Design decisions: concordance of designers and effects of the Arrow’s theorem on the collective preference ranking

Abstract

The problem of collective decision by design teams has received considerable attention in the scientific literature of engineering design. A much debated problem is that in which multiple designers formulate their individual preference rankings of different design alternatives and these rankings should be aggregated into a collective one. This paper focuses the attention on three basic research questions: (1) “How can the degree of concordance of designer rankings be measured?”, (2) “For a given set of designer rankings, which aggregation model provides the most coherent solution?”, and (3) “To what extent is the collective ranking influenced by the aggregation model in use?”. The aim of this paper is to present a novel approach that addresses the above questions in a relatively simple and agile way. A detailed description of the methodology is supported by a practical application to a real-life case study.

Appendix

1.1 Statistical meaning of W

This subsection analyzes the statistical significance of W. In general, the W distributions are available in tabular terms for small values of m and n (Kendall 1962). For higher values, ​​the Fisher distribution can be used:

$$F=\frac{{(m - 1) \times {W^{(m)}}}}{{[1 - {W^{(m)}}]}}$$

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with parameters ${\nu _1}$ and ${\nu _2}$ defined, respectively, as

$$\left\{ {\begin{array}{ll} {{\nu _1}=n - 1 - \frac{2}{m}} \\ {{\nu _2}=(m - 1)\left( {n - 1 - \frac{2}{m}} \right)} \end{array}} \right..$$

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When n > 7, ${W^{(m)}}$ can be described by a chi-square distribution $\chi _{r}^{2}=m\left( {n - 1} \right){W^{(m)}}$. $\chi _{r}^{2}$ is distributed as a $\chi _{{n - 1}}^{2}$ with $\nu =n - 1$ degrees of freedom.

For example, considering the data in Table 6, where $m=10$ designers and $n=4$ design concepts, ${W^{(m)}}=0.29$. Applying Eq. (8), it can be obtained:

$$F=\frac{{(10 - 1) \times 0.29}}{{(1 - 0.29)}}=3.67.$$

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The degrees of freedom are, respectively:

$$\left\{ {\begin{array}{ll} {{\nu _1}=4 - 1 - \frac{2}{{10}}=2.8 \approx 3} \\ {{\nu _2}=\left( {m - 1} \right)\left( {n - 1 - \frac{2}{m}} \right)=25.2 \approx 25} \end{array}} \right..$$

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From the tables of the Fisher distribution for a significance of 5%, it is obtained ${F_{5\% ;3;25}}=2.99$.

Since $F>{F_{5\% ;3;25}}$, the significance of the coefficient of concordance for the preference profile in Table 6 is confirmed.