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Media selection by mean-variance analysis
Media selection by mean-variance analysis C.A. DE K L U Y V E R * University of Canterbury. Christchurch, New Zealand Received 16 May 1978
Revised 8 June 1979 This article considers new formulations for media scheduling problems. By treating the size of the audience obtained as a random variable, the resulting formulations axe shown to be of the mean-variance type, and hence akin to Markowitz's familiar portfolio model. A binomial probability model is used t¢ derive expected values for gross and net audience segments obtained, and of average frequency realized. The proposed model minimizes the variance of gross audience obtained subject to a variety of constraints on gross audience, net audience, frequency, ant the budget allocated. A parametric solution structure is sought tracing the entire efficiency frontier of solutions, thus allowing trade-offs on a risk-return or budget-return basis. 0. Introduction
Media planning problems have received considexable attention during the last twenty years. Approaches taken to this problem include linear programming (see e.g. [2,5,8 ~23]), integer programming [25], incremental analysis [4], classical optimization [3,10,13-15,24] and, more recently, goal programruing [6,7,9]. A good survey of this area may be • found in [ 12]. Of these approaches, mathematical programming models have found the most wide-spread acceptance with media planners despite their sometimes relatively crude nature (consider e.g. approximations used to model audience duplication, discount structures and the fact that fractional exposure levels are admitted by most of these models). In defense of their use, media planners (correctly) point out that model answers rarely are taken at face value. Rather, they serve ~s starting points in the search for 'near-optimal' schedules in a process of negotiation involving clients and media representatives. This article considers new formulations for the media planning problem designed to relax a fairly restrictive assumption common to most mathematical • Current address: Purdue University, West-Lafayette, IN 47907, U.S.A. © North-Holland Publishing Company European Journal of Operational Research 5 (1980) 112-117
programming formulations, namely that of constant audience sizes (the goal programming model advanced by Charnes et al. [6,7] is an exception to this). ]By treating the size of the audience effectively reached as a random variable, the resulting formulation will be shown to be of the mean-variance variety, and hence akin to the familiar portfolio model due to Markowitz [ 18]. 1. Simple mean-variance representations for media
planning The simple mean-varience models described ifi this section take audience variability explicitly into account. The importance of this issue needs little emphasis. Media budgets and allocations are fixed long in advance of the actual exposure of the target audience. Often, competing television and radio shows have not been specified, or are new in which case their following cannot easily be predicted. News headlines are generally equally unpredictable and may affect circulation statistics for print media considerably. Morard, Poujaud end Tremolieres [22] cite several statistics on this issue. In any case, it seems logical to treat audience figures as random variables, with knowledge on~,y of their means and variances, obtained either from historical data, or estimated as one would task durations in a critical path analysis, i.e. assuming a beta distribution. Morard et al. [22] first considered the application of meen-vafiance analysis to media planning and postulated the following simple model. Let ai denote the mean (i.e. expected) audience effectively reached by medium i with one advertising unit (e.g. a page, half page, etc.) and oi its standard deviation. Further, letting es and bi denote respectively lower and upper bounds on the number of insertions in channel i, and A the mean total target audience desired by the advertiser, the following simple one-period allocation model results: i subject to i
=A
ei
(2)
(3)
CA. de Kluyver / Media selection by mean.variance analysis
113
In (1)-(3), we merely seek to find, for a given desired exposure Icy.elA, the configuration(s) with minimal variance. Of course, in practice one would solve (1)(3) parametrically for various values of A, thereby tracing the entire efficiency frontier of solutions, to locate an acceptable 'equilibrium' pair (r(A), A) where
Additionally, it should be observed that one of the most difficult problems in media planning, that of accurately representing media discounts, is avoided in the above formulation.
r(A ) = ~ / ( ~ o~" xl(A )),
While the concept of applying mean-variance type models to media scheduling problems, and the ensuing idea of tracing the entire efficiency frontier of solutions are appealing, the extreme simplicity of formulation (1)-(3) may cause media planners to question their value in practice. This section aims to partially overcome this deficiency by reformulating the basic model to include considerations of both gross and net audience (i.e. reach), and frequency of exposure. Additionally, initial efforts towards the development of a multiperiod equivalent model are reported.
(4)
i
the risk function associated with exposure level A. Morard et al. [22] continue to show that by reinterpreting the ai's as mean audiences reached per investment unit, and the bounds as limits on the number of investment units, budget considerations are implicitly taken into account in this model. With this interpretation, the mean audience A is reached with a budget Y~i xi(A), allowing sensitivity analysis on the budget allocated. The authors proceed by illustrating the approach using a simple, analytical technique based on Lagrangian arguments. While conceptually interesting, the above model has some serious shortcomings which may be recognized by examining its underlying assumptions: (i) the audience effectively reached in each channel is a linear function of the number of insertions. This assumption was common to early linear programming formulations of the media planning problem but has since been dealt with by using piecewise approximations for cumulative response, (ii) the total audience effectively reached is the sum of the audiences reached by the various media. This is probably the most serious shortcoming of the medel. The issue of audience overlap has received considerable attention in recent years (see e.g. [ 1,6, 19,21]) and most currently employed models for media planning distinguish between gross audience obtained, net audience or 'reach', and frequency considerations (see, e.g. [6]). Thus, to be useful, meanvariance models for media planning should take these considerations into account. (iii) the different audiences are uncorrelated. While this assumption is theoretically easily relaxed by recognizing a co-variance structure 02, we are then f~tced with the complicated problem of its estimation in practice, lIn a number of instances (Television, for example), this measurement may be practically impossible. (iv) Finally, along with most linear programming models for media planning, the model allows fractional answers.
2. More realistic mean-variance formulations
2.1. Gross audience, net audience and frequency
To develop explicit expressions for the mean (i.e. expected) gross kth audience segment obtained and its variance, consider the following binomial argument 1,2. Let p~(t) denote the probability that a particular individual in audience segment k is exposed to the ]th cumulative insertion in medium i in period t. The probability that an individual has not been exposed at least once through medium i following/ cumulative purchases in that medium is then given by q~(t) = ( i - p~l(t))" (1 - p~2(t)) ..... (1 - p~(t)) (5)
Hence, the distribution ofd/~(t), the gross kth audience segment obtained through medium i given ] cumulative purchases in that medium in period t is binomial, viz.
(6) Here, D~(t) represents the size of the total audience
I The author is indebted to Dr. D.C. McNickleof the U,aiversity of Canterbury for suggestingthisapproach and his assistance in detailing the argument. 2 A similar argument is used by Ellis [ 10].
C~4, d e Kluyver / Media selection by mean.varianceanalysis for m e d i u m / i n audience segment k in period t. 3 The mean and variance of (6), respectively, are given by
E~i (t) = E(d~i(t)) = D~(t) " (1 - q~i(t))
(7)
and V~(t) =
V(d~i(t))=D?(t). (1 -q~i(t)). q~(t).
(8)
Observe that as ] approaches.**, E~ (t) approaches the saturation level D~(t). The variance V~/(t)may initially increase depending on the magnitude of the impact probabilities p~i(t), but ultimately must decrease ,with limiting value zero. Also note that the definitions of E~(t) and V~(t) are conditional upon the numbelr of cumulative purchases/. To facilitate a representation as a mathematical programming problem, incremental (or, in the case of the variance, decremental) audience figures are desirable. Accordingly, by letting
Z~'~](t) = E~](t) - E~/_l (t) =D~ft).
p~.(t), q~,i_,(t)
(9)
and
V~(t)- Vik,i_,(t) =D~ (t)" { (q~,]_l(t)) 2 " (1 - (1 - p~(t)) 2) _ qk/_lq), pkq)} (10)
AVe(t) =
the expected gross kth audience segment, obtained through cumulative purchases in medium i in period t may be written as 4 E/C(t) = ~ z~7~(t), i
xti(t),
i
AVi~(t)" xq(t),
~xii(t)< l , i
xo(t)>~O.
(13)
Assuming independence with~a~ a given time period of the random variables d~(t)for the various media audiences, the expected gross kth audience segment, obtained by cumulative purchases in all media in period t may be written as 5
xo(t),
04)
AVe(t)" xo(t ) .
(15)
ek(t) - Z) Z: i
/
with variance Vk(t) - ~ ~ t /
The independence assumption employed here may be questioned. Ellis [ 10] defends its use on two principal grounds: computational simplicity and the lack of accurate information needed to estimate a covariance structure. Additionally, it should be noted that in the present context; i.e. in a mean-variance type analysis, the resulting bias will be towards more conservative decisions since Vk(t) as defined by (15) will overstate its true value. 6 Next, consider the expression for the net kth audience segment or 'reach'. Using the previously derived media probabilities q~i(t), i = 1, 2, ..., L the probability that an individual has not been exposed at least once by any medium in period t given that /(i) cumulative 'purchases have been made in medium i, is given by
(11)
Q~(t)(t) = H q~.(t)
(16)
i
with variance V~(t) = ~
approximated by requiring (see e.g. [6])
(12)
where xij(t) = 0 or 1, denoting the ]th cumulative purchase in medium i for period t. Generally, this is
3 As noted in [6], d~(i) might refer to the number of women aged 25 to 34 (k), obtained via a purchase of/issues in Time (f). D~(t) denotes the total expected audience of the medium (i) in the relevant audience segment (k). Section 3 discusses a model of audience change from period to period for this random variable. 4 Observe that A Vi~(t) is not the variance of the/th insertion in medium i directed at segment k. Rather, it represents the decrease (or, increase) in variance resulting from an additional insertiGn in channel i for audience segment k in period t.
Hence the distribution of d~i ) (t), the net kth audience segment, obtained by ](i) cumulative purchases in media i, i = 1, 2, ..., I in period t is again binomial, viz. Pr(d~(~-)(t) = d )
5 Note that independence over time is not assumed. A model of audience change over time is discussed in Section 3 of this article. Also observe that the derivation by Charnes et al. [6] does not require an independence assumption. This issue was resolved when the material in [6] was reprinted in [201. 6 Unlike in the Markowitz model, constraints on expected audiences sizes may be used to compensate for possible overest~nates in Vk(t).
C.Ao de Kluyver / Media selection by mean-variance analysis
with expected value • (1 -
(lS)
and variance P
--Q~o(t)).
Q~i)q).
(19)
Here, D ~ represents the union of the media total audiences D~(t), i = 1, 2, ...,L Again it is noted that the statistics E~i)(t) and V~i)(t) are conditioned on the number of cumulative purchases in each medium. The incremental representation used for gross audience statistics is not applicable in this case. However, concentrating on the expected net kth audience segment, E~0(t), note that (1
- FJ~i)(t)/Dk(t)) = Q~oq) = I] q~.(t) i
= H
(20)
from which it follows that
E~i)q)/Dkq))
= ~]) ~ Ln(l i l
Fk(t) = ~ ~ i i
p~(t), xq(t).
(23)
The distribution underlying (23) is a generalised binomial distribution in which the probability of 'success' is different for each 'trial' (cumulative purchase). (See [ 11 ].) The resulting formulation of a single-period meanvariance model for media planning thus seeks to mirimize the variance of all gross audience segments exposed, viz.
xq(t ~
(24)
H
= I'I l-[ (1-p~.(t)) xq(O , i /
-
(22)
from which an expression for average frequency may be derived as
Min ~ Z~ ~ AVe(t)k i ]
i iq)
Ln(l
probabilities p~.(t), it may be shown that the expected number of exposures given/cumulative purchases in medium i in period t (per individual) is given by
P~l(t)+ p~2(t)+ + p~(t),
i)(t) =
=Dkq) • (1
115
p~(t)), xq(t).
subject to a variety of constraints on the expected magnitude of gross audience segments, net audience segments, and average frequency realized. For instance, requiring that the gross kth audience segment be at least Mk(t) is written as
~ AEi~(e)" xq(t) >IMk(t). i i (21)
Observe that (21) is identical to the net audience representation derived by Charles et al. [6] using the idea of 'non-reach'. As a result, the above derivation provides a probabflistic justification for that net audience expression. The derivation of a variance expression for the net kth audience segment involving zero-one variables is mathematically less tractable. As a consequence, the mean-variance formulations proposed here concentrate on minimizing the variance of gross audience segments obtained, subject to constraints on the size of both gross and net audience segments exposed. Other constraints may include considerations of frequency. If average frequency is deemed adequate 7, the binomial argument employed throughout this development may again be invoked. Using the impact
7 Alternatively, constraints modeling an entire distribution o,"frequencies may be used. See Charnes et al. [6].
(25)
Similarly, for the kth net audience segment we might require
E~o >~Nk(t),
(26)
which would be represented as i
~ Ln( l - p~.(t)), ]
xii(t)
~>Ln(1-Nkq)/Dkq)).
(27)
Finally, average frequency controls may be of the form:
F_*(*)<~Yk(t)
(28)
Or
F__kq)<,~ ~ p~lq), xi]q)<~ffgq). i
l
(29)
Before discussing possible extensions toward multiperiod formulations, attention is drawn to the fact that the objective function (24) is linear in the decision variables, in contrast to the model due to Morard
C.A. de Kluyver / Media selection by mean.vorianceanalysis
116
et al, [22] discussed in the previous section. The fact that linear as oppcs~d to quadratic programming routines can be used with the formulations proposed here may well be a significant factor in their practical application.
2.2. Extension to multi-period mean-variance formulations The extension of (24)-(20) to a multi-period model merits further attention. This section considers the additional assumptions needed to produce a computationally convenient multi-period equivalent model, and, in the process, postulates a relationship for expected total media audiences, D~(O, between successive time periods. Starting with the latter, it seems reasonable to attribute changes in total media audiences (i.e. circulation or viewing statistics) both to random and nonrandom influences. Non-random fluctuations may be due to factors such as seasonality, conscious efforts of audience build-up, etc., whereas random disturbances reflect the uncertainty inherent to a competitive media climate. Accordingly, the following simple model of audience change is postulated:
b (t) =
3 '(t - 1) +
(30)
Here,/)~(t) denotes the random variable modeling total media audience, r~(t) represents an index of non-random audience fluctuation, and el(t) random error. Assuming that ~he error terms all are independent, identically distributed random variables with mean 0 and variance o 2 leads to the following relation between the expected total audiences
D~(t)=r~(t). D~(t- 1) t
=(1-1 r['(s)).
(30
S= t
from which estimates for AE/~(t) and AVe(t) for t = 1,2, ..., Tmay be determined using (9) and (10) respectively, and hence, a constraint structure for multiple periods similar to (24)-(20) specified. If, in a multi-period model, it is desirable to minimize the sum of the variances of gross audiences olbtained, exp:~ession (24) may simply be summed for each period t. if, on the other hand, the expressed objective is to minimize the variance of the sum of gross audiences obtained, recognition of a possible co-variance structure over time must be considered. Observe that the audience fluctuation model postu-
lated above merely shifts the binomial distribution (6) over time. Therefore, zero co-variance would be implied by an assumption of in0ependence of the deviations of true total audiences realized from their expected values D~(t) over time. The latter seems plausible if the probabilities {p~/(t)} remain fairly constant over time, i.e. when advertising effectiveness may be assumed constant over the planning period. This assumption is not unreasonable if the same set of agencies is employed over the entire planning horizon. In summary, a raulti-period equivalent model to (24)-(29) seeks to s Min ~ t
~ k
~ i
~ AV~/(t)" xi/(t)
(32)
/
subject to constraints on gross audience, net audience and frequency in each and across periods, where AVe(t) and AE~(t), t = 1, 2, ..., Tmay be estimated using the appropriate expressions given above.
3. Assessment of the mean-variance formulation: Conclusions
The media planning model developed in this article embodies two principal objectives: (i) a desire to obtain relattvely high values for expected gross and net audience segments and frequency, and (ii) a desire for this 'return' to be dependable, i.e. not subject to large amounts of uncertai:nty. As such, it recognizes that the media schedule with the highest: expected 'return' is not necessarily the one with the least 'uncertainty of return'. The most reliable schedule with an extremely high return may be subject to an unacceptably high degree of uncertainty, and.., conversely, the schedule with the least uncertainty may be characterized by an undesirably low re.'turn. By tracing the entire efficiency frontier of solut!~ons using parametric linear programruing techniques, the proposed mean-variance model allows a trade-off among these quantities of interest. If cost aspects of media schedules are modeled as e.g. in the LP I or LP II models due to Charnes et al. [6,7], the media planner has the option of balancing 8 Of course, the model formulation seeks allocations that are fixed over time. An alternative specification might include chance constrained programmingfeatures allowing ~:me-dependent ~onditional allocations.
CA. de Kluyver / Media selection by mean.variance analysis a schedule's return (i.e. expected audience), its variance, and the cost of the proposed schedule in arriving at final proposals. A convenient method to perform such comparisons consists of constructing both riskreturn and cost-return curves for the problem at hand. A final word concerns the use of the variance of a media schedule as a measure of risk. Critics may argue that the use of a symmetric measure such as the variance is not very descriptive since over-exposing a target audience is probably less undesirable to a media planner than under-exposing that same audience. Markowitz [18] recognized this problem in the context of portfolio selection and suggests the use of alternative measures such as the semi-variance to prevent 'over-achievements' from being penalized. In both instances, the usual assumptions of underlying normality and quadratic utility apply. However, in the case of the semi-variance, the conservative quadratic portion of the utility function corresponding to 'over-achievements' is replaced by a linear segment. While this conceptual advantage seems highly attractive, it may be outweighted when cost and convenience of computation are considered, particularly noting that an entire set of efficient media plans must be computed if full advantage is to be derived from the mean-variance approach.
Acknowledgements The author is indebted to Prof. W.W. Cooper of Harvard University for his constructive criticisms to an earlier draft of this article. His comm¢:nts and suggestions have allowed substantial refinements and improvements in the model.
References [ 1] M.M. Agostini, How to estimate unduplicated audiences, J. Advertising Res. 3 (1961) 11-14. [2] F.M. Bass and R.T. Lonsdale, An exploration of linear programming in media selection, J. Marketing Res. 3 (1966) 179-187. [3] E.M.L. Beale, P.A.B. Hughes and S.R. Broadbent, A computer assessment of media schedules, Operational Res. Quart. 17 (4) (1966) 381-411.
117
[4] D.B. Brown, A practical procedure for media selection, J. Marketing Res. 3 (1967) 267-269. [51 D.B. Brown and M.R. Warshaw, Media selection by linear programming, J. Marketing Res. 2 (1965) 83-88. [6] A. Charnes, W.W. Cooper, J.K. Devoe, D.B. Learner and W. Reinecke, A goal programming model for media planning, Management Sci. 14 (1968) 423-430. [7] A. Chames, W.W. Cooper, D.B. Learner and E.F. Snow, Note on an application of a goal programming model for media planning, Management Sci. 14 (1968) 431-436. [81 R.L. Day, Linear programming in media selection, J. Advertising Res. 4 (1962) 40-44. [91 C.A. De Kluyver, Hard and soft constraints in media scheduling, J. Advertising Res. 18 (1978) 27-33. [10l D.M. Ellis, Building up a sequence of optimum media schedules, Operational Res. Quart. 17 (1966) 413-424. 1111 W. Feller, An introduction to probability theory and its applications, Volume I (Wiley, New York, 1950). [121 D. Gensch, Computer models in advertising media selection, J. Marketing Res. 4 (1968) 414-424. 1131 A.M. Lee, Decision rules for media scheduling: Static campaigns, Operational Res. Quart. 13 (1962) 229- 241. [141 A.M. Lee, Decisioa rules for media scheduling: Dynamic campaigns, Operational Res. Quart. 14 (1963) 365-372. [15] A.M. Lee and A.J. Burkart, Some optimisation problems in advertising media scheduling, Operational Res. Quart. 11 (1960) 113-122. [16] J.D.C. Little and L.M. Lodish, A media selection model and its optimization by dynamic programming, Indust. Management Rev. 8 (1966) 15-24. [17] J.D.C. Little and L.M. Lodish, A media planning calculus, Operations Res. 18 (1969) 1-35. [ !8] H. Markowitz, Portfolio Selection (Wiley, New York, 1959). [ 19] R.A. Metheringham, Measuring the net cumulative coverage of a print campaign, J. Advertising Res. 3 (3) (1963) 23-28. [20] D.B. Montgomery and G.L. Urban (eds.), Applications of Management Science in Marketing (Prentice-Hall, Englewood Cliffs, NJ, 1970). [21] W. Kuhn, Net Audiences of German Magazines, J. Advertising Res. 3 (3) (1963) 30-33. [221 B. Morard, R. Poujaud and R. Tremolieres, A meanvariance approach to advertising media ,~election, Cornput. Operations Res. 3 (1976) 73-81. [23] S. Stasch, Linear programming and space-time considerations, J. Advertising Res. 4 (1965) 40-4.7. [241 C.J. Taylor, Some developments in the theory and application of media scheduling methods, Operational Res. Quart. 14 (1963) 291-305. [25] W.l. Zangwill, Media scheduling by decision programruing, J. Advertising Res. 5 (1965) 30-.';6.
Revised 8 June 1979 This article considers new formulations for media scheduling problems. By treating the size of the audience obtained as a random variable, the resulting formulations axe shown to be of the mean-variance type, and hence akin to Markowitz's familiar portfolio model. A binomial probability model is used t¢ derive expected values for gross and net audience segments obtained, and of average frequency realized. The proposed model minimizes the variance of gross audience obtained subject to a variety of constraints on gross audience, net audience, frequency, ant the budget allocated. A parametric solution structure is sought tracing the entire efficiency frontier of solutions, thus allowing trade-offs on a risk-return or budget-return basis. 0. Introduction
Media planning problems have received considexable attention during the last twenty years. Approaches taken to this problem include linear programming (see e.g. [2,5,8 ~23]), integer programming [25], incremental analysis [4], classical optimization [3,10,13-15,24] and, more recently, goal programruing [6,7,9]. A good survey of this area may be • found in [ 12]. Of these approaches, mathematical programming models have found the most wide-spread acceptance with media planners despite their sometimes relatively crude nature (consider e.g. approximations used to model audience duplication, discount structures and the fact that fractional exposure levels are admitted by most of these models). In defense of their use, media planners (correctly) point out that model answers rarely are taken at face value. Rather, they serve ~s starting points in the search for 'near-optimal' schedules in a process of negotiation involving clients and media representatives. This article considers new formulations for the media planning problem designed to relax a fairly restrictive assumption common to most mathematical • Current address: Purdue University, West-Lafayette, IN 47907, U.S.A. © North-Holland Publishing Company European Journal of Operational Research 5 (1980) 112-117
programming formulations, namely that of constant audience sizes (the goal programming model advanced by Charnes et al. [6,7] is an exception to this). ]By treating the size of the audience effectively reached as a random variable, the resulting formulation will be shown to be of the mean-variance variety, and hence akin to the familiar portfolio model due to Markowitz [ 18]. 1. Simple mean-variance representations for media
planning The simple mean-varience models described ifi this section take audience variability explicitly into account. The importance of this issue needs little emphasis. Media budgets and allocations are fixed long in advance of the actual exposure of the target audience. Often, competing television and radio shows have not been specified, or are new in which case their following cannot easily be predicted. News headlines are generally equally unpredictable and may affect circulation statistics for print media considerably. Morard, Poujaud end Tremolieres [22] cite several statistics on this issue. In any case, it seems logical to treat audience figures as random variables, with knowledge on~,y of their means and variances, obtained either from historical data, or estimated as one would task durations in a critical path analysis, i.e. assuming a beta distribution. Morard et al. [22] first considered the application of meen-vafiance analysis to media planning and postulated the following simple model. Let ai denote the mean (i.e. expected) audience effectively reached by medium i with one advertising unit (e.g. a page, half page, etc.) and oi its standard deviation. Further, letting es and bi denote respectively lower and upper bounds on the number of insertions in channel i, and A the mean total target audience desired by the advertiser, the following simple one-period allocation model results: i subject to i
=A
ei
(2)
(3)
CA. de Kluyver / Media selection by mean.variance analysis
113
In (1)-(3), we merely seek to find, for a given desired exposure Icy.elA, the configuration(s) with minimal variance. Of course, in practice one would solve (1)(3) parametrically for various values of A, thereby tracing the entire efficiency frontier of solutions, to locate an acceptable 'equilibrium' pair (r(A), A) where
Additionally, it should be observed that one of the most difficult problems in media planning, that of accurately representing media discounts, is avoided in the above formulation.
r(A ) = ~ / ( ~ o~" xl(A )),
While the concept of applying mean-variance type models to media scheduling problems, and the ensuing idea of tracing the entire efficiency frontier of solutions are appealing, the extreme simplicity of formulation (1)-(3) may cause media planners to question their value in practice. This section aims to partially overcome this deficiency by reformulating the basic model to include considerations of both gross and net audience (i.e. reach), and frequency of exposure. Additionally, initial efforts towards the development of a multiperiod equivalent model are reported.
(4)
i
the risk function associated with exposure level A. Morard et al. [22] continue to show that by reinterpreting the ai's as mean audiences reached per investment unit, and the bounds as limits on the number of investment units, budget considerations are implicitly taken into account in this model. With this interpretation, the mean audience A is reached with a budget Y~i xi(A), allowing sensitivity analysis on the budget allocated. The authors proceed by illustrating the approach using a simple, analytical technique based on Lagrangian arguments. While conceptually interesting, the above model has some serious shortcomings which may be recognized by examining its underlying assumptions: (i) the audience effectively reached in each channel is a linear function of the number of insertions. This assumption was common to early linear programming formulations of the media planning problem but has since been dealt with by using piecewise approximations for cumulative response, (ii) the total audience effectively reached is the sum of the audiences reached by the various media. This is probably the most serious shortcoming of the medel. The issue of audience overlap has received considerable attention in recent years (see e.g. [ 1,6, 19,21]) and most currently employed models for media planning distinguish between gross audience obtained, net audience or 'reach', and frequency considerations (see, e.g. [6]). Thus, to be useful, meanvariance models for media planning should take these considerations into account. (iii) the different audiences are uncorrelated. While this assumption is theoretically easily relaxed by recognizing a co-variance structure 02, we are then f~tced with the complicated problem of its estimation in practice, lIn a number of instances (Television, for example), this measurement may be practically impossible. (iv) Finally, along with most linear programming models for media planning, the model allows fractional answers.
2. More realistic mean-variance formulations
2.1. Gross audience, net audience and frequency
To develop explicit expressions for the mean (i.e. expected) gross kth audience segment obtained and its variance, consider the following binomial argument 1,2. Let p~(t) denote the probability that a particular individual in audience segment k is exposed to the ]th cumulative insertion in medium i in period t. The probability that an individual has not been exposed at least once through medium i following/ cumulative purchases in that medium is then given by q~(t) = ( i - p~l(t))" (1 - p~2(t)) ..... (1 - p~(t)) (5)
Hence, the distribution ofd/~(t), the gross kth audience segment obtained through medium i given ] cumulative purchases in that medium in period t is binomial, viz.
(6) Here, D~(t) represents the size of the total audience
I The author is indebted to Dr. D.C. McNickleof the U,aiversity of Canterbury for suggestingthisapproach and his assistance in detailing the argument. 2 A similar argument is used by Ellis [ 10].
C~4, d e Kluyver / Media selection by mean.varianceanalysis for m e d i u m / i n audience segment k in period t. 3 The mean and variance of (6), respectively, are given by
E~i (t) = E(d~i(t)) = D~(t) " (1 - q~i(t))
(7)
and V~(t) =
V(d~i(t))=D?(t). (1 -q~i(t)). q~(t).
(8)
Observe that as ] approaches.**, E~ (t) approaches the saturation level D~(t). The variance V~/(t)may initially increase depending on the magnitude of the impact probabilities p~i(t), but ultimately must decrease ,with limiting value zero. Also note that the definitions of E~(t) and V~(t) are conditional upon the numbelr of cumulative purchases/. To facilitate a representation as a mathematical programming problem, incremental (or, in the case of the variance, decremental) audience figures are desirable. Accordingly, by letting
Z~'~](t) = E~](t) - E~/_l (t) =D~ft).
p~.(t), q~,i_,(t)
(9)
and
V~(t)- Vik,i_,(t) =D~ (t)" { (q~,]_l(t)) 2 " (1 - (1 - p~(t)) 2) _ qk/_lq), pkq)} (10)
AVe(t) =
the expected gross kth audience segment, obtained through cumulative purchases in medium i in period t may be written as 4 E/C(t) = ~ z~7~(t), i
xti(t),
i
AVi~(t)" xq(t),
~xii(t)< l , i
xo(t)>~O.
(13)
Assuming independence with~a~ a given time period of the random variables d~(t)for the various media audiences, the expected gross kth audience segment, obtained by cumulative purchases in all media in period t may be written as 5
xo(t),
04)
AVe(t)" xo(t ) .
(15)
ek(t) - Z) Z: i
/
with variance Vk(t) - ~ ~ t /
The independence assumption employed here may be questioned. Ellis [ 10] defends its use on two principal grounds: computational simplicity and the lack of accurate information needed to estimate a covariance structure. Additionally, it should be noted that in the present context; i.e. in a mean-variance type analysis, the resulting bias will be towards more conservative decisions since Vk(t) as defined by (15) will overstate its true value. 6 Next, consider the expression for the net kth audience segment or 'reach'. Using the previously derived media probabilities q~i(t), i = 1, 2, ..., L the probability that an individual has not been exposed at least once by any medium in period t given that /(i) cumulative 'purchases have been made in medium i, is given by
(11)
Q~(t)(t) = H q~.(t)
(16)
i
with variance V~(t) = ~
approximated by requiring (see e.g. [6])
(12)
where xij(t) = 0 or 1, denoting the ]th cumulative purchase in medium i for period t. Generally, this is
3 As noted in [6], d~(i) might refer to the number of women aged 25 to 34 (k), obtained via a purchase of/issues in Time (f). D~(t) denotes the total expected audience of the medium (i) in the relevant audience segment (k). Section 3 discusses a model of audience change from period to period for this random variable. 4 Observe that A Vi~(t) is not the variance of the/th insertion in medium i directed at segment k. Rather, it represents the decrease (or, increase) in variance resulting from an additional insertiGn in channel i for audience segment k in period t.
Hence the distribution of d~i ) (t), the net kth audience segment, obtained by ](i) cumulative purchases in media i, i = 1, 2, ..., I in period t is again binomial, viz. Pr(d~(~-)(t) = d )
5 Note that independence over time is not assumed. A model of audience change over time is discussed in Section 3 of this article. Also observe that the derivation by Charnes et al. [6] does not require an independence assumption. This issue was resolved when the material in [6] was reprinted in [201. 6 Unlike in the Markowitz model, constraints on expected audiences sizes may be used to compensate for possible overest~nates in Vk(t).
C.Ao de Kluyver / Media selection by mean-variance analysis
with expected value • (1 -
(lS)
and variance P
--Q~o(t)).
Q~i)q).
(19)
Here, D ~ represents the union of the media total audiences D~(t), i = 1, 2, ...,L Again it is noted that the statistics E~i)(t) and V~i)(t) are conditioned on the number of cumulative purchases in each medium. The incremental representation used for gross audience statistics is not applicable in this case. However, concentrating on the expected net kth audience segment, E~0(t), note that (1
- FJ~i)(t)/Dk(t)) = Q~oq) = I] q~.(t) i
= H
(20)
from which it follows that
E~i)q)/Dkq))
= ~]) ~ Ln(l i l
Fk(t) = ~ ~ i i
p~(t), xq(t).
(23)
The distribution underlying (23) is a generalised binomial distribution in which the probability of 'success' is different for each 'trial' (cumulative purchase). (See [ 11 ].) The resulting formulation of a single-period meanvariance model for media planning thus seeks to mirimize the variance of all gross audience segments exposed, viz.
xq(t ~
(24)
H
= I'I l-[ (1-p~.(t)) xq(O , i /
-
(22)
from which an expression for average frequency may be derived as
Min ~ Z~ ~ AVe(t)k i ]
i iq)
Ln(l
probabilities p~.(t), it may be shown that the expected number of exposures given/cumulative purchases in medium i in period t (per individual) is given by
P~l(t)+ p~2(t)+ + p~(t),
i)(t) =
=Dkq) • (1
115
p~(t)), xq(t).
subject to a variety of constraints on the expected magnitude of gross audience segments, net audience segments, and average frequency realized. For instance, requiring that the gross kth audience segment be at least Mk(t) is written as
~ AEi~(e)" xq(t) >IMk(t). i i (21)
Observe that (21) is identical to the net audience representation derived by Charles et al. [6] using the idea of 'non-reach'. As a result, the above derivation provides a probabflistic justification for that net audience expression. The derivation of a variance expression for the net kth audience segment involving zero-one variables is mathematically less tractable. As a consequence, the mean-variance formulations proposed here concentrate on minimizing the variance of gross audience segments obtained, subject to constraints on the size of both gross and net audience segments exposed. Other constraints may include considerations of frequency. If average frequency is deemed adequate 7, the binomial argument employed throughout this development may again be invoked. Using the impact
7 Alternatively, constraints modeling an entire distribution o,"frequencies may be used. See Charnes et al. [6].
(25)
Similarly, for the kth net audience segment we might require
E~o >~Nk(t),
(26)
which would be represented as i
~ Ln( l - p~.(t)), ]
xii(t)
~>Ln(1-Nkq)/Dkq)).
(27)
Finally, average frequency controls may be of the form:
F_*(*)<~Yk(t)
(28)
Or
F__kq)<,~ ~ p~lq), xi]q)<~ffgq). i
l
(29)
Before discussing possible extensions toward multiperiod formulations, attention is drawn to the fact that the objective function (24) is linear in the decision variables, in contrast to the model due to Morard
C.A. de Kluyver / Media selection by mean.vorianceanalysis
116
et al, [22] discussed in the previous section. The fact that linear as oppcs~d to quadratic programming routines can be used with the formulations proposed here may well be a significant factor in their practical application.
2.2. Extension to multi-period mean-variance formulations The extension of (24)-(20) to a multi-period model merits further attention. This section considers the additional assumptions needed to produce a computationally convenient multi-period equivalent model, and, in the process, postulates a relationship for expected total media audiences, D~(O, between successive time periods. Starting with the latter, it seems reasonable to attribute changes in total media audiences (i.e. circulation or viewing statistics) both to random and nonrandom influences. Non-random fluctuations may be due to factors such as seasonality, conscious efforts of audience build-up, etc., whereas random disturbances reflect the uncertainty inherent to a competitive media climate. Accordingly, the following simple model of audience change is postulated:
b (t) =
3 '(t - 1) +
(30)
Here,/)~(t) denotes the random variable modeling total media audience, r~(t) represents an index of non-random audience fluctuation, and el(t) random error. Assuming that ~he error terms all are independent, identically distributed random variables with mean 0 and variance o 2 leads to the following relation between the expected total audiences
D~(t)=r~(t). D~(t- 1) t
=(1-1 r['(s)).
(30
S= t
from which estimates for AE/~(t) and AVe(t) for t = 1,2, ..., Tmay be determined using (9) and (10) respectively, and hence, a constraint structure for multiple periods similar to (24)-(20) specified. If, in a multi-period model, it is desirable to minimize the sum of the variances of gross audiences olbtained, exp:~ession (24) may simply be summed for each period t. if, on the other hand, the expressed objective is to minimize the variance of the sum of gross audiences obtained, recognition of a possible co-variance structure over time must be considered. Observe that the audience fluctuation model postu-
lated above merely shifts the binomial distribution (6) over time. Therefore, zero co-variance would be implied by an assumption of in0ependence of the deviations of true total audiences realized from their expected values D~(t) over time. The latter seems plausible if the probabilities {p~/(t)} remain fairly constant over time, i.e. when advertising effectiveness may be assumed constant over the planning period. This assumption is not unreasonable if the same set of agencies is employed over the entire planning horizon. In summary, a raulti-period equivalent model to (24)-(29) seeks to s Min ~ t
~ k
~ i
~ AV~/(t)" xi/(t)
(32)
/
subject to constraints on gross audience, net audience and frequency in each and across periods, where AVe(t) and AE~(t), t = 1, 2, ..., Tmay be estimated using the appropriate expressions given above.
3. Assessment of the mean-variance formulation: Conclusions
The media planning model developed in this article embodies two principal objectives: (i) a desire to obtain relattvely high values for expected gross and net audience segments and frequency, and (ii) a desire for this 'return' to be dependable, i.e. not subject to large amounts of uncertai:nty. As such, it recognizes that the media schedule with the highest: expected 'return' is not necessarily the one with the least 'uncertainty of return'. The most reliable schedule with an extremely high return may be subject to an unacceptably high degree of uncertainty, and.., conversely, the schedule with the least uncertainty may be characterized by an undesirably low re.'turn. By tracing the entire efficiency frontier of solut!~ons using parametric linear programruing techniques, the proposed mean-variance model allows a trade-off among these quantities of interest. If cost aspects of media schedules are modeled as e.g. in the LP I or LP II models due to Charnes et al. [6,7], the media planner has the option of balancing 8 Of course, the model formulation seeks allocations that are fixed over time. An alternative specification might include chance constrained programmingfeatures allowing ~:me-dependent ~onditional allocations.
CA. de Kluyver / Media selection by mean.variance analysis a schedule's return (i.e. expected audience), its variance, and the cost of the proposed schedule in arriving at final proposals. A convenient method to perform such comparisons consists of constructing both riskreturn and cost-return curves for the problem at hand. A final word concerns the use of the variance of a media schedule as a measure of risk. Critics may argue that the use of a symmetric measure such as the variance is not very descriptive since over-exposing a target audience is probably less undesirable to a media planner than under-exposing that same audience. Markowitz [18] recognized this problem in the context of portfolio selection and suggests the use of alternative measures such as the semi-variance to prevent 'over-achievements' from being penalized. In both instances, the usual assumptions of underlying normality and quadratic utility apply. However, in the case of the semi-variance, the conservative quadratic portion of the utility function corresponding to 'over-achievements' is replaced by a linear segment. While this conceptual advantage seems highly attractive, it may be outweighted when cost and convenience of computation are considered, particularly noting that an entire set of efficient media plans must be computed if full advantage is to be derived from the mean-variance approach.
Acknowledgements The author is indebted to Prof. W.W. Cooper of Harvard University for his constructive criticisms to an earlier draft of this article. His comm¢:nts and suggestions have allowed substantial refinements and improvements in the model.
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