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Advertising channel selection in a segmented market
Automatica 42 (2006) 1343 – 1347 www.elsevier.com/locate/automatica
Advertising channel selection in a segmented market夡 A. Buratto, L. Grosset, B. Viscolani ∗ Department of Pure and Applied Mathematics, Via Belzoni 7, I-35131 Padova, Italy Available online 15 May 2006
Abstract We consider a market with a finite number of segments and assume that several advertising channels are available, with different diffusion spectra and efficiencies. The problem of the choice of an advertising channel to direct the pre-launch campaign for a new product is analyzed in two steps. First, an optimal control problem is solved explicitly in order to determine the optimal advertising policy for each channel. Then a maximum profit channel is chosen. In a simulation example we consider the choice of a newspaper among six available and analyze the relations among the firm target market and the advertising channels environment which induce the optimal decision. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 49N90; 90B60 Keywords: Optimal control; Marketing; Advertising; Market segmentation
1. Introduction We consider the communication aspects which are related to the introduction of a new product. We assume that the consumer population can be partitioned into distinct and homogeneous groups, so that the members of one group exhibit the same needs and behaviors, whereas the members of different groups can be distinguished with respect of their needs and behaviors. Such a partitioning is known as market segmentation (Kotler, Armstrong, Saunders, & Wong, 1999, p. 386) and different instances of it can be achieved by considering different attributes to describe the consumers attitudes. When planning the product introduction campaign, one of the first phases consists in choosing the way of segmenting the market and defining the target market. Several methods can be used to tackle the segmentation process: they rely on multivariate statistical techniques, such as cluster analysis (see e.g. Arabie & Hubert, 2000), or on qualitative approaches, as the geographic, demographic, and behavioral segmentations 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ashutosh Prasad under the direction of Editor Suresh Sethi. Supported by MIUR-Italy and University of Padova. ∗ Corresponding author. Tel.: +39 049 827 5897; fax: +39 049 827 5892. E-mail addresses: [email protected] (A. Buratto), [email protected] (L. Grosset), [email protected] (B. Viscolani).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.03.015
(see e.g. Kotler et al., 1999, p. 386). For a general reference on market segmentation see Wedel and Kamakura (2000). We assume that the firm has already chosen a partition of the market into segments. Hence we do not care about this phase and we focus on the media selection problem, i.e. the problem of choosing among the available advertising channels the best for the firm goals. Only a few papers deal with dynamic advertising models for segmented markets (see e.g. Grosset & Viscolani, 2005; Little & Lodish, 1969; Seidmann, Sethi, & Derzko, 1987; Tragler, 2000). In particular, in Little and Lodish (1969) a discrete time stochastic model of multiple media selection in a segmented market has been studied. The main features of it are discussed in Lilien, Kotler, and Moorthy (1992, p. 313) in a deterministic framework. In Buratto, Grosset, and Viscolani (to appear), we find a formal approach to represent a segmented market, precisely in the context of advertising for the introduction of a new product. The deterministic model discussed in it exploits the same basic ideas as used in Little and Lodish (1969), but excludes the possibility of the simultaneous use of different media. The dynamics of the system is modelled in the linear framework of Nerlove and Arrow (1962), (see e.g. Sethi & Thompson, 2000, p. 186), which is one of the fundamental ideas of the optimal control models in advertising (Feichtinger, Hartl, & Sethi, 1994; Sethi, 1977). According to Nerlove and Arrow, the effect of advertising on sales is mediated by the goodwill variable.
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A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
Their simple model captures the goodwill features of spontaneous decay over time as well as of the positive and lasting response to advertising investment. It is still a basic element of a variety of advertising models which are used both in empirical studies, through its discrete time version which is known from Little (1979) and Zielske and Henry (1980) (see e.g. Mariel and Sandonís, 2004), and in theoretical ones, as documented in JZrgensen and Zaccour (2004, Section 3.5) and Buratto, Grosset, and Viscolani (2005). In Section 2, we describe the effects of advertising on the goodwill vector variable and explain how the firm revenue is related to the final goodwill value. In Section 3, we formulate and solve the optimal control problem of determining the advertising expenditure that maximizes the firm profit for any chosen channel. In Section 4, we deal with the channel selection problem, i.e. the choice of the advertising channel, among the available ones, that maximizes the firm profit. In Section 5, we apply the model to the case of the selection of a newspaper, taking the diffusion and cost data from the “Accertamenti Diffusione Stampa (ADS)”, an agency which certificates the Italian press data.1 We consider six Italian newspapers and evaluate which one is the most convenient for some advertising campaigns with different and suitably defined target markets. 2. Segmented goodwill evolution Let the consumer population be partitioned into segments a , a ∈ A, where A is the finite set of segment labels. A firm wants to advertise a new product which may encounter different attention levels among the consumer segments and it may use one advertising channel. Let T > 0 be the finite product introduction time, which either has been already fixed by the firm, or is exogenously given. We consider only the advertising activity of the firm in the time interval [0, T ] which precedes the introduction time T. This is exactly the situation which occurs when an organization plans the advertising campaign for an event, e.g. a soccer match or a concert: the advertising campaign takes place in the programming interval [0, T ], whereas the revenue from the tickets sales occurs only at time T, or in a narrow neighborhood of it. On the other hand, when the sale period cannot be approximated by the time T, the advertising activity after T might be planned separately, assuming as given the goodwill level at time T and taking into account other factors, such as word of mouth. We consider the goodwill stock, in the Nerlove–Arrow’s (1962) sense (Sethi and Thompson, 2000, p. 186), as a variable that summarizes the effects of past and current advertising flow on the demand. Let G(t, a) be the goodwill of the product at time t ∈ [0, T ] for the consumers in the segment a , a ∈ A. The goodwill evolution is driven by the channel activation intensity u(t) (the control function) according to ˙ a) = (a)(u(t)) − (a)G(t, a), G(t, G(0, a) = (a)0, 1 http://www.adsnotizie.it/.
a ∈ A,
(1) (2)
where (a) > 0 represents the goodwill depreciation rate for the members of the consumer group a and (a) is the goodwill level at the initial time. The assumption that (a) is not constant is consistent with the fact that consumers belonging to different segments have inhomogeneous behaviors. For example, in an age-segmented market, older consumers segments may have a reduced memory as compared with younger ones, possibly because of cognitive abilities decline (see Law et al., 1998). The possibility that the goodwill depreciation rate changes across the different segments was suggested also in Little and Lodish (1969). Each advertising channel is characterized by a pair of functions ((a), (u)). The function (a) 0 is called channel spectrum and it is a function such that a∈A (a) = 1. We call channel segment any segment a , a ∈ A, such that (a) > 0. We may say that (a) represents the conditional probability that the advertising message reaches one member of segment a , given that one member of some segment has been reached. The function (u)0 represents the effective advertising level of the channel. Different instances of the non-linear function (u) provide different estimates of the advertising message efficiency. The need for the function (u) in the model is due to the fact that the effects of different advertising intensities on the goodwill variable may vary with the advertising channel. This may occur also for channels with the same spectrum. We assume that ∈ C 2 , (u)0, limu→0 (u) = +∞, limu→+∞ (u) = 0 and (u) < 0, according to the empirical assumption that the goodwill, considered as consumer awareness, increases with the advertising intensity and with a decreasing rate (Kotler, 1997, p. 649). Later on we use the function √ (3) (u) = s u, where the parameter s > 0 is called the strength of the channel and may be represented, for example, by the diffusion of a magazine/newspaper or by the audience of a TV programme. The function (u) above is an element of a family of functions considered in Buratto and Viscolani (2002) for the same purpose. 3. Single channel advertising problem The segment-dependent product introduction problem requires to find a channel activation intensity function u(t) 0, in order to maximize the firm profit given by the functional T J =− pu(t) dt + r (a)G(T , a), (4) 0
a∈A
where p is the price of the channel activation intensity and r(a) is the marginal revenue of the final goodwill, from the customers of the segment a . In particular, r(a) is the revenue from the customers of the segment a when 1 is their goodwill level at time T. Roughly speaking, we may say that r(a) represents how positively an individual of segment a responds by expressing a demand for the product, when 1 is the goodwill level for his segment. In fact the above assumption on the linear
A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
dependence of the revenue on the final goodwill level G(T , a) is restrictive. Nevertheless, it is acceptable as long as the consumers are allowed to buy the product in a small neighborhood of the final time only. This is in agreement with the analysis of the advertising effects in a new product introduction campaign done in Buratto and Viscolani (2002), Buratto et al. (to appear). After integrating the motion equation (1)–(2), and substituting the final goodwill level G(T , a) into (4), we obtain that the objective functional takes the following form: J =r (a)(a)e−(a)T a∈A T − [pu(t) − rP (T − t; , , )(u(t))] dt, (5) 0
where P (; , , ) =
a∈A
(a)(a)e−(a) 0.
(6)
We call P (; , , ) the -delayed mean target coverage of the advertising channel. Either using the Pontryagin’s Maximum Principle (Sethi and Thompson, 2000, p. 33) or exploiting the monotonicity of the integral, we obtain the optimal control (channel activation intensity) function: u∗ (t) = (p/(rP (T − t; , , ))) = p r (a)(a)e−(a)(T −t) , a∈A
(7)
where (·) is the inverse function of the derivative (·). 4. Channel selection problem The results of the previous section can be profitably used when the firm has n channels available and needs to choose one of them to direct the pre-launch campaign for a new product. In order to obtain a closed-form solution, in the following we use the square root form (3) for the effective advertising level function. The firm will first determine the optimal channel activation intensity under the assumption that the ith channel has been chosen, for all i. This is the problem of maximizing T Ji = − pi ui (t) dt + r (a)G(T , a) (8) 0
a∈A
subject to
˙ a) = si i (a) ui (t) − (a)G(t, a), G(t,
a ∈ A,
(9)
where pi , si and i are the ith channel activation price, strength and spectrum. After denoting by Ji∗ the maximum value of the objective functional Ji in the above ith problem, the firm will choose the channel i ∗ for which Ji∗ is maximum: Ji∗∗ = max Ji∗ . i
(10)
If the ith channel is the optimal choice, then the firm activates it optimally, in view of (7), with the channel activation intensity 2 1 u∗i (t) = 2 rs i (a)i (a)e−(a)(T −t) . (11) a∈A 4pi
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We observe that the optimal expenditure is a positive and monotonically strictly increasing function, unless the set of the optimal channel segments is disjoint from the firm target market, so that a∈A (a)i (a) = 0. In this degenerate case the advertising channel would be useless for the firm objective. We observe that the firm, while using optimally the ith channel (i.e. with the channel activation intensity (11)), attains to the profit value Ji∗ = r (a)(a)e−(a)T a∈A 2 r 2 si2 T + (a)i (a)e−(a)(T −t) dt. (12) a∈A 4pi 0 From the special form of Ji∗ we obtain that solving the channel selection problem is equivalent to finding the maximum of the finite set of the positive numbers si i = √ P (·; , i , ), i = 1, 2, . . . , n, (13) pi where P (·; , i , ) is the L2 -norm of the delayed mean target coverage of the channel. We observe that 1/pi is the channel activation intensity value of money: the channel activation intensity bought by one money unit. The positive numbers i , i = 1, 2, . . . , n, are the values of a channel preference index, which is the product of the strength si , the square root of the √ advertising intensity value of money 1/ pi , and the norm of the delayed mean target coverage of the channel. In the unlikely case that the maximum value of the index is reached at two advertising channels or more, the firm may choose any one of the first classified channels. We remark that we have excluded the possibility of activating more than one advertising channel, in our problem definition. 4.1. Uniform goodwill decay Let us consider the special case of a uniform goodwill decay ¯ This is a rather reasonable asamong segments, i.e. (a) = . sumption for a geographical segmentation, as that considered in the next section. Here, the market segmentation still concerns the representation of revenue, through the function (·), and of advertising, through the function (·). The norm of the delayed mean target coverage has the product form ¯ e−T ¯ )· P (·; , , ) = (a)i (a) (14) sinh(T a∈A ¯ and the channel preference index i may be substituted equivalently by the simpler index si U (a)i (a), i = 1, 2, . . . , n, (15) i = √ a∈A pi ¯ In fact which does not depend on . ¯ ¯ )/, ¯ i = U e−T sinh(T i
(16)
so that the optimal advertising channel is associated with the maximum U i .
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A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
5. A newspaper selection case
Table 3 Advertising prices (page costs, ×1000 Euro)
We assume that the firm has planned an advertising campaign for the introduction of a new product with a given advertising message. The firm has to choose where to print the advertisement and must select it among six Italian newspapers. In order to present a sufficiently wide class of situations, we have chosen four newspapers with spectra that cover all the Italian regions well, namely Corriere della Sera (Co), Repubblica (Re), Stampa (St), Unità (Un), and two others which are essentially diffused in particular regions, namely Mattino (Ma) and Messaggero (Me). We consider the geographical segmentation given by the 20 Italian regions and use the newspapers diffusion data from the ADS agency. Table 1 displays the average daily numbers of sold copies and Table 2 displays the regional diffusion shares, i.e. the percentages of the newspapers diffusion which reach each segment. In this situation the segmentation variable is “Italian Region”, A = {Piemonte, V. Aosta, Lombardia, Trentino A.A., Veneto, Friuli V.G., Liguria, Emilia R., Toscana, Umbria, Marche, Lazio, Abruzzi, Molise, Campania, Puglia, Basilicata, Calabria, Sicilia, Sardegna}, while the advertising channels are the six newspapers mentioned above.
Newspaper (i)
pi
Co Ma Me Re St Un
39.7 18.0 25.0 35.0 28.0 15.0
Table 1 Daily diffusion (sold copies, year 2003) Newspaper (i)
si
Co Ma Me Re St Un
620,859 93,267 250,709 586,938 362,418 68,484
Table 2 Regional diffusion (%, year 2003), spectrum Region (a)
Co
Ma
Me
Re
St
Un
Piemonte V. Aosta Lombard. Trent. Veneto Friuli Liguria Emilia Toscana Umbria Marche Lazio Abruzzi Molise Campan. Puglia Basilic. Calabria Sicilia Sardegna
3.42 0.15 44.08 1.47 7.34 1.86 2.92 5.54 4.14 0.77 1.86 10.99 1.08 0.13 4.22 2.95 0.49 1.26 2.90 2.10
0.06 0.00 0.29 0.01 0.03 0.00 0.03 0.07 0.17 0.05 0.10 2.85 0.57 0.11 91.04 0.48 1.49 2.27 0.13 0.18
0.19 0.02 1.49 0.20 0.29 0.08 0.13 0.42 0.80 3.80 7.10 66.61 4.19 0.72 0.51 7.68 0.14 4.73 0.35 0.54
4.98 0.19 15.78 2.08 5.91 1.91 4.25 10.39 10.13 1.04 1.75 19.32 1.41 0.19 7.50 3.80 0.49 1.50 4.68 2.31
60.10 2.01 5.48 0.28 0.88 0.24 9.04 6.78 1.17 0.13 0.25 3.49 0.16 0.02 3.31 0.25 0.06 0.22 0.41 0.47
5.42 0.11 17.81 0.94 4.60 1.79 4.25 23.62 11.77 1.60 2.92 13.45 1.39 0.15 2.66 1.82 0.36 1.37 1.91 2.06
Table 4 Target markets (%) a
1 (a)
2 (a)
3 (a)
4 (a)
Piemonte V. Aosta Lombard. Trentino Veneto Friuli Liguria Emilia Toscana Umbria Marche Lazio Abruzzi Molise Campan. Puglia Basilic. Calabria Sicilia Sardegna
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0 0 0 10 15 10 10 15 10 10 10 10 0
12 10 14 10 14 5 3 5 6 3 5 5 2 2 2 0 0 2 0 0
4.98 0.19 15.78 2.08 5.91 1.91 4.25 10.39 10.13 1.04 1.75 19.32 1.41 0.19 7.50 3.80 0.49 1.50 4.68 2.31
As far as the advertising costs are concerned, we refer to the black&white page pricing-list, which provides the average price per page. The relevant data are presented in Table 3. We consider a few scenarios which represent meaningful firm target markets (see Table 4) and find the scenario-dependent optimal newspapers, along with the orderings of the six newspapers. The analysis is carried out under the uniform goodwill decay assumption, which is reasonable under the present geographical segmentation. Therefore, the goodwill depreciation rate is irrelevant for the channel selection problem and we do not need to specify it: we are concerned with the channel preference index U i . The first column of Table 4 represents the uniform target market, namely a target market for which the segmentation is not meaningful. For this scenario the newspaper selection decision only depends on the newspapers diffusions and prices. In particular, the newspaper (channel) preference index is propor√ tional to the ratio si / pi and we obtain the following newspapers ordering: Re Co Ma St Un Me. In the second column, a non-uniform firm target market is shown; it represents the typical target market of an event (e.g. a soccer match) which takes place in the South of Italy. It leads
A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
to the following preferences: Re Me Co Ma St Un. We can observe that Messaggero (Me) has a better rank than in the previous scenario, as it is the most diffused in the South of Italy. The target market 3 is typical of some ski equipment and the preferences related to such a target market are Co St Re Me Un Ma. The last scenario is defined by a target market, given by 4 , which is distributed over all the regions (segments). Differently from the previous scenarios, here the distribution represented by the target spectrum 4 is not uniform, nor is specially concentrated in the northern or southern regions. In such a case it is not apparent which the optimal newspaper (channel) must be. However, we can solve the problem and the preferences induced by the index (15) are Co Re Me St Un Ma. We observe that in some cases the choice between two advertising channels is apparent on inspection of their spectra and the target one: see e.g. “Me St” in the second scenario (target 2 ), or “St Ma” in the third scenario (target 3 ). On the other hand, there are cases in which such easy comparisons cannot be made: see e.g. “Re Me” with targets 2 or 3 , or “Co Me” with target 4 . In all cases the channel preference index is an easy and useful tool to this purpose. The practical application above shows how the previous results can be used with a set of real data. This rather simple model can be implemented in a spreadsheet straightforwardly to give a useful support to a decision maker. Acknowledgements We would like to thank Antonio Boscardin for contributing to the development of the paper with his M.Sc. thesis and also the anonymous referee and the editor Suresh Sethi for valuable comments and suggestions. References Arabie, P., & Hubert, L. (2000). Cluster analysis in marketing research. In R.P. Bagozzi (Ed.), Advanced methods of marketing research (pp. 160–189). Oxford: Basil, Blackwell. Buratto, A., Grosset, L., & Viscolani, B. Advertising a new product in a segmented market. European Journal of Operational Research, to appear. Buratto, A., Grosset, L., & Viscolani, B. (2005). Linear models and advertising. Rendiconti per gli Studi Economici Quantitativi, unico, 107–120. Buratto, A., & Viscolani, B. (2002). New product introduction: Goodwill, time and advertising cost. Mathematical Methods of Operations Research, 55, 55–68. Feichtinger, G., Hartl, R. F., & Sethi, S. P. (1994). Dynamic optimal control models in advertising: Recent developments. Management Science, 40, 195–226. Grosset, L., & Viscolani, B. (2005). Advertising for the introduction of an age-sensitive product. Optimal Control Applications and Methods, 26, 157–167.
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JZrgensen, S., & Zaccour, G. (2004). Differential games in marketing. Boston: Kluwer Academic Publishers. Kotler, P. (1997). Marketing management: Analysis, planning, implementation, and control. Upper Saddle River: Prentice-Hall. Kotler, P., Armstrong, G., Saunders, J., & Wong, V. (1999). Principles of marketing. Upper Saddle River: Prentice-Hall. Law, S., Hawkins, S. A., & Craik, F. I. M. (1998). Repetition-induced belief in the elderly: Rehabilitating age-related memory deficits. Journal of Consumer Research, 25, 91–107. Lilien, G. L., Kotler, P., & Moorthy, K. S. (1992). Marketing models. Englewood Cliffs: Prentice-Hall. Little, J. D. C. (1979). Aggregate advertising models: The state of the art. Operations Research, 27, 629–667. Little, J. D. C., & Lodish, L. M. (1969). A media planning calculus. Operations Research, 17, 1–35. Mariel, P., & Sandonís, J. (2004). A model of advertising with application to the German automobile industry. Applied Economics, 36, 83–92. Nerlove, M., & Arrow, K. J. (1962). Optimal advertising policy under dynamic conditions. Economica, 29, 129–142. Seidmann, T. I., Sethi, S. P., & Derzko, N. (1987). Dynamics and optimization of a distributed sales-advertising model. Journal of Optimization Theory and Applications, 52, 443–462. Sethi, S. P. (1977). Dynamic optimal control models in advertising: A survey. SIAM Review, 19(4), 685–725. Sethi, S. P., & Thompson, G. L. (2000). Optimal control theory: Applications to management science and economics. Boston: Kluwer Academic Publishers. Tragler, G. (2000). Optimal controls in spatial advertising diffusion models. In E.J. Dockner, R.F. Hartl, M. Luptacik, & G. Sorger (Eds.), Optimization, dynamics, and economic analysis (pp. 288–297). Heidelberg: PhysicaVerlag. Wedel, M., & Kamakura, W. A. (2000). Market segmentation. Boston: Kluwer Academic Publishers. Zielske, H. A., & Henry, W. A. (1980). Remembering and forgetting television ads. Journal of Advertising Research, 20, 7–13. Alessandra Buratto received an M.Sc. in Mathematics from the University of Padova and a Ph.D. in Mathematics for Economic Applications from the University of Trieste. She is a Research fellow in the Faculty of Economics of Padova University. Her main research area is optimal control applied to marketing. She teaches courses in Mathematics for Economic Decisions and Operations Research. Luca Grosset studied Mathematics at the University of Padova. He obtained his Ph.D. in Computational Mathematics from the same University in 2001. In 2003, he won a two-years fellowship in the Department of Pure and Applied Mathematics, University of Padova, to study some applications of the optimal control theory to marketing. His main research interests are the applications of the deterministic and stochastic optimal control theory to Economics and Management Science. Bruno Viscolani studied Mathematics at the University of Padova (1971–1976). He has been professor of Mathematics for Economics at the same University since 1995. Previously was research fellow in Padova (1980–1987) and professor at the University “Ca’ Foscari” of Venice (1987–1995). His research activity concerns mainly some applications of Optimal Control Theory and Mathematical Programming to Economics and Management Science, in particular to Marketing. He serves as an Associate Editor of the Journal of Information and Optimization Sciences.
Advertising channel selection in a segmented market夡 A. Buratto, L. Grosset, B. Viscolani ∗ Department of Pure and Applied Mathematics, Via Belzoni 7, I-35131 Padova, Italy Available online 15 May 2006
Abstract We consider a market with a finite number of segments and assume that several advertising channels are available, with different diffusion spectra and efficiencies. The problem of the choice of an advertising channel to direct the pre-launch campaign for a new product is analyzed in two steps. First, an optimal control problem is solved explicitly in order to determine the optimal advertising policy for each channel. Then a maximum profit channel is chosen. In a simulation example we consider the choice of a newspaper among six available and analyze the relations among the firm target market and the advertising channels environment which induce the optimal decision. 䉷 2006 Elsevier Ltd. All rights reserved. MSC: 49N90; 90B60 Keywords: Optimal control; Marketing; Advertising; Market segmentation
1. Introduction We consider the communication aspects which are related to the introduction of a new product. We assume that the consumer population can be partitioned into distinct and homogeneous groups, so that the members of one group exhibit the same needs and behaviors, whereas the members of different groups can be distinguished with respect of their needs and behaviors. Such a partitioning is known as market segmentation (Kotler, Armstrong, Saunders, & Wong, 1999, p. 386) and different instances of it can be achieved by considering different attributes to describe the consumers attitudes. When planning the product introduction campaign, one of the first phases consists in choosing the way of segmenting the market and defining the target market. Several methods can be used to tackle the segmentation process: they rely on multivariate statistical techniques, such as cluster analysis (see e.g. Arabie & Hubert, 2000), or on qualitative approaches, as the geographic, demographic, and behavioral segmentations 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ashutosh Prasad under the direction of Editor Suresh Sethi. Supported by MIUR-Italy and University of Padova. ∗ Corresponding author. Tel.: +39 049 827 5897; fax: +39 049 827 5892. E-mail addresses: [email protected] (A. Buratto), [email protected] (L. Grosset), [email protected] (B. Viscolani).
0005-1098/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.03.015
(see e.g. Kotler et al., 1999, p. 386). For a general reference on market segmentation see Wedel and Kamakura (2000). We assume that the firm has already chosen a partition of the market into segments. Hence we do not care about this phase and we focus on the media selection problem, i.e. the problem of choosing among the available advertising channels the best for the firm goals. Only a few papers deal with dynamic advertising models for segmented markets (see e.g. Grosset & Viscolani, 2005; Little & Lodish, 1969; Seidmann, Sethi, & Derzko, 1987; Tragler, 2000). In particular, in Little and Lodish (1969) a discrete time stochastic model of multiple media selection in a segmented market has been studied. The main features of it are discussed in Lilien, Kotler, and Moorthy (1992, p. 313) in a deterministic framework. In Buratto, Grosset, and Viscolani (to appear), we find a formal approach to represent a segmented market, precisely in the context of advertising for the introduction of a new product. The deterministic model discussed in it exploits the same basic ideas as used in Little and Lodish (1969), but excludes the possibility of the simultaneous use of different media. The dynamics of the system is modelled in the linear framework of Nerlove and Arrow (1962), (see e.g. Sethi & Thompson, 2000, p. 186), which is one of the fundamental ideas of the optimal control models in advertising (Feichtinger, Hartl, & Sethi, 1994; Sethi, 1977). According to Nerlove and Arrow, the effect of advertising on sales is mediated by the goodwill variable.
1344
A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
Their simple model captures the goodwill features of spontaneous decay over time as well as of the positive and lasting response to advertising investment. It is still a basic element of a variety of advertising models which are used both in empirical studies, through its discrete time version which is known from Little (1979) and Zielske and Henry (1980) (see e.g. Mariel and Sandonís, 2004), and in theoretical ones, as documented in JZrgensen and Zaccour (2004, Section 3.5) and Buratto, Grosset, and Viscolani (2005). In Section 2, we describe the effects of advertising on the goodwill vector variable and explain how the firm revenue is related to the final goodwill value. In Section 3, we formulate and solve the optimal control problem of determining the advertising expenditure that maximizes the firm profit for any chosen channel. In Section 4, we deal with the channel selection problem, i.e. the choice of the advertising channel, among the available ones, that maximizes the firm profit. In Section 5, we apply the model to the case of the selection of a newspaper, taking the diffusion and cost data from the “Accertamenti Diffusione Stampa (ADS)”, an agency which certificates the Italian press data.1 We consider six Italian newspapers and evaluate which one is the most convenient for some advertising campaigns with different and suitably defined target markets. 2. Segmented goodwill evolution Let the consumer population be partitioned into segments a , a ∈ A, where A is the finite set of segment labels. A firm wants to advertise a new product which may encounter different attention levels among the consumer segments and it may use one advertising channel. Let T > 0 be the finite product introduction time, which either has been already fixed by the firm, or is exogenously given. We consider only the advertising activity of the firm in the time interval [0, T ] which precedes the introduction time T. This is exactly the situation which occurs when an organization plans the advertising campaign for an event, e.g. a soccer match or a concert: the advertising campaign takes place in the programming interval [0, T ], whereas the revenue from the tickets sales occurs only at time T, or in a narrow neighborhood of it. On the other hand, when the sale period cannot be approximated by the time T, the advertising activity after T might be planned separately, assuming as given the goodwill level at time T and taking into account other factors, such as word of mouth. We consider the goodwill stock, in the Nerlove–Arrow’s (1962) sense (Sethi and Thompson, 2000, p. 186), as a variable that summarizes the effects of past and current advertising flow on the demand. Let G(t, a) be the goodwill of the product at time t ∈ [0, T ] for the consumers in the segment a , a ∈ A. The goodwill evolution is driven by the channel activation intensity u(t) (the control function) according to ˙ a) = (a)(u(t)) − (a)G(t, a), G(t, G(0, a) = (a)0, 1 http://www.adsnotizie.it/.
a ∈ A,
(1) (2)
where (a) > 0 represents the goodwill depreciation rate for the members of the consumer group a and (a) is the goodwill level at the initial time. The assumption that (a) is not constant is consistent with the fact that consumers belonging to different segments have inhomogeneous behaviors. For example, in an age-segmented market, older consumers segments may have a reduced memory as compared with younger ones, possibly because of cognitive abilities decline (see Law et al., 1998). The possibility that the goodwill depreciation rate changes across the different segments was suggested also in Little and Lodish (1969). Each advertising channel is characterized by a pair of functions ((a), (u)). The function (a) 0 is called channel spectrum and it is a function such that a∈A (a) = 1. We call channel segment any segment a , a ∈ A, such that (a) > 0. We may say that (a) represents the conditional probability that the advertising message reaches one member of segment a , given that one member of some segment has been reached. The function (u)0 represents the effective advertising level of the channel. Different instances of the non-linear function (u) provide different estimates of the advertising message efficiency. The need for the function (u) in the model is due to the fact that the effects of different advertising intensities on the goodwill variable may vary with the advertising channel. This may occur also for channels with the same spectrum. We assume that ∈ C 2 , (u)0, limu→0 (u) = +∞, limu→+∞ (u) = 0 and (u) < 0, according to the empirical assumption that the goodwill, considered as consumer awareness, increases with the advertising intensity and with a decreasing rate (Kotler, 1997, p. 649). Later on we use the function √ (3) (u) = s u, where the parameter s > 0 is called the strength of the channel and may be represented, for example, by the diffusion of a magazine/newspaper or by the audience of a TV programme. The function (u) above is an element of a family of functions considered in Buratto and Viscolani (2002) for the same purpose. 3. Single channel advertising problem The segment-dependent product introduction problem requires to find a channel activation intensity function u(t) 0, in order to maximize the firm profit given by the functional T J =− pu(t) dt + r (a)G(T , a), (4) 0
a∈A
where p is the price of the channel activation intensity and r(a) is the marginal revenue of the final goodwill, from the customers of the segment a . In particular, r(a) is the revenue from the customers of the segment a when 1 is their goodwill level at time T. Roughly speaking, we may say that r(a) represents how positively an individual of segment a responds by expressing a demand for the product, when 1 is the goodwill level for his segment. In fact the above assumption on the linear
A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
dependence of the revenue on the final goodwill level G(T , a) is restrictive. Nevertheless, it is acceptable as long as the consumers are allowed to buy the product in a small neighborhood of the final time only. This is in agreement with the analysis of the advertising effects in a new product introduction campaign done in Buratto and Viscolani (2002), Buratto et al. (to appear). After integrating the motion equation (1)–(2), and substituting the final goodwill level G(T , a) into (4), we obtain that the objective functional takes the following form: J =r (a)(a)e−(a)T a∈A T − [pu(t) − rP (T − t; , , )(u(t))] dt, (5) 0
where P (; , , ) =
a∈A
(a)(a)e−(a) 0.
(6)
We call P (; , , ) the -delayed mean target coverage of the advertising channel. Either using the Pontryagin’s Maximum Principle (Sethi and Thompson, 2000, p. 33) or exploiting the monotonicity of the integral, we obtain the optimal control (channel activation intensity) function: u∗ (t) = (p/(rP (T − t; , , ))) = p r (a)(a)e−(a)(T −t) , a∈A
(7)
where (·) is the inverse function of the derivative (·). 4. Channel selection problem The results of the previous section can be profitably used when the firm has n channels available and needs to choose one of them to direct the pre-launch campaign for a new product. In order to obtain a closed-form solution, in the following we use the square root form (3) for the effective advertising level function. The firm will first determine the optimal channel activation intensity under the assumption that the ith channel has been chosen, for all i. This is the problem of maximizing T Ji = − pi ui (t) dt + r (a)G(T , a) (8) 0
a∈A
subject to
˙ a) = si i (a) ui (t) − (a)G(t, a), G(t,
a ∈ A,
(9)
where pi , si and i are the ith channel activation price, strength and spectrum. After denoting by Ji∗ the maximum value of the objective functional Ji in the above ith problem, the firm will choose the channel i ∗ for which Ji∗ is maximum: Ji∗∗ = max Ji∗ . i
(10)
If the ith channel is the optimal choice, then the firm activates it optimally, in view of (7), with the channel activation intensity 2 1 u∗i (t) = 2 rs i (a)i (a)e−(a)(T −t) . (11) a∈A 4pi
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We observe that the optimal expenditure is a positive and monotonically strictly increasing function, unless the set of the optimal channel segments is disjoint from the firm target market, so that a∈A (a)i (a) = 0. In this degenerate case the advertising channel would be useless for the firm objective. We observe that the firm, while using optimally the ith channel (i.e. with the channel activation intensity (11)), attains to the profit value Ji∗ = r (a)(a)e−(a)T a∈A 2 r 2 si2 T + (a)i (a)e−(a)(T −t) dt. (12) a∈A 4pi 0 From the special form of Ji∗ we obtain that solving the channel selection problem is equivalent to finding the maximum of the finite set of the positive numbers si i = √ P (·; , i , ), i = 1, 2, . . . , n, (13) pi where P (·; , i , ) is the L2 -norm of the delayed mean target coverage of the channel. We observe that 1/pi is the channel activation intensity value of money: the channel activation intensity bought by one money unit. The positive numbers i , i = 1, 2, . . . , n, are the values of a channel preference index, which is the product of the strength si , the square root of the √ advertising intensity value of money 1/ pi , and the norm of the delayed mean target coverage of the channel. In the unlikely case that the maximum value of the index is reached at two advertising channels or more, the firm may choose any one of the first classified channels. We remark that we have excluded the possibility of activating more than one advertising channel, in our problem definition. 4.1. Uniform goodwill decay Let us consider the special case of a uniform goodwill decay ¯ This is a rather reasonable asamong segments, i.e. (a) = . sumption for a geographical segmentation, as that considered in the next section. Here, the market segmentation still concerns the representation of revenue, through the function (·), and of advertising, through the function (·). The norm of the delayed mean target coverage has the product form ¯ e−T ¯ )· P (·; , , ) = (a)i (a) (14) sinh(T a∈A ¯ and the channel preference index i may be substituted equivalently by the simpler index si U (a)i (a), i = 1, 2, . . . , n, (15) i = √ a∈A pi ¯ In fact which does not depend on . ¯ ¯ )/, ¯ i = U e−T sinh(T i
(16)
so that the optimal advertising channel is associated with the maximum U i .
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A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
5. A newspaper selection case
Table 3 Advertising prices (page costs, ×1000 Euro)
We assume that the firm has planned an advertising campaign for the introduction of a new product with a given advertising message. The firm has to choose where to print the advertisement and must select it among six Italian newspapers. In order to present a sufficiently wide class of situations, we have chosen four newspapers with spectra that cover all the Italian regions well, namely Corriere della Sera (Co), Repubblica (Re), Stampa (St), Unità (Un), and two others which are essentially diffused in particular regions, namely Mattino (Ma) and Messaggero (Me). We consider the geographical segmentation given by the 20 Italian regions and use the newspapers diffusion data from the ADS agency. Table 1 displays the average daily numbers of sold copies and Table 2 displays the regional diffusion shares, i.e. the percentages of the newspapers diffusion which reach each segment. In this situation the segmentation variable is “Italian Region”, A = {Piemonte, V. Aosta, Lombardia, Trentino A.A., Veneto, Friuli V.G., Liguria, Emilia R., Toscana, Umbria, Marche, Lazio, Abruzzi, Molise, Campania, Puglia, Basilicata, Calabria, Sicilia, Sardegna}, while the advertising channels are the six newspapers mentioned above.
Newspaper (i)
pi
Co Ma Me Re St Un
39.7 18.0 25.0 35.0 28.0 15.0
Table 1 Daily diffusion (sold copies, year 2003) Newspaper (i)
si
Co Ma Me Re St Un
620,859 93,267 250,709 586,938 362,418 68,484
Table 2 Regional diffusion (%, year 2003), spectrum Region (a)
Co
Ma
Me
Re
St
Un
Piemonte V. Aosta Lombard. Trent. Veneto Friuli Liguria Emilia Toscana Umbria Marche Lazio Abruzzi Molise Campan. Puglia Basilic. Calabria Sicilia Sardegna
3.42 0.15 44.08 1.47 7.34 1.86 2.92 5.54 4.14 0.77 1.86 10.99 1.08 0.13 4.22 2.95 0.49 1.26 2.90 2.10
0.06 0.00 0.29 0.01 0.03 0.00 0.03 0.07 0.17 0.05 0.10 2.85 0.57 0.11 91.04 0.48 1.49 2.27 0.13 0.18
0.19 0.02 1.49 0.20 0.29 0.08 0.13 0.42 0.80 3.80 7.10 66.61 4.19 0.72 0.51 7.68 0.14 4.73 0.35 0.54
4.98 0.19 15.78 2.08 5.91 1.91 4.25 10.39 10.13 1.04 1.75 19.32 1.41 0.19 7.50 3.80 0.49 1.50 4.68 2.31
60.10 2.01 5.48 0.28 0.88 0.24 9.04 6.78 1.17 0.13 0.25 3.49 0.16 0.02 3.31 0.25 0.06 0.22 0.41 0.47
5.42 0.11 17.81 0.94 4.60 1.79 4.25 23.62 11.77 1.60 2.92 13.45 1.39 0.15 2.66 1.82 0.36 1.37 1.91 2.06
Table 4 Target markets (%) a
1 (a)
2 (a)
3 (a)
4 (a)
Piemonte V. Aosta Lombard. Trentino Veneto Friuli Liguria Emilia Toscana Umbria Marche Lazio Abruzzi Molise Campan. Puglia Basilic. Calabria Sicilia Sardegna
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
0 0 0 0 0 0 0 0 0 0 10 15 10 10 15 10 10 10 10 0
12 10 14 10 14 5 3 5 6 3 5 5 2 2 2 0 0 2 0 0
4.98 0.19 15.78 2.08 5.91 1.91 4.25 10.39 10.13 1.04 1.75 19.32 1.41 0.19 7.50 3.80 0.49 1.50 4.68 2.31
As far as the advertising costs are concerned, we refer to the black&white page pricing-list, which provides the average price per page. The relevant data are presented in Table 3. We consider a few scenarios which represent meaningful firm target markets (see Table 4) and find the scenario-dependent optimal newspapers, along with the orderings of the six newspapers. The analysis is carried out under the uniform goodwill decay assumption, which is reasonable under the present geographical segmentation. Therefore, the goodwill depreciation rate is irrelevant for the channel selection problem and we do not need to specify it: we are concerned with the channel preference index U i . The first column of Table 4 represents the uniform target market, namely a target market for which the segmentation is not meaningful. For this scenario the newspaper selection decision only depends on the newspapers diffusions and prices. In particular, the newspaper (channel) preference index is propor√ tional to the ratio si / pi and we obtain the following newspapers ordering: Re Co Ma St Un Me. In the second column, a non-uniform firm target market is shown; it represents the typical target market of an event (e.g. a soccer match) which takes place in the South of Italy. It leads
A. Buratto et al. / Automatica 42 (2006) 1343 – 1347
to the following preferences: Re Me Co Ma St Un. We can observe that Messaggero (Me) has a better rank than in the previous scenario, as it is the most diffused in the South of Italy. The target market 3 is typical of some ski equipment and the preferences related to such a target market are Co St Re Me Un Ma. The last scenario is defined by a target market, given by 4 , which is distributed over all the regions (segments). Differently from the previous scenarios, here the distribution represented by the target spectrum 4 is not uniform, nor is specially concentrated in the northern or southern regions. In such a case it is not apparent which the optimal newspaper (channel) must be. However, we can solve the problem and the preferences induced by the index (15) are Co Re Me St Un Ma. We observe that in some cases the choice between two advertising channels is apparent on inspection of their spectra and the target one: see e.g. “Me St” in the second scenario (target 2 ), or “St Ma” in the third scenario (target 3 ). On the other hand, there are cases in which such easy comparisons cannot be made: see e.g. “Re Me” with targets 2 or 3 , or “Co Me” with target 4 . In all cases the channel preference index is an easy and useful tool to this purpose. The practical application above shows how the previous results can be used with a set of real data. This rather simple model can be implemented in a spreadsheet straightforwardly to give a useful support to a decision maker. Acknowledgements We would like to thank Antonio Boscardin for contributing to the development of the paper with his M.Sc. thesis and also the anonymous referee and the editor Suresh Sethi for valuable comments and suggestions. References Arabie, P., & Hubert, L. (2000). Cluster analysis in marketing research. In R.P. Bagozzi (Ed.), Advanced methods of marketing research (pp. 160–189). Oxford: Basil, Blackwell. Buratto, A., Grosset, L., & Viscolani, B. Advertising a new product in a segmented market. European Journal of Operational Research, to appear. Buratto, A., Grosset, L., & Viscolani, B. (2005). Linear models and advertising. Rendiconti per gli Studi Economici Quantitativi, unico, 107–120. Buratto, A., & Viscolani, B. (2002). New product introduction: Goodwill, time and advertising cost. Mathematical Methods of Operations Research, 55, 55–68. Feichtinger, G., Hartl, R. F., & Sethi, S. P. (1994). Dynamic optimal control models in advertising: Recent developments. Management Science, 40, 195–226. Grosset, L., & Viscolani, B. (2005). Advertising for the introduction of an age-sensitive product. Optimal Control Applications and Methods, 26, 157–167.
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JZrgensen, S., & Zaccour, G. (2004). Differential games in marketing. Boston: Kluwer Academic Publishers. Kotler, P. (1997). Marketing management: Analysis, planning, implementation, and control. Upper Saddle River: Prentice-Hall. Kotler, P., Armstrong, G., Saunders, J., & Wong, V. (1999). Principles of marketing. Upper Saddle River: Prentice-Hall. Law, S., Hawkins, S. A., & Craik, F. I. M. (1998). Repetition-induced belief in the elderly: Rehabilitating age-related memory deficits. Journal of Consumer Research, 25, 91–107. Lilien, G. L., Kotler, P., & Moorthy, K. S. (1992). Marketing models. Englewood Cliffs: Prentice-Hall. Little, J. D. C. (1979). Aggregate advertising models: The state of the art. Operations Research, 27, 629–667. Little, J. D. C., & Lodish, L. M. (1969). A media planning calculus. Operations Research, 17, 1–35. Mariel, P., & Sandonís, J. (2004). A model of advertising with application to the German automobile industry. Applied Economics, 36, 83–92. Nerlove, M., & Arrow, K. J. (1962). Optimal advertising policy under dynamic conditions. Economica, 29, 129–142. Seidmann, T. I., Sethi, S. P., & Derzko, N. (1987). Dynamics and optimization of a distributed sales-advertising model. Journal of Optimization Theory and Applications, 52, 443–462. Sethi, S. P. (1977). Dynamic optimal control models in advertising: A survey. SIAM Review, 19(4), 685–725. Sethi, S. P., & Thompson, G. L. (2000). Optimal control theory: Applications to management science and economics. Boston: Kluwer Academic Publishers. Tragler, G. (2000). Optimal controls in spatial advertising diffusion models. In E.J. Dockner, R.F. Hartl, M. Luptacik, & G. Sorger (Eds.), Optimization, dynamics, and economic analysis (pp. 288–297). Heidelberg: PhysicaVerlag. Wedel, M., & Kamakura, W. A. (2000). Market segmentation. Boston: Kluwer Academic Publishers. Zielske, H. A., & Henry, W. A. (1980). Remembering and forgetting television ads. Journal of Advertising Research, 20, 7–13. Alessandra Buratto received an M.Sc. in Mathematics from the University of Padova and a Ph.D. in Mathematics for Economic Applications from the University of Trieste. She is a Research fellow in the Faculty of Economics of Padova University. Her main research area is optimal control applied to marketing. She teaches courses in Mathematics for Economic Decisions and Operations Research. Luca Grosset studied Mathematics at the University of Padova. He obtained his Ph.D. in Computational Mathematics from the same University in 2001. In 2003, he won a two-years fellowship in the Department of Pure and Applied Mathematics, University of Padova, to study some applications of the optimal control theory to marketing. His main research interests are the applications of the deterministic and stochastic optimal control theory to Economics and Management Science. Bruno Viscolani studied Mathematics at the University of Padova (1971–1976). He has been professor of Mathematics for Economics at the same University since 1995. Previously was research fellow in Padova (1980–1987) and professor at the University “Ca’ Foscari” of Venice (1987–1995). His research activity concerns mainly some applications of Optimal Control Theory and Mathematical Programming to Economics and Management Science, in particular to Marketing. He serves as an Associate Editor of the Journal of Information and Optimization Sciences.