Analysis of the effectiveness of the theseus multi-criteria sorting method: theoretical remarks and experimental evidence

Abstract

The effectiveness of the THESEUS multi-criteria sorting method is characterized, here, by (i) its capacity for suggesting precise and appropriate assignments; (ii) the probability of suggesting imprecise assignments; and (iii) the probability of suggesting incorrect assignments. We study how these important features are influenced by the number of criteria and categories, the cardinality of the reference set and the level of decision-maker consistency. We present a theoretical characterization and a wide range of experimental results that confirm and complement the formal analysis. The proposed way of analyzing effectiveness may be applied to other multi-criteria sorting methods.

Appendix

1.1 Deriving Equations (14) and (15)

Suppose that $k'<k^{*}$.

Equations (12) and (13) can be arranged as:

$$\begin{aligned} \displaystyle { n }_{ i }(k')= & {} \sum _{ k=k'+1 }^{ { k }^{ * } }{ n(k)f({ k }^{ * }-k) } +\sum _{ k={ k }^{ * }+1 }^{ M }{ n(k)f({ k }^{ * }-k) } +\sum _{ k=1 }^{ k'-1 }{ n(k)f(k-{ k }^{ * }) } \\ \displaystyle { n }_{ i }({ k }^{ * })= & {} \sum _{ k={ k }^{ * }+1 }^{ M }{ n(k)f({ k }^{ * }-k) } +\sum _{ k=1 }^{ k'-1 }{ n(k)f(k-{ k }^{ * }) } +\sum _{ k=k' }^{ { k }^{ * }-1 }{ n(k)f(k-{ k }^{ * }) } \end{aligned}$$

Then,

$$\begin{aligned} \displaystyle { n }_{ i }({ k }')-{ n }_{ i }({ k }^{ * })=\sum _{ k={ k }'+1 }^{ { k }^{ * } }{ n(k)f({ k }^{ * }-k) } -\sum _{ k=k' }^{ { k }^{ * }-1 }{ n(k)f(k-{ k }^{ * }) } \end{aligned}$$

or

$$\begin{aligned} \displaystyle { n }_{ i }({ k }')-{ n }_{ i }({ k }^{ * })= & {} \sum _{ k={ k }'+1 }^{ { k }^{ * }-1 }{ n(k)f({ k }^{ * }-k) } +n({ k }^{* })f(0) \\&-\sum _{ k=k'+1 }^{ { k }^{ * }-1 }{ n(k)f(k-{ k }^{ * }) } -n(k')f(k'-{ k }^{ * }) \end{aligned}$$

Grouping both sums, we obtain Eq. (14).

If $k'>k^{*}$, Eqs. (12) and (13) can be arranged as:

$$\begin{aligned} \displaystyle { n }_{ i }({ k }')= & {} \sum _{ k={ k }'+1 }^{ { M } }{ n(k)f({ k }^{ * }-k) } +\sum _{ k=1 }^{ { k }^{ * }-1 }{ n(k)f(k-{ k}^{ * }) } +\sum _{ k={ k }^{ * } }^{ k'-1 }{ n(k)f(k-{ k}^{ * })}\\ \displaystyle { n }_{ i }({ { k }^{ * } })= & {} \sum _{ k={ k }'+1}^{ { M } }{ n(k)f({ k }^{ * }-k) } +\sum _{ k={ k }^{ * }+1 }^{ { k }' }{ n(k)f({ k }^{ * }-k) } \\&+\sum _{ k=1 }^{ { k }^{ * }-1 }{ n(k)f(k-{ k}^{ * }) }\\ \displaystyle { { n }_{ i }({ { k }' })-n }_{ i }({ { k }^{ * } })= & {} \sum _{ k={ k }^{ * } }^{ { k'-1 } }{ n(k)f(k-{ k }^{ * }) } -\sum _{ k={ k }^{ * }+1 }^{ { k }' }{ n(k)f({ k }^{ * }-k) } \end{aligned}$$

or

$$\begin{aligned} \displaystyle { { n }_{ i }({ { k }' })-n }_{ i }({ { k }^{ * } })= & {} n({ k }^{ * })f(0)+\sum _{ k={ k }^{ * }+1 }^{ { k'-1 } }{ n(k)f(k-{ k }^{ * }) } \\&-\sum _{ k={ k }^{ * }+1 }^{ { k }'-1 }{ n(k)f({ k }^{ * }-k) } -n(k')f({ k }^{ * }-k') \end{aligned}$$

Equation (15) comes from grouping both sums.