# Extended Transportation Models Based on Picture Fuzzy Sets

Authors: Muhammad Athar Mehmood; Shahida Bashir
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Picture fuzzy set (PFS) is an extension of intuitionistic fuzzy set, and it is capable to analyze the transportation problems that contain uncertain and vague information. PFSs are applicable in situations where human decisions demand more variety of responses: yes, abstain, no, and rejection which cannot be addressed in the traditional FSs and IFSs. In this paper, we propose a technique to solve fully picture fuzzy transportation problems (FPFTPs) which are based on picture fuzzy linear programming formulation. Fully picture fuzzy transportation problems are developed by considering all the variables and parameters as nonnegative trapezoidal picture fuzzy numbers. A ranking function is used to transform picture fuzzy numbers into crisp numbers. A model is presented to explain the suggested scheme. Finally, comparison analysis of fully picture fuzzy transportation model with fully intuitionistic fuzzy transportation model and fully fuzzy transportation model is presented with pictorial illustrations.


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## Body

## 1. Introduction

Zadeh [1] proposed the notion of fuzzy set (FS) theory to tackle the problems involving vague information. Atanassov [2] introduced intuitionistic fuzzy set (IFS), which is characterized by a membership function as well as a nonmembership function. Although IFSs have vast applications in many fields, it cannot provide all the information. In the voting process, we can vote in favor of someone, abstain, against someone, and even refuse to cost the vote, which cannot be handled by IFSs. To tackle such type of situations, Cuong [3] gave the idea of PFS, which is generalized structure of FS and IFS and further examined their basic properties and laws. In PFSs theory, we study about positive, neutral, and negative membership degrees of each element belonging to set. In practical life, PFSs theory contribute a significant role in medical diagnosis, career selection, decision making (DM), engineering, and networking.Dubois and Prade [4] discussed basic arithmetic operations related to fuzzy numbers. Bellman and Zadeh [5] proposed the idea of decision making in fuzzy environment. Tanaka et al. [6] presented fuzzy linear programming (FLP) problems. Zimmerman [7] analyzed multiobjective functions in FLP problems. Ganesan and Veeramani [8] studied FLP problems by using trapezoidal fuzzy numbers. Lotfi et al. [9] suggested the lexicography method to solve fully FLP problems and obtained approximate solutions. Allahviranloo et al. [10] solved fully fuzzy linear system and achieved the general solutions. Kaur and Kumar [11] proposed Mehar’s method to solve fully FLP problems by using LR fuzzy numbers. Pérez-Cañedo et al. [12] used lexicographical method and solved fully FLP problems having inequality constraints. Akram et al. [13] proposed Pythagorean FLP problems with equality constraints. Akram et al. [13] introduced LR-type Pythagorean fuzzy numbers and introduced a scheme to solve LR-type fully Pythagorean FLP problems with equality constraints. Mehmood et al. [14, 15] developed different techniques to find optimal solutions of fully bipolar FLP problems. Ahmed et al. [16] solved fully FLP problems in bipolar neutrosophic environment. Akram et al. [17] presented a scheme and obtained optimal solutions of fully FLP problems. For more information about FLP, the reader may study [18–20].In real life, we have to minimize the transportation cost in transportation problems (TPs) because the companies have to deliver the products from different sources to numerous destinations [21]. Gani and Abbas [22] suggested a new technique to obtain optimal solution of TPs in intuitionistic fuzzy environment. TPs are being applied in many areas like networking [23], shipping [24], production [25], and shortest route problems [26]. Singh and Yadev [27] solved fuzzy transportation problems (FTPs) and obtained optimal solution by applying accuracy function. Kumar et al. [28] have proposed two algorithms to achieve initial basic solution and optimal cost of FTPs. Nagoorgani and Razak [29] developed a method to minimize the cost function by using trapezoidal fuzzy numbers (TrFNs). Kaur and Kumar [30] solved FTPs and determined optimal solution. Singh and Yadav [31] proposed a method to solve FTPs by using intuitionistic fuzzy numbers as supply and demand. Basirzadeh [32] worked on FTPs and presented three different kinds of problems. Kumar and Hussain [33] used a computational approach to tackle fully intuitionistic fuzzy real life TPs. Abhishekh and Nishad [34] solved LR-intuitionistic FTPs by using ranking approach. Kaur et al. [35] proposed different schemes to solve fully FTPs by using TrFNs. Wang et al. [36] introduced geometric operators in picture fuzzy environment and studied decision-making problems (DMPs).PFSs are applied to handle real life TPs that involve uncertainty and have been effectively used in communication, management, and DM. Akram et al. [37] proposed the idea of complex PFS, which is generalized form of complex IFS and developed a DM model. Shit et al. [38] used harmonic operators with TrPFNs and showed its significance in multicriteria decision-making (MCDM) problems. Akram et al. [39] investigated shortest route problems by using trapezoidal picture fuzzy numbers (TrPFNs). Geetha and Selvakumari [40] solved picture fuzzy transportation problems and obtained minimum transportation cost. Mahmoodirad et al. [41] studied the existing shortcomings and proposed a method to handle TPs involving intuitionistic fuzzy numbers. Kane et al. [42] suggested a FLP scheme to solve TPs by using triangular fuzzy numbers. Veeramani et al. [43] proposed a technique regarding multiobjective fractional TP on the basis of NGP approach. Ali and Ansari [44] presented Fermatean fuzzy bipolar soft set (FFBSS) model and studied its basic properties. Tchier et al. [45] combined PFSs and soft expert sets and introduced a hybrid model which is used to analyze DMPs. Das [46] defined score function to get IBFS and solved neutrosophic TPs. Kané et al. [47] proposed two-step scheme to solve fully FTP by using all parameters as trapezoidal fuzzy numbers. Ali et al. [48] studied group DMPs and developed a hybrid model by considering bipolar soft expert sets. Ashraf et al. [49] explored MSM operator in the form of IVPFS and interpreted its applications. On the basis of CIVPFSs, Ali et al. [50] introduced Einstein operational laws by applying t-norm. Sahu et al. [51] analyzed student’s career selection in PFS environment. Yildirim and Yıldırım [52] used picture fuzzy VIKOR technique and evaluated satisfaction level of people regarding municipality services. The motivation of this manuscript is described as follows:(1)
PFSs manage the problems involving uncertainty more expeditiously as compared with FSs and IFSs.(2)
No one has yet introduced this particular concept of FPFTPs which is based on picture fuzzy linear programming (PFLP) formulation.The main contributions of this article are depicted as follows:(1)
We propose a scheme to solve FPFTPs based upon PFLP formulation(2)
We apply suggested technique to solve FPFTPs by considering all the variables as nonnegative TrPFNs(3)
We obtain picture fuzzy transportation cost/optimal value in the form of TrPFNs(4)
The supremacy of the proposed scheme is investigated by comparative analysis with existing approachesThe rest of the article is arranged as follows. Introductory concepts are depicted in Section2. The proposed scheme is elucidated in Section 3. A model regarding FPFTPs is considered in Section 4. Comparison analysis is discussed in Section 5, and conclusion is given in the last section.


## 2. Preliminaries

In this section, we review some preliminary concepts regarding PFSs.Definition 1.
(see [3]). A PFS P on a universal set X is an object of the form as follows:(1)P=x,μPx,ηPx,νPx|x∈X,where μPx∈0,1,ηPx∈0,1,νPx∈0,1 denote positive, neutral, and negative membership degrees, respectively, of element x∈P with 0≤μPx+ηPx+νPx≤1, ∀x∈X, and ΠPx=1−μPx−ηPx−νPx is said to be refusal degree of x in set P.Definition 2.
(see [53]). A TrPFN P=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ is a PFS defined on ℝ, whose positive μP, neutral ηP, and negative νP membership functions are, respectively, defined as follows:(2)μPx=x−u1ωs−u1,u1≤x≤s,ω,s≤x≤t,v1−xωv1−t,t≤x≤v1,0,otherwise,ηPx=x−u2ϑs−u2,u2≤x≤s,ϑ,s≤x≤t,v2−xϑv2−t,t≤x≤v2,0,otherwise,νPx=s−x+ζx−u3s−u3,u3≤x≤s,ζ,s≤x≤t,x−t+ζv3−xv3−t,t≤x≤v3,1,otherwise,where u3≤u2≤u1≤s≤t≤v1≤v2≤v3 and the values ω,ϑ,andζ indicate maximum degree of μP, maximum degree of ηP, and minimum degree of νP, respectively, such that ω,ϑ,ζ∈0,1, with 0≤ω+ϑ+ζ≤1.Definition 3.
(see [53]). A TrPFN P=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ is said to be nonnegative (respectively, nonpositive) if u3≥0 (respectively, v3≤0) and P is unrestricted TrPFN if u3 is any real number.Definition 4.
(see [53]). Let P1=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ and P2=u3′,u2′,u1′,s′,t′,v1′,v2′,v3′;ω′,ϑ′,ζ′ be two TrPFNs and λ be real number, then(1)
P1⊕P2=u3+u3′,u2+u2′,u1+u1′,s+s′,t+t′,v1+v1′,v2+v2′,v3+v3′;ω+ω′−ωω′,ϑϑ′,ζζ′(2)
−P1=−v3,−v2,−v1,−t,−s,−u1,−u2,−u3;ω,ϑ,ζ(3)
P1⊖P2=u3−v3′,u2−v2′,u1−v1′,s−t′,t−s′,v1−u1′,v2−u2′,v3−u3′;ω+ω′−ωω′,ϑϑ′,ζζ′(4)
λP1=λu3,λu2,λu1,λs,λt,λv1,λv2,λv3;ωλ,ϑλ,1−1−ζλ,λ≥0λv3,λv2,λv1,λt,λs,λu1,λu2,λu3;ω−λ,ϑ−λ,1−1−ζ−λ,λ<0(5)
P1⊗P2=U3,U2,U1,S,T,V1,V2,V3;ωω′,ϑϑ′,ζ+ζ′−ζζ′where(3)U1=minu1u1′,v1u1′,u1≥0,minu1v1′,v1u1′,u1<0,v1≥0,minu1v1′,v1v1′,v1<0,U2=minu2u2′,v2u2′,u2≥0,minu2v2′,v2u2′,u2<0,v2≥0,minu2v2′,v2v2′,v2<0,U3=minu3u3′,v3u3′,u3≥0,minu3v3′,v3u3′,u3<0,v3≥0,minu3v3′,v3v3′,v3<0,S=minss′,ts′,s≥0,minst′,ts′,s<0,t≥0,minst′,tt′,t<0,T=maxst′,tt′,s≥0,maxss′,tt′,s<0,t1≥0,maxss′,ts′,t1<0,V1=maxu1v1′,v1v1′,u1≥0,maxu1u1′,v1v1′,u1<0,v1≥0,maxu1u1′,v1u1′,v1<0,V2=maxu2v2′,v2v2′,u2≥0,maxu2u2′,v2v2′,u2<0,v2≥0,maxu2u2′,v2u2′,v2<0,V3=maxu3v3′,v3v3′,u3≥0,maxu3u3′,v3v3′,u3<0,v3≥0,maxu3u3′,v3u3′,v3<0.Definition 5.
(see [53]). Two TrPFNs P1=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ and P2=u3′,u2′,u1′,s′,t′,v1′,v2′,v3′
;ω′,ϑ′,ζ′ are said to be equal if u3=u3′, u2=u2′, u1=u1′, s=s′, t=t′, v1=v1′, v2=v2′, v3=v3′, ω=ω′, ϑ=ϑ′, and ζ=ζ′.Definition 6.
(see [53]). A TrPFN P1=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ is said to be zero if u3=0, u2=0, u1=0, s=0, t=0, v1=0, v2=0, v3=0, ω=0, ϑ=0, and ζ=0.Definition 7 (see [53]).
Letp=u3,u2,u1,s,t,v1,v2,v3;ω,ϑ,ζ be a TrPFN, then ranking of P is symbolized as ℜP and defined as(4)ℜP=ωs+t+u1+v1+ϑs+t+u2+v2+1−ζs+t+u3+v34.


## 3. Fully Picture Fuzzy Transportation Problems

In this section, we present a scheme to solve FPFTPs based on PFLP formulation. The steps are explained as follows.


### 3.1. Steps to Find Picture Fuzzy Optimal Solution

Consider a FPFTP containingq sources and r destinations in which cost, supply, and demand are used as TrPFNs CklP˜, A˜kP, and B˜lP, respectively.(5)MinimizeZ=∑k=1q∑l=1rCklP˜⊗XklP˜,subject to(6)∑l=1rXklP˜=A˜kP,∀k=1,2,3,…,q,∑l=1rXklP˜=A˜kP,∀k=1,2,3,…,q,X˜klP≥0,∀k=1,2,3,…,q,∀l=1,2,3,…,r,where CklP˜,XklP˜,A˜kP, and B˜lP are all nonnegative TrPFNs.To solve FPFTP equation (1), we give a criterion for picture fuzzy optimal solution (PFOS).Definition 8.
A PFOS of the FPFTP equation (1) with TrPFNs will be TrPFNs X˜klP if(i)
X˜klP are nonnegative TrPFNs.(ii)
ℜ∑l=1rXklP˜=ℜA˜kP, ∀k=1,2,3,…,q.(iii)
ℜ∑k=1qXklP˜=ℜB˜lP, ∀l=1,2,3,…,r.(iv)
If there exists any TrPFNsX˜kl′P satisfying the above three conditions, then(7)ℜ∑k=1q∑l=1rCklP˜⊗XklP˜<ℜ∑k=1q∑l=1rCklP˜⊗XklP˜.
Now, we explain the steps to determine the PFOS of FPFTP as given in equation (2).Step 1.
Calculate total picture fuzzy supply and total picture fuzzy demand.
If(8)∑l=1rB˜lP=∑k=1qA˜kP,it is a balanced FPFTP.
If(9)∑l=1rB˜lP≠∑k=1qA˜kP,it is an unbalanced FPFTP.
That is,(10)a,b,c,d,e,f,g,h;α,β,γ≠a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′.
Then, one of the following case arises:Case (a):(11)a≤a′,b−a≤b′−a′,c−b≤c′−b′,d−c≤d′−c′,e−d≤e′−d′,f−e≤f′−e′,g−f≤g′−f′,h−g≤h′−g′.Case (b):(12)a≥a′,b−a≥b′−a′,c−b≥c′−b′,d−c≥d′−c′,e−d≥e′−d′,f−e≥f′−e′,g−f≥g′−f′,h−g≥h′−g′.Case (c): when the above two cases do not hold, then there may exists infinitely many nonnegative TrPFNs:(13)a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,such that(14)a,b,c,d,e,f,g,h;α,β,γ⊕a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,but we have to determine such nonnegative TrPFNs as(15)a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,which satisfy the following conditions.(i)
a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1 and a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′ are nonnegative TrPFNs.(ii)
It satisfies(16)a,b,c,d,e,f,g,h;α,β,γ⊕a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′.(iii)
Further, if there exists two nonnegative TrPFNs,(17)s1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1,s1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′,such that(18)a,b,c,d,e,f,g,h;α,β,γ⊕s1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕s1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′,then(19)ℜs1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1≥ℜa1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,ℜs1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′≥ℜa1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′.Step 2.
Suppose(20)CklP˜=ckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl,XklP˜=xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl,A˜kP=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,B˜lP=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,then FPFTP equation (1) can be transformed as follows:(21)Minimize Z=∑k=1q∑l=1rckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl⊗xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl,subject to(22)∑l=1rxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,∑k=1qxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 3.
By applying arithmetic operations as described in Section2 and putting(23)ckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl⊗xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=dkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,then the fully picture fuzzy linear programming problem (FPFLPP) equation (2) can be transformed as follows:(24)Minimize Z=∑k=1q∑l=1rdkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,subject to(25)∑l=1rxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,∑k=1qxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 4.
Now, by applying Definitions5 and 7, then the FPFLPP equation (3) can be transformed as follows:(26)MinimizeZ=∑k=1q∑l=1rℜdkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,subject to(27)∑l=1rxkl1=ak1,∀k=1,2,3,…,q,∑l=1rxkl2=ak2,∀k=1,2,3,…,q,∑l=1rxkl3=ak3,∀k=1,2,3,…,q,∑l=1rxkl4=ak4,∀k=1,2,3,…,q,∑l=1rxkl5=ak5,∀k=1,2,3,…,q,∑l=1rxkl6=ak6,∀k=1,2,3,…,q,∑l=1rxkl7=ak7,∀k=1,2,3,…,q,∑l=1rxkl8=ak8,∀k=1,2,3,…,q,∑l=1rσkl=κk,∀k=1,2,3,…,q,∑l=1rτkl=λk,∀k=1,2,3,…,q,∑l=1rυkl=θk,∀k=1,2,3,…,q,∑k=1qxkl1=bl1,∀l=1,2,3,…,r,∑k=1qxkl2=bl2,∀l=1,2,3,…,r,∑k=1qxkl3=bl3,∀l=1,2,3,…,r,∑k=1qxkl4=bl4,∀l=1,2,3,…,r,∑k=1qxkl5=bl5,∀l=1,2,3,…,r,∑k=1qxkl6=bl6,∀l=1,2,3,…,r,∑k=1qxkl7=bl7,∀l=1,2,3,…,r,∑k=1qxkl8=bl8,∀l=1,2,3,…,r,∑k=1qσkl=ηl,∀l=1,2,3,…,r,∑k=1qτkl=ϵl,∀l=1,2,3,…,r,∑k=1qυkl=χl,∀l=1,2,3,…,r,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 5.
To obtain PFOS, solve the following crisp linear/nonlinear programming problem (LPP):(28)MinimizeZ=∑k=1q∑l=1r14μkldkl3+dkl4+dkl5+dkl6+δkldkl2+dkl4+dkl5+dkl7+1−ηkldkl1+dkl4+dkl5+dkl8,subject to(29)∑l=1rxkl1=ak1,∀k=1,2,3,…,q,∑l=1rxkl2=ak2,∀k=1,2,3,…,q,∑l=1rxkl3=ak3,∀k=1,2,3,…,q,∑l=1rxkl4=ak4,∀k=1,2,3,…,q,∑l=1rxkl5=ak5,∀k=1,2,3,…,q,∑l=1rxkl6=ak6,∀k=1,2,3,…,q,∑l=1rxkl7=ak7,∀k=1,2,3,…,q,∑l=1rxkl8=ak8,∀k=1,2,3,…,q,∑l=1rσkl=κk,∀k=1,2,3,…,q,∑l=1rτkl=λk,∀k=1,2,3,…,q,∑l=1rυkl=θk,∀k=1,2,3,…,q,∑k=1qxkl1=bl1,∀l=1,2,3,…,r,∑k=1qxkl2=bl2,∀l=1,2,3,…,r,∑k=1qxkl3=bl3,∀l=1,2,3,…,r,∑k=1qxkl4=bl4,∀l=1,2,3,…,r,∑k=1qxkl5=bl5,∀l=1,2,3,…,r,∑k=1qxkl6=bl6,∀l=1,2,3,…,r,∑k=1qxkl7=bl7,∀l=1,2,3,…,r,∑k=1qxkl8=bl8,∀l=1,2,3,…,r,∑k=1qσkl=ηl,∀l=1,2,3,…,r,∑k=1qτkl=ϵl,∀l=1,2,3,…,r,∑k=1qυkl=χl,∀l=1,2,3,…,r,xkl1≥0,xkl2−xkl1≥0,xkl3−xkl2≥0,xkl4−xkl3≥0,xkl5−xkl4≥0,xkl6−xkl5≥0,xkl7−xkl6≥0,xkl8−xkl7≥0,σkl≥0,τkl≥0,υkl≥0,σkl+τkl+υkl≤1,∀k=1,2,…,q,∀l=1,2,…,r.Step 6.
Solve crisp linear/non-LPP equation (28) to get optimal solution:(30)x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl,σkl∗,τkl∗,υkl∗.Step 7.
Find the PFOSX˜klP of the FPFTP (1) by putting values of x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl,σkl∗,τkl∗, and υkl∗ in X˜klP=x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl;σkl∗,τkl∗,υkl∗.Step 8.
Find picture fuzzy transportation cost/optimal value of FPFTP equation (1) by assigning values of X˜klP, as obtained in Step (7), in ∑k=1q∑l=1rCklP˜⊗XklP˜.


## 3.1. Steps to Find Picture Fuzzy Optimal Solution

Consider a FPFTP containingq sources and r destinations in which cost, supply, and demand are used as TrPFNs CklP˜, A˜kP, and B˜lP, respectively.(5)MinimizeZ=∑k=1q∑l=1rCklP˜⊗XklP˜,subject to(6)∑l=1rXklP˜=A˜kP,∀k=1,2,3,…,q,∑l=1rXklP˜=A˜kP,∀k=1,2,3,…,q,X˜klP≥0,∀k=1,2,3,…,q,∀l=1,2,3,…,r,where CklP˜,XklP˜,A˜kP, and B˜lP are all nonnegative TrPFNs.To solve FPFTP equation (1), we give a criterion for picture fuzzy optimal solution (PFOS).Definition 8.
A PFOS of the FPFTP equation (1) with TrPFNs will be TrPFNs X˜klP if(i)
X˜klP are nonnegative TrPFNs.(ii)
ℜ∑l=1rXklP˜=ℜA˜kP, ∀k=1,2,3,…,q.(iii)
ℜ∑k=1qXklP˜=ℜB˜lP, ∀l=1,2,3,…,r.(iv)
If there exists any TrPFNsX˜kl′P satisfying the above three conditions, then(7)ℜ∑k=1q∑l=1rCklP˜⊗XklP˜<ℜ∑k=1q∑l=1rCklP˜⊗XklP˜.
Now, we explain the steps to determine the PFOS of FPFTP as given in equation (2).Step 1.
Calculate total picture fuzzy supply and total picture fuzzy demand.
If(8)∑l=1rB˜lP=∑k=1qA˜kP,it is a balanced FPFTP.
If(9)∑l=1rB˜lP≠∑k=1qA˜kP,it is an unbalanced FPFTP.
That is,(10)a,b,c,d,e,f,g,h;α,β,γ≠a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′.
Then, one of the following case arises:Case (a):(11)a≤a′,b−a≤b′−a′,c−b≤c′−b′,d−c≤d′−c′,e−d≤e′−d′,f−e≤f′−e′,g−f≤g′−f′,h−g≤h′−g′.Case (b):(12)a≥a′,b−a≥b′−a′,c−b≥c′−b′,d−c≥d′−c′,e−d≥e′−d′,f−e≥f′−e′,g−f≥g′−f′,h−g≥h′−g′.Case (c): when the above two cases do not hold, then there may exists infinitely many nonnegative TrPFNs:(13)a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,such that(14)a,b,c,d,e,f,g,h;α,β,γ⊕a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,but we have to determine such nonnegative TrPFNs as(15)a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′,which satisfy the following conditions.(i)
a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1 and a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′ are nonnegative TrPFNs.(ii)
It satisfies(16)a,b,c,d,e,f,g,h;α,β,γ⊕a1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕a1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′.(iii)
Further, if there exists two nonnegative TrPFNs,(17)s1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1,s1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′,such that(18)a,b,c,d,e,f,g,h;α,β,γ⊕s1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1=a′,b′,c′,d′,e′,f′,g′,h′;α′,β′,γ′⊕s1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′,then(19)ℜs1,t1,u1,v1,w1,x1,y1,z1;α1,β1,γ1≥ℜa1,b1,c1,d1,e1,f1,g1,h1;α1,β1,γ1,ℜs1′,t1′,u1′,v1′,w1′,x1′,y1′,z1′;α1′,β1′,γ1′≥ℜa1′,b1′,c1′,d1′,e1′,f1′,g1′,h1′;α1′,β1′,γ1′.Step 2.
Suppose(20)CklP˜=ckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl,XklP˜=xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl,A˜kP=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,B˜lP=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,then FPFTP equation (1) can be transformed as follows:(21)Minimize Z=∑k=1q∑l=1rckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl⊗xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl,subject to(22)∑l=1rxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,∑k=1qxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 3.
By applying arithmetic operations as described in Section2 and putting(23)ckl1,ckl2,ckl3,ckl4,ckl5,ckl6,ckl7,ckl8;ξkl,ψkl,ωkl⊗xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=dkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,then the fully picture fuzzy linear programming problem (FPFLPP) equation (2) can be transformed as follows:(24)Minimize Z=∑k=1q∑l=1rdkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,subject to(25)∑l=1rxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=ak1,ak2,ak3,ak4,ak5,ak6,ak7,ak8;κk,λk,θk,∑k=1qxkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl=bl1,bl2,bl3,bl4,bl5,bl6,bl7,bl8;ηl,ϵl,χl,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 4.
Now, by applying Definitions5 and 7, then the FPFLPP equation (3) can be transformed as follows:(26)MinimizeZ=∑k=1q∑l=1rℜdkl1,dkl2,dkl3,dkl4,dkl5,dkl6,dkl7,dkl8;μkl,δkl,ηkl,subject to(27)∑l=1rxkl1=ak1,∀k=1,2,3,…,q,∑l=1rxkl2=ak2,∀k=1,2,3,…,q,∑l=1rxkl3=ak3,∀k=1,2,3,…,q,∑l=1rxkl4=ak4,∀k=1,2,3,…,q,∑l=1rxkl5=ak5,∀k=1,2,3,…,q,∑l=1rxkl6=ak6,∀k=1,2,3,…,q,∑l=1rxkl7=ak7,∀k=1,2,3,…,q,∑l=1rxkl8=ak8,∀k=1,2,3,…,q,∑l=1rσkl=κk,∀k=1,2,3,…,q,∑l=1rτkl=λk,∀k=1,2,3,…,q,∑l=1rυkl=θk,∀k=1,2,3,…,q,∑k=1qxkl1=bl1,∀l=1,2,3,…,r,∑k=1qxkl2=bl2,∀l=1,2,3,…,r,∑k=1qxkl3=bl3,∀l=1,2,3,…,r,∑k=1qxkl4=bl4,∀l=1,2,3,…,r,∑k=1qxkl5=bl5,∀l=1,2,3,…,r,∑k=1qxkl6=bl6,∀l=1,2,3,…,r,∑k=1qxkl7=bl7,∀l=1,2,3,…,r,∑k=1qxkl8=bl8,∀l=1,2,3,…,r,∑k=1qσkl=ηl,∀l=1,2,3,…,r,∑k=1qτkl=ϵl,∀l=1,2,3,…,r,∑k=1qυkl=χl,∀l=1,2,3,…,r,where xkl1,xkl2,xkl3,xkl4,xkl5,xkl6,xkl7,xkl8;σkl,τkl,υkl are nonnegative TrPFNs.Step 5.
To obtain PFOS, solve the following crisp linear/nonlinear programming problem (LPP):(28)MinimizeZ=∑k=1q∑l=1r14μkldkl3+dkl4+dkl5+dkl6+δkldkl2+dkl4+dkl5+dkl7+1−ηkldkl1+dkl4+dkl5+dkl8,subject to(29)∑l=1rxkl1=ak1,∀k=1,2,3,…,q,∑l=1rxkl2=ak2,∀k=1,2,3,…,q,∑l=1rxkl3=ak3,∀k=1,2,3,…,q,∑l=1rxkl4=ak4,∀k=1,2,3,…,q,∑l=1rxkl5=ak5,∀k=1,2,3,…,q,∑l=1rxkl6=ak6,∀k=1,2,3,…,q,∑l=1rxkl7=ak7,∀k=1,2,3,…,q,∑l=1rxkl8=ak8,∀k=1,2,3,…,q,∑l=1rσkl=κk,∀k=1,2,3,…,q,∑l=1rτkl=λk,∀k=1,2,3,…,q,∑l=1rυkl=θk,∀k=1,2,3,…,q,∑k=1qxkl1=bl1,∀l=1,2,3,…,r,∑k=1qxkl2=bl2,∀l=1,2,3,…,r,∑k=1qxkl3=bl3,∀l=1,2,3,…,r,∑k=1qxkl4=bl4,∀l=1,2,3,…,r,∑k=1qxkl5=bl5,∀l=1,2,3,…,r,∑k=1qxkl6=bl6,∀l=1,2,3,…,r,∑k=1qxkl7=bl7,∀l=1,2,3,…,r,∑k=1qxkl8=bl8,∀l=1,2,3,…,r,∑k=1qσkl=ηl,∀l=1,2,3,…,r,∑k=1qτkl=ϵl,∀l=1,2,3,…,r,∑k=1qυkl=χl,∀l=1,2,3,…,r,xkl1≥0,xkl2−xkl1≥0,xkl3−xkl2≥0,xkl4−xkl3≥0,xkl5−xkl4≥0,xkl6−xkl5≥0,xkl7−xkl6≥0,xkl8−xkl7≥0,σkl≥0,τkl≥0,υkl≥0,σkl+τkl+υkl≤1,∀k=1,2,…,q,∀l=1,2,…,r.Step 6.
Solve crisp linear/non-LPP equation (28) to get optimal solution:(30)x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl,σkl∗,τkl∗,υkl∗.Step 7.
Find the PFOSX˜klP of the FPFTP (1) by putting values of x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl,σkl∗,τkl∗, and υkl∗ in X˜klP=x1∗kl,x2∗kl,x3∗kl,x4∗kl,x5∗kl,x6∗kl,x7∗kl,x8∗kl;σkl∗,τkl∗,υkl∗.Step 8.
Find picture fuzzy transportation cost/optimal value of FPFTP equation (1) by assigning values of X˜klP, as obtained in Step (7), in ∑k=1q∑l=1rCklP˜⊗XklP˜.


## 4. Numerical Examples

In this section, to explain the proposed methodology, we present a model related to FPFTPs.Example 1.
(FPFTP based on PFLP formulation). A company containing two plants produces urea fertilizer with picture fuzzy availabilities ofProof.
15,35,45,60,75,95,110,200;0.92,0.02,0.01ton and 10,25,50,70,90,120,130,190;0.80,0.01,0.02ton, respectively, and supply it to two cities. The picture fuzzy demand at two cities is 20,40,50,65,85,100,140,160;0.92,0.01,0.02ton and 30,35,40,50,100,110,150,170;0.76,0.02,0.01ton. The price per ton by delivering the urea fertilizer at the two cities is presented in Table 1.
Find the minimum picture fuzzy transportation cost.(31)Minimize110,150,190,210,270,300,350,390;0.7,0.1,0.1⊗x˜11⊕130,180,220,250,290,340,370,410;0.8,0.1,0.1⊗x˜12⊕150,250,290,350,400,440,460,500;0.6,0.1,0.2⊗x˜21⊕190,210,250,270,310,330,380,430;0.6,0.1,0.2⊗x˜22,subject to(32)x˜11⊕x˜12=15,35,45,60,75,95,110,200;0.92,0.02,0.01,x˜21⊕x˜22=10,25,50,70,90,120,130,190;0.80,0.01,0.02,x˜11⊕x˜21=20,40,50,65,85,100,140,160;0.92,0.01,0.02,x˜12⊕x˜22=30,35,40,50,100,110,150,170;0.76,0.02,0.01,where x˜11, x˜12, x˜21, and x˜22 are nonnegative TrPFNs.
Now,Total supply =25,60,95,130,165,215,240,390;0.9840,0.0002,0.0002Total demand =50,75,90,115,185,210,290,330;0.9808,0.0002,0.0002
For an unbalanced FPFTP, we add dummy row and dummy column to make a balanced picture fuzzy TP.Dummy row =25,25,25,25,60,60,115,115;0.0000,0.0000,0.0000Dummy column =0,10,30,40,40,65,65,175;0.1666,0.0000,0.0000
Therefore, by supposing picture fuzzy transportation cost of unit quantity of product from dummy source to all destinations and from all sources to dummy destination to be zero TrPFNs, then FPFLPP equation (31) can be transformed as follows:(33)Minimize110,150,190,210,270,300,350,390;0.7,0.1,0.1⊗x˜11⊕130,180,220,250,290,340,370,410;0.8,0.1,0.1⊗x˜12⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗x˜13⊕150,250,290,350,400,440,460,500;0.6,0.1,0.2⊗x˜21⊕190,210,250,270,310,330,380,430;0.6,0.1,0.2⊗x˜22⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗x˜23⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗x˜31⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗x˜32⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗x˜33x˜11⊕x˜12=15,35,45,60,75,95,110,200;0.92,0.02,0.01,subject to(34)x˜11⊕x˜12⊕x˜13=15,35,45,60,75,95,110,200;0.92,0.02,0.01,x˜21⊕x˜22⊕x˜23=10,25,50,70,90,120,130,190;0.80,0.01,0.02,x˜31⊕x˜32⊕x˜33=25,25,25,25,60,60,115,115;0.0000,0.0000,0.0000,x˜11⊕x˜21⊕x˜31=20,40,50,65,85,100,140,160;0.92,0.01,0.02,x˜12⊕x˜22⊕x˜32=30,35,40,50,100,110,150,170;0.76,0.02,0.01,x˜13⊕x˜23⊕x˜33=0,10,30,40,40,65,65,175;0.1666,0.0000,0.0000,where x˜11, x˜12, x˜13, x˜21, x˜22, x˜23, x˜31, x˜32, and x˜33, are nonnegative TrPFNs.
By supposing,(35)x˜11=χ11,δ11,ϵ11,η11,κ11,ϑ11,ω11,ζ11;α11,β11,γ11,x˜12=χ12,δ12,ϵ12,η12,κ12,ϑ12,ω12,ζ12;α12,β12,γ12,x˜13=χ13,δ13,ϵ13,η13,κ13,ϑ13,ω13,ζ13;α13,β13,γ13,x˜21=χ21,δ21,ϵ21,η21,κ21,ϑ21,ω21,ζ21;α21,β21,γ21,x˜22=χ22,δ22,ϵ22,η22,κ22,ϑ22,ω22,ζ22;α22,β22,γ22,x˜23=χ23,δ23,ϵ23,η23,κ23,ϑ23,ω23,ζ23;α23,β23,γ23,x˜31=χ31,δ31,ϵ31,η31,κ31,ϑ31,ω31,ζ31;α31,β31,γ31,x˜32=χ32,δ32,ϵ32,η32,κ32,ϑ32,ω32,ζ32;α32,β32,γ32,x˜33=χ33,δ33,ϵ33,η33,κ33,ϑ33,ω33,ζ33;α33,β33,γ33,where x˜11, x˜12, x˜13, x˜21, x˜22, x˜23, x˜31, x˜32, and x˜33 are nonnegative TrPFNs, then the FPFLPP equation (33) can be transformed as follows:(36)Minimize110,150,190,210,270,300,350,390;0.7,0.1,0.1⊗χ11,δ11,ϵ11,η11,κ11,ϑ11,ω11,ζ11;α11,β11,γ11⊕130,180,220,250,290,340,370,410;0.8,0.1,0.1⊗χ12,δ12,ϵ12,η12,κ12,ϑ12,ω12,ζ12;α12,β12,γ12⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗χ13,δ13,ϵ13,η13,κ13,ϑ13,ω13,ζ13;α13,β13,γ13⊕150,250,290,350,400,440,460,500;0.6,0.1,0.2⊗χ21,δ21,ϵ21,η21,κ21,ϑ21,ω21,ζ21;α21,β21,γ21⊕190,210,250,270,310,330,380,430;0.6,0.1,0.2⊗χ22,δ22,ϵ22,η22,κ22,ϑ22,ω22,ζ22;α22,β22,γ22⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗χ23,δ23,ϵ23,η23,κ23,ϑ23,ω23,ζ23;α23,β23,γ23⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗χ31,δ31,ϵ31,η31,κ31,ϑ31,ω31,ζ31;α31,β31,γ31⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗χ32,δ32,ϵ32,η32,κ32,ϑ32,ω32,ζ32;α32,β32,γ32⊕0,0,0,0,0,0,0,0;0.0,0.0,0.0⊗χ33,δ33,ϵ33,η33,κ33,ϑ33,ω33,ζ33;α33,β33,γ33,subject to(37)χ11,δ11,ϵ11,η11,κ11,ϑ11,ω11,ζ11;α11,β11,γ11⊕χ12,δ12,ϵ12,η12,κ12,ϑ12,ω12,ζ12;α12,β12,γ12⊕χ13,δ13,ϵ13,η13,κ13,ϑ13,ω13,ζ13;α13,β13,γ13=15,35,45,60,75,95,110,200;0.92,0.02,0.01χ21,δ21,ϵ21,η21,κ21,ϑ21,ω21,ζ21;α21,β21,γ21⊕χ22,δ22,ϵ22,η22,κ22,ϑ22,ω22,ζ22;α22,β22,γ22⊕χ23,δ23,ϵ23,η23,κ23,ϑ23,ω23,ζ23;α23,β23,γ23=10,25,50,70,90,120,130,190;0.80,0.01,0.02χ31,δ31,ϵ31,η31,κ31,ϑ31,ω31,ζ31;α31,β31,γ31⊕χ32,δ32,ϵ32,η32,κ32,ϑ32,ω32,ζ32;α32,β32,γ32⊕χ33,δ33,ϵ33,η33,κ33,ϑ33,ω33,ζ33;α33,β33,γ33=25,25,25,25,60,60,115,115;0.0000,0.0000,0.0000χ11,δ11,ϵ11,η11,κ11,ϑ11,ω11,ζ11;α11,β11,γ11⊕χ21,δ21,ϵ21,η21,κ21,ϑ21,ω21,ζ21;α21,β21,γ21⊕χ31,δ31,ϵ31,η31,κ31,ϑ31,ω31,ζ31;α31,β31,γ31=20,40,50,65,85,100,140,160;0.92,0.01,0.02χ12,δ12,ϵ12,η12,κ12,ϑ12,ω12,ζ12;α12,β12,γ12⊕χ22,δ22,ϵ22,η22,κ22,ϑ22,ω22,ζ22;α22,β22,γ22⊕χ32,δ32,ϵ32,η32,κ32,ϑ32,ω32,ζ32;α32,β32,γ32=30,35,40,50,100,110,150,170;0.76,0.02,0.01χ13,δ13,ϵ13,η13,κ13,ϑ13,ω13,ζ13;α13,β13,γ13⊕χ23,δ23,ϵ23,η23,κ23,ϑ23,ω23,ζ23;α23,β23,γ23⊕χ33,δ33,ϵ33,η33,κ33,ϑ33,ω33,ζ33;α33,β33,γ33=0,10,30,40,40,65,65,175;0.1666,0.0000,0.0000.
By applying arithmetic operations as described in Section2, the FPFLPP equation (37) can be transformed as follows:(38)Minimize110χ11,150δ11,190ϵ11,210η11,270κ11,300ϑ11,350ω11,390ζ11;0.7α11,0.1β11,0.1+γ11−0.1γ11⊕130χ12,180δ12,220ϵ12,250η12,290κ12,340ϑ12,370ω12,410ζ12;0.8α12,0.1β12,0.1+γ12−0.1γ12⊕0χ13,0δ13,0ϵ13,0η13,0κ13,0ϑ13,0ω13,0ζ13;0.0α13,0.0β13,0.0+γ13−0.0γ13⊕150χ21,250δ21,290ϵ21,350η21,400κ21,440ϑ21,460ω21,500ζ21;0.6α21,0.1β21,0.2+γ21−0.2γ21⊕190χ22,210δ22,250ϵ22,270η22,310κ22,330ϑ22,380ω22,430ζ22;0.6α22,0.1β22,0.2+γ22−0.2γ22⊕0χ23,0δ23,0ϵ23,0η23,0κ23,0ϑ23,0ω23,0ζ23;0.0α23,0.0β23,0.0+γ23−0.0γ23⊕0χ31,0δ31,0ϵ31,0η31,0κ31,0ϑ31,0ω31,0ζ31;0.0α31,0.0β31,0.0+γ31−0.0γ31⊕0χ32,0δ32,0ϵ32,0η32,0κ32,0ϑ32,0ω32,0ζ32;0.0α32,0.0β32,0.0+γ32−0.0γ32⊕0χ33,0δ33,0ϵ33,0η33,0κ33,0ϑ33,0ω33,0ζ33;0.0α33,0.0β33,0.0+γ33−0.0γ33,subject to(39)χ11+χ12+χ13,δ11+δ12+δ13,ϵ11+ϵ12+ϵ13,η11+η12+η13,κ11+κ12+κ13,ϑ11+ϑ12+ϑ13,ω11+ω12+ω13,ζ11+ζ12+ζ13;α11+α12−α11α12+α13−α11α13−α12α13+α11α12α13,β11β12β13,γ11γ12γ13=15,35,45,60,75,95,110,200;0.92,0.02,0.01χ21+χ22+χ23,δ21+δ22+δ23,ϵ21+ϵ22+ϵ23,η21+η22+η23,κ21+κ22+κ23,ϑ21+ϑ22+ϑ23,ω21+ω22+ω23,ζ21+ζ22+ζ23;α21+α22−α21α22+α23−α21α23−α22α23+α21α22α23,β21β22β23,γ21γ22γ23=10,25,50,70,90,120,130,190;0.80,0.01,0.02χ31+χ32+χ33,δ31+δ32+δ33,ϵ31+ϵ32+ϵ33,η31+η32+η33,κ31+κ32+κ33,ϑ31+ϑ32+ϑ33,ω31+ω32+ω33,ζ31+ζ32+ζ33;α31+α32−α31α32+α33−α31α33−α32α33+α31α32α33,β31β32β33,γ31γ32γ33=25,25,25,25,60,60,115,115;0.0000,0.0000,0.0000χ11+χ21+χ31,δ11+δ21+δ31,ϵ11+ϵ21+ϵ31,η11+η21+η31,κ11+κ21+κ31,ϑ11+ϑ21+ϑ31,ω11+ω21+ω31,ζ11+ζ21+ζ31;α11+α21−α11α21+α31−α11α31−α21α31+α11α21α31,β11β21β31,γ11γ21γ31=20,40,50,65,85,100,140,160;0.92,0.01,0.02χ12+χ22+χ32,δ12+δ22+δ32,ϵ12+ϵ22+ϵ32,η12+η22+η32,κ12+κ22+κ32,ϑ12+ϑ22+ϑ32,ω12+ω22+ω32,ζ12+ζ22+ζ32;α12+α22−α12α22+α32−α12α32−α22α32+α12α22α32,β12β22β32,γ12γ22γ32=30,35,40,50,100,110,150,170;0.76,0.02,0.01χ13+χ23+χ33,δ13+δ23+δ33,ϵ13+ϵ23+ϵ33,η13+η23+η33,κ13+κ23+κ33,ϑ13+ϑ23+ϑ33,ω13+ω23+ω33,ζ13+ζ23+ζ33;α13+α23−α13α23+α33−α13α33−α23α33+α13α23α33,β13β23β33,γ13γ23γ33=0,10,30,40,40,65,65,175;0.1666,0.0000,0.0000.
By applying Definition7, the FPLPP equation (39) can be transformed as follows:(40)Minimizeℜ110χ11+130χ12+150χ21+190χ22,150δ11+180δ12+250δ21+210δ22,190ϵ11+220ϵ12+290ϵ21+250ϵ22,210η11+250η12+350η21+270η22,270κ11+290κ12+400κ21+310κ22,300ϑ11+340ϑ12+440ϑ21+330ϑ22,350ω11+370ω12+460ω21+380ω22,390ζ11+410ζ12+500ζ21+430ζ22;0.7α11+0.8α12−0.56α11α12+0.6α21−0.42α11α21−0.48α12α21+0.336α11α12α21+0.6α22−0.42α11α22−0.48α12α22+0.336α11α12α22−0.36α21α22+0.252α11α21α22+0.288α12α21α22−0.2016α11α12α21α22,0.0001β11β12β21β22,0.1+γ11−0.1γ110.1+γ12−0.1γ120.2+γ21−0.2γ210.2+γ22−0.2γ22,subject to(41)χ11+χ12+χ13,δ11+δ12+δ13,ϵ11+ϵ12+ϵ13,η11+η12+η13,κ11+κ12+κ13,ϑ11+ϑ12+ϑ13,ω11+ω12+ω13,ζ11+ζ12+ζ13;α11+α12−α11α12+α13−α11α13−α12α13+α11α12α13,β11β12β13,γ11γ12γ13=15,35,45,60,75,95,110,200;0.92,0.02,0.01χ21+χ22+χ23,δ21+δ22+δ23,ϵ21+ϵ22+ϵ23,η21+η22+η23,κ21+κ22+κ23,ϑ21+ϑ22+ϑ23,ω21+ω22+ω23,ζ21+ζ22+ζ23;α21+α22−α21α22+α23−α21α23−α22α23+α21α22α23,β21β22β23,γ21γ22γ23=10,25,50,70,90,120,130,190;0.80,0.01,0.02χ31+χ32+χ33,δ31+δ32+δ33,ϵ31+ϵ32+ϵ33,η31+η32+η33,κ31+κ32+κ33,ϑ31+ϑ32+ϑ33,ω31+ω32+ω33,ζ31+ζ32+ζ33;α31+α32−α31α32+α33−α31α33−α32α33+α31α32α33,β31β32β33,γ31γ32γ33=25,25,25,25,60,60,115,115;0.0000,0.0000,0.0000χ11+χ21+χ31,δ11+δ21+δ31,ϵ11+ϵ21+ϵ31,η11+η21+η31,κ11+κ21+κ31,ϑ11+ϑ21+ϑ31,ω11+ω21+ω31,ζ11+ζ21+ζ31;α11+α21−α11α21+α31−α11α31−α21α31+α11α21α31,β11β21β31,γ11γ21γ31=20,40,50,65,85,100,140,160;0.92,0.01,0.02χ12+χ22+χ32,δ12+δ22+δ32,ϵ12+ϵ22+ϵ32,η12+η22+η32,κ12+κ22+κ32,ϑ12+ϑ22+ϑ32,ω12+ω22+ω32,ζ12+ζ22+ζ32;α12+α22−α12α22+α32−α12α32−α22α32+α12α22α32,β12β22β32,γ12γ22γ3230,35,40,50,100,110,150,170;0.76,0.02,0.01χ13+χ23+χ33,δ13+δ23+δ33,ϵ13+ϵ23+ϵ33,η13+η23+η33,κ13+κ23+κ33,ϑ13+ϑ23+ϑ33,ω13+ω23+ω33,ζ13+ζ23+ζ33;α13+α23−α13α23+α33−α13α33−α23α33+α13α23α33,β13β23β33,γ13γ23γ33=0,10,30,40,40,65,65,175;0.1666,0.0000,0.0000.
Now, we solve the following crisp LPP:(42)Minimize140.7α11+0.8α12−0.56α11α12+0.6α21−0.42α11α21−0.48α12α21+0.336α11α12α21+0.6α22−0.42α11α22−0.48α12α22+0.336α11α12α22−0.36α21α22+0.252α11α21α22+0.288α12α21α22−0.2016α11α12α21α22210η11+250η12+350η21+270η22+270κ11+290κ12+400κ21+310κ22+190ϵ11+220ϵ12+290ϵ21+250ϵ22+300ϑ11+340ϑ12+440ϑ21+330ϑ22+0.0001β11β12β21β22210η11+250η12+350η21+270η22+270κ11+290κ12+400κ21+310κ22+150δ11+180δ12+250δ21+210δ22+350ω11+370ω12+460ω21+380ω22+1−0.1+γ11−0.1γ110.1+γ12−0.1γ120.2+γ21−0.2γ210.2+γ22−0.2γ22210η11+250η12+350η21+270η22+270κ11+290κ12+400κ21+310κ22+110χ11+130χ12+150χ21+190χ22+390ζ11+410ζ12+500ζ21+430ζ22,subject to(43)χ11+χ12+χ13=15,χ21+χ22+χ23=10,δ11+δ12+δ13=35,δ21+δ22+δ23=25,ϵ11+ϵ12+ϵ13=45,ϵ21+ϵ22+ϵ23=50,η11+η12+η13=60,η21+η22+η23=70,κ11+κ12+κ13=75,κ21+κ22+κ23=90,ϑ11+ϑ12+ϑ13=95,ϑ21+ϑ22+ϑ23=120,ω11+ω12+ω13=110,ω21+ω22+ω23=130,ζ11+ζ12+ζ13=200,ζ21+ζ22+ζ23=190,χ31+χ32+χ33=25,χ11+χ21+χ31=20,δ31+δ32+δ33=25,δ11+δ21+δ31=40,ϵ31+ϵ32+ϵ33=25,ϵ11+ϵ21+ϵ31=50,η31+η32+η33=25,η11+η21+η31=65,κ31+κ32+κ33=60,κ11+κ21+κ31=85,ϑ31+ϑ32+ϑ33=60,ϑ11+ϑ21+ϑ31=100,ω31+ω32+ω33=115,ω11+ω21+ω31=140,ζ31+ζ32+ζ33=115,ζ11+ζ21+ζ31=160,χ12+χ22+χ32=30,χ13+χ23+χ33=0,δ12+δ22+δ32=35,δ13+δ23+δ33=10,ϵ12+ϵ22+ϵ32=40,ϵ13+ϵ23+ϵ33=30,η12+η22+η32=50,η13+η23+η33=40,κ12+κ22+κ32=100,κ13+κ23+κ33=40,ϑ12+ϑ22+ϑ32=110,ϑ13+ϑ23+ϑ33=65,ω12+ω22+ω32=150,ω13+ω23+ω33=65,ζ12+ζ22+ζ32=170,ζ13+ζ23+ζ33=175,α11+α12−α11α12α31+α32−α31α32+α13−α11α13−α12α13+α33−α31α33−α32α33+α11α12α13=0.92,+α31α32α33=0.00,β11β12β13=0.02,β31β32β33=0.00,γ11γ12γ13=0.01,γ31γ32γ33=0.00,α21+α22−α21α22α11+α21−α11α21+α23−α21α23−α22α23+α31−α11α31−α21α31+α21α22α23=0.80,+α11α21α31=0.92,β21β22β23=0.01,β11β21β31=0.01,γ21γ22γ23=0.02,γ11γ21γ31=0.02,α12+α22−α12α22α13+α23−α13α23+α32−α12α32−α22α32+α33−α13α33−α23α33+α12α22α32=0.76,+α13α23α33=0.1666,β12β22β32=0.02,β13β23β33=0.00,γ12γ22γ32=0.01,γ13γ23γ33=0.00,δ11−χ11≥0,ϵ11−δ11≥0,η11−ϵ11≥0,κ11−η11≥0,ϑ11−κ11≥0,ω11−ϑ11≥0,ζ11−ω11≥0,δ12−χ12≥0,ϵ12−δ12≥0,η12−ϵ12≥0,κ12−η12≥0,ϑ12−κ12≥0,ω12−ϑ12≥0,ζ12−ω12≥0,δ21−χ21≥0,ϵ21−δ21≥0,δ13−χ13≥0,ϵ13−δ13≥0,η13−ϵ13≥0,κ13−η13≥0,ϑ13−κ13≥0,ω13−ϑ13≥0,ζ13−ω13≥0,η21−ϵ21≥0,κ21−η21≥0,ϑ21−κ21≥0,ω21−ϑ21≥0,ζ21−ω21≥0,δ22−χ22≥0,ϵ22−δ22≥0,η22−ϵ22≥0,κ22−η22≥0,ϑ22−κ22≥0,ω22−ϑ22≥0,ζ22−ω22≥0,δ23−χ23≥0,ϵ23−δ23≥0,η23−ϵ23≥0,κ23−η23≥0,ϑ23−κ23≥0,ω23−ϑ23≥0,ζ23−ω23≥0,δ31−χ31≥0,ϵ31−δ31≥0,η31−ϵ31≥0,κ31−η31≥0,ϑ31−κ31≥0,ω31−ϑ31≥0,ζ31−ω31≥0,δ32−χ32≥0,ϵ32−δ32≥0,η32−ϵ32≥0,κ32−η32≥0,ϑ32−κ32≥0,ω32−ϑ32≥0,ζ32−ω32≥0,δ33−χ33≥0,ϵ33−δ33≥0,η33−ϵ33≥0,κ33−η33≥0,ϑ33−κ33≥0,ω33−ϑ33≥0,ζ33−ω33≥0,χ11≥0,χ12≥0,χ13≥0,χ21≥0,χ22≥0,χ23≥0,χ31≥0,χ31≥0,χ32≥0,χ33≥0,α12≥0,α13≥0,α21≥0,α22≥0,α23≥0,β11≥0,β12≥0,β21≥0,β22≥0,β23≥0,β31≥0,β32≥0,β33≥0,γ11≥0,γ13≥0,γ21≥0,γ22≥0,γ23≥0,γ31≥0,γ32≥0,γ33≥0,α11≥0,β13≥0,γ12≥0,α11+β11+γ11≤1,α12+β12+γ12≤1,α13+β13+γ13≤1,α21+β21+γ21≤1,α22+β22+γ22≤1,α23+β23+γ23≤1,α31+β31+γ31≤1,α32+β32+γ32≤1,α33+β33+γ33≤1.
By using Software Maple, we get optimal solution.(44)χ11=15,δ11=35,ε11=45,η11=60,κ11=75,ϑ11=90,ω11=105,ζ11=125,α11=0.8000,β11=0.1000,γ11=0.1000,χ12=0.0000,δ12=0.0000,ε12=0.0000,η12=0.0000,κ12=0.0000,ϑ12=0.0000,ω12=0.0000,ζ12=20,α12=0.6000,β12=0.2000,γ12=0.1000,χ13=0.0000,δ12=0.0000,ε13=0.0000,η13=0.0000,κ13=0.0000,ϑ13=5,ω13=5,ζ13=55,α13=0.1666,β13=0.0000,γ13=0.1000,χ21=0.0000,δ21=0.0000,ε21=0.0000,η21=0.0000,κ21=0.0000,ϑ21=0.0000,ω21=0.0000,ζ21=0.0000,α21=0.6000,β21=0.1000,γ21=0.2000,χ22=10,δ22=15,ε22=20,η22=30,κ22=50,ϑ22=60,ω22=70,ζ22=70,α22=0.5000,β22=0.1000,γ22=0.1000,χ23=0.0000,δ23=10,ε23=30,η23=40,κ23=40,ϑ23=60,ω23=60,ζ23=120,α23=0.0000,β23=0.0000,γ23=0.0000,χ31=5,δ31=5,ε31=5,η31=5,κ31=10,ϑ31=10,ω31=35,ζ31=35,α31=0.0000,β31=0.0000,γ11=0.0000,χ32=20,δ32=20,ε32=20,η32=20,κ32=50,ϑ32=50,ω32=80,ζ32=80,α32=0.0000,β32=0.0000,γ32=0.0000,χ33=0.0000,δ33=0.0000,ε33=0.0000η33=0.0000,κ33=0.0000,ϑ33=0.0000,ω33=0.0000,ζ33=0.0000,α33=0.0000,β33=0.0000,γ33=0.0000.
The PFOS is(45)x˜11=15,35,45,60,75,90,105,125;0.8000,0.1000,0.1000,x˜12=0.00,0.00,0.00,0.00,0.00,0.00,0.00,20;0.6000,0.2000,0.1000,x˜13=0.00,0.00,0.00,0.00,0.00,5,5,55;0.1666,0.0000,0.0000,x˜21=0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00;0.6000,0.1000,0.2000,x˜22=10,15,20,30,50,60,70,70;0.5000,0.1000,0.1000,x˜23=0.00,10,30,40,40,60,60,120;0.0000,0.0000,0.0000,x˜31=5,5,5,5,10,10,35,35;0.0000,0.0000,0.0000,x˜32=20,20,20,20,50,50,80,80;0.0000,0.0000,0.0000,x˜33=0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00;0.0000,0.0000,0.0000.
The picture fuzzy optimal value/transportation cost of FPFTP is
3550,8400,13550,20700,35750,46800,63350,87050;0.8974976,0.00000002,0.00467856 and is shown graphically in Figure 1.Table 1
Input data for FPFTP.
CityT1CityT2PlantS1110,150,190,210,270,300,350,390;0.7,0.1,0.1130,180,220,250,290,340,370,410;0.8,0.1,0.1PlantS2150,250,290,350,400,440,460,500;0.6,0.1,0.2190,210,250,270,310,330,380,430;0.6,0.1,0.2Figure 1
Graphical representation of picture fuzzy transportation cost.Example 2.
(fully intuitionistic fuzzy transportation problem (FIFTP) based on FLP formulation).(46)Minimize190,210,270,300;110,210,270,390⊗x˜11⊕220,250,290,340;130,250,290,410⊗x˜12⊕0,0,0,0;0,0,0,0⊗x˜13⊕290,350,400,440;150,350,400,500⊗x˜21⊕250,270,310,330;190,270,310,430⊗x˜22⊕0,0,0,0;0,0,0,0⊗x˜23⊕0,0,0,0;0,0,0,0⊗x˜31⊕0,0,0,0;0,0,0,0⊗x˜32⊕0,0,0,0;0,0,0,0⊗x˜33,subject to(47)x˜11⊕x˜12⊕x˜13=45,60,75,95;15,60,75,200,x˜21⊕x˜22⊕x˜23=50,70,90,120;10,70,90,190,x˜31⊕x˜32⊕x˜33=25,25,60,60;25,25,60,115,x˜11⊕x˜21⊕x˜31=50,65,85,100;20,65,85,160,x˜12⊕x˜22⊕x˜32=40,50,100,110;30,50,100,170,x˜13⊕x˜23⊕x˜33=30,40,40,65;0,40,40,175,where x˜11, x˜12, x˜13, x˜21, x˜22, x˜23, x˜31, x˜32, and x˜33, are nonnegative TrIFNs.
The intuitionistic fuzzy optimal value/transportation cost of FIFTP [41] is
13550,20700,35750,46800;3550,20700,35750,87050 and is shown graphically in Figure 2.Figure 2
Graphical representation of intuitionistic fuzzy transportation cost.Example 3.
(fully fuzzy transportation problem (FFTP) based on FLP formulation).(48)Minimize190,210,270,300⊗x˜11⊕220,250,290,340⊗x˜12⊕0,0,0,0⊗x˜13⊕290,350,400,440⊗x˜21⊕250,270,310,330⊗x˜22⊕0,0,0,0⊗x˜23⊕0,0,0,0⊗x˜31⊕0,0,0,0⊗x˜32⊕0,0,0,0⊗x˜33,subject to(49)x˜11⊕x˜12⊕x˜13=45,60,75,95x˜21⊕x˜22⊕x˜23=50,70,90,120x˜31⊕x˜32⊕x˜33=25,25,60,60x˜11⊕x˜21⊕x˜31=50,65,85,100x˜12⊕x˜22⊕x˜32=40,50,100,110x˜13⊕x˜23⊕x˜33=30,40,40,65,where x˜11, x˜12, x˜13, x˜21, x˜22, x˜23, x˜31, x˜32, and x˜33 are nonnegative TrFNs.
The fuzzy optimal value/transportation cost of FFTP [35] is 13550,20700,35750,46800 and is shown graphically in Figure 3.Figure 3
Graphical representation of fuzzy transportation cost.


## 5. Discussion

The picture fuzzy optimal value/transportation cost obtained in Example1 by using the proposed method as discussed in Section 3 is 3550,8400,13550,20700,35750,46800,63350,87050 and can be illustrated as follows:(i)
The lowest amount of picture fuzzy transportation cost is 3550 units(ii)
The feasible amount of picture fuzzy transportation cost lies in20700,35750(iii)
The highest amount of picture fuzzy transportation cost is 87050 unitsSo, the least transportation cost will always be higher than 3550 units and lower than 87050 units, and the maximum chances of minimum transportation cost will belong to20700,35750.Furthermore, Examples2 and 3 are considered in an intuitionistic fuzzy environment and fuzzy environment, respectively. The minimum intuitionistic fuzzy transportation cost and minimum fuzzy transportation along with picture fuzzy transportation cost are shown in Table 2.Table 2
Comparison of optimal values.
FPFTP3550,8400,13550,20700,35750,46800,63350,87050FIFTP [41]3550,13550,20700,35750,46800,87050FFTP [35]13550,20700,35750,46800Obviously, Table2 demonstrates that the FPFTP gives the best optimal value/transportation cost as compared with FIFTP and FFTP.


## 6. Conclusion

PFS is the most comprehensive and generalized structure of FS and IFS because it is characterized by membership function, neutral membership function, and nonmembership function. In this paper, we have proposed a new scheme to discuss the fully picture fuzzy transportation problems. We have introduced FPFTPs by considering all the variables as nonnegative TrPFNs. The FPFTPs have been developed on the basis of picture fuzzy linear programming formulation. In order to transform the FPFTPs into crisp linear/non-LPPs, a ranking function is practised. The PFLP formulation technique has been applied to obtain the PFOSs in the form of TrPFNs. Further, we have investigated and compared the fully picture fuzzy transportation model with fully intuitionistic fuzzy transportation model and fully fuzzy transportation model to demonstrate that the proposed approach is more comprehensive and reliable as compared with the existing FIFTP approach [41] and FFTP approach [35].In future, this work can be extended to(1)
LR-type fully picture fuzzy transportation problems(2)
LR-type bipolar single-valued neutrosophic transportation problems



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*Source: 6518976-2022-06-27.xml*