An extension of the FlowSort sorting method to deal with imprecision

Abstract

We propose an extension of the FlowSort sorting method to the case when there is imprecision on the input data. Within multicriteria decision aid, a lot of attention has been paid to sorting problems where a set of actions has to be assigned to completely ordered categories. However, few methods suit when the data or the parameters of the model are not precisely defined. In this paper, instead of reducing the imprecise data to single values, we consider that the sorting parameters or the data are defined by intervals. We analyse the properties usually required for a sorting method and illustrate this extension on a practical example.

Appendix

1.1 The case of defining the categories by means of central profiles

Let us now suppose that a category, noted $C_h^*(h = 1, \ldots ,K)$, is defined by a central profile $r_h^*$ from the set $R^{*}=\left\{ {r_1^*,\ldots ,r_K^*}\right\} \!.$ The central profiles or the actions to be assigned may be imprecise and defined by means of interval performances. Actions will be assigned by comparing their interval flows with regard to the central profiles.

As the categories are completely ordered (with $C_1^*$ the best category), we will impose the following on the central profiles:

Condition 1:

$\forall r_h^*,r_l^*\in R\,with\,h<l:\,\forall j\in \{1,\ldots ,q\}:\,\underline{g_j }\left( {r_h^*} \right)\ge \overline{g_j } \left( {r_l^*} \right)\,and\,\exists k\!:\underline{g_k}\left( {r_h^*} \right)>\overline{g_k } \left( {r_l^*} \right)$ Condition 2:

$$\begin{aligned} \forall r_h^*,r_l^*\in R\, with\,h<l:\,\underline{\pi }\left( {r_h^*,r_l^*} \right)>0\quad and\quad \overline{\pi }\left( {r_l^*,r_h^*} \right)=0 \end{aligned}$$

When assigning an action to a category, we compare its flows with the flows of the central profiles and assign it to the category whose central profile has similar (or the most similar) flows. However, when working with imprecise information, it is difficult to express this similarity or the distance between intervals (such as in the crisp case). Therefore, we use the equivalent assignment rules (Nemery 2008) which use virtual limiting profiles.

The virtual limiting profiles define the virtual ‘boundary’ between two categories induced by their central profiles. Concretely, we obtain the virtual limiting profiles by computing the mean of two consecutive central profiles in terms of upper and lower flow bounds.

Let us remark that this virtual limiting profile ($r_h^L$) depends on the behaviour of $a_i$ in regards to the central profile and is thus not constant:

$$\begin{aligned} \underline{{\phi } _{R_i^*}^+ }\left( {r_h^L } \right)&= \frac{\underline{{\phi } _{R_i^*}^+ }\left( {r_{h-1}^*} \right)+\underline{{\phi } _{R_i^*}^+ }\left( {r_h^*} \right)}{2}\\ \overline{{\phi } _{R_i^*}^+ } \left( {r_h^L } \right)&= \frac{\overline{{\phi } _{R_i^*}^+ } \left( {r_{h-1}^*} \right)+\overline{{\phi } _{R_i^*}^+ } \left( {r_h^*} \right)}{2} \end{aligned}$$

This is illustrated in Fig. 5.

Fig. 5

Representation of two categories $C_h^*$ and $C_{h-1}^*$ defined by two central profiles $r_h^*$ and $r_{h-1}^*$ which lead to the virtual limiting profile $r_h^L$ where $\underline{{\upphi } _{R_i^*}^+ }\left( {r_h^L } \right)=\frac{\underline{{\upphi } _{R_i^*}^+ }\left( {r_{h-1}^*} \right)+\underline{{\upphi } _{R_i^*}^+ }\left( {r_h^*} \right)}{2}$ and $\overline{{\upphi } _{R_i^*}^+ } \left( {r_h^L } \right)=\frac{\overline{{\upphi } _{R_i^*}^+ } \left( {r_{h-1}^*} \right)+\overline{{\upphi } _{R_i^*}^+ } \left( {r_h^*} \right)}{2}$

If Conditions 1 and 2 are verified, we may use analogous assignment rules based on the leaving flows as based on the virtual limiting profiles:

$\widehat{C_{{\phi } ^{+}}^*}\left( {a_i } \right)=[\underline{C_{{\phi } ^{+}}^*}\left( {a_i } \right),\overline{C_{{\phi } ^{+}}^*} \left( {a_i } \right)],$ where:

$$\begin{aligned} \overline{C_{{\phi } ^{+}}^*} \left( {a_i } \right)=l,&\Leftrightarrow&\frac{\overline{{\phi } _{R_i^*}^+ } \left( {r_{l-1}^*} \right)+\overline{{\phi } _{R_i^*}^+ } \left( {r_l^*} \right)}{2}>\underline{{\upphi } _{R_i^*}^+ }\left( {a_i } \right)\ge \frac{\overline{{\phi } _{R_i^*}^+ } \left( {r_l^*} \right)+\overline{{\phi } _{R_i^*}^+ } \left( {r_{l+1}^*} \right)}{2}\\ \mathop {C_{{\phi } ^{+}}^*} \left( {a_i } \right)=h&\Leftrightarrow&\frac{\underline{{\phi } _{R_i^*}^+ }\left( {r_{h-1}^*} \right)+\underline{{\phi } _{R_i^*}^+ }\left( {r_h^*} \right)}{2}>\mathop {{\upphi } _{R_i^*}^+ } \left( {a_i } \right)\ge \frac{\underline{{\phi } _{R_i^*}^+ }\left( {r_h^*} \right)+\underline{{\phi } _{R_i^*}^+ }\left( {r_{h+1}^*} \right)}{2} \end{aligned}$$

(Assignment Rule 2bis)

$\widehat{C_{{\phi } ^{-}}^*}\left( {a_i } \right)=[\underline{C_{{\phi } ^{-}}^*}\left( {a_i } \right),\overline{C_{{\phi } ^{-}}^*} \left( {a_i } \right)],$ where:

$$\begin{aligned} \overline{C_{{\phi } ^{-}}^*} \left( {a_i } \right)=l,&\Leftrightarrow&\frac{\overline{{\phi } _{R_i^*}^- } \left( {r_{l-1}^*} \right)+\overline{{\phi } _{R_i^*}^- } \left( {r_l^*} \right)}{2}>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {a_i } \right)\ge \frac{\overline{{\phi } _{R_i^*}^- } \left( {r_l^*} \right)+\overline{{\phi } _{R_i^*}^- } \left( {r_{l+1}^*} \right)}{2}\\ \underline{C_{{\phi } ^{-}}^*}\left( {a_i } \right)=h&\Leftrightarrow&\frac{\underline{{\phi } _{R_i^*}^- }\left( {r_{h-1}^*} \right)+\underline{{\phi } _{R_i^*}^- }\left( {r_h^*} \right)}{2}>\underline{{\phi } _{R_i^*}^- }\left( {a_i } \right)\ge \frac{\underline{{\phi } _{R_i^*}^- }\left( {r_h^*} \right)+\underline{{\phi } _{R_i^*}^- }\left( {r_{h+1}^*} \right)}{2} \end{aligned}$$

(Assignment Rule 3bis)

Let us remark that an action $a_i$ which is ${\upphi }$+-indifferent (resp. ${\upphi }$–indifferent, ${\upphi }$-indifferent) to strictly one central profile $r_h^*$ will not necessarily be univocally assigned to the category $C_h^*$. This is illustrated in Fig. 6.

Fig. 6

In this situation, we have that $a_i$ is ${\upphi } ^{+}$- indifferent to strictly $r_h^*$ but will however be assigned to $[C_{h-1}^*,C_h^*]$

It seems reasonable to take the flow scale into account since there might be situations where we have ${\phi } _{R_i^*}^+ \left( {a_i^*} \right)>\frac{{\phi } _{R_i^*}^+ \left( {r_h^*} \right)+{\phi } _{R_i^*}^+ \left( {r_{h-1}^*} \right)}{2}$ or ${\phi } _{R_i^*}^+ \left( {a_i^*} \right)<\frac{{\phi } _{R_i^*}^+ \left( {r_h^*} \right)+{\phi } _{R_i^*}^+ \left( {r_{h-1}^*} \right)}{2}$.

In the same manner, even if an action is in “between” two central profiles according to the leaving flow intervals, it might be assigned to a unique category. This will be the case if ${\upphi } _{R_i^*}^+ \left( {a_i^*} \right)>\frac{{\upphi } _{R_i^*}^+ \left( {r_h^*} \right)+{\upphi } _{R_i^*}^+ \left( {r_{h-1}^*} \right)}{2}$ (see Fig. 7).

Fig. 7

In this situation, we have that $a_i$ is “between” two central profiles (i.e.$r_{h-1}^*P^{+}a_j$, $a_j P^{+}r_h^*)$ but will however be univocally assigned to $C_{h-1}^*$

Previous assignment results verify the monotonicity and the consistency property such as the limiting profiles do. Indeed, similar assignments rules may be defined on the basis of the net flows.

Furthermore, as in the FlowSort method (Prop. 5 in Nemery and Lamboray 2008), we have a relationship between the central and limiting profiles assignment procedures. Consider that the $K$ completely ordered categories are defined either by limiting profiles $R_i$ or by central profiles $R_i^*$ which verify Condition 6. Suppose moreover that their performances are such that:

$\forall g_j ,\forall r_h \in R:g_j \left( {r_{h+1} } \right)\le g_j \left( {r_h^*} \right)\le g_j \left( {r_h } \right)$ (ie., the central profile of a category is between the corresponding limiting profiles). On the basis of Proposition 5 in Huenaerts and Nemery (2007), we have thus $\forall a_i$ that $\overline{C_{{\phi } ^{+}} } \left( {a_i } \right)-\overline{C_{{\phi } ^{+}}^*} \left( {a_i } \right)|\in \{0,1\}\, (|\underline{C_{{\phi } ^{+}} }\left( {a_i } \right)-\underline{C_{{\phi } ^{+}}^*}\left( {a_i } \right)|\in \{0,1\})$ and $\overline{C_{{\phi } ^{-}} } \left( {a_i } \right)-\overline{C_{{\phi } ^{-}}^*} \left( {a_i } \right)|\in \{0,1\}$ ($|\underline{C_{{\phi } ^{-}} }\left( {a_i } \right)-\underline{C_{{\phi } ^{-}}^*}\left( {a_i } \right)|\in \{0,1\})$. In other words, an action will be assigned to the same category or to an adjacent category when defining the categories by limiting or central profiles.

1.2 Proof of Proposition 3

Having $\underline{C_{{\phi } ^{+}} }\left( {a_i } \right)=C_h$ and $\overline{C_{{\phi } ^{+}} } \left( {a_i } \right)=C_l$, we have thus to prove that:

$$\begin{aligned} {\phi } _{R_i }^+ \left( {r_{l+1}^c } \right)\le {\phi } _{R_i }^+ \left( {a_i^c } \right)<{\phi } _{R_i }^+ \left( {r_h^c } \right) \end{aligned}$$

(9)

According to the assignment rules, we have that $\widehat{C_{{\phi } ^{+}}}\left( {a_i } \right)=[\underline{C_{{\phi } ^{+}} }\left( {a_i } \right),\overline{C_{{\phi } ^{+}} } \left( {a_i } \right)]$:

$$\begin{aligned} \overline{C_{{\phi } ^{+}} } \left( {a_i } \right)=l,&\Leftrightarrow&\overline{{\phi } _{R_i }^+ } \left( {r_l } \right)>\underline{{\phi } _{R_i }^+ }\left( {a_i } \right)\ge \overline{{\phi } _{R_i }^+ } \left( {r_{l+1} } \right)\\ \underline{C_{{\phi } ^{+}} }\left( {a_i } \right)=h&\Leftrightarrow&\underline{{\phi } _{R_i }^+ }\left( {r_h } \right)>\overline{{\phi } _{R_i }^+ } \left( {a_i } \right)\ge \underline{{\phi } _{R_i }^+ }\left( {r_{h+1} } \right) \end{aligned}$$

This implies with (9) that we need to prove that

• ${{\phi }_{R_i}^+} \left( {a_i^c } \right)\in [\underline{{\phi }_{R_i }^+ }\left( {a_i } \right),\overline{{\phi } _{R_i }^+ } \left( {a_i } \right)]$,

• $\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_{l+1} } \right)\ge {\upphi } _{\mathrm{R}_\mathrm{i} }^+ \left( {r_{l+1}^c } \right)$

• ${\upphi } _{\mathrm{R}_\mathrm{i} }^+ \left( {r_h^c } \right)\ge \underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {r_h } \right)$.

Since $\forall k\in \left\{ {1,\ldots ,q} \right\} ,\forall \widehat{x}\in \widehat{\text{ R}_\mathrm{i}}:g_k \left( {x^{c}} \right)\in \widehat{g_k (x)}$ and since the preference functions are monotone, we necessarily have that: $\underline{P_k }\left( {a_i^c ,r_j^c } \right)\le P_k^c (a_i^c ,r_j^c )\le \overline{P_k } \left( {a_i^c ,r_j^c } \right)$ (11). We have therefore the same result as (11) for the preference degrees and the flows which implies that

$$\begin{aligned} {\phi } _{R_i }^+ \left( {a_i^c } \right)\in [\underline{{\phi } _{R_i }^+ }\left( {a_i } \right),\overline{{\phi } _{R_i }^+ } \left( {a_i } \right)]. \end{aligned}$$

We obtain the same for the leaving flows of $r_h$ and $r_l$ which proves the proposition.

Moreover, the proof is similar when working with the entering and net flows.

1.3 Proof of Proposition 4

We will suppose that the action $a_{i}$ has been assigned according to the leaving and entering flows to the following categories: $\widehat{C_{{\phi } ^{+}}}\left( {a_i } \right)=[C_h,C_l]$ and $\widehat{C_{{\phi } ^{-}}}\left( {a_i } \right)=[C_j ,C_i ]$.

This means thus that:

$$\begin{aligned}&\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_l } \right)>\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {a_i } \right)\ge \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_{l+1} } \right)\end{aligned}$$

(10)

$$\begin{aligned}&\underline{{\upphi }_{\mathrm{R}_\mathrm{i} }^+ }\left( {r_h } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {a_i } \right)\ge \underline{{\upphi }_{\mathrm{R}_\mathrm{i} }^+ }\left( {r_{h+1} } \right)\end{aligned}$$

(11)

$$\begin{aligned}&\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_{i+1} } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {a_i } \right)\ge \underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_i } \right)\end{aligned}$$

(12)

$$\begin{aligned}&\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_{j+1} } \right)>\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {a_i } \right)\ge \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_j } \right) \end{aligned}$$

(13)

We have thus moreover that $h \le l$ and $j \le i$. By subtracting (12) from (10) and (13) from (11), we obtain respectively (14) and (15):

$$\begin{aligned}&\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_l } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_{i+1} } \right)>\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {a_i } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {a_i } \right)\ge \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_{l+1} } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_i } \right)\end{aligned}$$

(14)

$$\begin{aligned}&\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {r_h } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_j } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {a_i } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {a_i } \right)\ge \underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {r_{h+1} } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_{j+1} } \right) \end{aligned}$$

(15)

If l$<$j:

We have thus that h $\le$ l $<$ j $\le$ i and with $Conditions\,2\,and\,3$ we know that:

$$\begin{aligned} \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_l } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_{l+1} } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_i } \right) \end{aligned}$$

(16)

which implies that:

$$\begin{aligned} \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_{l+1} } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_i } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_i } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_i } \right). \end{aligned}$$

(17)

Combining (17) with the right inequality of (14), leads to:

$$\begin{aligned} \underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {a_i } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {a_i } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_i } \right)-\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- }\left( {r_i } \right)\Leftrightarrow \overline{{\upphi } _{\mathrm{R}_\mathrm{i} } } \left( {a_i } \right)>\underline{{\upphi } _{\mathrm{R}_\mathrm{i} } }\left( {r_i } \right). \end{aligned}$$

(18)

Consequently, from (18) the upper net flow assignment must lie between $C_i$ and $C_l$.

Analogously, we have that:

$$\begin{aligned} \overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_{j+1} } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_j } \right)>\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_h } \right). \end{aligned}$$

(19)

Therefore:

$$\begin{aligned} \underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {r_h } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_h } \right)>\underline{{\upphi } _{\mathrm{R}_\mathrm{i} }^+ }\left( {r_h } \right)-\overline{{\upphi } _{\mathrm{R}_\mathrm{i} }^- } \left( {r_{j+1} } \right). \end{aligned}$$

(20)

Combining (20) with the left inequality of (15), we have that:

$$\begin{aligned} \overline{{\phi } _{\mathrm{R}_\mathrm{i} } } \left( {a_i } \right)>\underline{{\phi } _{\mathrm{R}_\mathrm{i} } }\left( {r_h } \right). \end{aligned}$$

(21)

Consequently, from (21) the upper net flow assignment must lie between $C_i$ and $C_l$. We have thus finally that $\widehat{C_{{\phi } ^{+}}}\left( {a_i } \right)=[C_h ,C_l ]$.

The other cases are similar.

1.4 Property of monotonicity

We need to prove that:

$$\begin{aligned} \forall a_i ,a_j \in A:a_i >^{D}a_j \Rightarrow \quad \overline{C_{{\phi } ^{+}} } \left( {a_i } \right)\le \overline{C_{{\phi } ^{+}} } \left( {a_j } \right)\, \text{ and} \, \underline{C_{{\phi } ^{+}} }\left( {a_i } \right)\le \underline{C_{{\phi } ^{+}} }\left( {a_j } \right) \end{aligned}$$

We have that $\forall a_i >^{D}a_j \Rightarrow \forall l=1,\ldots ,q:\underline{g_l \left( {a_i } \right)}\ge \overline{g_l \left( {a_j } \right)}$ which implies that $\forall r_h \in R\,and\,\forall l=1,\ldots ,q:\overline{P_l } \left( {r_h ,a_j } \right) \,\ge \,\overline{P_l } \left( {r_h ,a_i } \right)$ and $\underline{P_l }\left( {a_i ,r_h } \right) \ge \underline{P_l }\left( {a_j ,r_h } \right)$.

Based on this, we have necessarily that $\underline{\pi }\left( {a_{i,} r_h } \right) \quad \ge \quad \underline{\pi }\left( {a_j ,r_h } \right)$ and $\overline{\pi }\left( {r_h ,a_j } \right) \ge \overline{\pi }\left( {r_h ,a_i } \right)$. As a consequence, we have finally that $\underline{{\phi } _{R_j }^+ }\left( {a_j } \right)\le \underline{{\phi } _{R_i }^+ }\left( {a_i } \right)$ and $\forall r_h \!\in \! R \, \overline{{\phi } _{R_i }^+ } (r_h) \!\le \! \overline{{\phi } _{R_j }^+ } (r_h )$. Assuming that $\overline{C_{{\phi } ^{+}} } \left( {a_i } \right)=k$, we know that (assignment rule 2):

$$\begin{aligned} \overline{{\phi } _{R_i }^+ } \left( {r_k } \right)>\underline{{\phi } _{R_i }^+ }\left( {a_i } \right) \end{aligned}$$

And, with the above developments:

$$\begin{aligned} \mathop {{\phi } _{\mathrm{R}_\mathrm{j} }^+ }\left( {r_k } \right)\ge \mathop {{\phi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {r_k } \right)>\mathop {{\phi } _{\mathrm{R}_\mathrm{i} }^+ } \left( {a_i } \right)\ge \mathop {{\phi } _{\mathrm{R}_j }^+ } \left( {a_j } \right) \end{aligned}$$

Therefore we have that $\overline{C_{{\phi } ^{+}} } \left( {a_i } \right)\le \overline{C_{{\phi } ^{+}} } \left( {a_j } \right)$

A similar development can be followed to show that $\underline{C_{{\phi } ^{+}} }\left( {a_i } \right)\le \underline{C_{{\phi } ^{+}} }\left( {a_j } \right)$.