The Scientific World Journal

Volume 2015 (2015), Article ID 831396, 9 pages

http://dx.doi.org/10.1155/2015/831396

## Description and Application of a Mathematical Method for the Analysis of Harmony

College of Water Conservancy and Environment, Zhengzhou University, No. 100, Science Road, Zhengzhou 450001, China

Received 18 September 2014; Revised 9 March 2015; Accepted 30 March 2015

Academic Editor: Haibo He

Copyright © 2015 Qiting Zuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Harmony issues are widespread in human society and nature. To analyze these issues, harmony theory has been proposed as the main theoretical approach for the study of interpersonal relationships and relationships between humans and nature. Therefore, it is of great importance to study harmony theory. After briefly introducing the basic concepts of harmony theory, this paper expounds the five elements that are essential for the quantitative description of harmony issues in water resources management: harmony participant, harmony objective, harmony regulation, harmony factor, and harmony action. A basic mathematical equation for the harmony degree, that is, a quantitative expression of harmony issues, is introduced in the paper: , where is the uniform degree, is the difference degree, is the harmony coefficient, and is the disharmony coefficient. This paper also discusses harmony assessment and harmony regulation and introduces some application examples.

#### 1. Introduction

With the exception of “goodwill competition,” living in harmony (in terms of the relationships between people) is recommended, and the resulting community of people living in harmony is often called a “harmony society,” “harmony community,” “harmony city,” “harmony home,” and “harmony team.” From the point of view of relationships between humans and nature, it is impossible for human beings to dominate nature because people would be forced to live in harmony with nature as a result of a nature counterattack. Therefore, there is no doubt that human beings and nature should be harmonious.

When the word “harmony” is mentioned, it is often associated with the word “games.” Game theory is concerned with the behavior of absolutely rational decision makers with unlimited capabilities for reasoning and memorization [1]. Games are defined mathematical objects that consist of a set of players, a set of strategies (i.e., options or moves) that are available to the players, and a specification of the payoff that each player receives for each combination of strategies (i.e., possible outcomes of the game) [2]. Game theory has been used in a variety of fields, and it includes many contents in each field. For example, in water resources research, it reflects in lots of ways, including allocation of water resources [3, 4], water rights [5], water resources development [6], optimal allocation of water resources [7–9], problems of water environment [10], water resources management [11–13], and water conflicts [14, 15]. Game theory is used to represent the “struggle or competition” phenomenon and can be frequently encountered in practice, such as bargaining, offensive and defensive battles, horse racing, and auctions. However, it is insufficient just considering the games. Games can only be used to represent a struggle or competitive phenomenon. In contrast, it is necessary to build a harmony balance in many situations, and game theory cannot be applied for common harmony issues. In addition, there are some extraordinarily difficult problems, such as the “tragedy of the commons” [16, 17], which cannot be solved by game theory alone.

In game theory, “the tragedy of the commons” has been mentioned in the literature through various expressions, but the meaning is basically the same. The “tragedy of the commons” roughly means as follows: if there is a set piece of grassland that is shared by two homes for sheep grazing, the total number of sheep is limited due to the limited grass. From the point of view of the individual, a home that raises more sheep will have a better profit. To maximize his/her profits, each individual attempts to increase his/her number of sheep, which results in an increasingly high number of total sheep and thus an increasingly excessive use of the grass. This excess leads to grassland degradation and even destruction, that is, the “tragedy of the commons.” Therefore, in some cases, it is insufficient to only consider game theory; there is a need to consider harmony issues in these cases. As a result, harmony theory should also be established.

This paper has three objectives: to introduce the concepts of harmony theory and the five essential elements of harmony theory in water resources management based on the above analysis and previous studies [18]; to discuss the mathematical description of harmony theory by proposing a function for the harmony degree, introducing a mathematical approach for the assessment of harmony, and developing a method for harmony regulation; to illustrate the mathematical description of harmony by a series of typical examples.

#### 2. Concepts

Although the word “harmony” is widely used, a unifying concept has not yet been defined. Harmony in this paper is defined as follows: harmony is the action taken to achieve “coordination, accordance, balance, integrity, and adaptation.” Because people rely on nature to survive, it is necessary for human society to live in harmony with nature.

The theory and methodology of studies on harmony behavior are termed harmony theory, which is further defined as follows: harmony theory is a method through which various participants work together to achieve harmony. Harmony theory, which is of broad application prospect, is a significant theory that reveals the harmonious relationships in nature and is also a concrete manifestation of dialectical materialism on the assertion of “the coordinated development between humans and nature.” Firstly, it should be recognized that “harmony is an important concept in addressing interpersonal relationships and relationships between humans and nature, and it is also a major guarantee and a concrete manifestation to build a harmony society, harmony community, harmony team, and harmony nature.” Secondly, it is important to gradually establish the concept of harmony and adhere to the ideological philosophy of harmony. In addition, humans should take the initiative to coordinate the marvelous relationships between people, which is the basis for the coordination of relationships between humans and nature. Furthermore, it is a new theory, and it can provide an appropriate pathway for water resources management in China [19]. The main arguments of harmony theory are the following.(1)Harmony theory advocates the philosophy that “harmony is the most precious” to address a variety of relationships, and harmony ideology is the cornerstone of harmony theory.(2)Harmony theory advocates a rational understanding of various contradictions and conflicts existing in various types of relationships, allowing the existence of differences and promoting a harmonious attitude to address various factors of disharmony and problems. Instead of ignoring the disharmony factors, it is necessary to consider all of the harmony factors and disharmony factors.(3)Harmony theory advocates the concept of harmony between humans and nature and has very pronounced views on the coordinated development of these relationships. It asserts that human beings should take the initiative to coordinate the marvelous relationships among people. There is a possibility to achieve the coordination of the relationships between humans and nature based on this theory.(4)Harmony theory adheres to the system perspective by promoting system-wide theoretical methods to study the issues of harmonious relationship.

#### 3. Five Factors of Harmony Theory

To obtain a reasonable expression of harmony and a quantitative description of the harmony degree, the following five elements, which are the “five essential factors of harmony theory,” need to be defined [18].

*(1) Harmony Participant*. The term “harmony participant” refers to the parties (generally two or more) involved in the harmony relationship, which are known as “the harmony party.” The collection of harmony participants can be represented as , where is the number of participants in the harmony party, which is also named “-participant harmony.” For a certain harmony party, this variable can be expressed as . For instance, the participants of a harmonious couple are the two spouses, and the harmony participants of a family are all of the family members.

*(2) Harmony Objective*. This term refers to the target that the harmony participants have to achieve a state of harmony. If not, it is impossible to arrive at a state of harmony. In addition, attaining this goal might only lead to a partial state of harmony. For example, if there are families sharing a piece of meadow for sheep, it is imperative to ensure that the total number of sheep does not exceed a certain amount (i.e., stocking rate) to avoid grass damage; the certain amount is thus the harmony target of the households that share a piece of grassland.

*(3) Harmony Regulation*. This term refers to all of the rules or constraints established by the participants for the purpose of achieving the harmony goals. For example, in order to ensure rationality, a harmony regulation for the abovementioned households sharing a piece of grassland could be that the amount of the increase in sheep for each household should be proportional to their population. Thus, according to the conditions of these harmony rules, it is appropriate to study harmony problems.

*(4) Harmony Factor*. This term refers to the factor that should be considered by harmony participants to achieve overall harmony. Its collection is represented as , where the th harmony factor is and the total number of factors is . When , it indicates single-factor harmony, and the harmony factor can be directly expressed as . If , the harmony relationship is called multiple-factor harmony.

*(5) Harmony Action*. The term “harmony action” refers to the general name of the concrete behavior of the harmony participants for the harmony factors. For example, if households jointly own a field of grass, the specific action is the quantity of sheep that are raised on that land. The collection of harmony actions taken by the participants in the -participant harmony and the harmony factors can be expressed as a matrix:

A single-factor harmony action is represented as .

#### 4. Calculation of the Harmony Degree

The harmony degree is used for the quantitative expression of the harmony degree [18]. In this section, the harmony degree equation of a given factor will be introduced, (i.e, Zuo-harmony degree equation). Then, the calculations of the harmony degree in multifactor harmony and multilevel harmony will be discussed.

##### 4.1. Harmony Degree Equation of a Factor

The harmony degree of a given factor is defined by the following equation:where is the harmony degree corresponding to a certain factor and . A higher value of (closer to 1) indicates a higher harmony degree. If the result of (2) shows that , then is set to 0.

The variables and are the unity degree and the difference degree, respectively. The unity degree expresses the proportion of harmony participants in accordance with harmony rules with the same goal. The difference degree is the expression of the proportion of harmony participants with divergent harmony rules and goals. Note that , , and . In the presence of “neither unity nor differences” (i.e., “waiver” phenomenon), ; otherwise, . If the harmony actions of a given factor in -participant harmony are “,” it is assumed that the harmony actions of the -participant harmony with the same target are “”; thus, . If there is no waiver, then . For example, if the harmony rule is = 2 : 1 and and are 100 and 40, respectively, then and equal 80 and 40, respectively, , and . If and are 100 and 80, respectively, then and equal 100 and 50, respectively, , and .

The variable , which is the harmony coefficient, represents the satisfaction degree of the harmony goals and can be determined based on the calculation of the harmony goals, . If the harmony goals are absolutely achieved, then . In contrast, if the goals are not achieved, then . The harmony coefficient curve or function can be determined based on the satisfaction degree.

The variable , which is the disharmony coefficient that reflects the divergent harmony participants, can be calculated and determined according to the difference degree. Note that . If the harmony participants are completely opposed, then . In contrast, if the harmony participants are not opposed, then . In all other cases, the value of is within the range of 0 to 1. The disharmony coefficient curve or function can be determined based on the difference degree; that is, the disharmony coefficient depends on the extent of opposition.

In single-factor harmony (i.e., ), the harmony degree equation is expressed as the following equation:

##### 4.2. Harmony Degree Equation for Multifactor Harmony

If there are a number of factors in a harmony problem, a comprehensive multifactor harmony degree should be calculated based on the single-factor harmony degree. This can be accomplished through two methods: weighted average calculation and exponential weighted calculation.

###### 4.2.1. Weighted Average Calculation

Consider the following:where is the comprehensive harmony degree, , is the weight of each harmony degree, , and . The other variables have the same definition as above.

###### 4.2.2. Exponential Weighted Calculation

Consider the following:where is the index weight of each harmony degree, , and . The other variables have the same definition as before.

##### 4.3. Calculation of Multilevel Harmony Degree

There are complex multilevel harmony problems in real life, and a higher-level harmony problem (i.e., a more comprehensive harmony problem) includes or implies a set of lower-level harmony problems (i.e., single harmony problems). Therefore, the calculation of the harmony degree of harmony problems with different levels is essential. Figure 1 shows a harmony problem with two levels. The first level is the highest and the harmony degree is HD, and the second level is a lower level that includes several harmony problems, which are expressed as ( is the number of second-level harmony problems). Each lower-level harmony problem has corresponding indexes; that is, the indicators of HD_{21}, HD_{22}, and are , and , respectively.