Fuzzy relations have been widely applied in decision making process. However, the application process requires people to have a high level of ability to compute and infer information. As people usually have limited ability of computing and inferring, the fuzzy relation needs to be adapted to fit the abilities of people. The bounded rationality theory holding the view that people have limited rationality in terms of computing and inferring meets such a requirement, so we try to combine the fuzzy relation with the bounded rationality theory in this study. To do this, first of all, we investigate four properties of fuzzy relations (i.e. reflexivity, symmetry, transitivity and reciprocity) within the bounded rationality context and find that these properties are not compatible with the bounded rationality theory. Afterwards, we study a new property called the bounded rational reciprocity of fuzzy relations, to make it possible to combine a fuzzy relation with the bounded rationality theory. Based on the bounded rational reciprocity, the bounded rational reciprocal preference relation is then introduced. A rationality visualization technique is proposed to intuitively display the rationality of experts. Finally, a bounded rationality net-flow-based ranking method is presented to solve real decision-making problems with bounded rational reciprocal preference relations, and a numerical example with comparative analysis is given to demonstrate the advantages of the proposed methods.

The fuzzy relation (Zadeh,

Usually, the determination of a fuzzy preference relation requires people to have enough skills in particular aspects. For instance, the fuzzy preference ordering requires people to obey the property of transitivity. The weak transitivity means that, if A is better than B and B is better than C, A should be better than C. Constructing a fuzzy preference ordering needs people to have the perfect ability of computing and inferring data. There are technologies to repair the fuzzy preference ordering which does not obey the transitivity (Saaty,

In this study, the usability of the four properties of a fuzzy preference relation is firstly discussed under the bounded rationality situation. It is found that all four properties do not satisfy the bounded rationality. Because the transitivity might be violated in real decision-making problems (Świtalski,

The contributions of this study are highlighted as follows:

We introduce the bounded rational reciprocity. Since the reflexivity, symmetry, transitivity, and reciprocity do not suit the bounded rationality theory, we propose the rationality value to define a new property called the bounded rational reciprocity. Compared with the transitivity and reciprocity, the bounded rational reciprocity explains rather than restricts the membership degree, which reduces the difficulty of collecting information and improves the flexibility of decision making.

The bounded rational reciprocal preference relation is proposed to model the limited ability of people. By combining the idea of the bounded rational reciprocity and fuzzy preference relation, we introduce the bounded rational reciprocal preference relation. The rationality radius is presented to explain experts’ rationality. The rationality visualization technique is introduced to intuitively display the rationality of experts.

A bounded rationality net-flow-based method is presented to rank alternatives in decision-making problems. With the weights of experts, the aggregated bounded rational reciprocal preference relation is calculated. The positive and negative bounded rationality flow is introduced to get the bounded rationality net flow, which can be further used to rank alternatives. A numerical example is given to demonstrate the bounded rationality net-flow-based decision-making method. Comparative analyses with the reciprocal preference relation and non-reciprocal preference relation are given to show the advantages of the proposed decision-making method.

This study is organized as follows: In Section

For the convenience of presentation, in this section, the relevant theories are introduced.

To analyse the incompatibility between fuzzy preference relations and the bounded rationality, this section introduces the fuzzy preference relation and its developments. We begin with the concept of binary relation.

For two sets of evaluations on alternatives

In the above Example

For fuzzy relations, they have four properties, namely, reflexivity, symmetry, transitivity, and reciprocity (Zadeh,

Different combinations of these four properties yielded different developments of the fuzzy relation. These developments could be divided into two categories. Regarding the first group, Zadeh (

The relations of the developments of the fuzzy relation.

Among the developments of fuzzy relations, the fuzzy preference ordering attracted the attention of many researchers. Different kinds of uncertainty were considered to improve the fuzzy preference ordering. Xu (

In the traditional economic theory, there is a postulate of economic man, that experts are familiar with, who has related knowledge and a stable system of preference (Simon,

To make the postulate of economic man compatible with experts’ abilities, Simon (

Motivated by the bounded rationality theory, a mass of researches has been done, which can be grouped into two categories. The first group considered probability models with the bounded rationality theory. Mattsson and Weibull (

For the existing researches on fuzzy preference relations under the bounded rationality circumstance (Wang and Fu,

As mentioned in Section

Different relations “

The symmetry means that the membership degree of

There is a mass of rules concerning transitivity which requires experts to have a stable preference system (Zadeh,

According to the analyses in Section

Most of the time,

For two alternatives

For a fuzzy relation, the membership degree

Transitivity was usually used to judge experts’ rationality. If the preference did not satisfy the transitivity, some consistency adjustment methods (Herrera-Viedma

Compared with the transitivity and reciprocity, the bounded rational reciprocity tries to explain rather than restrict membership degrees. As the bounded rational reciprocity does not limit the value of

Because the reflexivity, symmetry, transitivity and reciprocity do not satisfy the bounded rationality, the existing preference relations reviewed in Section

For a given alternative set

The bounded rational reciprocal preference relation on an alternative set

In practical decision-making processes, experts may have different understandings about different alternatives, causing various

For a given alternative set

The developments of fuzzy relations based on the reciprocity, like the fuzzy preference ordering (Tanino,

The membership degrees in the bounded rational reciprocal preference relation can be represented by the triples

For all pairs

However, in the 3D coordinate system, the midpoints are not easy to be visually distinguished, so it is necessary to make a transformation. Let

The matrix in Example

The 3D coordinate system of the bounded rational reciprocal preference relation in Example

In Fig.

The rationality chart of Example

To use the bounded rational reciprocal preference relation in order to solve decision-making problems, we propose a bounded rationality net-flow-based method to rank alternatives.

Before proposing the bounded rationality net-flow-based ranking method, we first make a description of the decision making framework. There are

If the rationality of expert

Then,

For alternatives

For an alternative

For any

This section gives a numerical example to illustrate the bounded rationality net-flow-based ranking method. Suppose that three experts

In

The rationality chart of

From Fig.

By Eq. (

In this section, we make a comparative analysis to demonstrate the advantages of the bounded rationality net-flow-based ranking method and the bounded rational reciprocal preference relations.

We use the aggregated bounded rational reciprocal preference relation

Then, we use the net flow to rank the alternatives (Fodor and Roubens,

The result here is the same as the ranking obtained in Section

Parreiras

After that,

Afterwards, we remove

In the non-reciprocal preference relation (Parreiras

Compared with the bounded rationality net-flow-based ranking method, the multi-stage decision-making method here is time-consuming because it decides only one alternative at one stage. Besides, the weights in the multi-stage decision-making method are given by the decision-maker, which are not objective. For the multi-stage decision-making method, a small change on weights might influence the final ranking. For instance, if we do a small change on the weights

From the comparative analyses in Section

Compared with the reciprocal preference relation, the bounded rational reciprocal preference relation does not need to change the original preference information when expressing the information given by experts.

In the non-reciprocal preference relation, one of the membership and non-membership degrees should be 0 or 1, which has restrictions on preference relations. On the contrary, the bounded rational reciprocal preference relation allows experts to give preference information without restrictions, so it is convenient for experts to give the preference information in the bounded rational reciprocal preference relation.

Compared with the multi-stage decision-making method, the bounded rationality net-flow-based ranking method can get the final ranking of alternatives with one computation.

The ranking reversal says that when we add an alternative to the decision making problem, the ranking of the previous alternatives might be reversed (Belton and Gear,

Using the bounded rationality net-flow-based ranking method, the final ranking is

The four properties of fuzzy relations (i.e. reflexivity, symmetry, transitivity, and reciprocity) were not compatible with the bounded rationality theory. In this regard, the rationality value and rationality radius were introduced in this paper to explain experts’ rationality levels. Then, the bounded rational reciprocity of fuzzy relations was proposed. Based on this property, the bounded rational reciprocal preference relation, which can explain the rationalities of experts, was defined. A rationality visualization technique and a bounded rationality net-flow-based ranking method were given to help experts in using the bounded rational reciprocal preference relation. Comparative analysis showed the advantages of the bounded rational reciprocal preference relation and the bounded rationality net-flow-based ranking method.

There are still some unsolved issues. This paper focuses on the theory study of the bounded rational reciprocal preference relation. The reasonableness of the proposed theory needs to be certified by practical examples. The bounded rational reciprocal preference relation can be considered in other multiple criteria decision-making methods. Moreover, how to avoid the ranking reversals in the bounded rationality net-flow-based ranking method is an interesting research question that can be investigated in the future.