1 The representation of dynamic trajectories with the ...

The representation of dynamic trajectories with the Kolm's triangle for multiple-factor decision problems Pierre L. Kunsch(1) and Alain Chevalier(2) (1) Corresponding author, Free Universities of Brussels, avenue A. Buyl 12, BE - 1050 Brussels, Belgium; E-mail: [email protected] (2) ESCP-EAP, avenue de la République 79 Cedex 11, FR 75543 Paris, France E-mail: [email protected] Abstract: Decision problems often have a multidimensional character: they appear to depend on three, four, or more key factors responsible for their internal dynamics. These factors can be attributes, criteria, bargaining powers, interactions between agents, etc. The paper proposes to use the simple but insight-providing concept of Kolm's triangles to represent in an easy visual way the relative importance of these factors, and their possible dynamic evolution. This possibility results from interesting properties of equilateral triangles. The authors provide 3-D and 4-D examples of the use of Kolm’s triangles for visualising in a plane the importance of the key factors and their dynamic changes. Higher-dimension problems can be represented as well. The discussion is restricted to cases in which a full balance between the factors is sought for. A balance indicator is defined with respect to the ideal positioning of the multiple factors. Keywords: decision theory, multiple factors, importance, Kolm's triangle, dynamic trajectories

1. Introduction In the following paper we present how multiple-factor dynamics can be visualised thanks to the representation of so-called equilateral Kolm’s triangles. With this technique the relative importance expressed in percents of three or more factors important for understanding any decision systems can be directly represented in the plane. These factors may be express the relative importance of attributes, the weights in a multicriteria problems, the relative bargaining powers, the concentrations in a mixture, the components of a measure or an index, etc., as long as they sum up to one. This type of representation has been occasionally used in Multicriteria Decision Aid.

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In 3-D multi-objective linear programming problems Clímaco et al. (1989) have used the triangular representation, called TRIMAP, to show all efficient solutions in the weight space. Indifference regions corresponding to the same basic efficient solutions are identified. In this way the decision-makers can make experiments on additional constraints or trade-offs, which facilitates the selection of compromises. But our purpose is here different from such static representations. It is to evidence the performances of multi-factor trajectories. In addition, we will only consider cases in which a necessary condition for the harmonious functioning of the dynamical system under scrutiny imposes a near balance between the relevant factors or attributes. This is equivalent to say that an ideal point in Kolm’s triangle lies at its centre of gravity, or alternately that the factors have the same weight value. A balance indicator for the system will be elaborated by comparing the dynamic trajectory with this reference point. An easy extension can be found to four and more factors by considering the equivalent properties of regular tetrahedrons and 3-D projections onto one of their planar facets. In section 2, we describe the approach for the 3-D case, and how it can be extended in a straightforward way to 4-D and more dimensions. In section 3 examples are given of the application of the Kolm’s triangle for cases akin to the ‘Classical Triangle’ in the agricultural food sector. Another case previously published in Kunsch and Chevalier (1998) is described in the control of the different management levels of companies. We also describe how to represent the 4-D quality scoring of the divestiture of companies. So-called ‘balance indicators’ are elaborated in each case. Conclusions are given in section 4. 2. The use of the Kolm’s triangle 2.1 The representation of 3-D problems In his work "La Bonne Economie” (The Good Economy), Kolm (1984) claims that three basic forces shape the socio-economic systems, i.e. the “Market” (M), the “State” (S) and the “Autonomy” (A). The "Autonomy" is defined as the set of unstructured microagents and citizens, often referred to as "civil society". The interactive system of the three forces is represented in the so-called "Kolm's Triangle”. References to the interpretation of the Kolm's Triangle presented here are given in De Roose and Van Parijs (1991), and Chevalier and Kunsch (1997). In Fig. 1, the internal points of the equilateral triangle represent all existing and possible societies

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resulting from the interaction of the three forces (M, S, A), each of them being located at a vertex. For example, the oil spot market in Rotterdam is clearly situated entirely in the vertex "M", so that its importance is IM=1. The work of the national or European civil servant is in the vertex "S", so that its importance is IS=1. The individual working in the garden on a Saturday afternoon is in "A", so that its importance is IA=1. Each internal point in the triangle represents the A AUTONOMY A% +S% +M%

= 1 A M

S%

M% A%

M

S

STATE

S

MARKET

combination of the three influences (A, M, S). Fig.1. Multi-agent representation of Kolm's triangle in the initial macroeconomic setting

The importance IM , IS or IA in this particular point is directly given by the ratio in % between the height H i , drawn from this point to the edge opposite to the corresponding vertex A, M, or S, and the full height H. Assuming that the full height of the equilateral triangle is chosen as the unit length, i.e., H=1, it then comes: Ii = Hi where i= M, S, A

(2.1)

Under this condition a useful property of the equilateral triangle can be used which states that:

∑

i = A , B ,C

Hi =

∑

i = A , B ,C

I i = 1 = 100%

(2.2)

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Thanks to the property (2.2), this triangular diagram is used in several scientific fields whenever three key factors have to be represented. Examples are frequent in chemistry for the analysis of phase diagrams with three compounds, in ecology for representing soil textures, in geography for the analysis of evolutionary data, etc. In general, this technique is applicable for the interpretation of three variables linked by relations of the type xi = 1 (Schärlig 1994). As

∑ i

said above we will concentrate on the performance of systems characterised by a near equilibrium between factors or attributes. An equal importance (1/3) of each of the three factors (M,S,A) is of course found at the triangle's centre of gravity (COG) G, because the three heights intersect at a distance of 1/3 of the total height above the edge opposed to each vertex. An internal point close to G will thus represent a ‘well-balanced solution’ under those conditions. Under this constraint, a straightforward 3-D balance indicator “BI” will be the following expression, vanishing in the COG, seen as the ideal equilibrium point:

BI 3 = ∑

i =1,3

1 − Ii 3

(2.3)

2.2 The representation of 4-D problems The property (2.2) of the Kolm triangle can be immediately transposed to the regular tetrahedron in the 4-D space (see Fig. 2.a), in which four key factors are adequate in the representation and are settled at each of the four vertices. As in the equilateral triangle, the regular polyhedron has the property that, for any internal point, the sum of the distances perpendicular to each of the four triangular facets is equal to the total height. These distance can again be used to represent the importance’s (or the weights) of all four key factors. Clímaco and al. (1997) have indeed extended in this way TRIMAP to SOMMIX for representing efficient solutions of 4-D MOLP problems. A cut in the fourdimensional weight space by giving a defined value to one of the weight brings back to the 3-D representation of TRIMAX. We proceed in a different way, as now explained. If the total height drawn as a perpendicular from each vertex to the opposite facet of the polyhedron is normalised to 1 (of course all four heights are equal for symmetry reasons), then the sum of the importance’s will be equal to one as in equation (2.2). As a special case it can be shown that the centre of gravity (COG) of the regular

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tetrahedron settles at (¼, ¼, ¼, ¼ ) height above each facet. This corresponds to a situation in which all four vertices indeed have the same importance. This 3-D representation is however not very convenient, and it is better to go back to a planar representation which can easily be drawn and understood at a glimpse. For obtaining this result, one vertex, for example vertex D in Fig. 2.a, can be selected. Doing so, the tetrahedron (A,B,C,D) can be unfolded to four identical Kolm’s triangles in the horizontal plane defined by the internal triangle (A,B,C), as shown in Fig. 2.b. We keep the convention that the heights of all four triangles in Fig. 2.b are equal to one. The three external triangles have the common vertex D. For any internal point of the tetrahedron, the importance of D is represented by a percentage ID% of the total height H of the tetrahedron. The horizontal plane passing through any internal point will intersect the three external triangles along horizontal lines building an equilateral triangle (A’,B’,C’) at the distance ID * H above the (A,B,C)-plane. As the height of all four Kolm’s triangle is one by definition, so that ID%*1= ID, the distances (AB,A’B’), (BC,B’C’) and (CA,C’A’) in the three lateral facets are equal to ID. Calling θ the angle between the lateral facets and the plane (A,B,C), it is well known that cos θ=1/3 (θ=70° 32’). The projected equilateral triangle (A’,B’,C’) onto the (ABC) plane thus has sides parallel to the triangle (A,B,C) at a distance of ID cos θ= ID /3. It is easy to see from Fig. 2.c, how this new Kolm’s triangle (A’,B’,C’) can be constructed: the initial triangle (A,B,C) is contracted in the direction of its COG by the factor (1- ID ). The vertices (A’,B’,C’) are lying at the distance 2/3ID from the corresponding (A,B,C) vertices. Note that the projection onto the plane of the centre of gravity (COG) G of the polyhedron will come to lie exactly at the COG of the (A,B,C) and (A’,B’,C’) triangles, as can easily be seen. Moreover ID %(G)= ID (G) =0.25, so that the side B’C’ is parallel to BC at a distance of 0.25/3; A’ is at 0.5/3 away from A; the heights of (A’,B’,C’) are thus 0.75; the COG is located at 1/3 of the heights. Calling H i' , i = A' , B ' , C ' the three heights in triangle A’B’C’, the 3-D property (2.2) is immediately transposed to the 4-D case:

∑

i = A ', B ',C '

H i' + I D = 1

(2.4)

5

1

C

D

D

2 0.5 D

1

0

A

B

C 0 1

-0.5 0.5 0

-1 A

1

B

0

-1 -2

1

-1

0 D

1

2

1 C

0.5

C'

0

G'

C,C'' 0.5 G''

A'

0 B'

A -0.5 -1

-0.5

0

0.5

B''

A '' ID

B

A 1

-0.5 -1

-0.5

B 0

0.5

1

Fig. 2. Representation for m=4 2.a The Kolm’s tetrahedron (left above). 2.b. Unfolding of the tetrahedron in the plane to give four Kolm’s triangles (right above). 2c. Projection onto the (A,B,C) plane of the intersection (A’,B’,C’) of the plane at height equal to the importance ID with the Kolm’s tetrahedron. 2.d. Reorganisation of the two Kolm’s triangles. G” is the reference COG of the ‘ideal’ triangle for ID=0.25. It is located at 1/2 from the basis.

The presentation of Fig. 2.c is still not the best one, as the importance of ‘D’ as compared to ‘A’,’B’,’C’ are not immediately clear. In order to display the full importance of ‘D’ value on one side, it is indicated to shift the internal triangle along its vertical height to (A”,B”,C”), as shown in Fig. 2.d, so as to have CC”=0 (of course shifting the triangle towards A or B would give equivalent representations). The importance ID then fully appears at the distance between the two bases and it can be compared with I A , I B , I C , which are the perpendicular distances

H i" , i = A ", B ", C " towards the edges of the (A”,B”,C”) triangle. Calling H D" the (AB, A”B”) distance, one gets the full equivalent of (2.2):

∑

i = A , B ,C , D

H i' =

∑

i = A, B ,C , D

I i = 100%

(2.5)

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The position of the COG of (A”,B”,C”) now depends on I D , so that this point is not directly useful for assessing the ‘quality’ of the positioning within the triangle. The reference point will be the displaced projection of the COG of the tetrahedron (A, B, C, D), say G’’. As in this point, I D = H D" =1/4=0.25 and I C = H C" = 0.25 , G’’ is located at mid-value of the vertical height of the (A,B,C) triangle. Any point close enough to G’’ will thus represent the ‘ideal’ well-balanced situation, under the additional condition to have at the same time equal values of I C and I D . This approach permits also to represent the dynamic evolution within the Kolm’s triangle by representing two trajectories: one for the internal point in triangle (A”,B”,C”) and one for the position of the edge A”B”. From each time snapshot both points are set on the same vertical straight line, as shown in Fig. 3.

1

0.8 C 0.6

0.4 IB ' IB P II

0.2

IA ' IA PI

G'' 0 IC'

IC

ID'

ID

-0.2 A -0.4 -0.8

-0.6

-0.4

-0.2

0

0.2

B 0.4

0.6

Fig. 3. 4-D case. Dynamic evolution of point PI to point PII represented by four importance’s in the Kolm’s triangle. G’’ is the reference COG located at ½ vertical height for obtaining a well-balanced interaction of four factors.

An upper and a lower point on the same vertical straight-line represent any combination of four importance values. The 4-D balance indicator of any given combination Ii , I=1,4 is given by the following expression, equivalent to (2.3):

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BI 4 =

1

∑ 4−I

i =1,4

i

(2.6)

This expression is fully symmetrical in all four components, although the three first components are computed differently: each of them is equal to the distance between the centre of gravity G’’ and the corresponding straight- line parallel to each side of the internal triangle passing through the upper point. The fourth line is equal to the distance between the lower point and the straight-line parallel to the basis of the external triangle at distance ¼. 2.3 Representation of higher dimensional problems Multi-factor systems with dimensions higher than four can also be represented with the Kolm’s triangle or its generalisation to a tetrahedron. First, it must be remarked that the tetrahedron is the only polyhedron in which all vertices are pair-wise connected. A simple combinatorial analysis shows that for m factors to be connected pairwise Cm2 equilateral triangles must be connected, i.e., -

for m=3, 1 triangle, i.e., the original Kolm’s (A,B,C) triangle; for m=4, 4 triangles, i.e., the Kolm’s tetrahedron projected on the (A,B,C) plane; for m= 5, 10 triangles, impossible to join in the 3-D space; for m>5, this impossibility remains.

Nevertheless, it is possible to apply the Kolm’s representation to any dimension, with a growing loss of visibility, however, and it is why it is not recommend it for decision problems of dimensions much greater than four. This can be done in the following way: As we have seen in section 2.2 and Fig. 2, the Kolm’s tetrahedron is a convenient representation for 4-D problems, as it can be visualised in the plane by selecting one vertex, e.g. ‘D’, and then projecting onto the (A,B,C)plane. In this way a 4-D problem is reduced to a 2-D problem, in which two similar embedded equilateral (A, B, C) and (A’, B’, C’) Kolm’s triangles are produced. Therefore it must be possible to reduce a 5-D problem to a 3-D problem by producing two similar embedded regular (A, B, C, D) and (A’, B’, C’, D’) Kolm’s tetrahedrons with identical COG. This can easily be done by contracting the polyhedron by a factor equal to (1-IE) in the direction towards the COG. IE represents the importance of the fifth factor located in E. As an immediate result, the distances AA’, BB’, CC’, DD’ are equal to three fourth of the IE, i.e., to

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0.75* IE. Remember that, in a regular polyhedron, the COG is located along the height at ¾ of its total from each vertex. Of course the COG of the polyhedron projects perpendicularly onto the COG’s of each facet. It is however more difficult to visualise in a 3-D space that in a 2D plane. Therefore, again a projection onto the (A,B,C) plane can do it. A first internal and similar triangle (A’,B’,C’) is obtained from which the importance of E can be read, exactly as in Fig. 2.c. Because the COG’s are all preserved, this triangle is obtained by contracting triangle (A,B,C) in the direction of the COG of (A,B,C) by a factor (1-IE) in the same way, but not for the same reason, as explained in section 2.2. The distances AA’, BB’, CC’ are 2/3 IE. A second similar triangle (A’’,B’’,C’’) of height H” is obtained by a contraction of (A,B,C) in the direction of the COG by a factor (1-IE-ID), so that: IA+IB+IC=H’’/H=H (1-IE-ID) /H

(2.7)

So that we obtain, as it should be: IA+IB+IC +ID+IE = 1 = 100% (2.8) Now of course, the two new triangles are again shifted towards corner C of the first triangle, so as to obtain two lateral bands of breadth IE, ID as in Fig. 4, to be compared with Fig. 2.d in the 4-D case. 1

0.8 C C" 0.6

0.4 G" 0.2

0

B"

A" ID

-0.2 IE

A -0.4 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 4 Representation of the m=5-D case showing three embedded similar triangles. G” is the reference COG at (m-2)/m= 3/5 from the basis.

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Of course this process can be applied successively for 6-D, 7-D, and finally m-D problems, generating (m-2) Kolm’s triangles. The importance of weights is made visible by looking at the breadths of (m3) importance bands between the bases of the successive triangles. If the multiple factors are ordered such as: IA>IB>IC>ID>IE> …

(2.9)

The quality of a positioning can be measured by the distance between a given internal point and the projected and displaced COG(m) of the ‘ideal’ (A,B,C,D) polyhedron used as a reference. It is easy to show that: m H COG =

m−2 m

(2.10)

m where H COG represents the perpendicular distance of the COG(m) from the edge AB of the facet triangle (A, B, C). A necessary condition for m . This is not having a well-balanced situation is to come close to H COG a sufficient condition however, as the additional factors in excess of three may compensate. In addition, the breadths of the (m-3) importance bands located between the successive triangles, representing the importance’s I D , I E ,... for m>4, must all be comparable (see Fig. 4 for the case with two bands). The balance indicator is given by:

BI m =

∑

i =1, m

1 − Ii m

(2.11)

Note that this expression is fully symmetrical in all m-components, although again the three first components are computed differently. 3. Examples of use of the Kolm’s triangle in the 3-D and 4-D cases 3.1. 3-D problems The original Kolm’s triangle (Kolm 1984) is just a prototype of the socalled ‘classical triangle’ which is massively present in the literature. Let us quote here, in the ecological literature, the triangle of (economy, environment, social system), and in quality assurance the triangle of (producer, client, regulator). In the classical triangle the importance’s represent the balance of strengths, or the bargaining powers of the three key factors.

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3.1.1.

The champagne and yoghurt markets

The dynamic properties are very important as shown in the following unpublished results of the authors in the yoghurt and champagne market (Chevalier and Kunsch 1997). The Yoghurt market Over the past fifteen years about, fresh products, and most particularly yoghurt, have experienced a change in their design and manufacturing, both with respect to production techniques, operating conditions and distribution network. Former family or co-operative production has become very competitive and dominated by the "Market" after waves of mergers and acquisitions made possible by increased cash flows of large companies. However, the "State" and the "Autonomy" have been exerting counteracting forces as follows. Firstly, a successful fulfilment of commercial objectives has developed a constant relationship of the producers and distributors with the State authorities in order to acquire influence on the development of national and European regulations which threaten to become increasingly severe. Secondly, continuous communication with the "Autonomy" and with its agents (mainly consumers and ecologists) has become a stringent necessity to maintain sale figures. Fig. 5 shows a representation in the Kolm’s triangle of the dynamic evolution of the balance of power between the three agents over the past fifteen years. This figure shows a clear reinforcement of both “Market” and “State”, accompanied by a substantial weakening of “Autonomy”.

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A

m

I w m

w S

II

S

S

M

Fig. 5. Trajectory of the yoghurt market (from I to II) in Kolm’s triangle. Balance of strength for the three agents {A,M,S} (w = weak; m = average; S = strong)

The Champagne market Champagne has also recently experienced a rapid evolution. It is no longer just a rare, prestigious, but costly quality product, put on the market by its original manufacturer. It rather became a product that is more ordinary, but up market, less of a craft and more industrialised, to be purchased world-wide. This development required the use of modern management techniques with a view on tighter commercial objectives. However, this orientation meant increased risks of imitations and lower standard versions, bringing in turn a loss of quality and a growing indifference towards origin. Therefore at the same time as the influence of the "Market" have been strengthening, enterprises decided to reclaim some traditional roots and consideration from "Autonomy" while exerting pressures on national States and on the authorities in the European Union to get more protection for their original brands. Fig. 6 echoes Fig. 5 for yoghurt. It presents a qualitative interpretation of this evolution of power relations over the past fifteen years. It shows the strengthening of the “Market” becoming herewith the most powerful agent, accompanied by a limited reinforcement of the “State” (including European Union) requested to provide brand or origin protection to manufacturers. At the same time, the “Autonomy” has been constantly losing ground as ancient patrimonial approaches gave way to modern management techniques.

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A

S

I

w m w m

S

II

S

M

Fig. 6. Trajectory of the champagne market (from I to II) in Kolm’s triangle; Balance of strength for the three agents {A,M,S} (w = weak; m = average; S = strong)

In both cases the balance indicators are rather poor and have undergone further degradation during the described evolution. 3.1.2 An application in the control of companies Other applications are to be found in multi-criteria and in multi-attribute problems. In those problems the importance Ii of any vertex (see equation (2.1)) will correspond to the normalised weights given to this particular criterion or attribute. The originality of the approach also results in the dynamics properties of the complex trajectories. An example, which is akin to a dynamic multi-attribute problem, is given in (Kunsch and Chevalier 1998). It evidences the importance for the management of a company to keep an adequate balance between the importance given to three levels of management: Operational, Strategic and Normative Management levels are represented in the Kolm’s triangle (O,S,N). As in the previous cases discussed in this paper, the ideal lies near the COG of this triangle. The corresponding representation is shown in Fig. 7 to be compared to Fig. 1. In this representation the authors have shown that Kolm's triangle is equivalent to an artificial neuronal network (ANN) with learning capabilities integrating past experience. The action of management will consist in keeping the micro-economic system within the boundaries of a stable ‘viability polygon’ visible in Fig. 7. The latter is close to the COG. Coming to close to the edge of

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this polygon would endanger the survival of the company in the market. The dynamic evolution paths represented by arrows in the figure correspond to the modes of ruin, e.g. bankruptcy, due to the unbalance in management.

N (1)

(2)

(6)

(3) (5)

S

(4)

O

Fig. 7. Kolm's triangle in the firm (from Kunsch and Chevalier, 1998). The vertices correspond to the three management levels O (Operational); S (Strategic); N (Normative). The stability polygon (full lines), close to the center of gravity, and the 6 modes of ruin (broken lines) are represented.

The balance indicator BI3 should keep its relatively small value within the stability polygon to have a viable configuration. 3.2 Development of a 4-D quality scoring for divestiture operations Mergers and acquisitions have been frequent in recent years in the developed economies, like described in Chevalier and Kunsch (2003). They are frequently followed by divestiture operations, in which some companies are first restructured and resold because they are not profitable, or they do not comply with the strategic line of the acquiring company (see Chevalier et al. for a yet unpublished case study). These operations often have important consequences, also for the employees and managers. The objective of the procedure developed at the ESCPEAP Business School in Paris during the period 2001-2002 is to provide a quality scoring of the capacity of the divesting company in managing these operations, which are often accompanied by important

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reengineering steps. The idea is to attribute a global score between [0, 1] to each divestiture operation, and to verify the adequate balance between four main diagnostics: I Managerial Diagnostic, II Human Diagnostic, III Social Diagnostic, and IV Financial-Risk Diagnostic. The procedure for evaluating the four scores is as follows: -

For each diagnostic two sub-criteria are used. For example, the two sub-criteria ‘respect of persons’ and ‘happiness of persons’ are used for establishing the Human Diagnostic. There are thus in total 4x2=8 sub-criteria. Each of them is evaluated by means of six nonambiguous questions, the answer to which being grades on 5-point scales.

-

Three sub-criteria: ‘quality of social dialogue’ (diagnostic III), ‘operational risk’, and ‘financial risk’ (both diagnostic IV), can be directly evaluated on three [0 1] intervals on the basis of existing reports. The latter are available and produced at least once a year according to national and international standards.

-

The remaining five out of eight sub-criteria: ‘conformity with laws’, ‘quality of the managers’ behaviour’ (diagnostic I); ‘respect of persons’, ‘happiness of persons’ (diagnostic II); ‘flexibility and adaptive capability’ (diagnostic III) are indirectly evaluated by enquiries. In each industrial group, four groups of ten employees each (respectively support staff and managers of divested and notdivested companies) are requested to answer questionnaires. Each of the employees has to provide 5x6=30 five-point scale evaluations, so that in total 1200 evaluations are collected per company. These answers are translated into five scores in [0 1] intervals.

-

For each company and each of the four diagnostics, a score in the interval [0,1] is derived, by simply taking the arithmetic average of both sub-criteria score. A global score for each company is calculated as being ¼ of the sum of the four diagnostic scores. Global scores result, giving a quite aggregated first comparison. Equal weights are used, because it is the intention to consider that a good balance between all sub-criteria and diagnostics must be granted. Preference weights would be irrelevant, as they would largely vary according to the multiple stakeholders.

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What was missing so far is the direct comparison of the correct balance between the four aspects analysed in the diagnostics. For doing so, a graphical representation of the 4-D evaluations with the technique of Kolm's triangles has been considered by the authors. Also a ‘balance indicator’ has been calculated according to formula (2.6). For the sake of this representation complementing the global score for each company (j), the relative importance of each diagnostics (i=1,4) was calculated by normalisation as follows:

I i ( j ) = Score(i, j ) /(4* Total Score( j ))

(3.1)

Fig. 8 shows the results for six companies. These data are illustrative of main results found in the ESCP-EAP investigation (2002). Only the balance indicator BI4 will be discussed here. Arbitrarily the external triangle was attributed the diagnostics I to III. The ‘ideal’ band of width 0.25 was attributed to diagnostic IV. These evaluations show that there are three groups of companies with comparable characteristics: -

The first group is visible in the right part of the triangles. It consists of three companies for which there are some elements of weakness related to ‘respect of law’ in diagnostic I. For the cluster an average value of BI4= 0.35 is computed, which confirms the unbalance.

-

The second group includes a unique company which appears to be well balanced: the upper point is very close to the centre-of-gravity G’’, and the lower point is close to the 0.25 band. This good result is confirmed by the global score. A value BI4= 0.06 is computed.

-

The third group of two companies shows a significant unbalance with respect to diagnostics II (flexibility and adaptive capacity) and III (happiness of persons). The average BI4= 0.50 delivers the worst value for the whole sample.

An additional dynamic result is available here. One of the companies is common to the first and third group as indicated by the two arrows, but at different moments in time. It was first located in the third group and later moved to the first one, with some improvement of its performances as can be noticed, thanks to a learning process. Unfortunately diagnostic I decayed somewhat. The full results of this study are presented in an internal ESP-EAP report (2002). A later publication will be considered.

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1

0.8 III 0.6

0.4

0.2

G''

0 IV -0.2 II

I -0.4 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 8. Divestiture balance indicator. The diagram shows the positioning of six companies for the four macro-criteria I, II, II (internal triangle, upper point) and IV (lower band, lower point). The two ideal Kolm’s triangles are drawn representing ‘ideal’, i.e., well-balanced, diagnostics. The centre of gravity of the internal triangle is in G’’. Three sets of companies with comparable characteristics are well visible, as explained in the text. The dynamic evolution of the company symbolised by ‘*’ is indicated by an arrow between two moments in time.

4.

Conclusion

In this paper we have presented the simple but powerful technique of Kolm’s triangles. The purpose is to display the positioning on relative scales of multiple key factors m ≥ 3 , which are important for studying the evolution of dynamic decision systems. It must be understood that these factors represent balances of strength between different agents or

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influences, rather than weights related to preferences of decisionmakers. All examples we have developed assume that the ideal strength positioning gives the same importance to all factors, as a necessary condition for the well-balanced evolution of the system under scrutiny. If factors are thought to represent criteria, their weights should ideally be equal. The changes in performance of trajectories can easily be analysed graphically. A balance indicator for each combination of factors is computed with respect to the ideal well-balanced solution. Each point of a trajectory of m factors is represented by a combination of (m-2) points lying on the same vertical lines in m-2 similar Kolm’s triangles. The balance indicator is calculated as the sum of m distances to the ideal values (centre of gravity of the most internal Kolm’s triangle and bases of Kolm’s triangles with equal importances). The usefulness was shown in examples in application of the classical triangle, which was at the basis of Kolm’s basic idea when studying well-balanced economies. One trajectory is compared to the centre of gravity of the unique Kolm’s triangle. A 4-D case of quality assessment of divestitures has also been sketched. It results in two trajectories plotted in two Kolm’s triangles. The position of first trajectory is compared to the centre of gravity of the ideal internal Kolm’s triangle; the position of the second trajectory is compared to the basis of the same triangle. Similar problems of dimension equal or larger than four can also be considered. The advantages of this approach, as compared to other types of display, like for example histograms, or cobwebs (see Clímaco et al., op. cit.), is that trajectories are visualised at a glance by comparison with reference ideals. A disadvantage is that this approach is not really suitable for problems, which do not correspond to the dynamic characteristics we have described. If some factors are permitted to have much larger weights than other ones, the symmetry properties of equilateral triangles are no longer useful. Another, in our opinion not so severe, drawback is that the transparency of this planar representation with the Kolm’s triangle might suffer somewhat when the number of factors becomes quite large. For m=6, for example, four trajectories have to be compared, respectively to one centre of gravity and three edges of three ideal Kolm’s triangles. In all cases a symmetrical formula delivers a unique balance indicator, however. It is why we think that advantages are more important than drawbacks for the type of decision problems we have in mind. This is true even when m becomes quite larger than four.

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References Clímaco, J., and Antunes, C.H. (1989), Implementation of a userfriendly package – a guided tour of TRIMAP, Mathematical and Computer Modelling, Vol. 12, 1299-1309. Clímaco, J., Antunes, C.H., and Alves, M.J. (1997) From TRIMAP to SOMMIX – Building Effective Interactive MOLP Computational Tools, in, G. Fandel and T. Gal (eds.) Multiple Criteria decision Making, Proceedings of the XII International Conference on Multiple Criteria Decision Making. Lecture Note Series in Economics and Mathematical Systems, 448, 285-296, Springer-Verlag, 1997. Chevalier, A., and Kunsch, P.L. (1997) Multiple Attractor Dynamics in Socio-economic Decision Problems. ESCP working paper, 98/139, 35 p., Paris. Chevalier, A., Kunsch, P.L., and Brans, J.P. (2003) The Belin Frozen Pastries Divestiture Case. A contribution to the development of a strategic control and planning instrument in corporations. Submitted for publication to Operational Research. Chevalier, A., Kunsch, P.L., and Brans, J.P. (2003) A contribution to the development of strategic control and planning instruments. An acquisition case study. Accepted for publication in International Transactions in Operational Research (ITOR), Special issue on IFORS 2002 Edinburgh, 8-12 July 2002. ESCP-EAP (2003) Quality scoring of divestiture operations in six industrial groups. ESCP-EAP Working paper, Paris. De Roose F. and Van Parijs Ph. (1991). La Pensée Ecologiste, De Boeck Université, Brussels. Kolm S-C. (1984). La Bonne Economie. La Réciprocité Générale, PUF, Paris. Kunsch, P.L., and Chevalier, A. (1998). A Multiattractor Model for the Dynamic Control of Companies, European Journal of Operational Research, 109, 403-413.

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Schärlig, Alain (1994) Lecture in Statistics, private communication, University of Genève, Geneva.

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