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A note on product positioning
90
Short Communications
A note on product positioning K. B R O C K H O F F lnstitut fiir Betriebswirtschaftslehre. Christian-AlbrechtsUniversitiit Kiel. D-2300 Kiel. German); Fed. Rep. Received August 1981
Critical remarks are added to a paper that appeared in this journal [4]. We try to clarify assumptions on the objective function, and we refer to the problem of comparing algorithms if the problem structure differs.
In their recent paper, Bachem and Simon [4] have presented a model to position optimally one new product in a space that is spanned by product characteristics. The objective function may incorporate costs and prices to calculate new product positions that maximize expected profits. This has been suggested in [6] but was not made explicit. The authors suggest an assumption on buying probability that incorporates two others: those used by Shocker and Srinivasan [6] and those based on the single choice axiom, used by various authors [5,8,3,1]. Furthermore, two algorithms are suggested to solve the planning problem that resuits. A first point of criticism refers to the solution space that is used by the authors. Product attributes are "represented as dimensions of an mdimensional normed linear space L (e.g. R")". Both, consumers' perceptions of each attributes of existing brands as well as perceptions of most preferred attribute levels, called 'ideal points', are represented as elements of L [4, p. 362]. However, possible realizations of the new product y = (y~ ..... y,,) are chosen such as yERCL. It is assumed: "Surely R must be discrete" [4, p. 364]. In general, perceptual spaces are derived from multidimensional scaling measurements, while actionable attribute spaces result from conjoint analyses.
North-Holland Publishing Company o European Journal of Operational Research 9 (1982) 90-91
This assumption on R is crucial for the performm~ce of the enumeration procedure that is suggested as the first algorithm in the paper [4] to solve the product positioning rwoblem. Clearly, if y* ~ R is an optimal solutioa with respect to some objective function and constraints, this is not necessarily optimal in L. This could be taken lightly, if optimal product positions had in general flat optima. Unfortunately, this is not true as can be read from numerous simulation runs. If, however, R is constrained to discrete values for a simplification of product planning, this raises two questions: Firstly, what is the transformation of 'objective' planning characteristics to 'subjective' perceptual characteristics.'? Secondly, why should have R the same dimensionality as has L? By contrast, it is plausible to assume R C Rs and L C Rm, m < s . Consider the perception of the attribute 'roominess' of a car. It could be aggregated from leg space, inside height, trunk volume, etc., which are actionable product characteristics to be measured on a discrete scale. This would establish m < s. However, m < s would lower the speed of calculation. Leaving aside other problems that arise with the theoretical considerations given on CPU-time (such as neglect of comparison and assign statements [4, p. 365]) to solve the problem, this is of crucial importance for the economy of the algorithm. The second algorithm presented is of nonlinear programming type. Interestingly, it is noted that it "has the advantage thai the set of feasible solutions need not be discrete" [4, p. 367]! However, this algorithm does not guarantee global optima [4, p. 367]. A comparison of computing times with the algorithm presented by Albers [1] for the single choice model cannot be considered adequate. The latter algorithm is not constrained to discrete solutions for y*, and it leads to globally optimal solutions rather than to local optima. Recent improvements and comparisons with grid searches show its superior performance on a problem of positioning a political party among voters [2]. With respect to the consideration of costs and prices it is hard to share the view that "most... re-
0377-2217/82/0000-0000/$02.75 © 1982 North-Holland
K. Brockhoff / ,4 note on product positioning
lationships are objectively measurable" [4, p. 369]. No empirical evidence is presented to this point. However, if relationships were not linear as assumed, this would complicate the objective function. If local optima are considered sufficient, this may not add severely to solution problems. Otherwise it could preclude the construction of unimodal or even convex problem structures. This problem is furthermore related to the choice of solution spaces. It is argued convincingly that price should be considered as one of the perceptual characteristics [7]. If so, the problem appears to be different from considering price as an 'objective' product characteristic. A mixed conception of subjective and objective attributes (hybrid space) apparently and unfortunately cannot be derived from multidimensional scaling. From this short discussion it appears to be fruitful to follow a two stage approach. In the first stage optimal product positions fo maximize sales are sought in a perceptual space. Various procedures have been presented to solve this problem, among them [1]. In a second stage (that would rely on further research of transformation functions between 'objective' and 'perceptual characteristics') minimum cost solutions could be sought for the sales maximizing product position. If profit maximization is to be achieved these procedures Would have to be repeated in an iterative process for second best, third best.., sales maxima to check for global profit optima. Another point of criticism refers to the Bachem/Simon objective function. It optimizes expected profits by use of the purchase probability Pk(Y), where k = 1,2...... n is a set of consumers. The probability Pk(Y) is defined such as to encompass behavioral assumptions that underlie both the Shocker/Srinivasan [6] and the single choice models [5]. However, Shocker/Srinivasan use Pk(Y) only to calculate another parameter, namely the product's share in the k th consumer's purchases (or "the individual's likelihood of choosing prod-
91
ucts of the firm to his likelihoods of purchasing all products" [6,p. 932]). This parameter is used in their objective function rather than Pk(Y). As both parameters are not related linearly, it remains to be shown that the Bachem/Simon objective function would encompass the Shocker/Srinivasan objective function. Let us finally note that Bachem/Simon give a very general formulation of the single choice model by assuming that purchase probability for a product by any one customer equals one, if its distance to the ideal point of this customer is smaller than some given value a k (or--with reference to their fig. l-ilk). While this is correct in general, a~ has been specified to equal the minimum distance between the ideal point and any product known to the kth customer in [5,8.3,1]. This assumption saves to specify one extra parameter.
References [I] S. Albers, An extended algorithm for optimal product positioning. European J. Operational Res. 3 ( 1979l 222-231. [2] S. AIbers, Optimal positioning of political parties, Prec. Operations Res. I I (1981) to appear. [3] S. Albers and K. Brockhoff, Optimal product attributes in single choice models. J. Operational Res. Soc. 31 (1979) 647-655. [4] A. Bachem and H. Simon, A product positioning model with costs and prices, European J, Operational Res. 7 fl981) 362-370. [5] K Brockhoff and H. Rehder. Analytische Planun~g yon PJodukten im Raum der Produkteigenschaften, in: E. "Fopritzhofer, Ed., Marketing--Neue Ergebnisse aus Forsehung und Praxis (Gabler. Wiesbaden, 1978L [6] A.D. Shocker and V. Srinivasan, A consumer based methodology for the introduction of new product ideas, Management Sci 20 (19%~1 ~]!~ t}37. [7] V. Srinivasan, Comments on the role of price in individual utility judgments. Grad. School of Bus., Stanford Univ., Res. Pap. Ser. 569 (1980). [8] F. Zufryden. ZIPMAP--a zero-one integer programming model for markets segmentation and product positioning, J. Operational Res. Soc. 30 (1979) 63-70.
Short Communications
A note on product positioning K. B R O C K H O F F lnstitut fiir Betriebswirtschaftslehre. Christian-AlbrechtsUniversitiit Kiel. D-2300 Kiel. German); Fed. Rep. Received August 1981
Critical remarks are added to a paper that appeared in this journal [4]. We try to clarify assumptions on the objective function, and we refer to the problem of comparing algorithms if the problem structure differs.
In their recent paper, Bachem and Simon [4] have presented a model to position optimally one new product in a space that is spanned by product characteristics. The objective function may incorporate costs and prices to calculate new product positions that maximize expected profits. This has been suggested in [6] but was not made explicit. The authors suggest an assumption on buying probability that incorporates two others: those used by Shocker and Srinivasan [6] and those based on the single choice axiom, used by various authors [5,8,3,1]. Furthermore, two algorithms are suggested to solve the planning problem that resuits. A first point of criticism refers to the solution space that is used by the authors. Product attributes are "represented as dimensions of an mdimensional normed linear space L (e.g. R")". Both, consumers' perceptions of each attributes of existing brands as well as perceptions of most preferred attribute levels, called 'ideal points', are represented as elements of L [4, p. 362]. However, possible realizations of the new product y = (y~ ..... y,,) are chosen such as yERCL. It is assumed: "Surely R must be discrete" [4, p. 364]. In general, perceptual spaces are derived from multidimensional scaling measurements, while actionable attribute spaces result from conjoint analyses.
North-Holland Publishing Company o European Journal of Operational Research 9 (1982) 90-91
This assumption on R is crucial for the performm~ce of the enumeration procedure that is suggested as the first algorithm in the paper [4] to solve the product positioning rwoblem. Clearly, if y* ~ R is an optimal solutioa with respect to some objective function and constraints, this is not necessarily optimal in L. This could be taken lightly, if optimal product positions had in general flat optima. Unfortunately, this is not true as can be read from numerous simulation runs. If, however, R is constrained to discrete values for a simplification of product planning, this raises two questions: Firstly, what is the transformation of 'objective' planning characteristics to 'subjective' perceptual characteristics.'? Secondly, why should have R the same dimensionality as has L? By contrast, it is plausible to assume R C Rs and L C Rm, m < s . Consider the perception of the attribute 'roominess' of a car. It could be aggregated from leg space, inside height, trunk volume, etc., which are actionable product characteristics to be measured on a discrete scale. This would establish m < s. However, m < s would lower the speed of calculation. Leaving aside other problems that arise with the theoretical considerations given on CPU-time (such as neglect of comparison and assign statements [4, p. 365]) to solve the problem, this is of crucial importance for the economy of the algorithm. The second algorithm presented is of nonlinear programming type. Interestingly, it is noted that it "has the advantage thai the set of feasible solutions need not be discrete" [4, p. 367]! However, this algorithm does not guarantee global optima [4, p. 367]. A comparison of computing times with the algorithm presented by Albers [1] for the single choice model cannot be considered adequate. The latter algorithm is not constrained to discrete solutions for y*, and it leads to globally optimal solutions rather than to local optima. Recent improvements and comparisons with grid searches show its superior performance on a problem of positioning a political party among voters [2]. With respect to the consideration of costs and prices it is hard to share the view that "most... re-
0377-2217/82/0000-0000/$02.75 © 1982 North-Holland
K. Brockhoff / ,4 note on product positioning
lationships are objectively measurable" [4, p. 369]. No empirical evidence is presented to this point. However, if relationships were not linear as assumed, this would complicate the objective function. If local optima are considered sufficient, this may not add severely to solution problems. Otherwise it could preclude the construction of unimodal or even convex problem structures. This problem is furthermore related to the choice of solution spaces. It is argued convincingly that price should be considered as one of the perceptual characteristics [7]. If so, the problem appears to be different from considering price as an 'objective' product characteristic. A mixed conception of subjective and objective attributes (hybrid space) apparently and unfortunately cannot be derived from multidimensional scaling. From this short discussion it appears to be fruitful to follow a two stage approach. In the first stage optimal product positions fo maximize sales are sought in a perceptual space. Various procedures have been presented to solve this problem, among them [1]. In a second stage (that would rely on further research of transformation functions between 'objective' and 'perceptual characteristics') minimum cost solutions could be sought for the sales maximizing product position. If profit maximization is to be achieved these procedures Would have to be repeated in an iterative process for second best, third best.., sales maxima to check for global profit optima. Another point of criticism refers to the Bachem/Simon objective function. It optimizes expected profits by use of the purchase probability Pk(Y), where k = 1,2...... n is a set of consumers. The probability Pk(Y) is defined such as to encompass behavioral assumptions that underlie both the Shocker/Srinivasan [6] and the single choice models [5]. However, Shocker/Srinivasan use Pk(Y) only to calculate another parameter, namely the product's share in the k th consumer's purchases (or "the individual's likelihood of choosing prod-
91
ucts of the firm to his likelihoods of purchasing all products" [6,p. 932]). This parameter is used in their objective function rather than Pk(Y). As both parameters are not related linearly, it remains to be shown that the Bachem/Simon objective function would encompass the Shocker/Srinivasan objective function. Let us finally note that Bachem/Simon give a very general formulation of the single choice model by assuming that purchase probability for a product by any one customer equals one, if its distance to the ideal point of this customer is smaller than some given value a k (or--with reference to their fig. l-ilk). While this is correct in general, a~ has been specified to equal the minimum distance between the ideal point and any product known to the kth customer in [5,8.3,1]. This assumption saves to specify one extra parameter.
References [I] S. Albers, An extended algorithm for optimal product positioning. European J. Operational Res. 3 ( 1979l 222-231. [2] S. AIbers, Optimal positioning of political parties, Prec. Operations Res. I I (1981) to appear. [3] S. Albers and K. Brockhoff, Optimal product attributes in single choice models. J. Operational Res. Soc. 31 (1979) 647-655. [4] A. Bachem and H. Simon, A product positioning model with costs and prices, European J, Operational Res. 7 fl981) 362-370. [5] K Brockhoff and H. Rehder. Analytische Planun~g yon PJodukten im Raum der Produkteigenschaften, in: E. "Fopritzhofer, Ed., Marketing--Neue Ergebnisse aus Forsehung und Praxis (Gabler. Wiesbaden, 1978L [6] A.D. Shocker and V. Srinivasan, A consumer based methodology for the introduction of new product ideas, Management Sci 20 (19%~1 ~]!~ t}37. [7] V. Srinivasan, Comments on the role of price in individual utility judgments. Grad. School of Bus., Stanford Univ., Res. Pap. Ser. 569 (1980). [8] F. Zufryden. ZIPMAP--a zero-one integer programming model for markets segmentation and product positioning, J. Operational Res. Soc. 30 (1979) 63-70.