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Fractional Multi-objective Capacitated Transportation Problem with Different Membership Functions
1Sheema Sadia, 2Qazi Mazhar Ali, 3Zainab Asim, 4*Ahteshamul Haq
24Department of Statistics & Operations Research Aligarh Muslim University, Aligarh-202002 3Faculty of Commerce & Management, SGT University, Gurgaon, Haryana [email protected], [email protected], [email protected], 4*[email protected] ^Corresponding author
Abstract
This Fractional Transportation Problem arises when an enterprise has to face the issue of maintaining a good ratio of some critical parameters. These parameters are directly concerned with product(s) transportation from sources to destination. This paper considers a multi-objective Capacitated Transportation Problem with Fractional Objectives. A fuzzy goal programming approach with different membership functions is applied to generate a different set of solutions. We also use Chebyshev's Goal Programming for obtaining the solutions. Finally, a numerical illustration is provided to validate our proposed model.
Keywords: Multi-objective programming, Quadratic membership function, Mixed constraints, Fuzzy normal membership function, Fuzzy Cauchy membership function, Fuzzy programming
1. Introduction
A transportation problem (TP) occurs when a product (or products) must be transported from multiple sources (also known as origin, supply, or capacity centres) to multiple sinks (also called destination, demand or requirement centres). The fundamental TP was devised by Alfred Hitchcock [9]. The TP with fractional objective function is known as a fractional transportation problem (FTP). Swarup [14] was the first to propose it. It is crucial in logistics, supply chain management, stock cutting problems, resource allocation problems, ship and plane routing problems, cargo loading problems, and inventory problems. The FTP arises when an enterprise faces the challenge of maintaining a good ratio between critical parameters. These parameters are directly concerned with transporting a product or products from their origin to their destination. Fractional programming can optimize actual/standard transportation costs or total return/total investment on machines delivered from factories to workshops. In linear fractional TPs with mixed constraints, Gupta et al. [5] presented a paradox. Gupta and Arora [6-7] and Liu [10] are two other authors written about FTPs. In general, real-world TPs are modelled with multiple, conflicting objectives.
Furthermore, combining all objective functions into a single overall utility function is difficult for the decision-maker. So it is better to formulate a multi-objective TP. The capacitated TP are the TPs with bounds on general availabilities at assets and general vacation spot requirements. It may benefit telecommunication networks, production-distribution systems, rail and concrete street systems.
The capacitated TPs have also been discussed by authors like Arora and Gupta [2] and Gupta and Bari [8]. Zadeh [15] first delivered the idea of a fuzzy set concept. Then Zimmermann [16] carried out the fuzzy set concept with a few suitable membership functions to resolve linear programming problems with numerous goal functions. Bit et al. [3] implemented a fuzzy programming approach with a linear membership function to resolve the multi-objective TP. El-Wahed [4] gave the idea of a fuzzy programming approach to determine the optimal compromise solution of a MOTP with a fuzzy membership function. Akkapeddi [1] discussed the quadratic membership for the multi-objective TP. Singh [13] worked on multiple objective fractional costs TP with bottleneck time and impurities. Sadia et al. [12] presented a fuzzy approach to obtain the solution of multi-objective capacitated FTP. Fuzzy normal and fuzzy Cauchy membership functions were used by Mon and Cheng [11]. Gupta et al. [17] discussed two stage transportation problem with the different types of fuzzy environments. Kamal et al. [18] described the parameters estimation and goodness of fit for the multi-objective transportation problem under type-2 trapezoidal fuzzy numbers. The purpose of using FTP is to make the problem more realistic. It can prove more beneficial for the decision-maker to consider the proportion of transporting cost due to the covered path and favoured path because the transportation cost may vary due to the travelled and favoured path. Likewise, the proportion of exact and standard transportation time and transporting damage cost due to covered path and favoured path are also measured.
In this paper, we have taken mixed constraints of MOCFTP with fractional type objectives. As it is a multi-objective problem and the objectives are conflicting in nature. So a compromise solution is obtained by using the fuzzy programming approach. We have tried to use three membership functions: quadratic, fuzzy normal, and fuzzy Cauchy. As far as our knowledge, these membership functions have never been used to deal with TPs. The rest of the paper is organized as follows: Assumptions, notations and formulation are discussed in Section 2. In Section 3, we have discussed the algorithm using a fuzzy optimization approach with different membership functions and Chebyshev's Goal Programming. In section 4, an example of the proposed method is illustrated. The conclusion is presented in section 5.
2. Assumptions, notations and formulation of the problem We consider mixed constraints MOCFTP under the following notations
2.1 Notations
m Number of origins
n Number of destinations
a-i Units of supply (i = 1,2, ...,m)
bj Units of demands j = 1,2, ...,n
cij Unit transporting cost due to travelled endpoint.
Tij Unit transporting cost due to preferred endpoint
tfj Actual transportation time from the ith starting point to jth endpoint
tij Standard transportation time from the ith starting point to jth endpoint
dij Damage transporting cost due to travelled origin from the ith path to jth endpoint
Xij Units transported from the ith starting point to jth endpoint
lij Minimum quantity transported from the ith starting point to jth endpoint
Sij Maximum transported quantity from the ith starting point to jth endpoint
route from the ith starting point to jth route from the ith starting point to jth
2.2 Problem's statement
Consider a TP of fractional type objective function with m numbers of starting points having aj(i = 1,2,..., m) units of supply to be transported among n numbers of endpoints with ¿y(y = 1,2,..., n) units of demand. The problem is determining the best transportation schedule for transporting the available quantity of products to meet demand while minimizing total transportation costs, time, and damage charges.
Let Xjy be the number of units transported from ith starting point to the jth endpoint. The mathematical model of the MOCFTP with mixed constraints can be expressed as:
ym yn r..r.. ..... f yi=lyy=lcty^i7
Minimize/! =
yi=iyy=i'iyAiy
{tfAxi} > 0)
Minimize/2 = ^axi^-}
I ^¿yniy > 0J ym y« =l =l
Minimize/3 = y m y; r r
yi=1yy=l' ¿y^iy m n
subject to: ^ Xjy < aj; ^ Xjy > fy
¿=1 y=i
Z< x-■<<?■■■ x- > 0
2.3 Interpretation of objectives function
1. The proportion of unit transporting cost Cy and ry due to travelled path and a preferred route respectively.
2. The proportion of the actual transportation time ty and a standard transportation time ty.
3. The proportion of unit transporting damage cost dy (loss of quantity and quality transportation) and ry due to the travelled and a preferred path, respectively.
3. Solution Approach for MOCFTP
3.1 Fuzzy Optimization Approach-Algorithm:
In order to solve the multiobjective fractional capacitated TP with mixed constraints, we use the following algorithm
Step 1:- Firstly, we will formulate the payoff matrix as:-Payoff Matrix
r(2) „CO
/i(xj)) /1(xJ)) /1(xJ))"
/i^) /i(^J)) /i^)
LTi(xJ)) /i(x«) /i(xj)).
/1 /2 /3
( fc)
where, x( ); k = 1,2,..., K are the kfh individual optimal solutions that optimize the kth objective.
Sheema Sadia, Qazi Mazhar Ali, Zainab Asim, Ahteshamul Haq
FRACTIONAL MULTI-OBJECTIVE CAPACITATED TRANSPORTATION RT&A, No 2 (68)
PROBLEM WITH DIFFERENT MEMBERSHIP FUNCTIONS_Volume 17, June 2022
Step 2:- We will apply the fuzzy approach with the following membership functions defined below:
A. Quadratic membership function: To derive the compromise solution of MOCFTP, we used fuzzy programming. The membership functions for the cost objective are: fki and fku be the achieved aspired level and the highest acceptance level of the kth objective function, respectively. The membership function of the kth objective function is represented as follows:
Vk(fk) = Rkifk + Rkifk + qs The membership values of the kth objective function at the aspiration level and the highest acceptable level is 1 and 0, respectively. We used the equations as: ^k(fki) = qkifki + qkifki + qs = i
v-ktfku) = qkJku + qk2fkU + qs = 0 The above linear system of equations qk2 and qk3 are expressed in terms of qk1. Thus, the quadratic membership function for the kth objective function is used in the following form:
^k(Fk) = ikU ik + qk1fk - qk1(fkl + fku)fk + qk1flkflu Jku Jkl
B. Fuzzy Normal: The membership function for Fuzzy Normal will take the following form:
1 i ffk < fkl M-k(tI7t)2}iffki<fk<fku
0 i ffk > fku^ndk > 0
C. Fuzzy Cauchy: The membership function for Fuzzy Cauchy will take the following form:
1 i ffk < fkl
=
i
J tffkl < fk < fku
0 i ffk > fku
a > Oand/ffispositiveeven
Step 3: The MOCFTP with mixed constraints can now be converted into equivalent non-linear models for the above-defined membership functions as follow:
A. Quadratic Membership function: The proposed model for MOCFTP with mixed constraints on applying quadratic membership function will be of the following form:
Minimized
Subjectto f - f
iu 1 + qiif? - qiiifii + fiu)fi + qiifiufii < * + q2if2 - q2 i(Í2l + f2u)f2 + q2if2uf2l < * + qsifi - qs i(fsi + fsu)fs + qsifsufsi < *
fiu -fu
f2 u -f2
f2 u - f2
fs u -fk
fs u - fs
m n
^ xij[</=/>}ai; ^ xij{</=/>}bj i=i j=i 0 < x^ < Sif; x^ > 0;A> 0
B. Fuzzy Normal Membership Function: The proposed model for MOCFTP with mixed constraints on applying fuzzy normal membership function will be of the form:
Minimize X
subject to:
exp
k f /1 - fu N2 < ^ _k ifi^fnV2
Vlu - f1i' ' -
exp|-fc(ff3-f3;)
< X, exp <X
f2u /2f)
< X
^ xy < aj; ^ Xy > fy; 0 < xy < Sy; xy > 0; X > 0
i=1 7 = 1
We will solve it for k=1
C. Fuzzy Cauchy Membership Function: The proposed model for MOCFTP with mixed constraints on applying fuzzy Cauchy membership function will be of the form:
Minimize
Subjectto-ä < X
1 + g(fi-fu)^
Vf1u - f1i/
1
<X
1 + a (f2 f2')^
,f2u f2i/ 1
<X
1 + a (f3 f3'V
v3u /y
Z|=1 Xy < aj; 27=1 XU > 6,- ;0 < xu < Sj - xy > 0; X > 0
D. Chebyshev's Goal Programming: Chebyshev's Goal Programming is considered a particular case of the weighted Goal Programming technique. It seeks a solution that minimizes the worst deviation from each objective. The mixed constraints of the MOCFTP using Chebyshev's Goal Programming will be represented as:
MaximizeX
Subjecttof1 + X < f1u
f2+X<f2u
f3 + X < f3 u
n
^ Xy{</=/>)aj; ^ xy{</=/>}6,-
v¿7 !</=/> (G, ¿=1 7=1
0<xy <Sy;Xy>0;X>0
where, the worst deviation level (X) and aspiration levels for the upper bound is /¿u(i = 1,2,3).
m
4. Numerical Illustration
A case study is discussed to demonstrate and utility of the approaches. The numerical problem of simulated data (Sadia et al. [12]) is presented. The discussed models are defined to solve the problem. We consider three starting points and three endpoints. The fractional transportation cost, time and damage charges (both quantity and quality damage) are represented in Table [1-3].
Table 1: Cost charges matrix
bi b2 b3 Supply
ai 5/3 7/4 15/13 < 12
a2 8/12 17/14 12/7 = 15
a3 19/15 10/6 13/8 > 20
Demand > 9 = 13 < 21
Table 2: Time charges matrix
bi b2 b3 Supply
ai 17/9 5/2 10/3 < 12
a2 1/2 11/4 6/5 = 15
a3 13/8 16/12 10/11 > 20
Demand > 9 = 13 < 21
Table 3: Damage charges matrix
b1 b2 b3 Supply
a1 13/8 15/9 8/11 < 12
a2 11/15 14/6 19/7 = 15
a3 9/7 15/6 8/17 > 20
Demand > 9 = 13 < 21
The mixed constraints of the MOCFTP will be as follows:
_ SX11 + 7X12 + 1SX13 + 8X21 + 17X22 + 12X23 + 19X31 + 10X32 + 13X33 3x11+4x12 + 13x13 + 12x2l + 14x22 + 7x23 + 15x31 + 6x32 + 8x33 13X11 + 15X12 + 8X13 + 15x2! + 14X22 + 19X23 + 9X31 + 15X32 + 8X33
Minf2 =-
8X11 + 9Xi2 + 11Xi3 + 15x21 + 6X22 + 7X23 + 7X31 + 6X32 + 17X33 17X11 + 5X12 + 1OX13 + X21 + 11X22 + 6X23 + 13X31 + 16X32 + 1OX33
Minf =-
9X11 + 2X12 + 3X13 + 2x21 + 4X22 + 5X23 + 8X31 + 12X32 + 11X33
3 3 3
Subjectto ^ x1j < x2j < 15 ; ^ x3j <20
j=1 j=1 j=1 3 3 3
^xi1<9;^xi2<13;^xi3 <210<x 11 < 6,0 < x12 < 7,0 < x13 < 13,0 < x21 < 6, j=1 j=1 j=1 0 <x22< 2,0 < x23 < 13,0 < x31 < 4,0 < x32 < 7,0 < x33 < 14.
A. Different membership functions for fuzzy programming approach
The payoff matrix for [ lij = 0] is obtained after solving the problem as a single objective (ignoring the other objectives) using the LINGO optimization software will be as follows:
Payoff Matrix = A [2 f3
x(i)
l' 1.316832 1.16129 1.34472 ' x(2) 1.37988 1.068410 1.79661 (3) 11.406433 1.170886 1.168285i
xl
Sheema Sadia, Qazi Mazhar Ali, Zainab Asim, Ahteshamul Haq
FRACTIONAL MULTI-OBJECTIVE CAPACITATED TRANSPORTATION RT&A, No 2 (68)
PROBLEM WITH DIFFERENT MEMBERSHIP FUNCTIONS_Volume 17, June 2022
/1u = 1.406433, fi; = 1.316832,/2u = 1.170886, /2 = 1.068410,f3u = 1.79661andf3i = 1.168285
Individual optimum solutions are obtained by solving the above problem separately for each objective using the optimizing software LINGO in Table 4.
Table 4: Individual optimum solution
Objectives Objective Values Cost Damage Time
1.316832 1.068410 1.168285
X11 0 0 0
X12 4 7 6
« X13 5 0 0
c 0 X21 2 6 6
PS u X22 6 2 0
X23 7 7 9
< X31 4 3 3
X32 7 4 7
X33 9 14 12
The compromise solution obtained for Quadratic Membership Function is as follows: x^ =
0, X12 = 7, X13 = 2,X21 = 6, X22 = 2,X23 = 7, X31 = 4, X32 = 4, X33 = 12
The optimal compromise solution obtained using the Fuzzy normal Membership Function will be as follows: x^ = 0,xl2 = 4,xl3 = 1,x2*i = 5,x2*2 = 2,x2*3 = 8,X3*i = 4^2 = 7,X3% = 9
The crisp problem for fuzzy Cauchy has been obtained after setting a = 0.5and/ff = 2. The compromise solution obtained for Fuzzy Cauchy Membership Function is as follows: x^ =
4, X12 = 4, X13 = 4, X21 = 5,X22 = 2,X23 = 8, X31 = 4, X32 = 7, X33 = 9
The compromise solution obtained for Chebyshev's Goal Programming is as follows: x^ =
0, X12 = 7, X13 = 2,X21 = 6, X22 = 2,X23 = 7, X31 = 4, X32 = 4, X33 = 12
5. Conclusion
This article represents the optimal compromise solution with mixed constraints for a multiobjective fractional capacitated TP. Fuzzy programming with three different membership functions viz. quadratic, fuzzy normal and fuzzy Cauchy is used to obtain a compromise solution using a fuzzy programming approach, and Chebyshev's Goal Programming is also discussed to solve the problem multiobjective fractional capacitated TP. Finally, a comparative study is done with the results obtained in the paper and the results from Sadia et al. [12]. The results are summarized in Table 5.
This paper proposes fuzzy programming models by applying different membership functions to solve multiobjective fractional capacitated TP. Table 5 also compares the results obtained through different procedures to obtain and compare their efficiency. The methods used in the paper can also be applied for transportation, assignment and transhipment problems.
Table 5: Compromise optimum solution
Approach Membership/ Objective Values
Methods Cost Damage Charges Time
<N Linear 1.359296 1.238494 1.389058
i-H *« H-a 03 Fuzzy Programming Exponential Hyperbolic 1.349030 1.359296 1.229213 1.238494 1.314935 1.389058
-ö OS cn Goal Programming Lexicographic, Distance 1.353103 1.129854 1.237344
Quadratic 1.349869 1.097561 1.257840
-ö Fuzzy Programming Fuzzy Normal Fuzzy Cauchy 1.371758 1.359296 1.249359 1.238494 1.264085 1.389058
« 0J P s Goal Programming Chebyshev's 1.349869 1.097561 1.257840
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Sheema Sadia, Qazi Mazhar Ali, Zainab Asim, Ahteshamul Haq
FRACTIONAL MULTI-OBJECTIVE CAPACITATED TRANSPORTATION RT&A, No 2 (68) PROBLEM WITH DIFFERENT MEMBERSHIP FUNCTIONS_Volume 17, June 2022
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