A novel approach to interval-valued variables using new interval arithmetic to solve an intuitionistic fuzzy transportation problem

Abstract

This paper is concerned with the solution mechanism to solve the transportation problem under unpredictability by using interval valued intuitionistic fuzzy parameters. The parameters are chosen as intervals in which costs are modeled by intuitionistic fuzzy numbers, whereas source and destination are taken as crisp values. Various methods of transportation problem like VAM, Monalisha’s Approximation method, Zero point method are used to illustrate the cost in interval numbers by using the interval arithmetic operations. For each method, a solution is derived without converting into crisp expression followed by a graphical representation.

Keywords

Interval valued intuitionistic fuzzy numbers inteval valued intuitionistic fuzzy transportation problem interval arithmetic interval VAM interval ZPM

1 Introduction

The fuzzy set theory has been implemented in various domains like image processing, power systems engineering, optimization etc. It is highly beneficial in interpreting indefinite data. The concept of fuzzy set theory is found by Zadeh [30] and it is a useful, effectual and operative tool that can deal with the problems relating to ambiguous, indeterminate and impressive judgements. In this state, the main flaw is that the accomplishment of the concept of fuzzy set is related to the exact real number. Recently, the data are more imprecise and complex with high uncertainty and so Interval valued fuzzy set introduced by Zadeh becomes a more significant tool to deal with the optimization problems. Further, the fuzzy set is not suitable to all conditions under uncertainty and hesitation. On that account, Atanassov [5] proposed the concept of “Intuitionistic fuzzy set”, a realistic tool to deal with the problem having both unpredictability and hesitation. Here both the degree of membership (acceptance) and non membership (rejection) is considered which eases out the real life decision making problems. For the betterment of handling unpredictability and hesitancy, the concept of “interval valued intuitionistic fuzzy set” has been introduced by Atanassov and Gargov [6], where the data are in intervals instead of stable real number.

Transportation problem is a well known optimization technique, considered to be the vital characteristic which has been deliberated in a broad span of operations especially in research field. It is helpful in replicating many real life problems. The main purpose is to cut down the expenditure on transporting products between origin and destination. Initially the formulation of transportation problem in a linear programming problem is investigated by Hitchcock [14] in 1941. Fuzzy transportation problem is developed by many authors [3 , 22] to handle with the uncertainty in which only membership degrees are taken into account to obtain optimal solution. Fuzzy sets don’t always pave solution for real time problems as it cannot accept inconsistent or less information. Multiple authors [1 , 12] carried out a study in solving the intuitionistic fuzzy TP being generalized from fuzzy TP. Multiple objective TP carries an important role in obtaining optimal solution since one objective can’t be inhibited. Many on-going researchers are focusing on IVIFTP, as the expenditure can be represented in intervals rather than real numbers. These researches reveal a fascinating fact that the transportation expenditure rolls up on many unreliable aspects such as climatic change and weather forecasting, traffic jams etc.

The degree of membership and non membership should fall between 0 and 1 in IFS. Contrarily, we get into a situation where the degree can’t be defined within very least, least, moderate, severe, very high etc. Resolving this, “Interval valued Intuitionistic Fuzzy set” has been proposed by‘Atanassov and Gargov’, where the degree values are subsets of [0,1] instead being the exact value in [0,1]. Allocating the parameters as Interval valued, improves the process of fuzzy and IF sets. Bharati [7] presented an illustration of solving the transportation problem under IVIF triangular number by using a ranking method. Umamageswari and Uthra [29] presented the paper solving the Interval valued intuitionistic Linguistic fuzzy TP using a new ranking technique. Bharati and Singh [8] proposed an advanced method of solving Transportation problem with “Interval valued intuitionistic fuzzy numbers” to get the optimal solution by defining the degrees of membership and non membership functions in interval instead of fixed valued function. Sujeet Kumar Singh, Shiv Prasad Yadav [27] formulated the transportation problem under triangular intuitionistic fuzzy numbers and defined the accuracy function of membership and non membership functions by using the score functions. Intuitionistic fuzzy MODI method is used for applying the optimal solution. A well organized mathematical result in deciding intui fuzzy trans problem had been suggested by Ali Ebrahimnejad, Jose Luis Verdegay [10] drew on traditional transportation process and the prices are represented as tri intui fuzzy numbers. Hunwisaia et al. [15] introduced the new method for clearing up the IFTP is accessed by Intuitionistic fuzzy generalized trapezoidal parameters. The optimal solution for intuitionistic fuzzy transportation problem is found by NWC rule and MODI technique. A fresh ranking procedure for intuitionistic fuzzy generalized trapezoidal numbers is designed by Aggarwal et al. [2] to overcome the restriction of current method and final optimal solution of TP is acquired. With the help of nanogonal intuitionistic fuzzy numbers the TP is explained by Santhi et al. [24] by employing highest cost method and it is transformed into a crisp number by exploiting ranking process.

Numerous methods of TP are introduced by many authors. Among the three basic methods of transportation problem namely North west corner rule, Least cost method and VAM, VAM gives the most optimal solution. In this proposal, various methods like VAM [4], Harmonic mean method [21] which is a method close to VAM, Zack’s Algorithm [23] is employed to find the fundamental possible solution for theTP. Monalisha’s Approximation method [28] is developed to calculate both initial possible solution alongside optimal solution for the given TP and Zero point method [26], a similar method to VAM which is evolved to identify the minimal expenditure of the transportation problem. By using these methods, the comparative study is undertaken to trade in with the formulation of new interval arithmetic operations for solving interval-valued IF Transportation problem. The main aim of this proposal is to succeed an Iterative method of “IVIFTP” without transforming into classic representation, i.e., without converting into crisp number. The subsequent are the formation of this article:

(i) In segment 2, the recollection of the basic concepts that are correlated to our study are specified.

(ii) In segment 3, the detailed explanation concerning the mathematical formulating connected to IV intuitionistic fuzzy TP is stated.

(iii) The algorithms to recognize several methods of TP like VAM, Harmonic Mean Approach, Zack Algorithm, Monalisha’s Approximation method, Zero point method are validated in the segment 4.

(iv) In segment 5, Numerical illustration is carried out to examine the concepts related to this current paper.

2 Preliminaries

In this segment, the basic concepts like FS, IVFS, IFS, IVIFS and arithmetic operations are explained in a detailed way.

2.1 Fuzzy set

[19] Let A be the universal set and the elements are associated with membership values between 0 to 1, then the fuzzy set $\tilde{F}$ is interpreted as a set of 2-tuples such that, $\tilde{F} = {a, μ^{\tilde{F}} (a) : a \in A, μ^{\tilde{F}} (a) \in [0, 1]}$ (1) where $μ^{\tilde{F}} (a)$ is the membership function of a fuzzy set $\tilde{F}$ .

2.2 Interval valued fuzzy set

[19] An IV fuzzy set $\tilde{F}$ over a universal set A is a mapping such that $\tilde{F} = A \to Int ([0, 1])$ (2)].

The explanation of this generalization lies because occasionally it is not relevant to presume that the degree of membership for some elements are precisely prescribed, hence we confess a kind of further uncertainty, therefore the degree of membership is not a single number, but a whole interval.

2.3 Intuitionistic fuzzy set

[7] Let A be the universal set and an‘intuitionistic fuzzy set’ $\tilde{F}$ is represented by $\tilde{F} = {a, μ^{\tilde{F}} (a), ν^{\tilde{F}} (a) : a \in A, μ^{\tilde{F}} (a), ν^{\tilde{F}} (a) \in [0, 1]}$ (3) where $μ^{\tilde{F}} (a)$ and $ν^{\tilde{F}} (a)$ represents the degree of validity and degree of non-validity respectively, such that 0 $\leq μ^{\tilde{F}} (a) + ν^{\tilde{F}} (a) \leq 1 \forall a \in A$ .

The value of $π^{\tilde{F}} (a) = 1 - μ^{\tilde{F}} (a) - ν^{\tilde{F}} (a)$ is called the “intuitionistic fuzzy set index or hesitation margin” of a in $\tilde{F}$ and $π^{\tilde{F}} (a) : A \to [0, 1] and 0 \leq π^{\tilde{F}} (a) \leq 1 \forall a \in A$ , which shows the absence of knowledge whether‘a’ is joined to IFS $\tilde{F}$ or not.

2.4 Intuitionistic fuzzy number

[29] An IF integer $\tilde{F}$ is defined as an intuitionistic fuzzy subset of the real line R as follows,

i) $\tilde{F}$ is normal, i.e., for any a₀ ∈ R is described as for membership function, $μ^{\tilde{F}} (a_{0}) = 1$ and for non membership function, $ν^{\tilde{F}} (a_{0}) = 0$ .

ii) $\tilde{F}$ is convex set for the membership function $μ^{\tilde{F}} (a)$ such that $\begin{matrix} μ^{\tilde{F}} (λ a^{1} + (1 - λ) a^{2}) \geq m i n (μ^{\tilde{F}} (a^{1}), μ^{\tilde{F}} (a^{2})) \\ \forall a^{1}, a^{2} \in R, λ \in [0, 1] \end{matrix}$

iii) $\tilde{F}$ is concave set for the non-membership function $ν^{\tilde{F}} (a)$ such that

$\begin{matrix} υ^{\tilde{F}} (λ a^{1} + (1 - λ) a^{2}) \leq m i n (μ^{\tilde{F}} (a^{1}), μ^{\tilde{F}} (a^{2})) \\ \forall a^{1}, a^{2} \in R, λ \in [0, 1] \end{matrix}$

2.5 Interval valued intuitionistic fuzzy set

[7] Let A be the universal set and an IVIFS $\tilde{F}$ in A is exposed as,

$\tilde{F} = {(a, [μ^{{\tilde{F}}^{-}} (a), μ^{{\tilde{F}}^{+}} (a)] [ν^{{\tilde{F}}^{-}} (a), ν^{{\tilde{F}}^{+}} (a)]) : a \in A}$ (4) where $μ^{{\tilde{F}}^{-}} (a) : A \to [0, 1], μ^{{\tilde{F}}^{+}} (a) : A \to [0, 1]$ represents the beneath and overhead degrees of membership respectively and $ν^{{\tilde{F}}^{-}} (a) : A \to [0, 1], ν^{{\tilde{F}}^{+}} (a) : A \to [0, 1]$ represents the inferior and higher degrees of non-membership respectively and for every $a \in A, 0 \leq μ^{{\tilde{F}}^{+}} (a) + ν^{{\tilde{F}}^{+}} (a) \leq 1$ and it is shown in (Fig. 1).

Fig. 1

Interval valued IFS.

2.6 Interval valued intuitionistic fuzzy number

[29] Let Y be a finite non-empty set such that, Y = y₁, y₂, . . . . . , y_n. Let R[0,1] be all subintervals of [0,1]. An IVIFS $\tilde{X}$ in Y is expressed as, $\tilde{X} = {(y, μ^{\tilde{F}} (y), ν^{\tilde{F}} (y) : y \in Y)}$ (5) where $μ^{\tilde{F}} : Y \to R [0, 1]$ which satisfies the condition of $0 \leq \sup μ^{\tilde{F}} (y) + \sup ν^{\tilde{F}} (y) \leq 1 \forall y \in Y$ and $μ^{\tilde{F}} (y) and ν^{\tilde{F}} (y)$ represents the degree of validity and non-validity of y to $\tilde{X}$ respectively.

For an IVIFS $\tilde{A}$ , the pair ${μ^{\tilde{F}} (y), ν^{\tilde{F}} (y)}$ is called an ’Interval valued Intuitionistic fuzzy number’ and is described as ([α, β] , [γ, δ]) ; where [α, β] ⊂ [0, 1] , [γ, δ] ⊂ [0, 1] , 0 ≤ β + δ ≤ 1.

2.7 Arithmetic operations of interval valued intuitionistic fuzzy numbers

[13] Let $\tilde{A} = ([α_{1}, β_{1}], [γ_{1}, δ_{1}]) and \tilde{B} = ([α_{2}, β_{2}], allow [γ_{2}, δ_{2}])$ be two interval valued IF numbers, using these two IVIF numbers, the following expressions are defined:

$1 . \tilde{A} \oplus \tilde{B} = ([α_{1} + α_{2} - α_{1} α_{2}, β_{1} + β_{2} - β_{1} β_{2}], allow [γ_{1} γ_{2}, δ_{1} δ_{2}])$

$2 . \tilde{A} \otimes \tilde{B} = ([α_{1} α_{2}, β_{1} β_{2}], [γ_{1} + γ_{2} - γ_{1} γ_{2}, δ_{1} + δ_{2} - δ_{1} δ_{2}])$

$3 . λ \tilde{A} = ([1 - (1 - α_{1})^{λ}], [1 - (1 - β_{1})^{λ}], [γ_{1}^{λ}], [δ_{1}^{λ}])$

$4 . {\tilde{A}}^{λ} = ([α_{1}^{λ}, β_{1}^{λ}], [1 - (1 - γ_{1})^{λ}], [1 - (1 - δ_{1})^{λ}])$

$5 . \tilde{A} ⊖ \tilde{B} = ([\frac{α_{1} - α_{2}}{1 - α_{2}}, \frac{β_{1} - β_{2}}{1 - β_{2}}], [\frac{γ_{1}}{γ_{2}}, \frac{δ_{1}}{δ_{2}}])$

2.8 Interval number

[18] An interval is considered on the real line R and it can be represented by, r = [r¹, r²] = {a ∈ R : r¹ ≤ a ≤ r² and r¹, r² ∈ R} If $\tilde{r} = r^{1} = r^{2} = r$ , then $\tilde{r} = [r^{1}, r^{2}] = r$ is a real number or degenerate interval, then the midpoint and width of an interval number $\tilde{r} = [r^{1}, r^{2}]$ can be expressed as ${\tilde{r}}_{m} = (\frac{r^{1} + r^{2}}{2}), {\tilde{r}}_{w} = (\frac{r^{2} - r^{1}}{2})$ respectively. This interval number $\tilde{r}$ can be redefined with regard to its midpoint and width as $\tilde{r} = [r^{1}, r^{2}] = (({\tilde{r}}_{m}), ({\tilde{r}}_{w}))$ .

2.9 Relation of intervals

Sengupta and Pal [25] introduced a simple and effective index for linking any two intervals on real number and it is based on the satisfaction of decision makers. For a given two intervals $\tilde{r} = [r^{1}, r^{2}] and \tilde{s} = [s^{1}, s^{2}]$ in real numbers, the extended order relation between these intervals is denoted by ⪯, then

(i) $If ({\tilde{r}}_{m}) < ({\tilde{r}}_{w}),$ then $\tilde{r} < \tilde{s}$ (or) $\tilde{r}$ is inferior to $\tilde{s}$ .

(ii) $If ({\tilde{r}}_{m}) > ({\tilde{r}}_{w}),$ then $\tilde{r} > \tilde{s}$ (or) $\tilde{r}$ is superior to $\tilde{s}$ .

(iii) $If ({\tilde{r}}_{m}) = ({\tilde{r}}_{w}),$ then $\tilde{r} = \tilde{s}$ (or) $\tilde{r}$ is equal to $\tilde{s}$ .

2.10 New interval arithmetic

Ma et al. [20] projected a new interval fuzzy arithmetic which is built on both “location index and fuzziness index functions”. The location index number is considered as the usual arithmetic, meanwhile the fuzziness index function is observed to trail the lattice rule which is least upper bound in the lattice L. Let $\tilde{r}, \tilde{s} \in L$ , then we state $\tilde{r} \lor \tilde{s} = \max {\tilde{r}, \tilde{s}} and \tilde{r} \land \tilde{s} = \min {\tilde{r}, \tilde{s}}$ (6) For any two intervals, $\tilde{R} = [r^{1}, r^{2}], \tilde{S} = [s^{1}, s^{2}]$ , the arithmetic operations on $\tilde{R}$ and $\tilde{S}$ are well established and justified as,∗ ∈ {+ , - , × , ÷} $\tilde{R} * \tilde{S} = 〈 {\tilde{r}}_{m}, {\tilde{r}}_{w} 〉 * 〈 {\tilde{s}}_{m}, {\tilde{s}}_{w} 〉 = 〈 {\tilde{r}}_{m} * {\tilde{s}}_{m}, \max ({\tilde{r}}_{w}, {\tilde{s}}_{w}) 〉$ (7)

3 Interval valued intuitionistic fuzzy TP

An IVIFTP is a very unique event of‘IVIF LP problem’ and this IVIFLPP can be solved by simplex method, but it allows a frail basic feasible solution and also it takes a huge time to compute. Hence, basic feasible solution can be attained by using Interval val intui fuzzy Vogel’s Approxi method (IVIFVAM) and the solution is compared with various methods such as zero point method, Monalisha’s Approximation method etc. In this paper, we contemplate an IVIFTP with m stocks and n requirements. The Cost ${\tilde{C}}_{ij}$ of IVIF numbers is given by, ${\tilde{C}}_{ij} = ([C_{a}^{L}, C_{b}^{U}], [C_{c}^{L}, C_{d}^{U}])$

In this mode, the cost are represented in interval numbers but, the availability and requirement values are in regular crisp number. We conquire the solution of Initial BFS of TP numerous methods.

3.1 Mathematical formulation

Let us consider ${\tilde{C}}^{ij} = ([μ_{{CL}^{ij}}], [μ_{{CU}^{ij}}], [ν_{{CL}^{ij}}], allow [ν_{{CU}^{ij}}])$ be the interval val intui fuzzy numbers which represents the unit fuzzy transportation price from i^th stock to j^th demand. Let us assume, ${\tilde{a}}^{i}$ be the fuzzy supply at stock i and ${\tilde{b}}^{j}$ be the fuzzy requirement at destination j and ${\tilde{X}}^{ij}$ be the number of components carried from origin i to destiny j and the following supply and demand is exposed in (Fig. 2). Then IVIFTP can be mathematically expressed as,

Fig. 2

Supply and demand.

Minimize \tilde{z} = [μ_{{\tilde{C}}_{L}^{ij}}, μ_{{\tilde{C}}_{U}^{ij}}], [ν_{{\tilde{C}}_{L}^{ij}}, ν_{{\tilde{C}}_{U}^{ij}}] {\tilde{X}}^{ij}

subject to,

\sum_{j = 1}^{t} {\tilde{X}}^{ij} = [μ_{{\tilde{a}}_{L}^{i}}, μ_{{\tilde{a}}_{U}^{i}}], [ν_{{\tilde{a}}_{L}^{i}}, ν_{{\tilde{a}}_{U}^{i}}] i = 1, 2, 3, . . . . . . s

\sum_{i = 1}^{s} {\tilde{X}}^{ij} = [μ_{{\tilde{b}}_{L}^{j}}, μ_{{\tilde{b}}_{U}^{j}}], [ν_{{\tilde{b}}_{L}^{j}}, ν_{{\tilde{b}}_{U}^{j}}] j = 1, 2, 3, . . . . . . t

and {\tilde{X}}^{ij} \geq 0, i = 1, 2, 3, . . . . s, j = 1, 2, 3, . . . t \forall i and j

Hitchcock [14] discovered the basic TP with static parameters, and it was exhibited by standard LP except handling uncertainty and hesitancy referred in (Table 1). Many transportation models have seemed in the works in which unpredictability was a major cause to share out. In this paper, we challenge this unreliability and hesitancy by using interval valued IFS.

Table 1

Indeterminate transportation

Destinations →	Customer ₁	Customer ₂	Customer ₃	${\tilde{a}}^{i}$
Sources ↓
Factory ₁	${\tilde{C}}_{IV}^{11}$	${\tilde{C}}_{IV}^{12}$	${\tilde{C}}_{IV}^{13}$	${\tilde{a}}^{1}$
Factory ₂	${\tilde{C}}_{IV}^{21}$	${\tilde{C}}_{IV}^{22}$	${\tilde{C}}_{IV}^{23}$	${\tilde{a}}^{2}$
Factory ₃	${\tilde{C}}_{IV}^{31}$	${\tilde{C}}_{IV}^{32}$	${\tilde{C}}_{IV}^{33}$	${\tilde{a}}^{3}$
${\tilde{b}}^{j}$	${\tilde{b}}^{1}$	${\tilde{b}}^{2}$	${\tilde{b}}^{3}$

4 Computational method

In this segment, we calculate the IBFS using IV Intuitionistic fuzzy VAM, IVIF Harmonic mean approach, IVIF Zack algorithm, IVIF Monalisha’s Approximation method and IVIF zero point method.

4.1 IVIF Vogel’s Approximation method

Vogel’s Approximation method (VAM) is the familiar method which is used to identify the initial basic feasible solution of Transportation problem.

Step 1: The cost of transportation is revealed as IVI fuzzy numbers and examine whether the given fuzzy transportation problem is well-adjusted or not.

If it is finely adjusted (i.e., the total of quantity available is identical to the total of requirement) then proceed to step 3.

If it is not in a equilibrium state, (i.e.,the total of availability is not identical to the total of necessary values) then proceed to step 2.

Step 2: If it is not an equalized type, then initiate dummy rows or dummy columns with fuzzy costs as zero to assemble into a uniformity one.

Step 3: In each row and column of Interval valued intuitionistic fuzzy price Transportation table, select the minimum and next minimum IVIF value and then evaluate the penalty (i.e., subtracting least and the consequent least value) p^r, p^s, r=1,2,3,...,i; s=1,2,3,..., j in between each row and column.

Step 4: Associate the top most penalty p^r, p^s, r=1, 2,3,...,i; s=1,2, 3,...,j enclosed by all the rows and columns and then allocate as great as possible. If a bind up arise, then select the cell in which the maximum allocation can be done.

Step 5: Tune the available and wanted values, and strike out the satisfied row or column in which the cells are allocated in the particular row or column.

Step 6: Recur these steps as far as all supply and requirement values come by zero.

4.2 IVIF harmonic mean approach

Harmonic Mean Approach is the new statistical method used to discover the optimal solution. It is very nearer to VAM method. The arithmetical instance is specified to resemble with the existing methods. The steps are enumerated below:

Step 1: The charge of transportation is revealed as interval val intuit fuzzy numbers and examine whether the given fuzzy transport problem is leveled up or not.

If it is in a equilibrium one (i.e., the sum of stock is similar to sum of needed value) then proceed to step 3.

If it is not an adjusted type (i.e., the sum of available values is not uniform to the sum of desired cost) then move to step 2.

Step 2: If it is not a satisfied one, then initiate dummy rows or dummy columns with fuzzy costs as zero to assemble into a equilibrium one.

Step 3: Identify the harmonic mean for entire row and every column and the formula for calculating harmonic mean is, $H . M = \frac{Total number of observations}{sums of the reciprocal of the number}$

Step 4: After the calculation of Harmonic mean value, identify the highest price surrounded by them.

Step 5: After identifying the maximum value, allot the lowest supply or demand to the place of the allied row or column.

Step 6: Recur the steps 3-5 as far as all the supply and demand values are exhausted.

Step 7: Total minimum cost can be concluded by adding the multiplying the cost price and its subsequent allotted values of stock or requirement.

4.3 IVIF Zack Algorithm

Zack Algorithm is proposed to find an ini basic feasible solution to the TP. The main concept of Zack Algorithm is to occupy zero in the cell which has the highest cost and occupy the maximum cost in the cell which has the lowest cost. The steps are listed below:

Step 1: Choose the cell having the highest interval cost and permeate the cell with the number 0.

Step 2: In that cell having the greatest interval cost, pick the cell which has the least cost in the selected row and allocate with the maximum cost.

Step 3: Select the cell which has the least cost in the selected column and allocate with the maximum cost. If there is more than one cell having the least cost, then choose the cell that can be allocated with the extreme amount.

Step 4: If there are even so cells left over, the lack in the least cost cell has been filled with the utmost number and so on.

Step 5: At last the total transportation cost can be calculated by, Total cost = $\sum_{i = 1}^{m} \sum_{j = 1}^{n} {\tilde{C}}_{ij} {\tilde{X}}_{ij}$

4.4 IVIF maximum range column method

In solving problems, MRCM is a better tool to identify IBFS as analyzed with other methods along with, it can be updated based upon its attainment.

Step 1: From the given problem, select the interval cost values and test whether the total demand equals to the total supply. If satisfies, drive it to step 3.

Step 2: If it does not compensate, add a dummy row or column and the value of the expense elements are zero, whereas the upcoming value of availability and requirement can be given as the unsatisfied dissimilarity of amount available and amount wanted.

Step 3:Trace the range of each and every column by proceeding the difference among largest and least price

Step 4: With the values got in step 3, allocate the minimum value of (s,d) to the element having the minimum transportation price in the chosen row or column.

Step 5: In the obtained column transportation table, there is no need of considering the satisfied row or column. Consider only the containing row or column.

Step 6: Carry on the same process until entire rows and colums are fulfilled.

Step 7: Atlast, compute the final IBFS using the algorithm of the original transportation technique.

4.5 IVIF zero point method

The feasible solution is acquired by Zero point method by solving transportation problem to get minimized cost.

Step 1: The cost is represented in Interval valued intuitionistic fuzzy number and from the specified transportation problem, check whether it is in a balanced condition or not. If it is satisfied, proceed further. If it is not satisfied, substitute a dummy row or column and make it in a equilibrium state.

Step 2: From the transportation table, select and subtract each little row member with the comparable row entries.

Step 3: Also, pick every column constituents gathered in step 2 along with deduct the members from the smallest value of the interrelated columns.

Step 4: Examine whether each column requirement is lesser than the total of the row stocks in which the shortened costs in that column having the value as zero. Also, scrutinize if each row stock is shorter than the sum of the column requirements in which the reduced prices in that row are nought. If it is applicable, move to step 7 (in this case, the reduced transportation table is denominated as allotment table) If it is not relevant, move to step 5.

Step 5: Sketch the least possible number of parallel lines and perpendicular lines to laminate all the zeros of the shortened transportation table in which there will be some entries in rows or columns which doesnot satisfy the rule in step 4 are not to be shaded.

Step 6: Enlarge the just out reformed reduced transportation table in accordance with:

(i) Identify the least cell of the reduced cost members which is not shaded by any boundaries.

(ii) Subtract that cell from all the unshaded entries and sum up the least value at the meeting of any two boundaries, then move to step 4.

Step 7: Select an element in the row^α or column^β of the shortened transportation table in which the unique cell whose shortened time is zero and then allot the utmost attainable value to that cell. If such that cell does not happen for the top-most value, then find the next top value of the table. Even if such cell doesnot arise for any estimate, then from the table choose any entry to whom reduced time is nought.

Step 8: Refine the compressed transportation table by subsequently removing the entire stock and demand values and as well, change the not included stock and requirements points into the perfectly inserted.

Step 9: Iterate from step 7 to step 9 up to all stock points are utilized and all destiny points are completely received.

Step 10: This allocation effects a solution to the costs transportation calculation.

5 Numerical illustration

In this segment, numerical illustration of [29] is considered in Table 2 to prove the proposed computational method of the IVIFTP. In this case, the cost is viewed as a IVIFN and this number was acquired by the linguistic variables, whereas the supply and demand are taken as a fixed crisp numbers.

Table 2
Interval valued Intuitionistic fuzzy cost table

X Sri Thaanvi Sri Krishna Sri Sivasakthi Supply

Traders Traders Traders

Journey time [0.8,0.9] [0.7,0.8] [0.6,0.8] 2

[0.0,0.1] [0.1,0.2] [0.0,0.2]

Cost [0.6,0.7] [0.5,0.7] [0.5,0.6] 2

[0.2,0.3] [0.2,0.3] [0.2,0.3]

Distance [0.4,0.5] [0.5,0.6] [0.4,0.6] 1

[0.2,0.4] [0.2,0.3] [0.1,0.2]

Safety [0.7,0.8] [0.6,0.8] [0.6,0.7] 1

[0.1,0.2] [0.0,0.1] [0.1,0.2]

Supply Capacity [0.5,0.7] [0.7,0.8] [0.4,0.5] 2

[0.1,0.3] [0.1,0.2] [0.2,0.4]

On time delivery [0.2,0.3] [0.3,0.5] [0.4,0.6] 2

[0.5,0.6] [0.4,0.5] [0.3,0.4]

Response to change [0.4,0.5] [0.2,0.5] [0.4,0.7] 1

[0.3,0.4] [0.3,0.5] [0.2,0.3]

Demand 3 4 4

X	Sri Thaanvi	Sri Krishna	Sri Sivasakthi	Supply
Journey time	[0.8,0.9]	[0.7,0.8]	[0.6,0.8]	2
	[0.0,0.1]	[0.1,0.2]	[0.0,0.2]
Cost	[0.6,0.7]	[0.5,0.7]	[0.5,0.6]	2
	[0.2,0.3]	[0.2,0.3]	[0.2,0.3]
Distance	[0.4,0.5]	[0.5,0.6]	[0.4,0.6]	1
	[0.2,0.4]	[0.2,0.3]	[0.1,0.2]
Safety	[0.7,0.8]	[0.6,0.8]	[0.6,0.7]	1
	[0.1,0.2]	[0.0,0.1]	[0.1,0.2]
Supply Capacity	[0.5,0.7]	[0.7,0.8]	[0.4,0.5]	2
	[0.1,0.3]	[0.1,0.2]	[0.2,0.4]
On time delivery	[0.2,0.3]	[0.3,0.5]	[0.4,0.6]	2
	[0.5,0.6]	[0.4,0.5]	[0.3,0.4]
Response to change	[0.4,0.5]	[0.2,0.5]	[0.4,0.7]	1
	[0.3,0.4]	[0.3,0.5]	[0.2,0.3]
Demand	3	4	4

There are three transportation delivering companies and seven boundaries engaged during transportation. Evidently, the precise data is unknown while the lower and upper limits of such boundaries are known. At the moment, basic feasible solution can be obtained by the methods discussed in segment 4 and it is resulted in IVI fuzzy numbers.

R.M Umamageswari et al. converted the IVIF number into crisp number by using ranking technique and the cost is acquired as 1.25. In this paper, we consider the same problem and it can be solved by several methods. To get the feasible solution, it can be solved by using VAM. Let us establish all the interval parameters as $\tilde{r} = [r^{1}, r^{2}]$ and $\tilde{s} = [s^{1}, s^{2}]$ in terms of midpoint and width as $\tilde{r} = 〈 ({\tilde{r}}_{m}), ({\tilde{r}}_{w}) 〉$ and $〈 ({\tilde{s}}_{m}), ({\tilde{s}}_{w}) 〉$ . This is a balanced TP. Now, the given interval valued intuitionistic fuzzy TP is expressed as follows:

By using the new interval Arithmetic operations, the interval valued IFN is transformed into the numbers representing in midpoint and width as it is mentioned in (Table 3). The values of capacities and requirements are also assumed to be in interval and converted in terms of midpoint and width. Hence after the allocation of the respective supply and requirement, the initial basic feasible solution of the IVIFTP by using Vogel’s Approximation method is:

Table 3

Interval valued Intuitionistic fuzzy membership ana non-membership costs in terms of midpoint and width

X	Shree Thaanvi	Sri Krishna	Sri Sivasakthi	Supply
Traders	Traders	Traders
Journey time	〈0.85, 0.05〉	〈0.75, 0.05〉	〈0.7, 0.1〉	〈2, 0〉
	〈0.05, 0.05〉	〈0.15, 0.05〉	〈0.1, 0.1〉
Cost	〈0.65, 0.05〉	〈0.6, 0.1〉	〈0.55, 0.05〉	〈2, 0〉
	〈0.25, 0.05〉	〈0.25, 0.05〉	〈0.25, 0.05〉
Distance	〈0.45, 0.05〉	〈0.55, 0.05〉	〈0.5, 0.1〉	〈1, 0〉
	〈0.3, 0.1〉	〈0.25, 0.05〉	〈0.15, 0.05〉
Safety	〈0.75, 0.05〉	〈0.7, 0.1〉	〈0.65, 0.05〉	〈1, 0〉
	〈0.15, 0.05〉	〈0.05, 0.05〉	〈0.15, 0.05〉
Supply Capacity	〈0.6, 0.1〉	〈0.75, 0.05〉	〈0.45, 0.05〉	〈2, 0〉
	〈0.2, 0.1〉	〈0.15, 0.05〉	〈0.3, 0.1〉
On time delivery	〈0.25, 0.05〉	〈0.4, 0.1〉	〈0.5, 0.1〉	〈2, 0〉
	〈0.55, 0.05〉	〈0.45, 0.05〉	〈0.35, 0.05〉
Response to change	〈0.45, 0.05〉	〈0.35, 0.15〉	〈0.55, 0.15〉	〈1, 0〉
	〈0.35, 0.05〉	〈0.4, 0.1〉	〈0.25, 0.05〉
Demand	〈3, 0〉	〈4, 0〉	〈4, 0〉

We acquire the interval valued intuitionistic cost for each allocated cells C₁₁, C₂₁, C₂₂, C₃₃, C₄₂, C₅₂, C₆₃, allowC₇₃. From the Table 4, the initial IVIF feasible total transportation cost for the allocated membership function is obtained by, = 〈2, 0〉 ∗ 〈0.85, 0.05〉 + 〈1, 0〉 ∗ 〈0.65, 0.05〉 + 〈1, 0〉 ∗ 〈0.6, 0.1〉 + 〈1, 0〉 ∗ 〈0.5, 0.1〉 + 〈1, 0〉 ∗ 〈0.7, 0.1〉 + 〈2, 0〉 ∗ 〈0.75, 0.05〉 + 〈2, 0〉 ∗ 〈0.5, 0.1〉 + 〈1, 0〉 ∗ 〈0.55, 0.15〉 = 〈7.2, 0.15〉 = [7.05, 7.35]

Table 4

Allocation of Interval valued Intuitionistic fuzzy TP using VAM

X	Shree Thaanvi Traders	Sri Krishna Traders	Sri Sivasakthi Traders	Supply
	〈2, 0〉
Journey time	〈0.85, 0.05〉	〈0.75, 0.05〉	〈0.7, 0.1〉	〈2, 0〉
	〈0.05, 0.05〉	〈0.15, 0.05〉	〈0.1, 0.1〉
	〈1, 0〉	〈1, 0〉
Cost	〈0.65, 0.05〉	〈0.6, 0.1〉	〈0.55, 0.05〉	〈2, 0〉
	〈0.25, 0.05〉	〈0.25, 0.05〉	〈0.25, 0.05〉
			〈1, 0〉
Distance	〈0.45, 0.05〉	〈0.55, 0.05〉	〈0.5, 0.1〉	〈1, 0〉
	〈0.3, 0.1〉	〈0.25, 0.05〉	〈0.15, 0.05〉
		〈1, 0〉
Safety	〈0.75, 0.05〉	〈0.7, 0.1〉	〈0.65, 0.05〉	〈1, 0〉
	〈0.15, 0.05〉	〈0.05, 0.05〉	〈0.15, 0.05〉
		〈2, 0〉
Supply Capacity	〈0.6, 0.1〉	〈0.75, 0.05〉	〈0.45, 0.05〉	〈2, 0〉
	〈0.2, 0.1〉	〈0.15, 0.05〉	〈0.3, 0.1〉
			〈2, 0〉
On time delivery	〈0.25, 0.05〉	〈0.4, 0.1〉	〈0.5, 0.1〉	〈2, 0〉
	〈0.55, 0.05〉	〈0.45, 0.05〉	〈0.35, 0.05〉
			〈1, 0〉
Response to change	〈0.45, 0.05〉	〈0.35, 0.15〉	〈0.55, 0.15〉	〈1, 0〉
	〈0.35, 0.05〉	〈0.4, 0.1〉	〈0.25, 0.05〉
Demand	〈3, 0〉	〈4, 0〉	〈4, 0〉

The initial basic IVIF feasible total transportation cost for the allocated non-membership function is acquired by, = 〈2, 0〉 ∗ 〈0.05, 0.05〉 +〈1, 0〉 ∗ 〈0.25, 0.05〉 + 〈1, 0〉 ∗ 〈0.25, 0.05〉 + 〈1, 0〉 ∗ 〈0.15, 0.05〉 +〈1, 0〉 ∗ 〈0.05, 0.05〉 + 〈2, 0〉 ∗ 〈0.15, 0.05〉 + 〈2, 0〉 ∗ 〈0.35, 0.05〉 + 〈1, 0〉 ∗ 〈0.25, 0.05〉 = 〈2.05, 0.05〉 = [2, 2.1].

Therefore, the degrees of membership and non membership are exhibited in closed Interval and the IVI fuzzy feasible solution for the transportation problem is uttered by [7.05,7.35, 7.05,7.35] [2,2.1, 2,2.1]. The various methods by HMA, ZA, MRCM, ZPM is utilized and the costs price are produced in Table 5.

Table 5

Discrimination of proposed method with existing methods\label tab:5

S.No.	Computational methods	Costs
1.	Proposed method using VAM	[7.05,7.35]
		[2,2.1]
2.	Proposed method using Harmonic mean	[7.05,7.35]
	Approach	[2.1,2.2]
3.	Proposed method using Zack Algorithm	[7.05,7.35]
		[2,2.1]
4.	Proposed methos using Maximum range	[7.05,7.35]
	column method	[2.1,2.2]
5.	Proposed method using Zero point method	[7.05,7.35]
		[2,2.2]

6 Conclusion

This paper suggests a new technique to formulate the TP by utilizing interval valued IF numbers. This method is recommended to identify the IBFS of IVIFNs derived on both requirements and availabilities, which are in real numbers. The cost in [29] is acquired by transforming the interval number into single crisp number and the price captured is 1.25. The computational methodology by using new interval arithmetic operations is undergone and finally the costs of VAM, HMM, Zack Algorithm, MRCM and ZPM are uniform, specifically pointed the degree of validity in interval as [7.05,7.35, 7.05,7.35], whereas there is a little modification in the degree of non-validity cost in the alternative methods. The Costs of IVIF membership and non-membership function are shown in Fig. 3. Moreover, IVIF sets are applicable to various domains like decision making, medicine, industry, artificial intelligence etc., where the paremeters are uncertain and not clear in nature.

Fig. 3

Interval valued intuitionistic fuzzy membership and non-membership costs after applying interval arithmetic operations.

References

Abhishekh and A. Nishad, A novel ranking approach to solving fullylr-intuitionistic fuzzy transportation problems, NewMathematics and Natural Computation 15(01) (2019), 95–112.

Aggarwal

, Gupta

Algorithm for solving intuitionistic fuzzytransportation problem with generalized trapezoidal intuitionisticfuzzy number via new ranking method, arXiv preprintarXiv:1401.3353 (2014).

Akilbasha

, Natarajan

and Pandian

, Solving transportationproblems with mixed constraints in rough environment, International Journal of Pure and Applied Mathematics 113(9) (2017), 130–138.

Amaliah

, Fatichah

, Suryani

Two highest penalties: Amodified vogels approximation method to find initial basic feasible solution of transportation problem, In 2021 13th InternationalConference on Information & Communication Technology and System(ICTS) pp. 318–323. IEEE, (2021).

Atanassov

K.T.

, More on intuitionistic fuzzy sets, Fuzzy Setsand Systems 33(1) (1989), 37–45.

Atanassov

K.T.

Interval valued intuitionistic fuzzy sets, In Intuitionistic Fuzzy Sets pp. 139–177 Springer (1999).

Bharati

S.K.

, Transportation problem with interval-valuedintuitionistic fuzzy sets: impact of a new ranking, Progress inArtificial Intelligence 10(2) (2021), 129–145.

Bharati

S.K.

and Singh

S.R.

, A new interval-valued intuitionisticfuzzy numbers: ranking methodology and application, NewMathematics and Natural Computation 14(03) (2018), 363–381.

Dharani

and Selvi

, Solving intuitionistic fuzzy transportationproblem with ranking method using matlab code, Applied Scienceand Computations 5(12) (2018), 20–26.

10.

Ebrahimnejad

and Verdegay

J.L.

, An efficient computationalapproach for solving type-2 intuitionistic fuzzy numbers basedtransportation problems, International Journal of ComputationalIntelligence Systems 9(6) (2016), 1154–1173.

11.

Ebrahimnejad

and Verdegay

J.L.

, A new approach for solving fullyintuitionistic fuzzy transportation problems, FuzzyOptimization and Decision Making 17(4) (2018), 447–474.

12.

, Sayed and M.A. Abo-Sinna, A novel approach for fullyintuitionistic fuzzy multi-objective fractional transportationproblem, Alexandria Engineering Journal 60(1) (2021), 1447–1463.

13.

Gou

, Xu

and Liao

, Exponential operations of interval-valuedintuitionistic fuzzy numbers, International Journal of MachineLearning and Cybernetics 7(3) (2016), 501–518.

14.

Hitchcock

F.L.

, The distribution of a product from several sourcesto numerous localities, Journal of Mathematics and Physics 20(1–4) (1941), 224–230.

15.

Hunwisai

, Kumam

and Kumam

, A new method for optimalsolution of intuitionistic fuzzy transportation problems viageneralized trapezoidal intuitionistic fuzzy numbers, FuzzyInformation and Engineering 11(1) (2019), 105–120.

16.

Juman

and Nawarathne

, An efficient alternative approach tosolve a transportation problem, Ceylon Journal of Science 48(1) (2019), 19–29.

17.

Kaur

and Kumar

, A new method for solving fuzzy transportationproblems using ranking function, Applied MathematicalModelling 35(12) (2011), 5652–5661.

18.

Keerthana

and Ramesh

, A new approach for solving integerinterval transportation problem with mixed constraints In Journal of Physics: Conference Series Volume 1377 pp.43, IOP Publishing (2019).

19.

Klir

and Yuan

, Prentice hall New Jersey, Fuzzy Sets and Fuzzy Logic Volume 4 (1995).

20.

, Friedman

and Kandel

, A new fuzzy arithmetic, FuzzySets and Systems 108(1) (1999), 83–90.

21.

Palanivel

and Suganya

, A new method to solve transportationproblem-harmonic mean approach, Engineering Technology OpenAccess Journal 2(3) (2018), 1–3.

22.

Pratihar

, Kumar

, Edalatpanah

and Dey

, Modified vogel’sapproximation method for transportation problem under uncertainenvironment, Complex & Intelligent Systems 7(1) (2021), 29–40.

23.

Rizqi

Z.U.

Zack algorithm: A heuristic approach to solvetransportation problem, In Proceedings of the InternationalConference on Industrial Engineering and Operations Management pp. 1127–1131, (2019).

24.

Santhi

and Kungumaraj

, Optimal solution of a transportationproblem using nanogonal intuitionistic fuzzy number, IndianJournal of Mathematics Research 7 (2019).

25.

Sengupta

and Pal

, Interval-valued transportation problem withmultiple penalty factors, VU Journal of Physical Sciences 9 (2003), 71–81.

26.

Sharma

, Abbas

and Gupta

V.K.

, Solving time minimizingtransportation problem by zero point method, InternationalJournal of Engineering and Science 5(7) (2015).

27.

Singh

S.K.

and Yadav

S.P.

, A new approach for solving intuitionisticfuzzy transportation problem of type-2, Annals of OperationsResearch 243(1) (2016), 349–363.

28.

Soundararajan

et al., Solving unbalanced intuitionistic fuzzytransportation problem, Journal of Computational Mathematica 4(1) (2020), 1–8.

29.

Umamageswari

and Uthra

, Ranking linguistic interval valuedintuitionistic fuzzy transportation problem, The InternationalJournal of Analytical and Experimental Modal Analysis 12(1) (2020), 1373–1378.

30.

Zadeh

L.A.

Fuzzy sets, InFuzzy sets, fuzzy logic, and fuzzysystems: selected papers by Lotfi A Zadeh pp. 394–432, World Scientific, (1996).