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A Stochastic model of optimal advertising pulsing policy
Compur. Opns Res. Vol. 14, No. 3, pp. 231-239, 1987 Printed in Great Britain. All rights reserved
A STOCHASTIC
Copyright0
MODEL OF OPTIMAL POLICY H.
0305-0548/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
ADVERTISING
PULSING
ARSHAM*
School of Business, University of Baltimore, Baltimore, MD 21201, U.S.A. (Received October 1985; revised July 1986)
Scope and Purpose-
Mesak considers deterministic models which deal with decisions concerning advertising pulsing policy (APP). Since there may be a multitude offactors affecting sales, some ofthem unknown to the advertiser, it is suggested to model APP under uncertainty. Moreover, recognizing that the decision makers are concerned with economic and risk implications of their decision alternatives, this paper presents the relevant analytical, statistical and numerical procedures to be used in APP decisions under uncertainty. Abstract-A descriptive model of stochastic sales response to advertising pulsing policy is presented. The prescribed strategy is the maximizer of a discounted profit function which includes the decision maker’s attitude towards risk. Markovian assumptions concerning the sales behavior are made and the randomization technique for computing the probabilistic levels of sales is used. Maximum likelihood procedures for the estimation of market parameters are also discussed.
INTRODUCTION
The term “Advertising Modeling” has been used to describe the decision processes of improving sales of a product or service. No general theoretically structured models have been proposed to guide managerial decision making in this area of marketing [ 11. Many models begin with the assumption of an advertising/sales response function. Vidale and Wolfe [2] is the earliest of empirical studies of such models which assume a constant rate of response to advertising per dollar spent. In their model, advertising is assumed to be operative only on those potential customers who are not customers at the present time, i.e. advertising merely generates new customers and does not make current customers increase their volume of purchase. Rao [3] is aware of this negligence and recently Mesak [4] considered the general concepts of Vidale-Wolfe and Rao to obtain a near optimal advertising pulsing policy by satisfying the first-order necessary conditions for an infinite planning horizon. The goal of this study is to present a sales response model for advertising pulsing policy (APPperiods of high advertising intensity alternating with periods of very little or no advertising) under uncertainty. The proposed model is an extension of Mesak’s deterministic model. This extension is important since there may be a multitude of factors affecting sales, some of them unknown to management so it is safer to assume that the sales are stochastic processes. Markovian assumption concerning the sales behavior will be made. We consider some more realistic factors including “forced decay” caused by competing messages in a competitive market [l] and the risk factor involving variation in sales. The article is divided into 7 parts. First, the assumptions of the model are presented. Then the probabilistic sales behavior equations, based on these assumptions are developed. This is followed by a discussion of numerical methods for obtaining the probabilistic levels of sales from these equations. In the following section the parameter estimation procedures are discussed. The objective function and the resulting optimal strategy are developed in the next section. A numerical example illustrates the model. The final section of the paper discusses the findings and examines the implications for future research works. *Hossein Arsham is the Graduate Program Director of Operations Research and Assistant Professor of Information and Quantitative Sciences at the University of Baltimore. He received a B.S. in Engineering Physics from Aryameher Technical University, Tehran, Iran in 1971; a MS. in Production Studies in 1977 from Cranfteld Institute of Technology, England and a D.S. in Operations Research in 1982 from the George Washington University. His research interests include statistical inference basedon empirical distribution functions and stochastic modeling of consumer behaviour. Dr Arsham has published in a variety of professional journals including the Journal of Statistical Computation and Simulation, Commun. Statist.-Theor. Meth., IEEE Transactions on Reliability and Statistica Neerlandica. He is a Fellow of the Royal Statistical Society. 231
H. ARSHAM
232 MODEL
ASSUMPTIONS
An advertising campaign is characterized as APP where advertising is active at a constant level A during a period of time (tl) followed by a period (T- rl) during which advertising is stopped, i.e. A(t) =
A,
O
0,
t,
(1)
It becomes active again at the same level, then is shut off and so on in a cyclic fashion. A is a constant measured in dollars per unit of time. Both A and its duration are the decision variables for the model. The advertising effort A can be any kind of promotion for a non-durable product or service. In the following sections we investigate the stochastic sales response to the single pulse given by (1) and obtain its optimal policy under the budget constraint given by I= At 1. This type of policy is also known as bang-bang strategy which is shown to be optimal in some advertising models, e.g. Sethi [S], and Naslund [6]. (For further empirical evidence of the advantages of APP see the relevant references given in Mesak [4].) The following assumptions are made concerning the model: (1) A direct measurable tabulated relationship exists between advertising and sales at steady state denoted by S = S, known with ~e~ainty for all S = 1,2,3, . . ., Q where Q is the saturation level of sales. By this assumption, we mean that when the advertising expenditure is maintained at level A, the corresponding steady state sales potential is SAwhich is measured in units per unit of time. Johansson [7] provides a class of S-shaped logistic functions to be used in advertising modeling. One of the most aggravating problems in marketing research is the assumption on the shape of the sales response function. This is due to the difficulties involved in establishing with precision the relationship of sales to advertising expenditure. For some controversies over the shape of sales response, see for example, Quandt [S], Little [I], and Simon and Arndt [9]. In the model under study, one needs to have only a tabulated form of S, which is more readily available. Therefore, we do not assume any specific functional form of S = S,, so there is no need of using any curve fitting procedures such as the one used in Holthausen and Assmus [lo] who assumed a modified exponential function, which implies diminishing return to advertising, for S = SA. Let S(t) be the sales at time t as measured in units per unit of time and P(n, t) = P[S(t) = n], then (2) The probability of a unit increase in sales in the interval (t, t -I- At) is proportional to the untapped potential sales at time t, i.e. P[n+l,t+At/n,t]=a[S,-n]At,
O,
(2)
where 06 n 6 S,. Furthermore, the probability of losing a unit of sales in the interval (t, t + At) caused by factors such as a decay effect is proportional to the existing level of sales, i.e. P[n-
l,t+At/n,t]=ot,nAt,
O
(3)
By equation (2) we mean that there is a chance of market expansion, i.e. possible increase in consumption of consumers who already prefer the advertised brand and new consumers who now prefer some other brands and probably will switch to the brand advertised, if the effect of competitors does not cancel out. The cancelling out phenomenon is shown, e.g. by Metwally [ 1l] in a study of advertising in a number of Australian industries. The type of decay given by (3) is termed as a “forced decay”. This is associated with the fact that if one brand can increase its sales by advertising, so can the others. Therefore, a successful advertiser often finds his competitors using promotional policies in effect proportional to his sales to recoup the loss. For empirical evidence of this phenomenon see t!e data given in Table 3 of Lambin et al. [12] showing that an increase in advertising of a company causes a decrease in sales of the other company. Therefore, this loss in sales can be viewed as the impact of competing messages on sales, (3) In the absence of advertising, over the period t, < t < T, sales decline and the probability of a unit loss in the interval (t, t + At) is proportional to the present level of sales, i.e. P[n-l,t+At~n,t]=~nAt,
t,‘ct
(4)
A stochastic model of optimal advertising pulsing policy
233
when there is no promotional effort. By this assumption we mean that the probability of a unit decline (caused by such factors as memory loss or continuation of competitor efforts) is proportional to the level of sales at time t. Notice that equation (3) is different from equation (4) in the sense that equation (3) incorporates the decay factor into the model for the period with advertising while equation (4) considers the period with no advertising. This may justify usage of different decay rates for each period. SALES
BEHAVIOR
As before, let P(n, t) be the probability of sales S(r) = n at time t. Under an advertising pulsing policy we distinguish the following two periods: (1) Advertising with High Intensity A(t) = A, 0 ri: t < t , . Assume that in a small interval of time AConly one event can occur-a gain or a loss in sales. Then by using the transition probabilities given by (2) and (3) the following probability process may be constructed:
Wn, t) ~
for n=l,2,...,
dt
= aAn + l)P(n -t 1, t) - [a,n + a(S,4 - n)]P(n, t) + a[SA - n-t l]P(n - 1, t)
(5)
S, - 1, with the following boundary conditions
WO, t) ~
dt
dP[S,,t] dt
= aJ(l,
t) - aS, P(0, t)
(6)
= -GV[S,, t] + ffP[sA-
1, t].
To obtain the mean and variance denoted by ml(t), vi(t) respectively, one can use equations (5), (6) and (7) to compute them directly using the de~nitions of these moments or by obtaining the generating function of this process and then computing the tirst two moments, see, for example, Farley and Tapiero [ 131 and references therein. Because mathematically rigorous treatment of these computations is readily available in marketing literature, see, for example, Massy et al. [ 14, Chap. 61, we present the following results:
dm,(t) - dt = m;(t) = aS, - (a, f a)m~~t), h(t) = u\(t) dt
mz (0) = S(O),
= as, + (a, - a)m,(t) - 2(a, + a)u,(t),
u,(O) = 0.
(9)
The initial conditions given in (8) and (9) suggest that sales at t = 0 is m, (0) = S(0) known with certaintv u,(O)= 0. These two equations have the following solutions as may be verified by substitution,’ or see, for example, Rainville and Bedient [15] m,(t)=S(O)+[-$j$-S(O)][l-exp[-(a,+a)t]],
OGttt,
(10)
and
1
exp[ - 2(a,fa)r]
x
exp[ - (a, f a)t] + __._fE_ (a,+a)2 “’
O
+
S,
1 (11)
H. ARSHAM
234
Notice that in a market with no forced decay factor (a, = 0), equation (10) reduces to equation (2) given in Mesak 141. Therefore, the mean evolution of the process is his deterministic model. It is important to note that mean value deduced from stochastic models does not always coincide with the value obtained from the corresponding deterministic model, see Bartholomew [16] for an example. (2) No Advertising, Aft) = 0, t, c t < T. By assumption 3, in the absence of advertising, sales decline stochastically which is a pure death process with death rate fin where n is the present level of sales. The probability distribution for this process in closed form is available (e.g. Taylor and Karlin [ 17, Chap. 61) having a Binomial distribution, i.e. P[n,tjN,t,]
=B(n; N, exp(-fit)),
1z=0, I, 2, . . .,N
(12)
where iVis the random sales at the time when advertising is shut off. The mean and variance evolution of this process is:
~t(~f=~l(tl)expC-P(t-~r)l,
tl
(13)
u2ft)=m,(r,)expC-p(r-t,)lEl-expE-P(t-tt)1]+v,(t1)expE-2P(t-t,)]
(14)
and
for t, < t< T Notice again that equation (3) in Mesak [4] can be deduced from equation (13).
NUMERICAL
METHODS
OF SALES BEHAVIOR
Since the campaign duration is finite we are interested in the transient solution p(n, t) of the process. Given the initial state, the probabilistic prediction of sales can be obtained by computing this transient sotution. Again, under an advertising pulsing policy we distinguish the following two periods. (i) Advertising with High Intensity, A(t) = A, 06 t < t,. By recognizing that the Markov process for this period is a birth-death state-dependent process, one realizes that the closed form solution is extremely difficult and in most cases impossible (unless the number of states S, is very small). The usual approach to compute the transient probabilities for this Markov process is by numerical solution of the system of equations generated by (5) (6) and (7). This is an initial value system with initial condition, for example P(n, 0) =
1, n=S(O) i 0,
n#S(O).
(15)
There are severai methods to solve this initial value system of equations: (i) numerical integration techniques such as Runge-Kutta, Predictor Corrector, etc. (see, for example, Arsham et al. [18]); (ii) exponentiation by computing the spectrum (see, for example, Medhi [19]); (iii) randomization technique (see, for example, Gross and Miller [20]). The randomization technique which is especially tailored for the transient solution of Markov processes has a distinct advantage over other approaches in that a bound on the global error can be set by the user and it is achieved. Another advantage is that it is a “computational probability” technique, an emerging discipline concerned with numerical solution of applied probability problems. In our numerical example we will use this technique with global error E= 10W3, this guarantees at least two decimal places accuracy in the transient solution for 0 < t < t,. (2) No Advertising, .4(t) = 0, tl < t < T. The transient solution for the period with no advertising, i.e. beyond t = E, can be found by
A stochastic model of optimal advertising pulsing policy
unconditioning
235
the conditional probability given by (12)
P[n, t] =
2 qn,t/N, ill&%
N=O
51
n=0,1,2
,...,
exp(-/3nt)[l-exp(-/3t)]N-“P[N,
N
ti].
(16)
(17)
Using this closed form solution, the transient probabilities for t, < t < T can be found easily. With regard to the binomial coefficients which appear in (17), we note that their straightforward computation through factorials can easily iead to arithmetic ovefflow (when S, is large). It is better to use the Pascal Triangle relations
n>N+l 2 which involve only the addition of integers. PARAMETER
ESTIMATION
Stochastic models by their nature enable the user to perform statistical testing of assumptions and estimating the parameters of the models. Suppose the firm wishes to estimate the three parameters (a, a,) and fi by observing a sample path of sales over two periods, one with advertising A(t) = A and one with no adve~ising. Several sampling design plans are possible for parameter estimation. We shall discuss one such plan. (1) Estimations of a and ae-suppose the firm is campaigning with A(t) = A > 0 for a period of unit length and observes the system continuously for this period starting at t = 0 and ending at t = 1. The observed sample function will be a step function which we consider right continuous. Let x1 be the state of the system (number of sales) at t, = 0 and let ti+ 1 for i = 1,2, . . . be the time at which the ith jump (+ 1 or - I) of the step function occurs. The state of the system at ti+ 1 is xi+ r. The sequence the step function. Let {(XrJr),..., (&l+i, r,+i )> where n f 1 = max(i 1ti < l> completelydete~ines Uj = number of + 1 jumps while in state j, 3 = the total time spent in state j during (0, l), Wj = number of - 1 jumps while in state j, then by using the procedure given by Wolff [21] the maximum likelihood estimate of a and a, is given by k-l
* ,pk_;=O j;.
C
uj
*
iwi
&.r+
(k-J5
(19) Cj? j=l
where K = S,. (2) Estimation of p- when the advertising is shut off suppose that sales equal L and we observe the sales continuously over a period starting at 0 and ending at 1. Suppose n losses occur at time o
H. ARSHAM
236
where z= i
ri+ (L-n).
i=l
For hypothesis testing, power studies and further statistical properties of the procedures discussed in this section, the reader is referred to the two references given above. OPTIMAL
STRATEGY
UNDER
APP
In this section, we consider a firm facing the problem of designing an APP with advertising rate A(t) = A* and its duration rr = t: such that to maximize a profit function. The choice of an advertising strategy denoted by (A, ti) must reflect both attitude towards profit maximization as well as an attitude towards risk factor measured by the sales variation. It is also noteworthy that advertising policies which call for maximization of a profit function without taking account of risk have been already recognized as inappropriate, see, for example, Yawitz et al. [23] and Tapiero [24]. Along this line, we define a discounted profit function of the following form
Y= v
i
1[yPm(t)- m(t) - d(t)] exp( -it)dt
(21)
where P is the unit price as measured in dollars, y is a positive fraction less than one given by y = 1 - 6 where 6 represents the ratio of cost, not considering expenditure, to sales revenues and r is the market value, a known constant in (dollar time) units. The risk market value r shows advertiser’s attitude towards risk. A larger value for r is associated with a higher cost of risk. In designing an APP, the economic and risk impli~tions can now be considers in dete~ining an optimal APP. The firm will seek an APP which can reduce the sales variation. The optimal strategy under an APP is the maximizer of the profit function Y =
s
Cl [yPml(t)
- r,u,(t) - A] exp( -it&
T[yPm,(t) exp( -it)dt. rzuz(t)]
+
(22)
s 11
0
Subject to the budget constraint 1 = At,. The mean and variance of sales used in the first and second integrands (22) are given in explicit form by (IO), (I 1) and (13), (14) respectively. Carrying out the integration, we obtain Yin the following closed form; Y = (yPC - A - r,F)
+r,D
1 - exp( - il/A) i
+ (yPB + r,E)
exp[ - (a, +ta f i)l/A] - 1 a,+a+i
exp[ - (2cz,+ 2a + i)l/A] - 1 + yP - r2 g+i CC exp~~/~) - B expW - a - a,)~/All 2o1,+2cr+i
* [exp[ - (p + @/A] - exp[ - (B + i)T]]
+&
[C - F - (I.3+ E) exp[ - (a + ~,)I/A] - D exp[ - 2(cr-i- E,)I/A]]
*[exp[ - (2p + i)Z/A] - exp[ - (2fi + i)T] exp(2PZ/A),
A> 0
(23)
where C = aSA/(a, + a) B = c - S(0) F = CaJa,
I- a)
E = (S(O) - C)(a, - a)/(a, + a) D = - aF/a, - (a, - a)S(O)/(a~ + a).
(24)
A stochastic model of optimal advertising pulsing policy
237
The global maximizer A = A* can be found by direct evaluation of Y for different levels of advertising A, (A > 0). The optimal duration therefore, is t: = I/A*. A FORTRAN coded program easily enables the user to evaluate Y for a given tabulated-values of S = S,, as is done in our numerical example below. NUMERICAL
EXAMPLE
Mesak [4] considered a monthly S-shaped advertising/sales form;
s, =
75A2 - 25A3,
response function of the following
A62 A>2
100,
(25)
in his analysis. This functional form of S = S, is not readily available but can be estimated by standard regression procedures given the advertising rate and sales response. In the model presented here, there is no need of assuming any specific shape and estimating its parameters to obtain the closed form of S = S,; in fact a tabulated form is more convenient to work with. Table 1 shows the sales response for some levels of advertising. Assuming the following parameters T=6,
a=p=o.o4,
a, = 0.01)
y = 0.7,
S(0) = 0, Ii = r2 = 0.001,
P= 1, i = 0.01
and assuming the monthly budget allocation is 0.2546 then I = 6(0.2546). These parameters are the same as Mesak’s example except for the additional prarameters, a,, the forced decay factor and (r,, r2) the attitude toward risk factors. Upon substitution of these parameters in (23) values for Y can be computed for all possible levels of advertising; some of these values are given in Table 1. By examination of these values the global maximum of profit function is Y* = 9.921 with A+ = 1.568, SA. = 88 and by using I = A*tf we obtain tr = 0.97 which is the optimal duration of advertising with high intensity. The mean and variance of the process are ml(t) = 70.4[1 - exp( -O.O5t)], m2(t)=3.33exp[-O.O4(t-0.97)],
m(t) =
0 6 I 6 0.97
ur (t) = - 56.32 exp( - O.lt) + 42.24 exp( - 0.05~) + 14.08, u,(t) = 0.12 exp[ - O.O8(t- 0.97)] + 3.33 exp[ - O.O4(r- 0.97)],
v(t) =
(26)
0.97
(27) Using the randomization technique we obtain the sales following the optimal policy for 0 6 t < 0.97 with a bound on the error E = 0.001. The results are accurate at least for two decimal places for these
Table Sales SA
I 2 3 4 5
47 48 49 50
I.
Discrete-sales
Advertising A 0.1 178 0.1681 0.2073 0.2408 0.2707
0.9600 0.9733 0.9867
I .cPxlo
response to advertising
with the corresponding
profits
Advertising A
Profits Y
Profits Y
Sales
- I.626 - 0.8255 - 0.2274 0.2690 0.7008
51 52
I.013 I.027
8.266 8.336
87 88 89
1.548 I.568 I.588
9.920 9.921 9.917
98 99 100
I.832 1.882 2.mO
9.518 9.354 8.897
7.916 8.004 8.090 8.174
SA
ARSHAM
Ii.
238
Table 2. Sales level probabilities sales=
at some points for the optimal
0
1
2
3
4
5
l.@OO 0.703 0.495 0.348 0.245 0.173 0.122 0.135 0.146 0.158 0.170 0.182 0.194
(O.ooo) 0.248 0.349 0.370 0.347 0.306 0.259 0.271 0.282 0.292 0.302 0.311 0.320
0.043 0.122 0.194 0.243 0.269 0.273 0.274 0.273 0.272 0.270 0.267 0.263
0.003 0.028 0.067 0.112 0.15s 0.190 0.182 0.175 0.167 0.160 0.151 0.143
(0.001) 0.004 0.016 0.038 0.067 0.097 0.089 0,082 0,075 0.070 0.063 0.057
(0.002) 0.003 0.010 0.022 0.039 0.034 0.030 0.027 0.023 0.020 0.01 R
strategy 6
7
8
Time 0 0.1 0.3 0.5 0.8 0.9 0.97 1 2 3 4 5 6 Figures in parentheses
are the exceeding
sales probabitities
of the corresponding
(0.002) (0.005) 0.006 0.013 0.011 0.009
(0.002) 0.003 0.003 0.002 0.002
(0.N) (0.001) (O.Ool) (O.ool)
gZ) ww (0.005)
levels.
transient probabilities, which are given (in Table 2) for some values oft. Upon using equation (17) the transient solution is obtained for the rest of the period, when the advertising is shut off. Results are also given in Table 2 for some values of t up to E= T= 6. CONCLUDING
REMARKS
This paper has presented a stochastic model of advertising pulsing policy (APP). The model we constructed has a mean evolution similar to the APP model introduced by Mesak [4]. This paper can be considered an extension of Mesak’s model to probabilistic systems. This extension is an important one as it allows a more realistic interpretation of advertising effects, and provides the basis on which empirical studies of APP effectiveness can be conducted. Recognizing that decision makers are concerned with the economic and the risk implications of their decisions alternatives suggests that APP must be constructed as stochastic. The stochastic APP we constructed, simplified by equations (lo), (1 I), (13) and (14) provides a framework for theoretically and empirically evaluating the implications an APP has on the probability distribution of profits. By reducing the probabilistic evolution of sales to an evolution of moments we maximized a profit function which includes the risk factor. This allowed us to obtain a closed form of the profit function, simplified by equation (23). This closed analytical form, however complex, provides considerable insight into the sensitivity of APP decisions. The managerial implication is the exploration of the effect due to the following factors on the optimal strategy: (i) the parameters which characterize the produce (e.g. y); (ii) the market forces of industry sales (e.g. a,); (iii) the managerially determined factors (e.g. f ). We introduce a simple sampling plan ideally suited for estimation of the model’s parameters. A numericaIly tractable framework was presented that is useful for computing the sales level in Markovian advertising models. The proposed model and the derived optimal policy were function of a forced decay factor which was assumed to be predictable and independent of the firm’s actions. A more realistic approach would be one which assumes that the competition would react to the firm’s actions with a specific response function or a game theoretic one. We have not addressed the “feedback effect”; i.e. the effect of past sales on future advertising budgets (see, for example, Aaker et al. [25]). This paper assumed the market is stationary, i.e. where the parameters are not time dependent. When there is a time lag between advertising and its initial impact, it is more appropriate to model APP as a dynamic stochastic model analogous to the Nerlove-Arrow deterministic model [26]. It is hoped that these issues wilt be addressed in future research. Acknowledgements--I am most appreciative of the comments and suggestions of the referees and of Professor S. Shao Jr which into the final version of this paper. This research was partially supported by the Educational Foundation of the University of Baltimore. were incorporated
A stochastic model of optimal advertising pulsing
policy
239
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A STOCHASTIC
Copyright0
MODEL OF OPTIMAL POLICY H.
0305-0548/87 $3.00 + 0.00 1987 Pergamon Journals Ltd
ADVERTISING
PULSING
ARSHAM*
School of Business, University of Baltimore, Baltimore, MD 21201, U.S.A. (Received October 1985; revised July 1986)
Scope and Purpose-
Mesak considers deterministic models which deal with decisions concerning advertising pulsing policy (APP). Since there may be a multitude offactors affecting sales, some ofthem unknown to the advertiser, it is suggested to model APP under uncertainty. Moreover, recognizing that the decision makers are concerned with economic and risk implications of their decision alternatives, this paper presents the relevant analytical, statistical and numerical procedures to be used in APP decisions under uncertainty. Abstract-A descriptive model of stochastic sales response to advertising pulsing policy is presented. The prescribed strategy is the maximizer of a discounted profit function which includes the decision maker’s attitude towards risk. Markovian assumptions concerning the sales behavior are made and the randomization technique for computing the probabilistic levels of sales is used. Maximum likelihood procedures for the estimation of market parameters are also discussed.
INTRODUCTION
The term “Advertising Modeling” has been used to describe the decision processes of improving sales of a product or service. No general theoretically structured models have been proposed to guide managerial decision making in this area of marketing [ 11. Many models begin with the assumption of an advertising/sales response function. Vidale and Wolfe [2] is the earliest of empirical studies of such models which assume a constant rate of response to advertising per dollar spent. In their model, advertising is assumed to be operative only on those potential customers who are not customers at the present time, i.e. advertising merely generates new customers and does not make current customers increase their volume of purchase. Rao [3] is aware of this negligence and recently Mesak [4] considered the general concepts of Vidale-Wolfe and Rao to obtain a near optimal advertising pulsing policy by satisfying the first-order necessary conditions for an infinite planning horizon. The goal of this study is to present a sales response model for advertising pulsing policy (APPperiods of high advertising intensity alternating with periods of very little or no advertising) under uncertainty. The proposed model is an extension of Mesak’s deterministic model. This extension is important since there may be a multitude of factors affecting sales, some of them unknown to management so it is safer to assume that the sales are stochastic processes. Markovian assumption concerning the sales behavior will be made. We consider some more realistic factors including “forced decay” caused by competing messages in a competitive market [l] and the risk factor involving variation in sales. The article is divided into 7 parts. First, the assumptions of the model are presented. Then the probabilistic sales behavior equations, based on these assumptions are developed. This is followed by a discussion of numerical methods for obtaining the probabilistic levels of sales from these equations. In the following section the parameter estimation procedures are discussed. The objective function and the resulting optimal strategy are developed in the next section. A numerical example illustrates the model. The final section of the paper discusses the findings and examines the implications for future research works. *Hossein Arsham is the Graduate Program Director of Operations Research and Assistant Professor of Information and Quantitative Sciences at the University of Baltimore. He received a B.S. in Engineering Physics from Aryameher Technical University, Tehran, Iran in 1971; a MS. in Production Studies in 1977 from Cranfteld Institute of Technology, England and a D.S. in Operations Research in 1982 from the George Washington University. His research interests include statistical inference basedon empirical distribution functions and stochastic modeling of consumer behaviour. Dr Arsham has published in a variety of professional journals including the Journal of Statistical Computation and Simulation, Commun. Statist.-Theor. Meth., IEEE Transactions on Reliability and Statistica Neerlandica. He is a Fellow of the Royal Statistical Society. 231
H. ARSHAM
232 MODEL
ASSUMPTIONS
An advertising campaign is characterized as APP where advertising is active at a constant level A during a period of time (tl) followed by a period (T- rl) during which advertising is stopped, i.e. A(t) =
A,
O
0,
t,
(1)
It becomes active again at the same level, then is shut off and so on in a cyclic fashion. A is a constant measured in dollars per unit of time. Both A and its duration are the decision variables for the model. The advertising effort A can be any kind of promotion for a non-durable product or service. In the following sections we investigate the stochastic sales response to the single pulse given by (1) and obtain its optimal policy under the budget constraint given by I= At 1. This type of policy is also known as bang-bang strategy which is shown to be optimal in some advertising models, e.g. Sethi [S], and Naslund [6]. (For further empirical evidence of the advantages of APP see the relevant references given in Mesak [4].) The following assumptions are made concerning the model: (1) A direct measurable tabulated relationship exists between advertising and sales at steady state denoted by S = S, known with ~e~ainty for all S = 1,2,3, . . ., Q where Q is the saturation level of sales. By this assumption, we mean that when the advertising expenditure is maintained at level A, the corresponding steady state sales potential is SAwhich is measured in units per unit of time. Johansson [7] provides a class of S-shaped logistic functions to be used in advertising modeling. One of the most aggravating problems in marketing research is the assumption on the shape of the sales response function. This is due to the difficulties involved in establishing with precision the relationship of sales to advertising expenditure. For some controversies over the shape of sales response, see for example, Quandt [S], Little [I], and Simon and Arndt [9]. In the model under study, one needs to have only a tabulated form of S, which is more readily available. Therefore, we do not assume any specific functional form of S = S,, so there is no need of using any curve fitting procedures such as the one used in Holthausen and Assmus [lo] who assumed a modified exponential function, which implies diminishing return to advertising, for S = SA. Let S(t) be the sales at time t as measured in units per unit of time and P(n, t) = P[S(t) = n], then (2) The probability of a unit increase in sales in the interval (t, t -I- At) is proportional to the untapped potential sales at time t, i.e. P[n+l,t+At/n,t]=a[S,-n]At,
O,
(2)
where 06 n 6 S,. Furthermore, the probability of losing a unit of sales in the interval (t, t + At) caused by factors such as a decay effect is proportional to the existing level of sales, i.e. P[n-
l,t+At/n,t]=ot,nAt,
O
(3)
By equation (2) we mean that there is a chance of market expansion, i.e. possible increase in consumption of consumers who already prefer the advertised brand and new consumers who now prefer some other brands and probably will switch to the brand advertised, if the effect of competitors does not cancel out. The cancelling out phenomenon is shown, e.g. by Metwally [ 1l] in a study of advertising in a number of Australian industries. The type of decay given by (3) is termed as a “forced decay”. This is associated with the fact that if one brand can increase its sales by advertising, so can the others. Therefore, a successful advertiser often finds his competitors using promotional policies in effect proportional to his sales to recoup the loss. For empirical evidence of this phenomenon see t!e data given in Table 3 of Lambin et al. [12] showing that an increase in advertising of a company causes a decrease in sales of the other company. Therefore, this loss in sales can be viewed as the impact of competing messages on sales, (3) In the absence of advertising, over the period t, < t < T, sales decline and the probability of a unit loss in the interval (t, t + At) is proportional to the present level of sales, i.e. P[n-l,t+At~n,t]=~nAt,
t,‘ct
(4)
A stochastic model of optimal advertising pulsing policy
233
when there is no promotional effort. By this assumption we mean that the probability of a unit decline (caused by such factors as memory loss or continuation of competitor efforts) is proportional to the level of sales at time t. Notice that equation (3) is different from equation (4) in the sense that equation (3) incorporates the decay factor into the model for the period with advertising while equation (4) considers the period with no advertising. This may justify usage of different decay rates for each period. SALES
BEHAVIOR
As before, let P(n, t) be the probability of sales S(r) = n at time t. Under an advertising pulsing policy we distinguish the following two periods: (1) Advertising with High Intensity A(t) = A, 0 ri: t < t , . Assume that in a small interval of time AConly one event can occur-a gain or a loss in sales. Then by using the transition probabilities given by (2) and (3) the following probability process may be constructed:
Wn, t) ~
for n=l,2,...,
dt
= aAn + l)P(n -t 1, t) - [a,n + a(S,4 - n)]P(n, t) + a[SA - n-t l]P(n - 1, t)
(5)
S, - 1, with the following boundary conditions
WO, t) ~
dt
dP[S,,t] dt
= aJ(l,
t) - aS, P(0, t)
(6)
= -GV[S,, t] + ffP[sA-
1, t].
To obtain the mean and variance denoted by ml(t), vi(t) respectively, one can use equations (5), (6) and (7) to compute them directly using the de~nitions of these moments or by obtaining the generating function of this process and then computing the tirst two moments, see, for example, Farley and Tapiero [ 131 and references therein. Because mathematically rigorous treatment of these computations is readily available in marketing literature, see, for example, Massy et al. [ 14, Chap. 61, we present the following results:
dm,(t) - dt = m;(t) = aS, - (a, f a)m~~t), h(t) = u\(t) dt
mz (0) = S(O),
= as, + (a, - a)m,(t) - 2(a, + a)u,(t),
u,(O) = 0.
(9)
The initial conditions given in (8) and (9) suggest that sales at t = 0 is m, (0) = S(0) known with certaintv u,(O)= 0. These two equations have the following solutions as may be verified by substitution,’ or see, for example, Rainville and Bedient [15] m,(t)=S(O)+[-$j$-S(O)][l-exp[-(a,+a)t]],
OGttt,
(10)
and
1
exp[ - 2(a,fa)r]
x
exp[ - (a, f a)t] + __._fE_ (a,+a)2 “’
O
+
S,
1 (11)
H. ARSHAM
234
Notice that in a market with no forced decay factor (a, = 0), equation (10) reduces to equation (2) given in Mesak 141. Therefore, the mean evolution of the process is his deterministic model. It is important to note that mean value deduced from stochastic models does not always coincide with the value obtained from the corresponding deterministic model, see Bartholomew [16] for an example. (2) No Advertising, Aft) = 0, t, c t < T. By assumption 3, in the absence of advertising, sales decline stochastically which is a pure death process with death rate fin where n is the present level of sales. The probability distribution for this process in closed form is available (e.g. Taylor and Karlin [ 17, Chap. 61) having a Binomial distribution, i.e. P[n,tjN,t,]
=B(n; N, exp(-fit)),
1z=0, I, 2, . . .,N
(12)
where iVis the random sales at the time when advertising is shut off. The mean and variance evolution of this process is:
~t(~f=~l(tl)expC-P(t-~r)l,
tl
(13)
u2ft)=m,(r,)expC-p(r-t,)lEl-expE-P(t-tt)1]+v,(t1)expE-2P(t-t,)]
(14)
and
for t, < t< T Notice again that equation (3) in Mesak [4] can be deduced from equation (13).
NUMERICAL
METHODS
OF SALES BEHAVIOR
Since the campaign duration is finite we are interested in the transient solution p(n, t) of the process. Given the initial state, the probabilistic prediction of sales can be obtained by computing this transient sotution. Again, under an advertising pulsing policy we distinguish the following two periods. (i) Advertising with High Intensity, A(t) = A, 06 t < t,. By recognizing that the Markov process for this period is a birth-death state-dependent process, one realizes that the closed form solution is extremely difficult and in most cases impossible (unless the number of states S, is very small). The usual approach to compute the transient probabilities for this Markov process is by numerical solution of the system of equations generated by (5) (6) and (7). This is an initial value system with initial condition, for example P(n, 0) =
1, n=S(O) i 0,
n#S(O).
(15)
There are severai methods to solve this initial value system of equations: (i) numerical integration techniques such as Runge-Kutta, Predictor Corrector, etc. (see, for example, Arsham et al. [18]); (ii) exponentiation by computing the spectrum (see, for example, Medhi [19]); (iii) randomization technique (see, for example, Gross and Miller [20]). The randomization technique which is especially tailored for the transient solution of Markov processes has a distinct advantage over other approaches in that a bound on the global error can be set by the user and it is achieved. Another advantage is that it is a “computational probability” technique, an emerging discipline concerned with numerical solution of applied probability problems. In our numerical example we will use this technique with global error E= 10W3, this guarantees at least two decimal places accuracy in the transient solution for 0 < t < t,. (2) No Advertising, .4(t) = 0, tl < t < T. The transient solution for the period with no advertising, i.e. beyond t = E, can be found by
A stochastic model of optimal advertising pulsing policy
unconditioning
235
the conditional probability given by (12)
P[n, t] =
2 qn,t/N, ill&%
N=O
51
n=0,1,2
,...,
exp(-/3nt)[l-exp(-/3t)]N-“P[N,
N
ti].
(16)
(17)
Using this closed form solution, the transient probabilities for t, < t < T can be found easily. With regard to the binomial coefficients which appear in (17), we note that their straightforward computation through factorials can easily iead to arithmetic ovefflow (when S, is large). It is better to use the Pascal Triangle relations
n>N+l 2 which involve only the addition of integers. PARAMETER
ESTIMATION
Stochastic models by their nature enable the user to perform statistical testing of assumptions and estimating the parameters of the models. Suppose the firm wishes to estimate the three parameters (a, a,) and fi by observing a sample path of sales over two periods, one with advertising A(t) = A and one with no adve~ising. Several sampling design plans are possible for parameter estimation. We shall discuss one such plan. (1) Estimations of a and ae-suppose the firm is campaigning with A(t) = A > 0 for a period of unit length and observes the system continuously for this period starting at t = 0 and ending at t = 1. The observed sample function will be a step function which we consider right continuous. Let x1 be the state of the system (number of sales) at t, = 0 and let ti+ 1 for i = 1,2, . . . be the time at which the ith jump (+ 1 or - I) of the step function occurs. The state of the system at ti+ 1 is xi+ r. The sequence the step function. Let {(XrJr),..., (&l+i, r,+i )> where n f 1 = max(i 1ti < l> completelydete~ines Uj = number of + 1 jumps while in state j, 3 = the total time spent in state j during (0, l), Wj = number of - 1 jumps while in state j, then by using the procedure given by Wolff [21] the maximum likelihood estimate of a and a, is given by k-l
* ,pk_;=O j;.
C
uj
*
iwi
&.r+
(k-J5
(19) Cj? j=l
where K = S,. (2) Estimation of p- when the advertising is shut off suppose that sales equal L and we observe the sales continuously over a period starting at 0 and ending at 1. Suppose n losses occur at time o
H. ARSHAM
236
where z= i
ri+ (L-n).
i=l
For hypothesis testing, power studies and further statistical properties of the procedures discussed in this section, the reader is referred to the two references given above. OPTIMAL
STRATEGY
UNDER
APP
In this section, we consider a firm facing the problem of designing an APP with advertising rate A(t) = A* and its duration rr = t: such that to maximize a profit function. The choice of an advertising strategy denoted by (A, ti) must reflect both attitude towards profit maximization as well as an attitude towards risk factor measured by the sales variation. It is also noteworthy that advertising policies which call for maximization of a profit function without taking account of risk have been already recognized as inappropriate, see, for example, Yawitz et al. [23] and Tapiero [24]. Along this line, we define a discounted profit function of the following form
Y= v
i
1[yPm(t)- m(t) - d(t)] exp( -it)dt
(21)
where P is the unit price as measured in dollars, y is a positive fraction less than one given by y = 1 - 6 where 6 represents the ratio of cost, not considering expenditure, to sales revenues and r is the market value, a known constant in (dollar time) units. The risk market value r shows advertiser’s attitude towards risk. A larger value for r is associated with a higher cost of risk. In designing an APP, the economic and risk impli~tions can now be considers in dete~ining an optimal APP. The firm will seek an APP which can reduce the sales variation. The optimal strategy under an APP is the maximizer of the profit function Y =
s
Cl [yPml(t)
- r,u,(t) - A] exp( -it&
T[yPm,(t) exp( -it)dt. rzuz(t)]
+
(22)
s 11
0
Subject to the budget constraint 1 = At,. The mean and variance of sales used in the first and second integrands (22) are given in explicit form by (IO), (I 1) and (13), (14) respectively. Carrying out the integration, we obtain Yin the following closed form; Y = (yPC - A - r,F)
+r,D
1 - exp( - il/A) i
+ (yPB + r,E)
exp[ - (a, +ta f i)l/A] - 1 a,+a+i
exp[ - (2cz,+ 2a + i)l/A] - 1 + yP - r2 g+i CC exp~~/~) - B expW - a - a,)~/All 2o1,+2cr+i
* [exp[ - (p + @/A] - exp[ - (B + i)T]]
+&
[C - F - (I.3+ E) exp[ - (a + ~,)I/A] - D exp[ - 2(cr-i- E,)I/A]]
*[exp[ - (2p + i)Z/A] - exp[ - (2fi + i)T] exp(2PZ/A),
A> 0
(23)
where C = aSA/(a, + a) B = c - S(0) F = CaJa,
I- a)
E = (S(O) - C)(a, - a)/(a, + a) D = - aF/a, - (a, - a)S(O)/(a~ + a).
(24)
A stochastic model of optimal advertising pulsing policy
237
The global maximizer A = A* can be found by direct evaluation of Y for different levels of advertising A, (A > 0). The optimal duration therefore, is t: = I/A*. A FORTRAN coded program easily enables the user to evaluate Y for a given tabulated-values of S = S,, as is done in our numerical example below. NUMERICAL
EXAMPLE
Mesak [4] considered a monthly S-shaped advertising/sales form;
s, =
75A2 - 25A3,
response function of the following
A62 A>2
100,
(25)
in his analysis. This functional form of S = S, is not readily available but can be estimated by standard regression procedures given the advertising rate and sales response. In the model presented here, there is no need of assuming any specific shape and estimating its parameters to obtain the closed form of S = S,; in fact a tabulated form is more convenient to work with. Table 1 shows the sales response for some levels of advertising. Assuming the following parameters T=6,
a=p=o.o4,
a, = 0.01)
y = 0.7,
S(0) = 0, Ii = r2 = 0.001,
P= 1, i = 0.01
and assuming the monthly budget allocation is 0.2546 then I = 6(0.2546). These parameters are the same as Mesak’s example except for the additional prarameters, a,, the forced decay factor and (r,, r2) the attitude toward risk factors. Upon substitution of these parameters in (23) values for Y can be computed for all possible levels of advertising; some of these values are given in Table 1. By examination of these values the global maximum of profit function is Y* = 9.921 with A+ = 1.568, SA. = 88 and by using I = A*tf we obtain tr = 0.97 which is the optimal duration of advertising with high intensity. The mean and variance of the process are ml(t) = 70.4[1 - exp( -O.O5t)], m2(t)=3.33exp[-O.O4(t-0.97)],
m(t) =
0 6 I 6 0.97
ur (t) = - 56.32 exp( - O.lt) + 42.24 exp( - 0.05~) + 14.08, u,(t) = 0.12 exp[ - O.O8(t- 0.97)] + 3.33 exp[ - O.O4(r- 0.97)],
v(t) =
(26)
0.97
(27) Using the randomization technique we obtain the sales following the optimal policy for 0 6 t < 0.97 with a bound on the error E = 0.001. The results are accurate at least for two decimal places for these
Table Sales SA
I 2 3 4 5
47 48 49 50
I.
Discrete-sales
Advertising A 0.1 178 0.1681 0.2073 0.2408 0.2707
0.9600 0.9733 0.9867
I .cPxlo
response to advertising
with the corresponding
profits
Advertising A
Profits Y
Profits Y
Sales
- I.626 - 0.8255 - 0.2274 0.2690 0.7008
51 52
I.013 I.027
8.266 8.336
87 88 89
1.548 I.568 I.588
9.920 9.921 9.917
98 99 100
I.832 1.882 2.mO
9.518 9.354 8.897
7.916 8.004 8.090 8.174
SA
ARSHAM
Ii.
238
Table 2. Sales level probabilities sales=
at some points for the optimal
0
1
2
3
4
5
l.@OO 0.703 0.495 0.348 0.245 0.173 0.122 0.135 0.146 0.158 0.170 0.182 0.194
(O.ooo) 0.248 0.349 0.370 0.347 0.306 0.259 0.271 0.282 0.292 0.302 0.311 0.320
0.043 0.122 0.194 0.243 0.269 0.273 0.274 0.273 0.272 0.270 0.267 0.263
0.003 0.028 0.067 0.112 0.15s 0.190 0.182 0.175 0.167 0.160 0.151 0.143
(0.001) 0.004 0.016 0.038 0.067 0.097 0.089 0,082 0,075 0.070 0.063 0.057
(0.002) 0.003 0.010 0.022 0.039 0.034 0.030 0.027 0.023 0.020 0.01 R
strategy 6
7
8
Time 0 0.1 0.3 0.5 0.8 0.9 0.97 1 2 3 4 5 6 Figures in parentheses
are the exceeding
sales probabitities
of the corresponding
(0.002) (0.005) 0.006 0.013 0.011 0.009
(0.002) 0.003 0.003 0.002 0.002
(0.N) (0.001) (O.Ool) (O.ool)
gZ) ww (0.005)
levels.
transient probabilities, which are given (in Table 2) for some values oft. Upon using equation (17) the transient solution is obtained for the rest of the period, when the advertising is shut off. Results are also given in Table 2 for some values of t up to E= T= 6. CONCLUDING
REMARKS
This paper has presented a stochastic model of advertising pulsing policy (APP). The model we constructed has a mean evolution similar to the APP model introduced by Mesak [4]. This paper can be considered an extension of Mesak’s model to probabilistic systems. This extension is an important one as it allows a more realistic interpretation of advertising effects, and provides the basis on which empirical studies of APP effectiveness can be conducted. Recognizing that decision makers are concerned with the economic and the risk implications of their decisions alternatives suggests that APP must be constructed as stochastic. The stochastic APP we constructed, simplified by equations (lo), (1 I), (13) and (14) provides a framework for theoretically and empirically evaluating the implications an APP has on the probability distribution of profits. By reducing the probabilistic evolution of sales to an evolution of moments we maximized a profit function which includes the risk factor. This allowed us to obtain a closed form of the profit function, simplified by equation (23). This closed analytical form, however complex, provides considerable insight into the sensitivity of APP decisions. The managerial implication is the exploration of the effect due to the following factors on the optimal strategy: (i) the parameters which characterize the produce (e.g. y); (ii) the market forces of industry sales (e.g. a,); (iii) the managerially determined factors (e.g. f ). We introduce a simple sampling plan ideally suited for estimation of the model’s parameters. A numericaIly tractable framework was presented that is useful for computing the sales level in Markovian advertising models. The proposed model and the derived optimal policy were function of a forced decay factor which was assumed to be predictable and independent of the firm’s actions. A more realistic approach would be one which assumes that the competition would react to the firm’s actions with a specific response function or a game theoretic one. We have not addressed the “feedback effect”; i.e. the effect of past sales on future advertising budgets (see, for example, Aaker et al. [25]). This paper assumed the market is stationary, i.e. where the parameters are not time dependent. When there is a time lag between advertising and its initial impact, it is more appropriate to model APP as a dynamic stochastic model analogous to the Nerlove-Arrow deterministic model [26]. It is hoped that these issues wilt be addressed in future research. Acknowledgements--I am most appreciative of the comments and suggestions of the referees and of Professor S. Shao Jr which into the final version of this paper. This research was partially supported by the Educational Foundation of the University of Baltimore. were incorporated
A stochastic model of optimal advertising pulsing
policy
239
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