Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem

Complex & Intelligent Systems https://doi.org/10.1007/s40747-020-00251-3 ORIGINAL ARTICLE Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem Shyamali Ghosh1·Sankar Kumar Roy1·Ali Ebrahimnejad2·José Luis Verdegay3 Received: 12 January 2020 / Accepted: 4 December 2020 © The Author(s) 2021 Abstract During past few decades, fuzzy decision is an important attention in the areas of science, engineering, economic system, business, etc. To solve day-to-day problem, researchers use fuzzy data in transportation problem for presenting the uncontrollable factors; and most of multi-objective transportation problems are solved using goal programming. However, when the problem contains interval-valued data, then the obtained solution was provided by goal programming may not satisfy by all decision-makers. In such condition, we consider a fixed-charge solid transportation problem in multi-objective environment where all the data are intuitionistic fuzzy numbers with membership and non-membership function. The intuitionistic fuzzy transportation problem transforms into interval-valued problem using (α, β)-cut, and thereafter, it reduces into a deterministic problem using accuracy function. Also the optimum value of alternative corresponds to the optimum value of accuracy function. A numerical example is included to illustrate the usefulness of our proposed model. Finally, conclusions and future works with the study are described. Keywords Fixed-Charge transportation problem ·Fuzzy programming ·Intuitionistic fuzzy programming ·Goal programming ·Multi-objective decision-making ·Pareto-optimal solution Introduction In the last few decades, the traditional transportation problem (TP) considers only single objective function. When a homogeneous product is transferred from a source to different destinations in competitive economic condition, there exist more than single criterion such as the transportation BSankar Kumar Roy [email protected] Shyamali Ghosh [email protected] Ali Ebrahimnejad [email protected] José Luis Verdegay verde[email protected].es 1Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, West Bengal 721102, India 2Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran 3Department of Computer Science and Artificial Intelligence, University of Granada, Granada, Spain cost, average delivery time of product, deterioration rate of goods, fixed charge for an open route, etc. Therefore, in such a condition, the traditional TP is not sufficient to accommodate such real-life decision-making problem which contains single objective. To overcome such situation, we include TP with multi-objective functions that are contradict to each other. As a result, the single objective TP sets off into multi-objective transportation problem (MOTP). Apart from the transportation cost, a fixed cost, sometimes called set-up cost, is taken only when solution appears with positive level and such a problem is termed as fixedcharge transportation problem (FCTP) and it is associated with 0-1 variable. For transporting some quantity, there exist landing fees at an airport, toll charges on a highway, renting cost of a vehicle, set up cost for machines in manufacturing environment, etc. which are called as fixed-charge. In the presence of such costs, the TP is called FCTP. FCTP corresponds with two types of cost, one is direct cost and another is fixed-charge where fixed-charge occurs for transportation activity in source–destination pair which is independent on transportation amount, and for direct cost, it is dependent on transportation amount for each source to each destination. 123 Complex & Intelligent Systems Aside from source constraints and destination constraints in classical TP, another type of constraints named as conveyance constraints are added in TP, and then, new TP is entitled as solid transportation problem (STP). Haley [10] analyzed STP formerly. Different types of conveyances, e.g., trucks, goods train, ships, cargo flights, etc. are used for transporting a homogeneous product from one source to another destination in many circumstances. Therefore, when a single type conveyance is used in a FCTP, then the TP is refereed as fixed-charge solid transportation problem (FCSTP). For analyzing TP based on real-life situation, the transportation cost, fixed cost, supply, demand parameters, and conveyance are not always precise due to incomplete information. Therefore, uncertainty appears in various applications, such as: (i) Market situation fluctuates for all time, so the demand cannot be clearly determined at any stage. (ii) Decision-maker (DM) has some limitations in ability to tackle the related transportation cost whenever uncertainty occurs. (iii) Sometimes, DM cannot make up the delivery time for uncertain situation. Fuzzy system provides only the degree of membership function of the objective function and constraints. Zadeh [29] first initiated fuzzy set (FS), and Zimmermann [30] defined fuzzy linear programming for multi-objective decision-making problem. When the existence of hesitation is occurred, then the classical fuzzy TP is not capable to tackle the situation. Therefore, to analyze this situation, we incorporate intuitionistic fuzzy (IF) environment in our proposed method. Angelov [1] first brought out the optimization abstraction in IF environment. Intuitionistic Fuzzy Set (IFS) is an extension of FS and an important fact that clearly defines the difference between the degree of acceptance function and the degree of non-acceptance function of an element in the set. Again, the total sum of acceptance value and nonacceptance value of objective function and constraints always lies between 0 and 1. To sketch the imprecise concept and to integrate hesitancy of membership function, Atanassov [2] analyzed the concept of IFS. Wan and Li [28] represented Atanassov’s intuitionistic fuzzy programming (IFP) with truth degrees for heterogeneous multi-attribute group decision-making problem. All the required abbreviations are presented in Table 1. The major contributions of our presented approach are as follows: (i) In FCSTP, all the parameters and variables are considered as intuitionistic fuzzy numbers (IFNs). Table 1 Abbreviations Abbreviation Full name DM Decision-maker FCSTP Fixed-charge solid transportation problem FCTP Fixed-charge transportation problem FP Fuzzy programming FS Fuzzy set GP Goal programming IF Intuitionistic fuzzy IFN Intuitionistic fuzzy number IFP Intuitionistic fuzzy programming IFS Intuitionistic fuzzy set IFTP Intuitionistic fuzzy transportation problem IVIF Interval-valued intuitionistic fuzzy IVIFTP Interval-valued intuitionistic fuzzy transportation problem IVTP Interval-valued transportation problem LR flat fuzzy Left and right flat fuzzy MOFCSTP Multi-objective fixed-charge solid transportation problem MOTP Multi-objective transportation problem NIS Negative ideal solution PIS Positive ideal solution STP Solid transportation problem TIFN Triangular intuitionistic fuzzy number TP Transportation problem TC, DC, TT Transportation cost, deterioration cost, transportation time (ii) For solving the presented multi-objective fixed-charge solid transportation problem (MOFCSTP), we use linear membership and non-membership functions. (iii) (α, β)-cut is applied to convert the intuitionistic fuzzy transportation problem (IFTP) into an interval-valued transportation problem (IVTP). (iv) IVTP transforms into crisp TP by utilizing the accuracy function of the objective function. (v) Since different values of αand βallocate different solutions, therefore, to find a better solution, DM selects the values of αand β, such that α+β≤1. (vi) For finding best Pareto-optimal solutions, we use three methods, such as fuzzy programming (FP), IFP, and goal programming (GP). The remaining paper is depicted as follows. “Related work” interprets the related work of our proposed model. The basic preliminaries with IFS are defined in “Preliminaries”. “Mathematical model” represents the mathematical model of MOFCSTP with fully IF and thereafter interval problem and crisp problem defined in two models. Three methods, 123 Complex & Intelligent Systems namely FP, IFP, and GP, with related models are illustrated in “Solution procedure”. A numerical example is described in “Numerical example”. “Results and discussion” provides the results and discussion. Sensitivity analysis is interpreted in “Sensitivity analysis”. “Drawbacks of existing methods and contributions with limitations of our method” specifies about the drawbacks of the existing methods and the advantages with limitations of the proposed study. “Conclusion and future research scopes” outlines the conclusions with future research scopes. Related work Several papers are available with inexplicit data for solving MOTP. A few of them are included with the works. TP with linear programming problem is called Hitchcock– Koopmans TP as Hitchcock [12] in his famous paper where he described transportation model. Gupta et al. [9] displayed a TP with multiple objectives that optimized by parameter estimation. They inserted gamma distribution on this stochastic capacitated problem. Maity and Roy [14] represented a TP by considering interval goal and utility function which also extended with multiple objectives. In uncertain situation, an MOTP that included cost reliability was analyzed by Maity et al. [15]. Maity and Roy [16] represented multi-choice programming on a fuzzy MOTP. Malik and Gupta [17]solved a problem of transportation system with multiple objectives which was initiated on fully interval-valued IF environment. Roy and Maity [22] solved a single objective multi-choice TP which incorporated cost and time function. In the presence of multiple objectives, multi-choice, and interval goal, a TP was interpreted by Roy et al. [23] and the solution of the problem was covered by conic scalarization approach. The literature review provides some exact methods for solving FCTP. FCTP was first introduced by Hirsch and Dantzig [11]. Midya and Roy [18] applied interval programming to a TP with fixed-charge in the situation of interval and rough interval environment. Midya et al. [19] extended an FCTP in IF environment for green supply chain. Also, the problem was augmented by considering multiple stages and conveyance constraints. A TP with fixed-charge and multiple objectives was interpreted by Roy and Midya [25]inIFenvironment, and they solved the problem by comprising product blending constraints and conveyance constraints. From literature, we see that there exist various extensions of STP. Das et al. [5] provided a green STP-location problem with multiple objectives that analyzed by fuzzy and nonfuzzy techniques for carbon emission tax, cap, and offset policy including an extra condition as dwell time in type2 IF environment. Rani and Gulati [21] proposed STP with uncertain environment. A good number of researchers worked in MOTP in different uncertain situations such as fuzzy environment, IF environment, etc. Atanassov and Gargov [3] represented interval-valued IFSs. Ebrahimnejad [6] represented fuzzy TP with Left and Right (LR) flat fuzzy numbers. Ebrahimnejad and Verdegay [7] newly proposed a TP in fully IF environment. Garg [8] proposed a new ranking approach on normal intuitionistic sets that applied for the ranking of multi-attribute decision-making process and the approach completed on crisp and interval environment. Kumar and Hussian [13] briefly explained a method to solve TP with fully IF background. Niu et al. [20] developed a multiple criteria decision-making approach in IF situation by the consideration of interval-valued and double risk parameters. Roy et al. [24] analyzed an MOTP with IF uncertainty. Singh and Yadav [26] proposed a new method for finding solution of IF type-2 TP. Ulucay et al. [27] introduced IF multiple numbers on multi-criteria decision-making problems in trapezoidal fuzzy number. Some remarkable research works on MOTP are depicted in Table 2. Preliminaries Some related definitions and basic elementary operations are introduced here. Also, these definitions and operations are all based on IFNs. Definition 1 [2]LetXbe a universal set, and then, an IFS, ˜ AI in Xisgivenby: ˜ AI={x,μ˜ AI(x), γ ˜ AI(x):x∈X},where μ˜ AI(x), γ ˜ AI(x):X→[0,1]are the degrees of membership and of non-membership that satisfy: 0 ≤μ˜ AI(x)+γ˜ AI(x)≤ 1, x∈X. Again the degree of hesitation of an element xin the set ˜ AIis defined as function π˜ AI(x)=1−μ˜ AI(x)−γ˜ AI(x). When π˜ AI(x)=0,x∈X, then the IFS transforms into an FS. Definition 2 [2] Consider a non-empty set Xand two IFSs ˜ AI,˜ BIin Xwhich are given by ˜ AI={x,μ˜ AI(x), γ ˜ AI(x): x∈X}and ˜ BI={x,μ˜ BI(x), γ ˜ BI(x):x∈X}, respectively. Then, the following properties hold: 3.2.1: ˜ AI⊆˜ BIif and only if μ˜ AI(x)≤μ˜ BI(x)and γ˜ AI(x)≥γ˜ BI(x)∀x∈X. 3.2.2: ˜ AI˜ BI={x,min(μ ˜ AI(x), μ ˜ BI(x)), max(γ ˜ AI (x), γ ˜ BI(x)):x∈X}. 3.2.3: ˜ AI˜ BI={x,max(μ ˜ AI(x), μ ˜ BI(x)), min(γ ˜ AI (x), γ ˜ BI(x)):x∈X}. Definition 3 [2] The Atanassov’s interval-valued intuitionistic fuzzy (IVIF) set can be defined as: AI={x,[μl ˜ AI(x), μu ˜ AI(x)],[γl ˜ AI(x), γ u ˜ AI(x)] : x∈X}, where 0 ≤μl ˜ AI(x)≤ μu ˜ AI(x)≤1, 0 ≤γl ˜ AI(x)≤γu ˜ AI(x)≤1, 0 ≤ 123 Complex & Intelligent Systems Table 2 Some remarkable research works of TP in tabulated form References Environment Objective Fixed charge TC DC TT Solution method Ebrahimnejad [6] Fuzzy Single No Yes No No Simplex algorithm Ebrahimnejad and Verdegay [7] IF Single No Yes No No Standard linear programming algorithm Gupta et al. [9] Type-2 fuzzy Multi No Yes Yes Yes Fuzzy GP Maity and Roy [14] Crisp Multi No Yes No No Utility function approach Maity et al. [15] Uncertain Multi No Yes No No Fuzzy multi-choice GP Maity and Roy [16] Fuzzy Multi No Yes No No Multi-choice GP Midya and Roy [18] Rough Single Yes Yes No No Interval programming Midya et al. [19] IF Multi Yes Yes No Yes Min-max GP, weighted Tchebycheff metrics programming Rani and Gulati [21] Uncertain Multi No Yes No No FP Roy and Maity [22] Interval-valued multi-choice Single No Yes No Yes Multi-choice programming Roy et al. [23] Crisp Multi No Yes No No Conic scalarization approach Roy et al. [24] IF Multi No Yes No Yes IFP Singh and Yadav [26] IF Single No Yes No No IF modified distribution method This investigation Fully IF Multi Yes Yes Yes Yes FP, IFP and GP μl ˜ AI(x)+γu ˜ AI(x)≤1. μl ˜ AI(x), γ l ˜ AI(x)are lower bounds and μu ˜ AI(x), γ u ˜ AI(x)are upper bounds of membership and non-membership function, respectively, of the IFS ˜ AI. Definition 4 When the membership value and nonmembership value of an IFS ˜ AIare equal to 1 and 0, respectively, for any point x0, then the set is said to be normal. That is, for any point x0, such that μ˜ AI(x0)=1 and γ˜ AI(x0)=0. Definition 5 An IF subset ˜ AIof real numbers is said to be IFN ˆ AIthat satisfies the following results: 3.5.1: ˜ AIis normal, i.e., ∃x∈X, such that μ˜ AI(x)=1. 3.5.2: ˜ AIis convex, i.e., for the membership function μ˜ AI(x)withμ˜ AI[λx1+(1−λ)x2]≥min{μ˜ AI(x1), μ ˜ AI(x2)} for x1,x2∈R,λ∈[0,1]. 3.5.3: ˜ AIis concave, i.e., for the non-membership function γ˜ AI(x)with γ˜ AI[λx1+(1−λ)x2]≤max{γ˜ AI(x1), γ˜ AI(x2)}for x1,x2∈R,λ∈[0,1]. Definition 6 Consider two Atanassov’s IVIF sets AIand BI in the universal set X, where AIand BIare defined as AI={x,[μl ˜ AI(x), μu ˜ AI(x)],[γl ˜ AI(x), γ u ˜ AI(x)] : x∈X} and BI={x,[μl ˜ BI(x), μu ˜ BI(x)],[γl ˜ BI(x), γ u ˜ BI(x)] : x∈ X}.The operations of IVIF sets AIand BIare given as: 3.6.1: AI+BI={x,[μl ˜ AI(x)+μl ˜ BI(x)−μl ˜ AI(x)μl ˜ BI(x), μu ˜ AI(x)+μu ˜ BI(x)−μu ˜ AI(x)μu ˜ BI(x)],[γl ˜ AI(x)γ l ˜ BI(x), γu ˜ AI(x)γ u ˜ BI(x)] : x∈X}. 3.6.2: AI.BI={x,[μl ˜ AI(x)μl ˜ BI(x), μu ˜ AI(x)μu ˜ BI(x)], [γl ˜ AI(x)+γl ˜ BI(x)−γl ˜ AI(x)γ l ˜ BI(x), γ u ˜ AI(x)+γu ˜ BI(x)− γu ˜ AI(x)γ u ˜ BI(x)] : x∈X}. 3.6.3: r.AI={x,[1−(1−μl ˜ AI(x))r,1−(1−μu ˜ AI(x))r], [γl ˜ AI(x)r,γl ˜ AI(x)r] : x∈X},r≥0. 3.6.4: AI=BIif and only if μl ˜ AI(x) =μl ˜ BI(x), μu ˜ AI(x)=μu ˜ BI(x), γ l ˜ AI(x)=γl ˜ BI(x), γ u ˜ AI(x) =γu ˜ BI(x). Definition 7 If a Triangular Intuitionistic Fuzzy Number (TIFN) is of the form ˆ AI=(a1,a2,a3;a1,a2,a3), where (a1≤a1≤a2≤a3≤a3), then the membership and non-membership functions of ˆ AIare defined as: μˆ AI(x)=⎧ ⎪ ⎨ ⎪ ⎩ x−a1 a2−a1,if a1≤x≤a2, a3−x a3−a2,if a2≤x≤a3, 0,otherwise, and γˆ AI(x)=⎧ ⎪ ⎨ ⎪ ⎩ a2−x a2−a1,if a1≤x≤a2, x−a2 a3−a2,if a2≤x≤a3, 1,otherwise. The graphical presentation of membership and nonmembership function of TIFN is interpreted by Fig. 1. 123 Complex & Intelligent Systems Fig. 1 Graphical presentation of TIFN Arithmetic operations on TIFNs: Let two TIFNs be ˆ AI=(a1,a2,a3;a1,a2,a3)and ˆ BI= (b1,b2,b3;b1,b2,b3). Then, the arithmetic operations are defined as follows: Addition ˆ AI+ˆ BI=(a1+b1,a2+b2,a3+b3;a1+b1,a2+ b2,a3+b3). Subtraction ˆ AI−ˆ BI=(a1−b3,a2−b2,a3−b1;a1− b3,a2−b2,a3−b1). Multiplication ˆ AI.ˆ BI=(min{a1b1,a1b3,a3b1,a3b3},a2b2, max{a1b1,a1b3,a3b1,a3b3}; min{a1b1,a1b3,a3b1,a3b3}, a2b2,max{a1b1,a1b3,a3b1,a3b3}). Scalar multiplication Scalar multiplication for any real kis defined as kˆ AI=(ka1,ka2,ka3;ka1,ka2,ka3)if k≥0, and (ka3,ka2,ka1;ka3,ka2,ka1)if k<0. Inequality of TIFNs ˆ AI=(a1,a2,a3;a1,a2,a3)≤ˆ BI= (b1,b2,b3;b1,b2,b3)if and only if a1≤b1,a2≤b2,a3≤ b3,a1≤b1,a3≤b3,where ˆ AIand ˆ BIare TIFNs. Definition 8 (α, β)-cutof a TIFN ˆ AI=(a1,a2,a3;a1,a2,a3) is the set of all xwhose degree of membership is greater than or equal to αand degree of non-membership is less than or equal to β. That is defined by ˆ AI (α,β) ={x:μˆ AI(x)≥αand γˆ AI(x)≤β,(α +β) ≤1:x∈X}. Now, μˆ AI(x)≥α, which implies that x−a1 a2−a1≥α, a3−x a3−a2≥ α.Andx≥a1+α(a2−a1), x≤a3−α(a3−a2). Therefore, the α-cut of ˆ AIis [a1+α(a2−a1), a3−α(a3−a2)]. Again γˆ AI(x)≤β, which implies that a2−x a2−a1≤ β, x−a2 a3−a2≤β.Andx≥a2−β(a2−a1), x≤ a2+β(a3−a2). Therefore, the β-cut of ˆ AIis [a2−β(a2− a1), a2+β(a3−a2)]. Denoting α-cut of ˆ AIas [μl ˆ AI,μ u ˆ AI]=[a1+α(a2− a1), a3−α(a3−a2)]and β-cut of ˆ AIas [γl ˆ AI,γu ˆ AI]= [a2−β(a2−a1), a2+β(a3−a2)].(α, β)-cut of a TIFN is explained graphically in Fig. 2. Fig. 2 (α, β)-cut of a TIFN Definition 9 Consider a TIFN ˆ AI=(a1,a2,a3;a1,a2,a3). Then, the accuracy function is M(ˆ AI):X(ˆ AI)→R, where X(ˆ AI)is the collection of all IVIF sets obtained from (α, β)- cut. This accuracy function is defined in terms of the IVIF set X(ˆ AI)=[μl ˆ AI,μ u ˆ AI],[γl ˆ AI,γu ˆ AI] as follows: M(ˆ AI)= μl ˆ AI+μu ˆ AI+γl ˆ AI+γu ˆ AI 2. Mathematical model We consider an MOFCSTP (here three objective functions) where the transportation cost with fixed-charge from each source to each destination is represented by first objective function. The second objective function is the deterioration rate of goods and the third one is the transporting time of goods. Here, we assume that all the parameters are IFNs for realistic situation. The shipping cost is ˆcI ijk per unit item for transforming a homogeneous product from ith source to jth destination using any of the kth conveyance. The optimal solutions are obtained by optimizing all the objective functions concurrently based on real situation. The fixed-charge is ˆ fI ijk for shipping product from supplier ito customer jby means of kconveyance. Each supplier (i=1,2,...,m)has ˆaI iunits of supply, each customer (j=1,2,...,n)has ˆ bI j units of demand, and each conveyance (k=1,2,...,l)has ˆeI kunits of capacity. The following notations and assumptions are considered to describe our proposed mathematical model as: Notations ˆxI ijk :IF amount of product that transported from ith source to jth destination through kth conveyance, ˆcI ijk :IF cost for unit quantity of the product that transported from ith source to jth destination through kth conveyance, 123 Complex & Intelligent Systems ˆ fI ijk :IF fixed-charge for unit quantity of the product that transported from ith source to jth destination through kth conveyance, ˆ dI ijk :IF deterioration rate for unit quantity of the product that transported from ith source to jth destination through kth conveyance, ˆ tI ijk :IF time of transportation for unit quantity of the product that transported from ith source to jth destination through kth conveyance, ys ijk :Binary variable taking the value“1” if the source iused and “0” otherwise, for s=1,2,3,1,3, ηs ijk :Binary variable taking the value“1” if the source iused and “0” otherwise, for s=1,2,3,1,3, ˆaI i:The IF supply at ith source, ˆ bI j:The IF demand at jth destination, ˆeI k:The IF capacity of kth conveyance for the TP, ˆ ZI r:The IF objective function (r=1,2,3), ZI r:The interval-valued objective function (r= 1,2,3), Zr:The objective function (r=1,2,3)in crisp nature, where Zr=M(ˆ ZI r). Assumptions •ˆxI ijk =(x1 ijk,x2 ijk,x3 ijk;x1 ijk,x2 ijk,x3 ijk),ˆyI ijk =(y1 ijk,y2 ijk,y3 ijk;y1 ijk,y2 ijk,y3 ijk), •ˆηI ijk =(η1 ijk,η 2 ijk,η 3 ijk;η1 ijk,η 2 ijk,η 3 ijk),ˆcI ijk =(c1 ijk,c2 ijk,c3 ijk;c1 ijk,c2 ijk,c3 ijk), •ˆ dI ijk =(d1 ijk,d2 ijk,d3 ijk;d1 ijk,d2 ijk,d3 ijk),ˆ tI ijk =(t1 ijk,t2 ijk,t3 ijk;t1 ijk,t2 ijk,t3 ijk), •ˆ fI ijk =(f1 ijk,f2 ijk,f3 ijk;f1 ijk,f2 ijk,f3 ijk), •xs ijk ≥0,ys ijk ≥0,η s ijk ≥0,cs ijk ≥0,ds ijk ≥0,ts ijk ≥ 0,fs ijk ≥0, (s=1,2,3,1,3), •ys ijk =1,if xs ijk >0, 0,otherwise, and ηs ijk =1,if xs ijk >0, 0,otherwise. The mathematical model for MOFCSTP with fully IFN is presented here as: Model 1 minimize ˆ Z1 I= m  i=1 n  j=1 l  k=1 [(ˆcI ijk ⊗ˆxI ijk)⊕(ˆ fI ijk ⊗ˆyI ijk)] (4.1) minimize ˆ Z2 I= m  i=1 n  j=1 l  k=1 (ˆ dI ijk ⊗ˆxI ijk)(4.2) minimize ˆ Z3 I= m  i=1 n  j=1 l  k=1 (ˆ tI ijk ⊗ˆηI ijk)(4.3) subject to n  j=1 l  k=1 ˆxI ijk ≤ˆaI i(i=1,2,...,m), (4.4) m  i=1 l  k=1 ˆxI ijk ≥ˆ bI j(j=1,2,...,n), (4.5) m  i=1 n  j=1 ˆxI ijk ≤ˆeI k(k=1,2,...,l), (4.6) xs ijk ≥0,∀i,j,kand s=1,2,3,1,3.(4.7) The feasibility conditions of TP are as follows: m  i=1 ˆaI i≥ n  j=1 ˆ bI j; l  k=1 ˆeI k≥ n  j=1 ˆ bI j. This problem is of IF nature. Therefore, transforming the above problem into interval-valued intuitionistic fuzzy transportation problem (IVIFTP) using (α, β)-cut of the objective function and with the help of membership and nonmembership functions. Also utilizing inequality of IFNs, we transform all the IF constraints of Model 1 into crisp constraints of Model 2. Therefore, the IVIFTP is described as: Model 2 minimize ZI 1= m  i=1 n  j=1 l  k=1μl ˆcI ijk⊗ˆxI ijk ,μ u ˆcI ijk⊗ˆxI ijk ]; γl ˆcI ijk⊗ˆxI ijk ,γu ˆcI ijk⊗ˆxI ijk +μl ˆ fI ijk⊗ˆyI ijk ,μ u ˆ fI ijk⊗ˆyI ijk ]; γl ˆ fI ijk⊗ˆyI ijk ,γu ˆ fI ijk⊗ˆyI ijk minimize ZI 2= m  i=1 n  j=1 l  k=1μl ˆ dI ijk⊗ˆxI ijk ,μ u ˆ dI ijk⊗ˆxI ijk ]; γl ˆ dI ijk⊗ˆxI ijk ,γu ˆ dI ijk⊗ˆxI ijk minimize ZI 3= m  i=1 n  j=1 l  k=1μl ˆ tI ijk⊗ˆηI ijk ,μ u ˆ tI ijk⊗ˆηI ijk; γl ˆ tI ijk⊗ˆηI ijk ,γu ˆ tI ijk⊗ˆηI ijk ] subject to n  j=1 l  k=1 xs ijk ≤as i(i=1,2,...,m), (4.8) m  i=1 l  k=1 xs ijk ≥bs j(j=1,2,...,n), (4.9) m  i=1 n  j=1 xs ijk ≤es k(k=1,2,...,l), (4.10) x1 ijk ≥x1 ijk,x2 ijk ≥x1 ijk,x3 ijk ≥x2 ijk,x3 ijk ≥x3 ijk,(4.11) xs ijk ≥0,∀i,j,kand s=1,2,3,1,3.(4.12) 123 Complex & Intelligent Systems Here, [μl ˆcI ijk⊗ˆxI ijk ,μ u ˆcI ijk⊗ˆxI ijk ]and [γl ˆcI ijk⊗ˆxI ijk ,γu ˆcI ijk⊗ˆxI ijk ]are (α, β)-cuts, respectively, of (ˆcI ijk ⊗ˆxI ijk)which is defined in Def. 3.8. Hence, these are interval-valued numbers in the form of IFN (ˆcI ijk ⊗ˆxI ijk). This interval-valued MOFCSTP cannot be solved in any simple way, and therefore, we transform this interval-valued problem into crisp problem by utilizing the accuracy function of the objective function using Def. 3.9. The accuracy function corresponds to each alternative which provides the minimum value. Therefore, the interval-valued problem becomes a crisp problem and the equivalent crisp problem is given in Model 3 as: Model 3 minimize Z1=1 2 m  i=1 n  j=1 l  k=1(1−α)c1 ijkx1 ijk +2(α −β+1)c2 ijkx2 ijk +(1−α)c3 ijkx3 ijk +βc1 ijkx1 ijk +βc3 ijkx3 ijk +(1−α) f1 ijky1 ijk +2(α −β+1)f2 ijky2 ijk +(1−α) f3 ijky3 ijk +βf1 ijky1 ijk +βf3 ijky3 ijk minimize Z2=1 2 m  i=1 n  j=1 l  k=1(1−α)d1 ijkx1 ijk +2(α −β+1)d2 ijkx2 ijk +(1−α)d3 ijkx3 ijk +βd1 ijkx1 ijk +βd3 ijkx3 ijk minimize Z3=1 2 m  i=1 n  j=1 l  k=1(1−α)t1 ijkη1 ijk +2(α −β+1)t2 ijkη2 ijk +(1−α)t3 ijkη3 ijk +βt1 ijkη1 ijk +βt3 ijkβη3 ijk subject to constraints (4.8)−(4.12). Now, we solve Model 3 by considering each objective function separately with subject to the constraints. However, there exist distinct solutions for different objective functions which are contradict to each other. To derive the best Pareto-optimal solution, we solve Model 3 with help of three methods which are FP, IFP, and GP. These methods transform the MOTP into single objective TP. Determine the upper bound as Positive Ideal Solution (PIS) and lower bound as Negative Ideal Solution (NIS) for each objective function in the pay-off matrix, displaying in Table 3. PIS and NIS are defined as PIS = Zr∗=min {Zr(X1∗), Zr(X2∗), Zr(X3∗)}(r=1,2,3) and NIS = Zr=max {Zr(X1∗), Zr(X2∗), Zr(X3∗)}(r= 1,2,3), respectively. Table 3 Pay-off matrix Z1Z2Z3 X1∗Z1(X1∗)Z2(X1∗)Z3(X1∗) X2∗Z1(X2∗)Z2(X2∗)Z3(X2∗) X3∗Z1(X3∗)Z2(X3∗)Z3(X3∗) Definition 10 Pareto-optimal solution of Model 3 is a feasible solution x∗=(x∗ ijk :i=1,2,...,m;j= 1,2,...,n;k=1,2,...,l), such that there exists no other feasible solution x=(xijk :i=1,2,...,m;j= 1,2,...,n;k=1,2,...,l)with Zr(x)≤Zr(x∗), r= 1,2,3 and Zr(x)<Zr(x∗)for at least one r. Solution procedure Our proposed Model 1 is in IF nature. Transforming the IFTP into crisp problem and then find Model 3. Therefore, Model 3 is equivalent to Model 1, and hence, to obtain Pareto-optimal solution, we utilize three methods as: •FP, •IFP and •GP Now, we describe the necessary steps to solve Model 3 for each method in the following subsections. FP Several methods are available for solving MOTP. Among these methods, FP method is a useful method that finds the Pareto-optimal solution. Therefore, to acquire the best Pareto-optimal solution, we take the advantage of FP which is used to solve intuitionistic MOFCSTP. FP was introduced by Zimmermann [30] for solving any multi-objective linear programming problem and it is very easy for solving this type of problem. Therefore, to solve the proposed Model 3 in FP, we depict the steps as: •Step 5.1.1: Transform the IFTP into crisp problem using (α, β)-cut and then utilizing accuracy function. •Step 5.1.2: Solve each problem independently with subject to all constraints. •Step 5.1.3: Select the tolerance of each objective function. •Step 5.1.4: Determine the PIS and NIS from pay-off Table 3and formulate the membership function corresponding to each objective function defined as: find x=(x1,x2,...,xn)T, such that minimize Zrand subject to gj(X)≤0,(j=1,2,...,m)and xi≥0,(i= 123 Complex & Intelligent Systems 1,2,...,n), with tolerance pr,(r=1,2,3). The membership function is μI r(Zr(x)) which is defined as: μI r(Zr(x)) =⎧ ⎪ ⎨ ⎪ ⎩ 1,if Zr≤LT r, 1−Zr−LT r UT r−LT r ,if LT r≤Zr≤UT r,(r=1,2,3) 0,if Zr≥UT r. Here, UT rand LT rare NIS and PIS Zr,(r=1,2,3), respectively. •Step 5.1.5: Our goal is to maximize the degree of acceptance of each objective function, and if we denote the degree of acceptance as α, then with the help of FP, Model 3 can be formulated as: Model 4A maximize α subject to μI r(Zr(x)) ≥α, (r=1,2,3), α∈[0,1], constraints (4.8)−(4.12). Now, Model 4A is transformed into a simplified form which is noted as Model 4B: Model 4B maximize α subject to Zr(x)+(UT r−LT r)α ≤UT r,(r=1,2,3), α∈[0,1], constraints (4.8)−(4.12). •Step 5.1.6: Solve Model 4B by mathematical programming with parameter αand report the solution. Theorem 1 If x∗=(xijk :i=1,2,...,m;j= 1,2,...,n;k=1,2,...,l)is an optimal solution of Model 4B, then it is also a Pareto-optimal (non-dominated)solution of Model 3. Proof Let x∗is not a Pareto-optimal (non-dominated) solution of Model 3. Therefore, from Definition 10, we consider that there exists at least one x, such that Zr(x)≤Zr(x∗) for r=1,2,3 and Zr(x)<Zr(x∗)for at least one r. Therefore, membership function μI r(Zr(x)) is strictly decreasing with respect to the corresponding objective function Zr(x)in [0,1]. Hence, μI r(Zr(x)) ≥μI r(Zr(x∗)) ∀r and μI r(Zr(x)) > μI r(Zr(x∗)) for at least one r.Now,α= min {μI r(Zr(x))}≥min {μI r(Zr(x∗))}=α∗which is a contradiction that x∗is an optimal solution of Model 4B. Here, α∗is the value of αat x∗. This completes the proof of the theorem.  IFP IFP is used to solve multi-objective decision-making problem and to derive Pareto-optimal solution from this problem. In our IFP, we consider three objective functions ˆ ZI 1,ˆ ZI 2 and ˆ ZI 3. These objective functions are all IF nature, and therefore, to find the crisp form of the objective function ˆ ZI r,(r=1,2,3), we use the accuracy function. For each crisp objective function Zr,(r=1,2,3), we calculate the lower bound LT rand the upper bound UT r,(r=1,2,3). However, there exist distinct solutions for different objective functions which are contradict to each other. To get the best Pareto-optimal solution, we solve the problem using IFP which transforms the MOTP into single objective TP. Therefore with appropriate acceptance limit, violations of the constraints, and degree of acceptance, we formulate a crisp model. The steps of this method are depicted as follows: •Step 5.2.1: Transform the multi-objective intuitionistic fuzzy problem into interval problem using (α, β)-cut. •Step 5.2.2: With help of accuracy functions, intervalvalued problem of Step 5.2.1 becomes crisp problem. •Step 5.2.3: Solve the crisp problem with single objective function at a time subject to the constraints with omitting the other objective functions. •Step 5.2.4: Derive the best value and the worst value of every objective function that are PIS and NIS from Table 3. •Step 5.2.5: Formulate membership and non-membership functions for every objective function and then establish Model 5A as: μr(Zr(x)) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1,if Zr≤LT r, 1−Zr−LT r UT r−LT r ,if LT r≤Zr≤UT r,(r=1,2,3) 0,if Zr≥UT r, and γr(Zr(x)) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0,if Zr≤LF r, 1−Zr−LF r UF r−LF r ,if LF r≤Zr≤UF r,(r=1,2,3) 1,if Zr≥UF r. Model 5A maximize μ−γ subject to μ≤UT r−Zr UT r−LT r ,(r=1,2,3), γ≥UrF−Zr UF r−LF r ,(r=1,2,3), μ+γ≤1,μ≥γ, μ ≥0,γ≥0, constraints (4.8)−(4.12). Here, UT r= NIS for Zrand LT r= PIS for Zr.UF r= UT r,LF r=LT r+pr(UT r−LT r), (r=1,2,3);also, 123 Complex & Intelligent Systems μand γare the degrees of acceptance and rejection, respectively. pr,(r=1,2,3)is the acceptance limit of non-membership function. The equivalent simplified model of Model 5A is as: Model 5B maximize μ−γ subject to Zr(x)+(UT r−LT r)μ ≤UT r, Zr(x)−(UF r−LF r)γ ≤UF r, μ+γ≤1,μ≥γ, μ ≥0,γ≥0, 0≤pr≤UT r−LT r,r=1,2,3, constraints (4.8)−(4.12). •Step 5.2.6: Solve Model 5B using LINGO iterative scheme and obtain the Pareto-optimal solution of Model 3. Theorem 2 If x∗=(xijk :i=1,2,...,m;j= 1,2,...,n;k=1,2,...,l)is an optimal solution of Model 5B, then it is also a Pareto-optimal (non-dominated)solution of Model 3. Proof Let x∗is not a Pareto-optimal (non-dominated) solution of Model 3. Therefore, from Definition 4.1, we consider that there exists at least one x, such that Zr(x)≤Zr(x∗) for r=1,2,3 and Zr(x)<Zr(x∗)for at least one r. Therefore, membership function μ(Zr(x)) is strictly decreasing with respect to the corresponding objective function Zrin [0,1]. Again, the non-membership function γ(Zr(x)) strictly increases with respect to the objective function Zrin [0,1]. Hence, μ(Zr(x)) ≥μ(Zr(x∗)) ∀r and μ(Zr(x)) > μ(Zr(x∗)) for at least one r. Similarly, γ(Zr(x)) ≤γ(Zr(x∗)) ∀rand γ(Zr(x)) < γ (Zr(x∗)) for at least one r.Now,(μ −γ)=min {μ(Zr(x)), γ (Zr(x))} ≥min {μ(Zr(x∗)), γ (Zr(x∗))}=(μ∗−γ∗)which is a contradiction that x∗is an optimal solution of Model 5B. Here, μ∗and γ∗are the values of μand γat x∗, respectively. This completes the proof of the theorem.  GP GP is generally used for solving multi-objective decisionmaking problems and first presented by Charnes and Cooper [4]. This method is widely used to study the conflicting objective functions mainly and minimizes the deviation among respective goals and aspiration level of all the objective functions. DMs always choose their goals. The following steps are considered to solve the proposed model by GP. •Step 5.3.1: Solve the MOFCSTP with single objective function separately and then omitting the other objective functions at that time. •Step 5.3.2: Find the corresponding value of every objective function and fix the goal of every objective function. •Step 5.3.3: Formulate Model 6 using GP. Model 6 minimize 3  r=1 (d+ r+d− r) subject to Zr−d+ r+d− r=Zg r,(r=1,2,3), constraints (4.8)−(4.12). Here, d+ rand d− r(r=1,2,3)are the positive and negative deviations of the objective functions, respectively, from target values. Zg ris the corresponding goal of the objective function Zr,(r=1,2,3). •Step 5.3.4: Solve Model 6 using LINGO iterative scheme and derive the Pareto-optimal solution of Model 3. Now, we depict the solution methodology of the three methods in a flowchart by Fig. 3. Numerical example A fruit supply company in two states S1 and S2 supply a specific type of fruits. There also exist two companies in states D1 and D2, that are received the fruits. Each state represents a supply point and a demand point. The main aim is to transporting the fruits from supply point to destination point by minimizing total transportation cost with fixed-charge, transporting time, and deterioration rate. Due to the imprecise data of market condition, transportation time, rate of deterioration, all the cost coefficients of the objective functions, supply, demand, and conveyance are TIFNs. Also whenever shipping the materials from source to destination, a fixed-charge is added in the objective function. Hence, the transportation cost in hundred dollar ($) per unit, transportation time in hour per unit, and loss of deterioration in dollar ($) are considered by three objective functions ˆ ZI 1,ˆ ZI 2, and ˆ ZI 3, respectively. The aim of this problem is to determine the unknown quantity ˆxI ijk, that are transported from ith source to jth destination with kth conveyance with IF transportation cost ˆcI ijk, fixed-charge ˆ fI ijk, deterioration rate ˆ dI ijk, and time ˆ tI ijk which are given in Tables 4,5and 6. Source ˆaI iand demand ˆ bI jare given in each table. We choose the values of conveyances such as ˆeI 1= (200,240,260;180,240,280)and ˆeI 2=(200,230,250;180, 230,270)which are not given in tabulated form. For (α, β)- cut, we arbitrarily assume the values α=0.8 and β= 0.1. Now, the mathematical form of this problem which is obtained from Model 3 is as follows: 123