A NEW APPROACH TO PORTFOLIO MATRIX ANALYSIS FOR STRATEGIC MARKETING PLANNING 1 2 Vladimir Dobric , Boris Delibasic Faculty of organizational science, [email protected] rs 2 Faculty of organizational science, delibasic. [email protected] rs 1 Abstract: Portfolio matrix is probably the most important tool for strategic marketing planning, especially in the strategy selection stage. Position of the organization in the portfolio matrix and it’s corresponding marketing strategy depends on the aggregation of values of relevant strategic factors. Traditional approach to portfolio matrix analysis uses averaging function as an aggregation operator.

This approach is very limited in realistic business environment characterized by complex relations between strategic factors. An innovative approach to portfolio matrix analysis, presented in this paper, can be used to express complex interaction between strategic factors. The new approach is based on the logical aggregation operator, a generalized aggregation operator from which other aggregation operators can be obtained as special cases. Example of traditional approach to portfolio matrix analysis given in this paper clearly shows it’s inherited limitations.

The new approach applied to the same example eliminates weaknesses of traditional one and facilitates strategic marketing planning in realistic business environment. Key words: Portfolio matrix analysis, strategic marketing planning, logical aggregation, aggregation operator. 1. INTRODUCTION The portfolio matrix analysis is widely used in strategic management [2, 3, 6]. It offers a view of the position of the organization in its environment and suggests generic strategies for the future. Some of the most frequently used portfolio matrices are the ADL (developed by Arthur D.

The difference between aforementioned matrices lies in number and meaning of factors used in the analysis process as well as in the number and generality of recommended strategies. It is common for all the portfolio matrices that the position of the organization in a portfolio matrix is based on estimated values of two factors: the one describing external environment (market attractiveness in DPM) and the other describing inner characteristics of the organization compared to the major competitors (business strengths/position in DPM).

On the basis of portfolio matrix analysis , a generic marketing strategy is recommended based on an organization’s position in the portfolio matrix. In the portfolio matrix analysis, values of two factors describing external and internal environment are estimated as aggregations of values of strategic factors influencing respective environment. The choice of the most adequate aggregation functions depends on the condition in which organization operates, i. e. an aggregation functions describing external and internal environment should have a behaviour which models organization’s external and internal environment conditions respectively.

In the traditional approach to portfolio matrix analysis, weighted arithmetic mean is commonly used as an aggregation function. This aggregation operator describes an averaging behaviour, thus, it can be used to model business environment in which high and low values of strategic factors average each other. In the realistic business environment strategic factors can interact in a more complex way, i. e. they can average each other, reinforce or weaken each other (disjunctive or conjunctive behaviour), or exhibit various forms of mixed interactions [2, 3, 6].

It is clear that the use of weighted arithmetic mean as an aggregation operator can’t express all the possible interactions between strategic factors that exist in a realistic business environment. This explains why the traditional approach to portfolio matrix analysis is highly limited, with the inherited weaknesses that can’t be overcome without substantial modification. Therefore, under previous conditions, it is obvious that a new approach to portfolio matrix analysis is needed.

This new approach must take in consideration all the possible forms of interactions between strategic factors that can occur in a realistic business environment. These interactions can be expressed with a logical aggregation operator, so a new approach to portfolio matrix analysis can be based on this operator. W eighted arithmetic mean and other known aggregation operators are just, as we will see in the following sections, special cases of logical aggregation operator. 2. THE MCDONALD’S DIRECTIONAL POLICY MATRIX (DPM)

Although the DPM, like other models of portfolio matrices, attempts to define an organization’s strategic position and strategy alternatives, this objective can’t be met without considering what is meant by the term „organization“. The accepted level at which an organization can be analysed using the DPM is that of the „strategic business unit“. The most common definition of an SBU is as follows [3]: (1) It will have common segments and competitors for most of the products; (2) It will be a competitor in an external market; (3) It is a discrete, separate and identifiable „unit“; 4) Its manager will have control over most of the areas critical to success. DPM has two dimensions each built up from a number of factors: (1) Market attractiveness and (2) Business strengths/position. Using these factors, and some scheme for weighting them according to their importance, strategic business units are classified into one of nine cells in a 3 X3 matrix. Each cell is connected to a generic strategy recommended by the DPM. Factors used to form aggregated dimensions of DPM vary according to concrete circumstances in which SBU operates. Notice that previous explanations taken rom [3] suggest weighted arithmetic mean as an aggregation operator, thus, traditional approach to DPM analysis only considers a case of averaging behaviour between strategic factors. That is only one of the possible interactions between strategic factors that can occur in realistic business environment. Other possible interactions like conjunction, disjunction or mixed interaction can’ t be modelled by using weighted sum of factors as an aggregation operator. Definitions of market attractiveness and business strengths/positions dimensions are g iven in [3].

Market attractiveness is a measure of the marketplace potential to yield growth in sales and profits. It is important to highlight the need for an objective assessment of market attractiveness using data from the organization’s external environment. The criteria themselves will, of course, be determined by the organization carrying out the exercise and will be relevant to the objectives the organization is trying to achieve, but they should be independent of the organization’s position in its m arkets [3]. Business strengths/position is a measure of organization’s actual strengths in the marketplace (i. . the degree to which it can take advantage of a market opportunity). Thus, it is an objective assessment of an organization’s ability to satisfy market needs relative to competitors. DPM, together with generic marketing strategy options is shown in Picture 1. Picture 1: Directional policy matrix 3. TRADITIONAL APPROACH TO DIRECTIONAL POLICY MATRIX ANALYSIS In this section, traditional approach to DPM analysis using simple example will be presented, highlighting it’s inherited limitations originating from using non-adequate aggregation functions.

Tables 1 and 2 are slight modification of tables that are used in DPM analysis example in [3] on pages 202 and 203, where market attractiveness and business strengths/position are evaluated by using weights and scores of relevant strategic factors. The only modification applied on tables in [3] is the normalization of weights, scores and corresponding evaluations to [0, 1] interval. This is done with simple transformation, which is covered in the following sections. Table 1: Market attractiveness evaluation Strategic factor (Fi) Score (si) Total (M) 0. 25 0. 25 0. 5 0. 15 0. 1 0. 1 1. Growth 2. Profitability 3. Size 4. Vulnerability 5. Competition 6. Cyclicality W eight (wi) 0. 6 0. 9 0. 6 0. 5 0. 8 0. 25 0. 15 0. 225 0. 09 0. 075 0. 08 0. 25 Total 1 0. 645 Table 2: Business strengths/position evaluation Strategic factor (Fi) 7. Price 8. Product 9. Service 10. Image Total W eight (wi) 0. 5 0. 25 0. 15 0. 1 1 You company Competitor A Competitor C Score (si) Total (B) Score Total (A) Score Total (C) 0. 5 0. 6 0. 8 0. 6 0. 25 0. 15 0. 12 0. 06 0. 6 0. 8 0. 4 0. 5 0. 3 0. 2 0. 06 0. 05 0. 4 1 0. 6 0. 3 0. 2 0. 25 0. 09 0. 03 . 58 0. 61 0. 57 Market attractiveness (M) and business strengths/position (B) are evaluated using weighted arithmetic mean as an aggregation function of scores {s1, …, s6} and {s7, …, s10} given for relevant strategic factors {F1, …, F10} using weights {w1, …, w10}: M = w1 s1 + w2 s2 + w3 s3 + w4 s4 + w5 s5 + w6 s6 = 0. 645 (1) B = w7 s7 + w8 s8 + w9 s9 + w10 s10 = 0. 58 (2) The same equations can be given in matrix form: M = W M SM (3) B = W B SB (4) where M and B are market attractiveness and business strengths/position evaluation respectively, W M = [w1, T , w6] and SM = [s1, …, s6] are weighting and scoring vectors for market attractiveness strategic factors , T and W B = [w7, …, w10] and SB = [s7, …, s10] are weighting and scoring vectors for business strengths/position strategic factors. Notice that the exact position of the organization on the DPM is not given with business strengths/position value (B), but the relative business strengths/position value (BR), since business strengths/position is actually a measure of organizational abilities (B) (internal environment) relative to the competitors (i. e. respective abilities of market leader) [3].

In our example market leader is Competitor A (from Table 2), thus, organization’s relative business strengths/position value (BR) is calculated as: BR = B/A (5) Relative business strengths/position value (BR) is then plotted on the horizontal axis of the DPM using a logarithmic scale [3]. These explanations are not of importance for the domain of our investigation, so no futher considerations regarding relative business strengths/position value (BR) and DPM plotting are given. In the rest of this paper, the only consideration will be given to market attractiveness (M) and business strengths/position (B) evaluation.

W eighted arithmetic mean used for an aggregation function assumes that the interactions between strategic factors show averaging behavior, i. e. it is used to model business environment in which values of strategic factors average each other. This is the mayor drawback of traditional DPM analysis. Realistic business environment demands more modelling power for more complex factors interactions. Besides averaging, strategic factors can reinforce or weaken each other (disjunctive or conjunctive behaviour respectively), or exhibit various forms of interactions which are neither strictly averaging, conjunctive or disjunctive, but mixed, i. . aggregation function exhibits different behaviour on different parts of the domain (mixed behaviour). Under these circumstances, it is obvious that a new approach to portfolio matrix analysis demands an usage of different aggregation operator, the one capable of modelling all the possible interactions between strategic factors that can take place in a realistic business environment. The paper presents an approach to portfolio matrix analysis, using logical aggregation operator, which eliminates weaknesses of traditional one. If we return to ur example shown in Tables 1 and 2, we can restate possible business external and internal environment conditions in the following way: 1) It is possible that interactions between market attractiveness or business strengths/position strategic factors show averaging behaviour, i. e. scores {s1, …, s6} or {s7, …, s10} given to strategic factors {F1, …, F10} can average each other using weights {w1, …, w10}. In this case market attractiveness and business strengths/position are evaluated as shown in equations (1) and (2) , or in their matrix equivalents (3) and (4). ) It is possible that interactions between market attractiveness or business strengths/position strategic factors show conjunctive behaviour, i. e. scores {s1, …, s6} or {s7, …,s10} given to strategic factors {F1, …, F10} can weaken each other. In this case market attractiveness and business strengths/position evaluation depends upon the lowest score among the relevant factors: M = min(s1, …, s6) (6) B = min(s7, …, s10) (7) 3) It is possible that interactions between market attractiveness or business strengths/position strategic factors show disjunctive behaviour, i. e. cores { s1, …, s6} or {s7, …, s10} given to strategic factors {F1, …, F10} can reinforce each other. In this case market attractiveness and business strengths/position evaluation depends upon the highest score among the relevant factors: M = max(s1, …, s6) (8) B = max(s7, …, s10) (9) 4) It is possible that interactions between market attractiveness or business strengths/position strategic factors show mixed behaviour. For example, scores {s1, …,s6} or {s7, …,s10} given to strategic factors {F1, …, F10} can average, reinforce and weaken each other depending on their values.

Thus, the aggregation function can be conjunctive for low scores, disjunctive for high scores, and perhaps averaging when some scores are high and some are low (different behaviour of aggregation function on different parts of the domain). Example for this kind of aggregation function’s behaviour will be given in the following sections. Logical aggregation operator can express all previous types of interactions, so it naturally imposes itself as a replacement to weighted arithmetic mean aggregation operator in the new approach to portfolio matrix analysis.

Notice that interactions between strategic factors from organization’s external environment (market attractiveness factors) and those from organization’s internal environment ( business strengths/position factors) are not recognized in traditional approach to DPM analysis [3]. If those interactions can be recognized, they can easily be integrated into the model in the new approach. In the following section basic theory of logical aggregation will be briefly examined. After examining the theory, a simple example of new approach to portfolio matrix analysis using Tables 1 and 2 will be presented. . LOGICAL AGGREGATION Aggregation functions are functions with special properties. The purpose of aggregation functions (they are also called aggregation operators, both terms are used interchangeably in the existing literature) is to combine inputs and produce output, where the inputs are typically interpreted as degrees of preference, strength of evidence or support of hypothesis [1]. If we consider a finite set of inputs I = {i1, …, in}, we can aggregate them into single representative value by using infinitely many aggregation functions.

They are grouped in various families such as means, triangular norms and conor ms, Choquet and Sugeno integral, uninorms and nullnorms, and many others [1]. The question arises how to chose the most suitable aggregation function for a specific application. This question can be answered by choosing logical aggregation function – a generalized aggregation operator that can be reduced to any other known one. Logical aggregation is an aggregation method that combines inputs and produces output using logical aggregation operator [4, 5].

In a general case logical aggregation is carrried out in two distinct steps: 1) Normalization of input values which results in a generalized logical and/or [0, 1] value of analyzed input ij: ? ? ? : I > [0, 1] (10) 2) Aggregation of normalized values of inputs into resulting globaly representative value with a logical aggregation operator: n Aggr: [0, 1] > [0, 1] (11) The first step explains the reason for modification of tables from [3] in previous section, in order to obtain Tables 1 and 2 with normalized values of strategic factors’ scores on which logical aggregation operator can be applied.

Operator of logical aggregation in a general case (Aggr?? ) is a pseudo-logical function (???? ), a linear convex combination of generalized Boolean polynomials (?? ) [4, 5]: Aggr?? (? i1? , … , ? in? ) = ???? (? i1? , … , ? in? ) = ? wj? j? (? i1? , … , ? in? ) (12) where (? ) is a generalized product operator and (? ) is an aggregation measure as defined in [4, 5]. Generalized Boolean polynomial ?? is a value realization of Boolean logical function ?. Boolean logical function is an element of Boolean algebra of inputs ? (i1, … , in) ?

BA(I), to which corresponds uniquely a generalized Boolean polynomial ?? (? i1? , … , ? in? ) as it’s value: ?? : [0, 1] > [0, 1] n (13) Logical aggregation operator depends on the chosen measure of aggregation (? ) and operator of generalized product (? ). By a corresponding choice of the measure of aggregation (? ) and generalized product (? ) the known aggregation operators can be obtained as special cases [4, 5], e. g. for additive aggregation measure (? := ? add) and generalized product (? := min) logical aggregation operator reduces to weighted arithmetic mean: Aggradd in (? i1? , … , ? in? ) = ? wj (? ij? ) (14) After considering basic theory of logical aggregation, we can return to the domain of our investigation. In the following section the new approach to portfolio matrix analysis will be presented thoroughly using the same data from Tables 1 and 2. 5. A NEW APPROACH TO PORTFOLIO MATRIX ANALYSIS If we consider again Tables 1 and 2, and four cases of possible business environment conditions as defined in Section 3, we can design new aggregation functions that model all the aforementi oned conditions using logical aggregation operator.

In this section an example to all four types of strategic factors interactions will be given, together with logical functions modeling them. A starting point for the new approach to portfolio matrix anal ysis is a finite set of strategic factors F = {F1, … , F10} and a Boolean algebra BA(F), defined over it. The task of logical aggregation in DPM analysis is the fusion of strategic factors’ scores into resulting market attractiveness and business strengths/position values using logical tools. Logical aggregation has two steps: (1) Normalization of strategic factors’ scores (score Sj corresponds to factor Fj as its predefined value): ? ? : Sj > [0, 1] (15) that results in a logical and/or score sj ? [0, 1] of analyzed strategic factor Fj (j = 1.. |F|). Normalization of scores in S is done with simple transformation. In the original tables in [3], score (Sj) of strategic factor (Fj) belongs to interval [0.. 10], e. g. Strategic factor Growth (F1) has score S1 = 6 in the original table in [3]. The normalized score (s1) for this factor (F1) is given in Table 1 with the following equation: s1 = 6/10 = 0. 6 (16) The same transformation is applied to the rest of the strategic factors in tables in [3], resulting in Tables 1 and 2. 2) Aggregation of normalized scores {s1, … , s6} and {s7, … , s10} of factors {F1, … , F10} into resulting market attractiveness (M) and business strengths/position (B) values with a logical aggregation operator: M = Aggr?? (s1, … , s6) (17) B = Aggr?? (s7, … , s10) (18) Aggregation of scores {s1, … , s6} and {s7, … , s10} for strategic factors {F1, … , F10} is accomplished using generalized Boolean polynomials (? M? ) and (? B? ): Aggr?? (s1, … , s6) = ? M? (s1, … , s6) = [? M(F1, … , F6)]? (19) Aggr?? (s7, … , s10) = ? B? (s7, … s10) = [? B(F7, … , F10)]? (20) Generalized Boolean polynomials ? M? (s1, … , s6) and ? B? (s7, … , s10) are value realizations of Boolean logical functions ? M(F1, … , F6) and ? B(F7, … , F10), which belong to Boolean algebra of strategic factors BA(F). Notice that interactions between strategic factors from organization’s external environment (market attractiveness factors) and those from organization’s internal environment (business strengths/position factors) are not stated in [3]. If they exist, they can easily be integrated into the model.

Adequate generalized product operator (? ) in the domain of portfolio matrix analysis is min operator (? := min). If we return to the possible business environment conditions stated in Section 3, we can formulate logical functions to express corresponding types of interactions between the strategic factors: 1) If the interactions between market attractiveness or business strengths/position strategic factors show averaging behaviour, then the new approach to portfolio matrix analysis reduces to traditional one, as stated in equations (1) and (2), or matrix equivalents (3) and (4). ) If the interactions between market attractiveness or business strengths/position strategic factors show conjunctive behaviour, they are expressed in the following way: ? M = F1 ? F2 ? F3 ? F4 ? F5 ? F6 (21) ?B = F7 ? F8 ? F9 ? F10 (22) Market attractiveness and business strengths/position evaluation are given with corresponding generalized Boolean polynomial (? := and, ? := min): M = Aggrand (s1, … , s6) = ? M min B = Aggrand min = [F1 ? F2 ? F3 ? F4 ? F5 ? F6] min (s7, … , s10) = ? B min min = [F7 ? F8 ? F9 ? F10] min(s1, s2, s3, s4, s5, s6) = 0. 25 (23) min (24) = min(s7, s8, s9, s10) = 0. 5 3) If the interactions between market attractiveness or business strengths/position strategic factors show disjunctive behaviour, they are expressed in the following way: ? M = F1 ? F2 ? F3 ? F4 ? F5 ? F6 (25) ?B = F7 ? F8 ? F9 ? F10 (26) Market attractiveness and business strengths/position evaluation are given with corresponding generalized Boolean polynomial (? := or, ? := min): M = Aggror (s1, … , s6) = ? M min min = [F1 ? F2 ? F3 ? F4 ? F5 ? F6] min max(s1, s2, s3, s4, s5, s6) = 0. 9 (27) B = Aggror (s7, … , s10) = ? B min min = [F7 ? F8 ? F9 ? F10] min = max(s7, s8, s9, s10) = 0. 8 (28) 4) If the interactions between market attractiveness or business strengths/position strategic factors show mixed behaviour (aggregation function exhibits different behaviour on different parts of the domain), they can be modelled with the following logical functions, e. g. realistic external and internal business environment, where strategic factors show mixed behaviour, can be modelled as: ?

If the external environment conditions are that profitabilty (F2), size (F3) and cyclicality (F6) are important, but if the profitability (F2) is not high enough, growth (F1), vulnerability (F4) and competition (F5) are important, we can write the following expression: ?M = (F2 ? F3 ? F6) ? (c(F2) ? F1 ? F4 ? F5) (29) ? If the internal environment conditions are that price (F7) and product (F8) are important, but if the price (F7) and product (F8) are not competitive, service (F9) and image (F10) are important, we can write the following expression: ?B = (F7 ? F8) ? (c(F7 ? F8) ?

F9 ? F10) (30) Market attractiveness and business strengths/position evaluation, for organization’s external and internal environment conditions respectively, are given with corresponding generalized Boolean polynomial (? := min): M = Aggr? (s1, … , s6) = ? M = [(F2 ? F3 ? F6) ? (c(F2) ? F1 ? F4 ? F5)] = = s2 ? s3 ? s6 + (1 – s2) ? s1 ? s4 ? s5 – s2 ? s3 ? s6 ? (1 – s2) ? s1 ? s4 ? s5 = 0. 25 (31) B = Aggr? (s7, … , s10) = ? B = [(F7 ? F8) ? (c(F7 ? F8) ? F9 ? F10)] = = s7 ? s8 + (1 – (s7 ? s8)) ? s9 ? s10 – s7 ? s8 ? (1 – (s7 ? s8)) ? s9 ? s10 = 0. 6 (32) min min min min min min

Remember that when plotting the DPM, the exact position of the organization on the business strengths/position axis (horizontal) is calculated using relative business strengths/position value (BR) and logarithmic scale (see equation (5)), for all aforementioned types of strategic factors interactions . 5. CONCLUSION Traditional approach to portfolio matrix analysis uses weighted arithmetic mean as an aggregation function, thus, it can only be used to model business environment in which strategic factors’ interactions show averaging behavior. This is only one of the four cases of realistic business environment conditions, i. . strategic factors’ interactions showing conjunction, disjunction or mixed behavior are not covered in the traditional approach. The new approach uses generalized aggregation function – operator of logical aggregation. This operator can model all the possible business environment conditions – types of interactions between the strategic factors. This paper shows that traditional approach to portfolio matrix analysis is just a special case of the new one, since the weighted arithmetic mean is actually a special case of logical aggregation operator.

Usage of logical aggregation operator in the new approach clearly improves the traditional one, allowing more modeling power for complex relations among the strategic factors. Since the new approach to portfolio matrix analysis covers all four types of strategic factors’ interactions, it facilitates strategic marketing planning in a realistic business environment. 5. BIBLIOGRAPHY [1] Beliakov G. , Pradera A. , Calvo T. , Aggregation functions: A guide for practitioners , Springer-Verlag, Berlin Heilderberg, 2007. [2] Leibold M. Probst G. J. B. , Gibbert M. , Strategic Management in the Knowledge Economy”, Wiley VCH, 2005. [3] McDonald Malcolm, Marketing Plans (fourth edition), Butterworth-Heinemann, 1999. [4] Radojevic D. , “Logical aggregation based on interpolative Boolean algebra“, Mathware & Soft Computing, 15 (2008) 125 -141. [5] Radojevic D. , “(0,1) – valued logic: A natural generalization of Boolean logic“, Yugoslav Journal of operational Research, 10 (2000) 185 – 216. [6] Roney C. W. , Strategic Management Methodology, Praeger Publishers, 2004.