On LR-type fully intuitionistic fuzzy linear programming with inequality constraints: Solutions with unique optimal values

Highlights

• Singh and Yadav’s method cannot yield solutions with unique optimal values.

• A lexicographic criterion for ranking LR-type intuitionistic fuzzy numbers is given.

• A method to find solutions of FIFLP problems with unique optimal values is proposed.

• Intuitionistic fuzzy inequality constraints are defined lexicographically.

• A fully intuitionistic fuzzy production planning problem is solved as illustration.

Abstract

Singh and Yadav (2017) defined the product of unrestricted LR-type Intuitionistic Fuzzy Numbers (IFNs), and making use of the new product operation, the authors proposed a method to solve Fully Intuitionistic Fuzzy Linear Programming (FIFLP) problems. However, their method cannot be used to find the unique optimal value of FIFLP problems with inequality constraints. Recently, Pérez-Cañedo and Concepciõn-Morales (2019) presented a method to find the unique optimal fuzzy value of Fully Fuzzy Linear Programming (FFLP) problems with equality and inequality constraints based on the optimisation of a lexicographic criterion for ranking LR fuzzy numbers. The authors suggested that their method could be extended to find the unique optimal intuitionistic fuzzy value of FIFLP problems with inequality constraints as well. In this paper, we analyse Singh and Yadav’s method and modify it to find the unique optimal intuitionistic fuzzy value of FIFLP problems with equality and inequality constraints. Thus, a new method is obtained and is demonstrated by means of a fully intuitionistic fuzzy production planning problem. Results are compared with those obtained by using Singh and Yadav’s method and show that the proposed method overcomes the shortcomings and limitations of their method.

Introduction

Optimising a linear function over a set of linear equalities and inequalities is the task of Linear Programming (LP). Despite its apparent simplicity, LP has proved to be one of the most applicable operational research techniques. In its classical formulation, LP requires that all the problem parameters and decision variables be well-defined. In real-world LP problems, however, the values of the parameters are often imprecise because of incomplete or unobtainable information. In such cases, Fuzzy Linear Programming (FLP) may be a suitable alternative approach.

FLP is currently an active research area. However, the first contributions to FLP date back to the 1970s in the works by Tanaka, Okuda, and Asai (1974) and Zimmermann (1976), among others that were motivated by Bellman and Zadeh’s earlier research on decision-making in fuzzy environments (Bellman & Zadeh, 1970) based on Zadeh’s concept of a fuzzy set (FS) (Zadeh, 1965). There is an extensive literature dealing with FLP problems; the interested reader is referred to a recent survey by Ebrahimnejad and Verdegay (2018a) for a comprehensive exposition of classical and modern methodologies of FLP.

Atanassov (1986) generalised Zadeh’s fuzzy sets through his concept of an intuitionistic fuzzy set (IFS). This concept is useful in decision-making situations where there is uncertainty as well as hesitation on the values of some/all quantities involved. The first application of IFS theory to optimisation under uncertainty was given by Angelov (1997), and constituted a generalisation of previous results of FLP based on Bellman and Zadeh’s decision-making model (Bellman & Zadeh, 1970).

In this paper, we do not intend to give a thorough review of IFS theory or its application to Intuitionistic Fuzzy Linear Programming (IFLP); the reader may refer to Atanassov, 1986, Atanassov, 1999; Ebrahimnejad, Verdegay, 2017, Ebrahimnejad, Verdegay, 2018b; Kumar and Hussain (2016); Niroomand (2018); Roy, Ebrahimnejad, Verdegay, and Das (2018); Singh, Yadav, 2017, Singh, Yadav, 2018; Wan, Wang, Lin, and Dong (2015) and Sidhu and Kumar (2019) and references therein. Here, we analyse a recent method for solving FIFLP problems with unrestricted LR-type intuitionistic fuzzy parameters and decision variables proposed by Singh and Yadav (2017), and show that this method cannot be used to find the unique optimal intuitionistic fuzzy value of FIFLP problems with inequality constraints.

The uniqueness of the optimal value of an optimisation problem is a property of classical mathematical optimisation which is not necessarily guaranteed by the methods of FLP or IFLP. This difficulty arises from the fact that there is no natural order on the set of all fuzzy numbers and therefore, since most existing methods rely on order relations that cannot yield complete orderings on this set, decision-makers may not be able to properly identify the best solution by using those methods. There are several works addressing the uniqueness of optimal values in FFLP (see, e.g., Das, Mandal, Edalatpanah, 2017, Ezzati, Khorram, Enayati, 2015, Hosseinzadeh, Edalatpanah, 2016, Kaur, Kumar, 2012, Kaur, Kumar, 2016). In all those works, lexicographic criteria with the total order properties are used to rank fuzzy numbers, which guarantees the uniqueness of the optimal fuzzy value in FFLP problems having only equality constraints. However, the transformations proposed to handle the fuzzy inequality constraints (see, e.g., Das, Mandal, Edalatpanah, 2017, Ezzati, Khorram, Enayati, 2015) do not conform with the lexicographic criterion used in the objective function and may lead to suboptimal solutions or possibly unfeasible problems. We note here that transforming the fuzzy inequality constraints into fuzzy equality constraints by means of non-negative fuzzy slack and surplus variables, similarly to classical LP, is not mathematically correct (see, e.g., Bhardwaj, Kumar, 2015, Gupta, Kaur, Kumar, 2016).

Remarkably, the use of a lexicographic criterion to handle fuzzy inequality constraints in FFLP was first proposed by Hashemi, Modarres, Nasrabadi, and Nasrabadi (2006). It is worth mentioning that their approach is limited to FFLP problems with inequality constraints in which all the parameters and decision variables are symmetric LR-type fuzzy numbers. Pérez-Cañedo and Concepción-Morales (2019) proposed an approach to lexicographic modelling of fuzzy inequality constraints, and transformed the FFLP problem with arbitrary (in shape and sign) LR-type fuzzy parameters and decision variables into an equivalent crisp mixed 0–1 lexicographic non-linear programming (MLNLP) problem solvable by available optimisation algorithms.

To the best of our knowledge, there is no study on the uniqueness of the optimal value in the case of FIFLP with inequality constraints. Therefore, based on the previous work on the unique optimal fuzzy value of FFLP problems with inequality constraints (Pérez-Cañedo & Concepción-Morales, 2019), we propose a method to find the unique optimal intuitionistic fuzzy value of FIFLP problems with equality and inequality constraints, in which some/all problem parameters and/or decision variables can take on unrestricted LR-type IFNs. To this aim, we review some basic concepts of IFS theory which are fundamental for that purpose. Our contributions are summarised as follows:

• A lexicographic criterion for ranking LR-type IFNs is proposed. This ranking criterion has the total order properties and yields a complete ranking on the set of all LR-type IFNs of the same type.

• By using the proposed lexicographic ranking criterion, the FIFLP problems are transformed into equivalent MLNLP problems. This approach allows us to define lexicographically each intuitionistic fuzzy inequality relation and so to use the same lexicographic ranking criterion with the objective function and the inequality constraints; thus yielding solutions to FIFLP problems with unique optimal intuitionistic fuzzy values.

• Since IFS theory generalises FS theory, the proposed method for solving FIFLP problems generalises the method of Pérez-Cañedo and Concepción-Morales (2019) for solving FFLP problems.

In the rest of this section, we present some basic definitions concerning LR-type IFNs, the LR-type FIFLP problem and the solution method proposed by Singh and Yadav (2017); additionally, we highlight the shortcomings and limitations of their method. In Section 2, we propose a lexicographic criterion for ranking LR-type IFNs and a method to find the unique optimal intuitionistic fuzzy value of FIFLP problems with equality and inequality constraints. The proposed method for solving FIFLP problems is demonstrated in Section 3 by means of a fully intuitionistic fuzzy production planning problem. Concluding remarks and future research lines are given in Section 4.

To continue with our exposition, the following definitions and remarks are necessary.

Definition 1 Singh & Yadav, 2017

An IFN ${\tilde{a}}^I={(a,l,r,l^{\prime },r^{\prime })}_{LR}$ is said to be an LR-type IFN if its membership ${\mu }_{{\tilde{a}}^I}$ and non-membership ${\nu }_{{\tilde{a}}^I}$ functions are given as follows:

$$\begin{array}{cc}{\mu }_{{\tilde{a}}^I}\left(x\right)=\{\begin{array}{cc}L\left(\frac{a-x}{l}\right)\hfill & x\le a,l>0\hfill \\ R\left(\frac{x-a}{r}\right)\hfill & x\ge a,r>0\hfill \end{array}\hfill & {\nu }_{{\tilde{a}}^I}\left(x\right)=\{\begin{array}{cc}1-L\left(\frac{a-x}{l^{\prime }}\right)\hfill & x\le a,l^{\prime }>0\hfill \\ 1-R\left(\frac{x-a}{r^{\prime }}\right)\hfill & x\ge a,r^{\prime }>0\hfill \end{array}\hfill \end{array}$$

where l ≤ l′, r ≤ r′ and $0\le {\mu }_{{\tilde{a}}^I}\left(x\right)+{\nu }_{{\tilde{a}}^I}\left(x\right)\le 1$; L and R, called shape functions, are continuous and decreasing functions fulfilling $L\left(0\right)=R\left(0\right)=1$ and ${\lim }_{x\to \infty }R\left(x\right)={\lim }_{x\to \infty }L\left(x\right)=0$. a is called the mean value of ${\tilde{a}}^I,$ l and r are respectively the left and right spreads of ${\mu }_{{\tilde{a}}^I}$; l′ and r′ are respectively the left and right spreads of ${\nu }_{{\tilde{a}}^I}$. The set of all LR-type IFNs is denoted by $\mathcal{I}\left(\Re \right)$.

Definition 2 Singh & Yadav, 2017

An LR-type IFN ${\tilde{a}}^I={(a,l,r,l^{\prime },r^{\prime })}_{LR}$ is said to be non-negative (resp. non-positive), denoted ${\tilde{a}}^I\ge 0$ (resp. ${\tilde{a}}^I\le 0$), if $a-l^{\prime }\ge 0$ (resp. $a+r^{\prime }\le 0$); ${\tilde{a}}^I$ is said to be unrestricted if a is any real number.

The reader is referred to Singh and Yadav (2017) for the definition of addition, subtraction and multiplication operations on $\mathcal{I}\left(\Re \right)$.

Definition 3 Singh & Yadav, 2017

Let ${\tilde{a}}^I={(a,l,r,l^{\prime },r^{\prime })}_{LR}$ be an LR-type IFN, then the score and accuracy indices of ${\tilde{a}}^I$ are denoted by $\mathcal{S}\left({\tilde{a}}^I\right)$ and $\mathcal{A}\left({\tilde{a}}^I\right),$ respectively and are defined by:

$$\begin{array}{ccc}\hfill \mathcal{S}\left({\tilde{a}}^I\right)& :=& 1/2{\int }_0^1(a-l{L}^{-1}\left(\lambda \right)+a+r{R}^{-1}\left(\lambda \right))d\lambda \hfill \\ & & -1/2{\int }_0^1(a-l^{\prime }{L}^{-1}(1-\lambda )+a+r^{\prime }{R}^{-1}(1-\lambda ))d\lambda \hfill \\ \hfill \mathcal{A}\left({\tilde{a}}^I\right)& :=& 1/2{\int }_0^1(a-l{L}^{-1}\left(\lambda \right)+a+r{R}^{-1}\left(\lambda \right))d\lambda \hfill \\ & & +1/2{\int }_0^1(a-l^{\prime }{L}^{-1}(1-\lambda )+a+r^{\prime }{R}^{-1}(1-\lambda ))d\lambda \hfill \end{array}$$

Based on the score and accuracy indices, Singh and Yadav (2017) gave the following order relation on $\mathcal{I}\left(\Re \right)$:

Definition 4 Singh & Yadav, 2017

Let ${\tilde{a}}_1^I$ and ${\tilde{a}}_2^I$ be two LR-type IFNs, then:

• ${\tilde{a}}_1^I$ is defined less than ${\tilde{a}}_2^I$ and written as ${\tilde{a}}_1^I\prec {\tilde{a}}_2^I$ if $\mathcal{S}\left({\tilde{a}}_1^I\right)<\mathcal{S}\left({\tilde{a}}_2^I\right),$

• if $\mathcal{S}\left({\tilde{a}}_1^I\right)=\mathcal{S}\left({\tilde{a}}_2^I\right),$ then ${\tilde{a}}_1^I$ is defined less than ${\tilde{a}}_2^I$ and written as ${\tilde{a}}_1^I\prec {\tilde{a}}_2^I$ if $\mathcal{A}\left({\tilde{a}}_1^I\right)<\mathcal{A}\left({\tilde{a}}_2^I\right),$

• ${\tilde{a}}_1^I$ is defined as equal to ${\tilde{a}}_2^I$ and written as ${\tilde{a}}_1^I\approx {\tilde{a}}_2^I$ if $\mathcal{S}\left({\tilde{a}}_1^I\right)=\mathcal{S}\left({\tilde{a}}_2^I\right)$ and $\mathcal{A}\left({\tilde{a}}_1^I\right)=\mathcal{A}\left({\tilde{a}}_2^I\right)$.

Remark 1

The order relation given in Definition 4 cannot properly order any two LR-type IFNs; hence, it cannot be used as a criterion to find the unique optimal value of FIFLP problems. To show this, consider the LR-type IFNs ${\tilde{a}}^I={(4.75,1,1,2,4)}_{LR}$ and ${\tilde{b}}^I={(5,2,1,2,3)}_{LR},$ where $L\left(x\right)=\max \{0,1-x\}$ and $R\left(x\right)=\max \{0,1-x\}$. In this case, we have $\mathcal{S}\left({\tilde{a}}^I\right)=\mathcal{S}\left({\tilde{b}}^I\right)=-1/2$ and $\mathcal{A}\left({\tilde{a}}^I\right)=\mathcal{A}\left({\tilde{b}}^I\right)=10$; thus, according to item (iii) from Definition 4, ${\tilde{a}}^I$ and ${\tilde{b}}^I$ are defined (assumed) as equal but, obviously, they are not. The graphs of the membership and non-membership functions of ${\tilde{a}}^I$ and ${\tilde{b}}^I,$ depicted in Fig. 1, suggest that ${\tilde{a}}^I\prec {\tilde{b}}^I$. Also, from Fig. 1, it can be seen that we may use the mean value as a third criterion to properly order these two LR-type IFNs; thus, we conclude that ${\tilde{a}}^I\prec {\tilde{b}}^I$. We note here that ranking fuzzy numbers and IFNs is a rather controversial topic subject of extensive research, and there is no method that yields a satisfactory ranking in all cases, since, according to their subjectivity, decision-makers may rank them differently. The reader may refer to Lakshmana Gomathi Nayagam, Jeevaraj, and Sivaraman (2016); Wang and Wang (2014) and Lakshmana Gomathi Nayagam, Jeevaraj, and Dhanasekaran (2016) and references therein for an introduction and recent results on this topic.

Contrary to classical LP, it is not always possible to express an unrestricted LR-type intuitionistic fuzzy variable as the subtraction of two non-negative ones. This is so because the distributive law of the product operation does not hold in general. Therefore, a special product operation for unrestricted LR-type intuitionistic fuzzy parameters and decision variables needs to be defined. Singh and Yadav (2017) defined a new product operator ⊙ for that purpose and used it to solve FIFLP problems in which some/all parameters and/or decision variables may take on unrestricted LR-type IFNs. The new product operator ⊙, however, has a somewhat cumbersome mathematical expression involving several cases, and we do not present it in this paper; the reader is referred to Singh and Yadav (2017) for details.

The mathematical model of FIFLP is given by (1), and the aim is to find a vector of LR-type IFNs ${\tilde{x}}^{I*}=({\tilde{x}}_1^{I*},{\tilde{x}}_2^{I*},\dots ,{\tilde{x}}_n^{I*})$ satisfying the constraint set of (1) such that ${\sum }_{j=1}^n{\tilde{c}}_j^I\odot {\tilde{x}}_j^I\precsim {\sum }_{j=1}^n{\tilde{c}}_j^I\odot {\tilde{x}}_j^{I*}$ for all feasible ${\tilde{x}}^I\in \mathcal{I}{\left(\Re \right)}^n$.

$$\begin{array}{cc}\hfill \max & \sum _{j=1}^n{\tilde{c}}_j^I\odot {\tilde{x}}_j^I\hfill \\ \hfill \text{s.t.}& \sum _{j=1}^n{\tilde{a}}_{ij}^I\odot {\tilde{x}}_j^I\{\precsim ,\approx ,\succsim \}{\tilde{b}}_i^I,\text{for}i\in I:=\{1,2,\dots ,m\}\hfill \\ & {\tilde{x}}_j^I\in \mathcal{I}\left(\Re \right),\text{for}j\in J:=\{1,2,\dots ,n\}\hfill \end{array}$$

Singh and Yadav (2017) proposed the following steps to solve FIFLP problem (1):

• Step 1. Let ${\tilde{a}}_{ij}^I={(a_{ij},{\gamma }_{ij},{\delta }_{ij},{\gamma }_{ij}^{\prime },{\delta }_{ij}^{\prime })}_{LR},$ ${\tilde{x}}_j^I={(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR},$ ${\tilde{b}}_i^I={(b_i,{\eta }_i,{\vartheta }_i,{\eta }_i^{\prime },{\vartheta }_i^{\prime })}_{LR}$ and ${\tilde{c}}_j^I={(c_j,{\kappa }_j,{\lambda }_j,{\kappa }_j^{\prime },{\lambda }_j^{\prime })}_{LR},$ then FIFLP problem (1) can be written as:

  $$\begin{array}{cc}\hfill \max & \sum _{j=1}^n{(c_j,{\kappa }_j,{\lambda }_j,{\kappa }_j^{\prime },{\lambda }_j^{\prime })}_{LR}\odot {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}\hfill \\ \hfill \text{s.t.}& \sum _{j=1}^n{(a_{ij},{\gamma }_{ij},{\delta }_{ij},{\gamma }_{ij}^{\prime },{\delta }_{ij}^{\prime })}_{LR}\odot {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}\hfill \\ & \{\precsim ,\approx ,\succsim \}{(b_i,{\eta }_i,{\vartheta }_i,{\eta }_i^{\prime },{\vartheta }_i^{\prime })}_{LR},\text{for}i\in I\hfill \\ & {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}\in \mathcal{I}\left(\Re \right),\text{for}j\in J\hfill \end{array}$$

• Step 2. Let ${(c_j,{\kappa }_j,{\lambda }_j,{\kappa }_j^{\prime },{\lambda }_j^{\prime })}_{LR}\odot {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}={(s_j,{\tau }_j,{\omega }_j,{\tau }_j^{\prime },{\omega }_j^{\prime })}_{LR}$ and ${(a_{ij},{\gamma }_{ij},{\delta }_{ij},{\gamma }_{ij}^{\prime },{\delta }_{ij}^{\prime })}_{LR}\odot {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}={(m_{ij},l_{ij},r_{ij},l_{ij}^{\prime },r_{ij}^{\prime })}_{LR},$ then FIFLP problem (2) can be rewritten as follows:

  $$\begin{array}{cc}\hfill \max & \sum _{j=1}^n{(s_j,{\tau }_j,{\omega }_j,{\tau }_j^{\prime },{\omega }_j^{\prime })}_{LR}\hfill \\ \hfill \text{s.t.}& \sum _{j=1}^n{(m_{ij},l_{ij},r_{ij},l_{ij}^{\prime },r_{ij}^{\prime })}_{LR}\{\precsim ,\approx ,\succsim \}{(b_i,{\eta }_i,{\vartheta }_i,{\eta }_i^{\prime },{\vartheta }_i^{\prime })}_{LR},\text{for}i\in I\hfill \\ & {(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}\in \mathcal{I}\left(\Re \right),\text{for}j\in J\hfill \end{array}$$

• Step 3. Applying score ($\mathcal{S}$) and accuracy ($\mathcal{A}$) indices, FIFLP problem (3) is transformed into the following crisp non-linear optimisation problem:¹

  $$\begin{array}{cc}\hfill \max & w_1\mathcal{S}\left(\sum _{j=1}^n{(s_j,{\tau }_j,{\omega }_j,{\tau }_j^{\prime },{\omega }_j^{\prime })}_{LR}\right)\hfill \\ & +w_2\mathcal{A}\left(\sum _{j=1}^n{(s_j,{\tau }_j,{\omega }_j,{\tau }_j^{\prime },{\omega }_j^{\prime })}_{LR}\right)\hfill \\ \hfill \text{s.t.}& \mathcal{A}\left(\sum _{j=1}^n{(m_{ij},l_{ij},r_{ij},l_{ij}^{\prime },r_{ij}^{\prime })}_{LR}\right)\hfill \\ & \{\le ,=,\ge \}\mathcal{A}\left({(b_i,{\eta }_i,{\vartheta }_i,{\eta }_i^{\prime },{\vartheta }_i^{\prime })}_{LR}\right),\text{for}i\in I\hfill \\ & {\rho }_j\le {\rho }_j^{\prime },{\sigma }_j\le {\sigma }_j^{\prime },{\rho }_j\ge 0,\hfill \\ & {\sigma }_j\ge 0,{\rho }_j^{\prime }\ge 0,{\sigma }_j^{\prime }\ge 0,\text{for}j\in J\hfill \end{array}$$

  where w₁ ≥ 0, w₂ ≥ 0 and $w_1+w_2=1$.

• Step 4. Solve the crisp non-linear programming problem (4), obtained in Step 3, to find the optimal solution x_j, ρ_j, σ_j, ${\rho }_j^{\prime },$ ${\sigma }_j^{\prime }$ and put their values into ${\tilde{x}}_j^I={(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}$.

• Step 5. Find the optimal intuitionistic fuzzy value of FIFLP problem (1) by putting ${\tilde{x}}_j^I={(x_j,{\rho }_j,{\sigma }_j,{\rho }_j^{\prime },{\sigma }_j^{\prime })}_{LR}$ into ${\sum }_{j=1}^n{\tilde{c}}_j^I\odot {\tilde{x}}_j^I$.

Singh and Yadav’s method (Singh & Yadav, 2017) has the following shortcomings and limitations, which make it unsuitable for finding solutions of FIFLP problem (1) with unique optimal objective values:

• The order relation used with the objective function is not the same as the one used with the inequality constraints. Notice that, in the case of the objective function, a weighted sum of the score and accuracy indices is used, and in the case of the inequality constraints only the information provided by the accuracy index is used. Thus, Singh and Yadav (2017) did not adhere to their own order relation ≾.

• The order relations used with the objective function and the inequality constraints cannot rank any two LR-type IFNs; hence, the uniqueness of the optimal objective value of FIFLP problem (1) cannot be guaranteed.

• In general, the equality constraints are not satisfied exactly. Notice the  ≈  in place of = in FIFLP problems (1)–(3). Furthermore,  ≈  does not conform with the order relation given in Definition 4 since only the accuracy index is used in the constraints of problem (4). In Section 3.1, it is shown that this shortcoming may lead to unsatisfactory solutions of FIFLP problem (1).

In the next section, we propose some modifications to the method of Singh and Yadav (2017) to eliminate the above-mentioned shortcomings and limitations.