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Optimal control of the Vidale-Wolfe advertising model

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Optimal control of the Vidale-Wolfe advertising model
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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/228255232 Optimal Control of the Vidale-Wolfe AdvertisingModel  Article   in  Operations Research · May 1972 DOI: 10.1287/opre.21.4.998 CITATIONS 94 READS 753 1 author: Suresh SethiUniversity of Texas at Dallas 453   PUBLICATIONS   11,385   CITATIONS   SEE PROFILE All content following this page was uploaded by Suresh Sethi on 16 January 2017. The user has requested enhancement of the downloaded file.  OPTIMAL DYNAMICS OF THE VIDALE-WOLFE ADVERTISING MODEL PART I: FIXED TERMINAL MARKET SHARE by Suresh P. SethiTECHNICAL REPORT 72-9May 1972 DEPARTMENT OF OPERATIONS RESEARCH Stanford University Stanford, California Research and reproduction of this report was partially supported byOffice of Naval Research N-00014-67-A-0112-0011; National ScienceFoundation Grant GP 31393; and U.S. Atomic Energy Commission Contract AT(04-3)-326-PA #18. Reproduction in whole or in part is permitted for any purposes of the United States Government. This document has been approved for public release and sale; its distribution is unlimited.  OPTIMAL DYNAMICS OF THE VIDALE-WOLFE ADVERTISING MODEL PART I: FIXED TERMINAL MARKET SHARE by Suresh P. SethiStanford University I.   Introduction. As one of the earliest management science applications to marketing, Vidale and Wolfe [1.4] developed a simple model of the sales response to advertising that was consistent with their experi- mental observations. In the fifteen years since its inception, the model has had a wide appeal because of its extreme simplicity andcapacity to capture the interaction between advertising and sales in an intuitively satisfying manner. Vidale and Wolfe argued that changes in the rate of sales of a product depend on two effects; response to advertising whichacts (via the response constant p) on the unsold portion of themarket, and loss due to forgetting which acts (via the decay constant k) on the sold portion of the market: (1.1) x = pu(l-x) - kx;x(0) = x0 I would like to thank Professor J. V. Breakwell, R. E. Turner, and C. P. Neuman for their helpful suggestions. An earlier version of this paper with Part II was presented at the 41st ORSA Conference at New Orleans, April 1972. 1  where x is the market share (i.e0,,the rate of sales expressed as a 1 fraction cf the market potential or saturation level) and u is the rate of advertising expenditure (a control variable), at time t. Whereas Vidale and Wolfe offered their model primarily asa description of actual market phenomenar_epresented by cases whichthey had observed [14], the present paper (Part I) obtains the optimaladvertising schedule (optimal control) of the model; in the sense that it must maximize a certain objective function over horizon T while attaining a specified terminal market share XT at time T.2 In the next section, we complete the statement of the problem bydefining the objective function. 2.   Optimal Control Problem  , It is assumed that the objective to be maximized is the present value of the profit stream up to the horizon T. Morespecifically, if n denotes the maximum sales revenue potential (this assumes a constant margin per unit product) and i, the discount rate, then the optimal control problem is: T(2.1) max J = [nx(t) - u(t)] e -it dt u(t) = 0 0 1 This represents a slight departure from the Vidale-Wolfe model [14]. In terms of their formulation x(t) = S(t)/M, where S(t) is therate of actual sale at time t and M is the saturation level. 2 See Section 7(i) for the problems treated in Part II [12]. 2  subject to (1.1) and the terminal constraint x(T) = x T o  Note that the requirement 0 < x(t) < 1 for all t is automatically satisfied if 0 < x 0  < 1, x 0  being the initial market share. A convenient procedure to solve this problem, where there is only one state variable and where the control u appears linearlyin both the state equation (L1) and the objective function (2.1),is by using Green's theorem 181.   This is carried out in the following section. 3.   Solution by Green's Theorem . First we substitute (3.1) udt _ dx + kx dt p (l-x) derived from (1.1) into the objective function (2.1) to obtain the line integral -it (3.2) J =  [ 7x  _. kx ]   e t dt -dx p(1-x) p(1-x) Using Green's theorem, we can express the line integral above as a double integral in (t,x) space; -it . (3.3) J = f    [ - e  ]  - a  [Trx - kx ]   e-it dt dx a t  p (1-x) ax p ( i-x) = 1 a   i  f [ k + 1   -   Trp ] e-it dt dx p (1-x)2 1-x 9
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