NEW APPROACH TO SOLVE FUZZY MULTI-OBJECTIVE MULTI-ITEM SOLID TRANSPORTATION PROBLEM

. This paper explores the study of Multi-Objective Multi-item Solid Transportation Problem (MMSTP) under the fuzzy environment. Realizing the impact of real-life situations, here we consider MMSTP with parameters, e.g. , transportation cost, supply, and demand, treat as trapezoidal fuzzy numbers. Trapezoidal fuzzy numbers are then converted into nearly approximation interval numbers by using (P. Grzegorzewski, Fuzzy Sets Syst. 130 (2002) 321–330.) conversation rule, and we derive a new rule to convert trapezoidal fuzzy numbers into nearly approximation rough interval numbers. We derive different models of MMSTP using interval and a rough interval number. Fuzzy programming and interval programming are then applied to solve converted MMSTP. The expected value operator is used to solve MMSTP in the rough interval. Thereafter, two numerical experiments are incorporated to show the application of the proposed method. Finally, conclusions are provided with the lines of future study of this manuscript.


Introduction
Transportation Problem (TP) is mainly taken into consideration at early study in Operational Research to reduce the transportation cost from sources to various destinations.Hitchcock [20] initially modelled the basic TP, by modelling it as a conventional optimization problem with two-dimensional properties, i.e., supply and demand.In the classical sense of TP, the parameters, supply and demand are crisp numbers but in real situations these parameters are not always crisp.Several researchers considered TPs in a different environment and solved them by their proposed algorithm.Recently, Roy et al. [45] identified a new approach for solving the intuitionistic fuzzy multi-objective transportation problem.Ammar et al. [2] studied on multi-objective transportation problem with fuzzy numbers.Maity et al. [29] introduced a new approach for solving dualhesitant fuzzy transportation problem with restrictions.Anukokila et al. [6] used goal programming approach for solving multi-objective fractional transportation problem with fuzzy programming.Roy et al. [46] analyzed the random-rough variables in a multi-objective fixed-charge transportation problem.Maity et al. [30] analyzed the multi-modal transportation problem and they showed its applications to artificial intelligence.Giri and Roy [16] designed and solved multi-objective green four-dimensional fixed-charge transportation problem under neutrosophic environment.Kaur and Kumar [23] proposed a new approach for solving fuzzy transportation problem using generalized fuzzy numbers.Mahapatra et al. [27] solved multi-choice stochastic transportation problem involving extreme value distribution.Mardanya et al. [32] analyzed how to solve JTP with multi-item under multi-objective environment.Roy et al. [43] solved multi-choice multi-objective transportation problem with interval goal using conic scalarization approach.Ebrahimnejad [9] simplified a new approach for solving transportation problem with generalized trapezoidal fuzzy numbers.Mardanya et al. [33] depicted a study on bi-level multi-objective transportation problem in fuzzy environment.A good number of researches for solving multi-objective transportation problem in different directions were developed by several researchers such as (cf., [11,28,[44][45][46]). Mardanya and Roy [31] studied a time-variant multi-objective linear fractional transportation problem.Alharbi et al. [5] provided an interactive approach to solve the multi-objective minimum cost flow problem in the fuzzy environment.Kaur et al. [24] solved capacitated two-stage time minimization transportation problem with restriction flow.Ammar and Khalifa [3] solved fuzzy multi-objective multi-item solid transportation problems.Mardanya et al. [34] solved the MMTP via the rough interval approach.Recently, Tanksale and Jha [48] solved a hybrid fix and then optimized heuristic for integrated inventory transportation problem in a multi-region multi-facility supply chain.However, apart from supply and demand constraints, in real-world scenarios, often we need to consider the mode of transportation (e.g., goods train, cargo flights, and trucks), the kinds of goods, and so on.Under such circumstances, a TP is extended to a Solid Transportation Problem (STP), apart from source and destination constraints, additional constraints, related to the modes of transportation (conveyance) or types of goods, is chosen.
Originally the STP was stated by Schell [47].Haley [18] showed a comparison between the STP and the classical TP, and applied Modi-method to solve the STP.If more than one objectives are to optimize in an STP, then the problem is called a multi-objective solid transportation problem (MSTP).Uncertainty arises in an STP because of imprecise data and inexact information.Some of the most relevant works related to this are as follows: Jiménez and Verdegay [21] described two forms of uncertain STP in which the considered data are interval numbers and fuzzy numbers, respectively.Ghosh et al. [13] derived a multi-objective solid transportation model of waste management problem in agriculture field and forest department for urban or rural development.Ida et al. [10] chosen multi-criteria STP with fuzzy numbers.Ammar [1] studied on multi-objective solid transportation problems.Ghosh et al. [12] considered a fixed-charge STP in multi-objective environment where all the data are intuitionistic fuzzy numbers with membership and non-membership function.Li et al. [26] presented a genetic algorithm for solving an MSTP with coefficients of the objective function as fuzzy numbers.Midya et al. [36] formulated and solved fuzzy multiple objective fractional optimization in rough approximation and its aptness to the fixed-charge transportation problem.Nagarjan and Jeyaraman [39] studied an MSTP with parameters as stochastic intervals.Jiménez and Verdegay [22] applied an evolutionary algorithm based on parametric approach to solve the fuzzy solid transportation problem.Ghosh et al. [14] solved MSTP with preservation technology using Pythagorean fuzzy sets.Yang and Liu [51] presented the expected value model, chance-constrained programming model and dependent-chance programming for fixed charge STP in a fuzzy environment.Ammar et al. [4] developed a fuzzy solution approach to optimize water resources management problem.Yang and Yuan [50] investigated a bicriteria STP under a stochastic environment.Ghosh et al. [15] solved multi-objective fixed-charge STP under type-2 zigzag uncertain environment and made a comparison between time window and preservation technology.Recently Roy and Midya [42] studied multi-objective fixedcharge solid transportation problem with product blending under intuitionistic fuzzy environment and Midya et al. [35] solved intuitionistic fuzzy multi-stage multi-objective fixed-charge solid transportation problem in a green supply chain.In multi-item STP (MISTP), more than one items/products is transported through the conveyances.In spite of all the developments, there are several gaps in the literature.Previous researchers investigated multi-item two dimensional TP or multi-objective STP for a single item.Few shortcomings in the existing literature studies are pointed out as follows: 1) The fuzzy STP model using expected value operator may not yield provide feasible solutions in all cases.
2) Most investigations have been studied to derive the optimal crisp solution of the fuzzy objective function that does not give a proper idea about objective value according to fuzzy penalties of the objective function.The present investigation removes the above-mentioned lacuna.There are several methods available to deal with uncertain STP in a fuzzy environment, such as chance-constrained programming, expected value operator, dependent chance constrained programming and so on.In an STP the conditions that total supply (resources) and conveyance capacities are greater than or equal to the total demands must have to be satisfied.But these conditions have not been considered in the above-mentioned methods to obtain crisp equivalent form of an STP in fuzzy environment.In this study, we assume a multi-objective multi-item solid transportation problem with fuzzy data in which more than one objective are involved, and also several items are to be transported from sources to destinations.
In this paper, we present various models of MMSTP using the parameters as trapezoidal fuzzy numbers and extend the study of MMSTP by introducing interval and rough interval (RI) in the objective functions.Finally, we analyze the results extracted from different MMSTP models with interval and RI coefficients using presented methods and existing method with respect to the same real-life problem.The main contributions of the proposed study are summarized as follows: -Solve MMSTP with interval coefficients using fuzzy programming and compare the results with existing results mentioned in Kundu et al. [25].-Investigate the solution procedures between the presented method described in Section 4 and the existing method on MMSTP.-Design MMSTP model when the parameters in the objective functions are RIs.
-Expected value operator is introduced to tackle RI in MMSTP.
-A comparison is drawn between the solutions extracted from interval programming and RI programming for MMSTP.-Two real-life examples are incorporated in respect to all models to illustrate the applicability of the proposed model.
The rest of the paper is sorted out in the ordered as: In Section 2, we include the basic definitions and properties of a rough interval.In Section 3, the mathematical model is proposed for multi-objective multi-item solid transportation problem.Solution procedure of the proposed method is presented in Section 4. To demonstrate the utilization of the proposed model of the MMSTP, two numerical illustrations are incorporated into Section 5. Problem formulation for first numerical example using different techniques is presented in Subsection 5.1.In Section 6, we present results and discussion of the proposed method.Finally, this paper ends with conclusions in Section 7.

Preliminaries
In this paper, we assume that all parameters of considered problem are expressed as trapezoidal fuzzy numbers and then we convert them into nearest interval and nearest rough interval numbers.So we need to know about trapezoidal fuzzy numbers and nearest interval and nearest rough interval numbers.
Trapezoidal fuzzy number: A fuzzy number F = (, , , ) where  ≤  ≤  ≤  defined on the universal set of real numbers R is called a trapezoidal fuzzy number if its membership function  F () is defined as: where − − =  1 F () and − − =  2 F () are left and right hand sides of the membership function  F () (see Fig. 1).When  = , the trapezoidal fuzzy number becomes a triangular fuzzy number.Interval number: (Moore et al. [38]) An interval number is a number whose exact value is unknown distribution information, but the range of the value is known.Interval number is a number with both lower and upper bounds  ∈ [, ] where  ≤ .The main arithmetic operations can be expressed in interval numbers.Definition 2.1.(Moore et al. [38]) Let ũ1 = [ 1 ,  1 ] and ũ2 = [ 2 ,  2 ] be closed interval numbers.The following notations can be satisfied: where  ∈ [, ] is an interval number, its absolute value is the maximum of the absolute value of its endpoints: The center   and width   of an interval number of  ∈ [, ] are defined as follows: Rough set [40]: Suppose  ̸ =  is a finite set of objects and we define an equivalence relation  on  that partitioned  into a family of pairwise disjoined subsets  1 ,  2 , . . .,   each of which is an equivalence class of  and called elementary sets.The pair (, ) is called approximation space and it is denoted by ().
In the approximation space () = (, ), given an arbitrary set  ⊆ , one may represent  by a pair of lower approximation (LA) and upper approximation (UA): Where []  signifies the equivalence class containing .The LA and UA of  can equivalently describe as: The pair ((), ()) is called the rough set of .
Rough interval arithmetic [41]: The RI arithmetic operations are fundamentally in view of Moore's interval arithmetic [38].In the accompanying, some of these arithmetic operations of RIs are talked about.The detailed discussion of rough interval arithmetic (RIA) is in [41].Presently, as indicated by Hamzehee et al. [19], ) are two RIs.At that point, the RIA on two RIs is given by the following: 2.1) Addition: The set and logic operations with RIs are basic analyzed to similar operations with fuzzy set.Fuzzy sets must deal with continuous membership functions and cannot utilize Moore's interval calculus.
Order relation of a RI: Suppose  and  are any two RIs, the order relations '≤' and '<' between  and  is characterized as ; and  <  ⇔  ≤  with  ̸ = , respectively.The order relation ≤ or < demonstrates the inclination of Decision Makers (DMs) for the distinctive decisions in view of the upper most extreme midpoint in the ordinary case and in addition to the exceptional case circumstance, to concerning a maximization problem.Expected value maximization and vulnerability minimization are the choice reasoning.It is noticed that the order relations ≤ and < so-characterized are partially ordered relation.The -optimistic value is given by equation (2.1) and the -pessimistic value is shown in equation (2.2) as follows: , otherwise. (2.1) , otherwise. ( Expected value of a RI: Expected value operator is used to reduce rough interval to crisp interval.As our discussion is confined into transportation problem in rough intervals, so we have to define some important concepts on expected value operator. Definition 2.2.Suppose  = { ∈  : () ∈ }, where  :  → R is a real function,  ⊂ R; and  is approximated by (, ) according to the equivalence relation .Then, the lower expected, upper expected and expected value of  are defined as follows: The correlation among the expected value (), the lower expected value (), and the upper expected value () is placed into the following proposition.
Remark 2.5.For  = 0.5 the expected value of  is 1  4 ( 1 +  1 +  1 +  1 ).Nearest interval approximation of fuzzy number: According to Grzegorzewski [17], a fuzzy number is approximated to an equal crisp interval.The -cut of a trapezoidal fuzzy number  = (, , , ) is defined as [  ,   ].Thus, we have   =  + ( − ) and   =  + ( − ).We derive lower and upper bounds using the definition of nearest interval approximation as follows: Here, the nearest interval approximation is used to transform a trapezoidal fuzzy number  = (, , , ) into a crisp closed interval as Then we define . Now we define the nearest rough interval approximation of a trapezoidal fuzzy number  = (, , , ) as

Mathematical model
The mathematical formulation of a classical TP is given as: Here,  stands for the objective function to manipulate total transportation cost;   ( = 1, 2, . . ., ;  = 1, 2, . . ., ) is considered as transportation cost per unit commodity from  ℎ origin to  ℎ destination;   ( = 1, 2, . . ., ) and   ( = 1, 2, . . ., ) are taken as availability and demand in  ℎ origin and  ℎ destination, respectively, and the feasibility condition is as follows: The cost   is usually treated as being deterministic in nature.However, in real-life situations, the precise value of this transportation cost may not be known.To design the mathematical formulation of classical MMSTP, we use the following notations which are stated below: Notations of MMSTP: -   : Amount transported from  ℎ source to  ℎ destination for  ℎ item through  ℎ conveyance, -c  : Fuzzy shipping cost per unit amount for transporting  ℎ item from  ℎ source to  ℎ destination through  ℎ conveyance for  ℎ objective function, -c  : Fuzzy shipping cost per unit amount for transporting  ℎ item from  ℎ source to  ℎ destination through  ℎ conveyance for  ℎ objective function, -ã  : Fuzzy capacity of  ℎ source point for  ℎ item, -b  : Fuzzy demand of  ℎ destination point for  ℎ item, -ẽ  : Total fuzzy capacity of  ℎ item through  ℎ conveyance.A multi-objective multi-item solid transportation problem is formulated as follows: Model 5 is an MMSTP with parameters in RI.We solve Model 5 using expected value operator and the solution procedure is mentioned in Subsection 5.3.

Solution methodology
In order to solve the MMSTP with interval coefficients and with rough coefficients, we use the following techniques: -Fuzzy programming, -Interval programming, -Expected value operator.
Step 2: Then reducing Model 2 into two models Model 3 and Model 4, and due to presence of multi-objective in both the models we use Step 3 to make them single objective.Step From the results of Step 5, determine the corresponding value for every objective function at each solution derived.The equivalent linear programming problem for the minimization problem may then be written as: Here  = min{  () :  = ,  }.This linear programming problem can further be simplified as follows: maximize Now we maximize  subject to the constraints (3.3)-(3.6)using Lingo iterative scheme.

Interval programming to solve Model 2
We apply the following steps for solving Model 2 by using interval programming: Step Step 4: Thereafter we normalize the weights   ,   and for the objective functions   and   , for all  and .Step 6: We maximize   and   under the constraints described in MMSTP i.e., (3.3)-(3.6)using Lingo iterative scheme.

Expected value operator to solve Model 5
To solve Model 5 using expected value operator we go through the following steps: Step 1: Consider an MMSTP (Model 5) with  objective functions   ( = 1, 2, . . .,  ) of maximization type and  objective functions   ( = 1, 2, . . .,  ) of minimization type.Step 4: After that we consider normalize weights   ,   and for the objective functions   and   , for all  and , respectively.Step 6: We maximize   and   under the constraints depicted in MMSTP i.e., (3.3)-(3.6)using Lingo iterative scheme.

Numerical experiments
We present here two examples for showing the effectiveness of the proposed methodologies.The first example is taken from Kundu et al. [25] and the second example is considered due to our preference to explain the applicability and effectiveness of the suggested methodology.
Example 5.1.Considering the multi-objective multi-item solid transportation problem solved by Kundu et al. [25].In this problem, the number of destinations is three, while that of sources, items, conveyances, and objectives is two for each case.Authors proposed a method to find the crisp optimal compromise solution.Here we describe a new approach to find a crisp and interval solution to the same problem, we solve the same problem using the method proposed here in different approaches to extract the optimal compromise crisp solution and interval solution.The data of the problem is described in Tables 1 to 5.

Problem formulation
To make good sense of the proposed algorithms by fuzzy programming, interval programming and expected value operator here we design the problem formulation of the numerical example and solve them using LINGO iterative scheme.

Problem formulation by fuzzy programming
Using the steps from Step 1 to Step 5 mentioned in Section 4.1 we derive the following mathematical problem of the considered numerical example.

Step 3 :
Let    ( = 1, 2, . . .,  ) ( = ,  ) be an objective function which is of maximization type.Find the maximum value     ′  ′  ′ among all the cost parameters, then divide each of the cost parameters is by     ′  ′  ′ to obtain       ′  ′  ′ =     (say), where 0 <     ≤ 1.In the classical procedure, for solving TP of maximization type, the allocations are made in the cells according to values of cost parameters in descending order.This suggests that the allocation probably made at the values of     where it is maximum.Then the cost parameters in each maximization type objective function is reduced in the same scale.Again, considering    ( = 1, 2, . . .,  ) is objective function of minimization type.Find the maximum value     ′  ′  ′ among all cost parameters, then each of the cost parameters is divided by     ′  ′  ′ to obtain       ′  ′  ′ =     (say), where 0 >     ≥ −1.Since the objective function is minimization type so the allocations are made at the nodes where values of     are minimum.Then each of the cost parameters in the objective functions of minimization type is reduced in the same scale.Step 4: Thereafter considering normalize weights   ,   and for the objective functions   and   , individually for all  and .Step 5: Formulating the single objective function as follows: maximize   = ∑︁ constraints (3.3)-(3.6).

1 :
Consider an MMSTP (Model 2) with  objective functions   ( = 1, 2, . . .,  ) of maximization type and  objective functions   ( = 1, 2, . . .,  ) of minimization type.Step 2: Then reducing Model 2 into two models Model 3 and Model 4 ; and due to presence of multi-objective in both the models we use Step 3 to make them single objective.Step 3: Let    ( = 1, 2, . . .,  ) ( = ,  ) be an objective function which is of maximization type.Find the maximum value     ′  ′  ′ among all the cost parameters, then divide each of the cost parameters is by     ′  ′  ′ to obtain       ′  ′  ′ =     (say), where 0 <     ≤ 1.In the classical procedure, for solving TP of maximization type, the allocations are made in the cells according to values of cost parameters in descending order.This suggests that the allocation probably made at the values of     where it is maximum.Then the cost parameters in each maximization type objective function is reduced in the same scale.Again, considering    ( = 1, 2, . . .,  ) is objective function of minimization type.Find the maximum value     ′  ′  ′ among all cost parameters, then divide each of the cost parameters is by     ′  ′  ′ to obtain       ′  ′  ′ =     (say), where 0 >     ≥ −1.Since the objective function is minimization type so the allocations are made at the nodes where values of     are minimum.Then each of the cost parameters in the objective functions of minimization type is reduced in the same scale.

Step 5 :
Formulating the single objective function as follows: maximize   = ∑︁

Step 2 :
Convert the cost parameters of both the objective functions into crisp values by using expected value operator i.e., ([    ,     ], [    ,     ]) =    (say) and ([    ,     ], [    ,     ]) =    (say).Step 3: Let   ( = 1, 2, . . .,  ) be an objective function which is maximization type.Find maximum value    ′  ′  ′ among all cost parameters, then divide each of the cost parameters by    ′  ′  ′ to obtain       ′  ′  ′ =    (say), where 0 <    ≤ 1.In the classical procedure, for solving TP of maximization type, the allocations are made in the cells according to values of cost parameters in descending order.This suggests that the allocation probably made at the values of    where it is maximum.Then the cost parameters in each maximization type objective function are reduced in the same scale.Again, consider   ( = 1, 2, . . .,  ) is objective function of minimization type.Find maximum value    ′  ′  ′ among all cost parameters, then divide each of the cost parameters by    ′  ′  ′ to obtain       ′  ′  ′ =    (say), where 0 >    ≥ −1.Since the objective function is minimization type so the allocations are made at the nodes where values of    are minimum.Then each of the cost parameters in the objective functions of minimization type is reduced in the same scale.
6: Solve the bi-objective MMSTP as single objective STP using each time only one objective function and ignoring other.Step 7: We find lower bound   and upper bound   for  ℎ objective function   ( = ,  ), where   is the aspired levels of achievement for  ℎ objective function,   is the highest acceptable level of achievement for  ℎ objective function and   = [  −   ] is degradation allowance for  ℎ objective function.When the aspiration levels and degradation allowance for each objective function are specified, a fuzzy model is formed and then it converts into a crisp model.Step 8: From the results of Step 6, determine the corresponding values for every objective function at each solution derived.Step 9: From Step 8, we find the best   and the worst   values for each objective function corresponding to the set of solutions.The initial fuzzy model can then be stated, in terms of the aspiration levels of each objective function, as follows: Find    , so as to satisfy   ≤   :  = ,  with given constraints (3.3)-(3.6).For the bi-objective FCTP, a membership function   () corresponding to  ℎ objective function is defined as:

Table 1 .
Unit transportation penalties for item 1 in the first objective.

Table 2 .
Unit transportation penalties for item 2 in the first objective.

Table 3 .
Unit transportation penalties for item 1 in the second objective.

Table 4 .
Unit transportation penalties for item 2 in the second objective.
Solving Model 6 as a single objective STP using each time only one objective function and ignoring other, we get  1 = 113.102, 1 = 119.015, 2 = 127.35, 2 = 132.374andthenconstruct the membership function.Finally the mathematical model is designed as follows:maximize  subject to  1 + ( 1 −  1 ) ≤  1  2 + ( 2 −  2 ) ≤  2   ≥ 0 ∀ , .Now using solution procedure presented in Subsection 4.2 we derive Model 8 and Model 9 and then solve by LINGO iterative scheme.Example 5.2.A reputed mobile company has two production factories in two places namely  1 and  2 in India and distributed mobiles throughout the country.The mobiles are supplied into three markets namely  1 ,  2 , and  3 .The mobiles are transported by two types of conveyances ( = 2), such as large trucks and freight trains.DM desires to minimize the production cost, and transportation cost by different transportation ≥ 0 ∀ , .5.1.2.Problem formulation by interval programmingHere we derive MMSTP model from the numerical example when the parameters are interval numbers.