A Profit Maximisation Solid Transportation Problem Using Genetic Algorithm in Fuzzy Environment

Figures & data

Table 1. Comparison table among the existing models with proposed model.

References(s)	Unit Trans. Cost	Fixed Charge	Vehicle Cost	Profit	Solving Method
Bit et al. [13]	Fuzzy	No	No	No	Fuzzy programming
Sakawa et al. [21]	Fuzzy	No	No	Yes	Fuzzy programming
Liu et al. [29]	Fuzzy	Yes, Crisp	No	No	GA
Ojha et al.	Crisp AUD	Yes, Crisp	Yes	No	GA
Li et al. [12]	Fuzzy	No	No	No	GA
This model	Fuzzy AUD	Yes, Fuzzy	Yes with two different capacities	Yes	GA

Figure 1. Membership function of General fuzzy number $\tilde{A}$_.

Figure 2. Membership function of TFN $\tilde{A}$_.

Figure 3. Membership function of TrFN $\tilde{A}$

Figure 4. $\alpha$- cut of a fuzzy number $\tilde{A}$_.

Table 2. Input Values of ${\tilde{\mathit{C}}}_{\mathit{i}\mathit{j}\mathit{k}}$ (in $\mathrm{\$}$) under AUD.

C̃ijk	c̃ijk	AUD	C̃ijk	c̃ijk	AUD
C̃111	(4, 5, 6)	0 < xijk < 10	C̃111	(7, 8, 9)	0 < xijk < 10
	(2, 3, 4)(1, 2, 3)	10 ≤ xijk < 20xijk ≥ 20		(5, 6, 7)(3, 4, 5)	10 ≤ xijk < 20xijk ≥ 20
C̃121	(5, 6, 7)	0 < xijk < 10	C̃121	(6, 7, 8)	0 < xijk < 10
	(3, 4, 5)(1, 2, 3)	10 ≤ xijk < 20xijk ≥ 20		(4, 5, 6)(2, 3, 4)	10 ≤ xijk < 20xijk ≥ 20
C̃112	(7, 8, 9)	0 < xijk < 10	C̃112	(8, 9, 40)	0 < xijk < 10
	(5, 6, 7)(3, 4, 5)	10 ≤ xijk < 20xijk ≥ 20		(6, 7, 8)(4, 5, 6)	10 ≤ xijk < 20xijk ≥ 20
C̃122	(4, 5, 6)	0 < xijk < 10	C̃122	(6, 7, 8)	0 < xijk < 10
	(2, 3, 4)(1, 2, 3)	10 ≤ xijk < 20xijk ≥ 20		(4, 5, 6)(2, 3, 4)	10 ≤ xijk < 20xijk ≥ 20

Figure 5. Direct Transportation Cost Vs $\alpha$-Cut.

Table 3. Optimum Result of Case-1.

Value of α	Transportable Amounts x111 x121 x112 x122 x211 x221 x212 x222	PLα	PUα	Pc
1.0	8.3, 12.5, 27.5, 1.5, 5.1, 0.3, 12.0, 10.7	535.6	535.6	535.6
0.9	7.2, 12.2, 27.2, 2.3, 5.0, 0.2, 13.0, 10.1	529.2	542.3	535.7
0.8	7.7, 10.2, 27.3, 2.9, 3.9, 0.1, 13.5, 10.9	520.7	546.8	533.8
0.7	7.8, 14.5, 26.2, 0.0, 5.0, 10.1, 13.7, 0.0	529.1	570.3	549.7
0.6	10.5, 14.7, 24.0, 0.4, 6.6, 11.1, 11.0, 0.0	543.2	597.8	570.5
0.5	10.4, 13.0, 25.5, 0.0, 5.6, 0.4, 12.2, 11.8	548.0	616.8	582.4
0.4	11.2, 11.5, 25.6, 0.3, 4.5, 0.3, 12.0, 12.3	535.5	614.9	575.2
0.3	11.8, 11.6, 23.4, 2.0, 4.4, 0.0, 13.9, 10.9	533.5	628.5	581.0
0.2	11.8, 11.1, 26.7, 0.4, 4.3, 14.3, 11.2, 0.0	554.2	666.1	610.1
0.1	0.0, 12.2, 26.5, 10.4, 15.7, 2.6, 11.6, 0.0	535.4	661.5	598.4
0.0	3.6, 11.9, 23.2, 10.3, 17.2, 2.2, 10.6, 0.0	526.9	665.1	596.0

Table 4. Optimum Result of Case-2.

Value of α	Transportable Amounts x111x121x112x122x211x221x212x222	PLα	PUα	Pc
1.0	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	521.1	664.1	592.6
0.9	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	525.2	653.9	589.6
0.8	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	529.3	643.7	586.5
0.7	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	533.4	63.5	583.5
0.6	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	537.5	623.3	580.4
0.5	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	541.6	613.1	577.4
0.4	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	545.7	602.9	574.3
0.3	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	549.8	592.7	571.3
0.2	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	553.9	582.5	568.2
0.1	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	558	572.3	565.1
0.0	0.0, 17.5, 21.2, 11.3, 0.0, 7.5, 8.8, 13.7	562.1	562.1	562.1

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