Journal of Food Quality

Volume 2018, Article ID 2760907, 9 pages

https://doi.org/10.1155/2018/2760907

## Estimation of Food Security Risk Level Using *Z*-Number-Based Fuzzy System

^{1}Department of Computer Engineering, Applied Artificial Intelligence Research Centre, Near East University, Northern Cyprus, Mersin-10, Turkey^{2}Department of Economics, Near East University, Northern Cyprus, Mersin-10, Turkey

Correspondence should be addressed to Rahib H. Abiyev; rt.ude.uen@veyiba.bihar

Received 1 February 2018; Revised 4 April 2018; Accepted 16 April 2018; Published 9 May 2018

Academic Editor: Antonino Malacrinò

Copyright © 2018 Rahib H. Abiyev 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

Fuzzy logic systems based on If-Then rules are widely used for modelling of the systems characterizing imprecise and uncertain information. These systems are basically based on type-1 fuzzy sets and allow handling the uncertain and imprecise information to some degree in the developed models. Zadeh extended the concept of fuzzy sets and proposed *Z-*number characterized by two components, constraint and reliability parameters, which are an ordered pair of fuzzy numbers. Here, the first component is used to represent uncertain information, and the second component is used to evaluate the reliability or the confidence in truth. *Z-*number is an effective approach to solving uncertain problems. In this paper, *Z-*number-based fuzzy system is proposed for estimation of food security risk level. To construct fuzzy If-Then rules, the basic parameters cereal yield, cereal production, and economic growth affecting food security are selected, and the relationship between these input parameters and risk level are determined through If-Then fuzzy rules. The fuzzy interpolative reasoning is proposed for construction of inference mechanism of a *Z-*number-based fuzzy system. The designed system is tested using Turkey cereal data for assessing food security risk level and prediction periods of the food supply.

#### 1. Introduction

In the real world, the data are often imperfect because of their unreliability and nature. The complementary aspects of imperfect information are uncertainty and imprecision [1, 2]. Uncertainty characterizes the degree of truth, and imprecision characterizes the content of the data. Fuzzy set theory, introduced by Zadeh [1], can be used to deal with these factors in the modelling of the systems. Therefore, different industrial and nonindustrial problems that were characterized by uncertainty and imprecision were solved by means of the fuzzy set theory [2]. In this paper, the application of fuzzy sets to measurement of food security is considered.

The food security, as it was defined by United Nations’ Committee on World Food Security, is a social, physical, and economic access to the sufficient nutritious food that meets the needs of the pupil and also food preferences for an active and healthy life. Food security indicators are built on four pillars: availability, access, utilization, and stability [3]. Food security is highly affected by various factors such as a growing global population, changing the climate, rising food prices, and environmental stressors [4–6]. The possibility of negative actions causes a hazard and increases risk level of food security. The analyses of different factors having influences on food security have been done in different research works. Wu.et al. [6] combined the social, economic, and biophysical factors for estimation of the potential risks of food security. The two indicators, per capita food availability and per capita gross domestic product, were used to estimate the food availability, stability, accessibility, and affordability and to assess the potential future risk of food insecurity. Ehrlich et al. [7] examined two views on the causes of global food insecurity: biophysical and demographic. They concluded that the achieving food-secure future for all humanity is positively influenced by the enhancement of equality, the reduction of the human fertility, and improvement of the ecosystem health. Hubbard et al. [5] examined a number of factors that contribute to the food security. Wang et al. [8] designed a prediction model through the combination of the stepwise regression method with BP neural network for forecasting the total food yield. Xiao et al. [9] used BP neural network and considered the development of risk early warning system for food security under the supplied chain environment.

The impacts of factors affecting food security are imprecise, and it is difficult to describe them with certain numbers. For this reason, the fuzzy set theory is a very effective way of estimation of food security risk level. Uncertain availability of nutritious and safe foods is also called food insecurity that can be measured by risk level of food security. Nowadays, the fuzzy set theory is one of the important tools for determination of the risk level of food security and also for risk management [10–16]. Wang et al. [10] considered the combination of the fuzzy set theory and analytical hierarchy process for a risk assessment. Lee [11] evaluated the rate of aggregative risk by fuzzy set theory. Using fuzzy set theory and Dempster–Shafer evidence theory, Xu et al. [12] proposed a fuzzy evidential grain security warning method for food management. In the research works [10–14], using various factors the design of fuzzy rules is performed for management of the risk. Kadir et al. [15] designed a fuzzy system for assessment of food security risk level. The designed system is based on fuzzy If-Then rules that used three important factors cereal production, cereal yield, and economic growth for the determination of food security risk level. The values of these factors are estimated linguistically using fuzzy values. These fuzzy rules are developed using the knowledge of human experts. In the above research papers, the premise and the consequent parts of the fuzzy If-Then rules are basically designed from numeric data sets or linguistic information using type-1 membership functions. Abiyev et al. [16] designed the type-2 fuzzy system for assessment of food security risk level. Jiang et al. [17] presented the failure mode and effect analysis model based on a fuzzy evidential method for evaluation of the risk of failure modes. Ranking of the risk of the failure model is presented by fusing the occurrence, severity, and detection information and Dempster–Shafer theory.

The fuzzy knowledge-based systems used for decision-making are basically based on knowledge, experience, intuition, and assumption of humans. Sometimes, these qualities cannot completely cover all the complexity of the considered real-world problem. The modelling of fuzzy uncertainty inherent to the perception of human experts in evaluating the parameters of input and output variables of food security is very important in modelling the decision-making process. The fuzziness and reliability are associated with each other during evaluation of uncertainties related to the fuzzy values of input-output variables. The *Z-*number proposed by Zadeh allows modelling such kind of uncertainty [18]. As we know, the concepts in the human brain for evaluating natural events are vague and imprecise. The boundaries of the parameters low, average, or high level used for evaluating input-output variables cereal yield, cereal production, economic growth, and security level are not exactly defined. Therefore, the assertion that follows from them also becomes vague. The usage of *Z-*numbers allows estimation of input and output relationships using the concept of fuzzy information and partial reliability. Zadeh proposed *Z-*number to represent uncertain information using constraint and reliability information. *Z-*number as represented uses an ordered pair of fuzzy numbers (, ). Here, is the fuzzy restriction, and is the reliability used for valuation of [18]. *Z-*number allows us to use uncertainty measures to estimate the ambiguity associated with the estimating food risk security. The theoretical background of *Z-*number-based systems has been considered in [18, 19]. *Z-*number-based fuzzy set using fuzzy constraints and reliability factors describe the human knowledge. Based on the concept of *Z-*number, different research works have been done for solving different problems. Aliev et al. [20] and Kang et al. [21] used *Z-*number to solve multicriteria decision-making (MCDM). Kang et al. [21] solved the decision-making problem by converting *Z-*number to crisp numbers using the approach given in [22]. Xiao et al. [23] converted *Z-*number to type-2 fuzzy set to solve multicriteria decision-making. Here, by calculating the centroid type-2 fuzzy sets, it is converted to the crisp numbers for decision-making. Azadeh et al. [24] applied *Z-*number to solve the AHP. The research paper uses the approach described in [22] for problem-solving. Lorkowski et al. [25] considered a fair price approach for decision-making under interval, set-valued, fuzzy, and *Z-*number-based uncertainty. Abiyev et al. [26] proposed a *Z-*number-based interpolative reasoning for control of the dynamic plant. *Z-*numbers are used to solve the problems related to computing with words [27] and decision-making [28] problems. In [29–31], using discrete fuzzy numbers, a new vision of *Z-*numbers is described. An aggregation method is presented for group decision-making problems. In [32], a *Z-*number version of the data envelopment analysis (DEA) model is transformed into possible linear programming and then by applying an alternative -cut approach, a crisp linear programming model is obtained. The proposed model is used for a portfolio selection problem. In [33], a *Z-*number-based risk-minimization negotiation model is designed for a transmission company and a power purchaser under incomplete information. Here, *Z-*number is used to estimate the uncertainty distribution of the annual electricity transmission, and the benefit and the loss measured by the conditional value at risk is analyzed. Aliev et al. [34] presented an approximate reasoning with *Z-*rules on a basis of linear interpolation for evaluation of the job satisfaction and educational achievement of the students. Wang et al. [35] represented experts’ opinions by *Z-*numbers and presented a method based on the Choquet integral for MCDM problems using linguistic *Z-*numbers. Wu et al. [36] used experts’ opinions for representing a method for ranking *Z-*numbers. Based on this ranking method, the transformation of *Z-*numbers into basic probability assignments is presented. Two experiments on risk analysis and medical diagnosis illustrate the efficiency of the proposed methodology. In [37], the total utility of *Z-*number is applied to determine the ordering of *Z-*numbers. The approach is used in the application of MCDM under uncertain environments, and it is implemented using Gaussian and triangular *Z-*numbers. Jiang et al. [38] presented a method for ranking generalized fuzzy numbers. Here, the weight of centroid points, the spreads of fuzzy numbers, and degrees of fuzziness are taken into consideration. In this paper, the principles of ranking *Z-*numbers are considered. Using intuitive vectorial centroid, Ku Khalif et al. [39] presented a hybrid fuzzy MCDM model for *Z-*numbers. The presented model is applied to staff recruitment. Aliev et al. [40] presented the ranking of *Z-*numbers using a human-like fundamental approach. The considered approach is based on two main ideas: optimality degrees of *Z-*numbers and adjusting the obtained degrees using a degree of pessimism. Kang et al. [41] presented the stable strategies analysis based on the utility of *Z-*number to simulate the human’s competition in the evolutionary games.

The use of *Z-*numbers in the solution of different problems needs to use efficient inference mechanism. The interpolative reasoning is proposed by Koczy and Hirota [42, 43] for the sparse fuzzy rule base. The method can provide the logical interpretation of modus ponens. The proposed method is based on distance measure and used for approximating fuzzy reasoning [42, 43]. Johanyák et al. [44] showed that different distance measures can be used in the fuzzy sets. But these distance measures do not give full information about the shape of the membership functions. Kozcy et al. [43] mentioned that the Kozcy measure based on -cuts of two fuzzy sets can be used to solve this problem. Using -cuts, Koczy and Hirota proposed a fuzzy interpolative approximate reasoning [43]. Kovács et al. [45] and Hsiao et al. [46] used the interpolative reasoning for the solution of different practical problems. The novelty of this paper is emphasized in the following stages: using -cuts and Koczy and Hirota interpolative reasoning, the design of *Z-*number-based interpolative reasoning mechanism is presented; the development of the *Z-*number-based fuzzy inference system for estimation of food security risk level is proposed.

The paper is organized as follows: Section 2 presents the fuzzy rule interpolation. Section 3 describes the design *Z-*number-based fuzzy inference system. Section 4 presents the application of the developed *Z-*number-based interpolative reasoning mechanism for estimation of food security risk level. Section 5 presents the conclusions.

#### 2. Fuzzy Rule Interpolation

Fuzzy sets are the extension of classical sets whose elements have degrees of membership. Let’s give definition of fuzzy sets and fuzzy triangular membership functions that will be used in this paper.

*Definition 1*. Assume that is the universe of discourse. A fuzzy set in is defined as a set of ordered pairs . Here, is the membership function of , and is the degree of membership of in [1].

*Definition 2*. A triangular fuzzy number is a fuzzy subset defined by triplet on , where the membership function can be determined asThe fuzzy triangular membership function is often used for the numeric representation of linguistic terms. In the paper, triangular membership functions are used for the representation of the constraint and reliability of *Z-*number.

Fuzzy rule-based systems include fuzzy sets in antecedent and consequent parts of the rules and define relationships between input and output of the system. Using fuzzy rule base and input data, the fuzzy inference is implemented for making output decision. Different fuzzy reasoning mechanisms are suggested to process uncertain information. These mechanisms are mainly based on the analogy and similarity, compositional inference rule, interpolation, and the concept of distance. The processing capabilities, speed, and complexity of these inference mechanisms are important issues.

This paper considers the inference mechanism based on fuzzy interpolative reasoning proposed by Koczy and Hirota [42, 43]. The fuzzy interpolative reasoning can efficiently be used for processing sparse rule base and requires the satisfaction of the following conditions: the used fuzzy sets should be continuous, convex, and normal, with bounded support.

The interpolative reasoning is based on distance measure between two fuzzy sets. In the paper, the -cut is used to measure distance between fuzzy sets. Let’s consider two fuzzy sets: and . -cut of fuzzy sets and will be denoted as and . We say that fuzzy sets is less than , that is, < , ifwhere inf and inf are infimum of and , and sup and sup are supremum of and (Figure 1).

Let us consider interpolative reasoning mechanism. Consider the fuzzy controller based on single-input and single-output fuzzy rule base. Assume that, in the result of observation, the current input variable is . Let us determine the value of output of the rule base system corresponding to . Assume that is less than fuzzy sets and more than the fuzzy set , that is, < < and also < . Let us determine the system output for the input . The problem can be expressed as follows:In [42, 43] it was shown thatwhere is the distance between two fuzzy sets.

Koczy and Hirota [42, 43] determined the final fuzzy sets using distance based on cuts. Using , cut lower and upper distances between two fuzzy sets can be calculated as follows:In (5), the Hamming or Euclidean formula can be used to measure the distances. Using distance measure, Koczy and Hirota [42, 43] proposed interpolated conclusion for the output of 2k rules.where . In Section 3, we consider the use of interpolative reasoning for finding the output of the *Z-*number fuzzy system.