Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
Pareto Optimal Solutions to the Cost-Time Trade-Off
Bulk Transportation Problem through a Newly
Evolved Efficacious Novel Algorithm
*1
2
Satya Prakash , Rajesh Kumar Saluja , Pooja Singh
3
1
Department of Mathematics,
Department of Mechanical and Automation Engineering,
3
Department of Computer Science and Engineering, Amity School of Engineering and
Technology, New Delhi, INDIA
2
*1
2
3
Email Addresses: [email protected], [email protected], [email protected]
Abstract-The cost-time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk
transportation without according priorities to them is revisited. The entire requirement of each destination is to be met from one source
only; however a source can supply to any number of destinations subject to the availability of the commodity at it. An algorithm more
novel than the two algorithms developed earlier is provided to obtain the set of Pareto optimal solutions to the cost-time trade-off bulk
transportation problem. Both the algorithms developed earlier and the new algorithm consider a sequence of prioritized bicriterion bulk
transportation problems whose solutions provide the set of Pareto optimal solutions of the problem. Both the algorithms developed
earlier reduce the prioritized bicriterion bulk transportation problems to single-objective problems. In one case, it is done through the
introduction of preemptive priority factors, whereas in the other case, it is done by focusing on the first priority objective, keeping the
second priority objective in abeyance and finding all the alternative optimal solutions. Among all the alternative optimal solutions that
one is picked up for the optimal solution to the prioritized bicriterion bulk transportation problem for which the second priority
objective is minimum. The process of reducing the prioritized bicriterion bulk transportation problems into single-objective ones
involves a lot of computational work. The new algorithm simplifies the computational work by treating the prioritized bicriterion bulk
transportation problems as such and invoking the concept of lexicographic minimum.
Keywords- Multiobjective Programming; Optimization; Branch and Bound; Pareto Optimal; Transportation; Bulk Transportation
I
INTRODUCTION
The normal transportation problem, wherein the requirement of each destination can be met from one or more sources with the
single objective to minimize the total cost or duration of transportation or also with two objectives to minimize the total cost and
the duration of transportation with or without according priorities to them, has been studied in the past and is well known. There is
another version of the normal transportation problem, called the bulk transportation problem, which differs from the normal
transportation problem in that it stipulates that the requirement of each destination has to be met by one source only; however, a
source can supply to any number of destinations subject to the availability of the commodity at it. Maio and Roveda (1971) were
the first to formulate a bulk transportation problem with the single objective to minimize the total cost of bulk transportation and
devise a solution procedure based on zero-one implicit enumeration. Srinivasan and Thompson (1973) presented an algorithm
consisting of two phases to solve this problem. In the first phase, the problem is solved ignoring the constraint that stipulates that
the requirement of each destination has to be met by one source only. In the second phase, a branch and bound technique is used to
obtain a row unique solution. Murthy (1976) proposed a method based on the principle of lexicographic minimum to solve this
problem. Prakash and Ram (1995) considered a bulk transportation problem with the minimization of the total cost and duration of
bulk transportation as primary and secondary objectives, respectively.
Sometimes it happens that the decision maker is not able to assign priorities to his/her objectives. In such a situation, help can
be provided to the decision maker by presenting him/her with a set of Pareto optimal solutions. He/she can pick up the solution out
of this set that suits him/her most with regard to his/her objectives and liking. Prakash, Kumar, Prasad and Gupta (2008) attempted
to deal with this type of situation in the context of cost-time trade-off bulk transportation problem. For this purpose, Prakash and
Ram’s problem was considered with the alteration that the two objectives were not assigned priorities in contrast to the earlier
problem wherein they were assigned priorities. Two algorithms were developed for obtaining the set of Pareto optimal solutions to
this altered problem, which modify and extend the previous algorithm employed to solve the cost-time trade-off bulk transportation
problem with prioritized objectives, for obtaining the set of Pareto optimal solutions to the cost-time trade-off bulk transportation
problem without according priorities to the two objectives. As the set of Pareto optimal solutions to the altered problem contains
- 13 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
the optimal solution to Prakash and Ram’s problem, the altered problem extends and generalizes their work while providing
flexibility in decision making.
In the present paper, the cost-time trade-off bulk transportation problem with the objectives to minimize the total cost and
duration of bulk transportation without according priorities to them is revisited. A new algorithm, more efficacious and novel than
the two algorithms developed earlier, is presented for obtaining the set of Pareto optimal solutions to the cost-time-trade-off bulk
transportation problem. Both the algorithms developed earlier and the new algorithm solve a sequence of prioritized bicriterion
bulk transportation problems whose solutions provide the set of Pareto optimal solutions of the cost-time trade-off bulk
transportation problem. Both the algorithms developed earlier reduce the prioritized bicriterion problems to singe-objective
problems. In one case, the prioritized bicriterion problems are reduced to single-objective problems through the introduction of
preemptive priority factors before applying the branch and bound algorithm. In the other case, the prioritized bicriterion problems
are solved through a branch and bound algorithm focusing on the first priority objective, keeping the second priority objective in
abeyance and finding all the alternative optimal solutions. Among all the alternative optimal solutions, one is picked up for a
solution of the prioritized bicriterion problem for which the second priority objective is minimum. The process of reducing the
prioritized bicriterion problems to single-objective ones through the introduction of the preemptive priority factors or finding all
the alternative optimal solutions focusing on the first priority objective, keeping the second priority objective in abeyance and then
picking up the one among them whose second priority objective is minimum, which involves a lot of computational work. The
new algorithm reduces the computational work by treating the sequence of prioritized bicriterion problems, whose solutions yield
Pareto optimal solutions to the cost-time trade-off bulk transportation problem, as such and invoking the concept of lexicographic
minimum. The concept of lexicographic minimum is introduced in brief as follows. Among a set of ordered pairs, the pair that is
lexicographically minimum whose either first component is minimum or second component is minimum in case the first
components are equal in all the pairs. This implies that the ordered pair 큠䗠 ,
is lexicographically minimum among the set of all
ordered pairs 큠䗠 ,
, … , 큠䗠 ,
) if (i) 䗠
for all k=1, ..., n except k= r or (ii) 䗠
䗠 for all
1, . . . , but
for all k=1, ..., n except k= r.
A solution to a multi-objective optimization problem is Pareto optimal if there exists no solution to the multi-objective problem
that is superior to it with respect to at least one objective function but not inferior to it with respect to any objective function. The
terms efficient solution and nondominated solution are also used in the literature for Pareto optimal solution. Specifically, for the
cost-time trade-off bulk transportation problem with the total cost C and duration T of bulk transportation as the two minimizing
objective functions, a solution
y is Pareto optimal if there exists no solution x to the problem satisfying the conditions.
(i) C( y ) ³ C( x ) and (ii) T( y ) ³ T ( x ) with the inequality sign strictly holding in at least one of the conditions out of (i) and
(ii). Here C( x ), C(
respectively. Here
x= (
y ) refer to the total costs and T( x ), T( y ) refer to the durations of bulk transportation for the solutions x , y
: i=1,…,m; j=1,…,n) and y = ( 䋘 : i=1,…,m; j=1,…,n). The variables 䋘 ’s and 䋘 ’s and other notations are
explained hereinafter. A detailed discussion about Pareto optimal solution can be found in the works of Ignizio (1982), Steuer
(1986), Prakash, Agarwal and Shah (1988), Prakash, Balaji.and Tuteja (1999), Prakash and Gupta (2006), Prakash, Madhusudan
and Kunal (2007), Prakash, Kumar, Prasad and Gupta (2008), Prakash, Sharma and Singh (2009), Prakash, Tuli, Pant, Singh,
Agrawal and Gupta (2011), Prakash, Agrawal, Gupta, Garg, Jain, Sharma and Jamwal (2013).
䋘
II FORMULATION OF THE PROBLEM
It is supposed that there are m sources and n destinations. Given amounts of a commodity are available at the sources and
specified amounts of the commodity are required at the destinations. The entire requirement of each destination is to be met by one
source only; but a source can supply to any number of destinations subject to the availability of the commodity at it. The
transportation starts simultaneously. Let 䗠䋘 (i = 1,…,m) be the units of the commodity available at source i, (j = 1,…,n) the units
of the commodity required at destination j, 䋘 (i = 1,…,m; j = 1,…,n) the units of cost of bulk transportation of the commodity
from source i to destination j, 䋘 (i = 1,…,m; j = 1,…,n) the units of time of bulk transportation of the commodity from source i to
destination j, and 䋘 (i = 1,…,m; j = 1,…,n) the variable assuming the value 0 or 1 according as the entire requirement of
destination j is not met or met from source i. All the parameters 䗠䋘 ′s,
, 䋘 ′ 䗠 d 䋘 ′ are free to take any nonnegative real
values. Let C and T denote the total cost and duration of bulk transportation, respectively. The mathematical formulation of the
problem is as follows. Determine 䋘 ′ which minimize the two objective functions
- 14 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
m
n
C = åå cij xij
(1)
T = max{t ij xij : i = 1,..., m; j = 1,..., n}
(2)
i =1 j =1
without according priorities to them subject to the constraints
n
åb x
j =1
j
ij
m
åx
i =1
ij
£ a i (i = 1,..., m )
(3)
= 1 ( j = 1,..., n)
(4)
xij = 0 or 1 (i = 1,…,m; j= 1,…,n).
(5)
It is required to find the set of Pareto optimal solutions to the problem provided by Eqs. (1) – (5). For the purpose of listing the
(1)
Pareto optimal solutions of the formulated cost-time trade-off bulk transportation problem, we call a Pareto optimal solution x
the 1st Pareto optimal solution, if it is the optimal solution to the formulated problem with the minimization of C as the first priority
objective and that of T as the second priority objective. We call a Pareto optimal solution
there exists no Pareto optimal solution
(1)
x
(2)
the 2nd Pareto optimal solution, if
(1)
y of the formulated problem satisfying the conditions (i) C( x ) < C( y ) < C( x
( 2)
)
(2)
and (ii) T( x ) > T( y ) > T( x ) . Proceeding as we did to define the 2nd Pareto optimal solution, we defined the 3rd and
subsequent Pareto optimal solutions.
III SOLUTION PROCEDURE
The cost-time trade-off bulk transportation problem formulated above is an integer nonlinear problem. This is because the
objective function provided by Eq. (2) is nonlinear and the decision variables 䋘 ′ assume the integer value 0 or 1. The two
objectives are not prioritized. The set of Pareto optimal solutions to this problem is obtained by solving a sequence of prioritized
bicriterion bulk transportation problems. The total number of prioritized bicriterion bulk transportation problems to be solved for
obtaining the set of Pareto optimal solutions is only one more than the total number of the Pareto optimal solutions of the problem.
The two algorithms developed earlier, which reduce the sequence of prioritized bicriterion bulk transportation problems to singleobjective problems either through the introduction of preemptive priority factors or finding all the alternative optimal solutions
focusing on the first priority objective, keeping the second priority objective in abeyance and then picking up the one among them
whose second priority objective is minimum, involve a lot of computational work. In contrast to this, the new algorithm reduces the
computational work considerably by treating the sequence of prioritized bicriterion bulk transportation problems as such and
invoking the concept of lexicographic minimum. Procedures for obtaining the 1st, 2nd and subsequent Pareto optimal solutions are
outlined below.
A. Procedure for obtaining 1st Pareto optimal solution
Note that the 1st Pareto optimal solution of the cost-time trade-off bulk transportation problem is the optimal solution to the
problem with the minimization of C provided by Eq. (1) as the first priority objective and that of T provided by Eq. (2) as the
second priority objective subject to the constraints (3)-(5). This prioritized problem is designated as the 1st prioritized bicriterion
(1)
bulk transportation problem. Its optimal solution would yield the 1st Pareto optimal solution x
to the cost-time trade-off bulk
transportation problem. To explain the procedure to solve the 1st prioritized bicriterion bulk transportation problem through the
new algorithm, its tableau representation is needed and is shown in Table 1. In this Table, cells (i, j)’s correspond to the
variables x ′s
(i = 1,..,m; j = 1,...,n). The first and second components of an ordered pair (c ,tj )j in the cell (i, j) depict the units
j
of cost and time of bulk transportation from source S to destination Dj respectively. The marginal row and column entries bj
(j=1,2,...,n) and a (i=1,2,...,m) depict the units of the commodity required at the destinations and available at the sources
respectively. As the entire requirement of each destination Dj is to be met by one source only, all the variables x ′s
corresponding
j
to the cells in the column corresponding to the destination Dj except one would be zero. So a set consisting of one x j at level 1
- 15 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
(1)
corresponding to a cell (i, j) in each of the columns corresponding to the destinations would provide an optimal solution x to the
1st prioritized bicriterion bulk transportation problem provided that it satisfies the constraints (3)–(5) with the first and second
priorities to the minimization of C (x ) and T ( x ) respectively. This set would contain the number n of the variables x ′s
at
j
level 1, each with a different value of j €{1,...,n} and the remaining ones at level 0.
TABLE 1─ 1
ST
PRIORITIZED BULK TRANSPORTATION PROBLEM
Destinations
Sources
S1
S2
.
.
.
Sm
Requirement bj
D1
(c11, t11)
(c21, t21)
.
.
.
(cm1, cm1)
b1
D2
(c12, t12)
(c22, t22)
.
.
.
(cm2, tm2)
b2
…
…
…
…
…
…
…
…
Dn
(c1n, c1n)
(c2n, t2n)
.
.
.
(cmn, tmn)
bn
Availability
ai
a1
a2
.
.
.
am
Now a branch and bound algorithm is evolved for solving the 1st prioritized bicriterion bulk transportation problem using the
concept of lexicographic minimum. Different branch and bound methods are developed by choosing in different ways (i) the lower
bound of the objective function at a node, (ii) the node at each stage for branching the set of solutions, and (iii) the variable with
respect to which branching at the chosen node to be done.
A discussion about different branch and bound algorithms can be found in the works of Rao (1996), Rardin (1998), Wagner
(2001), Taha (2002), Kasana and Kumar (2003), Pant (2004), Bronson and Naadimuthu (2004), Hillier and Lieberman (2006),
Natarajan, Balasubramani and Tamilarasi (2006), and Sharma (2007). The evolved branch and bound algorithm for solving the 1st
prioritized bicriterion bulk transportation problem is a modification and extension of the branch and bound algorithm used for
solving the single-objective binary integer problems. Unlike one lower bound for the objective function of the single objective
-
-
problem, two lower bounds ─ one for the total cost C ( x) and another for the duration T ( x ) of bulk transportation ─ are provided
at each node in the tree. The various steps involved in solving the 1st prioritized bicriterion bulk transportation problem are as
follows:
1)
Lower bounds of the objective functions at the nodes.
-
-
Lower bounds LBC and LBT of the objective functions C ( x) and T ( x ) denoting total cost and duration of bulk
transportation at the topmost node S are the lower bounds of the objective functions of the set of all solutions to the problem. For
obtaining lower bounds LBC and LBT at the topmost node, we update the table of the topmost node S. By updating the table of
the topmost node S, we mean that the rows corresponding to the sources at which the commodity available is less than the
requirements of all the destinations are deleted, because these sources cannot meet the requirements of any of the destinations.
Also, each of the cells (i,j)’s corresponding to which the updated commodity available at the source Si is less than that required at
the destination is deleted; because the source corresponding to the row containing the cell cannot meet the requirement of the
destination corresponding to the column containing the cell. The Table thus obtained is the updated table of the topmost node S.
After this, ordered pairs with the lexicographically minimum values among all the ordered pairs in each of the columns
corresponding to the destinations are chosen. If one or more than one columns corresponding to a destination are missing from the
updated table of the topmost node S on account of deletion of all the cells therein, then the 1st prioritized bicriterion bulk
transportation problem has no solution, because then the requirements of the source corresponding to the missing column cannot be
met. However, if this does not happen, the lower bound LBC at the topmost node S is the sum of the first components of the
ordered pairs with the lexicographically minimum values among all the ordered pairs in each of the columns of its updated table. In
addition, the lower bound LBT at the topmost node S is obtained by setting it equal to 0. The computation of the lower bounds
LBC and LBT at the topmost node S in this way results in simplifying the computation LBC and LBT at the succeeding nodes.
Lower bounds at any node other than the topmost node S are obtained as follows. First, we update the table of the node at
which the lower bounds of the objective functions are to be obtained. By updating the table of the node, we here mean that all the
columns corresponding to the destinations associated with the variables 䋘
at level 1 along the branch from the topmost node S
to it, whose entire requirements have been met, are deleted from the table. Also, the cells associated with the variables 䋘
at
level 0 along the branch from the topmost node S to it are deleted from the Table. Thereafter, the amounts of the commodity that
have been used to meet the entire requirements of the destinations from the sources are subtracted from the amounts available at
- 16 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
the sources, resulting into updating the commodity available at the sources. Having deleted the columns associated with the
destinations whose entire requirements have been met and the cells corresponding to the 䋘
at level 0 and updating the
commodity available at the sources, rows corresponding to the sources at which the commodity available is less than the
requirements of all the destinations are deleted. Also, each of the cells (i,j)’s, corresponding to which the updated commodity
available at the source Si is less than that required at the destination , is deleted; because the source corresponding to the row
containing the cell cannot meet the requirement of the destination corresponding to the column containing the cell. The table thus
obtained is the updated table of the node. After this, ordered pairs with the lexicographically minimum values among all the
ordered pairs in each of the columns associated with the destinations are chosen. The lower bound LBC at the node other than the
topmost node S is the sum of the first components of the ordered pairs corresponding to the 䋘
along the branch from the
topmost node S to it and the first components of the ordered pairs with the lexicographically minimum values among all the
ordered pairs in each of the columns of its updated table. In addition, the lower bound LBT at the node is the maximum value
among the values of the second components of the ordered pairs corresponding to the 䋘
along the branch from the topmost
node S to it.
2)
The node at each stage for branching the set of solutions.
The node chosen at each stage for branching the set of solutions is the node containing the ordered pair with the
lexicographically minimum value among all the ordered pairs at the terminal unfathomed nodes. The first and second components
-
-
C ( x) and the lower bound LBT of T ( x ) respectively. In the case of
of the ordered pairs at the nodes are the lower bound LBC of
a tie, we branch at the node that contains more
3)
䋘
,
s at level 1 along the branch from the topmost node S to it.
The variable with respect to which branching at the chosen node to be done.
The variable 䋘 with respect to which branching at the chosen node is to be done is determined as follows. To do this, we
revisit the updated table of the chosen node. In the updated table of the chosen node, we enter penalties in the cells below the
ordered pairs with the lexicographically minimum values among all the ordered pairs in each of the columns corresponding to the
destinations. The penalty in a cell of a column corresponding to a destination is obtained by subtracting the ordered pair with the
lexicographically minimum value among all the ordered pairs in the column from an ordered pair with the next lexicographically
minimum value in the column. However, if more than one cell in a column has ordered pairs with the same lexicographically
minimum value, the penalty is the ordered pair (0,0). Furthermore, if there is only one cell having an ordered pair with the
lexicographically minimum value in a column and all the other cells in that column have been deleted, then the penalty is the
ordered pair (∞,∞) indicating that the requirement of the destination corresponding to the column containing the cell can be met
only by the source corresponding to the row containing the cell, because all other cells in the column have been deleted. The
penalty in a cell indicates the minimum possible loss to be incurred if the requirement of the destination containing the cell is not
met by the source corresponding to the row containing the cell. Branching is done at the chosen node with respect to the variable
䋘 corresponding to the cell (i,j) containing the greatest penalty among all the cells containing the ordered pairs with the
lexicographically minimum values among all the ordered pairs in each of the columns corresponding to the destinations in the
updated table of the chosen node.
4)
Termination of the procedure.
We shall terminate the procedure in either of the following two situations. The first situation wherein we will terminate the
procedure is that we have arrived at a terminal unfathomed node containing the number n of the variables 䋘 , s at level 1, each with
a different value of subscript j €{1,…,n}along the branch from the topmost node S to it with the lexicographically minimum lower
-
-
C ( x) and LBT of T ( x ) among all the terminal unfathomed nodes in the tree. That node has lexicographically
bounds LBC of
minimum lower bounds among the several nodes each having lower bounds LBC and LBT wherein the value of LBC is either least
or the value of LBT is least in the case of LBC having tied values. The unfathomed terminal node provides the optimal solution
̅ 큠 to the 1st prioritized bicriterion bulk transportation problem. ̅ 큠 consists of the number n of the variables 䋘 , at level 1,
each with a different value of j €{1,…,n} and the remaining 䋘 , s at level 0. Since the optimal solution to the 1st bicriterion bulk
transportation problem is the 1st Pareto optimal solution to the formulated problem, ̅ 큠 is the 1st Pareto optimal solution to the
formulated problem. The second situation wherein we terminate the procedure is that we find that all the terminal nodes in the tree
are fathomed without having arrived at a terminal unfathomed node containing the number n of 䋘 , at level 1 each with a
different value of j €{1,…,n} along the branch from the topmost node S to it, indicating that no solution to the 1st prioritized
bicriterion bulk transportation problem exists.
- 17 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
B. Procedure for obtaining 2nd and subsequent Pareto optimal solutions
(1 )
After having obtained the 1st Pareto optimal solution x
to the cost-time trade-off bulk transportation problem, its 2nd Pareto
optimal solution is obtained. For this purpose, we obtain the 2nd prioritized bicriterion bulk transportation problem from the 1st
prioritized bicriterion bulk transportation problem by deleting all the variables
䋘
,
s corresponding to the
䋘
,
³ T( x
(1)
) therein,
seeking to determine 䋘 , that minimize the objective functions C (x} and T( x ) provided by Eqs. (1) and (2) respectively with
the priorities in the order of their occurrence subject to the constraints (3)–(5). The problem thus obtained is designated as the 2nd
prioritized bicriterion bulk transportation problem. The optimal solution to this problem is obtained exactly in the same way as
(2)
done for the 1st prioritized bicriterion bulk transportation problem. This optimal solution yields the 2nd Pareto optimal solution x
. Now we explain why the optimal solution to the 2nd prioritized bicriterion bulk transportation problem would yield the 2nd Pareto
(2)
(1)
optimal solution x . The reason is this that all the 䋘 , corresponding to the 䋘 , ³ T( x
bicriterion bulk transportation problem. Thus, the optimal solution to this problem will have no
the tij’s ³ T( x
(1 )
). This will result in yielding the 2nd Pareto optimal solution
x
(2)
) are 0 in the 2nd prioritized
,
at level 1 corresponding to
䋘
for which the duration T( x
(2)
) of bulk
(1)
transportation will be less than T( x ). For obtaining the 3rd Pareto optimal solution, we obtain the 3rd prioritized bicriterion bulk
transportation problem from the 2nd prioritized bicriterion bulk transportation problem by deleing all the variables 䋘 ,
(2)
corresponding to the tij’s ³ T( x ) therein. The optimal solution to the 3rd prioritized bicriterion bulk transportation problem is
obtained exactly in the same way as done for the 1st prioritized bicriterion bulk transportation problem. This optimal solution
(3)
(3)
(2)
would yield the 3rd Pareto optimal solution x for which the duration T ( x ) of bulk transportation will be less than T ( x ) .
Subsequent Pareto optimal solutions are obtained by proceeding exactly in the same way as done for obtaining the Pareto optimal
(2)
(3)
solutions x
and x . This process of obtaining Pareto optimal solutions is terminated after encountering a prioritized
bicriterion bulk transportation problem whose tree contains all terminal nodes fathomed without having arrived at a node
containing the number n of the variables 䋘 , at level 1, each with a different value of j €{1,…,n}, along a branch from the
topmost node S to it, indicating that it is no longer possible to find a new Pareto optimal solution with lesser duration of bulk
transportation. Thus, the total number of the prioritized bicriterion bulk transportation problems to be solved for obtaining the set
of Pareto solutions is only one more than that of the Pareto optimal solutions to the problem.
IV NUMERICAL EXAMPLE
Now we illustrate the new algorithm explained above by obtaining the set of Pareto optimal solutions to a numerical problem
obtained by taking m = 4, n = 5 and assigning numerical values to all the other parameters in the problem formulated in Section 2.
The data in the numerical problem are fictitious. Nevertheless the problem has some bearing to real-life because the problems
similar to the one considered here are encountered in life. The tableau representation of the numerical problem is shown in Table 2.
In this Table, cells (i, j)’s correspond to the variables 䋘 , (i = 1,2,3,4; j = 1,2,3,4,5). The first and second entries of an ordered
pair inside a cell (i, j) depict the units of cost and time of bulk transportation from source 䋘 to destination respectively. The
marginal row and column depict the units of the commodity required at the destinations and available at the sources respectively.
TABLE 2:COST AND TIME OF BULK TRANSPORTATION TOGETHER WITH AVAILABILITIES AND REQUIREMENTS FOR NUMERICAL PROBLEM
Sources
S1
S2
S3
S4
Requirement bj
D1
(2,4)
D2
(3,4)
(4,4)
(1,7)
(1,8)
Destinations
D3
(3,10)
D4
(7,8)
D5
(1,7)
(1,12)
(2,14)
(8,8)
(7,2)
(11,4)
(1,4)
(5,4)
(20,4)
(30,6)
(10,7)
(2,2)
(5,2)
3
3
2
2
1
The numerical problem seeks to determine
䋘
,
which minimize the two objective functions
- 18 -
Availability
ai
5
4
3
2
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
ì2x11 + 3x12 + 3x13 + 7x14 + x15
ü
ï
ï
ï+ 4x 21 + x 22 + x 23 + 2x 24 + 8x 25
ï
C=í
ý
ï+ x 31 + 7x 32 + 11x 33 + x 34 + 5x 35
ï
ï+ 20x + 30x + 10x + 2x + 5x ï
41
42
43
44
45 þ
î
(6)
T = max{t ij xij : i = 1,2,3,4; j = 1,2,3,4,5}
(7)
without according priorities to them subject to the constraints (3)–(5) after assigning numerical values to all the parameters therein.
A. Solution to the numerical problem through the new algorithm
We apply the new algorithm outlined in Section 3 for obtaining the set of Pareto optimal solutions to the numerical problem. To
do this, we first obtain prioritized bicriterion bulk transportation problems of the numerical problem and then apply the branch and
bound algorithm evolved in Section 3 for obtaining their optimal solutions. We also explain in brief how the prioritized bicriterion
bulk transportation problems are obtained and how their optimal solutions are obtained. Optimal solutions to the prioritized
bicriterion bulk transportation problems yield Pareto optimal solutions of the numerical problem.
1)
First Pareto optimal solution to the numerical problem
The 1st Pareto optimal solution to the numerical problem is the optimal solution to the 1st prioritized bicriterion bulk
transportation problem, wherein the total cost C ( x ) provided by Eq. (6) and the duration T ( x ) provided by Eq. (7) of bulk
transportation are minimized with the first and second priorities respectively subject to the constraints (3)-(5) after assigning
numerical values to all the parameters therein. We apply the branch and bound algorithm evolved in Subsection 3.1 for solving the
1st prioritized bicriterion bulk transportation problem and draw a tree as shown in Fig. 1. In this Figure, 1st and 2nd entries in the
ordered pairs at the nodes containing LBC and LBT refer to the lower bounds of the total cost C ( x ) and the duration T ( x ) of
bulk transportation respectively. The topmost node S in the tree represents the set of all solutions to the 1st prioritized bicriterion
-
-
bulk transportation problem with the first and second priorities to the minimization of C ( x) and T ( x ) respectively. Any other
node in the tree represents the subset of the set of all solutions of the 1st prioritized bicriterion bulk transportation problem
containing the variables 䋘 , at level 1 or 0 along the branch from the topmost node S to it. The unfathomed terminal node with the
lexicographically minimum value ordered pair containing the entries of LBC and LBT among all the ordered pairs at the
unfathomed terminal nodes containing the number 5 of 䋘 , at level 1 along the branch from the topmost node S to it, yields the
optimal solution to the 1st prioritized bicriterion bulk transportation problem. Since the optimal solution to the 1st prioritized
bicriterion bulk transportation problem is the 1st Pareto optimal solution to the numerical problem, this solution would be the 1st
Pareto optimal solution to the numerical problem. The variables
, +,
, + ,
at level 1 along the branch from the
topmost node 8 to it provide the optimal solution
x
(1)
to the 1st prioritized bicriterion bulk transportation problem with the total
__ (1)
__ (1)
cost and duration of bulk transportation C ( x ) =1+3+1+1+2 =8 and T ( x
optimal solution yields the 1st Pareto optimal solution to the numerical problem.
- 19 -
) = max{7,10,7,8,2}=10 units respectively. This
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
Node S
LBC=5, LBT=0
1
=0
Node 1
(LBC =9, LBT = 0)
+=
Node 2
(LBC =5, LBT = 7)
0
+=
Node 3
(LBC =7, LBT = 7)
+=0
+=1
Node 5
(LBC =14, LBT = 7)
Node 6
(LBC= 7, LBT= 10)
=0
=1
Node 7
(LBC=13, LBT= 10)
Node 8
(LBC=7, LBT= 10)
+
1
Node 4
(LBC =7, LBT = 12)
=1
=0
Node 11
(LBC=11, LBT=12)
+
=0
+
Node 9
Fathomed
+
=1
Node 14
(LBC = 8, LBT=12)
Node 13
Fathomed
=0
Node 12
(LBC = 7, LBT =12)
=1
Node 10
(LBC = 8, LBT=10)
=0
=1
Node 15
(LBC= 13, LBT=10)
Node 16
(LBC = 8, LBT=10)
Fig.1: Tree showing the optimal solution to 1st prioritized bicriterion bulk transportation problem and yielding 1st Pareto optimal solution to the numerical problem.
B. Remaining Pareto optimal solutions to the numerical problem
The remaining Pareto optimal solutions to the numerical problem are obtained following the procedure outlined in Subsection
3.2. For obtaining the 2nd Pareto optimal solution, we obtain the 2nd prioritized bicriterion bulk transportation problem from the 1st
prioritized bicriterion bulk transportation problem of the numerical problem by deleting all the variables 䋘 ′ corresponding to the
(1)
′ ³ T( x ) = 10 therein. A tree is drawn for providing the solution to this problem in the same way as done for providing the
solution of the 1st prioritized bicriterion bulk transportation problem and is shown in Figure 2. Since the unfathomed terminal node
䋘
- 20 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
10 in the tree contains 5 variables
,
, + ,
, and + at level 1 along the branch from the topmost node S to it with the
lexicographically minimum value ordered pair containing the values of LBC=15 and LBT=7 at it among all the ordered pairs of
all the unfathomed terminal nodes, the node 10 yields the optimal solution to the 2nd prioritized bicriterion bulk transportation
problem. As the optimal solution to the 2nd prioritized bicriterion bulk transportation problem is the 2nd Pareto optimal solution
x
(2)
to the numerical problem, the variables
,
,
+
,
, and
+ provide
the 2nd Pareto optimal solution x
(2)
(2)
to the
(2)
numerical problem with the total cost and duration of bulk transportation C( x
)=1+1+1+2+10=15 and T( x
)=max{7,7,4,4,7}=7 units, respectively. For obtaining the 3rd Pareto optimal solution, we obtain the 3rd prioritized bulk
transportation bulk transportation problem from the 2nd prioritized bicriterion bulk transportation problem of the numerical problem
(2)
by deleting all the variables 䋘 ′ corresponding to the 䋘 ′ ³ T( x )=7 therein. A tree is drawn for providing solution to this
problem in the same way as done for providing the solution to the 1st prioritized bicriterion bulk transportation problem and is
shown in Figure 3. The unfathomed terminal node 10 yields the 3rd Pareto optimal solution x
variables
++
,
,
,
, and
+
provide the 3rd Pareto optimal solution x
(3)
(3)
to the numerical problem. The
(3)
of the numerical problem with the total cost
(3)
and duration of bulk transportation C( x )=11+3+4+2+5=25 and T( x ) = max{4,4,4,2,4}=4 units respectively. For obtaining
the further Pareto optimal solution, we obtain the 4th prioritized bicriterion bulk transportation problem of the numerical problem
from the 3rd prioritized bicriterion bulk transportation problem of the numerical problem by deleting all the variables
䋘 ′
(3)
corresponding to the 䋘 ′ ³ T( x ) =4 therein. A tree is drawn for providing the optimal solution to this problem and is shown
in Figure 4. Since the topmost node S in the tree in this figure is fathomed, it is not possible to have an unfathomed terminal node
in the tree containing 5 variables 䋘 ′ at level 1, each with a different value of j €{1,…,n}, along the branch from the
topmost node S to it. This indicates that there exists no Pareto optimal solution to the numerical problem for which the duration of
bulk transportation is less than 4 units and that the process of obtaining further Pareto optimal solutions should be terminated.
The numerical problem is found to have only three Pareto optimal solutions. The set of Pareto optimal solutions of the
numerical problem together with the variables 䋘 ′ at level 1, total cost and duration of bulk transportation are shown in Table 3.
V CONCLUSION
In this paper, we briefly explain the need for revisiting the cost-time trade-off bulk transportation problem after having already
developed two algorithms for obtaining the set of Pareto optimal solutions to it. The two previously developed algorithms
successfully obtain the set of Pareto optimal solutions to the cost-time trade-off bulk transportation problem by solving a sequence
of prioritized bicriterion bulk transportation problems by reducing them to single-objective bulk transportation problems through
the introduction of the preemptive priority factors or finding all the alternative optimal solutions focusing on the first priority
objective while keeping the second priority objective in abeyance and picking up that alternative optimal solution among them
whose second priority objective is minimum. This involves a lot of computational work. The new algorithm solves the prioritized
bicriterion bulk transportation problems as such without reducing them to single-objective bulk transportation problems by
invoking the concept of lexicographic minimum. This way, the new algorithm reduces the computational work substantially. A
look at the two algorithms developed earlier with their application in solving a numerical problem and the new algorithm
developed in this paper with its application in solving the same numerical problem brings out and confirms this clearly.
The present work could be useful to a decision maker who is not able to assign priorities to his/her objectives in the context of
the cost-time trade-off bulk transportation problem with the objectives to minimize the total cost and duration of bulk
transportation without according priorities to them when the entire requirement of each destination is to be met from one source
only. In such a situation, help can be provided to the decision maker by presenting him/her with a set of Pareto optimal solutions.
He/she can pickup that solution out of this set that suits him/her most with regard to his/her objective and liking. The only input
data needed to use the model are the amounts of commodity available at the sources and required at the destinations, costs and
times of bulk transportation from each source to each destination.
The model discussed in this work is versatile. The numerical problem of a small size is given in the paper to illustrate the
solution procedure but the model is capable of tackling large size problems after computerization.
- 21 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
Node S
(LBC=14, LBT=0)
x15=0 0
x15=1
Node 1
(LBC=18, LBT=0)
Node 2
(LBC=14, LBT=7)
x22= 0
x22= 1
Node 3
(LBC=16, LBT=7)
Node 4
(LBC=14, LBT=7)
x34=0
x34=1
Node 5
(LBC=15, LBT=7)
Node 6
(LBC=15, LBT=7)
x11=1
x11=0
Node 8
(LBC=15, LBT=7)
Node 7
Fathomed
x43=0
x43=1
Node 9
Fathomed
Node 10
(LBC=15, LBT=7)
Fig. 2. Tree showing the optimal solution to 2nd prioritized bicriterion bulk transportation problem and yielding
2nd Pareto optimal solution to the numerical problem.
- 22 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
Node S
(LBC=22, LBT=0)
x33=0
x33= 1
Node 1
Fathomed
Node 2
(LBC=23, LBT=4)
x12=0
x12=1
Node 3
Fathomed
Node 4
(LBC=25,
LBT4)
x21=1
x21=0
Node 5
Fathomed
Node 6
(LBC=25, LBT=4)
x44=0
x44=1
Node 8
(LBC=25, LBT=4)
Node 7
Fathomed
x35=0
x35=1
Node 9
Fathomed
Node 10
(LBC=25, LBT=4)
Fig. 3. Tree showing the optimal solution to 3rd prioritized bicriterion bulk transportation problem and yielding 3rd Pareto optimal solution to the numerical
problem.
Node S
Fathomed
Node S
Fathomed
Fig. 4. Tree showing no solution to 4th prioritized bicriterion bulk transportation problem and indicating that numerical problem has no further Pareto optimal
solutions.
- 23 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
TABLE 3. SET OF PARETO OPTIMAL SOLUTIONS TO THE NUMERICAL PROBLEM
Pareto optimal
solution
x
x
x
Variables at level 1
(1 )
(2)
(3)
,
+,
,
,
++ ,
,
,
+
+
Total cost of bulk transportation
,
,
,
,
,
+,
+
C(
x
C(
x
C(
x
Duration of bulk transportation
(1 )
) =1+3+1+1+2+= 8
(2)
T(
x
(1 )
) =1+1+1+2+10=15
T( x
) =11+3+4+2+5=25
T( x
(3)
)=max{7,10,7, 8, 2}=10
(2)
) =max{7,7,4,4,7}=7
(3)
) = max{4,4,4,2,4}=4
ACKNOWLEDGEMENTS
The authors profusely thank Honourable Dr. Ashok Kumar Chauhan, President, Prof. B.P.Singh, Senior Director and Prof. Dr.
Rekha Agarwal., Director, Amity School of Engineering and Technology, New Delhi for providing facilities to carry out this work.
The authors also express thanks to the Reviewer for reviewing the paper.
REFERENCES
[1] Bronson, R., Naadimuthu, G., 2004. Schaum’s Outline of Theory and Problems of Operations Research, 2nd Edition. Tata Mc-Graw Hill
Publishing Company, New Delhi, pp. 24-132.
[2] Hillier, F.S., Lieberman, G.J., 2006. Introduction to Operations Research, 8th Edition. Tata Mc-Graw Hill Publishing Company, New Delhi,
pp. 515-521.
[3] Ignizio, J.P., 1982. Linear Programming in Single- and Multiple-Objective Systems. Prentice Hall, Englewood Cliffs, NJ, pp. 374- 376,
475-483.
[4] Kasana, H.S., Kumar, K.D., 2003. Introductry Operations Research: Theory and Applications. Springer Verlag, Berlin, pp. 326-344.
[5] Maio, A.D., Roveda, C., 1971. An all zero-one algorithm for a certain class of transportation problems. Operations Research 19, 1406-1418.
[6] Murthy, M.S., 1976. A bulk transportation problem. Opsearch 13, 143-155.
[7] Natarajan, A.M., Balasubramani, P., Tamilarasi, A., 2006, Operations Research. Pearson Education (Singapore) Pte. Ltd., Indian branch,
New Delhi, pp. 151-155.
[8] Pant, J. C., 2004 Introduction to Optimization: Operation research, 6th Edition, Jain Brothers, New Delhi, pp. 213-237.
[9] Prakash, S., Agarwal, A.K., Shah, S., 1988. Nondominated solutions of cost-time trade-off
transportation and
assignment problems. Opsearch 25, 126-131.
[10] Prakash, S., Agrawal, A.K., Gupta, A., Garg, S., Jain, S., Sharma, S., Jamwal, S.S., 2013, A cost -time trade-off Köingsberg bridge problem
traversing all the seven bridges allowing repetition, OPSEARCH, DOI 10.1007/s12597-013-0143-4.
[11] Prakash, S., Balaji, B.V., Tuteja, D., 1999. Optimizing dead mileage in urban bus routes through a nondominated solution approach.
European Journal of Operational Research, 114, pp. 465-473.
[12] Prakash, S., Gupta, A., 2006. Nondominated solution of a site selection problem with two objectives without assigning priorities to them.
New Bull. Calcutta Math. Soc. 29 (1-3), 3-8.
[13] Prakash, S., Kumar, P., Prasad, B.V.N.S., Gupta, A., 2008. Pareto optimal Solutions of a cost-time trade-off bulk transportation problem,
Eur. J. Oper. Res. 188, 85-100.
[14] Prakash, S., Madhusudan, A., Kunal, 2007. Clustering offices of a company to internet service providers. Proc. Nat.Acad. Sci. India, Section
A, 77 (1), 31-35.
[15] Prakash, S., Ram, P.P., 1995. A bulk transportation problem with objectives to minimize total cost and duration of transportation. The
Mathematics Student. 64, 206-214.
[16] Prakash, S., Sharma, M.K., Singh, A., 2009. Selection of warehouse sites for clustering ration shops to them with two objectives through a
heuristic algorithm incorporating tabu search. OPSEARCH 46 (4), 449-460.
[17] Prakash, S., Tuli, R., Pant, P., Singh, L., Agrawal, A.K., Gupta, A., 2011. A bicriterion call center problem through an evolved branch and
bound algorithm. OPSEARCH 48 (4), 371-386.
[18] Rao, S.S.,1996. Engineering Optimization. New Age International Private Limited, New Delhi, pp. 690-697.
[19] Rardin, R.L., 1998. Optimization in Operation Research. Pearson Education (Singapore) Pte.. Ltd., Indian Branch, New Delhi, pp. 594-597,
651-676.
[20] Sharma, J.K., 2007. Operations Research, 3rd Edition. Macmillan India Ltd. New Delhi, pp. 275-281.
[21] Srinivasan, V., Thompson, G.L., 1973. An algorithm for assigning uses to sources in a special class of transportation problems. Operations
Research 21, 284-295.
- 24 -
Journal of Data and Information Processing
May 2014, Vol. 2 Iss. 2, PP. 13-25
[22] Steuer, R.E., 1986. Multiple Criteria Optimization: Theory, Computation, and Application. John Wiley and Sons, Inc., New York, pp. 138159.
[23] Taha, H.A., 2002. Operation Research: An Introduction. Pearson Education (Singapore) Pte. Ltd., Indian Branch, New Delhi, pp. 373-378.
[24] Wagner, H.M., 2001, Principles of Operations Research, 2nd Edition. Prentice Hall of India Pvt.. Ltd. New Delhi, pp. 484-490.
- 25 -