Classification of objects into quality categories in the presence of hierarchical decision-making agents

Abstract

In many practical contexts, it is often required to classify some objects of interest into predetermined unordered quality categories. This operation—referred to as quality classification problem—has received considerable attention in many fields of research, such as Analytical Chemistry, Materials Science, Medicine, Manufacturing, Quality Engineering/Management, Decision Analysis. Assuming that multiple agents perform subjective assignments of categories to the objects of interest, a further problem is that of fusing these assignments into global classifications. To this purpose, the mode and the weighted mode are very practical measures, as long as agents are equi-important or their (different) importance is expressed in the form of a set of weights. Unfortunately, these measures are not appropriate for quality classification problems where the agents’ importance is expressed in the form of a rank ordering (hierarchy). The aim of this article is to present a new method, which addresses the latter quality classification problem in a relatively simple and practical way. The peculiarity of this method is that the different importance of agents determines a different priority in considering their assignments and not a different weight of these assignments. A detailed description of the new method is supported by a realistic example in the Analytical Chemistry field.

Appendix

Further considerations on the range of no-voting-order-effect

Let us provide a formal proof that when $t \in \left] {{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2},m} \right]$ (i.e., t is included in the no-voting-order-effect range), the global quality classification does not depend on the voting sequence.

For a single object of interest, given that (1) the sum of the scores (s) assigned by agents (d) to the quality categories (c) is equal to m (see weight definition in Table 4), i.e.,

$$\sum\limits_{d} {\sum\limits_{c} {s\left( {c,d} \right)} = m} ,$$

(5)

and (2) m/2 < t ≤ m, if there exists a quality category (c*) whose total score (obtained cumulating the scores assigned by the m agents) exceeds t, i.e.,

$${\text{one}}\,{\text{element}}\,c^{*} \,{\text{exists}}|\frac{m}{2} < t \le \sum\limits_{d} {s\left( {c^{*} ,d} \right)} \le m,$$

(6)

then c* will be the only global quality category, since the total score of no other category will be able to reach t, no matter the agents’ rank ordering.

Assuming ad absurdum that there exists a second global quality category (c ^†), then we would have that

$$\sum\limits_{d} {s\left( {c^{*} ,d} \right) \ge t} > \frac{m}{2}\quad {\text{and}}\quad \sum\limits_{d} {s\left( {c^{\dag } ,d} \right)} \ge t > \frac{m}{2},$$

(7)

since the total scores of c* and c ^† would be both supposed to reach t. The sum of the total scores of the two quality categories c* and c ^† (neglecting the remaining ones) would therefore be

$$\sum\limits_{d} {s\left( {c^{*} ,d} \right)} + \sum\limits_{d} {s\left( {c^{\dag } ,d} \right)} \ge 2 \cdot t > m,$$

(8)

which is absurd, because it is incompatible with the condition in Eq. 5.

This demonstrates that if $t \in \left] {{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0pt} 2},m} \right]$ and a quality category c* has a total score exceeding t, this category will be the only global quality category, independently on the agents’ rank ordering.

Further considerations on the axiom of independence of irrelevant alternatives

This section provides a proof that the proposed method does not satisfy the axiom of independence of irrelevant alternatives (IIA).

Consider a quality classification problem where an object has been evaluated by three agents (d ₁ > d ₂ > d ₃) and classified into 4 quality categories (c ₁–c ₄) as follows:

$$\begin{aligned} & d_{1} :c_{3} ,c_{4} ; \\ & d_{2} :c_{1} ; \\ & d_{3} :c_{1} ,c_{2} ,c_{3} ,c_{4} . \\ \end{aligned}$$

In this case M = m = 3, so let us assume t = 1.25 and q = 1.5. Then, according to the proposed procedure, category c ₁ is selected at turn 3 with a cumulative score of 1.25. Let us comment this result. We notice that c ₁ is selected by d ₂ and d ₃, while quality categories c ₃ and c ₄ are selected by d ₁ and d ₃, but overall they are not selected (their cumulative scores are both 0.75), although they are chosen by a more important agent (d ₁). The reason is that the assignments by d ₁ and d ₃ are both multiple and inherently uncertain, and consequently the relevant scores are fractionalized; on the other hand, the single assignment by d ₂ entails a full score to c ₁.

Now assume that quality category c ₄ is removed from the analysis:

$$\begin{aligned} & d_{1} :c_{3} ; \\ & d_{2} :c_{1} ; \\ & d_{3} :c_{1} ,c_{2} ,c_{3} . \\ \end{aligned}$$

In this case, c ₁ and c ₃ are both selected, each having a score of 1.33 at turn 3. It is clear that the removal of the “non-global” quality category c ₄ changed the result, thus leading to a violation of the IIA axiom. It can be shown that, in the particular case in which agents classify the object into a single quality category (i.e., no multiple assignments), the IIA axiom is fulfilled.