Roy's comment on Bouyssou's example.
Bipolar [0-100]-valued outranking graph
Best choice recommendation, the robustness principle and the stability principle in question
Following Bouyssou's rejection of the stability principle (see Bouyssou's example, October 2005), Roy considers again four decision actions A={a,b,c,d} evaluated on a coherent family F={C_1,C_2,C_3,C_4,C_5} of five criteria of equal significance. On each criterion-function we apply a preference scale from 0 to 100 with an indifference threshold of 10, a preference threshold of 14, and a veto threshold of 50. The following performance tableau is given:
| Actions | C_1 | C_2 | C_3 | C_4 | C_5 |
|---|---|---|---|---|---|
| a | 30 | 85 | 80 | 60 | 70 |
| b | 40 | 60 | 60 | 80 | 75 |
| c | 75 | 60 | 60 | 25 | 75 |
| d | 85 | 40 | 70 | 60 | 55 |
Applying the standard concordance and non-veto principles as promoted in the Electre IS method, we obtain the following valued outranking graph:
| S (in %) | a | b | c | d |
|---|---|---|---|---|
| a | - | 80 | 80 | 0 (veto) |
| b | 60 | - | 80 | 80 |
| c | 40 | 0 (veto) | - | 80 |
| d | 60 | 40 | 60 | - |
On this outranking graph, the best choice would naturally be {a,d} as supported hereafter:
« ... Without even constructing the outranking graph (which, following the Electre IS method, coincides with the one considered by Bouyssou when a concordance level between 60% and 80% is required) it seems reasonable to me to:
1) Not select b once a is selected:
a is indifferent to b on criteria C_1 and C_5; a is preferred to b on criterion C_2 and C_3; b is preferred to a only on criterion C_4.
2) Not select c once a is selected:
a and c are indifferent on criterion C_5, a is significantly preferred to c on criteria C_2,C_3, and C_4; Only criterion C_1 favours c.
3) Selecting d, however, seems justified:
The performance profile of d is quite contrasted compared to a (cf. Criteria C_1 and C_2); d is indifferent to a on criteria C_3 and C_4 and the preference of a compared to d on criterion C_5 is hardly significant; I don't think thus that d may be considered to be the worst of the four decision alternatives. ....»
(Bernard ROY, private communication, November 4, 2005)
Applying the RuBy approach to the outranking graph above gives following results:
| Choice | a | b | c | d | Dom. | Abs. |
|---|---|---|---|---|---|---|
| {b} RuBy choice | 40 | 60 | 40 | 40 | 60 | 0 |
| {c} | 40 | 40 | 60 | 40 | 0 | 60 |
Action b represents the best choice and c represents the worst choice, but only with a concordance level of up to 60%. According to the RuBy methodology, {b} will be here the best choice recommendation, under the condition that a significance of 60% concordance is considered indeed sufficient for supporting it.
This result is not so surprising, as Roy's argument in favour of not selecting b once a is selected, may be mutatis mutandi, applied to the symmetric situation. Following the same scheme of reasonning as Roy, it seams indeed not unreasonable to:
When requiring now, as proposed by Roy above, a concordance level higher than 60%, the RuBy recommendation gets void. Indeed, the given performance tableau does not support any possible choice recommendations at such a high level of concordance.
Let us now drop for one moment the RuBy stability principle. The significant minimal outranking and outranked choices we get are the following:
| Choice | Dominance | Absorbency | Independence |
|---|---|---|---|
| {a,d} | 80 | 80 | 40 |
| {a,c} | 80 | 60 | 20 |
| {b} | 60 | 0 | 100 |
| Choice | Dominance | Absorbency | Independence |
|---|---|---|---|
| {a,d} | 80 | 80 | 40 |
| {b,d} | 60 | 80 | 20 |
| {c} | 0 | 60 | 100 |
As evidenced by Roy above, the choice {a,d} is indeed a highly significant minimal outranking choice. However, it is neither stable nor unstable as the corresponding independence statement is at that significance level logically undetermined. Furthermore it is not an effective best choice, as it is, with the same significance of 80% concordance, as well an outranked choice. A second, highly significant minimal outranking choice appears to be the couple {a,c}, but it is definitely not a stable choice, as a clearly outranks c at concordance level 80%. Naturally we recover here again at a 60% level of concordance both, the unique outranking kernel {b}, as well as the unique outranked kernel {c} previously shown.
It is worthwhile noticing finally that, contrary to Roy's idea, the crisp outranking graph one obtains when dropping all binary outranking statements which are not supported with a significance strictly between 60% and 80% of criterial concordance, is not quite the same as the one originally proposed by Denis Bouyssou. To be logically coherent one has indeed to apply the same symmetric dropping rule with respect to the significance of the falsity of some outranking statement.
| S_{> 60%} | a | b | c | d |
|---|---|---|---|---|
| a | - | 80 | 80 | 0 (veto) |
| b | 50 | - | 80 | 80 |
| c | 50 | 0 (veto) | - | 80 |
| d | 50 | 50 | 50 | - |
We thus get the symmetrically cut graph, shown above, where several pairwise comparisons, such as between b and a, are in fact logically undetermined at this required level of significance, and not strictly false as in the original Bouyssou example (see Bouyssou's example, October 2005).This graph supports, as expected, {a,d} with equally high significance, as both an outranking and an outranked prekernel, i.e a minimal outranking, respect. outranked choice with undetermined independence property.
Following the RuBy robustness principle, i.e. the stability of the best choice recommendation with respect to all possible high significance cuts, we must however conclude that the apparent highly significant minimal outranking choices, such as {a,d}, are in fact artifacts of the specifically proposed high significance cut. Neither a nor d present robust outranking properties that confirm their best choice potentiality when lowering the required significance level to 60%. Indeed, as shown above, at the 60% concordance level, it is b and not a which becomes definitely the unique best choice recommendation.
Let us conclude by noticing that on the one hand, - the bipolar valuation of the pairwise outranking via the concordance and non-veto principles and, on the other hand, - the high significance cutting approach, as proposed by Roy above, are not logically compatible in general. The high significance cut, via its logical polarising effect, may produce artificial results not effectively supported by the given performance tableau. That's why in the RuBy decision aid methodology, we avoid this kind of approach.