International Mathematical Forum, Vol. 6, 2011, no. 40, 1983 - 1992
Floating Point Method for Solving Transportation
Problems with Additional Constraints
P. Pandian and D. Anuradha
Department of Mathematics, School of Advanced Sciences
VIT University, Vellore-14, Tamilnadu, India
pandian61@rediffmail.com, anuradhadhanapal1981@gmail.com
Abstract
A new method namely, floating point method is proposed for finding an optimal
solution to a transportation problem with additional constraints. The proposed method is
easy to understand and to apply for finding an optimal solution to the above said
problem and it differs from the existing methods namely, simplex method [7] and
inverse matrix method [6]. With the help of real life examples, the proposed method is
illustrated. It can be very helpful for the decision makers to make economical oriented
managerial decisions.
Mathematics Subject Classifications: 90C08, 90C90
Keywords: Transportation problem, zero point method, additional constraints, floating
point method
1. Introduction
The classical transportation problem (TP) deals with shipping commodities from
sources to destinations at minimum cost. The basic TP was developed by Hitchcock [5].
In these problems it is assumed that the product is identical no matter which source
produces it and that the customers have no preference relating to the source of supply.
However, in many practical instances, the product does vary in some characteristics
according to its source and the final product mixture received by the destinations, may
1984
P. Pandian and D. Anuradha
then be required to meet known specifications. For example, it is assumed that the crude
ore contains different amounts of an impurity, phosphorous, according to its source and
the actual time to process the ore depends on both its source and destination. A TP of
this type is described by Haley [4]. The simplex method approach for solving
constrained TPs was considered by Klingman and Russell [7]. TP based on time
criterion with additional constraints was given by Nguyen Van Vy [8]. Kim [6] gave an
inverse matrix algorithm for solving a TP with additional constraints. Reeta Gupta [11]
obtained a solution procedure for solving the generalized TP with constraints. Fractional
TP with impurities and time minimizing TP with impurities was presented by Chandra
et al [1,2]. Singh and Saxena [10] developed a method for solving multi objective time
TP with additional impurity restrictions. Dutta and Murthy [3] developed a linear
fractional programming approach for solving fuzzy TP with additional restrictions.
In this paper, we propose a new method, namely floating point method to
determine an optimal solution to TPs with additional constraints. The floating point
method takes advantage of the fact that the problem has a special form and it does not
require the use of the simplex method [7] and inverse matrix method [6]. The proposed
method is illustrated with the help of two real life examples. The floating point method
provides an appropriate solution which helps the decision makers to analyze an
economic activities and to arrive at the correct managerial decisions.
2. Transportation problem with additional constraints
Consider the following TP with additional constraints:
(P) Minimize z = ∑m ∑n cij xij
i =1 j=1
subject to
n
∑ xij = ai
j=1
m
∑
i =1
m
∑
i =1
xij = b
ai
j
,
i = 1,2,..., m
(1)
j = 1,2,..., n
(2)
(k )
(k)
f ij xij ≤ p
j
xij ≥ 0
where
,
,
,
i = 1,2,..., m
j = 1,2,..., n ; k = 1,2,..., l
(3)
and j = 1,2,..., n
is the tons available at ith source;
unit of the product contains fij(k ) units of
l
b j is
(4)
the tons required at jth destination; one
impurities
k = 1,2,..., l when
it is sent from i to
Floating point method
j
1985
; customer j cannot receive more than
(k )
p j units
of impurity
transportation from the ith source to jth destination and
xij
k
;
cij
is the unit cost of
is the tonnage transported
from ith source to jth destination.
Any set of
{xijo ≥ 0, i = 1,2,..., m; j = 1,2,..n}
which satisfies the equations (1), (2), (3) and
(4) is called a feasible solution to the problem (P).
A feasible solution of problem (P) which minimizes the total shipping cost is
called an optimal solution to the problem (P).
2.1 Floating point method
Now, we propose a new method namely, floating point method for finding an
optimal solution to a TP with additional constraints. First, we prove the following
theorem which is going to be used in the floating point method.
Theorem 1: Let
U o = {uijo , i = 1,2,..., n and j = 1,2,..., m}
where
(Q) Minimize
is an optimal solution to the problem (Q)
m n
z = ∑ ∑ cij xij
i =1 j=1
subject to (1), (2) and (4) are satisfied,
o , i = 1,2,..., n and j = 1,2,..., m}
Then,
which
X o = {xij
U o = {uijo , i = 1,2,..., n and j = 1,2,..., m} ,
Proof: If
If not,
obtained
from
the
solution
is an optimal solution to the problem (P)
U o = {uijo , i = 1,2,..., n and j = 1,2,..., m}
problem (P), then
is
satisfies all the additional constraints of the
o , i = 1,2,..., n ; j = 1,2,..., m}
U o = {uij
o , i = 1,2,..., n and j = 1,2,..., m}
U o = {uij
is an optimal solution of the problem (P).
is not satisfied atleast one additional
constraint. Say, rth additional constraint. That is,
(k ) o
(k )
(k )
(k )
(k )
∑ f ir uir = f1r u1r + f 2r u 2r + ... + f mr u mr > p r
i
Now, we flow the excess quantity in the rth column to other quantity satisfied
column such that rth column and considered column are satisfied their additional
constraints and the allotment to rows and columns are not disturbed through a closed
rectangular loop.
1986
P. Pandian and D. Anuradha
Construct a rectangular loop ABCDA where AC are in the rth column and BD are in
the sth column which satisfies its additional constraint such that A and D are allotted
cells.
o = uo − θ )
A ( y tr
tr
C ( y opr = u opr + θ )
o = uo + θ )
B ( y ts
ts
D ( y ops = u ops − θ )
We flow the minimum amount θ through the loop ABCDA such that rth column and
sth column additional constraints are satisfied and the total allotment to rows
and columns are not disturbed. Then, we obtain a new allotment for A, B, C and D
are y tro , y tso , y opr and y ops and compute
⎛ c yo + c yo + c yo + c yo ⎞ − ⎛ c uo + c uo + c uo + c uo ⎞
⎜ tr tr
ts ts
pr pr
ps ps ⎟ ⎜ tr tr
ts ts
pr pr
ps ps ⎟
⎝
⎠ ⎝
⎠
called the excess flow of the loop ABCDA. Thus, we obtain a feasible solution
o , i = 1,2,..., n and j = 1,2,..., m} from U o = {u o , i = 1,2,..., n and j = 1,2,..., m} to the problem (Q)
Y o = { yij
ij
with the rth additional constraint where
⎧ u o : i ≠ t , p and j ≠ r , s
⎪ ij
⎪ u o − θ : i = t and j = r
tr
o = ⎪⎪ o
yij
⎨ uts + θ : i = t and j = s
⎪ o
⎪u ps − θ : i = p and j = s
⎪ o
⎪⎩u pr + θ : i = p and j = r
Construct all possible loops from an allotted cell in the rth column, flow the
minimum amount θ through each of the loops such that rth column additional constraint
is satisfied and the considered additional constraint satisfied column, the total allotment
to rows and columns are not disturbed and compute their excess flow. Such loops are
called admissible loops.
Select a loop in the set of admissible loops from an allotted cell in the rth column
having a minimum excess flow. After flowing the minimum amount through the
selected loop, a feasible solution to the problem (Q) with rth additional constraints can
be obtained. Since the flow in the loop is minimum, the obtained feasible solution to the
problem (Q) with rth additional constraint is optimal.
If the current solution is satisfied all the additional constraints of (P), it is optimal to
the given problem (P).
Floating point method
1987
If not, we continue the above process until the current solution is satisfied all the
additional constraints of (P).
Now, after a finite number of above said steps, we obtain an optimal solution
to the given problem (P).
o , i = 1,2,..., n ; j = 1,2,..., m} from U o = {u o , i = 1,2,..., n ; j = 1,2,..., m}
X o = {xij
ij
Hence the theorem.
Remark 1: If an admissible loop can not be formed for any allotted cell in any column
which is not satisfied its additional constraint, then the given problem (P) has no
feasible solution.
We now introduce the floating point method for finding an optimal solution to
the problem (P).
The proposed method proceeds as follows.
Step1: Identify all impossible allotment cells using additional constraints and remove
those cells from the given TP.
Step 2: Obtain an optimal solution to the reduced TP obtained from the Step 1. without
considering the additional constraints using the zero point method [9].
Step 3: Check that all additional constraints are satisfied for the optimal solution
obtained in the Step 2. If so, go to the Step 7. If not, go to the Step 4.
Step 4: Identify the column(s) which is (are) not satisfied the additional constraints.
Step 5: Select a loop in a column which is not satisfied the additional constraint and
flow the minimum quantity θ according to the Theorem 1.. Then, find a new solution to
the transportation problem with rth additional constraints.
Step 6: Check that the solution obtain from the Step 5. satisfies all additional
constraints. If so, go to the Step 7. If not, go to the Step 4.
Step 7: The current solution is an optimal solution to the TP with all additional
constraints by the Theorem 1..
Now, the proposed method can be illustrated with the help of following
examples.
1988
P. Pandian and D. Anuradha
Example 1: Coal Shipping Problem: Barat Coking Coal, a coal manufacturing unit of
the State has different types of coal pulverization units in each of the three work centers
(j) situated in various parts of the State. The work centers (j) are receiving a fixed
quantity of coal (i), which has three different grades. The basic goal is to determine a
feasible transportation cost, while satisfying the extra requirement that the amount of
sulphur impurity present in coal is less than a certain critical level. The following table
displays the transportation cost, availabilities, the impurities, requirements and the
maximum sulphur contents. .
j
Work centers
1
Coal i,
2
3
Tons required b j
Max. Sulphur p j
1
4
4
7
5
2
3
6
4
5
3
2
7
6
5
4
1
9
Tons
available ai
4
5
6
Sulphur
contents f i
2
1
0
Now, using the Step 1 we have the following reduced TP table
Work centers
Coal i,
1
2
3
Tons required b j
Max. Sulphur p j
j
1
4
4
7
2
X
6
4
3
2
7
6
5
5
5
4
1
9
Tons
available ai
4
5
6
Sulphur
contents f i
2
1
0
Now, we obtain the following optimal solution to the TP by the zero point method.
Work centers
j
1
2
3
1
Coal i,
2
3
Tons required b j
4
4 (5)
7
5
X
6
4 (5)
5
2 (4)
7
6 (1)
5
Max. Sulphur p j
4
1
9
Tons
available ai
4
5
6
Sulphur
contents f i
2
1
0
Now, the current solution does not satisfy the first additional constraint, but it satisfies
second and third additional constraints. Then, by using the Step 5, we have only one
minimum flow admissible loop L1 (2,1)-(2,3)-(3,3)-(3,1)-(2,1). After flowing 1unit
through L1, we obtain the following solution to the TP with first additional constraint:
Floating point method
1989
Work centers
1
Coal i,
2
3
Tons required b j
Max. Sulphur p j
j
1
4
4 (4)
7 (1)
5
2
X
6
4 (5)
5
3
2 (4)
7 (1)
6
5
4
1
9
Tons
available ai
Sulphur
contents f i
4
5
6
15
2
1
0
Now, the above solution satisfies all the additional constraints. Therefore, by the
Theorem 1., it is an optimal to the given TP with the additional constraints. Thus, the
optimal solution to the given problem is x13 = 4 , x21 = 4 , x23 = 1 , x31 = 1 , x32 = 5 and the
total minimum transportation cost is 58.
Example 2: A company making iron has a different type of furnace in each of three
works. The works must receive a fixed weight of ore which is available in three
different grades. For technical reasons the processing time of ore depends on its grade
and the works to which it is sent. The furnaces are only available for a certain time in
each week. The problem is to find the allocation which minimizes the production cost
while satisfying the extra requirement that the amount of phosphorus is less than a
certain critical level. The following table displays the cost cij (lower right corner) of
transport, purchase and processing ore per ton and process time tij (upper left corner) in
hours.
1
2
1
4
3
Ore , i
2
3
6
9
7
Works, j
2
4
2
5
1
8
8
3
7
Tons available
ai
7
Phos.
Content f i
12
0.8
6
0.7
0.4
9
1
5
6
3
Tons required b j
5
10
10
Max . phos p j
3.5
7
7
Total time available T j
30
60
40
Now, we obtain the following optimal solution to the TP by the zero point method.
1990
P. Pandian and D. Anuradha
1
1
2
3
Ore , i
5
Works, j
2
2
8
Tons
available ai
3
4
6
Tons required b j
5
10
10
Max . phos p j
3.5
7
7
Total time available T j
30
60
40
7
12
6
Phos.
Content fi
0.4
0.8
0.7
Now, the current solution does not satisfy the second and third impurity constraint, but
it satisfy the first impurity constraint and all the time constraints. Then, using Step 5, we
have only one minimum flow admissible loop L1 (2,2)-(1,2)-(1,1)-(2,1)-(2,2) .After
flowing 0.5 unit through L1, we obtain the following new solution to the problem.
Works,
2
1
1
2
3
j
3
4.5
0.5
2.5
7.5
Tons required b j
5
10
10
Max . phos p j
3.5
7
7
Total time available T j
30
60
40
Ore , i
4
6
Tons
available ai
7
12
6
Phos. Content
fi
0.4
0.8
0.7
Now, the current solution does not satisfy the third impurity constraint, but it satisfy
the first, second impurity constraints and all the time constraints. Repeating Step 3 to
the Step 7 by four times , we obtain the following optimum solution to the problem.
Works,
2
1
Ore , i
1
2
3
3.0535
0.1605
1.786
2.5
7.5
j
3
1.4465
4.3395
4.214
Tons required b j
5
10
10
Max . phos p j
3.5
7
7
Total time available T j
30
60
40
Tons available
ai
7
12
6
Phos.
Content f i
0.4
0.8
0.7
Floating point method
1991
Thus, the optimal solution to the given problem is x11 = 3.0535 , x12 = 2.5 , x13 = 1.4465 ,
x21 = 0.1605 , x22 = 7.5 , x23 = 4.3395 ,
x31 = 1.786 , x33 = 4.214 and the total minimum
transportation cost is 85.537.
Conclusion
In this paper, the transportation problems with additional constraints are solved
by the floating point method which differs from the existing methods namely, simplex
method and inverse matrix method. The optimal solution obtained by floating point
method is not necessarily to have (m + n − 1) alloted entries which is the main
advantage of this method. The proposed method enables the decision makers to
optimize the economical activities and make the correct managerial decisions. In near
future, we extend the proposed method to interval/fuzzy transportation problems with
additional constraints.
References
[1] S. Chandra and P.K. Saxena, Fractional transportation problem with impurities,
Advances in Management Studies, 2 , 1983, 335-349.
[2] S. Chandra, K.Seth and P.K. Saxena, Time minimizing transportation problem
with impurities, Asia-Pacific Journal of Operational Research, 4 ,1987, 19-27.
[3] D. Dutta and A. Satyanarayana Murthy, Fuzzy transportation problem with
additional restrictions, ARPN Journal of Engineering and Applied Sciences, Vol.
5, No.2, 2010.
[4] K.B. Haley and A. J. Smith, Transportation problems with additional restrictions,
JSTOR, 15, 1966, 116-127.
[5] F.L.Hitchcock, The distribution of a product from several sources to numerous
localities, J. Math. Phys. 20, 1941, 224-230.
[6] K.V. Kim, An inverse matrix algorithm for solving a transportation problem with
additional constraints, Ekonomika I matematicheskie melody, 1967, 588 – 592.
[7] D. Klingman and R. Russell, Solving constrained transportation problems,
Operations Research, Vol.23, No.1, 1975, 91-106.
1992
P. Pandian and D. Anuradha
[8] Nguyen Van Vy, Transportation problem based on time criterion with additional
constraints, Cybernetics and Systems Analysis, Vol. 14, No. 4, 569 – 575
[9] P. Pandian and G.Natarajan, A new method for finding an optimal solution of
fully interval integer transportation problems, Applied Mathematical Sciences,
4, 2010, 1819-1830.
[10] Preetvanti Singh and P.K. Saxena, The multiobjective time transportation
Problem with additional restrictions, European Journal of Operational Research,
146, 2003, 460- 476.
[11] Reeta Gupta , Solving the generalized transportation problem with constraints,
Journal of Applied Mathematics and Mechanics, Vol. 58, 451 – 458.
Received: February, 2011