According to the considerations and constructions in Sect. 38.2, each of our
problems can be written as a pair of primal-dual problems (38.9 and 38.10), which
is equivalent through the considerations in the same Sect. 38.2 with a system of
linear inequalities Ax b, where x ¼ [y
T , u
T ]
T
2 ℝ
49
 ℝ
21 , and A , b are constructed
as in (38.12).
For the transportation problem P1, the linear problems and the system of linear
inequalities Ax b are consistent, while for problem P2, they are inconsistent. The
linear problems were first solved with linprog Matlab R2011a implementation of
Simplex algorithm, whereas the associated system of linear inequalities with MH
algorithm presented in Sect. 38.2.
In our computations implemented in Matlab R2011a, all runs with respect to
MH algorithm are started with the initial approximations x 0 ¼ y
T
0 ; 0
À
Á T ,
b x 0 ¼ b y
T
0 ; 0
À
Á T , with y 0 ! 0, b y 0 ! 0, and are terminated if at the current iterations
x
k , b x
k satisfy
Table 38.2 The unbalanced inconsistent transportation problem
D 1
D 2
D 3
D 4
D 5
D 6
D 7
Supply(s)
S 1
3
3
4
12
20
5
9
850
S 2
7
1
5
3
6
8
4
3 5 0
S 3
5
4
7
6
5
1 2
3
4 7 0
S 4
4
5
14
10
9
8
7
400
S 5
8
2
1 2
9
8
4
2
3 0 0
S 6
6
1
8
7
2
3
1
3 8 0
S 7
9
1 0
6
8
7
6
5
3 5 0
Demand(d)
450
500
420
420
460
520
530
Table 38.3 The unbalanced consistent transportation problem
D 1
D 2
D 3
D 4
D 5
D 6
D 7
Supply(s)
S 1
3
3
4
12
20
5
9
1050
S 2
7
1
5
3
6
8
4
3 5 0
S 3
5
4
7
6
5
1 2
3
4 7 0
S 4
4
5
14
10
9
8
7
850
S 5
8
2
1 2
9
8
4
2
6 0 0
S 6
6
1
8
7
2
3
1
4 8 0
S 7
9
1 0
6
8
7
6
5
4 5 0
Demand(d )
450
500
420
420
460
520
530
638
D. Carp et al.