# TRALALA

###### Abstract

We derive computational formulas for the generalized Choquet integral based on
the novel survival function introduced by M. Boczek et al. [1]. We demonstrate
its usefulness on the Knapsack problem and the problem of accommodation
options. Moreover, we describe sufficient and necessary conditions under which
novel survival functions based on different parameters coincide. This is
closely related to the incomparability of input vectors (alternatives) in
decision-making processes.

Conditional aggregation-based Choquet integral
on discrete space

Stanislav Basarik, Jana Borzová, Lenka Halčinová, Jaroslav Šupina††Mathematics
Subject Classification (2010): Primary 28A12, 28E10
Institute of Mathematics, P. J. Šafárik University in Košice, Jesenná 5, 040
01 Košice, Slovakia

Keywords: Choquet integral; conditional aggregation operator; decision making,
survival function

##  1 Introduction

M. Boczek et al. [1], inspired by consumer’s problems, aggregation operators,
and conditional expectation, introduced a new notion of conditional
aggregation operators. Conditional aggregation operators cover many existing
aggregations, such as the arithmetic and geometric mean, or plenty of
integrals known in the literature [12, 14, 16]. They became the essence of the
generalization of survival function (a notion known from [7], from [9] known
as the decumulative distribution function or from [2] known as the strict
level measure) introduced by M. Boczek et al. [1]. Just as the survival
function is the basis of the definition of the famous Choquet integral, the
generalized survival function enabled the building of a new integral. The
generalized Choquet integral
$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)$ based on the
generalized survival function
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ has been naturally
introduced by

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\int_{0}^{\infty}\mu_{\mathscr{A}}(\mathbf{x},\alpha)\,\mathrm{d}\alpha.$

A discrete form of the famous Choquet integral is of great importance in
decision making theory, regarding a finite set
$\displaystyle[n]=\\{1,\dots,n\\}$ as criteria set, a vector
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$ as a score vector, and a
capacity $\displaystyle\mu\colon 2^{[n]}\to[0,+\infty)$ as the weights of
particular sets of criteria. We provide problems, which complement the ones
described in [1], and which stress the potential of the novel survival
function and integral based on it in decision theory. However, the main aim of
the present paper is to provide various computational formulas for
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$, and consequently for
$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)$, since the
computation of $\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)$ has
not yet been thoroughly investigated. Thus, we believe that our computational
algorithms improve the practical implementation of the novel concept of
aggregation in various problems.

The novel survival function
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ is based on an aggregation
of an input vector $\displaystyle\mathbf{x}$, applying each conditional
aggregation operator from the family $\displaystyle{\mathscr{A}}$ of size
$\displaystyle\kappa$ on $\displaystyle\mathbf{x}$ and corresponding set of
its indices. To illustrate our results we provide a sample formula for
$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)$. Indeed, we show
that

$\displaystyle\displaystyle{\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=0}^{\kappa-1}\mu(F_{\mathbf{i}(i)})(\mathsf{A}_{i+1}-\mathsf{A}_{i}),}$
(1)

where $\displaystyle\mathsf{A}_{0}\leq\dots\leq\mathsf{A}_{\kappa-1}$ is the
arrangement of all the values of the family of conditional aggregation
operators $\displaystyle{\mathscr{A}}$ applied on $\displaystyle\mathbf{x}$
and corresponding sets $\displaystyle E_{0},\dots,E_{\kappa-1}$, and
$\displaystyle\mu(F_{0})\leq\dots\leq\mu(F_{\kappa-1})$ are all the values of
the measure $\displaystyle\mu$. The index $\displaystyle\mathbf{i}(i)$
deciding the value of
$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)$ on the interval
$\displaystyle[\mathsf{A}_{i},\mathsf{A}_{i+1})$ is the minimum of the set
$\displaystyle\\{(0),(1),\dots,(i)\\}$ of the indices such that $\displaystyle
F_{(0)}=E_{0}^{c},\dots,F_{(i)}=E_{i}^{c}$. The proof of formula (1) as well
as many other similar formulas can be found in Theorem 5.2.

The paper is organized as follows. Section 2 contains necessary terminology on
measure and a family of conditional aggregation operators. Moreover, it
describes two problems in decision-making theory that can be modeled by the
generalized survival function. The first series of formulas for its
computation is presented in Section 3. The next section reduces the
minimization of the reals in the formulas to the minimization of indices, and
shows two graphical approaches to obtain the generalized survival function. In
addition, it contains a solution to the first of two mentioned problems in
decision-making theory. The formulas for the generalized Choquet integral
computation are listed in Section 5, including simplified fomulas for special
types of measures. The solution to the second problem from Section 2 is
presented here. The study of the conditions such that the novel survival
functions based on different parameters coincide is performed in Section 6.
Finally, the paper itself contains just proofs of selected results, the
remaining proofs are attached in the Appendix.

##  2 Background and interpretations

As we have already mentioned, we shall consider a finite set
$\displaystyle[n]:=\\{1,2,\dots,n\\}$, $\displaystyle n\in\mathbb{N}$,
$\displaystyle n\geq 1$. Let us denote $\displaystyle[n]_{0}:=\\{0\\}\cup[n]$.
By $\displaystyle 2^{[n]}$ we mean the power set of $\displaystyle[n]$. A set
function $\displaystyle\mu\colon\mathscr{S}\to[0,+\infty)$,
$\displaystyle\\{\emptyset\\}\subseteq\mathscr{S}\subseteq 2^{[n]}$, such that
$\displaystyle\mu(E)\leq\mu(F)$ whenever $\displaystyle E\subseteq F$, with
$\displaystyle\mu(\emptyset)=0$, we call monotone measure on
$\displaystyle\mathscr{S}$. Moreover, if $\displaystyle[n]\in\mathscr{S}$, we
assume $\displaystyle\mu([n])>0$, and if $\displaystyle\mu([n])=1$, the
monotone measure $\displaystyle\mu$ is called capacity or normalized monotone
measure. Further, we put $\displaystyle\max{\emptyset}=0$,
$\displaystyle\min{\emptyset}=+\infty$ and
$\displaystyle\textstyle\sum_{i\in\emptyset}x_{i}=0$.

We shall work with nonnegative real-valued vectors, we use the notation
$\displaystyle\mathbf{x}=(x_{1},\dots,x_{n})$, $\displaystyle
x_{i}\in[0,+\infty)$, $\displaystyle i\in[n]$. The family of all nonnegative
real-valued vectors on $\displaystyle[n]$ is the set
$\displaystyle[0,+\infty)^{[n]}$. By $\displaystyle\mathbf{1}_{E}$ we shall
denote indicator function of a set $\displaystyle E\subseteq[0,+\infty)$, i.e.
$\displaystyle\mathbf{1}_{E}(x)=1$, if $\displaystyle x\in E$, and
$\displaystyle\mathbf{1}_{E}(x)=0$, if $\displaystyle x\notin E$. Especially,
$\displaystyle\mathbf{1}_{\emptyset}(x)=0$ for each $\displaystyle x\in E$.
Let us consider a set $\displaystyle\mathscr{E}\subseteq 2^{[n]}$. Unless
otherwise stated, for application reasons we assume
$\displaystyle\\{\emptyset,[n]\\}\subseteq\mathscr{E}$. We call it the
collection. Let us denote the number of sets in $\displaystyle\mathscr{E}$ by
$\displaystyle\kappa$, i.e. $\displaystyle|\mathscr{E}|=\kappa$. Let
$\displaystyle\hat{{\mathscr{E}}}=\\{E^{c}:E\in{\mathscr{E}}\\}$, i.e.
$\displaystyle\hat{{\mathscr{E}}}$ contains the complements of the sets from
collection $\displaystyle\mathscr{E}$. The set of all monotone measures on
$\displaystyle\hat{\mathscr{E}}$ we shall denote by $\displaystyle\mathbf{M}$.

In the following, we present the definition of the conditional aggregation
operator. The inspiration for this concept can be found in probability theory,
specifically in conditional expectation. The idea of aggregating data not on
the whole, but on a conditional set is expressed in the terms of this
definition. The conditional aggregation operator generalizes the classical
definition of aggregation operator introduced by Calvo et al. in [3] and forms
the basic component of the definition of the generalized survival function.

###### Definition 2.1

(cf. [1, Definition 4.1.]) A family of conditional aggregation operators (FCA
for short) is a family

$\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in\mathscr{E}\\},$

such that each $\displaystyle\mathsf{A}(\cdot|E)$ is a map
$\displaystyle\mathsf{A}(\cdot|E)\colon[0,+\infty)^{[n]}\to[0,+\infty)$
satisfying the following conditions:

  1. (i)

$\displaystyle\mathsf{A}(\mathbf{x}|E)\leq\mathsf{A}(\mathbf{y}|E)$ for any
$\displaystyle\mathbf{x},\mathbf{y}$ such that $\displaystyle x_{i}\leq y_{i}$
for any $\displaystyle i\in E$, $\displaystyle E\neq\emptyset$;

  2. (ii)

$\displaystyle\mathsf{A}(\mathbf{1}_{E^{c}}|E)=0$, $\displaystyle
E\neq\emptyset$.

If $\displaystyle\mu$ is a monotone measure on
$\displaystyle\hat{{\mathscr{E}}}=\\{E^{c}:E\in{\mathscr{E}}\\}$, i.e.
$\displaystyle\mu\in\mathbf{M}$, then the generalized survival function with
respect to $\displaystyle{\mathscr{A}}$ is defined as

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha):=\min\left\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\right\\}$
(2)

for any $\displaystyle\alpha\in[0,+\infty)$.

If $\displaystyle E\neq\emptyset$, then $\displaystyle\mathsf{A}(\cdot|E)$ is
called the conditional aggregation operator w.r.t. $\displaystyle E$.
Moreover, we consider that each element of FCA satisfies
$\displaystyle\mathsf{A}(\cdot|\emptyset)=0$.

###### Remark 2.2

Note that the survival function
$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})=\mu(\\{i\in[n]:x_{i}>\alpha\\})$
can be rewritten as

$\displaystyle\displaystyle\mu(\\{\mathbf{x}>\alpha\\})$
$\displaystyle\displaystyle=\mu([n]\setminus\\{\mathbf{x}\leq\alpha\\})=\min\big{\\{}\mu(E^{c}):(\forall
i\in E)\,\,x_{i}\leq\alpha,\,E\in 2^{[n]}\big{\\}}$
$\displaystyle\displaystyle=\min\\{\mu(E^{c}):\max_{i\in
E}x_{i}\leq\alpha,E\in 2^{[n]}\\},$ (3)

therefore the introduction of generalized survival function consists in the
simple idea of replacing $\displaystyle\max_{i\in E}x_{i}$ in (2.2) with
another functional. Clearly, for $\displaystyle\mathscr{E}=2^{[n]}$ and
$\displaystyle{\mathscr{A}}^{\mathrm{max}}$ we get the original strict
survival function.

###### Example 2.3

Let $\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. Typical examples of the FCA
are:

  1. (i)

$\displaystyle{\mathscr{A}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in{{\mathscr{E}}}\\}$
with $\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|E)=\max_{i\in E}x_{i}$
for $\displaystyle E\neq\emptyset$;

  2. (ii)

$\displaystyle{\mathscr{A}}^{\mathrm{min}}=\\{\mathsf{A}^{\mathrm{min}}(\cdot|E):E\in{{\mathscr{E}}}\\}$
with $\displaystyle\mathsf{A}^{\mathrm{min}}(\mathbf{x}|E)=\min_{i\in E}x_{i}$
for $\displaystyle E\neq\emptyset$;

  3. (iii)

$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in{{\mathscr{E}}}\\}$
with $\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)=\sum_{i\in E}x_{i}$
for $\displaystyle E\neq\emptyset$.

Note that the FCA does not have to contain elements of only one type.

###### Example 2.4

Let $\displaystyle\mathbf{x}\in[0,+\infty)^{[3]}$, and
$\displaystyle{{\mathscr{E}}}=\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{1,2,3\\}\\}$.
We can consider the following FCA

$\displaystyle{\mathscr{A}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\\{\\{1\\},\\{2\\},\\{1,2,3\\}\\}\cup\\{\mathsf{A}^{\mathrm{min}}(\cdot|E):E\in\\{\\{3\\}\\}\\}\cup\\{\mathsf{A}(\cdot|\emptyset)\\}.$

For other examples of FCA we recommend [1]. On several places in this paper,
we shall work with the FCA that is nondecreasing w.r.t sets, i.e. the map
$\displaystyle E\mapsto\mathsf{A}(\cdot|E)$ will be nondecreasing. E.g., the
families $\displaystyle{\mathscr{A}}^{\mathrm{max}}$ and
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}$ in Example 2.3 are nondecreasing
w.r.t. sets.

In the following, we present problems that emphasize the need for the
generalized survival function and the generalized Choquet integral in real
situations. We stress that M. Boczek et al. in [1] introduced several problems
of this kind. However, we move beyond these examples. The solution to these
problems using our results is delayed to Subsection 4.1 and Section 5.

#### Knapsack problem

Let us imagine a person who is preparing for a holiday. The person plans to
travel by plane. Therefore while packing the suitcase, he must keep the rules
according to which it is allowed to carry less or equal to $\displaystyle 1$
liter of liquids in the suitcase. Moreover, liquids must be in containers with
a volume of up to $\displaystyle 100$ ml. The prices of these products (soap,
shampoo, cream, etc.) and their possible combinations (packages of products)
are more expensive in the destination country than at home. The person wants
to buy such products, or packages of products, while still at home, to
minimize the price of the holiday.

Let $\displaystyle[n]$, $\displaystyle n\geq 1$ be a set of liquid products
that the person needs. Then $\displaystyle\mathscr{E}\subseteq 2^{[n]}$
represents all possible combinations of products. Let us consider
$\displaystyle\mathbf{x}=(x_{1},\dots,x_{n})\in[0,+\infty)^{[n]}$, where
$\displaystyle x_{i}$ represents the volume of $\displaystyle i$-th container
and let a monotone measure $\displaystyle\mu\in\mathbf{M}$ represent the price
of a package of products. Note that the monotone measure $\displaystyle\mu$
need not be additive. It is often possible to buy a package of products that
is cheaper than the sum of the prices of the individual products. The task is
to choose such a combination $\displaystyle E\in\mathscr{E}$ of products that
their volume does not exceed the given limit, i.e.
$\displaystyle\textstyle\sum_{i\in E}x_{i}\leq 1000$ (milliliters), having in
mind that we want to minimize the price of those products that will no longer
fit in the suitcase and the person will have to buy them during the holiday.
In other words, we are faced to solve the optimization problem

$\displaystyle\min\left\\{\mu(E^{c}):\sum_{i\in E}x_{i}\leq
1000,\,E\in\mathscr{E}\right\\}\text{.}$

One can observe that the given formula is a special case of the generalized
survival function given in (2) with the conditional aggregation operator being
the sum.

Last, but not least let us point out the essence of the collection
$\displaystyle\mathscr{E}$. There are situations when instead of the whole
powerset one is forced to consider a subcollection
$\displaystyle\mathscr{E}\subset 2^{[n]}$. E.g. among the products that one is
considering can be shampoo and conditioner separately or shampoo containing
conditioner. Of course, these products should not be considered together. Thus
all sets consisting of these three products are disqualified and it makes
sense to take $\displaystyle\mathscr{E}\subset 2^{[n]}$ instead of the whole
powerset.

The full solution to the problem using our results is accomplished in
Subsection 4.1.

#### Problem of accommodation options

Three people, Anthony, Brittany, Charley, are going on three different
holidays and each of them is looking for accommodation in their own
destination. They use the same online search engine, which searches for
available accommodation based on three criteria, namely the distance of
accommodation from the destination (in km), the price per night (in euros),
and reviews (scale from $\displaystyle 1$ to $\displaystyle 10$). Each person
has a certain character (in the sense that the criteria have different
importance for them). Anthony is a person who saves money, Brittany does not
want to walk far and Charley is looking for high-quality accommodation. We can
model these characters using monotone measures as shown in Table 1 (assume
that the given values are chosen by these three people in the search engine).
Let us label the criteria as D (distance), P (price) and R (reviews).

| distance (D) | price (P) | reviews (R)
---|---|---|---
Anthony ($\displaystyle\mu$) | $\displaystyle 0.3$ | $\displaystyle 0.8$ | $\displaystyle 0.1$
Brittany ($\displaystyle\nu$) | $\displaystyle 0.75$ | $\displaystyle 0.4$ | $\displaystyle 0.2$
Charley ($\displaystyle\xi$) | $\displaystyle 0.2$ | $\displaystyle 0.5$ | $\displaystyle 0.7$
Table 1: Characters of people expressed by a monotone measures
$\displaystyle\mu$, $\displaystyle\nu$ and $\displaystyle\xi$

For the sake of simplicity, let us suppose that the search engine offered
Anthony, Brittany, and Charley two accommodation options, see Table 2. Let the
values be in the range that is acceptable for them.

|  D |  P |  R
---|---|---|---
opt. $\displaystyle 1$ |  $\displaystyle 4$ |  $\displaystyle 100$ |  $\displaystyle 7$
opt. $\displaystyle 2$ |  $\displaystyle 10$ |  $\displaystyle 84$ |  $\displaystyle 8$
Anthony

|  D |  P |  R
---|---|---|---
opt. $\displaystyle 1$ |  $\displaystyle 7$ |  $\displaystyle 100$ |  $\displaystyle 2$
opt. $\displaystyle 2$ |  $\displaystyle 10$ |  $\displaystyle 80$ |  $\displaystyle 10$
Brittany

|  D |  P |  R
---|---|---|---
opt. $\displaystyle 1$ |  $\displaystyle 3$ |  $\displaystyle 100$ |  $\displaystyle 8$
opt. $\displaystyle 2$ |  $\displaystyle 15$ |  $\displaystyle 95$ |  $\displaystyle 10$
Charley

Table 2: Accommodation options

The aim is to determine the accommodation for each person that would fit him
the best (based on character). In other words, the aim is to find a method
that will select (based on the character of a person) the accommodation we
would expect.

Clearly, this is a problem of the decision-making process. In the literature,
there is a known approach to solving this problem using the theory of
nonadditive measures and integrals, where Choquet integral is mainly used, see
[8]. The main advantage of this method is that by nonadditive measures one can
model interactions between criteria (e.g. usually with a higher review score
one can expect higher price). In this paper, we point out the advantages of
using a generalized version of the Choquet integral in (this) decision-making
process.

The full solution to the problem using our results is presented in Section 5.

##  3 Computational formulas

When working with a family of conditional aggregation operators
$\displaystyle{\mathscr{A}}$, an input vector $\displaystyle\mathbf{x}$ is
aggregated on sets $\displaystyle E$ from $\displaystyle{\mathscr{E}}\subseteq
2^{[n]}$. By this procedure, instead of $\displaystyle n$ input components of
$\displaystyle\mathbf{x}$ we obtain $\displaystyle\kappa=|{\mathscr{E}}|$
input values of the family of conditional aggregation operators
$\displaystyle{\mathscr{A}}$:

$\displaystyle\displaystyle
0=\mathsf{A}_{0}\leq\mathsf{A}_{1}\leq\dots\leq\mathsf{A}_{\kappa-2}\leq\mathsf{A}_{\kappa-1}<+\infty,$
(4)

with the convention $\displaystyle\mathsf{A}_{\kappa}=+\infty$. This “new”
input enters the computation of the generalized survival function. Now we
provide two approaches to compute generalized survival function values, the
first one is presented in Theorem 3.2 and the other one in Theorem 3.7. We
shall fix a notation.

Let $\displaystyle E_{*}$ be any bijection $\displaystyle
E_{*}\colon[\kappa-1]_{0}\to{{\mathscr{E}}}$ such that

$\displaystyle\displaystyle\mathsf{A}_{i}=\mathsf{A}(\mathbf{x}|E_{i}).$ (5)

The second bijection is a map $\displaystyle
F_{*}\colon[\kappa-1]_{0}\to{\hat{{\mathscr{E}}}}$ such that denoting
$\displaystyle\mu_{j}=\mu(F_{j})$ we have:

$\displaystyle\displaystyle
0=\mu_{0}\leq\mu_{1}\leq\dots\leq\mu_{\kappa-2}\leq\mu_{\kappa-1}<+\infty.$
(6)

It should be recalled that $\displaystyle\hat{{\mathscr{E}}}$ is the
collection of complements of $\displaystyle{\mathscr{E}}$, i.e. for every
$\displaystyle F_{j}\in\hat{{\mathscr{E}}}$ there exists $\displaystyle
E_{i}\in{\mathscr{E}}$ such that $\displaystyle F_{j}=E_{i}^{c}$. We denote
sets in collection $\displaystyle{\mathscr{E}}$ by $\displaystyle E_{i}$,
$\displaystyle i\in[\kappa-1]_{0}$, and their complements in
$\displaystyle\hat{{\mathscr{E}}}$ are denoted by $\displaystyle F_{j}$,
$\displaystyle j\in[\kappa-1]_{0}$, because of technical details. In the whole
paper for ease of writing, we shall use a shortcut notation

$\displaystyle\mathsf{A}_{{\langle
j\rangle}}:=\mathsf{A}(\mathbf{x}|F_{j}^{c}),\,\,\,\,\text{and}\,\,\,\,\mu_{(i)}:=\mu(E_{i}^{c}).$

One can notice that maps $\displaystyle E_{*}$ and $\displaystyle F_{*}$ need
not be unique (they are unique just in case they are injective on
$\displaystyle[\kappa-1]_{0}$), but this has no influence on presented
results.

In the following example, we only demonstrate the introduction of bijection
$\displaystyle E_{*}$ and $\displaystyle F_{*}$, respectively. We shall use
the input data below also in other examples in this paper.

###### Example 3.1

Let us consider the collection
$\displaystyle{\mathscr{E}}=\\{\emptyset,\\{1\\},\\{2\\},\\{3\\},\\{1,3\\},\\{1,2,3\\}\\}$,
the family of conditional aggregation operators
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in{\mathscr{E}}\\}$,
the vector $\displaystyle\mathbf{x}=(2,3,1)$, and the monotone measure
$\displaystyle\mu$ on $\displaystyle\hat{{\mathscr{E}}}$ with corresponding
values in the table.

$\displaystyle E$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{1\\}$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{1,2,3\\}$
---|---|---|---|---|---|---
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|E)$ |  $\displaystyle 0$ |  $\displaystyle 2$ |  $\displaystyle 3$ |  $\displaystyle 1$ |  $\displaystyle 3$ |  $\displaystyle 6$
$\displaystyle F$ |  $\displaystyle\\{1,2,3\\}$ |  $\displaystyle\\{2,3\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\emptyset$
$\displaystyle\mu(F)$ |  $\displaystyle 1$ |  $\displaystyle 0.7$ |  $\displaystyle 0.5$ |  $\displaystyle 0.5$ |  $\displaystyle 0.5$ |  $\displaystyle 0$

Using maps $\displaystyle E_{*}$ and $\displaystyle F_{*}$ we obtain the
following arrangement:

$\displaystyle i,j$ |  0 |  1 |  2 |  3 |  4 |  5
---|---|---|---|---|---|---
$\displaystyle E_{i}$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{1\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\\{1,2,3\\}$
$\displaystyle\mathsf{A}_{i}$ |  $\displaystyle 0$ |  $\displaystyle 1$ |  $\displaystyle 2$ |  $\displaystyle 3$ |  $\displaystyle 3$ |  $\displaystyle 6$
$\displaystyle F_{j}$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{2\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{1,3\\}$ |  $\displaystyle\\{2,3\\}$ |  $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu_{j}$ |  $\displaystyle 0$ |  $\displaystyle 0.5$ |  $\displaystyle 0.5$ |  $\displaystyle 0.5$ |  $\displaystyle 0.7$ |  $\displaystyle 1$

The first approach to computing generalized survival function values is based
on an (ascending) arrangement (4) of conditional aggregation operator values.
Let us note that this approach is directly derived from Definition 2.1. In the
computation process using this approach, we look for generalized survival
function values that are achieved at intervals
$\displaystyle[\mathsf{A}_{i},\mathsf{A}_{i+1})$ with the convention
$\displaystyle\mathsf{A}_{\kappa}=+\infty$.

###### Theorem 3.2

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. Then

  1. (i)

for any $\displaystyle i\in[\kappa-1]_{0}$ it holds that

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min_{k\leq
i}\mu_{(k)}\,\,\ \textrm{for
any}\,\,\alpha\in[\mathsf{A}_{i},\mathsf{A}_{i+1});$

  2. (ii)

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-1}\min_{k\leq
i}\mu_{(k)}\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}(\alpha)\,\,\
\textrm{for any}\,\,\alpha\in[0,+\infty).$ (7)

Proof. (i) Let us consider an arbitrary (fixed) $\displaystyle
i\in[\kappa-1]_{0}$ such that $\displaystyle\mathsf{A}_{i}<\mathsf{A}_{i+1}$,
and let us take $\displaystyle\alpha\in[\mathsf{A}_{i},\mathsf{A}_{i+1})$.
Since $\displaystyle
0=\mathsf{A}_{0}\leq\mathsf{A}_{1}\leq\dots\leq\mathsf{A}_{i}\leq\alpha$, then

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}=\min\\{\mu_{(k)}:k\in\\{0,\dots,i\\}\\}=\min_{k\leq
i}\mu_{(k)}.$

The case (ii) immediately follows from (i). $\displaystyle\Box\;\;$

###### Remark 3.3

Let $\displaystyle(x_{1},\dots,x_{n})\in[0,+\infty)^{[n]}$. Let us denote
$\displaystyle\mathbf{x}=(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)})$,
with $\displaystyle\sigma\colon[n]\to[n]$ being a permutation such that
$\displaystyle x_{\sigma(1)}\leq x_{\sigma(2)}\leq\dots\leq x_{\sigma(n)}$
with the convention $\displaystyle x_{\sigma(0)}=0$. Also let
$\displaystyle{\mathscr{A}}^{\mathrm{max}}$ be a FCA with the collection
$\displaystyle\mathscr{E}=\\{G_{\sigma{(i+1)}}^{c}:i\in[n]_{0}\\}$ where
$\displaystyle G_{\sigma{(i)}}=\\{\sigma{(i)},\dots,\sigma{(n)}\\}$ for
$\displaystyle i\in[n]$ and $\displaystyle G_{\sigma(n+1)}=\emptyset$. Then
from the previous proposition we get the standard formula of the survival
function [11]:

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max}}}(\mathbf{x},\alpha)=\sum_{i=0}^{n-1}\mu{(G_{\sigma(i+1)})}\mathbf{1}_{[x_{\sigma{(i)}},x_{\sigma{(i+1)}})}(\alpha)\,\,\
\textrm{for any}\,\,\alpha\in[0,+\infty)\text{.}$

Indeed,
$\displaystyle\mathsf{A}^{\mathrm{max}}(\mathbf{x}|G_{\sigma(i+1)}^{c})=x_{\sigma(i)}$
for any $\displaystyle i\in[n]_{0}$. Moreover, if $\displaystyle
x_{\sigma(i)}=\mathsf{A}^{\mathrm{max}}_{l_{i}}$, $\displaystyle
l_{i}\in[\kappa-1]_{0}$, then $\displaystyle\min_{k\leq
l_{i}}\mu_{(k)}=\min_{k\leq i}\mu(G_{\sigma(k)})=\mu(G_{\sigma(i)})$ because
of the monotonicity of $\displaystyle\mu$.

###### Remark 3.4

Since $\displaystyle[n]\in{\mathscr{E}}$, then the definition of generalized
survival function guarantees to achieve zero value. According to arrangement
in (4), for any
$\displaystyle\alpha\geq\mathsf{A}_{\kappa-1}\geq\mathsf{A}(\mathbf{x}|[n])$
it holds

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}\leq\mu([n]^{c})=\mu(\emptyset)=0\text{.}$

On the other hand $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)\geq 0$.
Thus, the generalized survival function achieves zero value on the interval
$\displaystyle[\mathsf{A}(\mathbf{x}|[n]),+\infty)\supseteq[\mathsf{A}_{\kappa-1},+\infty)$.

###### Example 3.5

Let us consider the same inputs as in Example 3.1. Using Theorem 3.2(i) let us
compute the generalized survival function for any
$\displaystyle\alpha\in[0,+\infty)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{(5)}=0$
for any $\displaystyle\alpha\in[\mathsf{A}_{5},\mathsf{A}_{6})=[6,+\infty)$,
which corresponds to Remark 3.4.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{(4)}=0.5$
for any $\displaystyle\alpha\in[\mathsf{A}_{4},\mathsf{A}_{5})=[3,6)$.

  * •

$\displaystyle[\mathsf{A}_{3},\mathsf{A}_{4})=[3,3)=\emptyset$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{(1)}=0.5$
for any $\displaystyle\alpha\in[\mathsf{A}_{2},\mathsf{A}_{3})=[2,3)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{(1)}=0.5$
for any $\displaystyle\alpha\in[\mathsf{A}_{1},\mathsf{A}_{2})=[1,2)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{(0)}=1$
for any $\displaystyle\alpha\in[\mathsf{A}_{0},\mathsf{A}_{1})=[0,1)$.

So we get

$\displaystyle\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)$
$\displaystyle\displaystyle=\mathbf{1}_{[0,1)}(\alpha)+0.5\cdot\mathbf{1}_{[1,2)}(\alpha)+0.5\cdot\mathbf{1}_{[2,3)}(\alpha)+0.5\cdot\mathbf{1}_{[3,6)}(\alpha)$
$\displaystyle\displaystyle=\mathbf{1}_{[0,1)}(\alpha)+0.5\cdot\mathbf{1}_{[1,6)}(\alpha)\text{,}$

$\displaystyle\alpha\in[0,+\infty)$.

The second approach to computing generalized survival function values is based
on an (ascending) arrangement (6) of measure values. Similarly, as in Remark
3.4, let us point out first where the generalized survival function achieves
zero value.

###### Remark 3.6

From arrangement of $\displaystyle\mu$, see (6), we have that
$\displaystyle\mathsf{A}_{{\langle 0\rangle}}\in\\{\mathsf{A}_{{\langle
j\rangle}}:\mu_{j}=0,\,j\in[\kappa-1]_{0}\\}$. Thus, for any
$\displaystyle\alpha\in[\mathsf{A}_{{\langle 0\rangle}},+\infty)$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{0}=0$.

Compared to the first approach presented in Theorem 3.2, here in the second
approach we look for intervals at which monotone measure values
$\displaystyle\mu(F_{j})$ (i.e. generalized survival function values) are
achieved. It is worth noting that the first approach is appropriate to use if
the number of aggregation operator values is much smaller than the number of
measure values. In the opposite case, it is more effective to use the second
approach.

###### Theorem 3.7

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. Then

  1. (i)

for any $\displaystyle j\in[\kappa-1]_{0}$ it holds that

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{j}\,\,\ \textrm{for
any}\,\,\alpha\in\Big{[}\min_{k\leq j}\mathsf{A}_{{\langle
k\rangle}},\min_{k<j}\mathsf{A}_{{\langle k\rangle}}\Big{)};$

  2. (ii)

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{j=0}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\min\limits_{k\leq
j}\mathsf{A}_{{\langle k\rangle}},\min\limits_{k<j}\mathsf{A}_{{\langle
k\rangle}}\big{)}}(\alpha)\,\,\ \textrm{for any}\,\,\alpha\in[0,+\infty).$ (8)

Proof. For $\displaystyle j=0$ we have, that $\displaystyle\Big{[}\min_{k\leq
0}\mathsf{A}_{{\langle k\rangle}},\min_{k<0}\mathsf{A}_{{\langle
k\rangle}}\Big{)}=[\mathsf{A}_{{\langle 0\rangle}},+\infty)$. Then, from
Remark 3.6 is easy to see, that for any
$\displaystyle\alpha\in[\mathsf{A}_{{\langle 0\rangle}},+\infty)$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{0}$. Let us take an
arbitrary (fixed) $\displaystyle j\in[\kappa-1]$. The case when
$\displaystyle\min_{k\leq j}\mathsf{A}_{{\langle
k\rangle}}=\min_{k<j}\mathsf{A}_{{\langle k\rangle}}$ is trivial. Let us
suppose that $\displaystyle\min_{k\leq j}\mathsf{A}_{{\langle
k\rangle}}\neq\min_{k<j}\mathsf{A}_{{\langle k\rangle}}$. Then from the fact
that

$\displaystyle\min\Big{\\{}\min_{k<j}\mathsf{A}_{{\langle
k\rangle}},\mathsf{A}_{{\langle j\rangle}}\Big{\\}}=\min_{k\leq
j}\mathsf{A}_{{\langle k\rangle}}<\min_{k<j}\mathsf{A}_{{\langle k\rangle}},$

we have $\displaystyle\min_{k\leq j}\mathsf{A}_{{\langle
k\rangle}}=\mathsf{A}_{{\langle j\rangle}}$. Further,
$\displaystyle\min_{k<j}\mathsf{A}_{{\langle
k\rangle}}\leq\mathsf{A}_{{\langle l\rangle}}$ for each $\displaystyle
l\in[j-1]$. Then for any $\displaystyle\alpha$ such that

$\displaystyle\mathsf{A}_{{\langle j\rangle}}=\min_{k\leq
j}\mathsf{A}_{{\langle k\rangle}}\leq\alpha<\min_{k<j}\mathsf{A}_{{\langle
k\rangle}}\leq\mathsf{A}_{{\langle l\rangle}}$

with $\displaystyle l\in[j-1]$ we have

  * •

$\displaystyle\mathsf{A}_{{\langle
j\rangle}}\in\\{\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}$,
therefore
$\displaystyle\mu_{j}\in\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}$.
Thus,

$\displaystyle\min\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}\leq\mu_{j}.$

  * •

$\displaystyle\mathsf{A}_{{\langle
l\rangle}}\notin\\{\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}$,
therefore
$\displaystyle\mu_{l}\notin\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}$
and from the ordering (6) we have $\displaystyle\mu(E^{c})\geq\mu_{j}$ for
each $\displaystyle E\in\mathscr{E}$ such that
$\displaystyle\mathsf{A}(\mathbf{x}|E)\leq\alpha$, therefore

$\displaystyle\min\\{\mu(E^{c}):\mathsf{A}(\mathbf{x}|E)\leq\alpha,\,E\in\mathscr{E}\\}\geq\mu_{j}.$

The formula in (ii) directly follows from (i).  $\displaystyle\Box\;\;$

###### Example 3.8

Let us consider the same inputs as in Example 3.1. According to Remark 3.6,
$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=0$ for any
$\displaystyle\alpha\in[\mathsf{A}_{{\langle 0\rangle}},+\infty)=[6,+\infty)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{1}=0.5.$
for any $\displaystyle\alpha\in\big{[}\mathsf{A}_{{\langle
1\rangle}},\mathsf{A}_{{\langle 0\rangle}}\big{)}=[3,6)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{2}=0.5$
for any $\displaystyle\alpha\in\big{[}\mathsf{A}_{{\langle
2\rangle}},\mathsf{A}_{{\langle 1\rangle}}\big{)}=[1,3)$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{3}=0.5$
for any $\displaystyle\alpha\in\big{[}\mathsf{A}_{{\langle
2\rangle}},\mathsf{A}_{{\langle 2\rangle}}\big{)}=\emptyset$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{4}=0.7$
for any $\displaystyle\alpha\in\big{[}\mathsf{A}_{{\langle
2\rangle}},\mathsf{A}_{{\langle 2\rangle}}\big{)}=\emptyset$.

  * •

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{5}=1$
for any $\displaystyle\alpha\in\big{[}\mathsf{A}_{{\langle
5\rangle}},\mathsf{A}_{{\langle 2\rangle}}\big{)}=[0,1)$.

Therefore, the generalized survival function has the form

$\displaystyle\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)$
$\displaystyle\displaystyle=0.5\cdot\mathbf{1}_{[3,6)}(\alpha)+0.5\cdot\mathbf{1}_{[1,3)}(\alpha)+\mathbf{1}_{[0,1)}(\alpha)$
$\displaystyle\displaystyle=\mathbf{1}_{[0,1)}(\alpha)+0.5\cdot\mathbf{1}_{[1,6)}(\alpha)\text{,}$

$\displaystyle\alpha\in[0,+\infty)$, compare with Example 3.5.

The following expressions of generalized survival functions are w.r.t. special
measures. Their detailed proofs are in Appendix.

###### Corollary 3.9

Let $\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$.

  1. (i)

Let $\displaystyle{\mathscr{A}}$ be FCA and $\displaystyle\bar{\mu}$ be the
greatest monotone measure, i.e., $\displaystyle\bar{\mu}(F)=0$, if
$\displaystyle F=\emptyset$ and $\displaystyle\bar{\mu}(F)=1$, otherwise. Then
the generalized survival function w.r.t. $\displaystyle\bar{\mu}$ takes the
form

$\displaystyle\bar{\mu}_{{\mathscr{A}}}({\mathbf{x}},{\alpha})=\mathbf{1}_{[0,\mathsf{A}(\mathbf{x}|[n]))}(\alpha).$

  2. (ii)

Let $\displaystyle{\mathscr{A}}$ be FCA and
$\displaystyle\,\underline{\smash{\\!\mu\\!}}\,$ be the weakest monotone
measure, i.e., $\displaystyle\,\underline{\smash{\\!\mu\\!}}\,(F)=1$, if
$\displaystyle F=[n]$ and
$\displaystyle\,\underline{\smash{\\!\mu\\!}}\,(F)=0$, otherwise. Then the
generalized survival function w.r.t.
$\displaystyle\,\underline{\smash{\\!\mu\\!}}\,$ takes the form

$\displaystyle\,\underline{\smash{\\!\mu\\!}}\,_{\mathscr{A}}({\mathbf{x}},{\alpha})=\mathbf{1}_{\big{[}0,\min\limits_{E\neq\emptyset}\mathsf{A}(\mathbf{x}|E)\big{)}}(\alpha).$

  3. (iii)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\mu$ be a symmetric
measure, i.e., $\displaystyle\mu(F)=\mu(G)$ for $\displaystyle F,G\in 2^{[n]}$
such that $\displaystyle|F|=|G|$ (see e.g. [13]). Let us set
$\displaystyle\mu^{i}:=\mu(F)$, if $\displaystyle|F|=i$, $\displaystyle
i\in[n]_{0}$. Then the generalized survival function w.r.t. $\displaystyle\mu$
takes the form

$\displaystyle\mu_{\mathscr{A}}({\mathbf{x}},{\alpha})=\sum_{i=0}^{n}\mu^{i}\mathbf{1}_{\big{[}\min\limits_{|E|=n-i}\mathsf{A}(\mathbf{x}|E),\min\limits_{|E|=n-i+1}\mathsf{A}(\mathbf{x}|E)\big{)}}(\alpha).$

  4. (iv)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\Pi$ be a possibility
measure given for any $\displaystyle F\subseteq[n]$ as
$\displaystyle\Pi(F)=\max_{i\in F}\pi(i)$. The function
$\displaystyle\pi\colon[n]\to[0,1]$, $\displaystyle\pi(i)=\Pi(\\{i\\})$ is
called a possibility distribution (of $\displaystyle\Pi$), see e.g. [18]. Let
$\displaystyle\sigma\colon[n]\to[n]$ be a permutation such that $\displaystyle
0=\pi(\sigma(0))\leq\pi(\sigma(1))\leq\dots\leq\pi(\sigma(n))=1$ and
$\displaystyle G_{\sigma(i)}=\\{\sigma(i),\dots,\sigma(n)\\}$ for
$\displaystyle i\in\\{1,\dots,n\\}$ with $\displaystyle
G_{\sigma(n+1)}=\emptyset$ and
$\displaystyle\mathsf{A}(\mathbf{x}|G_{\sigma(0)})=\infty$. Then the
generalized survival function w.r.t. $\displaystyle\Pi$ takes the form

$\displaystyle\Pi_{{\mathscr{A}}}(\mathbf{x},\alpha)=\sum_{i=0}^{n}\pi(\sigma(i))\mathbf{1}_{\big{[}\mathsf{A}(\mathbf{x}|G_{\sigma(i+1)}),\mathsf{A}(\mathbf{x}|G_{\sigma(i)})\big{)}}(\alpha).$

  5. (v)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\mathrm{N}$ be a
necessity measure given for any $\displaystyle F\subseteq[n]$ as
$\displaystyle\mathrm{N}(F)=1-\max_{i\notin F}\pi(i)$ with the convention that
the maximum of empty set is $\displaystyle 0$. Let
$\displaystyle\sigma\colon[n]\to[n]$ be a permutation given as in the previous
case. Then the generalized survival function w.r.t. $\displaystyle N$ takes
the form

$\displaystyle
N_{{\mathscr{A}}}(\mathbf{x},\alpha)=\sum_{i=0}^{n}\big{(}1-\pi(\sigma(i))\big{)}\mathbf{1}_{\big{[}\min\limits_{k\geq
i}\mathsf{A}(\mathbf{x}|\\{\sigma(k)\\}),\min\limits_{k>i}\mathsf{A}(\mathbf{x}|\\{\sigma(k)\\})\big{)}}(\alpha),$

where we need the convention $\displaystyle\min\limits_{k\geq
0}\mathsf{A}(\mathbf{x}|\\{\sigma(k)\\})=0$.

Let us compare both approaches to calculate the generalized survival function.
Comparing Example 3.5 and Example 3.8 one can observe that the domain
partition of the generalized survival function (the interval
$\displaystyle[0,\infty)$) in Example 3.5 is a superset of the domain
partition of the same generalized survival function in Example 3.8. This
observation, we show, holds in general. As a consequence, we get that the
number of nonzero summands in expression (8) is less than in expression (7).
The proof can be found in Appendix.

###### Proposition 3.10

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. For each $\displaystyle
i\in[\kappa-1]_{0}$ there exists $\displaystyle j$ such that
$\displaystyle[\mathsf{A}_{i},\mathsf{A}_{i+1})\subseteq[\min_{k\leq
j}\mathsf{A}_{{\langle k\rangle}},\min_{k<j}\mathsf{A}_{{\langle k\rangle}})$.

##  4 Permutations and visualization

Both formulas in Section 3, presented in Theorem 3.2, 3.7 ask to minimize the
values of either monotone measure or aggregation operator. The main aim of the
present section is to show that the computation of the generalized survival
function (and the minimization process) may be accomplished just on a set of
integer indices of the values of monotone measure and aggregation operator.
Moreover, the whole procedure may be visualized, so the computation of the
generalized survival function becomes easily accessible. The main tools are
several functions on a set of indices $\displaystyle[\kappa-1]_{0}$.

###  4.1 Computation via indices

Let us introduce the main tool that we shall work with in this subsection. In
accordance with (5), (6) one can see that each set from the collection
$\displaystyle{\mathscr{E}}$ appears in the basic enumeration $\displaystyle
E_{*}\colon[\kappa-1]_{0}\to{\mathscr{E}}$ once and then its complement
appears again in the basic enumeration $\displaystyle
F_{*}\colon[\kappa-1]_{0}\to\hat{{\mathscr{E}}}$. More precisely, if
$\displaystyle V\in{\mathscr{E}}$ then there are unique indices $\displaystyle
i,j\in[\kappa-1]_{0}$ such that

$\displaystyle V=E_{i}$ and $\displaystyle V^{c}=F_{j}$.

By going through all the sets from $\displaystyle{\mathscr{E}}$ we can define
a permutation $\displaystyle(\cdot)\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$
which will describe connections among these two indices. More precisely, the
basic permutation, essential for this subsection,
$\displaystyle(\cdot)\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$ is defined by

$\displaystyle(i)=j$ whenever $\displaystyle E_{i}=F_{j}^{c}$,

i.e., $\displaystyle F_{(i)}=E_{i}^{c}$. Thus, the permutation
$\displaystyle(\cdot)$ is in accordance with the previously adopted notation
since $\displaystyle\mu_{(i)}=\mu(F_{(i)})=\mu(E_{i}^{c})$. Let us illustrate
the computation of the permutation $\displaystyle(\cdot)$.

###### Example 4.1

Let us consider the same input data as in Example 3.1. Then the index space is
$\displaystyle[5]_{0}$. Thus $\displaystyle(\cdot)$ is a map from
$\displaystyle[5]_{0}$ to $\displaystyle[5]_{0}$. To calculate the value
$\displaystyle(0)$ we need the set $\displaystyle E_{0}=\emptyset$. Its
complement is $\displaystyle\\{1,2,3\\}$, which has index $\displaystyle 5$ in
the enumeration $\displaystyle F_{*}$, i.e., $\displaystyle
F_{5}=\\{1,2,3\\}$. Thus $\displaystyle(0)=5$. Similarly, to compute
$\displaystyle(1)$, we see that $\displaystyle E_{1}=\\{3\\}$ and
$\displaystyle E_{1}^{c}=\\{1,2\\}=F_{2}$. Thus $\displaystyle(1)=2$.
Continuing in a similar fashion, we obtain all the values,

$\displaystyle(2)=4$, $\displaystyle(3)=1$, $\displaystyle(4)=3$,
$\displaystyle(5)=0$.

The arrangement from previous example may be graphically represented by a
diagram or by graphs, see Figure 1. Each of these representations has certain
advantages. In diagram, the domain of $\displaystyle(\cdot)$ is the lower
axis, while the codomain is the upper one. Thus the indices of the aggregation
operator are in the lower axis, the indices in the upper axis correspond to
the values of the monotone measure. From a practical point of view, the axes
are reversed order. To describe all visualizations deeply, let us point out
that the index assignment process can be seen also reciprocally, i.e., let us
define
$\displaystyle{\langle\cdot\rangle}\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$

$\displaystyle{\langle j\rangle}=i$ whenever $\displaystyle E_{i}^{c}=F_{j}$,

i.e., $\displaystyle E_{{\langle j\rangle}}=F_{j}^{c}$. It is easy to check
that $\displaystyle{\langle\cdot\rangle}=(\cdot)^{-1}$.

(a) Diagram of $\displaystyle(\cdot)$, resp.
$\displaystyle{\langle\cdot\rangle}=(\cdot)^{-1}$

(b) Graph of $\displaystyle(\cdot)$

(c) Graph of $\displaystyle{\langle\cdot\rangle}=(\cdot)^{-1}$

Figure 1: Diagram and graphs of functions $\displaystyle(\cdot)$,
$\displaystyle{\langle\cdot\rangle}$ for data from Example 3.1

Once comparing Definition 2.1 of the generalized survival function and the map
$\displaystyle(\cdot)$, one can see that the assignment $\displaystyle(\cdot)$
is a part of the formula defining the generalized survival function. More
precisely, we have

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\left\\{\mu_{(i)}:\mathsf{A}_{i}\leq\alpha,\,i\in[\kappa-1]_{0}\right\\},\quad\alpha\in[0,+\infty).$

Then the whole computation in the previous formula may be visualised via
Figure 1(a). Indeed, once we need to compute
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ for a given
$\displaystyle\alpha$, we can find the largest index $\displaystyle i$ such
that $\displaystyle\mathsf{A}_{i}\leq\alpha$. Afterwards we need to consider
all the indices on the upper axis in Figure 1(a) adjacent with the indices
greater than or equal to $\displaystyle i$ (the right-hand side indices with
respect to $\displaystyle i$ on the lower axis). Finally, the minimization the
values of monotone measure of sets with the selected indices on the upper axis
leads to the value $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$.
However, the application of the assignment $\displaystyle(\cdot)$ goes far
beyond the latter observations.

To understand the real contribution of $\displaystyle(\cdot)$, we need to
analyse crossing-overs in Figure 1(a). The crossing-overs can be in only two
cases:

  * •

if $\displaystyle\textstyle\min_{k\leq i}\mu_{(k)}<\mu_{(i)}$, then the value
$\displaystyle\mu_{(i)}$ is not achieved by the generalized survival function
because of Theorem 3.2;

  * •

if $\displaystyle\mu_{(k)}=\mu_{(l)}$, $\displaystyle k,l\in[\kappa-1]_{0}$,
where $\displaystyle k<l$ and $\displaystyle(k)<(l)$, which corresponds to the
ambiguity of the arrangement (4), or (6).

These connections can be removed or redefined in an appropriate manner without
changing the formula of generalized survival functions. Thus let us redefine
the mapping $\displaystyle(\cdot)$ in a manner that will be beneficial for us.
Let us define a map
$\displaystyle\mathbf{i}\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$ by

$\displaystyle\displaystyle\mathbf{i}(i)=\min\\{(0),\dots,(i)\\}.$ (9)

The previous mapping will shorten a lot of further expressions, and
calculations. It will be useful in Section 5 in deriving the generalized
Choquet integral formulas.

Figure 2: Diagram of the map $\displaystyle\mathbf{i}$ for data from Example
3.1

Figure 2 contains a diagram describing function $\displaystyle\mathbf{i}$
computed for data from Example 3.1. Although one can use formula (9) directly
and compute $\displaystyle\mathbf{i}$ algebraically, it can be easily obtained
also from the diagram in Figure 1(a). Indeed, since $\displaystyle(4)=3$ and
$\displaystyle(3)=1$, the corresponding edges in Figure 1(a) are crossed-over.
It is just enough to eliminate this crossing-over by redefining
$\displaystyle\mathbf{i}(4)=1$. Similarly for crossing-over, which corresponds
to values $\displaystyle(1)$ and $\displaystyle(2)$. After its elimination we
obtain $\displaystyle\mathbf{i}$.

###### Lemma 4.2

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, and $\displaystyle\mathbf{i}$
be a map given in (9). Then for any $\displaystyle i\in[\kappa-1]_{0}$ it
holds

$\displaystyle\min_{k\leq i}\mu_{(k)}=\mu_{\mathbf{i}(i)}\text{.}$

Proof. Because of the definition of $\displaystyle(\cdot)$ and from the
arrangement (6) we get equalities

$\displaystyle\min_{k\leq i}\mu_{(k)}=\min\\{\mu_{l}:l=(k),k\leq
i\\}=\mu_{\min\\{(0),\dots,(i)\\}}=\mu_{\mathbf{i}(i)}\text{.}$

Thus we get the required result.  $\displaystyle\Box\;\;$

Via the assignment $\displaystyle\mathbf{i}$, the formula of the generalized
survival function can be described directly, without the need to minimize,
compare the following corollary with Theorem 3.2.

###### Corollary 4.3

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, and $\displaystyle\mathbf{i}$
be a map given in (9). Then

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-1}\mu_{\mathbf{i}(i)}\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}(\alpha)$
(10)

for any $\displaystyle\alpha\in[0,+\infty)$.

Proof. The required result follows from Lemma 4.2 and Theorem 3.2.
$\displaystyle\Box\;\;$

The difference between Theorem 3.2 and Theorem 3.7 is in the object being
minimized. In Theorem 3.2 are minimized the values of monotone measure, while
in Theorem 3.7 are minimized the values of aggregation operator. Thus
Corollary 4.3 is the counterpart of Theorem 3.2 and naturally, the need to
introduce a counterpart of Theorem 3.7 arises. Similarly, as the formula for
the generalized survival function can be rewritten via permutation
$\displaystyle(\cdot)$, it can be also rewritten via permutation
$\displaystyle{\langle\cdot\rangle}$ as follows

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min\left\\{\mu_{j}:\mathsf{A}_{\langle
j\rangle}\leq\alpha,\,j\in[\kappa-1]_{0}\right\\},\quad\alpha\in[0,+\infty).$

Let us recall that
$\displaystyle{\langle\cdot\rangle}=(\cdot)^{-1}\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$,
i.e., $\displaystyle{\langle(i)\rangle}=i,({\langle j\rangle})=j$, and
$\displaystyle E_{\langle j\rangle}=F_{j}^{c}$, i.e., $\displaystyle
F_{j}=E_{\langle j\rangle}^{c}$. The permutation
$\displaystyle{\langle\cdot\rangle}$ is in accordance with the previously
adopted notation since $\displaystyle\mathsf{A}_{\langle
j\rangle}=\mathsf{A}(\mathbf{x}|E_{\langle
j\rangle})=\mathsf{A}(\mathbf{x}|F_{j}^{c})$. Let us define a map
$\displaystyle\mathbf{j}\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$ as follows

$\displaystyle\displaystyle\mathbf{j}(j)=\min\\{{\langle
0\rangle},\dots,{\langle j\rangle}\\}.$ (11)

###### Remark 4.4

It is easy to see that for any $\displaystyle k\in[\kappa-1]_{0}$ it holds
$\displaystyle\mathbf{i}(k)\leq(k)$ and
$\displaystyle\mathbf{j}({k})\leq{\langle{k}\rangle}$. Moreover, the maps
$\displaystyle\mathbf{i},\mathbf{j}$ are nonincreasing.

Similarly as in the case of the map $\displaystyle\mathbf{i}$ we can shorten a
lot of expressions, and calculations with the map $\displaystyle\mathbf{j}$.

Figure 3: Diagram of the map $\displaystyle\mathbf{j}$ for data from Example
3.1.

The corresponding diagram of $\displaystyle\mathbf{j}$ based on data from
Example 3.1 is depicted in Figure 3. Note that similarly to the diagram in
Figure 2 it does not contain any crossing-over, and may be analogously created
either directly using formula (11) or by eliminating crossings-over in the
diagram in Figure 1.

###### Lemma 4.5

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, and $\displaystyle\mathbf{j}$
be a map given as in (11). Then for any $\displaystyle j\in[\kappa-1]_{0}$ it
holds

$\displaystyle\min\limits_{k\leq j}\mathsf{A}_{{\langle
k\rangle}}=\mathsf{A}_{\mathbf{j}(j)}\text{.}$

Proof. Because of the definition of $\displaystyle(\cdot)$ and from the
arrangement (4) we immediately have

$\displaystyle\min\limits_{k\leq j}\mathsf{A}_{\langle
k\rangle}=\min\\{\mathsf{A}_{l}:l={\langle k\rangle},k\leq
j\\}=\mathsf{A}_{\min\\{{\langle 0\rangle},\dots,{\langle
j\rangle}\\}}=\mathsf{A}_{\mathbf{j}(j)}\text{,}$

thus we get the required result.  $\displaystyle\Box\;\;$

From the formula (8) we get the new expression of the generalized survival
function in the following form.

###### Corollary 4.6

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, and $\displaystyle\mathbf{j}$
be a map given as in (11). Then

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{j=0}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\mathsf{A}_{\mathbf{j}(j)},\mathsf{A}_{\mathbf{j}(j-1)}\big{)}}(\alpha)$
(12)

for any $\displaystyle\alpha\in[0,+\infty)$ with the convention
$\displaystyle\mathsf{A}_{\mathbf{j}(-1)}=+\infty$.

Proof. The required result follows from a Lemma 4.5 and Theorem 3.7.
$\displaystyle\Box\;\;$

Let us conclude this subsection with the observation that if
$\displaystyle(\cdot)$ is decreasing, then formulas (7), (8) and (10), (12)
will be simplified.

###### Corollary 4.7

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. If the mapping
$\displaystyle(\cdot)$ is decreasing, then

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$
$\displaystyle\displaystyle=\sum_{i=0}^{\kappa-1}\mu_{\kappa-1-i}\cdot\mathbf{1}_{\big{[}\mathsf{A}_{i},\mathsf{A}_{i+1}\big{)}}(\alpha)=\sum_{j=0}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\mathsf{A}_{\kappa-1-j},\mathsf{A}_{\kappa-j}\big{)}}(\alpha)$

for any $\displaystyle\alpha\in[0,+\infty)$.

Proof.  If the map $\displaystyle(\cdot)$ is decreasing, then
$\displaystyle{\langle\cdot\rangle}$ is also decreasing and it holds
$\displaystyle(\cdot)=\mathbf{i}$,
$\displaystyle{\langle\cdot\rangle}=\mathbf{j}$. Then expressions (10), (12)
are simplified

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-1}\mu_{(i)}\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}(\alpha)=\sum_{j=0}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\mathsf{A}_{{\langle
j\rangle}},\mathsf{A}_{{\langle j-1\rangle}}\big{)}}(\alpha).$

Further, the map $\displaystyle(\cdot)$ is decreasing if and only if for any
$\displaystyle i\in[\kappa-1]_{0}$ it holds $\displaystyle(i)=\kappa-1-i$, or
equivalently for any $\displaystyle j\in[\kappa-1]_{0}$ it holds
$\displaystyle{\langle j\rangle}=\kappa-1-j$, thus
$\displaystyle(\cdot)={\langle\cdot\rangle}$. This completes the proof.
$\displaystyle\Box\;\;$

###### Example 4.8

Let us consider the collection
$\displaystyle{\mathscr{E}}=\\{\emptyset,\\{1\\},\\{3\\},\\{1,2\\},\\{1,2,3\\}\\}$,
the family of conditional aggregation operators
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in{\mathscr{E}}\\}$,
the vector $\displaystyle\mathbf{x}=(2,3,4)$, and the monotone measure
$\displaystyle\mu$ on $\displaystyle\hat{{\mathscr{E}}}$ with values
$\displaystyle\mu(\emptyset)=0$, $\displaystyle\mu(\\{3\\})=0.3$,
$\displaystyle\mu(\\{1,2\\})=0.5$, $\displaystyle\mu(\\{2,3\\})=0.8$,
$\displaystyle\mu(\\{1,2,3\\})=1$.

$\displaystyle i,j$ |  0 |  1 |  2 |  3 |  4
---|---|---|---|---|---
$\displaystyle E_{i}$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{1\\}$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{1,2,3\\}$
$\displaystyle\mathsf{A}_{i}$ |  $\displaystyle 0$ |  $\displaystyle 2$ |  $\displaystyle 4$ |  $\displaystyle 5$ |  $\displaystyle 9$
$\displaystyle F_{j}$ |  $\displaystyle\emptyset$ |  $\displaystyle\\{3\\}$ |  $\displaystyle\\{1,2\\}$ |  $\displaystyle\\{2,3\\}$ |  $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu_{j}$ |  $\displaystyle 0$ |  $\displaystyle 0.3$ |  $\displaystyle 0.5$ |  $\displaystyle 0.8$ |  $\displaystyle 1$

The diagram of functions $\displaystyle(\cdot)$ and
$\displaystyle{\langle\cdot\rangle}$ can be seen in the Figure 4.

Figure 4: Diagram of $\displaystyle(\cdot)$ and
$\displaystyle{\langle\cdot\rangle}$ from Example 4.8

It is easy to verify that the assumptions of Corollary 4.7 are satisfied.
According to this result, the generalized survival function has the form

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=1\cdot\mathbf{1}_{[0,2)}(\alpha)+0.8\cdot\mathbf{1}_{[2,4)}(\alpha)+0.5\cdot\mathbf{1}_{[4,5)}(\alpha)+0.3\cdot\mathbf{1}_{[5,9)}(\alpha)$

for any $\displaystyle\alpha\in[0,+\infty)$.

From the previous one can see that the assumption of Corollary 4.7 can be
graphically interpreted in the way that there are no crossing-overs in the
diagrams, see Figure 4.

In the following, we apply the obtained formulas to the solution of the
Knapsack problem described in Section 2.

#### Solution of the Knapsack problem

Let us assume that a total of $\displaystyle 800$ ml of liquids are already
packed in the knapsack. Thus, it is still possible to pack $\displaystyle 200$
ml of liquids. Let us choose from the products listed in Table 3. In the
table, except for the volume of the products also their price is indicated.

product | $\displaystyle a$ | $\displaystyle b$ | $\displaystyle c$ | $\displaystyle d$
---|---|---|---|---
volume | $\displaystyle 80$ | $\displaystyle 75$ | $\displaystyle 55$ | $\displaystyle 65$
price | $\displaystyle 1.2$ | $\displaystyle 1$ | $\displaystyle 0.6$ | $\displaystyle 0.8$
Table 3: List of products

As we have pointed out in Section 2, our goal is to minimize purchase costs
for products that we shall need to buy at the holiday destination (we suppose
prices are higher here). Moreover, we have to keep a limit of $\displaystyle
200$ ml of liquids we can carry, i.e., this is the limit for the total volume
of products, we shall buy at home. We shall solve the optimization problem

$\displaystyle\min\\{\mu(E^{c}):\mathsf{A}^{\text{sum}}(\mathbf{x}|E)\leq
200,\,E\in\mathscr{E}\\}\text{,}$

with $\displaystyle\mathbf{x}=(80,75,55,65)$, the collection
$\displaystyle\mathscr{E}$ consists of all possible combinations of elements
$\displaystyle a,b,c,d$, and the monotone measure $\displaystyle\mu$
represents the price of products. The values of $\displaystyle\mu$ together
with the values of the conditional aggregation operator
$\displaystyle\mathsf{A}^{\mathrm{sum}}$ can be seen in Table 4.

$\displaystyle\mu_{j}$ | $\displaystyle\mu(E^{c})$ | $\displaystyle E^{c}$ | $\displaystyle E$ | $\displaystyle\mathsf{A}^{\text{sum}}$ | $\displaystyle\mathsf{A}_{i}$ |  | $\displaystyle\mu_{j}$ | $\displaystyle\mu(E^{c})$ | $\displaystyle E^{c}$ | $\displaystyle E$ | $\displaystyle\mathsf{A}^{\text{sum}}$ | $\displaystyle\mathsf{A}_{i}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\displaystyle\mu_{15}$ | $\displaystyle 3.6$ | $\displaystyle\\{a,b,c,d\\}$ | $\displaystyle\emptyset$ | $\displaystyle 0$ | $\displaystyle\mathsf{A}_{0}$ |  | $\displaystyle\mu_{9}$ | $\displaystyle 1.8$ | $\displaystyle\\{a,d\\}$ | $\displaystyle\\{b,c\\}$ | $\displaystyle 130$ | $\displaystyle\mathsf{A}_{6}$
$\displaystyle\mu_{11}$ | $\displaystyle 2.4$ | $\displaystyle\\{b,c,d\\}$ | $\displaystyle\\{a\\}$ | $\displaystyle 80$ | $\displaystyle\mathsf{A}_{4}$ |  | $\displaystyle\mu_{7}$ | $\displaystyle 1.8$ | $\displaystyle\\{a,c\\}$ | $\displaystyle\\{b,d\\}$ | $\displaystyle 140$ | $\displaystyle\mathsf{A}_{8}$
$\displaystyle\mu_{13}$ | $\displaystyle 2.5$ | $\displaystyle\\{a,c,d\\}$ | $\displaystyle\\{b\\}$ | $\displaystyle 75$ | $\displaystyle\mathsf{A}_{3}$ |  | $\displaystyle\mu_{10}$ | $\displaystyle 2.2$ | $\displaystyle\\{a,b\\}$ | $\displaystyle\\{c,d\\}$ | $\displaystyle 120$ | $\displaystyle\mathsf{A}_{5}$
$\displaystyle\mu_{14}$ | $\displaystyle 3.0$ | $\displaystyle\\{a,b,d\\}$ | $\displaystyle\\{c\\}$ | $\displaystyle 55$ | $\displaystyle\mathsf{A}_{1}$ |  | $\displaystyle\mu_{2}$ | $\displaystyle 0.8$ | $\displaystyle\\{d\\}$ | $\displaystyle\\{a,b,c\\}$ | $\displaystyle 210$ | $\displaystyle\mathsf{A}_{13}$
$\displaystyle\mu_{12}$ | $\displaystyle 2.5$ | $\displaystyle\\{a,b,c\\}$ | $\displaystyle\\{d\\}$ | $\displaystyle 65$ | $\displaystyle\mathsf{A}_{2}$ |  | $\displaystyle\mu_{1}$ | $\displaystyle 0.6$ | $\displaystyle\\{c\\}$ | $\displaystyle\\{a,b,d\\}$ | $\displaystyle 220$ | $\displaystyle\mathsf{A}_{14}$
$\displaystyle\mu_{5}$ | $\displaystyle 1.4$ | $\displaystyle\\{c,d\\}$ | $\displaystyle\\{a,b\\}$ | $\displaystyle 155$ | $\displaystyle\mathsf{A}_{10}$ |  | $\displaystyle\mu_{3}$ | $\displaystyle 1.0$ | $\displaystyle\\{b\\}$ | $\displaystyle\\{a,c,d\\}$ | $\displaystyle 200$ | $\displaystyle\mathsf{A}_{12}$
$\displaystyle\mu_{8}$ | $\displaystyle 1.8$ | $\displaystyle\\{b,d\\}$ | $\displaystyle\\{a,c\\}$ | $\displaystyle 135$ | $\displaystyle\mathsf{A}_{7}$ |  | $\displaystyle\mu_{4}$ | $\displaystyle 1.2$ | $\displaystyle\\{a\\}$ | $\displaystyle\\{b,c,d\\}$ | $\displaystyle 195$ | $\displaystyle\mathsf{A}_{11}$
$\displaystyle\mu_{6}$ | $\displaystyle 1.4$ | $\displaystyle\\{b,c\\}$ | $\displaystyle\\{a,d\\}$ | $\displaystyle 145$ | $\displaystyle\mathsf{A}_{9}$ |  | $\displaystyle\mu_{0}$ | $\displaystyle 0$ | $\displaystyle\emptyset$ | $\displaystyle\\{a,b,c,d\\}$ | $\displaystyle 275$ | $\displaystyle\mathsf{A}_{15}$
Table 4: Values of the conditional aggregation operator and corresponding
monotone measure

We can notice that in Table 4 there are $\displaystyle 15$ different values of
the conditional aggregation operator and $\displaystyle 12$ different values
of the (nonadditive) monotone measure. To solve the knapsack problem we have
to find the value of the generalized survival function at point $\displaystyle
200$. This can be easily done using formula (7) or (10). The value of
$\displaystyle 200$ lies in the interval
$\displaystyle[\mathsf{A}_{12},\mathsf{A}_{13})$, where the generalized
survival function takes the value

$\displaystyle\min_{k\leq 12}\mu_{(k)}=\mu_{\mathbf{i}(12)}=\mu_{3}=1\text{.}$

The same result we get when we determine the whole formula of the generalized
survival function using formula (19)

$\displaystyle\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)$
$\displaystyle\displaystyle=3.6\cdot\mathbf{1}_{[0;55)}(\alpha)+3\cdot\mathbf{1}_{[55;65)}(\alpha)+2.5\cdot\mathbf{1}_{[65;80)}(\alpha)+2.4\cdot\mathbf{1}_{[80;120)}(\alpha)$
$\displaystyle\displaystyle+2.2\cdot\mathbf{1}_{[120;130)}(\alpha)+1.8\cdot\mathbf{1}_{[130;145)}(\alpha)+1.4\cdot\mathbf{1}_{[145;195)}(\alpha)+1.2\cdot\mathbf{1}_{[195;200)}(\alpha)$
$\displaystyle\displaystyle+\mathbf{1}_{[200;210)}(\alpha)+0.8\cdot\mathbf{1}_{[210;220)}(\alpha)+0.6\cdot\mathbf{1}_{[220;275)}(\alpha)\text{.}$

As we can see, the result coincides with the result from the previous
calculation. So, a traveler should buy products $\displaystyle a,c,d$ at home
and product $\displaystyle b$ at the destination.

###  4.2 How to draw a graph of a generalized survival function

In the previous section, we have presented visualizations of maps
$\displaystyle(\cdot)$ and $\displaystyle{\langle\cdot\rangle}$ using both a
diagram and a graph in the Cartesian coordinate system, see demonstratory
Figure 1 based on inputs from Example 3.1. We have already pointed out the
advantages of the first visualization, and in the following we deal with the
second one. In fact, we shall show how the graph of $\displaystyle(\cdot)$ or
$\displaystyle{\langle\cdot\rangle}$ can be transformed to the plot of the
generalized survival function.

The first step is to transform the graphs of $\displaystyle(\cdot)$ or
$\displaystyle{\langle\cdot\rangle}$, see Figure 1(b)1(c), into the graphs of
the mappings $\displaystyle\mathbf{i}$ or $\displaystyle\mathbf{j}$, see
Figure 5, decreasing some of the values of $\displaystyle(\cdot)$ and
$\displaystyle{\langle\cdot\rangle}$ to produce nonincreasing functions
$\displaystyle\mathbf{i}$ and $\displaystyle\mathbf{j}$, respectively. More
precisely, the formulas (9) and (11) are used to compute
$\displaystyle\mathbf{i}$ and $\displaystyle\mathbf{j}$ from
$\displaystyle(\cdot)$ and $\displaystyle{\langle\cdot\rangle}$, respectively.

Figure 5: Graphs of $\displaystyle\mathbf{i}$ and $\displaystyle\mathbf{j}$.

The second step is to extend the domain of $\displaystyle\mathbf{i}$ from
$\displaystyle[\kappa-1]_{0}$ to $\displaystyle[0,+\infty)$, and obtain the
graph of the indexed generalized survival function
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$. It is enough to
naturally define
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)=\mathbf{i}(\lfloor\beta\rfloor)$
for $\displaystyle\beta<\kappa-1$ and
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)=\mathbf{i}(\kappa-1)$
otherwise, as is done in Figure 6(a). There is similar way to obtain the
indexed generalized survival function from the graph of
$\displaystyle\mathbf{j}$, depicted in Figure 6(b), which we describe later.

(a) Indexed generalized survival function (using $\displaystyle\mathbf{i}$)

(b) Indexed generalized survival function (using $\displaystyle\mathbf{j}$)

Figure 6: Indexed generalized survival functions deriving by using maps
$\displaystyle\mathbf{i}$ and $\displaystyle\mathbf{j}$

Before we proceed to the last step, let us stress that Figure 6 of
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$ is very close to how
the real generalized survival function
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ depicted in Figure 7 looks
like. Vaguely speaking, the only difference is that the graph of
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ has values
$\displaystyle\mathsf{A}_{0},\mathsf{A}_{1},\dots,\mathsf{A}_{\kappa-1}$ on
the horizontal axis instead of values $\displaystyle 0,1,\dots,\kappa-1$, and
values $\displaystyle\mu_{0},\mu_{1},\dots,\mu_{\kappa-1}$ on the vertical
axis again instead of values $\displaystyle 0,1,\dots,\kappa-1$. Moreover,
some values $\displaystyle\mathsf{A}_{i},\mathsf{A}_{j}$ and
$\displaystyle\mu_{i},\mu_{j}$ may coincide even for different $\displaystyle
i,j$.

Figure 7: Generalized survival function

More formally, the similarities between
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$ and
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ may be understood once we
represent the latter one via the formula (10), and the first one in a similar
way, namely

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-1}\mu_{\mathbf{i}(i)}\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}(\alpha),\hskip
28.45274pt\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)=\sum_{i=0}^{\kappa-2}\mathbf{i}(i)\mathbf{1}_{[i,\,i+1)}(\beta).$
(13)

The equalities (13) are based on function $\displaystyle\mathbf{i}$, but we
may proceed similarly with function $\displaystyle\mathbf{j}$ using the
equality (12):

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-1}\mu_{\mathbf{i}(i)}\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}(\alpha),\hskip
28.45274pt\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$
$\displaystyle\displaystyle=\sum_{j=1}^{\kappa-1}j\mathbf{1}_{[\mathbf{j}(j),\,\mathbf{j}(j-1))}(\beta).$
(14)

The equality (14) gives clues how to obtain the graph of the indexed
generalized survival function
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$ from the graph of the
map $\displaystyle\mathbf{j}$. There is necessary one more step, namely, one
has to draw a graph of the generalized inverse $\displaystyle\mathbf{j}^{-}$
from the graph of $\displaystyle\mathbf{j}$ first. Then, similarly to the case
of $\displaystyle\mathbf{i}$, define
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)=\mathbf{j}^{-}(\lfloor\beta\rfloor)$
for $\displaystyle\beta<\kappa-1$ and
$\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)=\mathbf{j}^{-}(\kappa-1)$
otherwise, see Figure 6(b).

Let us summarize the calculation of the indexed generalized survival function.
First, we construct the bijections $\displaystyle E_{*}$ a $\displaystyle
F_{*}$ as it shown in (4) a (6). Then, using the permutation
$\displaystyle(\cdot)\colon[\kappa-1]_{0}\to[\kappa-1]_{0}$ described above,
we obtain the values of $\displaystyle\mathbf{i}(i)$ and
$\displaystyle\mathbf{j}(i)$ for each $\displaystyle i\in[\kappa-1]_{0}$, and
thus we can construct the indexed generalized survival function. Finally, for
each index $\displaystyle k\in[\kappa-1]_{0}$ on the horizotal, resp.
vertical, axes we assign the value $\displaystyle\mathsf{A}_{k}$, resp.
$\displaystyle\mu_{k}$. This entire process can be described by the following
Algorithm 1 and Algorithm 2. Note that these algorithms assume the fact that
the values of map $\displaystyle(\cdot)$, resp.
$\displaystyle{\langle\cdot\rangle}$ are known to the user, or for
completeness, we present algorithms for its calculation in Appendix.

Method  _the-graph-of-GSF
$\displaystyle(\mathscr{E},\mu,\mathbf{x},{\mathscr{A}})$_

$\displaystyle(0),\dots,(\kappa-1)$ $\displaystyle\leftarrow$
the-$\displaystyle(\cdot)$-map$\displaystyle(\mathscr{E},\mu,\mathbf{x},{\mathscr{A}})$;

$\displaystyle GSF:=0$;

for  _$\displaystyle(i=0$ , $\displaystyle i<\kappa-1$, $\displaystyle
i\mathrm{++})$_ do

if  _$\displaystyle(i+1) >(i)$_ then

$\displaystyle(i+1):=(i)$;

end if

$\displaystyle
GSF:=GSF+\mu_{(i)}\cdot\mathbf{1}_{[\mathsf{A}_{i},\mathsf{A}_{i+1})}$;

end for

_ _return _$\displaystyle GSF$ ;_

 __

end

Algorithm 1 Calculation of generalized survival function (GSF) using the map
$\displaystyle\mathbf{i}$

Method  _the-graph-of-GSF
$\displaystyle(\mathscr{E},\mu,\mathbf{x},{\mathscr{A}})$_

$\displaystyle{\langle 0\rangle},\dots,{\langle\kappa-1\rangle}$
$\displaystyle\leftarrow$
the-$\displaystyle{\langle\cdot\rangle}$-map$\displaystyle(\mathscr{E},\mu,\mathbf{x},{\mathscr{A}})$;

$\displaystyle GSF:=0$;

for  _$\displaystyle(j=1$ , $\displaystyle j<\kappa$, $\displaystyle
j\mathrm{++})$_ do

if  _$\displaystyle{\langle j-1\rangle} <{\langle j\rangle}$_ then

$\displaystyle{\langle j\rangle}:={\langle j-1\rangle}$;

end if

$\displaystyle GSF:=GSF+\mu_{j}\cdot\mathbf{1}_{[\mathsf{A}_{\langle
j\rangle},\mathsf{A}_{\langle j-1\rangle})}$;

end for

_ _return _$\displaystyle GSF$ ;_

 __

end

Algorithm 2 Calculation of generalized survival function (GSF) using the map
$\displaystyle\mathbf{j}$

Note that although the maps $\displaystyle\mathbf{i}$ and
$\displaystyle\mathbf{j}$ is not appear explicitly in the algorithms, their
generation is hidden in their for loops. In this part of algorithms, the
indexed generalized survival function is also generated, while its standard
version is immediately assigned to it. A zero value of generalized survival
function can also be marked in the graph (Figure 7), which corresponds to
Remark 3.4 and Remark 3.6, but it is not necessary, since this value does not
bring any new information.

###### Remark 4.9

The following relationships can be observed from formulas (13) and (14). Let
$\displaystyle I=\\{\mathbf{i}(i):i\in[\kappa-1]_{0}\\}$ and $\displaystyle
J=\\{\mathbf{j}(j):j\in[\kappa-1]_{0}\\}$, $\displaystyle j\in I$ and
$\displaystyle i\in J$, then

$\displaystyle\displaystyle\min\\{k\in[\kappa-1]_{0}:\mathbf{j}(k)=i\\}$
$\displaystyle\displaystyle=\mathbf{i}(i)\text{,}$
$\displaystyle\displaystyle\min\\{k\in[\kappa-1]_{0}:\mathbf{i}(k)=j\\}$
$\displaystyle\displaystyle=\mathbf{j}(j)\text{.}$

These equalities hold because of

$\displaystyle\displaystyle\min\\{k\in[\kappa-1]_{0}:\mathbf{j}(k)=i\\}$
$\displaystyle\displaystyle=\min\\{k\in[\kappa-1]_{0}:\min\\{{\langle
0\rangle},\dots,{\langle k\rangle}\\}=i\\}$
$\displaystyle\displaystyle=\min\\{k\in[\kappa-1]_{0}:{\langle
k\rangle}=i\\}=\min\\{k\in[\kappa-1]_{0}:k=(i)\\}$
$\displaystyle\displaystyle=\min\\{(0),\dots,(i)\\}=\mathbf{i}(i)\text{.}$

The penultimate equality follows from the fact that $\displaystyle(\cdot)$ is
nonincreasing, see Remark 4.4. Similarly for the second formula.

###### Remark 4.10

It is also possible to use (13) and (14) with analogy to the formula (2) for
calculating the indexed generalized survival function as follows

$\displaystyle\displaystyle\mu_{\mathscr{A}^{I}}(\mathbf{x},\beta)$
$\displaystyle\displaystyle=\min\left\\{(i):i\leq\beta,\,i\in[\kappa-1]_{0}\right\\}=\mathbf{i}(\beta)$
$\displaystyle\displaystyle=\min\left\\{j:{\langle
j\rangle}\leq\beta,\,j\in[\kappa-1]_{0}\right\\}=\mathbf{j}^{-}(\beta)$

for any $\displaystyle\beta\in[0,+\infty)$ with respect to extended domain of
$\displaystyle\mathbf{i}$ and $\displaystyle\mathbf{j}^{-}$, i.e.
$\displaystyle[0,+\infty)$, described above.

##  5 Generalized Choquet integral computation

The survival function measuring the $\displaystyle\alpha$-level set
$\displaystyle\\{f>\alpha\\}$222$\displaystyle\\{f>\alpha\\}=\\{x\in
X:f(x)>\alpha\\}$ of an arbitrary measurable function $\displaystyle f$ via
monotone measure $\displaystyle\mu$ is the essence of the concept of the
famous Choquet integral [5] defined by

$\displaystyle\mathrm{C}(f,\mu)=\int_{0}^{\infty}\mu(\\{f>\alpha\\})\,\mathrm{d}\alpha.$

On the finite spaces, the evaluation formula of the Choquet integral has the
simplified form

$\displaystyle\displaystyle\mathrm{C}(\mathbf{x},\mu)=\sum_{i=1}^{n}\mu{(G_{\sigma(i)})}(x_{\sigma{(i)}}-x_{\sigma{(i-1)}})=\sum_{i=1}^{n}x_{\sigma{(i)}}(\mu{(G_{\sigma(i)})}-\mu{(G_{\sigma(i+1)})}),$
(15)

where
$\displaystyle\mathbf{x}=(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)})$,
with $\displaystyle\sigma\colon[n]\to[n]$ being a permutation such that
$\displaystyle x_{\sigma(1)}\leq x_{\sigma(2)}\leq\dots\leq x_{\sigma(n)}$
with the convention $\displaystyle x_{\sigma(0)}=0$, $\displaystyle
G_{\sigma{(i)}}=\\{\sigma{(i)},\dots,\sigma{(n)}\\}$ for $\displaystyle
i\in[n]$, and $\displaystyle G_{\sigma(n+1)}=\emptyset$.

Based on generalized survival function, a new concept generalizing the Choquet
integral naturally arises. In this section, we aim to provide formulas for
discrete $\displaystyle{\mathscr{A}}$-Choquet integral.

###### Definition 5.1

(cf. [1, Definition 5.4.]) Let $\displaystyle{\mathscr{A}}$ be a FCA,
$\displaystyle\mu\in\mathbf{M}$ and
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. The Choquet integral with
respect to $\displaystyle{\mathscr{A}}$ and $\displaystyle\mu$
($\displaystyle{\mathscr{A}}$-Choquet integral, for short) of
$\displaystyle\mathbf{x}$ is defined as

$\displaystyle\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\int_{0}^{\infty}\mu_{\mathscr{A}}(\mathbf{x},\alpha)\,\mathrm{d}\alpha.$
(16)

As we have obtained the generalized survival function expressions, see
Sections 3, 4, the computation of the $\displaystyle{\mathscr{A}}$-Choquet
integral becomes a trivial matter. Note that the formulas in the first two
lines are obtained by means of mappings $\displaystyle\mathbf{i}$ and
$\displaystyle\mathbf{j}$, respectively. By introducing these mappings, we
managed to obtain expressions similar to the original one in (15).

###### Theorem 5.2

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$ and
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$. Then

  1. (i)

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=0}^{\kappa-2}\mu_{\mathbf{i}(i)}(\mathsf{A}_{i+1}-\mathsf{A}_{i})=\sum_{i=0}^{\kappa-1}\mathsf{A}_{i+1}\big{(}\mu_{\mathbf{i}(i)}-\mu_{\mathbf{i}(i+1)}\big{)}$,
[cf. Corollary 4.3]

  2. (ii)

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=1}^{\kappa-1}\mu_{i}\big{(}\mathsf{A}_{\mathbf{j}(i-1)}-\mathsf{A}_{\mathbf{j}(i)}\big{)}=\sum_{i=1}^{\kappa-1}\mathsf{A}_{\mathbf{j}(i-1)}(\mu_{i}-\mu_{i-1})$,
[cf. Corollary 4.6]

  3. (iii)

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=0}^{\kappa-2}\min_{k\leq
i}\mu_{(k)}(\mathsf{A}_{i+1}-\mathsf{A}_{i})=\sum_{i=0}^{\kappa-1}\mathsf{A}_{i+1}\big{(}\min_{k\leq
i}\mu_{(k)}-\min_{k\leq i+1}\mu_{(k)}\big{)}$, [cf. Theorem 3.2]

  4. (iv)

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=1}^{\kappa-1}\mu_{i}\big{(}\min_{k<i}\mathsf{A}_{\langle
k\rangle}-\min_{k\leq i}\mathsf{A}_{\langle
k\rangle}\big{)}=\sum_{i=1}^{\kappa-1}\min_{k<i}\mathsf{A}_{\langle
k\rangle}(\mu_{i}-\mu_{i-1})$. [cf. Theorem 3.7]

Let us recall that in Corollary 3.9 we listed computational formulas for
generalized survival function for special measures. Thus, Corollary 3.9 helps
us to substantially improve the formulas presented in Theorem 5.2 for special
measures.

###### Corollary 5.3

Let $\displaystyle x\in[0,+\infty)^{[n]}$.

  1. (i)

Let $\displaystyle{\mathscr{A}}$ be FCA and $\displaystyle\bar{\mu}$ be a
greatest monotone measure. Then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\,\underline{\smash{\\!\mu\\!}}\,)=\mathsf{A}(\mathbf{x}|[n]).$

  2. (ii)

Let $\displaystyle{\mathscr{A}}$ be FCA. Let
$\displaystyle\,\underline{\smash{\\!\mu\\!}}\,$ be a weakest monotone
measure. Then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\bar{\mu})=\min_{E\neq\emptyset}\mathsf{A}(\mathbf{x}|E).$

  3. (iii)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\mu$ be a symmetric
measure and set $\displaystyle\mu^{i}:=\mu(F)$, if $\displaystyle|F|=i$,
$\displaystyle i\in[n]\cup\\{0\\}$. Then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\sum_{i=1}^{n}(\mu^{i}-\mu^{i-1})\min\limits_{|E|=n-i+1}\mathsf{A}(\mathbf{x}|E).$

  4. (iv)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\Pi$ be a possibility
measure. Let $\displaystyle G_{\sigma(i)}=\\{\sigma(i),\dots,\sigma(n)\\}$ for
$\displaystyle i\in\\{1,\dots,n\\}$, where $\displaystyle\sigma$ is a
permutation as in Corollary 3.9. Then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\Pi)=\sum_{i=1}^{n}\left(\pi(\sigma(i))-\pi(\sigma(i-1))\right)\mathsf{A}(\mathbf{x}|G_{\sigma(i)}).$

  5. (v)

Let $\displaystyle{\mathscr{A}}$ be FCA monotone w.r.t. sets with
$\displaystyle{\mathscr{E}}=2^{[n]}$. Let $\displaystyle\mathrm{N}$ be a
neccessity measure and $\displaystyle\sigma$ be a permutation as in Corollary
3.9. Then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},N)=\sum_{i=1}^{n}\big{(}\pi(\sigma(i))-\pi(\sigma(i-1))\big{)}\min\limits_{k\geq
i}\mathsf{A}(\mathbf{x}|\\{\sigma(k)\\}).$

###### Remark 5.4

In the following, we aim to emphasize that the
$\displaystyle{\mathscr{A}}$-Choquet integral covers the famuous Choquet
integral. Let us consider a special family
$\displaystyle{\mathscr{A}}^{\mathrm{max}}$. Taking symmetric measures we
obtain

$\displaystyle\mathrm{C}_{{\mathscr{A}}^{\mathrm{max}}}(\mathbf{x},\mu)=\sum_{i=1}^{n}(\mu^{i}-\mu^{i-1})x_{\sigma(n-i+1)}=\sum_{i=1}^{n}(\mu^{n-i+1}-\mu^{n-i})x_{\sigma(i)},$

where $\displaystyle\sigma\colon[n]\to[n]$ is a permutation such that
$\displaystyle x_{\sigma(1)}\leq\dots\leq x_{\sigma(n)}$ with the convention
$\displaystyle x_{\sigma(0)}=0$. This is a formula of Yager’s ordered weighted
averaging (OWA) operator [17] as Grabisch has shown, see [10]. Therefore, the
generalized $\displaystyle{\mathscr{A}}$-Choquet integral w.r.t. symmetric
measures can be seen as a new type of the OWA operator.

The formula for possibility measure in Corollary 5.3 (iv) simplifies to the
form

$\displaystyle\displaystyle\mathrm{C}_{{\mathscr{A}}^{\mathrm{max}}}(\mathbf{x},\Pi)=\sum_{i=1}^{n}(\pi(\sigma(i))-\pi(\sigma(i-1)))\max_{k\geq
i}x_{\sigma(k)}=\sum_{i=1}^{n}(\pi(\sigma(i))-\pi(\sigma(i-1)))\max_{\pi(k)\geq\pi(\sigma(i))}x_{k},$

which is the famous Choquet integral with respect to possibility measure, cf.
[6], and permutation $\displaystyle\sigma$ as in Corollary 3.9. Similarly,
taking necessity measure in Corollary 5.3 (v) we obtain

$\displaystyle\displaystyle\mathrm{C}_{{\mathscr{A}}^{\mathrm{max}}}(\mathbf{x},\mathrm{N})=\sum_{i=1}^{n}(\pi(\sigma(i))-\pi(\sigma(i-1)))\min_{k\geq
i}x_{\sigma(k)}=\sum_{i=1}^{n}(\pi(\sigma(i))-\pi(\sigma(i-1)))\min_{\pi(k)\geq\pi(\sigma(i))}x_{k}.$

Due to formulas derived in this section, we are ready to solve the problem of
accomodation options introduced in Section 2, using the concept of the
generalized Choquet integral.

#### Solution of the Problem of accomodation options

The aim of this example is to determine the most suitable accommodation for
each character of the traveler (Anthony, Brittany, Charley). We use two
methods: the standard and the generalized Choquet integral. Since we are
deciding with respect to several criteria, the problem of choosing the best
accommodation is a multi-criteria decision-making problem.

As we can see in Table 2, the criteria of the multicriteria decision-making
process are different types: distance and price are minimization criteria,
while the review is maximization criteria. Thus values in tables in the recent
form are not suitable for working with, we have to reevaluate them using the
standard methods, see [15].

According to the previous let us adjust the column corresponding to distance:

  * •

In each table let us choose the minimum value in the column for the distance.

  * •

Let us divide the minimum from the previous step by each value from the
distance column.

Let us repeat this process for the price as well. For the reviews column let
us divide each value by the maximum value from the corresponding column. This
gives us the following input vectors for decision-making process
$\displaystyle\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{c}_{1},\mathbf{c}_{2}$:

|  D |  P |  R
---|---|---|---
$\displaystyle\mathbf{a}_{1}$ |  $\displaystyle 1$ |  $\displaystyle 0.84$ |  $\displaystyle 0.875$
$\displaystyle\mathbf{a}_{2}$ |  $\displaystyle 0.4$ |  $\displaystyle 1$ |  $\displaystyle 1$
Anthony

|  D |  P |  R
---|---|---|---
$\displaystyle\mathbf{b}_{1}$ |  $\displaystyle 1$ |  $\displaystyle 0.8$ |  $\displaystyle 0.2$
$\displaystyle\mathbf{b}_{2}$ |  $\displaystyle 0.7$ |  $\displaystyle 1$ |  $\displaystyle 1$
Brittany

|  D |  P |  R
---|---|---|---
$\displaystyle\mathbf{c}_{1}$ |  $\displaystyle 1$ |  $\displaystyle 0.95$ |  $\displaystyle 0.8$
$\displaystyle\mathbf{c}_{2}$ |  $\displaystyle 0.2$ |  $\displaystyle 1$ |  $\displaystyle 1$
Charley

Table 5: Accommodation options for people – adjusted data

To evaluate each input vector (alternative) by Choquet integrals we have to
know the values of monotone measures of each set $\displaystyle
E\subseteq\\{D,P,R\\}$. We already know the values of the monotone measures
$\displaystyle\mu$, $\displaystyle\nu$, $\displaystyle\xi$, that specify the
characters of persons, for singletons, see Table 1. Further, because of the
definition of the (normalized) monotone measure,

$\displaystyle\mu(\emptyset)=\nu(\emptyset)=\xi(\emptyset)=0$,
$\displaystyle\mu(\\{\text{D},\text{P},\text{R}\\})=\nu(\\{\text{D},\text{P},\text{R}\\})=\xi(\\{\text{D},\text{P},\text{R}\\})=1$.

It remains to determine the values of monotone measures of sets
$\displaystyle\\{\text{D},\text{P}\\}$,
$\displaystyle\\{\text{D},\text{R}\\}$,
$\displaystyle\\{\text{P},\text{R}\\}$. The question is: How to set this
values? Let us rephrase this question: If we set the values of the monotone
measure, how do we know that they are set well? Various approaches to verify
the correctness of the set values are known in the literature. Let us mention
e.g. interaction index [8], Banzhaf power index or Shapley value. We chose the
Shapley value and the methodology described in [8]. The Shapley value
expresses the significance of a given criterion by aggregating in a certain
way the values of the monotone measure of the sets to which the given
criterion belongs:

$\displaystyle\displaystyle\varphi_{\mu}(i)=\sum_{A\subseteq[n]\backslash\\{i\\}}\gamma_{[n]}(A)\cdot\left(\mu(A\cup\\{i\\})-\mu(A)\right)\text{,}$
(17)

where $\displaystyle i\in[n]$ is the index associated with the criterion and
$\displaystyle\textstyle\gamma_{[n]}(A)=\frac{(n-|A|-1)!\cdot|A|!}{n!}$.
Shapley value has the property that
$\displaystyle\textstyle\sum_{i=1}^{n}\varphi_{\mu}(i)=1$. So, we set the
monotone measures $\displaystyle\mu,\nu,\xi$ to obtain

$\displaystyle\mu(\\{i\\})=\varphi_{\mu}(i),\quad\nu(\\{i\\})=\varphi_{\nu}(i),\quad\xi(\\{i\\})=\varphi_{\xi}(i),$

i.e. we form the equations given by (17) and we calculate the remaining values
of the monotone measures (in the table are their approximated values to two
decimal places):

$\displaystyle E$ |  $\displaystyle\mu(E)$ |  $\displaystyle\nu(E)$ |  $\displaystyle\xi(E)$
---|---|---|---
$\displaystyle\\{\text{D},\text{P}\\}$ |  $\displaystyle 0.94$ |  $\displaystyle 0.85$ |  $\displaystyle 0.63$
$\displaystyle\\{\text{D},\text{R}\\}$ |  $\displaystyle 0.48$ |  $\displaystyle 0.76$ |  $\displaystyle 0.71$
$\displaystyle\\{\text{P},\text{R}\\}$ |  $\displaystyle 0.81$ |  $\displaystyle 0.59$ |  $\displaystyle 0.84$
Table 6: Monotone measures calculating by Shapley value

Finally, let us calculate the standard and generalized Choquet integral of
vectors $\displaystyle\mathbf{a}_{1},\mathbf{a}_{2},$
$\displaystyle\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{c}_{1},\mathbf{c}_{2}$
with respect to monotone measures $\displaystyle\mu$, $\displaystyle\nu$,
$\displaystyle\xi$. As a conditional aggregation operator let us use the
standard Choquet integral

$\displaystyle\mathsf{A}^{\mathrm{Ch}}(\mathbf{x}|E)=\mathrm{C}(\mathbf{x}\mathbf{1}_{E},m)$,

with $\displaystyle\mathbf{x}\in[0,\infty)^{[3]}$, $\displaystyle
E\in\mathscr{E}=2^{[3]}$ and monotone measure $\displaystyle m$ (we shall use
$\displaystyle\mu$, $\displaystyle\nu$, $\displaystyle\xi$ according to
character). Further, $\displaystyle\mathscr{E}=2^{[3]}$. We expect that

  * •

$\displaystyle\mathbf{a}_{2}$ should be preferred to
$\displaystyle\mathbf{a}_{1}$ with respect to character of Anthony,

  * •

$\displaystyle\mathbf{b}_{1}$ should be preferred to
$\displaystyle\mathbf{b}_{2}$, because of the character of Brittany,

  * •

$\displaystyle\mathbf{c}_{2}$ preferred to $\displaystyle\mathbf{c}_{1}$ for
Charley.

The results for both integrals are shown in Table 7. Let us look at the
highlighted diagonal of Table 7. In the highlighted cells, the first line, we
see the calculation of the standard Choquet integrals. As we can see, we did
not receive the expected accommodation option preferences. In the second line
of highlighted cells, the generalized Choquet integral is calculated. We see
that this method matches our expected preferences. This indicates its benefits
in decision-making processes. The generalized Choquet integral in the
calculation process takes into account the input vector with respect to
various conditional sets. If the collection contains more sets than the
cardinality of the basic set, this integral aggregates more input values than
when using the standard Choquet integral. Thanks to this, the calculation is
more detailed, ”smoother” and more accurate.

To be complete, we have calculated the overall scores of various combinations
of accommodation–person, see the other cells of Table 7.

accom. opt. | type of integral |  Anthony ($\displaystyle\mu$) |  Brittany ($\displaystyle\nu$) |  Charley ($\displaystyle\xi$)
---|---|---|---|---
$\displaystyle\mathbf{a}_{1}$, $\displaystyle\mathbf{a}_{2}$ | Choquet integral |  $\displaystyle\mathbf{a}_{1}\succ\mathbf{a}_{2}$ |  $\displaystyle\mathbf{a}_{1}\succ\mathbf{a}_{2}$ |  $\displaystyle\mathbf{a}_{2}\succ\mathbf{a}_{1}$
$\displaystyle 0.8943>0.8860$ |  $\displaystyle 0.9604>0.7540$ |  $\displaystyle 0.9040>0.8899$
generalized Choquet integral |  $\displaystyle\mathbf{a}_{2}\succ\mathbf{a}_{1}$ |  $\displaystyle\mathbf{a}_{1}\succ\mathbf{a}_{2}$ |  $\displaystyle\mathbf{a}_{2}\succ\mathbf{a}_{1}$
$\displaystyle 0.6857>0.6439$ |  $\displaystyle 0.6845>0.4554$ |  $\displaystyle 0.6566>0.6161$
$\displaystyle\mathbf{b}_{1}$, $\displaystyle\mathbf{b}_{2}$ | Choquet integral |  $\displaystyle\mathbf{b}_{2}\succ\mathbf{b}_{1}$ |  $\displaystyle\mathbf{b}_{2}\succ\mathbf{b}_{1}$ |  $\displaystyle\mathbf{b}_{2}\succ\mathbf{b}_{1}$
$\displaystyle 0.9430>0.8240$ |  $\displaystyle 0.8770>0.8600$ |  $\displaystyle 0.9520>0.6180$
generalized Choquet integral |  $\displaystyle\mathbf{b}_{2}\succ\mathbf{b}_{1}$ |  $\displaystyle\mathbf{b}_{1}\succ\mathbf{b}_{2}$ |  $\displaystyle\mathbf{b}_{2}\succ\mathbf{b}_{1}$
$\displaystyle 0.7127>0.6087$ |  $\displaystyle 0.6393>0.5861$ |  $\displaystyle 0.6766>0.3551$
$\displaystyle\mathbf{c}_{1}$, $\displaystyle\mathbf{c}_{2}$ | Choquet integral |  $\displaystyle\mathbf{c}_{1}\succ\mathbf{c}_{2}$ |  $\displaystyle\mathbf{c}_{1}\succ\mathbf{c}_{2}$ |  $\displaystyle\mathbf{c}_{1}\succ\mathbf{c}_{2}$
$\displaystyle 0.9560>0.8480$ |  $\displaystyle 0.9650>0.6720$ |  $\displaystyle 0.9045>0.8720$
generalized Choquet integral |  $\displaystyle\mathbf{c}_{1}\succ\mathbf{c}_{2}$ |  $\displaystyle\mathbf{c}_{1}\succ\mathbf{c}_{2}$ |  $\displaystyle\mathbf{c}_{2}\succ\mathbf{c}_{1}$
$\displaystyle 0.7069>0.6654$ |  $\displaystyle 0.6895>0.3532$ |  $\displaystyle 0.6433>0.6226$
Table 7: Results of preferences of accommodation options. The preference
relation is represented by Choquet integrals as follows:
$\displaystyle\mathbf{x}\succ\mathbf{y}$ ($\displaystyle\mathbf{x}$ is
preferred to $\displaystyle\mathbf{y}$) if only if
$\displaystyle\mathrm{C}(\mathbf{x},m)>\mathrm{C}(\mathbf{y},m)$, resp.
$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},m)>\mathrm{C}_{{\mathscr{A}}}(\mathbf{y},m)$,
where $\displaystyle m$ is $\displaystyle\mu$, $\displaystyle\nu$ or
$\displaystyle\xi$, respectively.

##  6 Searching optimal intervals, and indistinguishibility

The Choquet integral is a basic tool for multicriteria decision making and
modeling of decision under risk and uncertainty. In [6], Dubois and Rico
studied the equality conditions of Choquet integrals of particular input
vectors. They considered Choquet integrals with respect to possibility and
necessity measures. In [4], Chen et al. continued their research with a view
to a wider class of so-called universal integrals [12]. Universal integrals
form one class of utility functions in multicriteria decision making.

In this section, we formulate the equality conditions of generalized survival
functions considering arbitrary measures. Naturally, if generalized survival
functions coincide, then their Choquet integrals equal. In order to obtain the
equality conditions, we study the greatest possible intervals on which the
generalized survival function takes its possible values. For this purpose, for
any $\displaystyle j\in[\kappa-1]_{0}$ let us set

$\displaystyle\displaystyle{\varphi^{*}}(j):=\max\\{k\in[\kappa-1]_{0}:\mu_{k}=\mu_{j}\\}\quad\text{and}\quad{\varphi_{*}}(j)=\min\\{k\in[\kappa-1]_{0}:\mu_{k}=\mu_{j}\\}.$
(18)

The following proposition summarizes the basic properties of
$\displaystyle{\varphi^{*}}$ and $\displaystyle{\varphi_{*}}$.

###### Proposition 6.1

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, $\displaystyle{\varphi^{*}}$ and
$\displaystyle{\varphi_{*}}$ are defined as in (18) and $\displaystyle
a,b,j\in[\kappa-1]_{0}$. Then the following properties hold.

  1. (i)

$\displaystyle{\varphi_{*}}(j)\leq j\leq{\varphi^{*}}(j)$,

  2. (ii)

$\displaystyle\mu_{{\varphi^{*}}(j)}=\mu_{k}=\mu_{{\varphi_{*}}(j)}$ for any
integer $\displaystyle k\in\\{{\varphi_{*}}(j),\dots,{\varphi^{*}}(j)\\}$,

  3. (iii)

$\displaystyle\bigcup_{{\varphi_{*}}(j)\leq
l\leq{\varphi^{*}}(j)}\Big{[}\min_{k\leq l}\mathsf{A}_{{\langle
k\rangle}},\min_{k<l}\mathsf{A}_{{\langle
k\rangle}}\Big{)}=\Big{[}\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{{\langle
k\rangle}},\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{{\langle k\rangle}}\Big{)}$,

  4. (iv)

$\displaystyle{\varphi_{*}}$ and $\displaystyle{\varphi^{*}}$ are
nondecreasing.

Proof.  The statements (i) and (ii) follow directly from definitions of
$\displaystyle{\varphi^{*}}$, $\displaystyle{\varphi_{*}}$, and arrangement
(6). The validity of the statement (iii) follows from the fact that
$\displaystyle\min_{k<l}\mathsf{A}_{\langle k\rangle}=\min_{k\leq
l-1}\mathsf{A}_{\langle k\rangle}$ for $\displaystyle
l\in\\{{\varphi_{*}}(j),\dots,{\varphi^{*}}(j)\\},$ $\displaystyle l\neq 0$,
and because of

$\displaystyle\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle}\leq\min_{k<{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle}=\min_{k\leq{\varphi^{*}}(j)-1}\mathsf{A}_{\langle
k\rangle}\leq\dots\leq\min_{k<{\varphi_{*}}(j)+1}\mathsf{A}_{\langle
k\rangle}=\min_{k\leq{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle}\leq\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle k\rangle}$

with the convention already stated in this paper
$\displaystyle\min\limits_{k<0}\mathsf{A}_{\langle
k\rangle}=\min\emptyset=+\infty.$ Now we show (iv). If $\displaystyle a\leq
b$, then from arrangement (6) we have $\displaystyle\mu_{a}\leq\mu_{b}$ and
thus

$\displaystyle\max\\{k\in[\kappa-1]_{0}:\mu_{a}=\mu_{k}\\}\leq\max\\{k\in[\kappa-1]_{0}:\mu_{b}=\mu_{k}\\},$

hence $\displaystyle{\varphi^{*}}(a)\leq{\varphi^{*}}(b)$. The second part of
the statement (iv) can be shown analogously.  $\displaystyle\Box\;\;$

By means of $\displaystyle{\varphi^{*}}$ and $\displaystyle{\varphi_{*}}$ we
can determine the greatest possible intervals corresponding to given monotone
measure. Simultaneously, we show under what conditions the value of a given
monotone measure is not achieved by the generalized survival function.

###### Proposition 6.2

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\varphi_{*}}$
and $\displaystyle{\varphi^{*}}$ be given as in (18), and $\displaystyle
j\in[\kappa-1]_{0}$. Then

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{j}\text{ for any
}\alpha\in\Big{[}\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle k\rangle}\Big{)}$

with the convention $\displaystyle\min\limits_{k<0}\mathsf{A}_{\langle
k\rangle}=\min\emptyset=+\infty$ and this interval is the greatest possible.

Proof.  According to Theorem 3.7, Proposition 6.1(i), (ii), (iii) the
generalized survival function
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$ achieves the value
$\displaystyle\mu_{j}$ on interval
$\displaystyle\Big{[}\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle k\rangle}\Big{)}$.
However, it is not clear that this interval is the greatest possible. By
contradiction: So let exist $\displaystyle\tilde{j}\in[\kappa-1]_{0}$ such
that

$\displaystyle\min_{k\leq{\varphi^{*}}(\tilde{j})}\mathsf{A}_{\langle
k\rangle}<\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle}\quad\text{or}\quad\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle}<\min_{k<{\varphi_{*}}(\tilde{j})}\mathsf{A}_{\langle k\rangle},$

and $\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{j}$ for
$\displaystyle\alpha\in\Big{[}\min_{k\leq{\varphi^{*}}(\tilde{j})}\mathsf{A}_{\langle
k\rangle},\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle k\rangle}\Big{)}$ or
$\displaystyle\alpha\in\Big{[}\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min_{k<{\varphi_{*}}(\tilde{j})}\mathsf{A}_{\langle
k\rangle}\Big{)}$. Let us discuss these cases.

  * •

By Theorem 3.7(i) for
$\displaystyle\alpha=\min_{k\leq{\varphi^{*}}(\tilde{j})}\mathsf{A}_{\langle
k\rangle}$ it holds
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{{\varphi^{*}}(\tilde{j})}=\mu_{\tilde{j}}$,
where the last equality holds because of Proposition 6.1(i), (ii). However,
$\displaystyle\mu_{\tilde{j}}\neq\mu_{j}$, otherwise we get a contradiction.
Indeed, if $\displaystyle\mu_{\tilde{j}}=\mu_{j}$, then
$\displaystyle{\varphi^{*}}(j)={\varphi^{*}}(\tilde{j})$ and thus
$\displaystyle\min_{k\leq{\varphi^{*}}(\tilde{j})}\mathsf{A}_{\langle
k\rangle}=\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle k\rangle}$, which is
in conflict with choice of $\displaystyle\tilde{j}$.

  * •

Using the same arguments as in the previous case, for
$\displaystyle\alpha=\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle}=\min_{k\leq{\varphi_{*}}(j)-1}\mathsf{A}_{\langle k\rangle}$, we
have
$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{{\varphi_{*}}(j)-1}$.
Further,
$\displaystyle\mu_{{\varphi_{*}}(j)-1}<\mu_{{\varphi_{*}}(j)}=\mu_{j}$, thus
$\displaystyle\mu_{{\varphi_{*}}(j)-1}\neq\mu_{j}$.

This completes the proof.  $\displaystyle\Box\;\;$

From the above proposition, we immediately obtain the following result.

###### Corollary 6.3

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\varphi_{*}}$
and $\displaystyle{\varphi^{*}}$ be given as in (18), and $\displaystyle
j\in[\kappa-1]_{0}$. The value $\displaystyle\mu_{j}$ is not achieved if and
only if $\displaystyle\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle}=\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle k\rangle}$.

###### Lemma 6.4

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\varphi_{*}}$
and $\displaystyle{\varphi^{*}}$ be given as in (18).

  1. (i)

Let $\displaystyle j\in[\kappa-1]$. Then it holds

    1. (i1)

If $\displaystyle\mu_{j}>\mu_{j-1}$, then
$\displaystyle{\varphi_{*}}(j)-1={\varphi^{*}}(j-1)$ and
$\displaystyle\mu_{j}$ is achieved on
$\displaystyle\big{[}\min\limits_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k\leq{\varphi^{*}}(j-1)}\mathsf{A}_{\langle
k\rangle}\big{)}$. Moreover, this interval is the greatest possible.

    2. (i2)

If $\displaystyle\mu_{j}=\mu_{j-1}$, then
$\displaystyle{\varphi^{*}}(j)={\varphi^{*}}(j-1)$.

  2. (ii)

Let $\displaystyle j\in[\kappa-1]_{0}$. Then it holds

    1. (ii1)

If $\displaystyle\mu_{j}<\mu_{j+1}$, then
$\displaystyle{\varphi^{*}}(j)+1={\varphi_{*}}(j+1)$ and
$\displaystyle\mu_{j}$ is achieved on
$\displaystyle\Big{[}\min_{k<{\varphi_{*}}(j+1)}\mathsf{A}_{\langle
k\rangle},\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle k\rangle}\Big{)}$, with
$\displaystyle\min\limits_{k<{\varphi_{*}}(\kappa)}\mathsf{A}_{\langle
k\rangle}=0$ by convention. Moreover, this interval is the greatest possible.

    2. (ii2)

If $\displaystyle\mu_{j}=\mu_{j+1}$, then
$\displaystyle{\varphi_{*}}(j)={\varphi_{*}}(j+1)$.

Proof.  Let us prove part (i).

  1. (i1)

If $\displaystyle\mu_{j}>\mu_{j-1}$, then directly from definitions of
$\displaystyle{\varphi_{*}}$ and $\displaystyle{\varphi^{*}}$ we have
$\displaystyle{\varphi_{*}}(j)=j$ and $\displaystyle{\varphi^{*}}(j-1)=j-1$,
thus $\displaystyle{\varphi^{*}}(j-1)={\varphi_{*}}(j)-1$. Further, according
to Proposition 6.2, we have

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{j}$

for each
$\displaystyle\alpha\in\big{[}\min\limits_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k\leq{\varphi_{*}}(j)-1}\mathsf{A}_{\langle
k\rangle}\big{)}$ and this interval is the greatest possible. Using the above
equality we have

$\displaystyle\big{[}\min\limits_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k\leq{\varphi_{*}}(j)-1}\mathsf{A}_{\langle
k\rangle}\big{)}=\big{[}\min\limits_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k\leq{\varphi^{*}}(j-1)}\mathsf{A}_{\langle
k\rangle}\big{)}.$

  2. (i2)

It holds trivially.

Part (ii) can be proved analogously.  $\displaystyle\Box\;\;$

Let us notice that Proposition 6.2 and its corollaries are crucial to state
the sufficient conditions for indistinguishability of generalized Choquet
integral equivalent pairs, see Section 5. Using the above results, one can
obtain the improvement of formula (8).

###### Proposition 6.5

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, and $\displaystyle\varphi$ be
given as in (18). Then

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)$
$\displaystyle\displaystyle=\sum_{j=1}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\min\limits_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k\leq{\varphi^{*}}(j-1)}\mathsf{A}_{\langle
k\rangle}\big{)}}(\alpha)=\sum_{j=1}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\min\limits_{k<{\varphi_{*}}(j+1)}\mathsf{A}_{\langle
k\rangle},\min\limits_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle}\big{)}}(\alpha)\ $ (19)

for any $\displaystyle\alpha\in[0,\infty)$, with the conventions
$\displaystyle{\varphi_{*}}(\kappa):=\kappa$ (thus
$\displaystyle\min\limits_{k<{\varphi_{*}}(\kappa)}\mathsf{A}_{\langle
k\rangle}=0$), $\displaystyle\min\limits_{k<0}\mathsf{A}_{\langle
k\rangle}=\min\emptyset=+\infty$.

Proof.  Let us consider an arbitrary (fixed) $\displaystyle j\in[\kappa-1]$.
If $\displaystyle\mu_{j}>\mu_{j-1}$, then according to Lemma 6.4 (i1)
$\displaystyle\mu_{j}$ is achieved on
$\displaystyle\Big{[}\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min_{k\leq{\varphi^{*}}(j-1)}\mathsf{A}_{\langle k\rangle}\Big{)}$.
Moreover, this interval is the greatest possible. If
$\displaystyle\mu_{j}=\mu_{j-1}$, then
$\displaystyle{\varphi^{*}}(j)={\varphi^{*}}(j-1)$, see Lemma 6.4(i2), thus
$\displaystyle\Big{[}\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}_{\langle
k\rangle},\min_{k\leq{\varphi^{*}}(j-1)}\mathsf{A}_{\langle
k\rangle}\Big{)}=\emptyset$. This demonstrates that each value of generalized
survival function is included just once in the sum and the first formula is
right. The second formula can be proved analogously.  $\displaystyle\Box\;\;$

If we use the mapping $\displaystyle\mathbf{j}$ defined by (11), then formulas
in (19) can be rewritten similarly as in Corollary 4.6, i.e.,

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{j=1}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\mathsf{A}_{\mathbf{j}({\varphi^{*}}(j))},\mathsf{A}_{\mathbf{j}({\varphi^{*}}(j-1))}\big{)}}(\alpha)=\sum_{j=1}^{\kappa-1}\mu_{j}\mathbf{1}_{\big{[}\mathsf{A}_{\mathbf{j}({\varphi_{*}}(j+1)-1)},\mathsf{A}_{\mathbf{j}({\varphi_{*}}(j)-1)}\big{)}}(\alpha).$

The use of the approach described in the previous proposition is shown in the
following example.

###### Example 6.6

Let us consider the same inputs as in Example 3.1. Let us use the last formula
given in (19) to calculate the generalized survival function.

  * •

For $\displaystyle j=1$ we get
$\displaystyle\mu_{1}\cdot\mathbf{1}_{[\mathsf{A}_{\mathbf{j}(0)},\mathsf{A}_{\mathbf{j}(0)})}=0.5\cdot\mathbf{1}_{\emptyset}$.

  * •

For $\displaystyle j=2$ we get
$\displaystyle\mu_{2}\cdot\mathbf{1}_{[\mathsf{A}_{\mathbf{j}(0)},\mathsf{A}_{\mathbf{j}(0)})}=0.5\cdot\mathbf{1}_{\emptyset}$.

  * •

For $\displaystyle j=3$ we get
$\displaystyle\mu_{3}\cdot\mathbf{1}_{[\mathsf{A}_{\mathbf{j}(3)},\mathsf{A}_{\mathbf{j}(0)})}=\mu_{3}\cdot\mathbf{1}_{[\mathsf{A}_{1},\mathsf{A}_{5})}=0.5\cdot\mathbf{1}_{[1,6)}$.

  * •

For $\displaystyle j=4$ we get
$\displaystyle\mu_{4}\cdot\mathbf{1}_{[\mathsf{A}_{\mathbf{j}(4)},\mathsf{A}_{\mathbf{j}(3)})}=\mu_{4}\cdot\mathbf{1}_{[\mathsf{A}_{1},\mathsf{A}_{1})}=0.7\cdot\mathbf{1}_{\emptyset}$.

  * •

For $\displaystyle j=5$ we get
$\displaystyle\mu_{5}\cdot\mathbf{1}_{[\mathsf{A}_{\mathbf{j}(5)},\mathsf{A}_{\mathbf{j}(4)})}=\mu_{5}\cdot\mathbf{1}_{[\mathsf{A}_{0},\mathsf{A}_{1})}=1\cdot\mathbf{1}_{[0,1)}$.

Therefore, the generalized survival function has the form

$\displaystyle\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mathbf{1}_{[0,1)}(\alpha)+0.5\cdot\mathbf{1}_{[1,6)}(\alpha)\text{,}$

$\displaystyle\alpha\in[0,+\infty)$, compare with Example 3.8.

As in the whole paper let us introduce the greatest possible intervals on
which a value of monotone measure is achieved using bijection $\displaystyle
E_{*}\colon[\kappa-1]_{0}\to{{\mathscr{E}}}$, see (5). Since the proofs of the
following propositions are analogous, we omit them. Let $\displaystyle
i\in[\kappa-1]_{0}$, let us define

$\displaystyle\displaystyle\begin{split}{\psi_{*}}(i)&:=\min\\{l\in[\kappa-1]_{0}:\min_{k\leq
l}\mu_{(k)}=\min_{k\leq i}\mu_{(k)}\\}\text{,}\\\
\text{and}\quad{\psi^{*}}(i)&:=\max\\{l\in[\kappa-1]_{0}:\min_{k\leq
l}\mu_{(k)}=\min_{k\leq i}\mu_{(k)}\\}\text{.}\end{split}$ (20)

###### Proposition 6.7

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,\infty)^{[n]}$, $\displaystyle{\psi^{*}}$,
$\displaystyle{\psi_{*}}$ be defined as in (20) and $\displaystyle
a,b,i\in[\kappa-1]_{0}$. Then the following properties hold.

  1. (i)

$\displaystyle{\psi_{*}}(i)=\min\\{l\in[\kappa-1]_{0}:\mu_{(l)}=\min_{k\leq
i}\mu_{(k)}\\}\text{.}$

  2. (ii)

$\displaystyle{\psi_{*}}(0)=0$, $\displaystyle{\psi_{*}}(i)\leq
i\leq{\psi^{*}}(i)$.

  3. (iii)

For any integer $\displaystyle u\in\\{{\psi_{*}}(i),\dots,{\psi^{*}}(i)\\}$ it
holds $\displaystyle\min_{k\leq u}\mu_{(k)}=\min_{k\leq
i}\mu_{(k)}=\mu_{({\psi_{*}}(i))}=\mu_{\mathbf{i}(i)}$.

  4. (iv)

$\displaystyle\min_{k\leq a}\mu_{(k)}=\min_{k\leq b}\mu_{(k)}$ if only if
$\displaystyle{\psi_{*}}(a)={\psi_{*}}(b)$ if only if
$\displaystyle{\psi^{*}}(a)={\psi^{*}}(b)$.

  5. (v)

$\displaystyle\min_{k\leq a}\mu_{(k)}>\min_{k\leq b}\mu_{(k)}$ if only if
$\displaystyle{\psi_{*}}(a)<{\psi_{*}}(b)$ if only if
$\displaystyle{\psi^{*}}(a)<{\psi^{*}}(b)$.

  6. (vi)

$\displaystyle{\psi_{*}}$ and $\displaystyle{\psi^{*}}$ are nondecreasing.

Proof.  See Appendix. $\displaystyle\Box\;\;$

###### Remark 6.8

In general, we cannot determine the order of the values
$\displaystyle\mu_{(i)}$, $\displaystyle i\in[\kappa-1]_{0}$. However, it
holds

$\displaystyle\displaystyle\mu_{({\psi_{*}}(0))}\geq\mu_{({\psi_{*}}(1))}\geq\dots\geq\mu_{({\psi_{*}}(\kappa-1))}\text{,}$
and
$\displaystyle\displaystyle\mu_{({\psi^{*}}(0))}\geq\mu_{({\psi^{*}}(1))}\geq\dots\geq\mu_{({\psi^{*}}(\kappa-1))}\text{.}$

This property follows directly from the definitions of
$\displaystyle{\psi_{*}}$ and $\displaystyle{\psi^{*}}$. Indeed, we have
already shown that for $\displaystyle a\leq b$ we get
$\displaystyle{\psi_{*}}(a)\leq{\psi_{*}}(b)$. In case that
$\displaystyle{\psi_{*}}(a)={\psi_{*}}(b)$, the result is clear. If
$\displaystyle{\psi_{*}}(a)<{\psi_{*}}(b)$, then the result follows from
Proposition 6.7 (vi). Similarly for $\displaystyle{\psi^{*}}$.

In Theorem 3.2 and Corollary 4.3, we have described the value of the monotone
measure acquired on a given interval. However, the same value can be acquired
on several intervals. Using $\displaystyle{\psi^{*}}$ and
$\displaystyle{\psi_{*}}$, we can determine the greatest interval on which the
value of the monotone measure is achieved.

###### Proposition 6.9

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\psi_{*}}$ and
$\displaystyle{\psi^{*}}$ be given as in (20), and $\displaystyle
i\in[\kappa-1]_{0}$. Then

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\min_{k\leq
i}\mu_{(k)}\text{ for any
}\alpha\in\big{[}\mathsf{A}_{{\psi_{*}}(i)},\mathsf{A}_{{\psi^{*}}(i)+1}\big{)}$

and this interval is the greatest possible.

Proof.  See Appendix. $\displaystyle\Box\;\;$

###### Lemma 6.10

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\psi_{*}}$ and
$\displaystyle{\psi^{*}}$ be given as in (20).

  1. (i)

Let $\displaystyle i\in{[\kappa-2]_{0}}$ . Then it holds

    1. (i1)

If $\displaystyle\min_{k\leq i}\mu_{(k)}>\min_{k\leq i+1}\mu_{(k)}$, then
$\displaystyle{\psi^{*}}(i)+1={\psi_{*}}(i+1)$ and $\displaystyle\min_{k\leq
i}\mu_{(k)}$ is achieved on
$\displaystyle\big{[}\mathsf{A}_{{\psi_{*}}(i)},\mathsf{A}_{{\psi_{*}}(i+1)}\big{)}$.
Moreover, this interval is the greatest possible.

    2. (i2)

If $\displaystyle\min_{k\leq i}\mu_{(k)}=\min_{k\leq i+1}\mu_{(k)}$, then
$\displaystyle{\psi_{*}}(i)={\psi_{*}}(i+1)$.

  2. (ii)

Let $\displaystyle i\in{{[\kappa-1]}}$. Then it holds

    1. (ii1)

If $\displaystyle\min_{k\leq i-1}\mu_{(k)}<\min_{k\leq i}\mu_{(k)}$, then
$\displaystyle{\psi_{*}}(i)={\psi^{*}}(i-1)+1$ and $\displaystyle\min_{k\leq
i}\mu_{(k)}$ is achieved on
$\displaystyle\Big{[}\mathsf{A}_{{\psi^{*}}(i-1)+1},\mathsf{A}_{{\psi^{*}}(i)+1}\Big{)}$.
Moreover, this interval is the greatest possible.

    2. (ii2)

If $\displaystyle\min_{k\leq i-1}\mu_{(k)}=\min_{k\leq i}\mu_{(k)}$, then
$\displaystyle{\psi^{*}}(i-1)={\psi^{*}}(i)$.

###### Proposition 6.11

Let $\displaystyle{\mathscr{A}}$ be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x}\in[0,+\infty)^{[n]}$, $\displaystyle{\psi_{*}}$ and
$\displaystyle{\psi^{*}}$ be given as in (20), and $\displaystyle
i\in[\kappa-1]_{0}$. Then the formula of generalized survival function is as
follows

$\displaystyle\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-2}\mu_{({\psi_{*}}(i))}\mathbf{1}_{\big{[}\mathsf{A}_{{\psi_{*}}(i)},\mathsf{A}_{{\psi_{*}}(i+1)})}(\alpha)=\sum_{i={0}}^{\kappa-2}{\mu_{({\psi_{*}}(i))}}\mathbf{1}_{\big{[}\mathsf{A}_{{\psi^{*}}(i-1)+1},\mathsf{A}_{{\psi^{*}}(i)+1})}(\alpha)$
(21)

for any $\displaystyle\alpha\in[0,+\infty)$ with the convention
$\displaystyle{\psi^{*}}(-1)=0$.

Proof.  See Appendix.  $\displaystyle\Box\;\;$

If we use the mapping $\displaystyle\mathbf{i}$ defined by (9), then formulas
in (21) can be rewritten similarly as in Corollary 4.3, i.e.,

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\sum_{i=0}^{\kappa-2}\mu_{\mathbf{i}(i)}\mathbf{1}_{\big{[}\mathsf{A}_{{\psi_{*}}(i)},\mathsf{A}_{{\psi_{*}}(i+1)})}(\alpha)=\sum_{i={0}}^{\kappa-2}{\mu_{\mathbf{i}(i)}}\mathbf{1}_{\big{[}\mathsf{A}_{{\psi^{*}}(i-1)+1},\mathsf{A}_{{\psi^{*}}(i)+1})}(\alpha)\text{.}$

As we have already mentioned, searching for optimal intervals will be helpful
for studying the indistinguishability of generalized survival functions. In
the following, we state sufficient and necessary conditions under which the
generalized survival functions coincide. This is applicable to decision-making
problems. In fact, if the generalized survival functions of two alternatives
(e.g. two offers of accommodation) are the same, then their overall score will
be the same. Both alternatives will be in the same place in the ranking.

###### Definition 6.12

The triples $\displaystyle(\mu,{\mathscr{A}},\mathbf{x})$ and
$\displaystyle(\mu^{\prime},{\mathscr{A}}^{\prime},\mathbf{x}^{\prime})$,
where $\displaystyle\mu,\mu^{\prime}$ are monotone measures,
$\displaystyle{\mathscr{A}},{\mathscr{A}}^{\prime}$ are FCA and
$\displaystyle\mathbf{x},\mathbf{x}^{\prime}$ are vectors, are called integral
equivalent, if

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu^{\prime}_{\mathscr{A}^{\prime}}(\mathbf{x}^{\prime},\alpha).$

###### Proposition 6.13

Let $\displaystyle{\mathscr{A}}$ and $\displaystyle{\mathscr{A}}^{\prime}$ be
a FCA, $\displaystyle\mu,\mu^{\prime}\in\mathbf{M}$,
$\displaystyle\mathbf{x},\mathbf{x}^{\prime}\in[0,+\infty)^{[n]}$ and
$\displaystyle{\varphi^{*}}$ be given by (18). Then the following assertions
are equivalent:

  1. (i)

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu^{\prime}_{{\mathscr{A}}^{\prime}}(\mathbf{x}^{\prime},\alpha)$
for any $\displaystyle\alpha\in[0,\infty)$;

  2. (ii)

for each $\displaystyle j\in[\kappa-1]_{0}$ with
$\displaystyle\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}(\mathbf{x}|F_{k}^{c})<\min_{k<{\varphi_{*}}(j)}\mathsf{A}(\mathbf{x}|F_{k}^{c})$
there exists $\displaystyle j^{\prime}\in[\kappa^{\prime}-1]_{0}$ such that
$\displaystyle\mu_{j}=\mu^{\prime}_{j^{\prime}}$,
$\displaystyle\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{k\leq{\varphi^{*}}(j^{\prime})}\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|F_{k}^{c})$
and $\displaystyle\min_{k<{\varphi_{*}}(j)}\mathsf{A}_{\langle
k\rangle}=\min_{k<{\varphi_{*}}(j^{\prime})}\mathsf{A}^{\prime}_{\langle
k\rangle}$.

Proof.  It follows from Proposition 6.2.  $\displaystyle\Box\;\;$

###### Remark 6.14

Following Proposition 6.9, condition (ii) in the previous proposition can be
equivalently formulated as follows: for each $\displaystyle
i\in[\kappa-1]_{0}$ with
$\displaystyle\mathsf{A}_{{\psi_{*}}(i)}<\mathsf{A}_{{\psi^{*}}(i)+1}$ there
exists $\displaystyle i^{\prime}\in[\kappa^{\prime}-1]$ such that
$\displaystyle\mu_{{\psi_{*}}(i)}=\mu^{\prime}_{{\psi_{*}}(i^{\prime})}$,
$\displaystyle\mathsf{A}(\mathbf{x}|E_{{\psi_{*}}(i)})=\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|E_{{\psi_{*}}(i^{\prime})})$
and
$\displaystyle\mathsf{A}(\mathbf{x}|E_{{\psi^{*}}(i)+1})=\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|E_{{\psi^{*}}(i^{\prime})+1})$.

Fixing a collection $\displaystyle{\mathscr{E}}$ and a monotone measure
$\displaystyle\mu$ we obtain the following sufficient and necessary condition
for integral equivalence of triples $\displaystyle(\mu,\mathsf{A},\mathbf{x})$
and $\displaystyle(\mu,\mathsf{A}^{\prime},\mathbf{x}^{\prime})$.

###### Corollary 6.15

Let $\displaystyle{\mathscr{A}}=\\{\mathsf{A}(\cdot|E):E\in\mathscr{E}\\}$ and
$\displaystyle{\mathscr{A}}^{\prime}=\\{\mathsf{A}^{\prime}(\cdot|E):E\in\mathscr{E}\\}$
be a FCA, $\displaystyle\mu\in\mathbf{M}$,
$\displaystyle\mathbf{x},\mathbf{x}^{\prime}\in[0,+\infty)^{[n]}$, and
$\displaystyle{\varphi_{*}},{\varphi^{*}}$ and
$\displaystyle{\psi_{*}},{\psi^{*}}$ be given by (18), (20), respectively.
Then the following assertions are equivalent:

  1. (i)

$\displaystyle\mu_{\mathscr{A}}(\mathbf{x},\alpha)=\mu_{\mathscr{A}^{\prime}}(\mathbf{x}^{\prime},\alpha)$
for any $\displaystyle\alpha\in[0,+\infty)$;

  2. (ii)

$\displaystyle\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{k\leq{\varphi^{*}}(j)}\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|F_{k}^{c})$
for any $\displaystyle j\in[\kappa-1]_{0}$
(or equivalently,
$\displaystyle\mathsf{A}(\mathbf{x}|E_{{\psi_{*}}(i)})=\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|E_{{\psi_{*}}(i)})$
for each $\displaystyle i\in[\kappa-1]_{0}$).

###### Remark 6.16

In [4], the authors derived the necessary and sufficient condition of equality
$\displaystyle\mu(\\{\mathbf{x}\geq\alpha\\})=\mu(\\{\mathbf{y}\geq\alpha\\})$
with $\displaystyle\mu$ being a possibility and necessity measure,
respectively. Our result includes equality
$\displaystyle\mu(\\{\mathbf{x}>\alpha\\})=\mu(\\{\mathbf{y}>\alpha\\})$ with
$\displaystyle\mu$ being an arbitrary monotone measure.

###### Example 6.17

Let us consider the collection
$\displaystyle\mathscr{E}=\\{\emptyset,\\{1\\},\\{1,2,3\\}\\}$, the families
of conditional aggregation operators
$\displaystyle{\mathscr{A}}^{\mathrm{max}}=\\{\mathsf{A}^{\mathrm{max}}(\cdot|E):E\in\mathscr{E}\\}$
and
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in\mathscr{E}\\}$,
the vectors $\displaystyle\mathbf{x}=(2,5,9)$ and
$\displaystyle\mathbf{x}^{\prime}=(2,3,4)$, and the monotone measure
$\displaystyle\mu\in\mathbf{M}$ with corresponding values in table.

$\displaystyle j$ |  0 | 1 | 2
---|---|---|---
$\displaystyle F_{j}$ |  $\displaystyle\emptyset$ | $\displaystyle\\{2,3\\}$ | $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu_{j}$ |  $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 1$
$\displaystyle\mathsf{A}^{\textrm{max}}(\mathbf{x}|F_{j}^{c})=\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{j}^{c})$ |  $\displaystyle 9$ | $\displaystyle 2$ | $\displaystyle 0$

It is easy to see that for any $\displaystyle j\in[2]_{0}$ we have
$\displaystyle{\varphi^{*}}(j)=j$ and $\displaystyle\min_{k\leq
j}\mathsf{A}^{\mathrm{max}}(\mathbf{x}|F_{k}^{c})=\min_{k\leq
j}\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{k}^{c})$. Then according to
previous proposition one can expect the equality of corresponding generalized
survival functions. And, it is:

$\displaystyle\mu_{\mathscr{A}^{\mathrm{max}}}(\mathbf{x},\alpha)=\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x}^{\prime},\alpha)=1\cdot\mathbf{1}_{[0,2)}+0.5\cdot\mathbf{1}_{[2,9)}$

for any $\displaystyle\alpha\in[0,+\infty)$.

The following example demonstrates a standard situation in decision-making
processes. Different alternatives can be evaluated with the same score, thus
they are incomparable. It is not possible to decide which one is better.

###### Example 6.18

Let us consider the collection
$\displaystyle\mathscr{E}=\\{\emptyset,\\{2\\},\\{3\\},\\{1,2\\},\\{1,3\\},\\{1,2,3\\}\\}$,
the family of conditional aggregation operators
$\displaystyle{\mathscr{A}}^{\mathrm{sum}}=\\{\mathsf{A}^{\mathrm{sum}}(\cdot|E):E\in\mathscr{E}\\}$,
vectors $\displaystyle\mathbf{x}=(1,3,5)$ and
$\displaystyle\mathbf{x}^{\prime}=(2,4,3)$, and the monotone measure
$\displaystyle\mu\in\mathbf{M}$ with corresponding values in table.

$\displaystyle j$ |  0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
$\displaystyle F_{j}$ |  $\displaystyle\emptyset$ | $\displaystyle\\{1,3\\}$ | $\displaystyle\\{1,2\\}$ | $\displaystyle\\{3\\}$ | $\displaystyle\\{2\\}$ | $\displaystyle\\{1,2,3\\}$
$\displaystyle\mu_{j}$ |  $\displaystyle 0$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 0.5$ | $\displaystyle 1$
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}|F_{k}^{c})$ |  $\displaystyle 9$ | $\displaystyle 3$ | $\displaystyle 5$ | $\displaystyle 4$ | $\displaystyle 6$ | $\displaystyle 0$
$\displaystyle\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{k}^{c})$ |  $\displaystyle 9$ | $\displaystyle 4$ | $\displaystyle 3$ | $\displaystyle 6$ | $\displaystyle 5$ | $\displaystyle 0$

It is easy to see that
$\displaystyle{\varphi^{*}}(0)=0,{\varphi^{*}}(1)={\varphi^{*}}(2)={\varphi^{*}}(3)={\varphi^{*}}(4)=4,{\varphi^{*}}(5)=5$
and

$\displaystyle\min_{k\leq
0}\mathsf{A}^{\mathrm{max}}(\mathbf{x}|F_{k}^{c})=9=\min_{k\leq
0}\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{k}^{c})$,

$\displaystyle\min_{k\leq
4}\mathsf{A}^{\mathrm{max}}(\mathbf{x}|F_{k}^{c})=3=\min_{k\leq
4}\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{k}^{c})$,

$\displaystyle\min_{k\leq
5}\mathsf{A}^{\mathrm{max}}(\mathbf{x}|F_{k}^{c})=0=\min_{k\leq
5}\mathsf{A}^{\mathrm{sum}}(\mathbf{x}^{\prime}|F_{k}^{c})$.

Then according to previous proposition one can expect the equality of
corresponding generalized survival functions. And, it is:

$\displaystyle\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x},\alpha)=\mu_{\mathscr{A}^{\mathrm{sum}}}(\mathbf{x}^{\prime},\alpha)=1\cdot\mathbf{1}_{[0,3)}+0.5\cdot\mathbf{1}_{[3,9)}$

for any $\displaystyle\alpha\in[0,+\infty)$.

Immediately, we get the sufficient condition for equality of generalized
Choquet integrals defined w.r.t. different FCA and vectors.

###### Corollary 6.19

Let $\displaystyle{\mathscr{A}}$ and $\displaystyle{\mathscr{A}}^{\prime}$ be
a FCA, $\displaystyle\mu\colon\hat{{\mathscr{E}}}\to[0,+\infty)$,
$\displaystyle\mathbf{x},\mathbf{x}^{\prime}\in[0,+\infty)^{[n]}$ and
$\displaystyle\varphi$ be given by (18). If
$\displaystyle\min_{k\leq\varphi(i)}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{k\leq\varphi(i)}\mathsf{A}^{\prime}(\mathbf{x}^{\prime}|F_{k}^{c})$
for any $\displaystyle\varphi(i)\in[\kappa]$, then

$\displaystyle\mathrm{C}_{{\mathscr{A}}}(\mathbf{x},\mu)=\mathrm{C}_{{\mathscr{A}}^{\prime}}(\mathbf{x}^{\prime},\mu).$

## Conclusion

In the paper, we dealt with the concept of the generalized survival function
and the generalized Choquet integral related to it. Considering their
applications, see Section 2, we were mainly interested in their computational
formulas on discrete space. The derivation of these formulas required the
introduction of new notations whose idea and practical meaning can be visually
interpreted, as we described in Sections 3 and 4. Interesting results are
Proposition 3.2, Proposition 3.7, Corollary 4.3 and Corollary 4.6. In Section
4 we also pointed out the direct and efficient construction of the graph of
the generalized survival function.

Motivated by applications we solved the indistinguishability of generalized
survival functions. This question is interesting, especially in decision-
making processes, because the indistinguishability means incomparability of
two inputs (alternatives). The interesting results given in Proposition 6.5,
Proposition 6.11 and Proposition 6.13 are related to this.

In this paper, we also mentioned the formulas for calculating the generalized
Choquet integral with respect to special types of monotone measure, and we
solved the introductory problems serving as a motivation to study mentioned
concepts.

## Acknowledgments

The work was supported the grants APVV-21-0468, VEGA 1/0657/22, and grant
scheme VVGS-PF-2022-2143.

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## Appendix

Proof of Corollary 3.9

  1. (i)

Clearly, in accordance with (6) we have $\displaystyle
0=\mu_{0}=\mu(\emptyset)$ and according to Theorem 3.7 it is achieved on
$\displaystyle\Big{[}\min_{k\leq 0}\mathsf{A}_{{\langle
k\rangle}},+\infty\Big{)}=[\mathsf{A}_{{\langle
0\rangle}},+\infty)=[\mathsf{A}(\mathbf{x}|[n]),+\infty)$. Then $\displaystyle
1=\mu_{1}=\dots=\mu_{\kappa-1}$ is achieved on
$\displaystyle[0,\mathsf{A}(\mathbf{x}|[n]))$.

  2. (ii)

From Theorem 3.7, the value $\displaystyle 1=\mu_{\kappa-1}$ is achieved on
$\displaystyle\Big{[}0,\min_{k<\kappa-1}\mathsf{A}_{{\langle
k\rangle}}\Big{)}=\Big{[}0,\min_{E\neq\emptyset}\mathsf{A}(\mathbf{x}|E)\Big{)}.$

  3. (iii)

Let us consider the bijection $\displaystyle F_{*}$ given in (6) such that
$\displaystyle|F_{k}|<|F_{l}|$ implies $\displaystyle k<l$ and for any
$\displaystyle i\in[n]_{0}$ let us set

$\displaystyle\displaystyle\omega^{*}(i)$
$\displaystyle\displaystyle=\max\\{j\in[\kappa-1]_{0}:|F_{j}|=i\\},\,\,\omega_{*}(i)=\min\\{j\in[\kappa-1]_{0}:|F_{j}|=i\\}.$

It is easy to see that $\displaystyle\omega_{*}(i)-1=\omega^{*}(i-1)$ with the
convention $\displaystyle\omega^{*}(-1)=-1$. Further, because of monotonicity
of $\displaystyle\mu$ and FCA for any $\displaystyle i\in[n]_{0}$ we have

$\displaystyle\displaystyle\min_{k\leq\omega^{*}(i)}\mathsf{A}_{{\langle
k\rangle}}=\min_{k\leq\omega^{*}(i)}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{k\in[\omega_{*}(i),\omega^{*}(i)]}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{|E|=n-i}\mathsf{A}(\mathbf{x}|E).$

Indeed, for any $\displaystyle k<\omega_{*}(i)$ since
$\displaystyle\mathscr{E}=2^{[n]}$ there exists
$\displaystyle\tilde{k}\in[\omega_{*}(i),\omega^{*}(i)]$ such that
$\displaystyle F_{\tilde{k}}\supset F_{k}$. Then
$\displaystyle\mathsf{A}(\mathbf{x}|F_{\tilde{k}}^{c})\leq\mathsf{A}(\mathbf{x}|F_{k}^{c})$.
According to Theorem 3.7 we have that $\displaystyle\mu^{i}$ is achieved on

$\displaystyle\displaystyle\bigcup_{j\in\\{\omega_{*}(i),\dots,\omega^{*}(i)\\}}\Big{[}\min_{k\leq
j}\mathsf{A}_{{\langle k\rangle}},\min_{k<j}\mathsf{A}_{{\langle
k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\min_{k\leq\omega^{*}(i)}\mathsf{A}_{{\langle
k\rangle}},\min_{k<\omega_{*}(i)}\mathsf{A}_{{\langle k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\min_{k\leq\omega^{*}(i)}\mathsf{A}_{{\langle
k\rangle}},\min_{k\leq\omega^{*}(i-1)}\mathsf{A}_{{\langle k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\min_{E=n-i}\mathsf{A}_{{\langle
k\rangle}},\min_{E=n-i+1}\mathsf{A}_{{\langle k\rangle}}\Big{)}.$

  4. (iv)

Let us consider the bijection $\displaystyle F_{*}$ given in (6) such that
$\displaystyle F_{0}=\emptyset$ and for any $\displaystyle
i\in\\{2,3,\dots,n\\}$

$\displaystyle\\{\sigma(i-1)\\}\subseteq F_{k}\subseteq G^{c}_{\sigma(i)}$,
$\displaystyle\\{\sigma(i)\\}\subseteq F_{l}\subseteq G^{c}_{\sigma(i+1)}$,
implies $\displaystyle k<l$.

According to Theorem 3.7 it is clear that $\displaystyle\pi(\sigma(0))$ is
achieved on
$\displaystyle\Big{[}\mathsf{A}(\mathbf{x}|G_{\sigma(1)}),+\infty\Big{)}$.
Further, for any $\displaystyle i\in[n]$ let us set

$\displaystyle\displaystyle\tau^{*}(\sigma(i))$
$\displaystyle\displaystyle=\max\\{j\in[\kappa-1]_{0}:\Pi(F_{j})=\pi(\sigma(i))\,\,\text{and}\,\,\\{\sigma(i)\\}\subseteq
F_{j}\subseteq G^{c}_{\sigma(i+1)}\\},\,\,$
$\displaystyle\displaystyle\tau_{*}(\sigma(i))$
$\displaystyle\displaystyle=\min\\{j\in[\kappa-1]_{0}:\Pi(F_{j})=\pi(\sigma(i))\,\,\text{and}\,\,\\{\sigma(i)\\}\subseteq
F_{j}\subseteq G^{c}_{\sigma(i+1)}\\}.$

It is easy to see that
$\displaystyle\tau_{*}(\sigma(i))-1=\tau^{*}(\sigma(i-1))$ with the convention
$\displaystyle\tau^{*}(0)=0$. Further, because of monotonicity of
$\displaystyle\mu$ and FCA for any $\displaystyle i\in[n]_{0}$ we have

$\displaystyle\displaystyle\min_{k\leq\tau^{*}(\sigma(i))}\mathsf{A}_{{\langle
k\rangle}}=\min_{k\leq\tau^{*}(\sigma(i))}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\min_{k\in\\{\tau_{*}(\sigma(i)),\dots,\tau^{*}(\sigma(i))\\}}\mathsf{A}(\mathbf{x}|F_{k}^{c})=\mathsf{A}(\mathbf{x}|G_{\sigma(i+1)}).$

Indeed, for any $\displaystyle k<\tau_{*}(\sigma(i))$ we have $\displaystyle
F_{k}\subseteq G^{c}_{\sigma(i+1)}$, where $\displaystyle
G^{c}_{\sigma(i+1)}=F_{\tilde{k}}$ for some
$\displaystyle\tilde{k}\in[\tau_{*}(\sigma(i)),\tau^{*}(\sigma(i))]$, thus
explaining the second equality. Further, for any $\displaystyle
k\in[\tau_{*}(\sigma(i)),\tau^{*}(\sigma(i))]$

$\displaystyle F_{k}\subseteq G^{c}_{\sigma(i+1)}$

therefore
$\displaystyle\mathsf{A}(\mathbf{x}|G_{\sigma(i+1)})\leq\mathsf{A}(\mathbf{x}|F_{k}^{c})$.
According to Theorem 3.7 we have that $\displaystyle\pi(\sigma(i))$ is
achieved on

$\displaystyle\displaystyle\bigcup_{j\in\\{\tau_{*}(\sigma(i)),\dots,\tau^{*}(\sigma(i))\\}}\Big{[}\min_{k\leq
j}\mathsf{A}_{{\langle k\rangle}},\min_{k<j}\mathsf{A}_{{\langle
k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\min_{k\leq\tau^{*}(\sigma(i))}\mathsf{A}_{{\langle
k\rangle}},\min_{k<\tau_{*}(\sigma(i))}\mathsf{A}_{{\langle k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\min_{k\leq\tau^{*}(\sigma(i))}\mathsf{A}_{{\langle
k\rangle}},\min_{k\leq\tau^{*}(\sigma(i-1))}\mathsf{A}_{{\langle
k\rangle}}\Big{)}$
$\displaystyle\displaystyle=\Big{[}\mathsf{A}(\mathbf{x}|G_{\sigma(i+1)}),\mathsf{A}(\mathbf{x}|G_{\sigma(i)})\Big{)}.$

  5. (v)

It is clear that $\displaystyle N([n])=1-\Pi(\emptyset)=1$ and for each
$\displaystyle\\{\sigma(i)\\}\subseteq F^{c}\subseteq G_{\sigma(i+1)}^{c}$

$\displaystyle N(F)=1-\Pi(F^{c})=1-\pi(\sigma(i)),$

with $\displaystyle i\in[n]$ and

$\displaystyle 0=1-\pi(\sigma(n))\leq\dots\leq 1-\pi(\sigma(i))\leq\dots\leq
1-\pi(\sigma(0))=1.$

Let us consider the bijection $\displaystyle F_{*}$ given in (6) such that
$\displaystyle F_{\kappa-1}=\emptyset$ and for any $\displaystyle
i\in\\{2,3,\dots,n\\}$

$\displaystyle\\{\sigma(i-1)\\}\subseteq F_{k}\subseteq G^{c}_{\sigma(i)}$,
$\displaystyle\\{\sigma(i)\\}\subseteq F_{l}\subseteq G^{c}_{\sigma(i+1)}$,
implies $\displaystyle k>l$.

According to Theorem 3.7 it is clear that $\displaystyle 0=1-\pi(\sigma(n))$
is achieved on
$\displaystyle\Big{[}\mathsf{A}(\mathbf{x}|\\{\sigma(n)\\}),+\infty\Big{)}$.
Further, for any $\displaystyle i\in[n-1]$ let us set

$\displaystyle\displaystyle\rho^{*}(\sigma(i))$
$\displaystyle\displaystyle=\max\\{j\in[\kappa-1]_{0}:N(F_{j})=1-\pi(\sigma(i))\,\,\text{and}\,\,\\{\sigma(i)\\}\subseteq
F_{j}^{c}\subseteq G_{\sigma(i+1)}^{c}\\}$
$\displaystyle\displaystyle\rho_{*}(\sigma(i))$