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Transportation problem
The transportation problem seeks the determination of a
minimum cost transportation plan for a single commodity
from a number of sources to a number of destinations.
It requires the specification of the level of supply at each
source, the amount of demand at each destination, and
the transportation cost from each source to each
destination.
The transportation problem is linear if the cost on a route is
directly proportional to the amount transported; otherwise
it is nonlinear.
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Assume there are n sources and k destinations. The amount
of supply at source i is sour(i) and the demand at
destination j is dest(j). The unit transportation cost
between source i and destination j is cost(i,j). If x(i,j) is the
amount transported from source i to destination j then the
transportation problem is given as
min i,jf(i,j)
s.t. jx(i,j)sour(i)
i=1,2,,n
ix(i,j)dest(j)
j=1,2,,k
x(i,j)0 i=1,2,,n and j=1,2,,k
If f(i,j)=cost(i,j)*x(i,j) for all i and j, the problem is linear.
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The above problem implies that the total supply sour(i)
must at least equal total demand dest(j). When the total
supply equals the total demand, the resulting formulation
is called a balanced transportation problem. It differs from
the above only in that all the corresponding constraints
are equations. That is
jx(i,j)=sour(i)
i=1,2,,n
ix(i,j)=dest(j)
j=1,2,,k
If all sour(i) and dest(j) are integers any optimal solution to a
balanced linear transportation problem is an integer
solution, i.e., all x(i,j) are integers. Moreover the number of
positive integers among x(i,j) is at most k+n-1.
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An example
Assume 3 sources and 4 destinations. The supply is
sour(1)=15, sour(2)=25, sour(3)=5
The demand is
dest(1)=5, dest(2)=15, dest(3)=15, dest(4)=10
Note: the total supply and demand equal 45.
The unit transportation cost cost(i,j) is given as
1
2
3
4
1 10 0
20
11
2 12 7
9
20
3 0
16
18
14
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The optimal solution is shown below
5
15
15
10
15
0
5
0
10
25
0
10
15
0
5
5
0
0
0
The total cost is 315.
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Classical genetic algorithms
A straightforward approach in defining the chromosome for a
solution in the transportation problem is to create a vector
(v(1),,v(p)) where p=nk. Each component v(i)
represents an integer associated with row j and column m
in the allocation matrix, where j=(i-1)/k+1 and m=(i-1)
mod k+1.
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Constraint satisfaction
It is clear that every solution vector must satisfy the following:
v(q)0 for all q=1,2,, nk
v(ck+1)+v(ck+2)+ +v(ck+k)=sour(c+1) for c=0,1, ,n-1
v(m)+v(m+k)+v(m+2k)+ =dest(m) for m=1,2, ,k
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Evaluation function
fitness=v(i)cost(j,m)
where
j=(i-1)/k+1 and m=(i-1) mod k+1.
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Genetic operators
There is no natural definition of genetic operators the
transportation problem with the above representation.
Mutation is usually defined as a change in a single gene in a
solution vector. This, for our problems, would trigger a
series of changes in different places (at least three other
changes) in order to maintain the constraint equalities.
The situation is even worse if we try to define a crossover
operator.
Conclusion: the above vector representation is not the most
suitable for defining genetic operators in our problems.
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Incorporating problem-specific knowledge
Can we improve the representation of a solution while
preserving the basic structure of this vector representation?
The answer is yes, but we have to incorporate problemspecific knowledge into the representation.
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Initialization
The following procedure will create a matrix of at most k+n-1
non-zero elements such that all constraints are satisfied.
procedure initialization
set all numbers from 1 to kn as unvisited
repeat
select an unvisited random number q from 1 to kn and set it as visited
(row)i= (q-1)/k+1
(column)j=(q-1) mod k+1
val=min(sour[i],dest[j])
set v(i,j)=val
sour[i]=sour[i]-val
dest[j]=dest[j]-val
until all numbers are visited
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Assume 3 sources and 4 destinations.
The supply is
sour(1)=15, sour(2)=25, sour(3)=5
The demand is
dest(1)=5, dest(2)=15, dest(3)=15, dest(4)=10
5
15
15
10
15
0
0
0
0
25
0
0
0
0
5
0
0
0
0
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The first random number is 10. This means row number i=3
and column number j=2.
The val=min(sour[3],dest[2])=5, so v(3,2)=5. After the first
iteration, sour[3]=0 and dest[2]=10
5
10
15
10
15
0
0
0
0
25
0
0
0
0
0
0
5
0
0
The next three random numbers are 8,5, and 3.
What is the resulting matrix?
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5
10
15
10
15
0
0
0
0
25
0
0
0
0
0
0
5
0
0
The random numbers is 8 (corresponding to row 2 and
column 4).
5
10
15
0
15
0
0
0
0
15
0
0
0
10
0
0
5
0
0
14
5
10
15
0
15
0
0
0
0
15
0
0
0
10
0
0
5
0
0
The random numbers is 5 (corresponding to row 2 and
column 1).
0
10
15
0
15
0
0
0
0
10
5
0
0
10
0
0
5
0
0
15
0
10
15
0
15
0
0
0
0
10
5
0
0
10
0
0
5
0
0
The random numbers is 3 (corresponding to row 1 and
column 3).
0
10
0
0
0
0
0
15
0
10
5
0
0
10
0
0
5
0
0
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If the further sequence of random number is
1,11,4,12,7,6,9,2, the final matrix produced is
0
0
0
0
0
0
0
15
0
0
5
10
0
10
0
0
5
0
0
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This technique can generate any feasible solution that
contains at most k+n-1 non-zero integer elements.
It will not generate other solutions which, though feasible, do
not share this characteristic.
The initialization procedure would certainly have to be
modified when we attempt to solve non-linear versions of
the transportation problems.
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The knowledge of the problem and its solution characteristic
gives us an opportunity to represent a solution to the
transportation problem as a vector.
A solution vector will be a sequence of nk distinct integers
from the range <1, nk>, which (according to the
procedure initialization) would produce an acceptable
solution.
In other words we would view a solution-vector as a
permutation of numbers, and we would look for particular
permutations which correspond to the optimal solution.
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Constraint satisfaction
Any permutation of kn distinct numbers produces a unique
solution which satisfies all constraints. This is guaranteed
by procedure initialization.
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Evaluation function
This is relatively easy. Any permutation would correspond to
a unique matrix <v(i,j)>.
The evaluation function is i,jv(i,j)cost(i,j)
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Genetic operators
Inversion: any solution vector (x(1),x(2),, x(q)) (q=kn) can
be easily inverted into another solution vector (x(q), x(q-1),
, x(1))
Swap: any two elements of a solution vector, say x(i) and x(j)
can be swapped easily resulting in another solution vector.
Crossover: PMX, CX, OX
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A matrix as representation structure
Perhaps the most natural representation of a solution for the
transportation problem is a two dimensional structure.
After all, this is how the problem is presented and solved
by hand. In other words, a matrix V=(v(i,j)) (1ik, 1jn)
may represent a solution.
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Constraint satisfaction
Every solution matrix V=(v(i,j)) should satisfy the following:
jv(i,j)=sour(i)
i=1,2,,k
iv(i,j)=dest(j)
j=1,2,,n
v(i,j)0 i=1,2,,k and j=1,2,,n
This is similar to the set of constraints in the straightforward
approach, but the constraints are expressed in an easier
and more natural way.
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Mutation
Assume that {i(1), i(2),, i(p)} is a subset of {1,2, ,k} and
{j(1),j(2), ,j(q)} is a subset of {1,2, ,n} such that 2pk,
2qn.
Let us consider the matrix V=(v(i,j)). We can create a pq
submatrix W=(w(i,j)) from all elements of the matrix V in the
following way: an element v(i,j)V is in W if and only if i 
{i(1), i(2),, i(p)} and j  {j(1),j(2), ,j(q)} . If i=i(r) and j=j(s)
then the element v(i,j) is placed in the r-th row and s-th
column of the matrix W.
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We assign new values sour_w(i) and dest_w(j) (1ip, 1jq)
for matrix W:
sour_w(i)=j{j(1),j(q)}v(i,j)
1ip
dest_w(j)=i{i(1),i(p)}v(i,j)
1j q
We then use the procedure initialization to assign new values
too the matrix W such that all constraints sour_w(i) and
dest_w(j) are satisfied. After that we replace appropriate
elements of matrix V by a new elements from the matrix W.
In this way all global constraints (sour(i) and dest(j)) are
preserved.
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Example
Given a problem with 4 sources and 5 destinations and the
following constraints:
sour(1)=8, sour(2)=4, sour(3)=12, sour(4)=6
dest(1)=3, dest(2)=5, dest(3)=10, dest(4)=7, dest(5)=5
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The following matrix V is selected as a parent for mutation
0
0
5
0
3
0
4
0
0
0
0
0
5
7
0
3
1
0
0
2
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Select (at random) the two rows{2,4} and three columns
{2,3,5}. The corresponding submatrix W is
4 0 0
1 0 2
Note that sour_w(1)=4, sour_w(2)=3, dest_w(1)=5,
dest_w(2)=0, dest_w(3)=2.
After the reinitialization, the matrix W may get the following
values
2 0 2
3 0 0
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So finally the offspring of matrix V after mutation is
0 0 5 0 3
0 2 0 0 2
0 0 5 7 0
3 3 0 0 0
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Crossover
Assume that two matrices V1=(v1(i,j)) and V2=(v2(i,j)) are
selected as parents for the crossover operation.
Create two temporary matrices DIV=(div(i,j)) and
REM=(rem(i,j)) in the following way:
div(i,j)=(v1(i,j)+v2(i,j))/2
rem(i,j)=(v1(i,j)+v2(i,j)) mod 2
Matrix DIV keeps rounded average values from both parents,
the matrix REM keeps track of whether any rounding was
necessary.
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Matrix REM has some interesting properties: the number of
1s in each row and each column is even. In other words,
the values of sour_REM(i) and dest_REM(j) are even
integers. We use this property to transform the matrix
REM into two matrices REM1 and REM2 such that
REM=REM1+REM2
sour_REM1(i)=sour_REM2(i)=sour_REM(i)/2
dest_REM1(j)=dest_REM2(j)=dest_REM(j)/2
Then we produce two offspring of V1 and V2
V3=DIV+REM1
V4=DIV+REM2
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Example
Given a problem with 4 sources and 5 destinations and the
following constraints:
sour(1)=8, sour(2)=4, sour(3)=12, sour(4)=6
dest(1)=3, dest(2)=5, dest(3)=10, dest(4)=7, dest(5)=5
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Assume the following matrices V1 and V2 are selected as
parents
V1
V2
1 0 0 7 0
0 0 5 0
0 4 0 0 0
0 4 0 0
2 1 4 0 5
0 0 5 7
0 0 6 0 0
3 1 0 0
The matrices DIV and REM are
DIV
0 0
0 4
1 0
1 0
2
0
4
3
3
0
3
0
1
0
2
1
REM
1 0 1
0 0 0
0 1 1
1 1 0
1
0
1
0
3
0
0
2
1
0
1
0
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DIV
REM
0 0 2 3 1
1 0 1 1 1
0 4 0 0 0
0 0 0 0 0
1 0 4 3 2
0 1 1 1 1
1 0 3 0 1
1 1 0 0 0
The two matrices REM1 and REM2 are
REM1
0 0 1
0 0 0
0 1 0
1 0 0
0
0
1
0
1
0
0
0
REM2
1 0 0
0 0 0
0 0 1
0 1 0
1
0
0
0
0
0
1
0
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Finally two offspring V3 and V4 are
V3
0 0
0 4
1 1
2 0
3
0
4
3
3
0
4
0
2
0
2
1
V4
1 0
0 4
1 0
1 1
2
0
5
3
4
0
3
0
1
0
3
1
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Job shop problem
n/m JSP: It is a production scheduling problem in which n
jobs must be processed in m machines. A job is a set of
tasks. Every task can be labeled by three numbers: i is the
job number, j is the task number and k is the machine
number. The following is a 5/3 JSP:
Job1: 1,1,2 1,2,3 1,3,3 1,4,2, 1,5,2
Job2: 2,1,1 2,2,2 2,3,1 2,4,2
Job3: 3,1,1 3,2,3
Job4: 4,1,1 4,2,2 4,3,1 4,4,1 4,5,3
Job5: 5,1,3 5,2,2 5,3,3
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Representation
The schedule can be represented by a set of
permutations.Each permutation stands for the task carried
out by the machine.
chromosome={m1,m2,m3}
m1={3,1,1 2,1,1 2,3,1 4,1,1 4,3,1 4,4,1}
m2={1,1,2 1,4,2 2,2,2 1,5,2 4,2,2 2,4,2 5,2,2}
m3={1,2,3 3,2,3 5,1,3 1,3,3 5,3,3 4,5,3}
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Fitness function
The total cost according to the chromosome
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Initialization
while(not all the jobs are finished)
{
time forward;
for(every machine)
if (the machine is idle || the task is finished)
the machine is available;
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for(every machine)
if(the machine is available)
{
calculate the available task set;
if(the available task set is not empty)
choose a new task and set the machine as unavailable;
}
}
The available task set contains the tasks that all the pretasks are finished.
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Genetic operators
Swap: any two elements of a machine can be swapped
easily and check the feasibility
Modified-OX: choose the crossover points in one machine
and calculate the similar crossover points (the beginning
time is similar for the first point while the ending time is
similar for the second point) for other machines. Then do
like normal OX operator.
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