Decision Making Under Z-Information

Abstract

Rational decisions are based on information usually uncertain, imprecise and incomplete. The existing decision theories deal with three levels of generalization of decision making relevant information: numerical valuation, interval valuation and fuzzy number valuation. The classical decision theories, such as expected utility theory proposed by von Neumann and Morgenstern, and subjective expected utility theory proposed by Savage use the first level of generalization, i.e. numerical one. These approaches require that the objective probabilities or subjective probabilities and utility values be precisely known. But in real world in many cases it becomes impossible to determine the precise values of needed information. Interval analysis and classical fuzzy set theories have been applied in making decisions and many fruitful results have been achieved. But a problem is that in the mentioned above decision theories the reliability of the decision relevant information is not well taken into consideration. Prof. L. Zadeh introduced the concept of Z-numbers to describe the uncertain information which is more generalized notion closely related with confidence (reliability). Use of Z-information is more adequate and intuitively meaningful for formalizing information structure of a decision making problem. In this chapter we consider two approaches to decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. A fuzzy measure is calculated on a base of a given Z-information. The second approach is based on direct computation with Z-numbers. To illustrate a validity of suggested approaches to decision making with Z-information the numerical examples are used.

1 A Brief Review of Existing Decision Theories

For decision making the first axiomatic foundation of the utility paradigm was the expected utility (EU) theory of von Neumann and Morgenstern [34]. This model compares finite-outcome lotteries (alternatives) on the base of their utility values under conditions of precisely known utilities and probabilities of outcomes.

The assumptions of von Neumann and Morgenstern expected utility model stating that objective probabilities of events are known makes this model unsuitable for majority of real-world applications.

We have no representative experimental data or complete knowledge to determine objective probabilities. For such cases, Savage suggested a theory able to compare alternative actions on the base of a DM’s experience or vision [32]. Savage’s theory is based on a concept of subjective probability suggested by Ramsey [28] and de Finetti [13]. Subjective probability is DM’s probabilistic belief concerning occurrence of an event and is assumed to be used by humans when no data on objective (actual) probabilities of outcomes is available. Savage’s subjective expected utility (SEU) theory based on the use of subjective probabilities in the expected utility paradigm of von Neumann and Morgenstern instead of objective probabilities. SEU became a base of almost all the utility models for decision making under uncertainty.

Prospect Theory (PT) of Kahneman and Tversky [18, 33] is the one of the most famous theories in the new view on the utility concept. This theory is successful because it includes psychological aspects that form human behavior. Kahneman and Tversky uncovered series of features of human behavior in decision making and used them to construct their utility model.

Choquet Expected Utility (CEU) was suggested by Schmeidler [30] as a model with a new view on belief and representation of preferences in contrast to the SEU model. In CEU a belief is described by a capacity [10]—not necessarily additive measure.

Kahneman and Tversky, the authors of PT, suggested cumulative prospect theory (CPT) as a more advanced theory which can be applied, in contrast to PT, both for decisions under risk and uncertainty.

The principle of uncertainty aversion was formalized in form of an axiom by Gilboa and Schmeidler [17]. This axiom is one of the axioms underlying a famous utility model called Maximin Expected Utility (MMEU) [17]. According to the axiomatic basis of this model, there is a unique closed and convex set $C$ of priors (probability measures) over states of nature and overall utility of an act is a minimum among all its expected utilities each obtained for prior one $P\in C$.

Ghirardato, Maccheroni and Marinacci suggested a generalization of MMEU [15] consisting in using all its underlying axioms except uncertainty aversion axiom. The obtained model is referred to as $\alpha$ – MMEU.

The main disadvantages of the MMEU are that in real problems it is difficult to strictly constrain the set of priors and various priors should not be considered equally relevant to a problem at hand. From the other side, in MMEU each act is evaluated on the base of only one prior. In order to cope with these problems Klibanoff et al. suggested a smooth ambiguity model as a more general way to formalize decision making under ambiguity than MMEU [19].

There are many different approaches for describing imprecision of probability relevant information. One of the approaches is the use of hierarchical imprecise models. These models capture the second-order uncertainty inherent in real problems. According to this approach an expert opinion on probability assessments is usually imprecise [2, 11, 35].

The existing decision theories are not developed for applications in fuzzy environment and consequently require more deterministic information. Development of new theories is now possible due to an increased computational power of information processing systems which allows for computations with imperfect information, particularly, imprecise and partially true information, which are much more complex than computations over numbers and probabilities. There is the series of fuzzy approaches to decision making like fuzzy AHP [14, 22], fuzzy TOPSIS [22, 37], fuzzy Expected Utility [9, 16, 23]. However, they are mainly fuzzy generalizations of the mathematical structures of the existing theories used with intent to account for vagueness, impreciseness and partial truth. Direct fuzzification of the existing theories often leads to inconsistency and loss of properties of the latter.

Approaches that are based on fuzzy description of the most part of a decision problem are lack of mathematical proof of an existence of a utility function. From the other side, many of the existing fuzzy approaches follow too simple models like EU model.

In [1] authors present a fuzzy-logic-based decision theory with imperfect information. This theory is developed for the framework of mix of fuzzy information and probabilistic information and is based on a fuzzy utility function represented as a fuzzy-valued Choquet integral.

On basis of analysis of the existing decision theories described above we may conclude that the existing decision models yielded good results, but nowadays there is a need in generation of more realistic decision models.

These approaches require that the objective probabilities or subjective probabilities and utility values be precisely known. But in real world in many cases it becomes impossible to determine the precise values of needed information. Interval analysis and classical fuzzy set theories have been applied in making decisions and many fruitful results have been achieved. But a problem is that in the mentioned above decision theories the reliability of the decision relevant information is not well taken into consideration.

In [21] Zadeh introduced the concept of Z-numbers to describe the uncertain information which is more generalized notion. A Z-number is an ordered pair of fuzzy numbers $({\tilde{A}},{\tilde{R}})$. Here $\tilde{A}$ is a value of some variable and $\tilde{R}$ represents an idea of certainty or other closely related concept such as sureness, confidence, reliability, strength of truth, or probability [29]. It should be noted that in everyday decision making most decisions are in the form of Z-numbers. Zadeh suggests operations for computation with Z-numbers, using the extension principle. In [29] author uses Z-numbers to provide information about an uncertain variable in the form of Z-valuations, assuming that this uncertain variable is random. In [29] author also offers an illustration of a Z-valuation, showing how to make decisions and answer questions. Also an alternative formulation is used for the information contained in the Z-valuations in terms of a Dempster-Shafer belief structure that made use of type-2 fuzzy sets. Simplified version of Z-valuation of decision relevant information is considered in [8]. In [7] authors considered a multi-criteria decision making problem using Z-numbers. For this purpose the Z-numbers are converted to classical fuzzy number and a priority weight of each alternative is determined.

In this chapter we consider two approaches to decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. A fuzzy measure is calculated on a base of a given Z-information. The second approach is based on direct computation with Z-numbers. To illustrate a validity of suggested approaches to decision making with Z-information the numerical examples are used.

The study is organized as follows. In Sect. 2 we present required preliminaries and cover some prerequisite material. In Sect. 3 we consider a generalization of Expected Utility Theory using Z-number. In Sect. 4 we present a method of Choquet integral based decision making using Z-information. In Sect. 5 we consider the second approach based on direct computation with Z-numbers applying it to the same problem. In Sect. 6 we cover application of the suggested method to a real-life business problem of hotel management using the suggested approaches. Concluding comments are included in Sect. 7.

2 Preliminaries

Definition 1

Fuzzy sets [4].

Let $\mathcal{X}$ be a classical set of objects, called the universe, whose generic elements are denoted $x$. Membership in a classical subset $\mathcal{A}$ of $\mathcal{X}$ is often viewed as a characteristic function $\mu _\mathcal{A}$ from $\mathcal{X}\;\text {to}\;\{0,1\}$ such that

$$\begin{aligned} \mu _\mathcal{A} (x)=\left\{ {\begin{array}{l@{\quad }l} 1&{}{ iff}\,x \in \mathcal{A} \\ 0&{}{ iff}\,x \notin \mathcal{A} \\ \end{array}} \right. \end{aligned}$$

where $\{0,1\}$ is called a valuation set; 1 indicates membership while 0—non membership.

If the valuation set is allowed to be in the real interval [0,1], then $\mathcal{A}$ is called a fuzzy set, $\mu _\mathcal{A}$ is the grade of membership of $x$ in $A$: $\mu _\mathcal{A} (x)\,{:}\,\mathcal{X}\rightarrow [0{,}1]$.

Let $\mathcal{E}^{n}$ be a space of all fuzzy subsets of $\mathcal{R}^{n}$. These subsets satisfy the conditions of normality, convexity, and are upper semicontinuous with compact support.

Definition 2

A Z-number [21].

A Z-number is an ordered pair of fuzzy numbers,$(\tilde{A},\tilde{R})$. $\tilde{A}$-is a fuzzy restriction on the values which a real-valued uncertain variable is allowed to take. $\tilde{R}$ is a measure of reliability of the first component.

Example

(anticipated budget deficit, about three million USD, likely);

(price of oil in the near future, significantly over 50 dollars/barrel, very likely).

Denote $\Omega$ a universe of discourse and denote $\mathcal{F}$ a $\sigma$-algebra of subsets of $\Omega$.

Definition 3

Choquet integral [6, 23, 25, 31, 39, 40]. Let $\varphi \,{:}\,\Omega \rightarrow R$ be a measurable real-valued function on $\Omega$ and $\eta \,{:}\,\mathcal{F}\rightarrow [0{,}1]$ be a non-additive measure defined over $\mathcal{F}$. The Choquet integral of $\varphi$ with respect to $\eta$ is defined as

$$\begin{aligned} \int _\Omega \varphi d \eta =\sum _{i\,=\,1}^n {\left( \eta (B_{(i)})-\eta (B_{(i-1)})\varphi (\omega _{(i)})\right. } \end{aligned}$$

(1a)

where index $(i)$ implies that elements $\omega _i \in \Omega , i=1,\,\ldots ,\,n$ are permuted such that $\varphi (\omega _{(i)})\ge \varphi (\omega _{(i+1)})$, $\varphi (\omega )=0$ and $B_{(i)} =\{\omega _{(1)} ,\,\ldots ,\,\omega _{(i)} \}\subseteq \Omega$.

A value of fuzzy utility function for an action is determined as a fuzzy number-valued Choquet integral

$$\begin{aligned} \int _\Omega \tilde{\varphi }d\tilde{\eta }= \sum _{i\,=\,1}^n {\left( \tilde{\eta }(B_{(i)})-\tilde{\eta }(B_{(i-1)}) \tilde{\varphi }(\omega _{(i)})\right. } \end{aligned}$$

(1b)

$(i)$means that utilities are ranked such that $\tilde{\varphi }(\omega _{(1)})\ge \ldots \ge \tilde{\varphi }(\omega _{(n)}),\tilde{\varphi }(\omega )=0$.

Let $\tilde{\mathcal{F}}(\Omega )=\big \{ {\tilde{V}\left| {\mu _{\tilde{V}}\,{:}\,\Omega \rightarrow [0{,}1]} \right. } \big \}$ be the class of all fuzzy subsets of $\Omega$.

Definition 4

[1, 41]. A subclass $\tilde{\mathcal{F}}$ of $\tilde{\mathcal{F}}(\Omega )$ is called a fuzzy $\sigma$-algebra if it has the following properties:

• (1) $\varnothing ,\Omega \in \tilde{\mathcal{F}}$

• (2) if $\tilde{V}\in \tilde{\mathcal{F}}$, then $\tilde{V}^{c}\in \tilde{\mathcal{F}}$

• (3) if $\left\{ {\tilde{V}_n } \right\} \subset \tilde{\mathcal{F}}$, then $\bigcup _{n=1}^\infty {\tilde{V}_n \in \tilde{\mathcal{F}}}$

Definition 5

Fuzzy number-valued fuzzy measure [1, 41]. A fuzzy number-valued fuzzy measure ((z) fuzzy measure) on $\tilde{\mathcal{F}}$ is a fuzzy number-valued fuzzy set function $\tilde{\eta }\,{:}\,\tilde{\mathcal{F}} \rightarrow \mathrm{E}^{1}$ with the properties:

• (1) $\tilde{\eta }(\varnothing )=0$;

• (2) if $\tilde{V}\subset \tilde{W}$ then $\tilde{\eta }(\tilde{V})\le \tilde{\eta }(\tilde{W})$;

• (3) if $\tilde{V}_1 \subset \tilde{V}_2 \subset \ldots ,\tilde{V}_n \subset \ldots \in \tilde{\mathcal{F}}$, then $\tilde{\eta }( \bigcup _{n=1}^\infty {\tilde{V}_n}) =\mathop {\lim \limits _{n\rightarrow \infty }} \tilde{\eta }( \tilde{V}_n )$;

• (4) if $\tilde{V}_1 \supset \tilde{V}_2 \supset \ldots ,\tilde{V}_n \in \tilde{\mathcal{F}}$, and there exists $n_{0}$ such that $\tilde{\eta }(\tilde{V}_{n_{0}})\ne \tilde{\infty }$, then $\tilde{\eta }( \bigcap _{n=1}^\infty {\tilde{V}_n })=\mathop {\lim \limits _{n\rightarrow \infty }} \tilde{\eta }(\tilde{V}_n )$.

Definition 6

Lower prevision [2, 3, 12, 20, 24–26]. A coherent lower prevision is defined as a lower expectation functional from the set of gambles to the real numbers that satisfies some rationality criteria. This function is conjugate to another that is called a coherent upper prevision. When a coherent lower prevision coincides with its conjugate coherent upper prevision, we call it a linear prevision. An unconditional lower prevision $\underline{P}(B)$ is coherent if and only if it is the lower envelope of dominating linear previsions.

If the lower prevision $\underline{P}$ is represented as the lower envelope of a closed convex set $\mathrm{P}$ of linear previsions then

$$\begin{aligned} \underline{P}=\min \{P(B)\},B \subset S \end{aligned}$$

(2)

Lower prevision $\underline{P}$ is characterized by probability density function of each linear prevision in extreme points [36].

In particular case, when linear prevision is a probability measure the lower prevision is the lower envelope of multiple priors. In this work we use lower prevision as non-additive measure. So we can define $\eta$ as $\underline{P}$.

Example [1]

Let $\Omega$ be a nonempty set and $\Omega = [0{,}1]$. Consider values of a fuzzy number-valued fuzzy measure $\tilde{\eta }$ for some fuzzy subsets $s_i \subset \Omega ,i=\overline{1,3}$ and their unions. Then the corresponding values of the fuzzy number-valued fuzzy measure $\tilde{\eta }_{\tilde{P}^{l}}$ can be as the triangular fuzzy numbers given in Table 1:

Definition 7

Expected utility [23, 27]

Let $P:S\rightarrow R$ be any probability measure on a set of states $S$ such that $P(s)>0$ for all $s \in S$. For each $s \in S$ define $v:\mathcal{X}\rightarrow \mathcal{R}$. Then

$$\begin{aligned} U(f)=\sum _{s\in S} {P(s)v(f(s))} \end{aligned}$$

(3)

where $f$ is an act, $x=f(s)$ is an outcome, $v(f(s))$ is a utility in state $s$ and $U(f)$ is the expected value of utility.

Table 1 The values of the fuzzy number-valued fuzzy measure $\tilde{\eta }_{\tilde{P}^{l}}$

3 A Generalization of Expected Utility Theory Using Z-Number

A set of acts $f_1 ,f_2 ,\ldots ,f_n$ with a number of possible utilities $\tilde{Z}_{v(f_i (s_1 ))} ,\tilde{Z}_{v(f_i (s_2 ))} ,\ldots ,\tilde{Z}_{v(f_i (s_m ))}$ in states $s_1 ,s_2 ,\ldots ,s_m \in S$ and the corresponding state probabilities $\tilde{Z}_{P(s_1 )} ,\tilde{Z}_{P(s_2 )} ,\ldots ,\tilde{Z}_{P(s_m )}$ are given and described by Z-numbers (Tables 2 and 3). Then we can determine the value of expected utility function for each act.

Table 2 The payoff table with utilities as Z-numbers

In payoff Table 2 ${\tilde{R}}_1$ is a confidence degree for the value of utility.

As decision maker usually is uncertain about first-order imprecise probabilities, we describe the probabilities of states of nature as Z-numbers (Table 2).

Table 3 Probabilities of states as Z-numbers

In Table 3 ${\tilde{R}}_2$ is a confidence degree for the value of probability of the state of nature.

Now using the Z-valuations of the values of utilities and probabilities of states of nature we can determine the values of expected utility for any act, represented as Z-numbers.

$$\begin{aligned} {\tilde{Z}}_{U(f_1 )}&=({\tilde{P}}(s_1 ),\tilde{R}_2 )\times ({\tilde{v}}(f_1 (s_1 )),{\tilde{R}}_1 )+({\tilde{P}}(s_2 ),{\tilde{R}}_2 )\times ({\tilde{v}}(f_1 (s_2 )),{\tilde{R}}_1 )+,\ldots ,\nonumber \\&\quad \;+(\tilde{P}(s_m ),\tilde{R}_2 )\times (\tilde{v}(f_1 (s_m )),\tilde{R}_1 )\nonumber \\ \tilde{Z}_{U(f_2 )}&=(\tilde{P}(s_1 ),\tilde{R}_2 )\times (\tilde{v}(f_2 (s_1 )),\tilde{R}_1 )+(\tilde{P}(s_2 ),\tilde{R}_2 )\times (\tilde{v}(f_2 (s_2 )),\tilde{R}_1 )+,\ldots ,\nonumber \\&\quad \;+(\tilde{P}(s_m ),\tilde{R}_2 )\times (\tilde{v}(f_2 (s_m )),\tilde{R}_1 )\nonumber \\ \quad \\ {\tilde{Z}}_{U(f_n )}&=({\tilde{P}}(s_1 ),{\tilde{R}}_2 )\times ({\tilde{v}}(f_n (s_1 )),{\tilde{R}}_1 )+({\tilde{P}}(s_2 ),{\tilde{R}}_2 )\times ({\tilde{v}}(f_n (s_m )),{\tilde{R}}_1 )+,\ldots ,\nonumber \\&\quad \;+({\tilde{P}}(s_m ),{\tilde{R}}_2 )\times ({\tilde{v}}(f_n (s_m )),{\tilde{R}}_1 )\nonumber \end{aligned}$$

(4)

Now we have to choose the act with maximal expected utility i.e. the decision making problem in this case consists in the determination of an optimal action $f^{*}\in A$ as the following

$$\begin{aligned} \tilde{Z}_{U(f^{*})} =\mathop {\max }\limits _{f\in A} (\tilde{Z}_{U(f_1 )} ,\tilde{Z}_{U(f_2 )} ,\ldots ,\tilde{Z}_{U(f_n )}) \end{aligned}$$

(5)

Let outcomes $\tilde{Z}_{v(f_i (s_j ))} =(\tilde{v}(f_i (s_j )),\tilde{R}_1 )$ and the probabilities $\tilde{Z}_{P(s_j )} =(\tilde{P}(s_j ),\tilde{R}_2 )$ of the states $s_j \in S$, where $\tilde{R}_1 =\{(x_2 ,\mu _{\tilde{R}_1 } (x))\,{:}\,x_2 \in [0{,}1]\} , \tilde{R}_2 =\{(y_2 ,\mu _{\tilde{R}_2 } (y)){:}\,y_2 \in [0{,}1]\}$ are represented by trapezoidal and triangle fuzzy numbers respectively.

In this study it is assumed that it is given only NL-described reasonable knowledge about probability distribution over $S$. It means that a state $s_j$ is assigned a linguistic probability $\tilde{P}_j$ that can be described by Z-number. Initial data for the problem are represented by given linguistic probabilities for $m-1$ states of nature whereas for one of the given states the probability is unknown. So at first it is required to obtain the unknown probability. To determinate an unknown probability of state $s_j -\tilde{Z}_{P(s_j )}$ on a base of given probabilities $\tilde{Z}_{P(s_1 )} ,\tilde{Z}_{P(s_2 )} ,\ldots ,\tilde{Z}_{P(s_{j-1})} ,\ldots ,\tilde{Z}_{P(s_{j+1})} ,\ldots ,\tilde{Z}_{P(s_m )}$ we use the method suggested in [1]. In the framework of Computing with Words the problem of obtaining the unknown linguistic probability for state $\tilde{s}_j$ given linguistic probabilities of all other states is a problem of propagation of generalized constraints. Formally this problem is formulated as follows:

given

$$\begin{aligned} \tilde{P}\left( {\tilde{s}_i } \right) =\tilde{P}_i ; \tilde{s}_i \in \varepsilon ^{n} , \tilde{P}_i \in \varepsilon _{[0{,}1]}^1 , i \in \{ {1,\ldots ,j-1,j+1,\ldots ,n} \} \end{aligned}$$

(6)

find unknown

$$\begin{aligned} \tilde{P}\left( {\tilde{s}_j } \right) =\tilde{P}_j , \tilde{P}_j \in \varepsilon _{[0{,}1]}^1 \end{aligned}$$

This problem reduces to a variation problem of constructing the membership function $\mu _{\tilde{P}_j } (\cdot )$ of an unknown fuzzy probability $\tilde{P}_j$ [1]:

$$\begin{aligned} \mu _{\tilde{P}_{j}} (p_j )=\text {sup}_{\rho }\min \nolimits _{i=\left\{ {1,\ldots ,j-1,j+1,\ldots ,n} \right\} } (\mu _{\tilde{P}_i } (\mathop \smallint \limits _{S} {\mu _{\tilde{s}_i } (s)\rho (s)ds})) \end{aligned}$$

(7)

subject to $\mathop \smallint \limits _{S} {\mu _{\tilde{s}_j } (s)\rho (s)ds}= p_j , \mathop \smallint \limits _{S} {\rho (s)ds=1}$

Given the payoff table and the complete probability distribution we can evaluate the values of expected utility on base of (4). For this aim we use computation with Z-numbers which falls within the province of Computing with words. Computation with Z-information in this study is based on converting of Z-numbers to classical fuzzy numbers [8].

To convert the given Z-numbers on outcomes and probabilities first we determine the expected values of fuzzy numbers $R_1$ and $R_2$ describing reliability of variables of outcome and probability:

$$\begin{aligned} \alpha _1 =\frac{\int {x\mu _{\tilde{R}_1 } (x)dx} }{\int {\mu _{\tilde{R}_1 } (x)dx} }, \end{aligned}$$

(8)

$$\begin{aligned} \alpha _2 =\frac{\int {y\mu _{\tilde{R}_2 } (y)dy} }{\int {\mu _{\tilde{R}_y } (y)dy} } \end{aligned}$$

(9)

Now we can represent the values of outcome and probability variables as trapezoidal fuzzy numbers: $\tilde{Z}_{v_{s_j } (f_i (s))}^{\alpha _1 } =(\text {a}_1, \text {a}_2, \text {a}_3, \text {a}_4 ;\alpha _1 ),\tilde{Z}_{P(s_j )}^{\alpha _2 } =(c_1 ,c_2 ,c_3 ;\alpha _2 )$.

Then we convert this weighted Z-number to fuzzy number: ${\tilde{Z}}'_{v_{s_j } (f_i (s))} =(\sqrt{\alpha _1 }\text {a}_1, \sqrt{\alpha _1 }\text {a}_2, \sqrt{\alpha _1 }\text {a}_3, \sqrt{\alpha _1 }\text {a}_4 ;1), {\tilde{Z}}'_{P(s_j )} =(\sqrt{\alpha _2 }c_1 ,\sqrt{\alpha _2 }c_2 ,\sqrt{\alpha _2 }c_3 ;1)$

As we have an ordinary fuzzy numbers with trapezoidal and triangular membership functions then we can obtain fuzzy values of utility function $U(f(s))$ for each alternative by (4):

$$\begin{aligned} {\tilde{Z}}'_{U(f_i )} = \sum _{\begin{array}{l} i=1 \\ j=1 \\ \end{array}}^{k,n} {{\tilde{Z}}'_{v(f_i (s))}} \times {\tilde{Z}}'_{P(s_j )} =(\sqrt{\alpha _1 \alpha _2 }\times (\sum _{i,j=1}^{k,n} {\tilde{v}(f_i (s))} \times \tilde{P}(s_j ));1) \end{aligned}$$

(10)

An optimal action $f^{*}\in A$ is obtained in accordance with (5).

The value of Z-number for optimal utility function may be described as

$$\begin{aligned} \tilde{Z}_{U(f_i )} = ({\tilde{Z}}'_{U(f_i )} /\sqrt{\alpha _1 \alpha _2 });\tilde{R}_3 ). \end{aligned}$$

(11)

where $({\tilde{Z}}'_{U(f_i )} /\sqrt{\alpha _1 \alpha _2 });\tilde{R}_3 )$ describes the reliability of the utility function. More preferable act is determined by ranking $\tilde{Z}_{U(f_i )}$ using ranking procedure given in Sect. 5.

4 Choquet Integral Based Decision Making Using Z-Information

Formally the problem is formulated as follows. Decision-making under Z-information can be considered as 4-tuple $(S, \tilde{Z}_X , \mathcal{A},\underline{\succ })$, where $S=\{s_1 ,s_2 ,\ldots ,s_n \}$– a space of mutually exclusive and exhaustive states of nature, $\tilde{Z}_X$– a set of outcomes, described by Z-valuation. A is the set of actions that are functions $f\,{:}\,S \quad \rightarrow \tilde{Z}_X , \underline{\succ }$ is the non-additive preference relation on the set of actions. In decision-making under uncertainty, a probability over $S$ is imprecise. $\mathcal{F}_S$ is a $\sigma$—algebra of subsets $B$ of $S$. Denote by $\mathcal{A}_0$ the set of all $\mathcal{F}_S$-measurable step-valued functions from $S$ to $X$ and denote $\mathcal{A}_C$ the constant actions in $\mathcal{A}_0$. Let A be a convex subset of $X^{S}$ which includes $\mathcal{A}_C$. X can be considered as a subset of some linear space, and $X^{S}$ can then be considered as a subspace of the linear space of all functions from $S$ to the first linear space. The problem is to determine preferences among alternatives by means of a utility function.

The suggested decision-making methodology uses Choquet expected utility for description of preferences. The utility function used here is as follows

$$\begin{aligned} {\tilde{Z}}'_{U(f_i )} = \int \limits _{s} {{Z}'_{v(f_i (s))} d{Z}'_\eta } \end{aligned}$$

(12)

The decision making problem in this case consists in the determination of an optimal action $f^{*}\in A$ such that

$$\begin{aligned} {\tilde{Z}}'_{U(f_i^*)} = \mathop {\max }\limits _{f\in A} \Bigg \{\int \limits _{s} {{\tilde{Z}}'_{v(f_i (s))} d{\tilde{Z}}'_\eta \Bigg \}} \end{aligned}$$

(13)

As it was mentioned above the outcomes $\tilde{Z}_{v(f_i (s_j ))} =(\tilde{v}(f_i (s_j )),\tilde{R}_1 )$ and the probabilities $\tilde{Z}_{P(s_j )} =(\tilde{P}(s_j ),\tilde{R}_2 )$ of the states $s_j \in S$ where $\tilde{R}_1 =\{(x_2 ,\mu _{\tilde{R}_1 } (x))\,{:}\,x_2 \in [0{,}1]\}$ and $\tilde{R}_2 =\{(y_2 ,\mu _{\tilde{R}_2 } (y))\,{:}\,y_2 \in [0{,}1]\}$ are represented by trapezoidal and triangle fuzzy numbers. At first it is required to determine the unknown probability of state $s_j -\tilde{Z}_{P(s_j )}$ on a base of given probabilities $\tilde{Z}_{P(s_1 )} ,\tilde{Z}_{P(s_2 )},\ldots ,\tilde{Z}_{P(s_{j-1})} ,\ldots ,\tilde{Z}_{P(s_m )}$ by formulas (6 and 7).

Given the payoff Table 2 and the complete probability distribution we can evaluate the values of Choquet integral on base of (12) [7, 8, 30, 38].

Given the complete probability distribution we construct measure as lower prevision.

The determination of a lower prevision ${Z}'_\eta$ from linguistic probability distribution $\tilde{P}$ has a great role in the determination of the preferences in this model.

When the states of nature are just some elements, the measure is defined [1] as

$$\begin{aligned} \tilde{Z}^\prime _{{\eta _{{\tilde{P}}} }} \left( H \right) = \mathop \cup \limits _{{\alpha \in (0,1]}} a\cdot \Big [ {\tilde{Z}^{\prime ~{\alpha }} _{{\eta _{{\tilde{P}_{{left}} }} }} \left( H \right) ,\tilde{Z}^{\prime ~{\alpha }}_{{\eta _{{\tilde{P}_{{right}} }}}} \left( H \right) } \Big ],H \subset S = \left\{ {s_{1} , \ldots ,s_{m} } \right\} \end{aligned}$$

(14)

where

$$\begin{aligned} \tilde{Z}^{\prime ~{\alpha }} _{{\eta _{{\tilde{P}}}}} ( H )&= inf\bigg \{ {\sum \limits _{{{\text {s}}_{{\text {i}}} \in H}} {p( {s_{j} } ), \ldots ,} p( {s_{m} } )} \bigg \},\left( {p( {s_{1} } ), \ldots ,p( {s_{m} } )} \right) \in P^{\alpha } \\ P^{\alpha }&=\Big \{({p}{(s_1)},\ldots ,{p}{(s_m )})\in P_{1}^{\alpha } \times \ldots \times P_{m}^{\alpha } |\sum _{j=1}^{m} {p(s_j )=1}\Big \}, \end{aligned}$$

Here $P_1^\alpha ,\ldots ,P_{m}^{\alpha }$ are $\alpha$-cuts of fuzzy probabilities ${\tilde{P}}_1,\ldots ,{\tilde{P}}_{m}, p(s_1 ),\ldots ,p(s_m )$ are basic probabilities for ${\tilde{P}}_1 ,\ldots ,{\tilde{P}}_{m}, \times$ denotes the Cartesian product.

Now we can construct a fuzzy measure with triangle membership function from fuzzy set of possible probability distributions as its lower probability function (lower prevision) taking into consideration (14) and the method used in [1].

As we have an ordinary fuzzy numbers with trapezoidal and triangular membership functions then we can obtain the fuzzy values of utility function ${\tilde{U}}(f(s))$ for each alternative by (1b):

$$\begin{aligned} {\tilde{Z}}'_{U(f_i )}&= \int \limits _{s} {{Z}'_{v(f_i (s))} d{Z}'_\eta } =\int \limits _{s} {\sqrt{\alpha _1 }\times {\tilde{v}}(f(s);1)d{\tilde{Z}}'_{\eta _{\tilde{P}} }} \nonumber \\&=\sum _{j=1}^m {\big ({\tilde{Z}}'_{\eta _{\tilde{P}} } (B_{(j)})-{\tilde{Z}}'_{\eta _{\tilde{P}} } (B_{(j-1)})\big ){\tilde{Z}}_{v(f_i (s))} }\nonumber \\&=\sum _{j=1}^m {\big ({\tilde{Z}}'_{\eta _{\tilde{P}} } (B_{(j)})-{\tilde{Z}}'_{\eta _{\tilde{P}} } (B_{(j-1)})\big )\big (\sqrt{\alpha _1 }\times {\tilde{v}}(f_i (s);1)\big )} \end{aligned}$$

(15)

An optimal action $f^{*}\in A$ is obtained in accordance with (13) using ranking procedure given in Sect. 5.

5 Direct Computation with Z-Numbers

Let us now apply the second approach of direct computation with Z-numbers without transformation of Z-number into ordinal fuzzy number. A $Z$-number $({\tilde{A}},{\tilde{R}})$ can be interpreted as ${\Pr }\,{ob}(X\,is\,{\tilde{A}}) \text {is}\,{\tilde{R}}$. This expresses that we do not know the true probability density over $X$, but have a constraint in form of a fuzzy subset ${\tilde{P}}$ of the space $\mathbf{P}$ of all probability densities over $X$. This restriction induces a fuzzy probability ${\tilde{R}}$. Let p be density function over $X$. The probability ${\Pr }\,{ob}(X is{\tilde{A}})$ (probability that $X is{\tilde{A}})$ is determined on the base of the definition of the probability of a fuzzy subset as

$$\begin{aligned} {\Pr }\, {ob}(X\,is\,{\tilde{A}})=\int \limits _{-\infty }^{+\infty } {\mu _A (x) {p_{X}} (x)dx} . \end{aligned}$$

Then the degree to which $p$ satisfies the $Z$-valuation ${\Pr } \,{ob}(X\,is\,{\tilde{A}})\,\text {is}\,{\tilde{R}}$ is $\mu _P(p)=\mu _R ({\Pr }\, {ob}(X\,\text {is}\,{\tilde{A}})\,\text {is}\,{\tilde{R}}))=\mu _R (\int \limits _{-\infty }^{+\infty }{\mu _A (x) {p_{X}} (x)dx}).$

Here $p$ is taken as some a parametric distribution. The density function of a normal distribution is

$$\begin{aligned} p_X (x)=normpdf(x,m,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}\exp \left( {-\frac{(x-m)^{2}}{2\sigma ^{2}}} \right) . \end{aligned}$$

In this situation, for any $m, \sigma$ we have

$$\begin{aligned} {\Pr }\, {ob}_{m,\sigma }(X\,\text {is}\,{\tilde{A}})&=\int \limits _{-\infty }^{+\infty } {\mu _A (x)p_{m,\sigma } (x)dx} =\int _{-\infty }^{+\infty } {\mu _A (x)\frac{1}{\sigma \sqrt{2\pi }}} \exp \left( {\frac{(x-m)^{2}}{2\sigma ^{2}}} \right) dx\\&=(trapmf(x,[a_1,a_2 ,a_3 ,a_4])^*normpdf(x,m,\sigma ),-\inf ,+\inf ) \end{aligned}$$

Then the space $P$ of probability distributions will be the class of all normal distributions each uniquely defined by its parameters $m,\sigma$.

Let $X=({\tilde{A}}_X ,{\tilde{R}}_X )$ and $Y=({\tilde{A}}_Y ,{\tilde{R}}_Y )$ be two independent Z-numbers. Consider determination of $W=X+\hbox {Y}$. First, we need compute ${\tilde{A}}_X +{\tilde{A}}_Y$ using Zadeh’s extension principle:

$$\begin{aligned} \mu _{(A_X +A_Y )} (w)=\mathop {\sup }\limits _x (\mu _{A_X } (x)\wedge \mu _{A_Y } (w-x)), \quad \wedge =\min . \end{aligned}$$

As the sum of random variables involves the convolution of the respective density functions we can construct ${\tilde{P}}_W$, the fuzzy subset of $\mathbf{P}$, associated with the random variable $W$. Recall that the convolution of density functions $p_1$ and $p_2$ is defined as the density function

$$\begin{aligned} p=p_1 \oplus p_2 \end{aligned}$$

such that

$$\begin{aligned} p(w)=\int \limits _{-\infty }^{+\infty } {p_{1} (x)p_{2} (w-x)dx} =\int \limits _{-\infty }^{+\infty } {p_{1} (w-x)p_{2} (x)dx} \end{aligned}$$

One can then find the fuzzy subset ${\tilde{P}}_W$. For any$p_W \in \mathbf{P}$, one obtains

$$\begin{aligned} \mu _{P_{W}} (p_{W} )=\mathop {\max }\limits _{p_{U} ,p_{V} } [\mu _{P_{X} } (p_{X} )\wedge \mu _{P_{Y}} (p_Y )], \end{aligned}$$

subject to

$$\begin{aligned} p_W =p_X \oplus p_Y , \end{aligned}$$

that is,

$$\begin{aligned} p_W (w)=\int \limits _{-\infty }^{+\infty } {p_\mathrm{U} (x)p_\mathrm{V} (w-x)dx} =\int \limits _{-\infty }^{+\infty } {p_\mathrm{U} (w-x)p_\mathrm{V} (x)dx} . \end{aligned}$$

Given $\mu _{P_X } (p_X )=\mu _{P_X } (m_X ,\sigma _X )$ and $\mu _{P_Y } (p_Y )=\mu _{P_Y } (m_Y ,\sigma _Y )$ as

$$\begin{aligned}&\mu _{P_X } (m_X ,\sigma _X )=\mu _{B_X } \bigg (\int \limits _{-\infty }^{+\infty } {\mu _{A_X } (x)\frac{1}{\sigma _X \sqrt{2\pi }}\exp \left( {\frac{(x-m_X )^{2}}{2\sigma _X ^{2}}} \right) dx}\bigg ) ,\\&\mu _{P_Y } (m_Y ,\sigma _Y )=\mu _{B_Y } \bigg (\int \limits _{-\infty }^{+\infty } {\mu _{A_Y } (x)\frac{1}{\sigma _Y \sqrt{2\pi }}\exp \left( {\frac{(x-m_Y )^{2}}{2\sigma _Y ^{2}}} \right) dx}\bigg ) \end{aligned}$$

one can define ${\tilde{P}}_W$ as follows

$$\begin{aligned} p_W&=p_{m_X ,\sigma _Y } \oplus p_{m_X ,\sigma _Y } ,\\ p_W (w)&=p_{m_W ,\sigma _W } =normpdf[w,m_W ,\sigma _W ]\\&=(normpdf(x,m_X ,\sigma _X )*normpdf(w-x,m_Y ,\sigma _Y ),-\inf ,+\inf )\\&=\int \limits _{-\infty }^{+\infty } {\frac{1}{\sigma _X \sqrt{2\pi }}\exp \left( {\frac{(x-m_X )^{2}}{2\sigma _X ^{2}}} \right) \frac{1}{\sigma _Y \sqrt{2\pi }}\exp \left( {\frac{(w-x-m_Y )^{2}}{2\sigma _Y ^{2}}} \right) dx} \\ \end{aligned}$$

where

$$\begin{aligned} m_W&=m_X +m_Y\,\text {and}\,\sigma _W =\sqrt{\sigma _{_X }^2 +\sigma _Y^2},\\ \mu _{P_W } (p_W )&=\sup (\mu _{P_X } (p_X )\wedge \mu _{P_Y } (p_Y )) \end{aligned}$$

subject to

$$\begin{aligned} p_W =p_{m_X ,\sigma _Y } \oplus p_{m_Y ,\sigma _Y } \end{aligned}$$

$B_W$is found as follows.

$$\begin{aligned} \mu _{B_W } (b_W )=\sup (\mu _{{\tilde{P}}_W } (p_W )) \end{aligned}$$

subject to

$$\begin{aligned} b_W =\int \limits _{-\infty }^{+\infty } {p_W (w)\mu _{A_W } (w)dw} \end{aligned}$$

Let us now consider determination of $W=X\cdot Y$. ${\tilde{A}}_U \cdot {\tilde{A}}_V$ is defined by:

$$\begin{aligned} \mu _{(A_X \cdot A_Y )} (w)=\mathop {\sup }\limits _x (\mu _{A_X } (x)\wedge \mu _{A_Y } (\frac{w}{x})), \quad \wedge =\min . \end{aligned}$$

the probability density $p_W$ associated with $W$ is obtained as

$$\begin{aligned} p_W =p_{m_X ,\sigma _X } \otimes p_{m_Y ,\sigma _Y } , \end{aligned}$$

$p_W (w)=p_{m_W ,\sigma _W } =\int \limits _{-\infty }^{+\infty } {\frac{1}{\sigma _X \sqrt{2\pi }}\exp \left( {\frac{(x-m_X )^{2}}{2\sigma _X ^{2}}} \right) \frac{1}{\sigma _Y \sqrt{2\pi }}\exp \left( {\frac{(\frac{w}{x}-m_Y )^{2}}{2\sigma _Y ^{2}}} \right) dx}$ where

$$\begin{aligned} m_W =\frac{m_X m_Y }{\sigma _X \sigma _Y }+r, \end{aligned}$$

and

$$\begin{aligned} \sigma _W =\frac{\sqrt{m_{_X }^2 \sigma _Y^2 +m_{_Y }^2 \sigma _X^2 +2m_X m_Y \sigma _X \sigma _Y r+\sigma _X^2 \sigma _Y^2 +\sigma _X^2 \sigma _{Y}^{2} r^{2}}}{\sigma _{X} \sigma _{Y}}, \end{aligned}$$

where $r$ is correlation coefficient.

If $X$ and $Y$ are two independent random variables, then

$$\begin{aligned} m_W =\frac{m_X m_Y }{\sigma _X \sigma _Y },\text {and}\,\sigma _W =\frac{\sqrt{m_{_X }^2 \sigma _Y^2 +m_{_X }^2 \sigma _{X}^2 +\sigma _{X}^2 \sigma _{Y}^2 }}{\sigma _{X} \sigma _{Y}}, \end{aligned}$$

if take into account compatibility conditions $\sigma _{X} \sigma _{Y} =1$.

The other steps are analogous to those of determination of $W=X+\hbox {Y}$.

Assume we compare two courses of action $f_1$ and $f_2$ [29]. We must select between these two based on the objective of getting the largest utilities. Assume we have information about the utilities associated with these two courses of action expressed in terms of Z-valuations. These are

$$\begin{aligned}&{\tilde{Z}}_{U(f_1 )}\,\text {is}\,({\tilde{A}}_1 ,{\tilde{R}}_1 )\,\text {if we select}\,f_1\\&{\tilde{Z}}_{U(f_2 )}\,\text {is}\,({\tilde{A}}_2 ,{\tilde{R}}_2 )\,\text {if we select}\,f_2 \end{aligned}$$

A main problem is how to choose between these two alternatives, $f_1$ and $f_2$.

We have to represent the information obtained from Z-valuations on a space of probability distributions $P$. These two observations initiate the possibility distributions $G_1$ and $G_2$ over $P$. For any probability density $p_k \in P$, we have $G_i (p_k )=R_i (\int _S {A_i (x)p_k (x)dx)}$. We now have to select between these $G_i$. We obtain for each $p_k$ its expected value $E_k =\int _S {xp_k (x)dx}$ and using this we can determine two possibility distributions $F_1$ and $F_2$ as

$$\begin{aligned} F_i =\bigcup _{p_k } {\left\{ {\frac{G_i (p_k )}{E_k }} \right\} } \end{aligned}$$

The numerical value of each $F_i$ is determined as

$$\begin{aligned} e_i =\sum _{p_k } {E_k \cdot G_i (P_k )} \end{aligned}$$

and we choose the action with the largest value for $e_i$ .

6 Applications

We consider the business problem under imprecise information described by Z-valuation. Suppose a hotel is considering the construction of an additional wing. The possibility of adding 30 $(f_1)$, 40 $(f_2)$ and 50 $(f_3)$ rooms is evaluating. The success of the extension depends on a combination of local government legislation and competition in the field. There are three states of nature: positive legislation and low competition $(s_1)$, positive legislation and strong competition $(s_2 )$, no legislation and low competition $(s_3)$. Also we have the values anticipated payoffs (in percentage). The problem is to find how many rooms to build in order to maximize the return on investment. Z-valuation for the utilities of the each act taken at various states and probabilities on states are provided in Tables 4 and 5, respectively.

Table 4 The utility values of actions under various states

Table 5 The values of probabilities of states of nature

Here ${\tilde{Z}}_{v(f_i (s_j ))} = ({\tilde{v}}(f_i (s_j )),{\tilde{R}}_1 )$, where the outcomes are the trapezoidal fuzzy numbers and corresponding reliability is a triangular fuzzy number:

$\tilde{Z}_{{v(f_{1} (s_{1} ))}} = (\tilde{v}(f_{1} (s_{1} )),\tilde{R}_{1} ) =$ (high; likely) = [(7, 8, 9, 10;1), (0.6, 0.7, 0.8; 1)],

$\tilde{Z}_{{v(f_{1} (s_{2} ))}} = (\tilde{v}(f_{1} (s_{2} )),\tilde{R}_{1} ) =$ (below than high; likely) = [(6, 7, 8, 9;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_1 (s_3 ))} = ({\tilde{v}}(f_1 (s_3 )),{\tilde{R}}_1 )=$ (medium; likely) = [(4, 5, 6, 7;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_2 (s_1 ))} = ({\tilde{v}}(f_2 (s_1 )),{\tilde{R}}_1 )=$ (below than high; likely) = [(6, 7, 8, 9;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_2 (s_2 ))} = ({\tilde{v}}(f_2 (s_2 )),{\tilde{R}}_1 )=$ (low; likely) = [(3, 4, 5, 6;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_2 (s_3 ))} = ({\tilde{v}}(f_2 (s_3 )),{\tilde{R}}_1 )=$ (below than high; likely) = [(6, 7, 8, 9;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_3 (s_1 ))} = ({\tilde{v}}(f_3 (s_1 )),{\tilde{R}}_1 )=$ (below than high; likely) = [(6, 7, 8, 9;1), (0.6, 0.7, 0.8; 1)],

${\tilde{Z}}_{v(f_3 (s_2 ))} = ({\tilde{v}}(f_3 (s_2 )),{\tilde{R}}_1 )=$ (high; likely) = [(7, 8, 9, 10;1), (0.6, 0.7, 0.8; 1)],

$\tilde{Z}_{{v(f_{3} (s_{3} ))}} = (\tilde{v}(f_{3} (s_{3} )),\tilde{R}_{1} )=$ (medium; likely) = [(4, 5, 6, 7;1), (0.6, 0.7, 0.8; 1)].

Let the probabilities for $s_1$ and $s_2$ be Z-numbers ${\tilde{Z}}_{P(s_j )} = ({\tilde{P}}(s_j )),{\tilde{R}}_2 )$, where the probabilities and the corresponding reliability are the triangular fuzzy numbers:

${\tilde{Z}}_{P(s_1 )} = ({\tilde{P}}(s_1 )),{\tilde{R}}_2 )=$ (medium; quite sure) = [(0.25, 0.3, 0.35; 1), (0.8, 0.9, 1; 1)].

${\tilde{Z}}_{P(s_2 )} = ({\tilde{P}}(s_2 )),{\tilde{R}}_2 )=$ (more than medium; quite sure) = [(0.35, 0.4, 0.45; 1), (0.8, 0.9, 1; 1)].

In accordance with [1] we have calculated probability for$s_3$:

${\tilde{Z}}_{P(s_3 )} = ({\tilde{P}}(s_3 )),{\tilde{R}}_2 )=${low; quite sure} =[(0.2, 0.3, 0.4; 1), (0.8, 0.9, 1; 1)].

Then we convert the value of fuzzy reliability into a numerical number based on (8 and 9):

$$\begin{aligned} \alpha _1&=\frac{\int {x\mu _{{\tilde{R}}_1 } (x)dx} }{\int {\mu _{{\tilde{R}}_1 } (x)dx} }=0.7,\\ \alpha _2&=\frac{\int {y\mu _{{\tilde{R}}_2 } (y)dy} }{\int {\mu _{{\tilde{R}}_y } (y)dy} }\quad =0.9 \end{aligned}$$

Given the complete fuzzy probability distribution ${\tilde{P}}(s_j ),j=\overline{1,3}$ , we add the weight of the reliability to the restriction and have the weighted Z-number for the outcomes and the probabilities:

${\tilde{Z}}_{v(f_1 (s_1 ))}^{\alpha _1 } = (7, 8, 9, 10; 0.7), {\tilde{Z}}_{v(f_1 (s_2 ))}^{\alpha _1 } = (6, 7, 8, 9; 0.7), {\tilde{Z}}_{v(f_1 (s))}^{\alpha _1 } =(4, 5, 6, 7; 0.7),$

${\tilde{Z}}_{v(f_2 (s_1 ))}^{\alpha _1 } =(6, 7, 8, 9; 0.7), {\tilde{Z}}_{v(f_2 (s_2 ))}^{\alpha _1 } =(3, 4, 5, 6; 0.7),{\tilde{Z}}_{v(f_2 (s))}^{\alpha _1 } =(6, 7, 8, 9; 0.7),$

${\tilde{Z}}_{v(f_3 (s_1 ))}^{\alpha _1 } = (6, 7, 8, 9; 0.7), {\tilde{Z}}_{v(f_3 (s_2 ))}^{\alpha _1 } =(7, 8, 9, 10; 0.7), {\tilde{Z}}_{v(f_3 (s_3 ))}^{\alpha _1 } = (4, 5, 6, 7; 0.7),$

${\tilde{Z}}_{P(s_1 )}^{\alpha _2 } = (0.25, 0.3, 0.35; 0.9), {\tilde{Z}}_{P(s_2 )}^{\alpha _2 } = (0.35, 0.4, 0.45; 0.9), {\tilde{Z}}_{P(s_3 )}^{\alpha _2 } = (0.2, 0.3, 0.4; 0.9).$

Now we convert the obtained weighted numbers to fuzzy numbers:

$\tilde{Z}'_{{v(f_{1} (s_{1} ))}} = (5.85, 6.69, 7.52, 8.36; 1),$

$\tilde{Z}'_{{v(f_{1} (s_{2} ))}} = (5.01, 5.85, 6.69, 7.52; 1),$

$\tilde{Z}'_{{v(f_{1} (s_{3} ))}} = (3.34, 4.18, 5,01, 5,85; 1),$

$\tilde{Z}'_{{v(f_{2} (s_{1} ))}} = (5.01, 5.85, 6.69, 7.52; 1),$

$\tilde{Z}'_{{v(f_{2} (s_{2} ))}} = (2.50, 3.34, 4.18, 5.01;1),$

$\tilde{Z}'_{{v(f_{2} (s_{3} ))}} = (5.01, 5.85, 6.69, 7.52; 1),$

$\tilde{Z}'_{{v(f_{3} (s_{1} ))}} = (5.01, 5.85, 6.69, 7.52; 1),$

$\tilde{Z}'_{{v(f_{3} (s_{2} ))}} = (5.85, 6.69, 7.52, 8.36; 1),$

$\tilde{Z}'_{{v(f_{3} (s_{3} ))}} = 3.34, 4.18, 5.01, 5.85; 1).$

$\tilde{Z}'_{{P(s_{1} )}} = (0.23, 0.28, 0.33; 1),$

$\tilde{Z}'_{{P(s_{2} )}} = (0.33, 0.37, 0.42; 1),$

$\tilde{Z}'_{{P(s_{3} )}} = (0.18, 0.28, 0.37; 1).$

Given these data and following the proposed decision making method, we get the expected values of utility for acts $f_{1},\,f_{2},\,f_{3}$:

$$\begin{aligned} \tilde{Z}'_{{U\left( {f_{1} } \right) }}&= \tilde{Z}'_{{v\left( {f_{1} \left( {s_{1} } \right) } \right) }} * \tilde{Z}'_{{P\left( {s_{1} } \right) }} + \tilde{Z}'_{{v\left( {f_{1} \left( {s_{2} } \right) } \right) }} *\,\tilde{Z}'_{{P\left( {s_{2} } \right) }} + \tilde{Z}'_{{v\left( {f_{1} \left( {s_{3} } \right) } \right) }} *\tilde{Z}'_{{P\left( {s_{3} } \right) }} \\&= \left( {3.69,4.32,9.48,12.42;1} \right) , \\ \tilde{Z}'_{{U\left( {f_{2} } \right) }}&= \tilde{Z}'_{{v\left( {f_{2} \left( {s_{1} } \right) } \right) }} * \tilde{Z}'_{{P\left( {s_{1} } \right) }} + \tilde{Z}'_{{v\left( {f_{2} \left( {s_{2} } \right) } \right) }} *\,\tilde{Z}'_{{P\left( {s_{2} } \right) }} + \tilde{Z}'_{{v\left( {f_{2} \left( {s_{3} } \right) } \right) }} *\tilde{Z}_{{P\left( {s_{3} } \right) }} \\&= \left( {2.97,3.61,8.49,10.79;1} \right) . \\ \tilde{Z}'_{{U\left( {f_{3} } \right) }}&= \tilde{Z}'_{{v\left( {f_{3} \left( {s_{2} } \right) } \right) }} * \tilde{Z}'_{{P\left( {s_{1} } \right) }} + \tilde{Z}'_{{v\left( {f_{3} \left( {s_{2} } \right) } \right) }} *\,\tilde{Z}'_{{P\left( {s_{2} } \right) }} + \tilde{Z}'_{{v\left( {f_{3} \left( {s_{3} } \right) } \right) }} *\tilde{Z}'_{{P\left( {s_{3} } \right) }} \\&= \left( {3.77,4.40,10.08,12.54;1} \right) .\\ \end{aligned}$$

The value of Z-number for optimal utility function in accordance with (11) is determined as

$$\begin{aligned} {\tilde{Z}}_{U(f_{1} )}&= (4.65, 5.45, 11.95, 15.65; 0.79, 0.89, 0.99),\\ {\tilde{Z}}_{U(f_{2} )}&= (3.75, 4.55, 10.7, 13.6; 0.79, 0.89, 0.99),\\ {\tilde{Z}}_{U(f_{3} )}&= (4.75, 5.55, 12.7, 15.8; 0.79, 0.89, 0.99).\\ \end{aligned}$$

As we have the Z-number valued utility functions then we can select between the preferences ranking them.

Ranking of fuzzy values of utilities gives a preference to the third alternative, i.e.$f_{3} \succ f_{1} \succ f_{2}$.

Given these data and following the proposed decision making method, we can obtain an overall utility as a fuzzy-valued Choquet integral:

$$\begin{aligned} {\tilde{Z}}'_{U(f_i )}&= {\tilde{Z}}'_{\eta _{\tilde{P}} } (\{s_{(1)} \}-{\tilde{Z}}'_{\eta _{\tilde{P}} } \{s_{(0)} \})*{\tilde{Z}}'_{v_{s_{(1)} } (f_i (s))} +{\tilde{Z}}'_{\eta _{\tilde{P}} } (\{s_{(1)} ,s_{(2)} \}-{\tilde{Z}}'_{\eta _{\tilde{P}} } \{s_{(1)} \})\\&\quad \;*{\tilde{Z}}'_{v_{s_{(2)} } (f_i (s))} +{\tilde{Z}}'_{\eta _{\tilde{P}} } (\{s_{(1)} ,s_{(2)} ,s_{(3)} \}-{\tilde{Z}}'_{\eta _{\tilde{P}} } \{s_{(1)} ,s_{(2)} \})*{\tilde{Z}}'_{v_{s_{(3)} } (f_{i} (s))}\\ \end{aligned}$$

The states are ranked as:

For the first alternative ${\tilde{Z}}'_{v_{s_1 } (f_1 (s_1 ))} >{\tilde{Z}}'_{v_{s_2 } (f_1 (s_2 ))} > {\tilde{Z}}'_{v_{s_3 } (f_1 (s_3 ))}$,

For the second alternative ${\tilde{Z}}'_{v_{s_1 } (f_1 (s_1 ))} > {\tilde{Z}}'_{v_{s_3 } (f_1 (s_3 ))} > {\tilde{Z}}'_{v_{s_2 } (f_1 (s_2 ))}$,

For the third alternative ${\tilde{Z}}'_{v_{s_2 } (f_1 (s_2 ))} > {\tilde{Z}}'_{v_{s_1 } (f_1 (s_1 ))} > {\tilde{Z}}'_{v_{s_3 } (f_1 (s_3 ))}$.

The $\alpha$-cuts of $\tilde{Z}'_{{\tilde{\eta }_{{\tilde{P}}} }} (\left\{ {s_{1} ,s_{2} } \right\} ,\tilde{Z}'_{{\tilde{\eta }_{{\tilde{P}}} }} (\left\{ {s_{1} ,s_{3} } \right\}$ are found as the solutions of (14).

So we can determine the triangular fuzzy numbers

$$\begin{aligned} {Z}'_{\eta _{\tilde{P}} } (\{s_1 ,s_2 \}&=(0.62,0.71,0.71)\\ {Z}'_{\eta _{\tilde{P}} } (\{s_1 ,s_3 \}&=(0.57,0.62,0.62)\\ \end{aligned}$$

Given this, the values of the utility function for the alternatives are as follows:

$$\begin{aligned} {\tilde{Z}}'_{U(f_1 )}&= (-0.75,5.61,6.45,11.79)\\ {\tilde{Z}}'_{U(f_2 )}&= (-1.99,5.53,6.36,13.30)\\ {\tilde{Z}}'_{U(f_3)}&= (-0.11,6.29,7.21,12.66).\\ \end{aligned}$$

According to (11) we can determine the Z-number valued utility function for each alternative

$$\begin{aligned} {\tilde{Z}}_{U(f_1 )}&= (-0.94, 7.07, 8.13, 14.85; 0.79, 0.89,0.99),\\ {\tilde{Z}}_{U(f_2 )}&= (-2.51, 6.97, 8.02; 0.79, 0.89, 0.99),\\ {\tilde{Z}}_{U(f_3 )}&= (-0.14, 7.93, 9.08; 0.79, 0.89, 0.99).\\ \end{aligned}$$

Ranking of Z-number valued utility functions gives a preference to the third alternative, i.e.$f_3 \succ f_1 \succ f_2$.

We applied the suggested in Sect. 5 approach to the same hotel management problem and get $f_3 \succ f_1 \succ f_2$.

7 Conclusion

We first analyzed the main existing decision making theories and concluded that in almost all these theories a reliability of the decision relevant information is not well taken into consideration. We then recalled the concept of Z-numbers introduced by Zadeh and showed how we can use a Z-valuation to make decisions. We investigated two approaches to decision making with Z-information. The first approach is based on reducing of Z-numbers to classical fuzzy numbers, and generalization of expected utility approach and use of Choquet integral with an integrant represented by Z-numbers. A fuzzy measure is calculated on a base of a given Z-information. The second approach is based on direct computation with Z-numbers. To illustrate a validity of suggested approaches to decision making with Z-information the numerical examples were used.