A Fuzzy Goal Programming Approach to Multiobjective Transportation Problems

Figures & data

Figure 1. Fuzzy set $z_k\tilde{\le }{b}_k$.

Table 1. Optimal objective values obtained from model (3).

	Weights	$z_1,z_2$	$p_1,p_2$
1	$w_1$=0.1, $w_2$=0.9	186,79	43,0
2	$w_1$=0.2, $w_2$=0.8	176,83	33,4
3	$w_1$=0.3, $w_2$=0.7	156,98	13,19
4	$w_1$=0.4, $w_2$=0.6	156,98	13,19
5	$w_1$=0.5, $w_2$=0.5	156,98	13,19
6	$w_1$=0.6, $w_2$=0.4	156,98	13,19
7	$w_1$=0.7, $w_2$=0.3	156,98	13,19
8	$w_1$=0.8, $w_2$=0.2	143,163	0,84
9	$w_1$=0.9, $w_2$=0.1	143,163	0,84

Table 2. Optimal objective values obtained from model (4).

	Weights	$z_1,z_2$	$r(1-w_1),r(1-w_2)$
1	$w_1$=0.1, $w_2$=0.9	176,83	36,4
2	$w_1$=0.2, $w_2$=0.8	172,86	29,7.25
3	$w_1$=0.3, $w_2$=0.7	168,89	24.998,10.714
4	$w_1$=0.4, $w_2$=0.6	164,92	21,14
5	$w_1$=0.5, $w_2$=0.5	160,95	17,17
6	$w_1$=0.6, $w_2$=0.4	156,98	13,19.5
7	$w_1$=0.7, $w_2$=0.3	155,106	12,28
8	$w_1$=0.8, $w_2$=0.2	152,118	9.75,39
9	$w_1$=0.9, $w_2$=0.1	149,133	6,54

Table 3. Optimal objective values obtained from model (3).

	Weights	$z_1,z_2,z_3$	$p_1,p_2,p_3$
1	$w_1$= 0.2, $w_2$=0.2, $w_3$=0.6	129,126,64	27,54,0
2	$w_1$=0.3, $w_2$=0.2, $w_3$=0.5	129,126,64	27,54,0
3	$w_1$=0.2, $w_2$=0.1, $w_3$=0.7	129,126,64	27,54,0
4	$w_1$=0.4, $w_2$=0.2, $w_3$=0.4	129,126,64	27,54,0
5	$w_1$=0.3, $w_2$=0.4,$w_3$=0.3	127,104,76	25,32,12
6	$w_1$=0.2, $w_2$=0.5, $w_3$=0.3	153,89,75	51,17,11
7	$w_1$=0.4, $w_2$=0.3, $w_3$=0.3	127,123,66	25,51,2

Table 4. Optimal objective values obtained from model (4).

	Weights	$z_1,z_2,z_3$	$r(1-w_1),r(1-w_2),r(1-w_3)$
1	$w_1$=0.2, $w_2$=0.2, $w_3$=0.6	132,100,76	30,30,15
2	$w_1$=0.3, $w_2$=0.2, $w_3$=0.5	127,102,77	26.25,30,18.75
3	$w_1$=0.2, $w_2$=0.1, $w_3$=0.7	133,103,75	31,34.875,11.625
4	$w_1$=0.4, $w_2$=0.2, $w_3$=0.4	115,99,87	22.998,30.666,22.998
5	$w_1$=0.3, $w_2$=0.4,$w_3$=0.3	129,95,87	26.999,23.142,26.999
6	$w_1$=0.2, $w_2$=0.5, $w_3$=0.3	131,90,90	29.713,18.571,25.999
7	$w_1$=0.4, $w_2$=0.3, $w_3$=0.3	130,99,87	23.142,26.999,26.999

Table 5. Optimal objective values obtained from the weighted sum method.

	Weights	$z_1,z_2,z_3$
1	$w_1$=0.2, $w_2$=0.2, $w_3$=0.6	129,126,64
2	$w_1$=0.3, $w_2$=0.2, $w_3$=0.5	127,123,66
3	$w_1$=0.2, $w_2$=0.1, $w_3$=0.7	129,126,64
4	$w_1$=0.4, $w_2$=0.2, $w_3$=0.4	120,122,72
5	$w_1$=0.3, $w_2$=0.4,$w_3$=0.3	141,86,82
6	$w_1$=0.2, $w_2$=0.5, $w_3$=0.3	157,72,86
7	$w_1$=0.4, $w_2$=0.3, $w_3$=0.3	112,110,88

Table 6. Optimal objective values obtained from model (4) and the weighted sum method.

	Weights	$z_1,z_2(\text{model}(4))$	$z_1,z_2(\text{weighted sum method})$
1	$w_1$=0.1, $w_2$=0.9	150,230	143,265
2	$w_1$=0.2, $w_2$=0.8	154,210	156,200
3	$w_1$=0.3, $w_2$=0.7	156,200	156,200
4	$w_1$=0.4, $w_2$=0.6	160,195	156,200
5	$w_1$=0.5, $w_2$=0.5	164,190	176,175
6	$w_1$=0.6, $w_2$=0.4	168,185	176,175
7	$w_1$=0.7, $w_2$=0.3	172,180	176,175
8	$w_1$=0.8, $w_2$=0.2	176,175	186,171
9	$w_1$=0.9, $w_2$=0.1	186,171	208,167