# An Application of Possibilistic Moments of Nonlinear Type of Fuzzy Numbers in Supply Chain Management

Authors: Srimantoorao S. Appadoo; Yuvraj Gajpal; Amit Kumar
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The supply chain (SC) network is prone to disturbance due to various uncertainties associated with their subsystems. The COVID-19 outbreak has exposed the global vulnerability of the supply chain network. The current pandemic has severely affected almost every SC network because its members are situated at the international level. One of the reasons for SC network failure is the deterministic assumptions of different parameters. A realistic SC network model requires the use of the uncertain value of the parameters, which can be further captured by fuzzy numbers. This paper discusses the possibilistic moment of several nonlinear types of fuzzy numbers that are important for SC network modeling. We give closed-form possibilistic moments’ expression for various types of fuzzy numbers that are very similar to the moment’s properties in probability theory and stochastic process. We then illustrate the application of proposed fuzzy numbers by solving an inventory model. This paper also provides results related to the EPQ inventory model in a fuzzy possibilistic setup.


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## Body

## 1. Introduction

The COVID-19 pandemic has exposed the vulnerability of many organizations, especially the supply chain network that is heavily dependent on China. The improper management of the supplier chain network has left many companies without an adequate supply of raw materials or with unsold finished products. One of the reasons for supply chain failure is the traditional consideration of deterministic parameters, despite real-world supply chain management actually containing uncertainty in the model parameters. The uncertainties in different supply chain activities from raw material suppliers to the final customers make the supply chain network imprecise [1]. Thus, many researchers have implemented the fuzzy set theory to model various aspects of supply chain management. Some of the examples of fuzzy set application in supply chain management are supply chain innovation enabling process [2], the supplier selection decision [3, 4], the vehicle routing determination process [5], the domino effect of supply chain resilience [6], quality design process [7], etc. According to Zardani et al. [1], the system designers can use fuzzy set concepts to capture the uncertainty in a real-world environment. One of the major advantages of using fuzzy set theory is its simplicity and ability to deal with a nonlinear function of arbitrary complexity [1]. In light of this observation, this paper presents a possibility framework for supply chain decision-making problems. First, the paper provides the possibilistic moment of different fuzzy numbers and then illustrates the application of proposed fuzzy numbers through the economic production quantity (EPQ) inventory model.The fuzzy set theory has been applied in decision-making, often in the discipline of management. The interest in fuzzy set theory is growing because of its proximate use in capturing real-world problems. In the literature, fuzzy algebra-based possibility theory has been used in various decision-making environments [8–11]. Carlsson and Fuller [9] studied possibility theory and provided closed-form expression for possibilistic mean, possibilistic variance, and possibilistic covariance of fuzzy numbers. Fuller and Majlender [12] provided the concept of crisp weighted possibilistic moments of fuzzy numbers. Thavaneswaran et al. [13] and Thavaneswaran et al. [11] extended those concepts and provided expressions for crisp possibilistic rth moments of fuzzy numbers. Thavaneswaran et al. [11] applied the crisp possibilistic rth moments to dynamic time-series models and the option pricing models. Gu and Xuan [14] introduce a new method of the ranking method based on possibility theory, and some examples are given to illustrate the advantages of the method.In recent times, possibility theory has been applied to many decision-making processes. Appadoo et al. [15] derived possibilistic moments of a special type of fuzzy numbers. They showed that the linear trapezoidal and the linear triangular fuzzy numbers are special cases of the nonlinear fuzzy numbers. They also provided examples of weighted possibilistic moments for a nonlinear class of trapezoidal fuzzy numbers using some form of nonnegative weight function. Wang et al. [16] successfully applied possibility theory to a fuzzy control chart. Tsaur [17] used possibilistic mean-standard deviation theory in a portfolio selection model. An illustrative numerical example is provided towards the end of the paper to illustrate the proposed portfolio selection model. Rabbani et al. [18] evaluated the sustainability performance of suppliers using the interval-valued fuzzy group decision theory. They provided separation measure matrices using possibility theory. Verma et al. [19] used possibility theory to determine an optimal maintenance policy. They provided a detailed algorithm for the replacement maintenance interval optimization model. Borges et al [20]. presented a scenario-based real option valuation method using possibility distributions. Chen and Tan [21] applied possibility theory to the portfolio selection with an uncertain time horizon. Zhang et al. [22] introduced the concepts of possibilistic efficient portfolios using possibility distributions. They applied possibility theory to the multiperiod portfolio selection problem in which the degree of portfolio diversification is measured using the possibilistic entropy. Li et al. [23] used possibilistic moments in a portfolio selection model with value at risk constraint. An empirical study based on the Shanghai Stock Exchange validated the results.Some of the most essential studies into fuzzy inventory models have recently been conducted. Masae et al. [24] presented a systematic review of order picking routing models in warehouses. Ali et al. [25] considered a multiobjective inventory optimization model with intuitionistic fuzziness in the model parameters. They used a fuzzy goal programming-based approach to solve the proposed inventory model. In a recent article, Gautam et al. [26] performed an extensive literature review that considered the effect of imperfect quality items in inventory replenishment or production models. The findings offered state-of-the-art insights to scientists and business professionals in supply chain management. Similarly, Janssen et al. [27] provided a comprehensive literature review of perishable inventory models from January 2012 to December 2015. Ali et al. [28] developed a supply chain optimization model that minimizes the combination of transportation, deliveries, and ordering costs under uncertain situations. Ali et al. [25] used both a multiobjective vendor selection problem under fuzzy environment and a fuzzy goal programming approach in their model formulation. The proposed model in Ali et al. [25] can handle realistic vendor selection problems in a fuzzy setup and can serve as a useful approach for multicriteria decision-making in supply chain management. Medaglia et al. [29] provide intuitive methods to construct membership functions with desired shapes that can be very useful in supply chain applications models. Medasani et al. [30] provide an overview of various techniques for generating membership functions that can be easily tuned. Medasani et al. [30] also discuss the suitability of these membership function generation techniques to some particular situations. Ye and Li [31] consider a fuzzy single-period product inventory control in a distributed supply chain using the weighted possibilistic mean value method to rank the model parameters considered in the model.The motivation of this research is to reconsider the EPQ model using fuzzy sets theory assisted by possibility theory to capture the uncertainty embedded in the model. The EPQ model is illustrated in Figure1. This paper is an extension of the work carried out in [32, 33]. The economic production quantity (EPQ) deterministic inventory model with finite replenishment rate is one of the most commonly inventory model, used by many industries. The EPQ inventory model extends the economic order quantity (EOQ) model rests on crisp value assumptions of different parameters such as the setup cost, the holding cost, demand rate, and production rate. Due to unpredictable market conditions, these assumptions may not hold in practice. This is mainly due to those weaknesses of the current EPQ deterministic inventory models that call upon more realistic fuzzy and stochastic inventory model formulations. The main contributions of this paper are as follows:(i)
Generally, inventory model parameters may be imprecise. Therefore, we use the possibility theory to calculate the order quantity.(ii)
The fuzzy model discussed in this paper can be easily modified to account for multi-items inventory models with imprecise parameters.(iii)
We derived various closed-form expressions for possibilistic moments of different types of fuzzy numbers.(iv)
The fuzzy inventory model discussed in this paper is straightforwardly understandable and can be extended to accommodate nonlinear types of fuzzy numbers.(v)
The proposed fuzzy inventory model does not imply the rejection of other fuzzy inventory formulations, but rather complements the existing fuzzy inventory model formulation.Figure 1
Production vs. inventory level with uniform replenishment [34].This study follows an organizational structure. Section1 provides a detailed literature review on the application of possibility theory and a summary of the preliminaries and notations that will be used in the sequel. Section 2 discusses possibility theory along the line of [9]. In Section 3, we provide a mathematical derivation of possibilistic moments of two nonlinear classes of fuzzy numbers along with the necessary theorems and illustrative examples. Two applications of possibility theory are provided in Section 4 to discuss the applicability of this paper’s theory to weighted functions. Finally, the conclusion and remarks are given in Section 5.


### 1.1. Preliminaries and Notation

In this section, we provide some fuzzy definitions necessary to follow the discussion in the paper. Detail descriptions may be found in [35–39].Definition 1 (see [15]).
A fuzzy numberA=a1,a2,a3,a4Om,n, a1<a2<a3<a4 is said to be Om,n-trapezoidal-type fuzzy number (Om,n-Tr.F.N.) if its membership function is given as(1)μx=0,x≤a1,1−a2−xa2−a1m,a1≤x≤a2,1,a2≤x≤a3,1−a3−xa3−a4n,a3≤x≤a4,0,x≥a4.
Alternatively, following [36], defining the α-cut (interval of confidence at level-α) as Aα=a1α,a2α, we characterize Om,n-Tr.F.N. a1,a2,a3,a4Om,n as(2)Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n,∀α∈0,1,by setting 1−a2−x/a2−a1m=α and 1−a3−x/a3−a4n=α, respectively.
AnOm,n-Tr.F.N. is said to be symmetrical if it satisfied the following two conditions:(a)
a2−a1=a4−a3(b)
m=n
It is important to note that (Om,n-Tr.F.N.) can be easily tuned to desired shapes.Definition 2.
LetA=x:μ,σ,α be a Gaussian fuzzy number; then, the membership function can be written as(3)μAx=exp−12x−μσ2,where μ is the mean value and σ2 is the variance. The confidence interval will be Aα=a1α,a2α and setting will be exp−1/2x−μ/σ2=α; we get a1α=μ−σ−2lnα and a2α=μ+σ−2lnα. Given the above expressions, the α representation of the Gaussian fuzzy number is(4)Aα=a1α,a2α=μ−σ−2lnα,μ+σ−2lnα,∀α∈0,1.


## 1.1. Preliminaries and Notation

In this section, we provide some fuzzy definitions necessary to follow the discussion in the paper. Detail descriptions may be found in [35–39].Definition 1 (see [15]).
A fuzzy numberA=a1,a2,a3,a4Om,n, a1<a2<a3<a4 is said to be Om,n-trapezoidal-type fuzzy number (Om,n-Tr.F.N.) if its membership function is given as(1)μx=0,x≤a1,1−a2−xa2−a1m,a1≤x≤a2,1,a2≤x≤a3,1−a3−xa3−a4n,a3≤x≤a4,0,x≥a4.
Alternatively, following [36], defining the α-cut (interval of confidence at level-α) as Aα=a1α,a2α, we characterize Om,n-Tr.F.N. a1,a2,a3,a4Om,n as(2)Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n,∀α∈0,1,by setting 1−a2−x/a2−a1m=α and 1−a3−x/a3−a4n=α, respectively.
AnOm,n-Tr.F.N. is said to be symmetrical if it satisfied the following two conditions:(a)
a2−a1=a4−a3(b)
m=n
It is important to note that (Om,n-Tr.F.N.) can be easily tuned to desired shapes.Definition 2.
LetA=x:μ,σ,α be a Gaussian fuzzy number; then, the membership function can be written as(3)μAx=exp−12x−μσ2,where μ is the mean value and σ2 is the variance. The confidence interval will be Aα=a1α,a2α and setting will be exp−1/2x−μ/σ2=α; we get a1α=μ−σ−2lnα and a2α=μ+σ−2lnα. Given the above expressions, the α representation of the Gaussian fuzzy number is(4)Aα=a1α,a2α=μ−σ−2lnα,μ+σ−2lnα,∀α∈0,1.


## 2. Possibility Theory

In this section, we discussed the possibilistic moment properties of fuzzy numbers along the line of [9, 41].


### 2.1. Moment Properties of Fuzzy Numbers

As in [9], lower possibilistic mean ELA, upper possibilistic mean ERA, possibilistic mean EA, and interval value possibilistic mean IVPMA=ELA,ERA of a fuzzy number A defined by a1α,a2α, α∈0,1 are given below. It is important to note here that, in order to derive the results in this section, we implicitly use the following results:(5)PossibilityA≤a1α=π−∞,a1α=supu≤a1αAu=α,PossibilityA≥a2α=πa2α,∞=supu≥a2αAu=α.For more detail, an interested reader may refer to [9](6)ELA=∫01PosA≤a1αa1αdα∫01PosA≤a1αdα=2∫01αa1αdα,ERA=∫01PosA≥a2αa2αdα∫01PosA≥a2αdα=2∫01αa2αdα,where Pos denotes possibility and EA is the level-weighted average of the arithmetic means of all α-level sets and is given by(7)EA=∫01PosA≤a1αa1αdα2∫01PosA≤a1αdα+∫01PosA≥a2αa2αdα2∫01PosA≥a2αdα.The interval value possibilistic meanIVPMA of a fuzzy number A is defined as(8)IVPMA=∫01PosA≤a1αa1αdα∫01PosA≤a1αdα,∫01PosA≥a2αa2αdα∫01PosA≥a2αdα.According to Carlsson and Fuller [9], the zero centered possibilistic variance VarA of a fuzzy number A is defined as(9)VarA=∫0112αa2α−a1α2dα.The possibilistic covariance between fuzzy numbersA and B is defined as(10)CovA,B=12∫01αa2α−a1αb2α−b1αdα.Zhang and Nie [42] and Zhang and Xiao [43] defined the lower possibilistic variance of a fuzzy number A as(11)VarLA=∫01PosA≤a1αELA−a1α2dα∫01PosA≤a1αdα=2∫01αELA−a1α2dαand the upper possibilistic variance of a fuzzy number A as(12)VarRA=∫01PosA≥a2αERA−a2α2dα∫01PosA≥a2αdα=2∫01αERA−a2α2dα.Similar to the definition ofEA, the crisp possibilistic variance is as the arithmetic mean of its lower and upper possibilistic variances, i.e.,(13)VarA=12∫01PosA≤a1αELA−a1α2dα∫01PosA≤a1αdα+∫01PosA≥a2αERA−a2α2dα∫01PosA≥a2αdα.Zhang and Nie [42] and Zhang and Xia [43] defined the lower and upper possibilistic covariances between two fuzzy numbers A and B as given in expressions (14) and (15):(14)CovLA,B=∫01PosA≤a1α,B≤b1αELA−a1αERB−b1αdα∫01PosA≤a1α,B≤b1αdα=2∫01αELA−a1αELB−b1αdα,(15)CovRA,B=∫01PosA≥a2α,B≥b2αELA−a2αERB−b2αdα∫01PosA≥a2α,B≥b2αdα=2∫01αERA−a2αERB−b2αdα,and the crisp possibilistic covariances between fuzzy numbers A and B as the arithmetic mean of their lower and upper possibilistic covariances, i.e.,(16)CovA,B=∫01PosA≤a1α,B≤b1αELA−a1αERB−b1αdα2∫01PosA≤a1α,B≤b1αdα+∫01PosA≥a2α,B≥b2αELA−a2αERB−b2αdα2∫01PosA≥a2α,B≥b2αdα.We use the following results as in [22, 42–44]:(17)PosA≤a1α,B≤b1α=supu≤a1α,v≤b1αminAu,Bv=α,PosA≥a2α,B≥b2α=supu≥a2α,v≥b2αminAu,Bv=α.Alternatively, Fuller and Majlender [12] introduced the crisp possibilistic variance and covariance of a fuzzy number A as(18)VarA=∫01αEA−a1α2+EA−a2α2dα.(19)CovA,B=∫01αEA−a1αEB−b1α+EA−a2αEB−b2αdα.Remark 1.
We can define the lower possibilistic variance of fuzzy numberA as follows:(20)VarLA=∫01PosA≤a1αEA−a1α2dα∫01PosA≤a1αdα,where the upper possibilistic variance of fuzzy number A is as follows:(21)VarRA=∫01PosA≥a2αEA−a2α2dα∫01PosA≥a2αdα,Remark 2.
An alternative definition for therth possibilistic moments of a fuzzy number can be formulated as follows:(22)ErA=∫01αELA−a1αr+ERA−a2αrdα.Remark 3.
Along the lines of [42, 43], we can also defined the lower and upper possibilistic rth moments of fuzzy number A as follows:(23)ELrA=∫01PosA≤a1αELA−a1αrdα∫01PosA≤a1αdα=2∫01αELA−a1αrdα,ERrA=∫01PosA≥a2αERA−a2αrdα∫01PosA≥a2αdα=2∫01αERA−a2αrdα.In this section, we will discuss moments of Gaussian fuzzy numbers andOm,n−Tr.F.N.


### 2.2. Moment ofOm,n−Tr.F.N

Using the mathematical properties that linked the Gamma functionΓu with the Beta function Bu;v such as Bu,v=∫01tu−11−tv−1dt, it can easily mathematically shown that the symmetry property also holds true for the beta function such that Bu;v=Bv;u. The mathematical relationship that exists between the beta function and the gamma function is that Bu,v=ΓuΓv/Γu+v, Γu=u−1, and Bu,v=u−1!v−1!/u+v−1!. Now, if A is an Om,n−Tr.F.N whose α-cuts is given by Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n, then we have the lower possibilistic mean ELA and the upper possibilistic mean ERA as(24)ELA=a2−2a2−a1B2,1m+1,ERA=a3−2a3−a4B2,1n+1.The interval value possibilistic mean and possibilistic mean are(25)IVPMA=a2−2a2−a1B2,1m+1,a3−2a3−a4B2,1n+1.Ifm=n, then the interval value possibilistic mean and possibilistic mean are(26)IVPMA=ELA,ERA=a2−2a2−a1B2,1m+1,a3−2a3−a4B2,1m+1.The possibilistic mean is(27)EA=a2+a32−a2−a1B2,1m+1+a3−a4B2,1n+1.Ifm=n, then the possibilistic mean of A is(28)EA=a2+a32−B2,1m+1a2−a1+a3−a4.The zero centered variance ofA is(29)VarA=a3−a224+a3−a2a2−a1B2,1m+1−a3−a2a3−a4B2,1n+1+a2−a122B2,2m+1−a2−a1a3−a4B2,1m+1n+1+a3−a422B2,2n+1.The zero-centered variance ofA is when m=n is(30)VarA=a3−a224+a3−a2a2−a1−a3−a2a3−a4B2,1m+1+a2−a122−a2−a1a3−a4+a3−a422B2,2m+1.Thus, we have the following possibilistic expressions for a fuzzy number, wherem=2 and n=1/2 in expressions (24)–(29), respectively:(31)ELA=a2−2a2−a1Γ2Γ1+1/2Γ3+1/2=815a1+715a2,ERA=a3−2a3−a4Γ2Γ1+2Γ3+2=56a3+16a4,EA=415a1+730a2+512a3+112a4,IVPMA=815a1+715a2,56a3+16a4.The possibilistic variance of a fuzzy number, wherem=2 and n=1/2, is given by(32)VarA=112a1−a22+14a2−a32+160a3−a42+415a1−a2a2−a3+463a1−a2a3−a4+112a2−a3a3−a4.To find more details on possibility theory, any interested reader may refer to the works of Carlsson and Fuller [9], Appadoo et al. [41], and Georgescu [45].


## 2.1. Moment Properties of Fuzzy Numbers

As in [9], lower possibilistic mean ELA, upper possibilistic mean ERA, possibilistic mean EA, and interval value possibilistic mean IVPMA=ELA,ERA of a fuzzy number A defined by a1α,a2α, α∈0,1 are given below. It is important to note here that, in order to derive the results in this section, we implicitly use the following results:(5)PossibilityA≤a1α=π−∞,a1α=supu≤a1αAu=α,PossibilityA≥a2α=πa2α,∞=supu≥a2αAu=α.For more detail, an interested reader may refer to [9](6)ELA=∫01PosA≤a1αa1αdα∫01PosA≤a1αdα=2∫01αa1αdα,ERA=∫01PosA≥a2αa2αdα∫01PosA≥a2αdα=2∫01αa2αdα,where Pos denotes possibility and EA is the level-weighted average of the arithmetic means of all α-level sets and is given by(7)EA=∫01PosA≤a1αa1αdα2∫01PosA≤a1αdα+∫01PosA≥a2αa2αdα2∫01PosA≥a2αdα.The interval value possibilistic meanIVPMA of a fuzzy number A is defined as(8)IVPMA=∫01PosA≤a1αa1αdα∫01PosA≤a1αdα,∫01PosA≥a2αa2αdα∫01PosA≥a2αdα.According to Carlsson and Fuller [9], the zero centered possibilistic variance VarA of a fuzzy number A is defined as(9)VarA=∫0112αa2α−a1α2dα.The possibilistic covariance between fuzzy numbersA and B is defined as(10)CovA,B=12∫01αa2α−a1αb2α−b1αdα.Zhang and Nie [42] and Zhang and Xiao [43] defined the lower possibilistic variance of a fuzzy number A as(11)VarLA=∫01PosA≤a1αELA−a1α2dα∫01PosA≤a1αdα=2∫01αELA−a1α2dαand the upper possibilistic variance of a fuzzy number A as(12)VarRA=∫01PosA≥a2αERA−a2α2dα∫01PosA≥a2αdα=2∫01αERA−a2α2dα.Similar to the definition ofEA, the crisp possibilistic variance is as the arithmetic mean of its lower and upper possibilistic variances, i.e.,(13)VarA=12∫01PosA≤a1αELA−a1α2dα∫01PosA≤a1αdα+∫01PosA≥a2αERA−a2α2dα∫01PosA≥a2αdα.Zhang and Nie [42] and Zhang and Xia [43] defined the lower and upper possibilistic covariances between two fuzzy numbers A and B as given in expressions (14) and (15):(14)CovLA,B=∫01PosA≤a1α,B≤b1αELA−a1αERB−b1αdα∫01PosA≤a1α,B≤b1αdα=2∫01αELA−a1αELB−b1αdα,(15)CovRA,B=∫01PosA≥a2α,B≥b2αELA−a2αERB−b2αdα∫01PosA≥a2α,B≥b2αdα=2∫01αERA−a2αERB−b2αdα,and the crisp possibilistic covariances between fuzzy numbers A and B as the arithmetic mean of their lower and upper possibilistic covariances, i.e.,(16)CovA,B=∫01PosA≤a1α,B≤b1αELA−a1αERB−b1αdα2∫01PosA≤a1α,B≤b1αdα+∫01PosA≥a2α,B≥b2αELA−a2αERB−b2αdα2∫01PosA≥a2α,B≥b2αdα.We use the following results as in [22, 42–44]:(17)PosA≤a1α,B≤b1α=supu≤a1α,v≤b1αminAu,Bv=α,PosA≥a2α,B≥b2α=supu≥a2α,v≥b2αminAu,Bv=α.Alternatively, Fuller and Majlender [12] introduced the crisp possibilistic variance and covariance of a fuzzy number A as(18)VarA=∫01αEA−a1α2+EA−a2α2dα.(19)CovA,B=∫01αEA−a1αEB−b1α+EA−a2αEB−b2αdα.Remark 1.
We can define the lower possibilistic variance of fuzzy numberA as follows:(20)VarLA=∫01PosA≤a1αEA−a1α2dα∫01PosA≤a1αdα,where the upper possibilistic variance of fuzzy number A is as follows:(21)VarRA=∫01PosA≥a2αEA−a2α2dα∫01PosA≥a2αdα,Remark 2.
An alternative definition for therth possibilistic moments of a fuzzy number can be formulated as follows:(22)ErA=∫01αELA−a1αr+ERA−a2αrdα.Remark 3.
Along the lines of [42, 43], we can also defined the lower and upper possibilistic rth moments of fuzzy number A as follows:(23)ELrA=∫01PosA≤a1αELA−a1αrdα∫01PosA≤a1αdα=2∫01αELA−a1αrdα,ERrA=∫01PosA≥a2αERA−a2αrdα∫01PosA≥a2αdα=2∫01αERA−a2αrdα.In this section, we will discuss moments of Gaussian fuzzy numbers andOm,n−Tr.F.N.


## 2.2. Moment ofOm,n−Tr.F.N

Using the mathematical properties that linked the Gamma functionΓu with the Beta function Bu;v such as Bu,v=∫01tu−11−tv−1dt, it can easily mathematically shown that the symmetry property also holds true for the beta function such that Bu;v=Bv;u. The mathematical relationship that exists between the beta function and the gamma function is that Bu,v=ΓuΓv/Γu+v, Γu=u−1, and Bu,v=u−1!v−1!/u+v−1!. Now, if A is an Om,n−Tr.F.N whose α-cuts is given by Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n, then we have the lower possibilistic mean ELA and the upper possibilistic mean ERA as(24)ELA=a2−2a2−a1B2,1m+1,ERA=a3−2a3−a4B2,1n+1.The interval value possibilistic mean and possibilistic mean are(25)IVPMA=a2−2a2−a1B2,1m+1,a3−2a3−a4B2,1n+1.Ifm=n, then the interval value possibilistic mean and possibilistic mean are(26)IVPMA=ELA,ERA=a2−2a2−a1B2,1m+1,a3−2a3−a4B2,1m+1.The possibilistic mean is(27)EA=a2+a32−a2−a1B2,1m+1+a3−a4B2,1n+1.Ifm=n, then the possibilistic mean of A is(28)EA=a2+a32−B2,1m+1a2−a1+a3−a4.The zero centered variance ofA is(29)VarA=a3−a224+a3−a2a2−a1B2,1m+1−a3−a2a3−a4B2,1n+1+a2−a122B2,2m+1−a2−a1a3−a4B2,1m+1n+1+a3−a422B2,2n+1.The zero-centered variance ofA is when m=n is(30)VarA=a3−a224+a3−a2a2−a1−a3−a2a3−a4B2,1m+1+a2−a122−a2−a1a3−a4+a3−a422B2,2m+1.Thus, we have the following possibilistic expressions for a fuzzy number, wherem=2 and n=1/2 in expressions (24)–(29), respectively:(31)ELA=a2−2a2−a1Γ2Γ1+1/2Γ3+1/2=815a1+715a2,ERA=a3−2a3−a4Γ2Γ1+2Γ3+2=56a3+16a4,EA=415a1+730a2+512a3+112a4,IVPMA=815a1+715a2,56a3+16a4.The possibilistic variance of a fuzzy number, wherem=2 and n=1/2, is given by(32)VarA=112a1−a22+14a2−a32+160a3−a42+415a1−a2a2−a3+463a1−a2a3−a4+112a2−a3a3−a4.To find more details on possibility theory, any interested reader may refer to the works of Carlsson and Fuller [9], Appadoo et al. [41], and Georgescu [45].


## 3. Possibilistic Properties of Gaussian Fuzzy Numbers

In this section, we discuss certain moments of fuzzy numbers and their applications, such as skewness and kurtosis.Theorem 1.
Consider the Gaussian fuzzy number of the form given below such thatAα=μ−σ−2lnα,μ+σ−2lnα. Then, the lower possibilistic mean, upper possibilistic mean, possibilistic mean, and possibilistic variance are given below:(33)ELA=μ−12σπ,ERA=μ+12σπ,EA=μ,VarA=σ2.Proof.
The proof of (33) is easy and is omitted.Remark 4.
ConsiderA and B are two Gaussian fuzzy numbers as in Definition 2 such that the α-cuts for A and B are as follows:(34)Aα=μ−σ−2lnα,μ+σ−2lnα,Bα=μ1−σ1−2lnα,μ1+σ1−2lnα,where 0≤α≤1, μ≠μ1, and σ≠σ1>0. It can easily be shown that the possibilistic covariance between those two fuzzy numbers is(35)CovA,B=12∫01αa2α−a1αb2α−b1αdα=σσ1.Remark 5.
Note that if we used the definition of crisp possibilistic variance and crisp possibilistic covariance of [42], then(36)VarA=∫01α2∫01αa1αdα−a1α2+2∫01αa2αdα−a2α2dα,CovA,B=∫01α2∫01αa1αdα−a1α2∫01αb1αdα−a2α+2∫01αa2αdα−b1α2∫01αb2αdα−b2αdα.
Then,(37)VarA=14σ24−π,CovA,B=18σ2+σ12π−12σ2+σ12.Remark 6.
LetA be a Gaussian fuzzy number as in Definition 2; then, for 0≤α≤1, we have a1α=μ−σ−2lnα and a2α=μ+σ−2lnα. It can easily be shown that the rth possibilistic moments of A are given by ErA=σr−1r+1/2Γ2/r/B2/r,1. Then, the first four moments of the Gaussian fuzzy number are as follows: E1A=0, E2A=σ2, E3A=0, and E4A=2σ4. Notice that all the odd powers are zero which is also true for the Gaussian probability distribution in probability theory. As a result, it is observed that the possibilistic skewness is equal to zero and possibilistic kurtosis is equal to 2.Theorem 2.
Let A be a Gaussian fuzzy number of the form given in Definition2 whose α-cuts are(38)Aα=a1α,a2α=μ−σ−2lnα,μ+σ−2lnα,where a1α=μ−σ−2lnα and a2α=μ+σ−2lnα for 0≤α≤1. If we use the rth possibilistic moments formula given by(39)ErA=∫01αELA−a1αr+ERA−a2αrdα,where we explicitly used(40)ELA=μ−12σπ,ERA=μ+12σπ,then(41)ErA=−1r+12∑k=0rr!r−k!k!−σk12σπr−kΓk2+1.Proof.
(42)ErA=∫01αELA−a1αr+ERA−a2αrdα=−1r+1∑k=0rr!r−k!k!12σπr−k∫0∞e−2u−σ2ukdu=−1r+12∑k=0rr!r−k!k!−σk12σπr−kΓk2+1.
It can be observed thatE2A=−1/4σ2π+σ2, E3A=0, E4A=−3/16σ4π2+2σ4, E5A=0, E6A=−5/64σ6π3−15/16σ6π2+15/8σ6π+6σ6, and E7A=0. Note that all odd possibilistic moments are zeros. The possibilistic skewness and possibilistic kurtosis are as follows:(43)SA=E3AE2A3=0,KA=E4AE2A2=−3/16σ4π2+2σ4−1/4σ2π+σ22=3.2451.Theorem 3.
LetA be a Gaussian fuzzy number of the form as in Definition 2 whose α-cuts are(44)Aα=a1α,a2α=μ−σ−2lnα,μ+σ−2lnα,where a1α=μ−σ−2lnα and a2α=μ+σ−2lnα for 0≤α≤1. If we use the rth possibilistic moments formula given by(45)ErA=∫01αa1rα+a2rαdα,then(46)ErA=∑k=0r−1r−k+12r!r−k!k!μkσr−kΓr−k2+1.Proof.
(47)ErA=∫01αarα+a2rαdα=∫01αμ−σ2−lnαr+μ+σ2−lnαrdα=∫0∞e−2u∑k=0rr!r−k!k!μk−σ2u1/2r−kdu+∫0∞e−2u∑k=0rr!r−k!k!μkσ2u1/2r−kdu=∑k=0r−1r−k+12r!r−k!k!μkσr−kΓr−k2+1.
The possibilistic kurtosis is given by(48)KA=2σ4+6μ2σ2+μ4−4μ3μσ2+μ3+6μ2σ2+μ2−4μμ3+μ4σ22=2,and the possibilistic skewness is SA=0.Theorem 4.
LetA be an Om,n−Tr.T.N, whose α-cuts are explicitly given by(49)Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n,where a1α=a2−a2−a11−α1/m and a2α=a3−a3−a41−α1/n for 0≤α≤1. If we use the rth possibilistic moments formula given by(50)ErA=∫01αa1rα+a2rαdα,then(51)ErA=∑k=0rr!r−k!k!a2r−ka1−a2kΓ2Γk/m+1Γk/m+3+∑k=0rr!r−k!k!a3r−ka4−a3kΓ2Γk/n+1Γk/n+3.Proof.
The proof is obvious and is omitted.
For a Tr.F.N. withm=n=1, we have the following results:(52)E2A=14a22+a32+16a2a1+a3a4+112a12+a42,(53)E3A=15a23+a33+1203a22a1+a13+3a32a4+a43+110a2a12+a3a42,(54)E4A=115a2a13+2a33a4+a3a43+2a23a1+110a22a12+a32a42+16a24+a34+130a14+a44.
Note that if we seta1=a−α, a2=a, a3=b, and a4=b+β in (52)–(54), we obtained the same results as Thavaneswaran et al. [13] and Thavaneswaran et al. [11] as follows:(55)E2A=14a22+a32+16a2a1+a3a4+112a12+a42=a2+b22+bβ−aα3+α2+β212,E3A=15a23+a33+1203a22a1+a13+3a32a4+a43+110a2a12+a3a42=β3−α320+aα2+bβ24+b2β−a2α+a3+b32,E4A=115a2a13+2a33a4+a3a43+2a23a1+110a22a12+a32a42+16a24+a34+130a14+a44=α4+β430+bβ3−aα35+a2α2+b4+a4+b2β22+2b3β−2a3α3.Theorem 5.
LetA be an Om,n−Tr.T.N, whose α-cuts are explicitly given by(56)Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n,where a1α=a2−a2−a11−α1/m and a2α=a3−a3−a41−α1/n for 0≤α≤1 and(57)EA=a2+a32−m2a2−a11+m1+2m−n2a3−a4n+12n+1.
We use therth possibilistic moments formula given by(58)ErA=∫01αa1α−EAr+a2α−EArdα∑k=0rr!r−k!k!a2−a2+a32−m2a2−a11+m1+2m−n2a3−a4n+12n+1r−ka1−a2kΓ2Γk/m+1Γk/m+3+∑k=0rr!r−k!k!a3−a2+a32−m2a2−a11+m1+2m−n2a3−a4n+12n+1r−ka4−a3kΓ2Γk/n+1Γk/n+3.Proof.
The proof is easy and is omitted.Theorem 6.
LetA be an Om,n-Tr.T.N, whose α-cuts are explicitly given by(59)Aα=a2−a2−a11−α1/m,a3−a3−a41−α1/n,where a1α=a2−a2−a11−α1/m and a2α=a3−a3−a41−α1/n for 0≤α≤1. If we use the following formula for the rth possibilistic moments(60)ErA=∫01αELAa1αr+ERA−a2αrdα,where(61)ELA=a2−2m2a2−a11+m1+2m,ERA=a3−2n2a3−a4n+12n+1,then we have the rth moments as(62)ErA=∑k=0rr!r−k!k!2m2a1−a21+m1+2mr−ka2−a1kΓ2Γk/m+1Γk/m+3+∑k=0rr!r−k!k!2n2a4−a3n+12n+1r−ka3−a4kΓ2Γk/n+1Γk/n+3.Proof.
(63)ErA=∫01αELA−a1αr+ERA−a2αrdα=∫01αa2−2m2a2−a11+m1+2m−a2−a2−a11−α1/mr+a3−2n2a3−a4n+12n+1−a3−a3−a41−α1/nrdα=∫01α2m2a1−a21+m1+2m+a2−a11−α1/mr+2n2a4−a3n+12n+1+a3−a41−α1/nrdα.
Hence, the result(64)ErA=∑k=0rr!r−k!k!2m2a1−a21+m1+2mr−ka2−a1kΓ2Γk/m+1Γk/m+3+∑k=0rr!r−k!k!2n2a4−a3n+12n+1r−ka3−a4kΓ2Γk/n+1Γk/n+3.


### 3.1. Fuzzy Possibilistic Economic Production Quantity Inventory Model

In this section, the following fuzzy numbers are used to develop the fuzzy production inventory model, whereD˜ is the fuzzy yearly demand, C0˜ is the fuzzy annual ordering cost per run, d˜ is the fuzzy daily demand rate, and Ch˜ is the fuzzy holding cost. For further details, see [32, 33]:(65)Coα=C1α,C2α,Dα=D1α,D2α,Chα=h1α,h2α,dα=d1α,d2α.It is important to note here that the only parameter that we allow to be crisp in our model formulation is the daily production ratep. Thus, the fuzzy total production inventory cost can be written in terms of its α-level set as(66)TCα=C0αDαq+Chαq21−dαp.Using equation (3), we can rewrite the α-level set of TCα, for 0≤α≤1, as follows:(67)TC1α,TC2α=Co1α,Co2αD1αq,D2αq+Ch1α,Ch2αq21−d2αp,1−d1αp=Co1αD1αq,Co2αD2αq+Ch1αq21−d2αp,Ch2αq21−d1αp.We make the assumption at0<d1α/p<1 and 0<d2α/p<1 along with d2α/p>d1α/p. Simplifying expression (67) yields the following α-level set for fuzzy total cost as(68)TC1α,TC2α=Co1αD1αq+Ch1αq21−d2αp,Co2αD2αq+Ch2αq21−d1αp,where(69)TC1α=Co1αD1αq+Ch1αq21−d2αp,TC2α=Co2αD2αq+Ch2αq21−d1αp.The lower possibilistic total production inventory cost is given as(70)ELTC=∫01PossTC≤TC1αCo1αD1α/q+Ch1αq/21−d2α/pdα∫01PossTC≤TC1αdα.The upper possibilistic total production inventory cost is given as(71)ERTC=∫01PossTC≥TC2αCo2αD2α/q+Ch2αq/21−d1α/pdα∫01PossTC≥TC2αdα.The weighted average of (70) and (71) lead to ELRTC as follows:(72)ELRTC=∫01αCo1αD1αq+Ch1αq21−d2αp+Co2αD2αq+Ch2αq21−d1αpdα.(a)
In order to find the lower possibilistic optimal economic production quantity, we set(73)ddq∫01PossTC≤TC1αCo1αD1α/q+Ch1αq/21−d2α/pdα∫01PossTC≤TC1αdα=0and then solve forq. That value of q that minimizes (73) is the lower possibilistic production quantity which will be denoted by q1.(b)
In order to find the upper possibilistic optimal economic production quantity, we set(74)ddq∫01PossTC≥TC2αCo2αD2α/q+Ch2αq/21−d1α/pdα∫01PossTC≥TC2αdα=0and then solve forq. That value of q that minimizes (74) is the upper possibilistic production quantity which will be denoted by q2.(c)
In order to find the possibilistic optimal economic production quantity, we set(75)ddq∫01αCo1αD1αq+Ch1αq21−d2αp+Co2αD2αq+Ch2αq21−d1αpdα=0and then solve for q. That value of q that minimizes (75) is the possibilistic optimal economic production quantity which will be denoted by q∗.Remark 7.
It is important to remark here thatq∗≠q1+q2/2.Below, we give a theorem to find fuzzy total cost for the economic production quantity inventory model with finite replenishment rate in a fuzzy possibilistic setup. In what follows, we shall assume thatCo, D, Ch, and d are Tr.F.N.’s, where Co=c1,c2,c3,c4, D=D1,D2,D3,D4, Ch=h1,h2,h3,h4, and d=d1,d2,d3,d4. Paranoindent after equation (76).(76)Coα=Co1α,Co2α=c1+αc2−c1,c4+αc3−c4,Dα=D1α,D2α=D1+αD2−D1,D4+αD3−D4,Chα=Ch1α,Ch2α=h1+αh2−h1,h4+αh3−h4,dα=d1α,d2α=d1+αd2−d1,d4+αd3−d4,(77)TCLq=163c2D2+c2D1+c1D2+c1D1q+112q−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p,(78)TCRq=163c3D3+D4c3+c4D3+c4D4q−112q3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp,and we have the following values for lower q1 as(79)TCLRq=1123c2D2+c2D1+c1D2+c1D1q+3c3D3+D4c3+c4D3+c4D4q+124q−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp,Theorem 7.
For0≤α≤1, let the fuzzy ordering cost C0α, fuzzy demand Dα, fuzzy holding cost Chα, and fuzzy demand rate dα be given by(80)q1=2p3c2D2+c2D1+c1D2+c1D1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1.The upper possibilistic production lot sizeq2 is(81)q2=2p3c3D3+D4c3+c4D3+c4D44h3p+2h4p−3h3d2−h4d2−h3d1−h4d1.The possibilistic production lot sizeq∗ is(82)q∗=2p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p.Using (77), the total cost in a possibilistic setup at q1 is(83)TCLq=q1=3c2D2+c2D1+c1D2+c1D162p3c2D2+c2D1+c1D2+c1D1/−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1+1122p3c2D2+c2D1+c1D2+c1D1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p.Using (78), the total cost in a possibilistic setup at q2 is(84)TCRq=q2=3c3D3+D4c3+c4D3+c4D462p3c3D3+D4c3+c4D3+c4D4/4h3p+2h4p−3h3d2−h4d2−h3d1−h4d1+1122p3c3D3+D4c3+c4D3+c4D44h3p+2h4p−3h3d2−h4d2−h3d1−h4d13h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp.Using (79), the total cost in a possibilistic setup at q∗ is(85)=3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4122p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4/−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p+1242p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp.


## 3.1. Fuzzy Possibilistic Economic Production Quantity Inventory Model

In this section, the following fuzzy numbers are used to develop the fuzzy production inventory model, whereD˜ is the fuzzy yearly demand, C0˜ is the fuzzy annual ordering cost per run, d˜ is the fuzzy daily demand rate, and Ch˜ is the fuzzy holding cost. For further details, see [32, 33]:(65)Coα=C1α,C2α,Dα=D1α,D2α,Chα=h1α,h2α,dα=d1α,d2α.It is important to note here that the only parameter that we allow to be crisp in our model formulation is the daily production ratep. Thus, the fuzzy total production inventory cost can be written in terms of its α-level set as(66)TCα=C0αDαq+Chαq21−dαp.Using equation (3), we can rewrite the α-level set of TCα, for 0≤α≤1, as follows:(67)TC1α,TC2α=Co1α,Co2αD1αq,D2αq+Ch1α,Ch2αq21−d2αp,1−d1αp=Co1αD1αq,Co2αD2αq+Ch1αq21−d2αp,Ch2αq21−d1αp.We make the assumption at0<d1α/p<1 and 0<d2α/p<1 along with d2α/p>d1α/p. Simplifying expression (67) yields the following α-level set for fuzzy total cost as(68)TC1α,TC2α=Co1αD1αq+Ch1αq21−d2αp,Co2αD2αq+Ch2αq21−d1αp,where(69)TC1α=Co1αD1αq+Ch1αq21−d2αp,TC2α=Co2αD2αq+Ch2αq21−d1αp.The lower possibilistic total production inventory cost is given as(70)ELTC=∫01PossTC≤TC1αCo1αD1α/q+Ch1αq/21−d2α/pdα∫01PossTC≤TC1αdα.The upper possibilistic total production inventory cost is given as(71)ERTC=∫01PossTC≥TC2αCo2αD2α/q+Ch2αq/21−d1α/pdα∫01PossTC≥TC2αdα.The weighted average of (70) and (71) lead to ELRTC as follows:(72)ELRTC=∫01αCo1αD1αq+Ch1αq21−d2αp+Co2αD2αq+Ch2αq21−d1αpdα.(a)
In order to find the lower possibilistic optimal economic production quantity, we set(73)ddq∫01PossTC≤TC1αCo1αD1α/q+Ch1αq/21−d2α/pdα∫01PossTC≤TC1αdα=0and then solve forq. That value of q that minimizes (73) is the lower possibilistic production quantity which will be denoted by q1.(b)
In order to find the upper possibilistic optimal economic production quantity, we set(74)ddq∫01PossTC≥TC2αCo2αD2α/q+Ch2αq/21−d1α/pdα∫01PossTC≥TC2αdα=0and then solve forq. That value of q that minimizes (74) is the upper possibilistic production quantity which will be denoted by q2.(c)
In order to find the possibilistic optimal economic production quantity, we set(75)ddq∫01αCo1αD1αq+Ch1αq21−d2αp+Co2αD2αq+Ch2αq21−d1αpdα=0and then solve for q. That value of q that minimizes (75) is the possibilistic optimal economic production quantity which will be denoted by q∗.Remark 7.
It is important to remark here thatq∗≠q1+q2/2.Below, we give a theorem to find fuzzy total cost for the economic production quantity inventory model with finite replenishment rate in a fuzzy possibilistic setup. In what follows, we shall assume thatCo, D, Ch, and d are Tr.F.N.’s, where Co=c1,c2,c3,c4, D=D1,D2,D3,D4, Ch=h1,h2,h3,h4, and d=d1,d2,d3,d4. Paranoindent after equation (76).(76)Coα=Co1α,Co2α=c1+αc2−c1,c4+αc3−c4,Dα=D1α,D2α=D1+αD2−D1,D4+αD3−D4,Chα=Ch1α,Ch2α=h1+αh2−h1,h4+αh3−h4,dα=d1α,d2α=d1+αd2−d1,d4+αd3−d4,(77)TCLq=163c2D2+c2D1+c1D2+c1D1q+112q−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p,(78)TCRq=163c3D3+D4c3+c4D3+c4D4q−112q3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp,and we have the following values for lower q1 as(79)TCLRq=1123c2D2+c2D1+c1D2+c1D1q+3c3D3+D4c3+c4D3+c4D4q+124q−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp,Theorem 7.
For0≤α≤1, let the fuzzy ordering cost C0α, fuzzy demand Dα, fuzzy holding cost Chα, and fuzzy demand rate dα be given by(80)q1=2p3c2D2+c2D1+c1D2+c1D1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1.The upper possibilistic production lot sizeq2 is(81)q2=2p3c3D3+D4c3+c4D3+c4D44h3p+2h4p−3h3d2−h4d2−h3d1−h4d1.The possibilistic production lot sizeq∗ is(82)q∗=2p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p.Using (77), the total cost in a possibilistic setup at q1 is(83)TCLq=q1=3c2D2+c2D1+c1D2+c1D162p3c2D2+c2D1+c1D2+c1D1/−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1+1122p3c2D2+c2D1+c1D2+c1D1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p.Using (78), the total cost in a possibilistic setup at q2 is(84)TCRq=q2=3c3D3+D4c3+c4D3+c4D462p3c3D3+D4c3+c4D3+c4D4/4h3p+2h4p−3h3d2−h4d2−h3d1−h4d1+1122p3c3D3+D4c3+c4D3+c4D44h3p+2h4p−3h3d2−h4d2−h3d1−h4d13h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp.Using (79), the total cost in a possibilistic setup at q∗ is(85)=3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4122p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4/−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p+1242p3c2D2+c2D1+c1D2+c1D1+3c3D3+D4c3+c4D3+c4D4−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4p−3d3h2−h1d3−d4h2−h1d4+4ph2+2ph1p−3h3d2+h4d2+h3d1+h4d1−4h3p−2h4pp.


## 4. Findings and Practical and Managerial Implications

The fuzzy possibilistic inventory model derived in this paper can be considered as a more generalized version of the classical economic inventory model. We have determined the optimal order quantity in a fuzzy sense using the concept of possibility theory. Although the problem considered in this paper is geared towards a single-item and single-period inventory model, it can be easily extended to multiple periods with multiple items’ inventory models. These fuzzy possibilistic inventory models can be easily developed to accommodate other inventory models with various products. The solution of the fuzzy model does not require excessive computation; moreover, the fuzzy possibilistic approach can be extended to include budgetary, space, and weight constraints.Most of the methods in the literature are unfortunately limited in that sense. The possibilistic inventory model can be modified to compute future demand in a vague sense and thus generate dynamic reordering points in case of unexpected stock out of products. It is vital to reduce inventory costs during the ongoing COVID-19 pandemic; thus, it is our view that fuzzy inventory models can addressed the sudden rise in inventory cost, with this paper serving as an attempt to do so. It has been heavily documented that inventories consume a large part of budget, space, overheads, and maintenance. Therefore, we need a sophisticated and practical solution-based approach to address the rise in inventory costs.


## 5. Conclusion

The supply chain network contains many subsystems, and each subsystem usually includes uncertainties. Thus, the overall network becomes uncertain and imprecise. This paper advocates the use of fuzzy numbers to model the uncertainty underlying the SC network. In this paper, we derive the possibilistic moments of some nonlinear types of fuzzy numbers, and those proposed fuzzy numbers can be regarded as an extension of previous results for the linear fuzzy numbers. These new derivations and concepts are consistent with the definition of moments in probability theory. Furthermore, we have been able to give an alternative definition for possibilistic variance. We also define lower possibilisticrth moments and upper possibilistic rth moments of fuzzy numbers, which have numerous applications in supply chain management. Finally, a fuzzy EPQ inventory model is developed to demonstrate the application of proposed fuzzy sets. The proposed fuzzy inventory model does not reject other fuzzy inventory formulations, but rather complements those already commonly used. Future studies should also consider a modified version of the EPQ model for perishable products, making the model even more appealing for supply chain professionals. This approach can motivate new research in the other functional areas of the supply chain network as well. Another important aspect of this research is that, in the post-COVID era, most of the parameters in inventory models will be arbitrary and include a considerable degree of randomness. Researchers have started to realize the complicated natures of those inventory systems. Most inventory parameters fluctuate from one period to the next, and the classical deterministic inventory model will no longer be applicable. This research considers a fuzzy inventory model compared to a deterministic one, enabling the inventory personnel to adapt to demand fluctuations and thus replenish stock as required. In our view, this research partly addresses this gap by considering fuzzy inventory parameters.



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*Source: 5538185-2021-12-08.xml*