American Journal of Oper ations Research, 2011, 1, 100-117
doi:10.4236/ajor.2011.13013 Published Online September 2011 (http://www.SciRP.org/journal/ajor)
Copyright © 2011 SciRes. AJOR
Vertical Decomposition Approach to Solve
Single Stage Capacitated Warehouse Location
Problem (SSCWLP)
Priyanka Verma, Renduchintala Raghavendra Kumar Sharma
Department of Industrial and Management Engineering, Indian Institute of Technology,
Kanpur, India
E-mail: rrks@iitk.ac.in
Received May 4, 2011; revised May 25, 2011; accepted June 20, 2011
Abstract
Single Stage Capacitated Warehouse Location Problem (SSCWLP) has been attempted by few researchers in
the past. These are Geoffrion and Graves [1 ], Sharma [2 ], Sharma [3 ] and Sharma and Berry [4]. In this pa-
per we give a “vertical decomposition” approach to solve SSCWLP that uses Lagrangian relaxation. This
way SSCWLP is broken into two versions of capacitated plant location problem (the CPLP_L and CPLP_R)
by relaxing the flow balance constraints. For CPLP_R, we use well known Lagrangian relaxations given in
literature (Christofides and Beasley [5] and Nauss [6]); and adopt them suitably for solving CPLP_L. We
show theoretically in this paper that SSCWLP can be more efficiently solved by techniques of vertical de-
composition developed in this paper than the method available in literature (Sharma and Berry [4]). Encour-
aging computational study is reported in this paper.
Keywords: Single Stage Capacitated Warehouse Location Problem, Linear Programming Relaxation,
Lagrangian Relaxation, Vertical Decomposition
1. Introduction and Literature Review
The problem considered in this work is referred to as
the single stage capacitated warehouse location problem
(SSCWLP), which arises when the distances between
plants and markets are large and it becomes necessary to
route the supplies through warehouses (of limited ca-
pacities). A set of points are given where warehouses of
limited capacities can be located. Each of these points
has a known fixed cost associated with them. We are to
choose a sufficient number of points where warehouses
can be located so that sum tot al of location – distribution
co st i s minimized.
The study on the SSCWLP is motivated by its applica-
tion in many fields, especially in supply chains in which
hierarchical structure exists. Melo et al. [7] has reviewed
different variants of multistage facility location problem
and classified the types of problems attempted in this field
on the basis of number of layers, number of location lay-
ers, number of commodities, nature of the planning ho-
rizon (single/multi period) and the type of data (determi-
nistic/stochastic).
There are authors who have attempted for SSCWLP
and its variants, some of these are discussed here. Geof-
frion and Graves [1] gave a Bender’s decomposition ap-
proach to solve multi-commodity SSCWLP. Later Hindi
and Basta [8] address similar type of distribution design
problem with consideration of operating cost per unit of
commodity at warehouse along with the fixed cost asso-
ciated with it. They used a branch and bound (B&B) al-
gorithm based on weak Linear Programming (LP) re-
laxation of the problem. Sharma [3] have attempted for a
food distribution problem faced by Food Corporation of
India (FCI), which is closer to multistage uncapacitated
warehouse location problem. Later a solution procedure
based on Lagrangian relaxation was developed for this.
Klose [9] attempts for capacitated two-stage facility lo-
cation problem with single source constraints. The de-
veloped model is a single commod ity version of distribu-
tion design given by Geoffrion and Graves [1]. They
give LP formulation, which is iteratively refined using
valid inequalities and facets, and feasible solutions are
obtained using simple heuristics by the information from
solving LP. In this wa y good lower an d upper boun ds are
P. VERMA ET AL.
101
developed. Tragantalerngsak et al. [10] addresses for two
layer facility location problem, where second layer facil-
ity has limited cap acity and service of customers is done
by only one facility in the second layer. Simultaneously,
second layer facility can b e supplied by only one facility
in the first layer. A Lagrangian relaxation (LR) based
B&B algorithm is given which is compared with LP
based B&B and is shown to be efficient as it provide
smaller sized tree and requires less CPU time. A realistic
variant to the problem that measure custo mer satisfaction
is attempted by Eskigun et al. [11] where an outbound
supply chain network is designed considering operational
dynamics of lead times, etc. They provide a Lagrangian
based heuristic, which gives good quality solutions. Me-
lachrinoudis et al. [12] uses physical programming ap-
proach to reconfigure a warehouse network through
consolidation and elimination. Their model enables a
decision maker to consider multiple criteria (i.e., cost,
customer service, etc) and to express criteria preferences
not in a traditional form of weights, but in ranges of dif-
ferent degrees of desirability. Farahani and Asgari [13]
investigate locating some warehouses in a real-world
military logistics system. They use multiple objective
decision making techniques to determine the locations of
warehouses. To assign each supported center to only one
of the located warehouses, a set partitioning model is
used. Melachrinoudis and Min [14] develop a mixedin-
teger programming model to solve the warehouse redes-
ign problem to phase-out underutilized warehouses
without deteriorating customer services. They validate
the model by applying it to a real-world problem and by
its sensitivity analyses. Keskin and Üster [15] attempted
for SSCWLP and gave a scatter search-based heuristic
approach for the problem.
It is to be noted that Geoffrion and Graves [1] and
Sharma [2] have attempted the warehouse location prob-
lem very interestingly; their formulations are completely
different from each other. Formulation given by Sharma
[2] has reduced number of variables. Sharma and Berry
[4] discuss in detail the differences in the approach to
formulation o f SSCWLP due to Geoffrion an d Graves [1]
and Sharma [2]. It is to be noted that the primal problem
in the Bender’s decomposition used by Geoffrion and
Graves [1] and Sharma [2] has a min-cost-flow problem
involving flow from plants to warehouse to markets. In
this paper, we present a method of “Vertical Decomposi-
tion” that breaks the original problem into two pro blems.
One involving flow from plants to warehouse (CPLP_L),
two involving f l ow from warehouse to ma rket s (C PLP_R ).
Using this approach we use the theory developed for
capacitated plant location problem (CPLP) (Cornuejols,
Sridharan, & Thizy [16] ) (by suita bly extending i t to adopt
to new version of CPLP (referred to as CPLP_L)) to
solve the original SSCWLP by using a Lagrangian re-
laxation. We finally show that this results in a better ap-
proach to solve SSCWLP (than the one given by Sharma
and Berry [4]).
In Section 2 we give the formulation and different re-
laxations of SSCWLP. In Section 3 vertical decomposi-
tion approach is discussed. Section 4 gives the results of
empirical investigation used to verify the theorems of
Section 3. In Section 5 we give the detailed procedure to
determine lower bounds and feasible solution with the
use of these relaxations and the usage of the bounds ob-
tained in determination of exact solution to SSCWLP.
Section 6 provides computational results along with its
analysis. We finally provide our conclusion on this ap-
proach in Section 7.
2. Formulation and Relaxations of SSCWLP
Here we give a mathematical formulation of SSCWLP
using the style of Sharma and Sharma (2000). In later
section, different relaxations of SSCWLP are developed
and discussed.
2.1. Index
i: Index for supply points (plants); i = 1, ···, I; I =
number of plants.
j: Index for warehouses; j = 1, ···, J; J = number
of potential warehouse points.
k: Index for markets; k = 1, ···, K; K = number of
markets.
2.2. Definition of Constants
k
D
d
: Demand for the commodity at ‘k
k: Demand at market ‘k’ as a fraction of the total
market demand; kk
k
DD
i
S: Supply available at ‘i
i
s
: Supply available at point ‘i’ as a fraction of
the total market demand; ik
k
SD
.
j
f
: Fixed cost of locating a warehouse at ‘j
j
CAP : Capacity of a warehouse at ‘j
j
cap : Capacity of warehouse at ‘j’, as a fraction of
the total market demand;
j
k
k
CAP D
.
ij
cpw : Cost of transporting k quantity of goods
from ‘i’ to ‘j’. kD
j
k
cwm : Cost of transporting quantity of goods
from ‘j’ to ‘k’. k
kD
2.3. Definition of Variables
ij
X
PW : Quantity shipped from ‘i’ to ‘j’.
k
X
WM : Quantity shipped from ‘j’ to ‘k’.
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
Copyright © 2011 SciRes. AJOR
102
ij
x
pw : Quantity shipped from ‘i’ to ‘j’ as a fraction of
total market demand; ijij k
k
x
pw XPWD
: Quantity shipped from ‘j’ to ‘k’ as a fraction
of total market demand;
j
kjk
k
x
wm
j
k
k
x
wm XWMD.
j
y: Location variable (= 1 if warehouse is located
at point ‘j’ , 0 otherwise)
2.4. Mathematical Formulation
Minimize
ijijjkjkj j
jk jij
Z
cpw xpwcwmxwmfy 
 (1)
Subject to:
1
ij
ij
xpw
 (2)
1
jk
jk
xwm
 (2a)
ijj i
x
pwy s,i (3) j
j
kjk
x
wmyd (3a) ,jk
ijjj
i
x
pwy c
ap j (4)
j
kj
k
j
x
wmycap
(4a)
j
ij i
j
x
pw s
(5) i
j
kk
j
x
wm d
(5a) k
0xpw  
ij
0xwm
(6) ,ij
jk

0,1y
(6a) ,jk
j
cap y
(7) j
1
jj
j
ij jk
(8)
ik
x
pw xwm (9)
j
Just to emphasize that in the above formulati on in s te ad
of using single letter names of variable and constant, we
use multiple letter names (viz. xpw, xwm, cpw, cwm, etc)
so that one can easily recall their meanings, that is,
flow/cost between p lant to warehouse (pw) or warehouse
to market (wm) etc, while reading the paper.
First part of the objective function denotes the total
cost of transporting the commod ity from supply poin ts to
the warehouses. The second part is the transportation
cost from warehouses to markets; and the last part is the
fixed cost of locating warehouses. Constraint (2) and (5)
are the supply constraint which ensures that the total
commodity shipped out from a supply point to all ware-
houses can at most be equal to the total supp ly from that
supply point. Constraint (2a) and (5a) ensures the total
quantity received at any market to be equal to the de-
mand at that market. Constraint (3) and (3a) link 0-1 in-
teger variables (
j
y) and other distribution variables that
are real and greater than zero. These ensure that if a
warehouse is located at any po int then quantities ship ped
in or out of that warehouse point are positive, else the
quantities shipped will be zero, as the warehouse is not
located. Constraint (4) and (4a) ensures that total quan-
tity flown in and out of the warehouse does not exceed
its capacity if it’s located. We assume to
ensure feasibility of the problem. 1
j
is
Non-negativity constraint on ij
x
pw and
j
k
x
wm are
given by (7) and (7a); while (8) is the 0-1 integer con-
straint on
j
y. (11) is the flow balance constraint which
ensures that total incoming quantity at a particular ware-
house from all plants is equal to total outgoing quantity
from the same warehouse to all th e markets. (9) is a sur-
rogate constraint. With so many constraints available, we
can formulate SSCWLP in a variety of different ways. In
these if (8) is replaced by (12) we obtain various LP re-
laxations. In addition, various relaxations can be ob-
tained as Lagrangian relaxation (LR).
2.5. Relaxations of SSCWLP
We now develop different LP relaxations and LRs of
SSCWLP formulated above. The notation R*_O repre-
sent different relaxations of SSCWLP, where (*) is a
replacem en t for n u mbers 1 to 4.
When integrality restrictions on
j
y are relaxed we
obtain (10)
0
i
y1
for all j (10)
R1_O: (1); Subject to: (2)-(6), (2a)-(6a), (9) and (10).
Result 1: R1_O is equal to strong LP relaxation of
SSCWLP given by Sharma and Berry (2007). Here re-
striction on
j
y is relaxed by (10).
Proof: It is easy to see.
R2_O: In this LR a constraint introducing upper (U)
and lower (P
P) limits on the number of open plants is
added up. i.e.
1
n
L
j
j
Py
U
P
(11)
This LR is obtained by dualizing (2), (2a), (5) and (5a).
Let 0
x
pw
, 0
x
wm
be the Lagrangian multipliers as-
sociated with (2), (2a) and i
x
pw
, k
x
wm
with (5) and
5a) respect i v ely. (
P. VERMA ET AL.
103

00
2_ 00
,,
,,,
00
max min
R
Oijiijjk
xpw xwm y
xpwxwmxpw xwmij jk
jj iikk
ji k
kjk
Z
cpwxpwxpw xpwcwmxwmxwmxwm
fyxpw sxwmdxpwxwm
 
 


 
 
 
2
(12)
Hence objective (12); Subject to: (3), (3a), (4), (4a),
(6), (6a), (7), (9) and (11).
R3_O: This is a LR obtained by dualizing (2), (2 a), (6)
and (6a). Hence objective is (14); Subject to: (3), (3a),
(4), (4a), (6), (6a), (7), (8), (9).
R4_O: (1); Subject to: (2)-(6), (2a)-(6a), (7,8) and (9).
The original problem SSCWLP is referred for com-
parison.
3. Vertical Decomposition Approach for
SSCWLP
SSCWLP is a well known NP hard problem. Hence, in
order to solve large sized and realistic instances of
SSCWLP, we devise a decomposition procedure—called
“vertical decomposition approach”, which is described
below.
The objective function (1) may be rewritten as:
Minimize 1
Z
ff, where

1,,,
1
min 2
ijij j j
ijijj j
ij j
fxpwcpwyf
cpw xpwfy


2,,,
1
min 2
jkjkj j
jkjkj j
jk j
fxwmcwm yf
cwmxwmfy

This restructured objective function becomes a moti-
vation to introduce the vertical decomposition approach
for solving SSCWLP. If we give a close look to the for-
mulation of SSCWLP, it can be observed that (9) is the
only constraint which involves both the flow variables -
xpw and xwm. So, if (9) is relaxed, the problem decom-
poses into two versions of CPLP, each with one part (f1
or f2) of the objective function shown above. We name
this approach as vertical decomposition approach, be-
cause the full SSCWLP is decomposed into two parts –
left and right. The left part, CPLP_L, contains the vari-
ables and parameters of plants and warehouse only (xpw
and y). Similarly, the right part, CPLP_R, contains the
variables and parameters of warehouse and markets only
(xwm and y). Therefore, in a sense, by vertical decompo-
sition approach, the stages of the problem are decom-
posed to get smaller sized problems, which are relatively
easier to solve.
Now, (9) has to be relaxed in order to make the
SSCWLP formulation amenable to vertical decomposi-
tion approach. We use _
j
f
low
as the Lagrangian
multipliers to dualize (11). The resulting formulation of
CPLP_L, CPLP_R and SSCWLP, with (9) relaxed, are
as shown below:
CPLP_L:
_,
min _
1
2
CPLP Lijjij
xpw yij
jj
j
Z
cpwflowxpw
fy


(1a)
Subject to: (2) , (3), (4), (5), (6), (7), (8) and (9).
CPLP_R:
_,
min _
1
2
CPLP Rjkjjk
xwm yjk
jj
j
Z
cwmflow xwm
fy


(1b)
Subject to: (2a), (3a) , (4a), (5a), (6a), (7), and (8 ).
SSCWLP:


,,
_
max min_
_
SSCWLPijjij
xpw xwm y
flow ij
j
kjjk
jk j
jj
Z
cpwflowxpw
cwmflowxwmfy

 

 
or,
_
_
max
SSCWLPCPLP LCPLP R
flow
ZZZ

_
(1c)
Subject to: (2) to (9) and (2a) to (6a).
Here the problem SSC WLP is broken int o sub probl ems
CPLP_L and CPLP_R. This is vertical decomposition.
Sharma [2] decomposed a multi commodity and multi
period problem into single commodity and single period
problem representing flow from plants to warehouses to
markets. This is referred to as horizontal decomposition
to bring out the contrast. In the next sub-section different
relaxations of CPLP_L and CPLP_R are discussed in
detail. Further, procedure to solve SSCWLP using these
decomposed problems is discussed. Later a relationship
theorem indicating the strength of different relaxations of
SSCWLP is given.
3.1. Relaxations of CPLP_L
R1_L: (1a); Subject to: (2)-(6) and (10).
R
2_L: In this LR is obtained by dualizing (2) and (6).
Copyright © 2011 SciRes. AJOR
104 P. VERMA ET AL.

0
2_0 0
,
,
1
max min_2
RLijjiijjjii
xpw y
xpw xpwij ji
Z
cpwflowxpwxpwxpwfyxpw sxpw

 

  (13)
Subject to: (3), (4), (6), (7) and (11).
R3_L: (13); Subject to: (3), (4), (6)-(9).
R4_L: (1a); Subject to: (2), (3), (4), (5), (6), (7) and
(8).
The main problem i.e. ZCPLP_L is referred as R4_L for
comparison.
3.2. Relaxations of CPLP_R
R1_R: (1b); Subject to: (2a)-(6a) and (10). R1_R is a
strong LP relaxation (follows from Davis and Ray (1969)).
R2_R: LR proposed by Christofides and Beasley [5].
This LR is obtained by dualizing (2a) and (6a).

0
2_ 0 0
,
,
1
max min_2
k
RRjkjk jkjjkk
xwm y
xwm xwmjk jk
Z
cwmflowxwmxwmxwmfywmdxwm

 

  (14)
Subject to: (3a), (4a), (6a), (7), (1).
R3_R: (14); Subject to: (3a), (4a), (6a), (7), (8). It is
attempted by Nauss [6].
R4_R: (1 b); Subject to: (2a), (3a), (4a), (5a), (6a), (7),
and (8).
The main problem i.e. ZCPLP_R is referred as R4_R for
comparison.
3.3. Relationship between Relaxations of
CPLP_L
Note that CPLP_L is different from CPLP_R. In this
section, a comparison of the strength of the bounds given
by different relaxations of CPLP_L is given. We con-
structed the proofs and found that they are marginally
different from CPLP_R (Cornuejols, Sridharan, & Thizy
[16].
Theorem 1: 1_ 2_ 3_ 4_
R
LRLRLR
ZZZZ
L
.
This theorem provides the relative effectiven ess of the
bounds that may be obtained for these relaxations of
CPLP_L.
3.4. Relationship between Relaxations of CPLP_R
Cornuejols, Sridharan, & Thizy [16] have given relative
strength of different relaxations of standard CPLP which
is similar to CPLP_R. Hence, the same relaxations along
with their relative strength are used for CPLP_R here in
form of theorem 2 below.
Theorem 2: 1_ 2_ 3_ 4_
R
RRRRRR
ZZZZ
R
.
This theorem provides the relative effectiven ess of the
bounds that may be obta in fo r the se rela xat io ns of C PLP _R.
3.5. Relationship between Relaxations of
SSCWLP
Here a comparison of the strength of the bounds, given
by different relaxations of SSCWLP based on CPLP_L
and CPLP_R is given.
Proposition 1: The bounds obtained by relaxations
R1_O and R2_O are related as 1_ 2_
R
ORO
Proof: With the flow balance constraint (9) relaxed,
objective of R2_O can be written as:
ZZ.


0
0
2_ 0
,,
,,
,,_
000
max min_
_
ROijjiij
xpw xwmy
xpw xpwij
xwm xwmflow
jkjkjkjji ikk
jk ji k
Zcpwflowxpwxpw xpw
cwmflowxwmxwmxwmfyxpw sxwmdxpwxwm



 

 

 
Subject to: (4) , (4a), (6), (6 a), (7).


0
0
2_ 0
,,
_,,
,
000
max maxmin_
_
j
ROijji ij
xpw xwm y
flow xpw xpwij
xwm xwm
jkjkjkjjiikk
jk ji k
Zcpwflowxpwxpwxpw
cwmflowxwmxwmxwmfyxpw sxwmdxpwxwm



 



 
Subject to: (4), (4a), (6), (6a), (7).

2_2_ 2_
_
max
j
R
ORLR
flow
ZZZ

According to the proofs earlier shown, 1_ 2_
R
RR
ZZ
R
[for CPLP_R] and 1_ 2_
R
LR
ZZ
L
[for CPLP_L].
R
Hence,
2_1_ 1_
_
max
R
ORLR
flow Z
R
ZZ.
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
105
Note that here (9) is excluded. If (9) is also included,
then the problem become R1_O subjected to its usual
constraints. i.e. 1_ 2_
R
OR
ZZO
.
Proposition 2: 2_3_
R
ORO
Proof: We find that all the results of original problem
have identical proof. Hence proof for Proposition 1 has
been shown and the proofs for Proposition 2 being simi-
lar to it are omitted. When these propositions are com-
bined, following theorem is developed showing relative
strength of bounds given by different relaxations of
SSCWLP.
ZZ.
Theorem 3: .
1_O2_O3_O4_ORRR R
Strongest LP relaxation of SSCWLP proposed by
Sharma and Berry [4] is same as relaxation R1_O of
SSCWLP proposed. From theorem 3, we observe that the
lagrangian relaxations R2_O and R3_O can be better
than the strongest relaxations known of SSCWLP (that is
R1_O). Empirical study given below shows that differ-
ences in performances of relaxations considered in theo-
rem 3 are statistically significant.
ZZZZ
4. Empirical Investigation for CPLP_L and
CPLP_R
Here we give results of an empirical study. We find that
there is significant difference in the performance of dif-
ferent relaxations considered in theorem 3.
Sample problems for SSCWLP sized 50 × 50 × 50
were randomly created using C codes; a 50 × 50 × 50
problem corresp onds to a set of 50 pl ants, 50 warehouses
and 50 markets. Two categories of problems are consid-
ered—nil category (A) and abundance category (B); the
same is tabulated in Table 1 belo w.
For each of these categories, we created 25 problems
in which fixed location cost and the transportation cost
are uniformly distributed as shown in Table 1. For each
problem instance, we obtained the value of different re-
laxations of CPLP_R and CPLP_L using GAMS 22.3 in
a Pentium D 2.80 GHz, 1 GB RAM computer. Sub-gra-
dient optimization is used for solving Lagrangian relaxa-
tions of CPLP_L and CPLP_R. Lagrangian relaxation
method is powerful in the sense that it gives good lower
bounds (for the minimization problem) in competitive
computational time. Starting Lagrangian multiplier is
taken either to be 0 or maximum value of

0.5*
j
j
capf
from all j warehouses. Using trial and error approach
between these two choices, good results were obtained.
Details can be seen in appendix A. Here we take per-
centage improvement between a pair of relaxations (to
generate normalized data) for every problem instance;
and then t-test is performed.
t-tests for bounds of the different relaxations for dif-
ferent categories are shown in Tables 2 and 3. These
t-values are compared with those in Table 4.
Note that each cell of the above table shows the t-value
of the t-test between the two relaxations represented by
the respective row/column of the matrix. For e.g. t-value
between R1 and R2 shows the t-test done between 0 and
100*(R2-R1)/R1. Table 2 shows the t-values obtained
for differences in bounds of relaxations for category A
problems; and Table 3 is revealing the t-values for dif-
ferences in bounds of relaxations for category B prob-
lems.
4.1. Analysis for Category A Problems
When comparing t values for bounds of different relaxa-
tions of CPLP_L [Table 2], the LP relaxation R1_L is
giving significantly better bounds as compared to rest all
relaxations. We observe from Table 2 that linear relaxa-
tion R1_R is the best performing relaxation for category
A problems.
Table 1. Problem categories for SSCWLP.
Parameters
Problem category
i
i
s
j
j
cap
Fixed location cost Transportation Cost
A 1.0 1.0 U [100, 1000] U [1, 100]
B 5.0 5. 0 U [100-1000] U [1, 100]
Table 2. t-test for differences in bounds for category A problems.
CPLP_L CPLP_R
t-test R1 R2 R3 R4
t-test R1 R2 R3 R4
R1 –23.1+ –23.1+ –0.16 R1 –24.3+ –24.7+ 1.44
R2 1 23.35+ R2 2.2+++ 24.3+
R3 23.35+ R3 24.71+
Copyright © 2011 SciRes. AJOR
106 P. VERMA ET AL.
Table 3. t-test for differences in bounds for category B problems.
CPLP_L CPLP_R
t-test R1 R2 R3 R4
t-test R1 R2 R3 R4
R1 1.01 6.29+ 6.35 R1 –0.61 6.29+ 6.29+
R2 6.29+ 6.35+ R2 6.29+ 6.29+
R3 1.63 R3 3.18+
Table 4. t-critical value (Table value).
Significance level α = 0.05 α = 0.01 α = 0.005
t-critical (one tail) 1.68 2.40 2.68
+++: At α = 0.05, significance level = 1 tail; ++: At α = 0.01, significance
level = 1 tail; +: At α = 0.005, significance level = 1 tail.
4.2. Analysis for Category B Problems
When comparing t-values from Table 3 and Table 4, it
is found that most of the relaxations are following theo-
rem-1 and theorem-2 respectively for CPLP_L as well as
CPLP_R. Significant t-values indicate that the LR R3_L
of CPLP_L and R3_R of CPLP_R are providing signifi-
cantly better bounds. This results in faster execution of
the branch and bound procedure based procedure to
solve SSCWLP as shown in the next section.
5. Branch and Bound Procedure to Solve
SSCWLP
In this section we aim to use three best performing re-
laxations to determine bounds in a branch and bound
procedure. It is known that stronger the relaxations, bet-
ter the bounds, lesser will be the nodes traversed in an
enumeration tree and hence one achieves computational
advantage. In R2_O and R3_O we relax the flow balance
constraints and solve the associated CPLP_R and
CPLP_L by lagrangian relaxation procedure. Details of
lagrangian relaxation procedure is given in Sharma
(1991). The sketch of branch and bound procedure to
solve SSCWLP is given below.
5.1. Procedure Branch and Bound
Step 1: Initialization. Set current_best _solution = INFIN-
ITY;
best_lower _bound = - INFINITY;
Node [1]. Fixed Var List = {y1 = 0};
Node [1]. Free Var List = {2, ···, N};
Node [2]. Fixed Var List = {y1 = 1};
Node [2]. Free Var List = {2, ···, N};
Add nodes 1 and 2 to list A.
Step 2: If list A is empty
Then Stop. Optimal Solution Found
Else pick up top node (TN) from list A
Step 3: Compute lagrangian bound (associated with
R2_O or R3_O) or LP relaxation bound (associated with
R1_O) associated with node TN (referred to as TN.R_
Bound). If (TN_R_Bound > best_lower_bound), then
best_lower_bound = TN_R_Bound .
Solve associated min cost flow problem (with TN) and
generate a feasible solution for SSCWLP (referred as
TN_feasible_solution). If (TN_feasible_solution < cur-
rent_best_solution), then current_best_solution = TN_
feasible_solution.
If (TN.R_Bound > current_best _solution)
Then node TN gets fathomed. Go to step 2.
Else If free variables associated with TN >= 1
Then generate two more nodes by setting a chosen free
variable associated with TN at 0 and 1; make relevant
updates and add these nodes to list A.
Else No more branching possible; go to step 2.
The above procedure is repeated separately for R1_O,
R2_O and R3_O to facilitate a comparison. Empirical
investigation is given below.
6. Empirical Investigation for SSCWLP
In the earlier sections, the relationship theorem indicat-
ing the strength of different relaxations of CPLP_L,
CPLP_R and SSCWLP is shown. Also solution proce-
dures of some of the best performing relaxations of
SSCWLP using vertical decomposition approach are
discussed. In this section we focus on usage of these re-
laxations of SSCWLP in a branch and bound procedure
to obtain an optimal solution. Performance of these re-
laxations with its application in a branch and bound is
compared with a standard branch and bound procedure
(BB) using strong LP relaxation (R1_O) of SSCWLP as
its lower bound. We thus compare the new proposed
relaxations with that of the best possible known relaxa-
tion, R1_O (Sharma and Berry, 2007 ), and show its effi-
cacy by the reduction on nodes and time achieved. The
objective of this section is to study how well these re-
laxations do relative to each other. In particular, we are
interested in looking at the influence of supplies versus
capacities, capacities versus demands, and fixed costs
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
107
versus transportations costs on the bounds provided by
these relaxations, and the solutions provided by the heu-
ristics.
Sample problem s for SSC WLP, sized 50 × 50 × 50 and
100 × 100 × 100 (plants × warehouse × markets) were
created randomly. From the empirical investigations done
for CPLP_L and CPLP_R, we have observed that rela-
tionship theorems 1 and 2 are satisfied for the abundance
case more promptly. Hence for conducting empirical
experiments for complete problem of SSCWLP we have
considered varying categories of abundance in supply and
capacity limits. Four categories of problems with varying
problem size and supply – capacity limits were prepared
details of which are given in Table 5.
We solved 20 problem instances in each of the category
of P1, P2, P3 and P4 on Pentium D 2.80 GHz, 1 GB RAM
computer. All the algorithms are coded in MATLAB
software with calls to GAMS22.3 for solving the min-
cost-flow problem. Details are given in appendix B. We
compute percentage improvement for any two methods
for every problem instance (to generate normalized data)
and then t -tests are perform ed on the numb er of n odes and
the time required to solve them; these are shown in Table
6.
For each Rm-Rn (“m” and “n” are BB, 1, 2, 3 as shown
in column 3 of Table 6), “Nodes” is “t” calculated for the
difference between (Number of nodes taken with formu-
lation Rm/Number of nodes taken with formulation Rn)
and 1. Similarly, “Time” is “t” calculated for difference
between CPU time for Rm/CPU time for Rn) and 1. BB
refers to the complete enumeration for solving the prob-
lem. Nodes under BB refer to th e number of nodes taken
to solve the problem. Negative t value indicates reduced
number of node and reduced execution time.
6.1. Analysis for Category P1 and P3 Problems
Category P1 and P3 problem s have got the same variation
in supply and capacity limits, but their problem sizes are
different. For th e problem category P1, we note from the
Table 6 that the performance (in terms of reduction in
number of nodes) of relaxation R1_O is significantly
better than BB. Similarly R2_O is significantly better
performer as c ompared to R 1_O; however performance of
Table 5. Problem categories for SSCWLP.
Parameters
Problem category Probl e m Size
i
i
s
j
j
cap
F ixed Location CostTransportation C ost
P1 50 × 50 × 50 2.5 2.5 U [100, 1000] U [1, 100]
P2 50 × 50 × 50 10 10 U [100, 1 0 00] U [1, 100]
P3 100 × 100 × 100 2.5 2.5 U [1500, 2000] U [1, 100]
P4 100 × 100 × 100 10 10 U [1500, 2000] U [1, 100]
Table 6. t-test for Nodes and time taken to solve the problems.
Problem cat egory Problem Size Comparison between ‘Rm-Rn’ Nodes Time
R1_O – BB –5.02+ –5.24+
R2_O – R1_O –3.12+ –3.53+ P1 50 × 50 × 50
R3_O – R2_O –2.75+ –1.76+
R1_O – BB –2.67++ –2.92+
R2_O – R1_O –1.32 –1.8+++
P2 50 × 50 × 50
R3_O – R2_O 1 –1.96+++
R1_O – BB –5.07+ –5.1+
R2_O – R1_O –5.56+ –5.72+
P3 100 × 100 × 100
R3_O – R2_O –3.82+ –3.83+
R1_O – BB –2.22+++ –2.47++
R2_O – R1_O –3.83+ –3.82+ P4 100 × 100 × 100
R3_O – R2_O –3.1+ –3.28+
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
Copyright © 2011 SciRes. AJOR
108
R3_O is still better than R2_O. Also higher t values in-
dicate that superiority of R1_O over BB is most signifi-
cant and R3_O over R2_O is least (however it is still
significant). Exactly similar is the behavior of perform-
ance in terms of time taken to solve the problems. That is
R3_O is the best performer, superior to R2_O, which is
better than R1_O; BB is the worst performer in terms of
time.
Now when the problem size increases, that is for the
problem category P3, it can be observed that R3_O still
remains the best performer in terms of reduction in
number of n odes as well as time taken to s olve a problem .
The relaxations R2_O and R1_O (in that sequence) follow
R3_O; however BB still remains the worst performer,
both in terms of nodes as well as solution time. Also as
can be observed from the t values, the superior perform-
ance of R2_O over R1_O is most significant; the pairs
“R1_O over BB” and “R3_O over R2_O” follows in that
sequence.
So it can be inferred from here that the use of relaxation
R3_O is most suited to solve small as well as large sized
SSCWLP, when there is a moderate level (2.5 times) of
over-supply and over-capacity. Solving SSCWLP of
category P1 and P3 using Branch and Bound is the least
efficient method; however the use of R1_O, R2_O and
R3_O (in ascending order of better performance) to de-
termine bounds in a Branch and Bound method proves to
be very fruitf ul.
6.2. Analysis for Category P2 and P4 problems
Category P2 and P4 problem s have got the same variation
in supply and capacity limits, but their problem size is
different. It can be observed from the Table 6 that for
problem category P2, relaxation R1_O performs signifi-
cantly better, both in terms of reduction in number of
nodes traversed as well as solution time, than the usual
branch and bound me thod (BB). However, there is not any
significant improvement in the performance of “R2_O
over R1_O” or “R3_O over R2_O”.
However when the problem size increases, that is for
problem category P4, R1_O performs better than BB,
R2_O performs better than R1_O, and R3_O performs
significantly better than R2_O both in terms of reduction
in number of nodes as well as reduction in the time re-
quired to solve a problem. Also observing the t values, it is
evident that the significant performance of R1_O over BB
is not as noteworthy as the significance of “R2_O over
R1_O” or “R3_O over R2_O”.
So it can be inferre d that when level of over-supply and
over-capacity is high (10 times), it is better to solve a
smaller sized SSCWLP using R1_O (or R2_O or R3_O)
and a large sized SSCWLP using R3_O. Also in this
category for smaller sized problems, the use of R1_O,
R2_O or R3_O is equally better option compared to the
branch and bound method because both R2_O and R3_O
perform same as R1_ O. However for large sized p roblems
of this category, with the use of relaxations R1_O, R2_O
and R3_O (in ascending order of better performance) as
bounds in the branch and bound method perform better
than the usual branch and bound technique.
We have shown that lagrangian based branch and bound
procedures (R2_O and R3_O) are superior to R1_O
(Sharma and Berry [4]) by implementing these algorithm s
on the common platform of MATLAB. It may be noted
that BB and R1_O are sol vable by commercially available
packages as LINGO, CPLEX or GAMS; whereas la-
grangian relaxation based procedures (R2_O and R3_O)
are not directly solvable by these commercially available
packages . First aut hor of t his paper (a Ph D candida te then)
coded these algorithms on MATLAB; and there is enor-
mous scope for improvement before its performance be
compared to commercially available packages like CPLEX.
However it is to be noted that Sharma and Berry [4]
showed that R1_o is significantly superior to plain branch
and bound procedure (BB) compared on commercially
available LINGO software. Thus we provide preliminary
evidence that R2_O and R3_O have significant merit for
solving SSCWLP.
7. Conclusions
The contribution of this work in the existing vast literature
of location problems is two folds. First is the introduc-
tion of “vertical decomposition” approach for SSCWLP,
which can easily be extended to the multi stage ware-
house location problems, or to the problems of different
domains that are modeled in a manner similar to
SSCWLP. Vertical decomposition approach allows us to
decompose the complex and large sized SSCWLP into
two versions of the standard CPLP. One of the decom-
posed problems, CPLP_R, is well researched and has
different known relaxations. For the other problem,
CPLP_L, we provide different relaxation and show that
some of them are similar to CPLP_R. A relationship
theorem of different relaxations shows the superiority of
some relaxations over the others. In particular we show
theoretically that for SSCWLP better Lagrangian relaxa-
tion exists than the LP relaxation given by Sharma and
Berry [4]. Computational studies give support to our
theoretical propositions.
Second and the major contribution of this work is to
show the efficacy of “vertical decomposition” approach,
by using the best performing relaxations of the decom-
posed and original SSCWLP to advantageously deter-
mine an exact solution to the large sized SSCWLP.
P. VERMA ET AL.
109
Three best performing relaxations are selected based on
their relative efficiencies given in relationship theorems,
and a procedure to solve them is also provided. These
relaxations are used to determine lower bound and a fea-
sible solution, which in turn are used in a branch and
bound method. Computational study is done for a variety
of problems of different sizes. It is found that one of the
Lagrangian relaxations (R3_O) is performing best in
terms of time and nodes travelled in almost all the cases,
as compared to the remaining relaxations. This is a sig-
nificant finding as relaxation R1_O was found to be the
best performer for SSCWLP by Sharma and Berry [4];
that is we actually landed up finding a relaxation of
SSCWLP which is better than that existing in literature.
A future research possibility with huge potential could be
to extend the results of this paper to multistage location
distribution problems, which can be modeled using the
“vertical decomposition” approach.
8. References
[1] A. M. Geoffrion and G. W. Graves, “Multicommodity
Distribution System Design by Benders Decomposition,”
Management Science, Vol. 20, No. 5, 1974, pp. 822-844.
doi:10.1287/mnsc.20.5.822
[2] R. R. K. Sharma, “Modeling a Fertilizer Distribution Sys-
tem,” European Journal of Operational Research, Vol.
51, No. 1, 1991, pp. 24-34.
doi:10.1016/0377-2217(91)90142-I
[3] R. R. K. Sharma, “Food Grains Distribution in the Indian
Context: An Operational Study,” In: A. Tripathi and J.
Rosenhead, Eds., Operations Research for Development,
New Age International Publishers, New Delhi, 1996, pp.
212-227.
[4] R. R. K. Sharma and V. Berry, “Developing New Formu-
lations and Relaxations of Single Stage Capacitated Ware-
house Location Problem (SSCWLP): Empirical Investi-
gation for Assessing Relative Strengths and Computa-
tional Effort,” European Journal of Operational Research,
Vol. 177, No. 2, 2007, pp. 803-812.
doi:10.1016/j.ejor.2005.11.028
[5] N. Christofides and J. E. Beasley, “Extensions to a La-
grangean Relaxation Approach for the Capacitated Ware-
house Location Problem,” European Journal of Opera-
tional Research, Vol. 12, No. 1, 1983, pp. 19-28.
doi:10.1016/0377-2217(83)90179-0
[6] R. M. Nauss, “An Improved Algorithm for the Capaci-
tated Facility Location Problem,” Journal of Operational
Research Society, Vol. 29, No. 12, 1978, pp. 1195-1201.
[7] M. T. Melo, S. Nickel and F. Saldanha-da-Gama, “Facil-
ity Location and Supplychain Management—A Review,”
European Journal of Operational Research, Vol. 196, No.
2, 2009, pp. 401-412. doi:10.1016/j.ejor.2008.05.007
[8] K. S. Hindi and T. Basta, “Computationally Efficient
Solution of a Multiproduct, Two-Stage Distribution-Loca-
tion Problem,” The Journal of the Operational Research
Society, Vol. 45, No. 11, 1994, pp. 1316-1323.
[9] A. Klose, “An LP-Based Heuristic for Two-Stage Ca-
pacitated Facility Location Problems,” The Journal of the
Operational Research Society, Vol. 50, No. 2, 1999, pp.
157-166.
[10] S. Tragantalerngsak, J. Holt and M. Rönnqvist, “An Ex-
act Method for the Two-Echelon, Single-Source, Capaci-
tated Facility Location Problem,” European Journal of
Operational Research, Vol. 123, No. 3, 2000, pp. 473-489.
doi:10.1016/S0377-2217(99)00105-8
[11] E. Eskigun, R. Uzsoy, P. V. Preckel, G. Beaujon, S.
Krishnan and J. D. Tew, “Outbound Supply Chain Net-
work Design with Mode Selection, Lead Times and Ca-
pacitated Vehicle Distribution Centers,” European Jour-
nal of Operational Research, Vol. 165, No. 1, 2005, pp.
182-206. doi:10.1016/j.ejor.2003.11.029
[12] E. Melachrinoudis and H. Min, “Redesigning a Ware-
house Network,” European Journal of Operational Re-
search, Vol. 176, No. 1, 2007, pp. 210-229.
doi:10.1016/j.ejor.2005.04.034
[13] R. Z. Farahani and N. Asgari, “Combination of MCDM
and Covering Techniques in a Hierarchical Model for Fa-
cility Location: A Case Study,” European Journal of Op-
erational Research, Vol. 176, No. 3, 2007, pp. 1839-1858.
doi:10.1016/j.ejor.2005.10.039
[14] E. Melachrinoudis, A. Messac and H. Min, “Consolidat-
ing a Warehouse Network: A Physical Programming Ap-
proach,” International Journal of Production Economics,
Vol. 97, No. 1, 2005, pp. 1-17.
doi:10.1016/j.ijpe.2004.04.009
[15] B. B. Keskin and H. Üster, “A Scatter Search-Based Heu-
ristic to Locate Capacitated Transshipment Points,” Com-
puters & Operations Research, Vol. 34, No. 10, 2007, pp.
3112-3125. doi:10.1016/j.cor.2005.11.020
[16] G. Cornuejols, R. Sridharan and J. M. Thizy, “A Com-
parison of Heuristics and Relaxations for the Capacitated
Plant Location Problem,” European Journal of Opera-
tional Research, Vol. 50, No. 3, 1991, pp. 280-297.
doi:10.1016/0377-2217(91)90261-S
Copyright © 2011 SciRes. AJOR
110 P. VERMA ET AL.
Appendix A
Computational results for LHSCPLP, category A problem 50 × 50 × 50.
R1_LHS R2_LHS R3_LHS R4_LHS
S.No Objective Iteration Objective Iteration Objective Iteration Objective Iteration
1 15171.58 2227.00 15169.882 34 15169.885 37 15171.59 59
2 13103.62 1992.00 13102.209 35 13102.21 35 13103.62 81
3 13395.97 2158.00 13393.873 37 13393.874 37 13395.96 65
4 13141.67 1985.00 13139.668 37 13139.672 37 13141.67 78
5 12755.00 2030.00 12753.723 35 12753.727 35 12755 58
6 13424.41 2957.00 13422.896 35 13422.896 35 13424.41 67
7 11782.85 2320.00 11781.386 35 11781.386 35 11782.85 64
8 15200.70 2319.00 15198.726 37 15198.727 37 15200.7 76
9 12873.67 2529.00 12872.091 35 12872.094 35 12873.67 82
10 13231.09 2303.00 13229.122 37 13229.123 37 13231.09 59
11 11347.41 2638.00 11345.71 35 11345.71 35 11347.41 75
12 13626.14 2633.00 13624.432 37 13624.432 35 13626.14 79
13 12858.80 2350.00 12857.204 35 12857.204 35 12858.8 75
14 13592.69 2548.00 13591.355 35 13591.355 35 13592.69 41
15 13970.07 2257.00 13969.255 35 13969.255 35 13970.07 40
16 13348.74 2591.00 13347.718 35 13347.722 35 13348.74 87
17 13164.99 3069.00 13163.302 37 13163.314 35 13164.99 82
18 14130.20 2458.00 14128.861 35 14128.866 35 14130.2 88
19 13469.87 2459.00 13468.141 37 13468.141 37 13469.87 71
20 14493.84 2414.00 14491.808 37 14491.808 37 14493.84 65
21 13842.17 2105.00 13840.31 37 13840.31 37 13842.17 82
22 13700.39 1903.00 13698.911 35 13698.911 35 13700.39 81
23 14183.09 2225.00 14181.863 35 14181.863 35 14183.09 74
24 13956.53 2320.00 13955.301 35 13955.301 35 13956.53 77
25 14220.34 2192.00 14219.027 35 14219.027 35 14220.34 81
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
111
Computational results for RHSCPLP, category A problem 50 × 50 × 50.
R1_RHS R2_RHS R3_RHS R4_RHS
S.No Objective Iteration Objective Iteration Objective Iteration Objective Iteration
1 15172 1022 15170.34 43 15170.34 43 15172 75
2 13104.08 1242 13102.31 43 13102.31 43 13104.08 73
3 13395.2 1180 13393.64 43 13393.65 43 13395.2 59
4 13141.71 1372 13139.67 43 13139.67 43 13141.71 66
5 12754.94 1212 12753.82 41 12753.82 41 12754.94 70
6 13422.99 1047 13421.89 41 13421.89 41 13422.99 66
7 11782.99 793 11781.44 43 11781.44 43 11782.99 56
8 15201.93 1007 15199.67 43 15199.67 43 15201.93 79
9 12873.23 805 12871.22 43 12871.23 43 12873.24 68
10 13230.66 913 13229.05 43 13229.05 43 13230.66 60
11 11346.39 1055 11345.02 41 11345.02 41 11346.39 89
12 13625.79 1259 13623.66 43 13623.66 43 13625.79 77
13 12858.7 966 12857.22 43 12857.22 43 12858.71 55
14 13592.4 1058 13590.74 43 13590.74 43 13592.4 64
15 13971.45 925 13969.87 43 13969.87 43 13971.45 67
16 13348.72 1479 13347.35 41 13347.35 43 13348.72 77
17 13164.36 940 13163.07 41 13163.07 41 13164.36 73
18 14129.75 984 14128.09 43 14128.09 43 14129.75 55
19 13470.62 891 13468.43 43 13468.43 43 13470.62 68
20 14493.33 856 14491.95 43 14491.95 43 14493.33 65
21 13841.5 944 13839.65 43 13839.65 43 13841.5 66
22 13701.3 1438 13699.17 43 13699.17 43 13701.3 54
23 14182.93 908 14181.34 43 14181.34 43 14182.93 65
24 13957.9 985 13955.42 43 13955.43 43 13957.9 74
25 14221.12 1228 14219.6 43 14219.6 43 14221.12 65
Copyright © 2011 SciRes. AJOR
112 P. VERMA ET AL.
Computational results for LHSCPLP, category B problem 50 × 50 × 50.
R1_LHS R2_LHS R3_LHS R4_LHS
S.No Objective Iteration Objective Iteration Objective Iteration Objective Iteration
1 497.088 107 497.088 58 526.801 88 526.8437 282
2 563.076 105 563.076 59 595.003 34 595.0317 92
3 705.075 101 705.067 31 732.775 84 732.8163 330
4 798.315 116 798.32 37 818.037 16 818.0726 12
5 1030.828 107 1030.828 35 1062.68 117 1062.729 267
6 746.658 101 746.658 47 855.058 13 855.1125 541
7 894.089 125 894.089 60 930.367 23 930.4226 129
8 548.966 112 548.966 61 577.275 36 577.3666 74
9 801.892 106 801.892 38 816.326 57 816.3609 45
10 661.608 108 661.61 42 672.059 52 672.1011 13
11 797.476 127 797.476 60 839.853 96 839.9097 316
12 986.781 110 986.781 59 998.737 39 999.0034 141
13 961.018 115 961.018 40 971.421 17 971.4302 75
14 705.189 118 705.187 29 715.545 157 715.6213 246
15 1044.981 118 1044.981 6 1064 4 1068.014 1488
16 775.113 104 775.1 31 827.737 18 827.8167
130
17 513.108 109 513.108 24 514.347 29 514.3696 12
18 1148.365 91 1148.365 37 1168.512 53 1168.53 166
19 516.55 117 516.56 129 573.696 28 573.7134 122
20 808.334 108 808.334 117 839.348 60 839.4315 139
21 684.192 115 684.192 41 701.988 72 702.016 154
22 891.605 111 891.6 76 931.094 111 931.7247 282
23 731.887 112 731.887 44 784.454 28 784.5266 164
24 808.642 129 808.642 51 819.314 67 819.3718 47
25 652.084 115 652.08 23 684.047 24 684.0895 79
Copyright © 2011 SciRes. AJOR
P. VERMA ET AL.
113
Computational results for RHSCPLP, category B problem 50 × 50 × 50.
R1_RHS R2_RHS R3_RHS R4_RHS
S.No Objective Iteration Objective Iteration Objective Iteration Objective Iteration
1 509.67 2380 509.67 42 538.511 62 538.5373 1062
2 574.659 2468 574.65 74 606.901 73 607.0102 1611
3 719.101 1867 719.105 70 746.441 95 746.4609 1191
4 810.901 2722 810.86 81 830.328 70 830.362 17
5 1042.539 2456 1042.539 70 1074.908 50 1074.91 2037
6 755.642 2859 755.653 86 865.948 90 866.1968 5019
7 906.407 2610 906.4 83 942.529 46 942.5336 599
8 558.779 3002 558.779 81 585.691 77 585.7031 188
9 820.528 2086 820.5 76 833.97 67 833.9829 190
10 673.108 2475 673.076 70 683.48 54 683.486 206
11 805.905 2799 805.9 91 848.007 77 848.0675 892
12 993.454 2457 993.44 90 1005.197 84 1005.206 2672
13 974.165 2377 974.16 83 983.737 73 983.8542 145
14 718.786 1854 718.7 48 728.797 95 728.8193 2267
15 1057.976 2766 1057.96 83 1080.574 53 1080.589 3897
16 786.446 3070 786.44 81 838.286 49 838.2869
906
17 523.433 2614 523.45 80 524.643 61 524.6626 22
18 1157.567 2758 1157.567 83 1177.199 58 1177.203 1171
19 532.823 2395 532.77 78 587.822 69 587.8443 433
20 821.588 2436 821.57 69 854.055 58 854.0678 501
21 696.342 2614 696.342 76 713.355 62 713.3634 1328
22 899.603 2410 899.5 67 939.821 91 939.8355 124
23 744.329 2348 744.319 88 795.562 58 795.5632 2627
24 815.427 3415 815.422 89 827.091 50 827.1029 370
25 669.165 2064 669.16 94 700.632 63 700.6466 182
Copyright © 2011 SciRes. AJOR
114 P. VERMA ET AL.
Appendix B
Computational results for category P1 problems 50 × 50 × 50. (time in seconds).
BB R1 R2 R3
S.No Optimal
Nodes Time Nodes Time Nodes Time Nodes Time
1 3628.5500 319 5260 293 4786 269 4513 269 4516
2 4177.1810 137 1741 137 1666 125 1500 125 1525
3 3867.7100 65 909 61 789 57 711 49 706
4 5747.2630 117 1600 111 1525 111 1431 95 1339
5 4077.6970 187 3163 175 2971 175 2873 175 2905
6 3541.6310 69 888 65 811 43 467 43 457
7 5407.8320 109 1571 93 1297 93 1248 93 1436
8 5015.8550 57 662 45 428 45 426 31 382
9 4271.4190 121 1707 119 1598 119 1597 91 1374
10 3725.4900 131 2016 119 1747 119 1743 119 1727
11 4856.1350 211 2105 201 1920 163 1536 139 1372
12 3553.6670 615 7072 519 5883 499 5645 447 5240
13 4115.5330 65 703 45 416 45 409 45 406
14 5218.9520 13 261 13 272 13 264 13 240
15 5413.4630 297 3428 283 3179 271 3030 259 2880
16 4952.1770 167 2192 145 1822 145 1824 145 1830
17 4593.6770 483 5339 359 3879 359 3885 355 3837
18 4983.7330 297 4970 285 4676 235 3844 235 3848
19 4566.5570 364 4893 361 4759 243 3182 243 3202
20 4783.2700 401 4586 329 3682 271 3009 271 3019
Copyright © 2011 SciRes. AJOR
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115
Computational results for category P2 problems 50 × 50 × 50.
BB R1 R2 R3
S.No Optimal
Nodes Time Nodes Time Nodes Time Nodes Time
1 832.7146 33 748 33 742 33 736 33 735
2 656.6308 99 1844 89 1618 89 1622 89 1644
3 375.7824 17 376 13 264 13 264 13 260
4 547.0326 19 646 19 631 19 607 19 599
5 991.5337 57 1297 57 1254 57 1279 57 1254
6 548.5699 65 1594 65 1578 65 1555 65 1572
7 561.4253 23 673 23 675 23 641 23 628
8 670.8930 75 1427 67 1251 67 1259 67 1259
9 547.2954 51 1165 51 1155 51 1132 51 1147
10 632.5004 11 215 7 118 7 98 7 97
11 473.5373 11 219 9 132 9 134 9 129
12 566.0262 11 227 11 219 9 132 9 133
13 583.2800 35 577 35 575 35 574 35 546
14 521.6505 53 1129 53 1110 53 1102 53 1114
15 702.6561 15 310 15 326 15 328 15 270
16 573.2853 13 333 11 266 11 264 11 241
17 940.2433 119 2975 119 2972 119 2968 119 2919
18 572.7860 37 707 37 687 37 688 37 679
19 493.0821 27 855 25 752 25 747 25 743
20 558.8294 57 1421 57 1415 53 1284 53 1306
Copyright © 2011 SciRes. AJOR
116 P. VERMA ET AL.
Computational results for category P3 problems 100 × 100 × 100.
BB R1 R2 R3
S.No Optimal
Nodes Time Nodes Time Nodes Time Nodes Time
1 37229.89 1217 210810 1217 210780 1213 207760 1013 175297.9
2 40632.64 219 43232 203 39138 203 39114 171 33650.49
3 45733.24 383 77366 363 73386 346 69842 316 63670
4 44268.57 1539 283600 1517 279643 1503 276771 1399 257673.4
5 39611.71 878 170332 796 154451 760 147288 612 118623
6 40424.14 405 97605 390 94082 328 78889 328 78877
7 38088.52 571 121623 484 103126 465 98901 460 97881
8 42114.38 308 53284 305 52823 283 48770 268 46284
9 40042.96 631 163429 625 161924 612 158394 483 124922
10 40399.63 624 154128 590 145761 547 135022 531 131001
11 35108.18 935 201960 872 188436 801 172838 687 148314
12 41030.85 2571 732735 2151 613087 2124 605149 1915 545664
13 37272.51 342 77292 284 64212 241 54319 241 54354
14 41362.47 196 41944 188 40331 160 34163 160 34137
15 38363.8 1315 318230 1275 308621 1162 281116 1151 278457
16 37185.86 761 196338 710 183218 568 146358 568 146448
17 40360.54 2062 583546 1566 443250 1511 427479 1477 417826
18 37376.5 1275 336600 1209 319200 1005 265121 1005 265264
19 42302.74 1617 350889 1561 338820 1257 272619 1251 271292
20 39393.8 1714 368510 1394 299780 1260 270724 1260 270701
Copyright © 2011 SciRes. AJOR
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Copyright © 2011 SciRes. AJOR
117
Computational results for category P4 problems 100 × 100 × 100.
BB R1 R2 R3
S.No Optimal
Nodes Time Nodes Time Nodes Time Nodes Time
1 8709.583 425 106250 425 106112 373 93216 373 93210
2 9476.864 25 3995.1 25 3854.1 13 2026.452 13 1985.452
3 9674.657 71 15926 71 15823 71 15888 69 15413.38
4 10010.77 995 267790 995 267725 995 267693 932 250745.5
5 8624.385 189 44226 189 44117 142 33195 142 33151
6 9308.648 414 102672 414 102642 414 102618 384 95187
7 8012.688 370 85840 370 85794 274 63525 274 63526
8 9758.747 988 211432 883 188830 848 181444 786 168167
9 9351.761 587 102725 587 102666 440 76910 440 76904
10 9436.933 1080 190080 1067 187697 1067 187754 986 173438
11 9331.35 119 28203 97 22909 97 22935 97 22942
12 9282.44 1181 232657 1181 232619 934 183966 934 183930
13 7862.446 601 109983 601 109938 590 107879 532 97257
14 9388.992 547 126904 547 126768 547 126812 498 115482
15 9494.626 164 31488 164 31357 139 26636 139 26642
16 9327.953 1144 257400 968 217726 824 185346 824 185317
17 9417.17 1326 290394 1316 288068 1296 283760 1296 283744
18 8126.866 488 89792 481 88445 358 65822 358 65805
19 8095.194 1402 291616 1298 269873 1298 269934 1285 267208
20 9329.764 1205 256665 1202 255878 1094 232970 1094 232993