers; one crime drama--In the Heat of the Night; and one drama--Midnight Caller. For

C.M.L. Kelton, L.G. SchneiderStone~European Journal of Operational Research 104 (1998) 451-473 J Day

Mon. 2/11/91

J 7:00

] 7:30

Blossom

Fresh Prince of Bel-Air Top Cops

Major Dad Fantasies

Tues.

2/12/91

Wings

Thurs. 2/7/91

Fri. 2/8/91

I Cosby

Full House Dark Shadows Movie

[ Anything

Perfect Strangers Super Bloopers Murphy Brown Going Places

J Growing Pains

Night

J Seinfeld

Court 48 Hours Family Matters

[ 8:30

Unsolved Mysteries Rescue 911

but Love

Midnight Caller Sons and Daughters Wonder Years L.A. Law

[ Doogie

Howser MD

I MacGyver Cheers

Evening Shade Coach

I

9:00 In the Heat of the Night Jake and the Fatman Roseanne

thirtysomethin s

Flash

wed, 2/6/91

[ 8:00

Different World

] Net. I NBC CBS

[ RulesDavis

ABC

L.A. Law Special

NBC

"Fop Cops

CBS

Father Dowling Mysteries Law 8z Order Face to Face with Connie ChunK Special Barbara Waiters Special Matlock

ABe

Designing

Knots Landing Movie

] 9:30

471

I Good

Women

NBC CBS ABe NBC CBS

Sports

ABe

Movie

NBC

Dallas Prime Time

Who's the

Live

Boss

War Special

CBS

20/20

ABe

Fig. 6. Nash schedules for the week of February6, 1991.

I Strategy I 7:00 OptimM Cheers Coop.

I 7:30 Night Court

Night Court Roseanne

I 8:30

Midnight

9:00 In the Heat

19:30

Caller thirtysomething

of the Night

Cheers

Midnight Caller

Davis Rules

MacGyver

In the Heat of the Night thirtysomething

MacGyver

N~sh

l S:00

Roseanne

Davis Rules

I Net. I NBC ABe NBC

ABe

Fig. 7. Optimalcooperativeversus Nash schedules for Monday,February 11, 1991. A B e , we selected one adventure show--MacGyver; two comedies--Roseanne and Davis Rules; and one drama--thirtysomething. Then, for Monday night only, we obtained forecasted ratings using (15) and then solved ( 7 ) - ( 1 2 ) to obtain the optimal cooperative schedule presented in Fig. 7. A "monopolist" would schedule the NBC comedies from 7:00-8:00; both dramas on both networks from 8:00-9:00; the NBC crime drama from 9:00-10:00; the A B e adventure show from 7:00-8:00; and, finally, the two A B e comedies in the final prime-time hour from 9:00-10:00. These network schedules are not Nash-equilibrium schedules. The Nash-equilibrium schedules for this same problem are also given in Fig. 7. In this ex-

ample, under competition, A B e has an incentive to schedule its two comedies from 7:00 to 8:00--an example where a simple counterprogramming heuristic would not be optimal. We see that, under competition, both networks do best to schedule their comedy shows during the first prime-time hour. The networks do not do best to schedule their dramas opposite each other. Whereas the NBC drama Midnight Caller is scheduled for 8:00-9:00, the A B e drama thirtysomething appears between 9:00 and 10:00. Finally, A B e does best to schedule MacGyver between 8:00 and 9:00; NBC does best to schedule In the Heat of the Night at the end of the evening. Although this is a very simple illustration, we do gain some additional appreciation for the importance

472

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

Table 10 Total nightly ratings under cooperation versus Nash for February ll, 1991 Network NBC ABC Joint

Total for Nash schedule 94.8 92.6 187.4

Total for optimal cooperative schedule

189.9

of optimization in program scheduling since the cooperative solution with a forecasted nightly ratings total of 189.9 is not much larger than the competitive solution (with a total across the two networks of 187.4). (See Table 10.) Although our model suggests that there are clearly ratings gains available by reducing the competitive effects (particularly of comedy shows) across networks through cooperation, these gains may be small in comparison to other scheduling considerations. Nothing in this illustration contradicts our conclusion of potentially large gains from optimization regardless of competitive environment.

5. Discussion In this paper we have formulated as binary integer programs, under three different competitive environments, network television-scheduling problems, assuming a fixed number of shows to fill a given number of days and time slots. To provide input data for the models we developed a regression to forecast showpart ratings for each of the possible days and time slots. This regression is general enough to handle new shows, movies, and specials, as well as regular "mature" shows. With this model, we were able to explain 70% of the variation in ratings for twelve weeks of data during the period 1981-1982 through 1988-1989. We then presented two applications of our model--one for a week in our sample and one for an out-of-sample week. In each case, we found large gains, relative to the actual schedules, from optimal program scheduling. And, in each case, we found that there was relatively little difference between the optimal myopic schedule (no competition) and the Nash schedule under competition. Although there are competitive effects, there are also many other significant influences on television ratings--including lead-in and attribute effects.

Our model does not explicitly account for show quality. It indirectly takes quality into account through the show-maturity variables. This omission, however, except insofar as it affects scheduling of shows in the 9:30 slot (the last slot) of any evening, should not affect scheduling decisions for a given repertoire of shows to be scheduled. Quality would be a much more important factor in determining which shows to schedule in the first place--out of a bigger pool of shows than can be aired. We have not dealt with such a problem at all in this paper. Although Horen (1980) proxied show quality by a function of prior-year ratings, we elected not to do so primarily since such information would not generally be available for new shows, movies, or specials. Although not perfect, our showmaturity variables do capture quality, at least of regular shows, to some degree and are flexible enough not to preclude the scheduling of new shows. After all, shows of lesser quality would not be scheduled for more than one season. (The worst performers might be cancelled after just a few weeks.) We did not solve the week-long cooperative problem. Large integer programs are notoriously difficult to solve. It is possible that studying the polyhedral structure of the model's feasible region would reveal several classes of easy-to-compute facets. If so, we could then develop a relatively efficient branch-andcut method for solving the model. Since the model is very similar in structure to an assignment problem, there is, in fact, hope that such facets can be found. Alternatively, one could develop an heuristic to obtain a close-to-optimal solution for the large cooperative problem. For example, one could initially require that only variables associated with the longest programs have integer values. This would suggest where these programs should be positioned during the week. We could then fix the positions of these programs by removing any variables that could cause them to be moved to a different position. The next step of the heuristic would solve this reduced model requiring that only variables associated with the second longest programs have integer values. We could then fix the positions of these second longest programs by removing the appropriate variables from the model. The heuristic would proceed in this manner, fixing the positions of programs of shorter and shorter length, until all programs have been positioned.

C.M.L. Kelton, L.G. Schneider Stone~European Journal of Operational Research 104 (1998) 451-473

While there is much known about integer programming (see Nernhauser and Wolsey 1988), it is still part art and part science. Whether the approaches suggested here for dealing with large cooperative models would work well is a subject for future research. In any case, the small cooperative scheduling problem that we were able to solve led to two interesting results. First, the best cooperative schedule vector is not necessarily the best Nash schedule vector. Second, the value of the objective function, that is "total weekly ratings," is not very different for the cooperative as opposed to the Nash solution. The evidence suggests that big gains in scheduling come from optimization rather than from cooperation.

Acknowledgements Computational support was provided by grants from the Minnesota Supercomputer Institute. Research support was provided by a grant from the University of Minnesota Graduate School. We are very grateful for the theoretical and computational advice from David Kelton, Kipp Martin, Barry Nelson, Charles Reilly, Richard Steinberg, and Richard E. Stone. We also benefited tremendously from the feedback from the participants--particularly Peter Danaher, John Rossiter, and Roland Rust--in our session at the Marketing Science Conference, London, July 1992. We would, in addition, like to thank our graduate assistant Lu Qu and an anonymous referee. Of course, any mistakes are ours alone.

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