An Implementation of the Parallel Algorithm for solving Nonlinear
Multi-commodity Network Flow Problem
Shieh-Shing Lin, Huay Chang, Cheng-Chien Kuo, Chun-Liang Hsu
Dept. of Electrical Engineering
St. John’s University
#499, Sec. 4 Tam-King Rd., Tamsui
TAIWAN
sslin@mail.sju.edu.tw

Dept. of Information Management

Chihlee Institute of Technology
Abstract: - In this paper, we propose a method to solve the Nonlinear Multicommodity Network Flow (NMNF)
Problem and discuss the associated implementation. We have combined this method with a projected-Jacobi
(PJ) method and a dual-type method possessing decomposition effects. With the decomposition, our method can
be parallel processed and is computationally efficient. We have tested our method on several examples of
NMNF problem in two Processor-Network systems. As seen from the simulation results, the parallel version of
the proposed method achieved a constant CPU time saving in comparison to the sequential version in solving the
NMNF problem. And the efficiency of our algorithm is more significant as the number of commodity increase.
Key-Words: - NMNF problem, projected-Jacobi method, dual-type method, decomposition, Network.
The paper is organized in the following manner.
Section 2 states the problem of the NMNF problem.
Section 3 presents the method combining the
projected-Jacobi and the parallel dual-type methods
for solving NMNF problems. The simulation results
those are used to demonstrate the computational
efficiency are given in Section 4. Finally, Section 5
gives a brief conclusion
1 Introduction
There are many large practical systems formed by the
network-like mesh-interconnected buses or nodes
through tier-lines. For example, the Network system
is formed by a number of nodes interconnected with
each other through tier-lines.
Nonlinear multicommodity network flow
(NMNF) problem is computationally difficult
because of their large dimension and nonlinearity and
has important applications to traffic assignment
[1]-[4] and data network routing [5]-[8]. Such
problems are typical convex programming problems,
and the NMNF problem solution techniques mostly
originate from nonlinear programming algorithm that
are especially to exploit the linear constraint structure
with various approaches [9]-[11]. Recently, an
efficient method developed in [12], Dual Projected
Pseudo Quasi-Newton method takes advantage of the
special structure of inequality constraints and
network sparsity. The method in [12] abbreviated
DPPQN have achieved a dramatic speed-up ratio
over a typical method for NMNF problem.
Considering the trend about the number of
commodities in the networks is increasing in NMNF
problem. In this paper, we use the framework of the
DPPQN method and propose a parallel algorithm to
solve the NMNF with many commodities problem.
Furthermore, we implement the proposed method in
two real Processor-Network systems and
demonstrate the computational efficiency through the
simulation results.
2 Statement of the NMNF Problem
We first introduce the notation for the K-commodity
NMNF problem in the following:.
k : denotes the index of the commodity of the
network system.
K : denotes the total number of the commodity of the
network system.
i : denotes the index of the node of the network
system.
I : denotes the total number of the node of the
network system.

Lij : denotes the set of the node j which connected
with node i , from node i to node j .
Lh i : denotes the set of node h which connected with
node i , from node h to node i .
k
f ij : denotes the flow over the branch (i, j ) with the
destination node (commodity) k .
ri : denotes the flow requirement at node i with the
k
1
( f ) : denotes the set of f satisfying the
destination node k .
(i, j ) : denotes the branch from node i to node j .
(h,i ) : denotes the branch from node h to node i .
B : denotes the set of all network branches in the
network system.
|  | : denotes the cardinality of the set．.
inequality constraints of flow.
t, w : iteration index.
 ,  : step size.
 1 ,  2 : predetermined positive real values.
 ( ) : the dual function of constrained QP
I
Fk (  f ijk ) : denotes the total branch cost
sub-problem.
 ( ) : the dual function of unconstrained QP
sub-problem.
 : denotes the vector of the Lagrange multipliers,
ik , i  1,2,..., I , k  1,2,..., K and
u
i 1 jLij
I
associated with total branch flows
f
i 1 jLij
k
ij
for the commodity k , and Fk () is a convex
  [11 , 12 ,..., 1I , 1.2 , 22 ,..., 2I ,..., 1K , 2K ,..., KI ]T .
|| () ||  : denotes the infinite norm of the vector () .
function in () .
K
I
 Fk (
k 1
f
i 1 jLij
k
ij
) : represents the sum of all the
The NMNF with K-commodity problem can be
stated as follows:
commodity of all the branch costs of the
network system.
() : the increment of the vector () .
K
I
min F ( f ijk )   Fk (  f ijk )
k 1
()T : denotes the transpose of the vector () .
(1a)
i 1 jLij
subject to
diag[□]: a diagonal matrix formed by the diagonal
terms of the matrix [□].
2
 f k Fk (t ) : denotes the Hessian of Fk with respect
f
jLij
k
ij

f
hLhi
k
hi
 ri k  0, i  1,2,..., I ,k  1,2,..., K
ij
(1b)
to f ijk evaluated at f ijk (t ) .
 2 Fk (t )
f
k2
ij
0  fijk , (i, j )  B, k  1,2,..., K .
: denotes the diagonal entry corresponding
The object of NMNF with many commodities
problem is to find an optimal flow solution that
satisfies the flow balance constraints (1b), and the
nonnegative constraints (1c), while minimizing the
objective function (1a).
to the branch (i , j ) of the matrix  2f k Fk (t ) .
ij
 f k Fk (t ) : denotes the gradient of Fk with respect
ij
to f ijk evaluated at f ijk (t ) .
Fk (t )
: denotes the component corresponding to the
fijk
3 Solution Method
3.1 The Projected-Jacobi (PJ) Method
branch (i , j ) of the vector  f k Fk (t ) .
The projected-Jacobi method uses the following
iterations to solve the NMNF problem given in
Eqs.(1a)-(1c),
ij
f : denotes the vector of all
fijk (t ), (i, j )  B, k  1,2,..., K and
f  [ f11 , f 21 ,..., f|B1| , f12 , f 22 ,..., f|B2| ,...,
K
1
K
2
f (t + 1) = f (t ) +  (t )f * (t ) ,
K T
|B|
f , f ,..., f ] .
where
f (t ) : denotes the vector of
the
vector
of
all
following:
f ,..., f , f ,..., f ]
K
2
denotes
f (t ), (i, j )  B, k  1,2,..., K , at iteration t and
the | B | K  1 column vector f is described in the
f  [f11 , f 21 ,..., f|B1| , f12 , f 22 ,...,
K
1
f (t )
(2)
k
ij
all fijk (t ), (i, j )  B, k  1,2,..., K and
2
|B|
(1c)
f  [ f11 , f 21 ,.., f|B1| , f12 , f 22 ,.., f|B2| ,.., f1K , f 2K ,.., f|BK| ]T
;  (t )  0 is a step-size determined by the
K T
|B|
QP: quadratic programming.
2
centralized Amijo rule [13], and f * (t ) denotes the
where Dijk (t ) is a B  B diagonal matrix
vector of all f ijk , (i, j )  B, k  1,2,..., K , the
corresponding to commodity k , and Dijk (t ) =
| B | K  1 column increment vector  f is of the
f1K , f 2K ,..., f |BK| ]T , that solves the following
1
diag [2f k Fk (t )] + I , and the diagonal entry
ij
2
 2 Fk (t )
corresponding to the branch ( i, j ) is
..
2
f ijk
quadratic programming (QP) sub-problems:
. From Eqs.(1a)-(1c) and since D ij is a block
*
following form,
f  [f11 , f 21 ,..., f |B1| , f12 , f 22 ,..., f |B2| ,..., .
I
min
f ij
 
i 1
jLij
diagonal matrix, we can rewrite Eqs.(3a)-(3c) as
1
[ f ij (t ) T Dij (t )f ij (t )
2
K
min
  fij F (t ) f ij (t )]
T
f ij
(3a)
( f
jLij
1
 [ 2 f
k 1 i 1
jLij
(t )T Dijk (t )f ijk (t )
k
ij
  f k Fk (t )T f ijk (t )]
(5a)
ij
subject to
k
ij
I
subject to
(t )  f (t )) 
k
ij
( f
hLhi
k
hi
(t )  f (t ))
k
hi
( f
 ri k  0, i  1,2,..., I , k  1,2,..., K
jLij
(3b)
k
ij
(t )  f ijk (t )) 
( f
hLhi
k
hi
(t )  f hik (t ))
 ri k  0i  1,2,..., I , k  1,2,..., K
0  f ijk (t )  f ijk (t ), (i, j )  B, k  1,2,..., K ,(3c)
0  f ijk (t )  f ijk (t ) , (i, j )  B , k =1,2,…, K
(5c)
where Dij (t ), (i, j )  B is a B  K  B  K block
1
2
diagonal matrix and Dij (t )  diag [ 2f ij F (t )]  I ,
3.2 The Parallel Dual-type Method
Let  denote the set of  f satisfying the
inequality constraints of flows, that is, f  f  
represents f  f satisfying (3c). Therefore, we
see that if f  f   , then f  f   for
 , 0<   1 . Hence, we can rewrite (5a)-(5c) as
and
 fij F (t )T  [ f 1 F1 (t )T ,  f 2 F2 (t )T ,...,  f K FK (t )T ]
ij
,  F and  f F
2
f ij
ij
T
ij
denote the Hessian and the
ij
gradient of F with respect to f ij for the branch
(i, j ) , (i, j )  B , respectively, and f ij (t ) denotes
K
the vector of all fijk (t ), (i, j )  B , k =1,2,…,K.
( f ( t )  f ( t ))
k 1 i 1
jLij
k
ij
(t )T Dijk (t )f ijk (t )
(6a)
ij
subject to
diagonal matrix Dij , and
bik (t )   f ijk (t ) 



  1
+ I

  2

 DijK (t )

1
  f k Fk (t )T f ijk (t )]
Let Dijk be the block diagonal term of the block
 D (t )
0

2
0
Dij (t )
Dij (t )  
 



 0
I
 [ 2f
min
And  is a small real number but large enough to
make D ij positive definite.
1
ij
(5b)
jLij
0
 f
hLhi
k
hi
(t )  0,
i  1,2,..., I , k  1,2,..., K
where
bik (t ) 
(4)
f
jLij
k
ij
(t ) 
f
hLhi
k
hi
(t )  ri k ,
i  1,2,..., I , k  1,2,..., K ,
3
(6b)
g1
0

0 g 2

g 
 


0
 0
  {f ijk 0  f ijk (t )  f ijk (t ), (i, j )  B,
k  1,2,..., K } .
The dual problem of the QP sub-problems
Eqs.(6a),(6b) is
max  ( )
(9)
Each g k is an I  B matrix corresponding to
(7)

0 

 

 

 g K 


commodity k in the network system.
where the dual function,
 ( ) 
K
min
f ( t )  f ( t )
I
1
 [ 2f
k 1 i 1 jLij
K
kT
ij
The parallel dual-type method uses the
following iteration to solve (7):
Dijk (t )f ijk
k (w  1)  k (w)   (w)k (w), k  1,2,..., K
(10)
I
  f k Fk (t )T f ijk ]   ik [bik (t )
ij
(8)
where w denotes the iteration index,  ( w)  0 is a
step-size determined according to the centralized
Armijo’s rule [13] and k  [1k , k2 ,..., kI ]T .
Furthermore,
k 1 i 1
  f ijk 
 f
jLij
hLhi
k
hi
]
is a function of  , which is the vector of Lagrange
multipliers ik , i  1,2,..., I , k  1,2,..., K , and
  [11 , 12 ,..., 1I , 12 , 22 ,..., 2I ,...,
1K , 2K ,..., KI ]T
  [11 , 12 ,..., 1I , 1.2 , 22 ,..., 2I ,..., 1K , 2K ,..., KI ]T
can be obtained by solving the following linear
equations:
. It should be noticed that for the sake of clarity, we
neglect the index t from the vector of the variable
f (t ) in (8).
2 u ( ( w))  ( w)   ( ( w))  0
k
i
We define g be the vector of equality constraint
function on the LHS of (6b). Therefore,
g ik  bik (t )   f ijk 
jLij
 f
hLhi
k
hi
(11)
where the column vector  ( ( w)) with dimension
I  K  1 is the gradient of  ( ) with respect to 
at  (w) , and  ( ( w)) is also the vector of all
,
 k  ( ( w)) , i =1,2,…,I, k =1,2,…, K , and the
i  1,2,..., I , k  1,2,..., K
,
and
the
I  K  1 column vector g can be described in the
following:
i
g  [ g11 , g 12 ,..., g 1I , g12 , g 22 ,..., g I2 ,..., g1K , g 2K ,..., g IK ]T
Since
f  [ f11 , f 21 ,.., f 1B , f12 , f 22 ,.., f B2 ,.., f1K , f 2K ,.., f BK ]T
matrix  2 u ( ( w)) with dimension I  K  I  K
denotes the Hessian of the unconstrained dual
function  ( ) .
From the Duality theorem [13], we see that the
 ( ( w)) and  2 u ( ( w)) can be computed by
and
f  [f11 , f 21 ,..., f 1B , f12 , f 22 ,..., f B2 ,...,
 k  ( ( w))  bik (t ) 
f , f ,..., f ] , we see that  g , the
K
1
gradient
of
K
2
g
K T
B
with
respect
to
f
is
i
  fˆ
jLij
k
ij

  fˆ
hLhi
k
hi
,
i =1,2,…,I, k =1,2,…, K , (12)
an
I  K  B  K block diagonal matrix shown in the
2 u ( ( w))  gD(t )1g T .
following:
(13)
The fˆ in (12) is the optimal solution of the
constrained minimization problem on the RHS of (8).
4
Decomposition Effect:
fˆ (* ) be the optimal solution (6a), (6b), (7),
The first decomposition effect can be described in
the following:
Based on the structure of the  ( ( w)) and
and the minimization problem on the RHS of (8) with
 = * , respectively, then f   fˆ (* ) .
To compute k  ( ( w )) , we need to solve the
 2 u ( ( w)) in (12) and (13), D (t ) and
g in (4) and (9), the equation in (13) can be
decomposed into the following K -independent sets
minimization problem on the RHS of Eq.(8) to obtain
of linear equations:
two-stage algorithm:
[2 u ( (w))]k k (w)  k  ( (w)) ,
k  1,2,..., K
i
fˆijk . This can be achieved by the following
Stage 1: Solve the following unconstrained
minimization problem:
(14)
and [2 u ( ( w))]k can be constructed in the
following:
 u ( )  min
K
I
1
 [ 2f
k 1 i 1
jLij
kT
ij
Dijk (t )f ijk
  f k Fk (t )T f ijk ]
[2 u ( (w))]k  g k Dk (t )g k , k  1,2,..., K ,
(15)
k
2 u
and the I  I matrix [   ( ( w)) ] denotes the
T
ij
K

k 1
I
  [b
i 1
k
i
k
i
(t ) 
 f
jLij
k
ij

 f
hLhi
k
hi
]
(16)
By using a gradient method [14] to obtain an
approximately solution for the entry of branch (i , j )
kth diagonal sub-matrix of 2 u ( ( w))
corresponding to commodity k .
The K -independent sets of linear equation (15)
can be parallel processed.
The second decomposition effect can be described
in the following:
The inequality constraint set  can be decomposed
into K-independent sets of k , k  1,2,..., K , that is
~
in each commodity k , and each f ijk can be
computed in parallel described in the following:
 2 Fk (t ) Fk (t )
~
f ijk ( ( w))  
(
 ik ( w)  kj ( w)),
k
k2
f ij
f ij
  kK1 k , where
(i, j )  B, k  1,2,..., K (17)
k  {fijk | 0  fijk (t )  fijk (t ), (i, j )  B} .
~
Stage 2: Project fijk , (i, j )  B, k  1,2,..., K , the
These two decomposition effects also contribute
the computational efficiency of the proposed method.
solution obtained from Stage 1, onto  .
The
resulting projection is fˆ ( (w)) . It is the solution
of the minimization problem on RHS of Eq.(8). Due
to the decomposition effect, the resulting projection
Convergence:
Each Dijk , (i, j )  B is a B  B diagonal
fˆijk , (i, j )  B, k  1,2,..., K , can be parallel
matrix corresponding to commodity k , and the
diagonal term corresponding to commodity k is
computed by the following analytical formula,
1
diag [ 2f k Fk ( f ijk )]  I . Therefore, Dijk is
ij
2
positive and so is D . Since D is positive definite
and g is of full rank,  2 u shown in (13) is a
negative definite matrix. Thus,  (w) obtained
fˆijk ( ( w)) 
~k
k
k

 f ij ( ( w)) if f ij ( ( w))  f ij ( ( w))  0
 ~k

f ij ( ( w)) otherwise
from (11) is an ascent direction for the dual problem
(7), by the Strong Duality Theorem [13], solving (6a),
(6b) is equivalent to solving the dual problem (7).
Therefore, this implies that we let f * , * , and
(18)
It should be noticed that the computations in Stage 2
are the comparison check shown in Eg.(18).
5
3.3 The Complete Algorithm for solving
Nonlinear Multicommodity Network Flow
problem
MP
Our method for solving Nonlinear Multicommodity
Network Flow problem is using PJ method Eq.(2)
where f * (t ) is the solution of the QP sub-problem
Eqs.(3a)-(3c). The proposed parallel dual-type
method uses Eq.(10) to solve Eq.(7), the dual
problem of QP sub-problem, instead of solving
Eqs.(3a)-(3c) directly. The k (w) in Eq.(10) is
obtained from solving Eq.(14) using linear
programming technique.
And can be parallel
SP1
4 Simulation and Test Results.
The fˆ , (i, j )  B, k  1,2,..., K is
We have tested our method on several examples of
NMNF problem. In the experiment, we fixed the
network size and varied the number of commodities.
The test system shown in Fig. 3 is a 10-node,
42-branch network system. We consider two types of
tests of NMNF problem in this network system.
The first one is a two-commodity NMNF
problem. There are four different kind of flow
requirements in this experiment described in the
following:
(1a) r19  20, r24  20 , (1b) r29  20, r34  20 ,
needed to set up  k  ( ( w)) and can be computed
using the two-stage method. Consequently, the
parallel dual-type method converges to optimal
solution   and the solution  fˆ of Eq.(8) with
   is f , the solution of Eqs.(3a)-(3c). The
complete algorithm and a typical K+1
Processor-Network system for the K-commodity
NMNF problem are shown in Figs. 1 and 2.
For every commodity k , k  1,2,..., K , the
NMNF problem be processed in parallel in each
individual processor.
Set the parameters,
  0,  1 ,  2
(1c) r59  20, r64  20 , (1d) r39  20, r74  20 .
The second one of test is a four-commodity
NMNF problem in the network system. There are
four different kind flow of requirements in this
experiment described in the following:
(2a) r19  20, r34  20, r48  20, r102  20 ,

k
Update
( w  1)
by (10)
(Slave Processor)
the initial step-sizes,
 (t ),  ( w).
(Master Processor)
(2b) r29  20, r34  20, r58  20, r62  20 ,
Is
||  ( w)
. ( w) ||   1
Initially
k
(t )
guess f
and set t  0 .
(Slave Processor)
Initially
k
guess
(w)
and set w  0 .
(Slave Processor)

 ( w))
Compute fˆ k (
by (18).
(Slave Processor)
(2c) r39  20, r64  20, r78  20, r42  20 , and
No
(2d) r69  20, r84  20, r28  20, r52  20 .
The objective function of the above two types of
w  w 1
tests have the same form:
Set f
~
Compute f k (
by (17).
(Slave Processor)
Yes
(Master Processor)
Set
k*
 fˆ k ( ( w))
and update
f
k
(t  1)
||  (t ) f
Is
*
(t ) || ∞  2
Yes
(Master Processor)
  ( w))
Compute  k
(

by (12)
(Slave Processor)
No
Set
 u ( ( w))]k
t  t 1
Compute [
by (15).
(Slave Processor)
2
k
Solve 
(w)
by (14).
(Slave Processor)
1 K I
(  f ijk ) 2 .

2 k 1 i1 jLij
We use the following parameters in our
algorithm:  1   2  10 3 ,   0.01 , and initial zero
flow are set to solve the above two type of tests of
NMNF problem. To implement our algorithm in a
real dedicated network environment, we choose two
different types of Processor-Network, with Network
1 combined with 3 Processors, and Network 2
combined with 5 Processors (Fig. 4 and Fig.5) as our
experimental computer network. In Network 1, there
are three processors: including: a Master Processor
(MP-N1) and two Slave Processors (SP1-N1 and
SP2-N1). The MP-N1 executes the evaluation of the
convergence criteria and the step-size determination
during the iterative procedure. Each of the two Slave
by (2) (Slave Processor)
 ( w))
SPK
Fig. 2 A typical Processor-Network combined with one
Master Processor (MP) and K-Slave Processors
(SP1-SPK) for NMNF with K-commodity problem.
k
ij
processed.
SP2
f (t ) is the solution.
(Master Processor)
Fig. 1 Complete algorithm for NMNF problem.
6
Processors (SP1-N1 and SP2-N1) individually
executes the parallel algorithm for solving the
preassigned commodity NMNF problem.
In
Network 2, there are five processors including: a
Master Processor (MP-N2) and four Slave Processors
(SP1-N2, SP2-N2, SP3-N2, SP4-N2). The MP-N2
executes the evaluation of the convergence criteria
and the step-size determination during the iterative
procedure. Each of the four Slave Processors
(SP1-N2, SP2-N2, SP3-N2, SP4-N2) individually
executes the parallel algorithm for solving the
preassigned commodity NMNF problem. We use the
same model of PCs to simulate the processor in the
Networks 1 and 2. The model of each PC is Pentium
4, the processor speed of it is 3.2 GHzs, and the
memory RAM of each PC is 512 Mbytes. The
communication hardware used between the Master
Processor and the Slave Processors are the RS232
interface cards associated with a local network link
[15] and the language of the computer code is C   .
We called these experimental works as the parallel
version of test. We use Networks 1 and 2 to solve the
NMNF with two and four commodities problem
described above, respectively.
We solve 100
examples in each case in each type of NMNF
problem. The average consumption of CPU time and
the resulting final objective value in each case of test
for the NMNF with two and four-commodity
problem in parallel version are shown in Tables 1 and
2, respectively.
To investigate the computational efficiency of
the proposed parallel algorithm, we use a single PC
(Pentium 4) to solve the same test examples of the
above two types of NMNF problem with the same
initial guess and the same terminated criteria. We
call this experimental work as the sequential version
of test. The average consumption of CPU time and
the final objective value in each case in each type of
test in the sequential version are also shown in Tables
1 and 2. From Table 1, we see that the average
speed-up ratio of the proposed algorithm in parallel
version is about 1.44 times compared with the
sequential version of two-commodity NMNF
problem. Furthermore, from Table 2, we see that the
average speed-up ratio of the proposed algorithm in
parallel version is significant increasing about 3.68
times compared with the sequential version. This
indicated that our parallel algorithm is suitable for
solving the NMNF problem with many commodities
problem.
5
3
2
1
8
6
9
4
7
10
Fig.3 A 10-node, 42-branch network system for
NMNF problem.
MP-N1
SP1-N1
SP2-N1
Fig. 4 Network 1-A dedicated Processor-Network
system for two-commodity NMNF Problem.
MP-N2
SP1-N2
SP2-N2
SP3-N2
SP4-N2
Fig. 5 Network 2-A dedicated Processor-Network
system for four-commodity NMNF Problem.
5 Conclusion
We have developed a parallel algorithm for solving
NMNF problem.
The method combines the
projected-Jacobi method and a dual-type method
possessing commodity decomposition effects. With
the decomposition, the proposed method can be
parallel processed. We implement the proposed
algorithm in two real Processor-Network systems and
solve quite a few examples of NMNF with two and
four-commodity problem.
As seen from the
simulation results, the proposed algorithm achieve a
constant CPU time saving in parallel version with
respect to the sequential version in solving the
NMNF problem. The efficiency of the proposed
algorithm is more significant as the number of
commodity increase.
7
Table 1 Comparison the efficiency of the proposed algorithm in parallel version with the sequential version
in solving two-commodity NMNF problem.
Parallel version
Cases
CPU time (s)(i)
(1a)
(1b)
(1c)
(1d)
3.141
3.212
3.244
3.081
Final objective
value
9.35
10.12
10.14
9.89
Sequential version
CPU time (s)(ii)
4.481
4.662
4.713
4.424
Final objective
value
9.35
10.12
10.14
9.89
Speed up ratio (ii)/(i)
1.427
1.451
1.453
1.435
Table 2 Comparison the efficiency of the proposed algorithm in parallel version with the sequential version
in solving four-commodity NMNF problem.
Parallel version
Cases
CPU time (s)(i)
(2a)
(2b)
(2c)
(2d)
2.145
2.091
2.063
2.015
Final objective
value
19.24
18.92
19.16
18.78
Sequential version
CPU time (s)(ii)
7.682
7.602
7.654
7.589
Final objective
value
19.24
18.92
19.16
18.78
Speed up ratio (ii)/(i)
3.581
3.635
3.710
3.766
331-356.
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benders decomposition, ” Mgmt. Sci. Vol.20, No.
5, 1974, pp. 822-884.
[12] S.-Y. Lin and C.-H. Lin, “ A Computational
efficient method for nonlinear multicommodity
network problems, ” Networks, Vol.29, 1997,
pp. 255-244.
[13] D. Luenberger, Linear and nonlinear programming, 2nd ed. Addison-Wesley Reading,
MA,1984.
[14] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and
Distributed Computation: Numerical Methods.
Prentice-Hall, London. 1989.
[15] Mazidi, The 80x86 IBM PC Compatible
Computers Volume II Design & Interfacing of
the IBM PC & Compatible 3/e, 2000.
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8