Reply to the comments of Ganjavi et al
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Table 1 Computational results of 11 cases with different number of objectives Different combinations of f i(x) ( for i ¼ 1, 2, 3 and 4)
D x1 x2 x3 f1 b1 Ratio1 f2 b2 Ratio2 f3 b3 Ratio3 f4 b4 Ratio4
(1,2,3,4)
(1,2,3)
(1,2,4)
(1,3,4)
(2,3,4)
(1,2)
(1,3)
(1,4)
(2,3)
(2,4)
(3,4)
1.000 4.500 4.500 0.000 10.125 11.000 0.080 6.750 15.000 0.550 0.000 8.000 1.000 0.000 9.000 1.000
0.507 1.433 7.567 2.185 5.428 11.000 0.507 7.402 15.000 0.507 3.948 8.000 0.507
0.351 9.000 0.000 0.742 7.144 11.000 0.351 9.742 15.000 0.351
1.000 4.500 4.500 0.000 10.125 11.000 0.080
1.000 0.000 0.000 0.000
0.351 9.000 0.000 0.742 7.144 11.000 0.351 9.742 15.000 0.351
0.053 0.421 8.000 0.000 10.421 11.000 0.053
0.165 8.258 0.742 0.000 9.186 11.000 0.165
0.478 0.516 8.000 3.310
0.294 9.000 0.000 1.588
1.000 0.000 0.000 0.000
7.826 15.000 0.478 4.174 8.000 0.478
10.588 15.000 0.294
7.763 9.000 0.137
0.000 8.000 1.000 0.000 9.000 1.000
The four objective constraints fi ðxÞ=bi þ D 5 1 and 4) will then be formed as follows: f1 ðxÞ :
1 5 5 x1 þ x2 x3 þ D 5 1; 11 44 22
f2 ðxÞ :
1 1 1 x þ x þ x þ D 5 1; 15 1 30 2 15 3
0.000 15.000 1.000 0.000 8.000 1.000 0.000 9.000 1.000
(i ¼ 1, 2, 3
1 1 1 f3 ðxÞ : x1 þ x2 x3 þ D 5 1; 8 8 8 f4 ðxÞ :
1 1 5 x x x þ D 5 1: 9 1 9 2 27 3
By picking a combination of four, three or two of these four restrictions, the following 11 cases are created: ð f1 ðxÞ; f2 ðxÞ; f3 ðxÞ and f4 ðxÞÞ; ð f1 ðxÞ; f2 ðxÞ and f3 ðxÞÞ; ð f1 ðxÞ; f2 ðxÞ and f4 ðxÞÞ; ð f1 ðxÞ; f3 ðxÞ and f4 ðxÞÞ; ð f2 ðxÞ; f3 ðxÞ and f4 ðxÞÞ; ð f1 ðxÞ and f2 ðxÞÞ; ð f1 ðxÞ and f3 ðxÞÞ; ð f1 ðxÞ and f4 ðxÞÞ; ð f2 ðxÞ and f3 ðxÞÞ; ð f2 ðxÞ and f4 ðxÞÞ and ð f3 ðxÞ and f4 ðxÞÞ: These 11 cases are examined and the results are presented in Table 1. Since Wi ¼ 1 and Ki ¼ bi ð8 i 2 I Þ; the ratio ðbi fi ðxÞÞ=bi is computed as a measure of weighted and normalized deviation for all objectives of each case. It can be observed that for all six cases when two objectives are considered both ratios are the same. Out of four cases when three objectives are considered two cases result in a mismatch of ratios (ie the solution is not balanced). Finally, the only case with four objectives also resulted in a mismatch.
Conclusions This viewpoint has demonstrated that the optimal solution of the programme 1 does not necessarily produce a balanced
7.579 8.000 0.053 7.515 9.000 0.165
6.353 9.000 0.294
0.000 8.000 1.000 0.000 9.000 1.000
allocation among the achievement of the different objectives. So, the finding of Romero et al1 can hold only in some special cases. It also shows some necessary but not sufficient conditions for the existence of a balanced solution.
References 1 Romero C, Tamiz M and Jones DF (1998). Goal programming, compromise programming and reference point method formulations: linkages and utility interpretations. J Opl Res Soc 49: 986–991. 2 Martel J-M and Aouni B (1998). Diverse imprecise goal programming model formulations. J Global Opt 12: 24–34. 3 Ballestero E and Romero C (1991). A theorem conne
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