A New Approach to Solve Fully Fuzzy Multi-Objective Transportation Problem

Figures & data

Table 1. Fuzzy parameters of fully fuzzy multi-objective transportation problem.

	D1	D2	D3	D4	Supply
S1	(1,1.5,2)	(1,2,3)	(5,7,9)	(4,6,8)	(7,8,9)
	(3,4,5)	(2,4,6)	(2,3,4)	(1,3,5)
S2	(1,1.5,2)	(7,8.5,10)	(2,4,6)	(3,4,5)	(17,19,21)
	(4,5,6)	(7,8,9)	(7,8.5,10)	(9,10,11)
S3	(7,8,9)	(7,9,11)	(3,4,5)	(5,6,7)	(16,17,18)
	(4,6,8)	(1,2,3)	(3,4.5,6)	(1,1.5,2)
Demand	(10,11,12)	(2,3,4)	(13,14,15)	(15,16,17)

Table 2. Calculation of the values of parameters of fully fuzzy multi-objective transportation problem.

Parameter	Value
$c({\tilde{\mathrm{z}}}_1)$	$\frac{1}{2}\left(\begin{array}{l}{x}_{11}^1+3x_{11}^2+2x_{11}^3+x_{12}^1+4x_{12}^2+3x_{12}^3+5x_{13}^1+14x_{13}^2+9x_{13}^3+4x_{14}^1+12x_{14}^2+8x_{14}^3\\ +x_{21}^1+3x_{21}^2+2x_{21}^3+7x_{22}^1+17x_{22}^2+10x_{22}^3+2x_{23}^1+8x_{23}^2+6x_{23}^3+3x_{24}^1+8x_{24}^2+5x_{24}^3\\ 7x_{31}^1+16x_{31}^2+9x_{31}^3+7x_{32}^1+18x_{32}^2+11x_{32}^3+3x_{33}^1+8x_{33}^2+5x_{33}^3+5x_{34}^1+12x_{34}^2+7x_{34}^3\end{array}\right)$
$w({\tilde{\mathrm{z}}}_1)$	$\frac{1}{2}\left(\begin{array}{l}2x_{11}^3-x_{11}^1+3x_{12}^3-x_{12}^1+9x_{13}^3-5x_{13}^1+8x_{14}^3-4x_{14}^1+2x_{21}^3-x_{21}^1+10x_{22}^3-7x_{22}^1+6x_{23}^3-2x_{23}^1+5x_{24}^3-3x_{24}^1+9x_{31}^3-7x_{31}^1+11x_{32}^3-7x_{32}^1+5x_{33}^3-3x_{33}^1+7x_{34}^3-5x_{34}^1\end{array}\right)$
$c({\tilde{\mathrm{z}}}_2)$	$\frac{1}{2}\left(\begin{array}{l}3x_{11}^1+8x_{11}^2+5x_{11}^3+2x_{12}^1+8x_{12}^2+6x_{12}^3+2x_{13}^1+6x_{13}^2+4x_{13}^3+1x_{14}^1+6x_{14}^2+5x_{14}^3+4x_{21}^1+10x_{21}^2+6x_{21}^3+7x_{22}^1+16x_{22}^2+9x_{22}^3+7x_{23}^1+17x_{23}^2+10x_{23}^3+9x_{24}^1+20x_{24}^2+11x_{24}^{3}4x_{31}^1+12x_{31}^2+8x_{31}^3+1x_{32}^1+4x_{32}^2+3x_{32}^3+3x_{33}^1+9x_{33}^2+6x_{33}^3+1x_{34}^1+3x_{34}^2+2x_{34}^3\end{array}\right)$
$w({\tilde{\mathrm{z}}}_2)$	$\frac{1}{2}\left(\begin{array}{l}5x_{11}^3-3x_{11}^1+6x_{12}^3-2x_{12}^1+4x_{13}^3-2x_{13}^1+5x_{14}^3-x_{14}^1+6x_{21}^3-4x_{21}^1+9x_{22}^3-7x_{22}^1+10x_{23}^3-7x_{23}^1+11x_{24}^3-9x_{24}^1+8x_{31}^3-4x_{31}^1+3x_{32}^3-x_{32}^1+6x_{33}^3-3x_{33}^1+2x_{34}^3-x_{34}^1\end{array}\right)$
$c({\tilde{S}}_i)$	$\frac{1}{2}(x_{i1}^1+2x_{i1}^2+x_{i1}^3+x_{i2}^1+2x_{i2}^2+x_{i2}^3+x_{i3}^1+2x_{i3}^2+x_{i3}^3+x_{i4}^1+2x_{i4}^2+x_{i4}^3+l_i^1+2l_i^2+l_i^3)$
$w({\tilde{S}}_i)$	$\frac{1}{2}(x_{i1}^3-x_{i1}^1+x_{i2}^3-x_{i2}^1+x_{i3}^3-x_{i3}^1+x_{i4}^3-x_{i4}^1+l_i^3-l_i^1)$
$c({\tilde{D}}_j)$	$\frac{1}{2}(x_{1j}^1+2x_{1j}^2+x_{1j}^3+x_{2j}^1+2x_{2j}^2+x_{2j}^3+x_{3j}^1+2x_{3j}^2+x_{3j}^3-(e_j^1+2e_j^2+e_j^3))$
$w({\tilde{D}}_j)$	$\frac{1}{2}(x_{1j}^3-x_{1j}^1+x_{2j}^3-x_{2j}^1+x_{3j}^3-x_{3j}^1-(e_j^1-e_j^3))$

Table 3. The optimal values of fuzzy variables.

Variable	Value
${\tilde{x}}_{11}$	$(0,0,1)$
${\tilde{x}}_{12}$	$(1.25,1.25,1.25)$
${\tilde{x}}_{13}$	$(0,0,0)$
${\tilde{x}}_{14}$	$(2.5,2.5,2.5)$
${\tilde{x}}_{21}$	$(5.25,5.25,5.25)$
${\tilde{x}}_{22}$	$(0,0,1)$
${\tilde{x}}_{23}$	$(3.75,3.75,4.75)$
${\tilde{x}}_{24}$	$(0,0,0)$
${\tilde{x}}_{31}$	$(0,0,0)$
${\tilde{x}}_{32}$	$(0,0,0)$
${\tilde{x}}_{33}$	$(3,3,3)$
${\tilde{x}}_{34}$	$(5.25,5.25,6.25)$

Table 4. The optimal values of fuzzy objective functions.

Objective function	Value
${\tilde{\mathrm{z}}}_1$	$(59.25,83.875,133.5)$
${\tilde{\mathrm{z}}}_2$	$(66.5,92,143.5)$

Figure 1. The set of Pareto optimal solutions.

Table 5. The optimal values of fuzzy variables of the fuzzy transportation problem instance in [20].

	1	2	3	4	5	6	7	8
1	$(0,0,0)$	$(0,0,0)$	$(34,36,36)$	$(37,43,43)$	$(12.1,17.1,17.1)$	$(0,0,0)$	$(27,29.5,29.5)$	$(0,0,0)$
2	$(27.13,30.63,30.63)$	$(19,21,21)$	$(0,0,0)$	$(0,0,0)$	$(39.1,39.1,39.1)$	$(0,3.5,3.5)$	$(0,0,0)$	$(0,0,0)$
3	$(0,0,0)$	$(2.9,7.4,7.4)$	$(0,0,0)$	$(0,0,0)$	$(0,0,0)$	$(22.63,22.63,22.63)$	$(0,3.5,3.5)$	$(37.5,41.5,41.5)$

Table 6. The optimal values of fuzzy objective functions of the fuzzy transportation problem instance in [20].

Objective function	Value
${\tilde{\mathrm{z}}}_1$	$(1877.32,2377.9,2655.28)$
${\tilde{\mathrm{z}}}_2$	$(543.16,798.1,987.07)$
${\tilde{\mathrm{z}}}_3$	$(16402.4,21449.12,23998.56)$

Bagheri M, Ebrahimnejad A, Razavyan S, et al. Solving the fully fuzzy multi-objective transportation problem based on the common set of weights in DEA. J Intell Fuzzy Sys. 2020:39(3): 3099–3124.

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