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ON BICRITERIA LARGE SCALE TRANSSHIPMENT PROBLEMS
Dr. Jasem M.S. Alrajhi, Dr. Hilal A. Abdelwali, Dr. Mohsen S. Al-Ardhi, Eng. Rafik El Shiaty
In the following we will present an algorithm for determining all
nondominated extreme points of the (BMTSP3) model from which the
solution of the (BMTSP1) and the (BMTSP2) models can be deduced
as special cases from it.
Let: for independent constraints:
D k , k= 1,2,…,N be the technological matrix of the k th stage activity, D k
is (m k + n k ) * (m k + n k ) matrix, N is the number of stages, m k is the
number of sources at k th stage, n k is the number of destinations.
b k be the column vector consisting of the availabilities and
requirements of the k th subproblem, b k is (m k + n k ) * 1 column vector.
It follows that each set of independent constraints can be written as:
D k x k = b k , k = 1,2,…,N
x k represent the vector of the corresponding variables, x k is (m k + n k ) *1
column vector.
Let: For the common constraints:
A k be the technogical matrix of k th stage activity, A k is m 0 * (m k * n k )
matrix, m 0 be the number of common constraints.
b 0 be its corresponding common resources vector, b 0 is (m 0 * 1) column
vector.
This gives: A 1 x 1 + A 2 x 2 + …….+ A k x k + ……. +A N x N = b 0
Let: for the objective functions:
c K represent the vector of the first criterion coefficients for the k th stage
activity, c k is 1*(m k *n k ) row vector.
d k represent the vector of the second criterion coefficients for the k th
stage activity, d k is 1*(m k *n k ) row vector.
Let: For the master program:
B be the basic matrix associated with the current basic solution, B is
(m o *N) * (m o +N) matrix.
C B be the row vector of the corresponding coefficients in the objective
function, C B is 1*(m o +N) row vector.
R o is the matrix of size (m o + N)*m o consisting of the first m o columns
of B -1 , and
v j is the (m o + j) th column of the same matrix B -1
The algorithm presented here is divided into two phases.
Phase 1: Determine the nondominated extreme points in the objective
space. And the algorithm is validated by the following theorem [1].
Theorem:
A point z (q) q
q
= z1 , z2
is a nondominated extreme point is the
objective space if and only if z(q) is recorded by the algorithm.
Phase II: Is the decomposition algorithm which can be found in [7].
Since the special structure of the (BMTSP3) model may allow the
determination of the optimal solution by, first decomposing the
problem into small subproblems and then solving those subproblems
almost independently, then the decomposition algorithm for solving
large scale linear programming problems utilizing the special nature of
transshipment problem can be used to solve it.
Phase I:
Step 1: Go to phase II, find
z
(1 ) Min z / x M
1
And find
.
1
(1)
(1)
z
2
Min . z
2
/ z
1
z
1
and x M .
Step 2:
(1) (1)
Record ( z
1 , z
2 ) and set q = 1.
Similarly, go to phase II, find
z
( 2 )
2
And find
Min . z
2
/ x
( 2 )
( 2 )
z
1
Min . z
1
/ z
2
z
2
and x M .
( 2 ) ( 2 ) (1) (1)
( z
1
, z
2
) ( z
1
, z
2
), stop
M
If .
(2) (2)
Otherwise record ( z
1 , z
2 ) and set q = q+1
Defines sets L = {(1,2)} and E = , and go to step 2.
Choose an element (r,s) L and set
( r , s ) ( s ) ( r )
a z z and
a
1
( r , s )
2
z
2
( s )
1
z
2
( r )
1
k
Go to phase II to obtain the optimal solution ( x , k=1,2,..,N) to the
multistage transshipment problem.
Minimize
x
k
N
k 1
ik
, jk
and
( r,
s)
k
( r,
s)
k
(e
1
cik
j
a d
k 2 ik
j
)
k
Subject to
k
M , x o , k 1,2 ,.., N
x
k
ik
jk
If there are alternative optima, choose an optimal solution
k=1,2,.,N, for which
N
k 1
ik
, jk
Let z
1
=
z
2
=
N
k1
( c
N
k1
k
ik
jk
i , j
k
x
ik
, jk
k
k
ik
jk
d
c
k
ik
jk
k
i j
k k
min.)
x
x
k
ik
jk
k
i j
k k
; and
( r ) ( r )
If ( z
1
,
2
( z
1
, z
2
)
Set E = E {(r,s)} and go to step 3.
( q)
( q)
Otherwise record ( z
1
, z
2 ) such that
z
( s ) ( s )
z ) is equal to or ( z , z )
( q )
( q )
1
z1
, z
2
z 2 and set q q
Step 3:
L = L {(r,q)}, (q,s)} and go to step 3.
Set L = L - {(r-s)}. If L = , stop.
Otherwise go to step 2.
1
1,
2
k
x ,
Phase II:
Step 1: Reduce the original problem to the modified form in terms of
the new variables k
Step 2: Find an initial basic feasible solution to the modified problem.
Step 3: Solve the subproblems
k k
k
k k
w ( c OR d c
B
R
o
A ) x
Subject to
k k k
D x b ,
x k
o , k 1,2,..., N .
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