BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL SCIENCES, Vol. 64, No. 3, 2016
DOI: 10.1515/bpasts-2016-0056
Algorithms solving the Internet shopping optimization problem
with price discounts
J. MUSIAL1*, J.E. PECERO2, M.C. LOPEZ-LOCES3, H.J. FRAIRE-HUACUJA3,
P. BOUVRY2, and J. BLAZEWICZ1, 4
1
2
Institute of Computing Science, Poznan University of Technology, 2 Piotrowo St., 60-965 Poznan, Poland
Comp. Sci. and Commun. Res. Unit, University of Luxembourg, 6 rue Coudenhove Kalergi, L-1359 Luxembourg, Luxembourg
3
Instituto Tecnológico de Ciudad Madero, Tecnológico Nacional de México, 1º de Mayo S/N, 89440, Ciudad Madero, Mexico
4
Institute of Bioorganic Chemistry, Polish Academy of Sciences, 12/14 Noskowskiego St., 61-704 Poznan, Poland
Abstract. The Internet shopping optimization problem arises when a customer aims to purchase a list of goods from a set of web-stores with
a minimum total cost. This problem is NP-hard in the strong sense. We are interested in solving the Internet shopping optimization problem
with additional delivery costs associated to the web-stores where the goods are bought. It is of interest to extend the model including price
discounts of goods.
The aim of this paper is to present a set of optimization algorithms to solve the problem. Our purpose is to find a compromise solution between
computational time and results close to the optimum value. The performance of the set of algorithms is evaluated through simulations using
real world data collected from 32 web-stores. The quality of the results provided by the set of algorithms is compared to the optimal solutions
for small-size instances of the problem. The optimization algorithms are also evaluated regarding scalability when the size of the instances
increases. The set of results revealed that the algorithms are able to compute good quality solutions close to the optimum in a reasonable time
with very good scalability demonstrating their practicability.
Key words: e-commerce, Internet shopping, applications of operations research, approximations, algorithms, heuristics, combinatorial optimization.
1. Introduction
Internet shopping, fitting into a business-to-consumer subcategory, becomes more and more popular thanks to the facilities
provided by the new information and communication technologies such as cloud computing, mobile computing (mobile devices – smartphones, tablets), the availability of data centers, and
the improvement in the payment gateways on trusted computing
platforms. Products available in web-stores are often cheaper
than those offered by regular local retailers, and a wide choice
of offers is available just a click away from the customer. Based
on outstanding logistics, the delivery can usually be operated
within 48 hours or less. Customers often need to take into account that shipping cost are charged, so that it is a good idea to
group purchased products into sets and buy them from small
number of retailers to minimize these delivery costs. Automating such decisions requires three elements: information about
the product availability, price lists and finally a specialized analytical tool that could find the minimal subset of shops (in this
work we use shops and web-stores interchangeably) where all
products from the customer’s shopping list could be bought at
the lowest price. To automate these decisions one can prepare an
algorithm / computer tool to calculate the best possible solution
from the customers perspective. The issue could be perceived as
an Internet optimization problem connected with online shop*e-mail: [email protected]
ping. Partial solution to the most trivial version of the problem
(just one product, no discounts, no additional costs, and so on)
comes from software agents [1], so called price comparators.
However their current functionality is limited to building a price
rank for a single product among registered offers that fit to the
customer’s query (search phrase).
It is worth noting that beside all benefits of Internet shopping, it is still a type of shopping that may lead to problematic
behaviors. One can notice that some customers may be addicted
to shopping on the Internet. This kind of addiction is somehow similar to gaming and general Internet dependency. Many
types of addictions and negative forms of consumptions connected with shopping in general have been widely researched
and described in both medial and business journals. BusinessDictionary.com states that shopping is “the process of browsing and/or purchasing items in exchange for money’’. What
will be important here that the platform is not so important.
Therefore, Internet shopping experience may cause many of
these negative behaviors. Follow the publication [2] for more
information about different types of online shopping behaviors
and addictions.
State-of-the-art research on the problem we will tackle in
this paper consists of works that introduced Internet shopping
optimization problem (ISOP in short) and more complicated
versions of the mentioned problem, taking into account also
price discounts. [3] introduced ISOP with the following idea.
The addressed problem is to manage a multiple item shopping
list over several shopping locations. The objective is to have all
505
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J. Musial et al.
shopping done at minimum total expense. It is worth pointing
out that dividing the original shopping list into several sub-lists
with items being delivered by different providers increases delivery costs. These are counted and paid individually for each
package (sub-list) assigned to a specific Internet shop in the optimization process. Authors focused on the problem definition
and its complexity analysis (with some sub-cases of the main
problem). The first algorithms and computational results were
introduced in [4].
Much more complicated version of ISOP that takes into account
price discounts was introduced in [5, 6]. The Focus was on the
problem definition and complexity analysis. Some basic algorithms were introduced. The current paper is focused on the
introduced problem with new achievements and contributions
to the shopping optimization topic.
ISOP could be perceived as a base general name for Internet shopping problems since it leaves a lot of space for future
additional requirements and attributes. Introducing complicated
multi-objective decision-aided ISOP or a multi-objective problem could be one the upcoming interesting research steps. Many
additional decisions could be made by customers with respect to
their personal shops preferences, trust, delivery time, negative
factors, and all other significant elements. Since the number
of decisions could increase, it would be reasonable to support
the client with a specific system or tool. This program would
attempt to help them to make reasonable decisions that satisfy
the customer and do not imply significant price changes. An
interesting idea of providing a simple tool that helps decision
makers make good decisions in the financial market is described
in the literature [7].
In this paper we address the Internet shopping optimization
problem considering delivery costs. It is of interest from the
customer side to investigate an extended version of the problem
which also considers price discounts [5, 6] offered by different
sellers. The discount policy is based on real world observations
and similar to that used in [8], in the sense that is expresses the
discount as a function of the total quantity of money spent in
the web-store; the more money a customer spends, the bigger
discount he may obtain. In this work, we assume that there are
no differences between the quality of the goods the web-stores
sell beside the prices they charge for the different products.
We introduce a new set of heuristic approaches to solve
the problem. The set of heuristics is composed of a new lightweight metaheuristic based on a cellular optimization process,
a extended greedy algorithm and two state-of-the-art greedy
algorithms. We have designed the heuristics as a compromise
solution balancing computational time and results close to the
optimum solution. We empirically evaluate the heuristics on
large set of real world data. The set of data was collected from
32 stores and covered the largest United States-based stores,
including Amazon, BarnesandNoble.com, Borders.com, Buy.
com, Booksamillion and top sellers among Internet bookstores
in Poland such as empik.com and merlin.pl. We compare the
performance of the proposed heuristics to the optimal values.
The optimal solutions for small problem instances are computed
using a branch and bound algorithm. To evaluate the scalability
of the heuristics, we increased the problem size.
506
To enlighten the pure original content of the paper one
should notice such important elements as: new original cellular algorithm created especially to solve a problem, redesigned
forecasting algorithm, problem-specific modification of a stateof-the-art algorithm, new world data model as a data generator
for an experiment, original designed experiment with a vast
number of computational test, huge portion of computational
results carefully analyzed and described by the authors.
The paper is organized as follows. Section 2 describes the
problem model. In Section 3, some of the relevant work is presented. In Section 4, we consider the extended version of the
investigated problem. Section 5 provides a detailed description
of a new set of proposed heuristics. It is followed by experimental results presentation in Section 6. We conclude the paper
in Section 7.
2. Problem definition
Notation used throughout this paper is given in Table 1.
Table 1
Notation
Symbol
Explanation
M
set of products
N
set of shops
m
number of products
n
number of shops
i
product indicator
j
shop indicator
Mj
multiset of products available in shop j
dj
delivery price of all products from shop j
yj
usage indicator for shop j
pij
cost of product i in shop j
xij
usage indicator for product i in shop j
T
cumulative value of all products bought in all shops
Tj
cumulative value of all products bought in shop j
f j (Tj)
piecewise function for all products bought in shop j
X = (X1,…, Xn)
sequence of selections of products in shops 1,…, n
F(X)
sum of product and delivery costs
d(x)
X¤
0-1 indicator function for x = 0 and x > 0
F¤
optimal sequence of selections of products
optimal (minimum) total cost
Internet shopping optimization problem (ISOP) should be
described as follows. A single buyer is looking for a multiset
of products M = f1, …, mg to buy in n shops. A multiset of
available products M j, a cost pij of product i in store j, and a delivery cost dj of any subset of the products from shop j. It is assumed that pij = 1 if i 2
/ M j. The problem is to find a sequence
of disjoint selections (or carts) of products X = ( X1, …, X n),
which we call a cart sequence, such that X j µ M j, j = 1, …, n,
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Algorithms solving the Internet shopping optimization problem with price discounts
n
[ j=1
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as F(X):= Σjn=1 (δ(|Xj |)dj + Σi2Xj pij), is minimized. Here |Xj | amount of good k to be procured is defined. To each supplier
denotes the cardinality of the multiset Xj, and δ(x) = 0 if x = 0 i in S we associate a sequence of intervals Zi = f0, 1, …, maxig,
and δ(x) = 1 if x > 0. Its solution is denoted as X ¤, and its indexed by j. Furthermore, for each supplier i 2 S and interval
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and theircosts
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and costs facility
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Discussions
of
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can
be
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in
[13,
14,
15,
16].
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Discussions
FLPs
can be foundfacility
in [13–16].
The tradicustomer locations
and of
their
corresponding
locations
is
xi j ∈ {0, 1}, y j ∈ {0, 1}, i = 1, . . . , m, j = 1, . . . , n.
traditional
discrete
FLP
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[17]
in
the
strong
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tional
discrete
FLP
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in
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strong
sense.
Note,
0 ≤ xi j ≤ y j , i = 1, . . . , m, j = 1, . . . , n,
minimized.
Note,
however,
thatgeneral
the general
problem
dis- were a customer would like to buy products from a given set
however,
that the
problem
ISOPISOP
with with
priceprice
discounts
Discussions
of
FLPs
can
be
found
in
[13,
14,
15,
16].
The
. . {0,
. , m}
in ay jgiven
of iInternet
= 1,
{1,. .. .. ,. ,n.n}
counts
be treated
as a traditional
FLP because
cannot cannot
be treated
as a traditional
discretediscrete
FLP because
there is M =x{1,
,
1},
∈ {0,set1},
= 1, . .shops
. , m, Nj =
ij ∈
traditional
discrete
FLP
is
NP-hard
[17]
in
the
strong
sense.
at
the
minimum
total
final
price.
There
are
the
following
given
there
is
no
evident
motivation
for
a
discount
on
the
cumulative
no evident motivation for a discount on the cumulative cost in
a customer
would
like
buy
products
from
a given
Note, however,
that
theofgeneral
ISOP
price
disparameters
and decision
variables:
cost
in the
sense
distances.
The important
point
tohere
note
the sense
of
distances.
Theproblem
important
pointwith
to
note
ishere
that were
where
a customer
wants
to to
buy
products
from
a given
set set
M
=
{1,
.
.
.
,
m}
in
a
given
set
of
Internet
shops
N
=
{1,
.
. . , n}
counts cannot
be
treated
as
a
traditional
discrete
FLP
because
is
that
this
problem
and
ISOP
are
not
each
other’s
sub-cases,
this problem and ISOP are not each other’s sub-cases, while M = f1,
…, mgprice
in a given
set of Internet
shops
of all products
from shop
j, N = f1, …, ng
d j - delivery
while
the
traditional
discrete
FLP
is
a
special
case
of
any
of
at
the
minimum
total
final
price.
There
are
the
following
given
there is no
motivation
discount
the of
cumulative
theevident
traditional
discrete for
FLPa is
a specialoncase
any of these aty the
minimum total final price. There are the following given
j - usage indicator for shop j,
these
problems.
problems.
parameters
anddecision
decision
variables:
and
variables:
cost in the
sense
of distances. The important point to note here parameters
pi j - standard
price
of product
i in shop j,
the ISOP
Internet
Shopping
Optimization
Problem
Lookingatatand
Internet
shopping
optimization
problem
with d
–
delivery
price
of all products
shop j,
j
is that thisLooking
problem
are
not each
other’s sub-cases,
usage
indicator
for
product
i in from
shop from
j,
x
i
j
priceindicator
of all products
shop j,
dy
j - delivery
considering
price
discounts
one
can
notice
some
similarities
the
focus
on
price
discounts,
one
can
notice
some
similarities
–
usage
for
shop
j,
j
while the traditional discrete FLP is a special case of any of
f j (T j ) - piecewise function (discounting) for final price of
indicator price
for shop
j,
j - usage
with
Quantitydiscount
Discountproblem
Problem(TQD)
(TQD)[8].
[8].To To
show
with Total
total quantity
show
the yp
of product
i in shop j,
ij– standard
all products
T bought in shop
j.
these problems.
price
of product
i in shop
the
similarities
mostofofallalltotoshow
show distinct
distinct differences
similarities
andand
most
differenceswe
we p
x
indicator
for product
i in j,
shop j,
i j - standard
ij– usage
Looking
at enclose
the
Internet
Shopping
Optimization
Problem
functionfor
model
is(discounting)
non-linear
to itsof
should
mathematical
formulation
of TQD.
One
can deshould
enclose
mathematical
formulation
of TQD.
One
can xf
piecewise
function
for final price
usage
product
i in shop according
j,
j (T j)–indicator
iA
j -piecewise
considering
price
discounts
one
can
notice
some
similarities
original
nature
and
it
is
why
we
are
using
non-linear
model
fine
G as
set of
indexed
by k,by
and
as the
of set
n
define
G the
as the
setmofgoods,
m goods,
indexed
k, Sand
S assetthe
all products
T bought
in shop j. for final price of
function
(discounting)
f j (T j ) - piecewise
with Total
Quantity
Discount
Problem
(TQD)
the show
amount here.
suppliers,
indexed
by i. For
each good
k in G,[8].
dk asTo
all products T bought in shop j.
the similarities
of all tois show
distinct
differences
of good and
k to most
be procured
defined.
To each
supplier iwe
in S
P ROPOSITION 1. In [3] ISOP is demonstrated to be part
we
sequence
of intervals of
Zi =
{0, 1,One
. . . , maxi},
piecewise
function model is non-linear according507
to its
Bull.associate
Pol.mathematical
Ac.: aTech.
64(3)formulation
2016
should enclose
TQD.
can de-in- ofANP-Hard
problems. The ISOP can be reduced to the basic
byofj. m
Furthermore,
for each
supplier
S and
nature
and
it
is
why
we
are
using
non-linear
model
fine G asdexed
the set
goods, indexed
by k,
and S ias∈ the
setinterval
of n original
10.1515/bpasts-2016-0056
ISOPwD problem.
define
thegood
minimum
andd maximum
number here.
Zi , li j and
Downloaded from De Gruyter Online at 09/27/2016 01:29:57PM
suppliers,j ∈indexed
byui.i j For
each
k in G,
k as the amount
via ISOPwD
Gdansk University
of Technology
is NP-Hard
in the strong
S TATEMENT 1. The
respectively that needs to be ordered from supplier
of good ofk goods
to be procured
is defined. To each supplier i in S
P ROPOSITION 1. In [3] ISOP is demonstrated to be part
i to be in interval j. Finally, for each supplier i ∈ S, for each sense.
J. Musial et al.
5.2. Algorithm with forecasting. Observed weak points of
Greedy led to the creation of a new, upgraded version. The local step choice analysis is more complicated than in the basic
Proposition 1. In [3] ISOP is demonstrated to be part of Greedy. The forecasting method is looking for a step ahead
NP- Hard problems. The ISOP can be reduced to the basic [4]. Therefore, technically this algorithm is not a strict greedy
ISOPwD problem.
algorithm. Sometimes it proposes J.a current
which is
Musial etsolution
al.
not optimal for the current step (local solution) but for a better
2
order solution.
to hedge against
Statement 1. The ISOPwD is NP-Hard in the strong sense.
overallTable
solution
in hope of providing an optimalInglobal
Price structure for poor performance of Greedy
2, we
developed
anoth
In order to hedge against the instancesTable
similar
to that
in
1
prod
prod n another
delivery heuristic algorithm,
as Forecasting.
Basic ISOP problem (known as strongly NP-hard) can beprod Table
2,2we ...developed
denotedThe
as main ide
shop 1 forW − Forecasting.
ε
W −ε
...TheWmain
− ε idea0 is to check thestep
ahead one
(forecasting
bad s
transformed to the ISOPwD considering the sub-case
situation
step
shop
2
0
0
...
0
W
to
the
penultimate,
the
algori
which all piecewise functions (discounts for each shop n 2 N) ahead (forecasting bad situations). From the first step to the
antoeligible
j for produc
equals fj (Tj) = Tj. Therefore the extended ISOPwD problem penultimate, the algorithm calculates “rating”
pick an shop
eligible
local
optimum
it
looks
one ste
is NP-hard in the strong sense.
shop j for product i. Instead for looking for a local optimum it
Proof. Basic ISOP problem (known as strongly NP-hard) can case from occurring) and cal
looks one step ahead (which prevents a bad case from occurf j (pi j +pi j+1 )+d j
be transformed to the ISOPwD considering the sub-case for
ring) and calculates the choosing factor asTT + for every
j +
2
which all piecewise functions (discounts for each shop n ∈ N)
with
the calculated
lowest calculated v
for every shop j and picking the one with the
lowest
5. Proposed algorithms
equals f j (T j ) = T j . Therefore the extended ISOPwD problem
i isactaken into a
value. In each following step next productnext
i is product
taken into
is NP-hard in the strong sense.
T =j.TAfterwards,
+ f j (pi j ) +ddj j . Afterwa
The ISOPwD is strongly NP-hard. Moreover, to our best knowl- count, i = i + 1. T is set as T = T + fj (pij) + d
The last step of the algorit
edge it cannot be reduced to one of the known problems. There- is set to 0.
5.
Proposed
Algorithms
(forecast
not work beyo
fore, it is apparent right to propose heuristic solution, simple The last step of the algorithm works in a different
waycould
(forecast
The
ISOPwD
is
strongly
NP-hard.
Moreover,
to
our
best
last
product
is
selected
in its
efficient greedy algorithms [5, 22] that use local knowledge could not work beyond the set of products i). The last product
knowledge
it
cannot
be
reduced
to
one
of
the
known
probvalue
T
+
f
(p
).
j ji(p
j ij).
and do not allow any backtracking for efficiency purpose. It is selected in its eligible shop j with minimum value T + f
Therefore,
right
to propose
sois worth to notice that the greedy algorithmlems.
does not
always it is apparent
Piecewise
function
fj ( pijheuristic
+ pij+1) returns
total costs of prodlution, an
simple
efficientucts
greedy
22] that discount
use local for shop
Piecewise
function f j (pi j +
yield optimal solutions. However, it could provide
optimal
i andalgorithms
i + 1 after[5,applying
j.
knowledge
and
do
not
allow
any
backtracking
for
efficiency
products
i
and
i + 1 after apply
or close to optimal solution using much less resources and time
purpose.
It
is
worth
to
notice
that
the
greedy
algorithm
does
than other optimal working algorithms (i.e., full scan).
5.3. Cellular processing algorithm. The cellular processing
Cellular Processing Alg
not always yield optimal
solutions.
However,
it could
provide 5.3.
based
algorithm
is a new
pseudo-parallel
optimization
approcessing
based algorithm is
or close to
optimal
solution
usingmultiple
much less
re5.1. Greedy algorithm. In the first heuristic an
foroptimal
the ISOPwD,
proach
[23].
It includes
processing
cells that explore
tion approach [23]. It include
sources
and time
other optimal
working
algorithms
(i.e.,
denoted as Greedy [9], products are considered
in a certain
or-thandifferent
regions
of the search
space.
Each processing cell can
explore different regions of th
fullorders
scan).and the be implemented using population or search
der. The algorithm is run for various product
based heuristics
ing cell can be implemented u
best solution found is presented to the customer. Let us consid- or an hybridization of them. The main idea and the principle
5.1. Greedy algorithm - Greedy. In the first heuristic for the heuristics or an hybridization
er that the products are sorted in an ascending order 1, …, m. of the algorithm is to split a sequential algorithm into several
ISOPwD, denoted as Greedy [9], products are considered in a principle of the algorithm is to
Values of the total delivery for each store are initially set as dj, pseudo-parallel processing (i.e., cell) modules, so that each cell
certain order. The algorithm is run for various product orders several pseudo-parallel proces
j = f1, …, n. yj is the shipping indicator. The standard price for can explore different regions of the search
space. The main
and the best solution found is presented to the customer. Let each cell can explore different
each product i in shop j is set as pij. In iteration i of Greedy, feature of the new approach is that the iterative
verification
us consider that the products are sorted in an ascending order main feature of the new appr
product i is selected in its eligible shop j with
minimum
valof
the
stagnation
conditions
prevents
wasting
time
on unnec1, . . . , m. Values of the total delivery for each store are initially cation of the stagnation condi
ue T + f j (pij) + d j, and the corresponding T-value
is
re-set:
essary
tasks.
set as d j , j = 1, . . . , n. y j is the shipping indicator. The stan- unnecessary tasks.
T = T + fj (pij) + dj. Afterwards, dj is set to 0.dard price for each product
Wei in
design
based on
We design an opnew algorithm
i a pseudo-parallel
shop an
j isnew
set asalgorithm
pi j . In iteration
Piecewise function fj (pij) returns value of product
i after
applytimization
approach
introduced
by
[23].
The
new
algorithm
we introduced
approach
of Greedy, product i is selected in its eligible shop j with min- timization
ing discount for shop j.
design
is
a cellular
processing
approach,
which
includes
muldesign
is
a
cellular
processing
imum value T + f j (pi j ) + d j , and the corresponding T -value is
We observed that Greedy demonstrates very
good
perfortiple
processing
cells
that
helps
to
explore
different
regions
of
tiple
processing
cells
that help
re-set: T = T + f j (pi j ) + d j . Afterwards, d j is set to 0.
mance on the experimental data. However, it can
provide
a soluthe
search
optimization
space.
the
search
optimization
space.
Piecewise function f j (pi j ) returns value of product i after aption whose value is much worse than the optimum.
One can
candidate of the alg
The of
components
plying discount
for shop The
j. components of the algorithm are a pool
consider products and delivery prices in Table 2. The
experi-that Greedy
solutions,
generated either
a constructive
or a random
algo- either by a
generated
We first
observed
demonstrates
very by
good
perfor- solutions,
mental computation results of Greedy can be found
in [5]
andexperimental
[9]. rithm data.
and a cell
set that
is simple,
and a cell set that is sim
mance
on the
However,
it can
provide independent,
a so- rithmself-contained
For any product sequence algorithm Greedy
selects
all prodapplied
work
the subset
solutions
that with the s
and applied
to work
lution
whose
value is and
much
worseto
than
thewith
optimum.
One of
cancandidate
ucts in shop 1, which costs nW ¡ ε, while anconsider
optimal products
solution andwere
givenprices
to solve.
This2.process
continues
untilgiven
the cells
stall This proc
to solve.
delivery
in Table
The first
ex- were
is to select all products in shop 2, which costsperimental
W.
all results
solutions
in theircan
local
Afterall
that,
the solutions
solutions
in their local opt
computation
of Greedy
be optimum.
found in [5]
return to the pool, and the cells share information
withpool,
eachand the ce
return to the
and [9].
Table 2
other inalgorithm
order to Greedy
escape selects
from the
continue
other inand
order
to escape from
For any product sequence
all local
prod- optimum
Price structure for poor performance ofucts
Greedy
search
global
optimum.
in shop 1, which the
costs
nW −for
ε, the
while
an optimal
solution the search for the global optim
In thisatwork
we generated
this2,
work
wecosts
generated
random.
is to select all products inInshop
which
W . the candidate solutions
prod 1
prod 2
…
delivery
prod n
dom.
Weasdesigned
We designed an iterated local search algorithm
(ILS)
the core an Iterated
thesimplicity
core of theand
cells. This c
5.2.
withofforecasting
- Forecasting.
the cells. This
choice was Observed
made due toasthe
shop 1
…
0
W–ε
W–ε
W
– ε Algorithm
plicity
and
high
configurability
weak points of Greedy
led
to
the
creation
of
a
new,
upgraded
high configurability of this structure, which allows it to be highshop 2
0
0
…
0
W
it to configurations.
be highly scalable and to r
version. The local step
choice analysis
is more
complicated
ly scalable
and to run
in a variety
of hardware
than in the basic Greedy. The forecasting method is looking figurations. Moreover, the ILS
for a step ahead [4]. Therefore, technically this algorithm is metaheuristic that can be seen
technique
not a strict greedy algorithm. Sometimes it proposes a current
508
Bull. Pol. erful
Ac.: Tech.
64(3) for
2016extending s
The
algorithm
starts off by
solution which is not optimal for the current step (local
solu- 10.1515/bpasts-2016-0056
Then,
a
local
search
process i
tion) but for a better
overall
solution
in
hope
of
providing
an
Downloaded from De Gruyter Online at 09/27/2016 01:29:57PM
After that, following an it
optimal global solution.
via Gdansk University of tion.
Technology
A piecewise function model is non-linear according to its
original nature and it is why we are using non-linear model here.
Algorithms solving the Internet shopping optimization problem with price discounts














Fig. 1. Algorithm Cellular-ILS
Moreover, the ILS algorithm is a trajectory-based metaheuristic
that can be seen as a straight-forward, yet powerful technique
for extending simple local search algorithms.
The algorithm starts off by generating an initial solution.
Then, a local search process is applied to the candidate solution. After that, following an iteration based approach, it seeks
to improve the solutions from one iteration to the next. At each
iteration, a perturbation of the obtained local optimum is carried
out. The perturbation mechanism introduces a modification to
a given candidate solution to allow the search process to escape
from a local optimum. A local search is applied to the perturbed
solution. The new solution is then evaluated and accepted as
the new current solution under some conditions. The algorithm
finishes when the termination condition is met.
The proposed local search algorithm comprises the following methods:
● The criticalElements obtains a list of elements that may be
promising in finding a better search region. This case comprises products that are more expensive and are bought only
in one store, so the heuristic favors elements that are cheaper
and are bought in the same store.
● The evaluateChange method explores the entire neighborhood of search to find the permutation that improves the
current solution, but without perturbing it, and returns the
store in which it is more convenient to purchase the product
under consideration.
● The makeChange makes the proposed change given by evaluateChange.
● Once the change is performed, the applyDiscount method
applies the discounts on the price of the products (piecewise
function).
The number of cells used in this configuration was established to five, each one starting from a different region of the
search space and performing the process described beforehand.
The communication, once all were stagnated, was done by comparing the quality of the solutions obtained by each processing
cell.
The stagnation was determined by the number of consecutive iterations without improvement, that in the case of each
cell was established to ten iterations and for the whole process
to five iterations without improvement.
Coded algorithm was carefully designed and build especially to solve ISOPwD problem taking into account its specific
and original nature. The algorithm is original and created from
the basis especially for this purpose.
It is natural to try to relate Cellular to Hyper heuristic. However, it is worth pointing out that Cellular differs from the Hyper
heuristic approach in that each processing cell has complete
knowledge on the problem that is being solved, as opposed to
the Hyper heuristic, where exists a domain barrier among the
controller and the low level heuristics. Another main difference
resides in that the processing cells are adapted and tailored
to the problem that is being solved, while the Hyper heuristic
depends on generic low level heuristics that can be applied to
a greater variety of optimization problems, according to Burke
et al. [24].
5.4. Minimum-minimum algorithm. The min-min algorithm
is commonly and widely used in the context of scheduling independent tasks in distributed computing to minimize the total
completion time of the tasks [25, 26]. One of the main advantages of this approach is that in general is capable to obtain good
quality solutions with a relatively small computational cost, but
as it schedules those tasks or processes with minimum cost first,
it may result in an imbalanced solution [27]. [28] described
the process of the heuristic min-min approach. In this paper
we extended min-min to ISOPwD for the selection of a list of
products in a set of web-stores. The extended version is called
MinMin in this paper.
The process begins with the search of the product in the
list of unassigned products N, which minimizes the total cost
TC in the shopping cart among the different stores M, given
the cost of the product, Cij plus delivery cost, Di . In the case
of a tie, the product with the lower delivery cost is selected,
and if both stores have the same delivery cost, the product is
chosen randomly.
Once assigned, the total cost of the shopping cart is updated
with the selected product, and it is removed from the unassigned
product list N. The process continues until the unassigned product list is empty.
5.5. Branch and bound algorithm. To calculate the optimal
solution, we designed a branch and bound algorithm [29].
The branch and bound (BB) algorithm starts off by calculating an upper bound (UB) employing the solution given by
509
Bull. Pol. Ac.: Tech. 64(3) 2016
- 10.1515/bpasts-2016-0056
Downloaded from De Gruyter Online at 09/27/2016 01:29:57PM
via Gdansk University of Technology
J. Musial et al.
the Cell Processing Algorithm and proceeds to branch the first Internet bookstores in Poland such as empik.com and merlin. pl)
level of the search tree in the stack. Subsequently, the algorithm to create our own model with instances generator as close to
pops the top element of the stack and evaluates the objective reality as possible.
value it would have if it were part of the current solution. If
The working model was prepared as follows. In the
the partial solution exceeds the limit given by the upper bound computational experiments we assume that n 2 f20, 40g,
(UB), the current branch is fathomed. Otherwise, if it were not m 2 f2, 3, …, 10, 15, …, 100g. For each pair (n, m), 100 ina leaf of the tree, the algorithm would pile up the following stances were generated. In each instance, the following values
Musialwere
et al.randomly generated for all i and j in the corresponding
elements within the stack. If it were a leaf, it would meanJ.that
J.
Musial
et al.
the current solution is better than the best global solution found ranges. Delivery
price: d j 2 f5, 10, 15, 20, 25, 30g, publishmended
price ofprice
book
ri ∈
boundsowith
the
new
value
found.
10, 10, 15, 15, 20,20, 25},
and
far. Consequently it would update the upper bound with the er’s recommended
of i:
book
i: r{5,
25g,
andprice of
i 2 f5, mended
price
of
book
i:
r
bound
with
the
new
value
found.
∈
{5,
10,
15,
20,
25},
and pric
new
value found.
i in bookstore
aij ¸ 0.69r
∈ [aiji, i jb,ijb],i jwhere
], where
ai j ≥j,0.69r
bookof ibook
in bookstore
j: j:ppiijj 2 [a
This
process
continues as long as there are elements in the price
j,
∈
[a
,
b
],
where
a
≥
0.6
book
i
in
bookstore
j:
p
This
long
as
there
are elements
in the
,
and
the
structure
of
intervals
[a
, b
]
is
prepared
Thisprocess
processcontinues
continues
as long
as there
are been
elements
≤
1.47r
,
and
the
structure
of
intervals
[a
,
b
]
is
prepared
bij ∙ 1.47r
i
j
i
j
i
j
i
j
j
ij
ij
stack, which
means
that
the as
whole
search
tree
has
ex- inbthe
ij
j
ij ij
stack,
which
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plored. Founded upper bound is now considered as the optimal
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mal
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erature.
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minimum + (median−minimum)
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Computational
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and maximum = 1.47r.
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main memory.
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This kind of advertisement is well-known and very often
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iment
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compared
to
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the optimal
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exact BB algorithm
compared
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stores and
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compared
to
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channel. We
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heuristics to
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ksamillion
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withPHP
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el embed
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and
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mal solution)
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a given
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iment we chose books, because
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mal solution)
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from
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standard
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value is
mainly on electronic bookstores
definition,
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value the
BB algorithm
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as it computes
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cell the
standard
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nsame
∈ {20},
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for
the
instance
of
n
shops
and
m
products
ceptance factor, retailer brand
[31]
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what is important
for the standard deviation value
the
exact
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cell
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valuefollow
is presented
for the same instance of n shops for
1.04
1.04
m ∈ {2, 3, 4, 5, 6, 7, 8, 9, 10},
and discounts
the proposed
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definition,
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100
computational
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uscan
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very useful
presented
for theprice
same
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n shops discounting
and m products
for
and m products
for (n,
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us infunction.
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stores (i.e. Amazon,
formation
on
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stability
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the
algorithms’
results
or
the
lack
100Internet
computational
tests. It can provide
us
with
very
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in subsection 6.1.
with the
very
useful information
on the stability of the algorithms’
BarnesandNoble.com,
Borders.com, Buy.com, Booksamillion stances were generated using
1.02
1.02
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formation on the stability of the algorithms’ results or the
lackoptimal
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results
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lack thereof.
and top sellers among Internet 1.02bookstores in Poland such as The number of instances
thereof.
empik.com and merlin.pl) to
create our own model with in- BB is equal to 900.
Table 3 provides more detailed information
which is rep20 Shops
The algorithms were compared using three metrics: an ap1 1
1
stances generator
as close
to reality
as4 4possible.
Greedy
2 2
3 3
53 5
7 7 6
8 87
98 9
resented
by
the
of
variation
(CV)
normalized
mea2
46 6
5
9 10 10
10
0.09 coefficient
Forecasting
20 Shops
proximation factor, the run time spent by each heuristic
to comProducts
Cellular
Products
Products
The working model was prepared as follows.
In the comMinMin
Greedy
sure
(percentage
value).
Each
cell
represents
the
coefficient
of
7
8
9
10
pute
a
solution,
and
the
dispersion
analysis
based
on
the
stan0.08
0.09
Forecasting
putational experiments Fig.
we assume
that
nresults
∈ {20,comparison
40}, m ∈ - experiment with 20 shops (inCellular
2.
Algorithm
MinMin
dard
deviation.
The
approximation
factor
of
a
heuristic
is
deFig.
2.
Algorithm
results
comparison
–
experiment
with
20
shops
(invariation
(CV)
normalized
measure
(deviation
around
optimum
{2, 3, . . . , 10, 15, . . . , 100}. For each
F(X)
0.07
0.08 pair (n, m), 100 instances
F(X)
cluding solution).
the
optimalThe
solution).
The approximation
factor
is ρ , where
= F(X)
riment
20 shops
(in∗,
fined as
= F(X)
F(X)
represents
solution foundtests for 20 shops where every
cluding
the
optimal
approximation
is ρρ = value)
from
100 the
computational
∗
were with
generated.
In
each
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values were factor
F(X) where F(X)
∗
0.06
∗ denotes the optimal
represents
solution
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a and
heuristic
and
ation
factor generated
is
ρ = F(X)
∗), all
by abyheuristic
and)¤
F(X)F(X)
randomly
for
i and j in the
corresponding
ranges.
where
F(X
represents
solutionthe
found
by a heuristic
F(X
value
for each testsolution.
is presented as a correlation of the result to
0.05
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0.06 optimal solution.
25, 30},
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denotes
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the optimum.
A higher
CVvalues
value means more unstable work
(big difference in distance from the optimal solution) over 100
0.05
0.04
Bull. Pol. Ac.: Tech. XX(Y) 2016
Among all heuristics, Cellular provides the best quality of
solutions
(closest
to theprovides
optimum),
the of
size of the
des the best quality
of all
Among
heuristics,
Cellular
theregardless
best quality
20 Shops
20 Shops
instances.
In
the
worst
case,
the
solution
proposed
by this alardless thesolutions
size of the(closest
to the optimum), regardless the size of the
Greedy
Greedy
was
merely
1.47%
more
expensive
than
the
cheapest
on proposedinstances.
by this al-Ingorithm
0.09
0.09
Forecasting
Forecasting
the worst case, the solution proposed by this algoCellular
Cellular
one. For many instances Cellular algorithm computes the opti- MinMin
MinMin
nsive than the
cheapest
rithm
was merely
1.47% more expensive than the cheapest one.
0.080.08
mal
solution.
Cellular
computed
the
optimal
solution
for
62%
hm computes
opti-instances Cellular algorithm computes the optimal
Forthe
many
Fig. 3. Algorithm standard deviation chart - experiment with 20 shops
of the instances. It is worth noting that the algorithm is very
timal solution
for
62%
0.070.07
(including the optimal solution).
solution.
Cellular
computed
thestandard
optimal
solution
for- experiment
62% of thewith 20
Fig.
3.
Algorithm
deviation
chart
shops
t the algorithm is very stable as regards to the quality of solutions (for most cases the
instances. It is (including
worth noting
that the
algorithm is very stable as
the optimal
solution).
0.060.06
ons (for most cases the solution is between 1.24% - 1.47% worse than optimum).
regards to the
quality
of
solutions
(for
most
cases
the
solution
Both Greedy and Forecasting algorithms provide similar qualTable 3 provides more detailed information which is represe than optimum).
0.050.05
is between 1.24%solutions
– 1.47% (for
worselower
than optimum).
number information
of products which
m, the islatter
s provide similar qual- ity ofTable
3 providesa more
detailed
repre- sented by the coefficient of variation (CV) normalized mea0.04
Both
and
Forecasting
algorithms
better
andtheforcoefficient
a higher number
of similar
products
m > 5 the0.04
forproducts m,
theGreedy
latter was
sented
by
ofprovide
variation
(CV)quality
normalized
mea- sure (percentage value). Each cell represents the coefficient
of
solutions
(for
a lower
number
of
products m,
the
latter
was
mer
outperformed
Forecasting).
Moreover,
it
is
easily
noticeof variation (CV) normalized measure (deviation around optiroducts m > 5 the for- sure (percentage value). Each cell represents the coefficient
0.030.03
better
and
for
a higher
number
of
products
m > 5
the
former
able
that
the
quality
of
solution
degrades
with
the
increasing
ver, it is easily notice- of variation (CV) normalized measure (deviation around opti- mum value) from 100 computational tests for 20 shops where
0.020.02
Forecasting).
Moreover,
it is easily
noticeable
that
of products
m. Greedy
was able
to for
find20
the
optimal
es with theoutperformed
increasing number
mum value)
from
100
computational
tests
shops
where every value for each test is presented as a correlation of the
solution
degrades
with
number
of
solution
6%for
of the
instances
and Forecasting
computed
the
result to the optimum. A higher CV value means more unstable to find the
the quality
optimal of
every for
value
each
testthe
is increasing
presented
as
a correlation
of
0.01 the
0.01
optimal
solutions
forfind
7% the
of
instances.
products m.
able
to
optimal
solution
for 6%more unsta- ble work (big difference in distance from the optimal solution)
ecasting computed
the Greedy
resultwas
to the
optimum.
A the
higher
CV
value means
0 0
The
last
heuristic
algorithm
MinMin
provides
the worst
soover
1003 3test from
a 5given
instance
optimum
of the instances
and
Forecasting
computed
the optimal
solutions
5
7
2 2
4 4
6 6
7 (sometimes
8 8
9 9
10 10 or
es.
ble
work
(big difference
in- distance
from
the
optimal
solution)
Products
Products
lutions
for
a
lower
number
of
products
regarding
the
rest
of
close
to,
sometimes
quite
far
from
it).
We
used
the
results
forworst
7% of
provides the
so-the instances.
over 100 test from a given instance (sometimes optimum or
However,
the
algorithm
isthe
quite
stable
quality,
of the BB
algorithm
as a reference
value. Inwith
each20cell
the CV
s regardingThe
the last
rest heuristic
of algorithms.
close
to, sometimes
quite
far from
it).
We
usedin the
results
algorithm
– MinMin
provides
worst
soluFig.
3. Algorithm
standard
deviation
chart – experiment
shops
therefore
for
a
bigger
number
of
products
it
provides
better
sovalue
is
presented
for
the
same
instance
of
n
shops
and
m
prodquite stabletions
in quality,
of
the
BB
algorithm
as
a
reference
value.
In
each
cell
the
CV
for a lower number of products regarding the rest of algo(including the optimal solution)
Greedy and
Forecasting.
Theof
algorithm
wasmable
ts it provides better so- lutions
valuethan
is presented
for the
same instance
n shops and
prod- ucts for 100 computational tests.
solution of tests.
20% of the instances.
he algorithm was able to find
ucts the
for optimal
100 computational
If one is looking solely for the quality of solutions the undisAn important factor to consider in the context of online
he instances.
Bull. Pol. Ac.: Tech. 64(3) 2016
511
leader
amongfactor
algorithms
is Cellular.
thisofalgoof solutions the undis- putedAn
important
to consider
in theUsing
context
online shopping is the run time needed by any algorithm to compute
10.1515/bpasts-2016-0056
the customer
is able
save 5.88%
more
of the total
cost a solution. Figure 4 displays the results regarding the comparular. Using this algo- rithm
shopping
is the run
timetoneeded
by any
algorithm
toDownloaded
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than
using
MinMin,
6.3%
and
6.64%
more
than
Greedy
and ison of algorithms run time in microseconds [ms] (including
% more of the total cost a solution. Figure 4 displays the results regarding the comparvia Gdansk University of Technology
respectively.
more than Greedy and Forecasting,
ison of algorithms
run time in microseconds [ms] (including the optimal solution). Each cell represents the average value
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
2
3
4
5
6
Products
0
3
4
5
6
7
8
9
Products
Standard Deviation
Standard Deviation
2
10
7
8
9
10
Greedy pres
casting by prov
more than m >
J. Musial et al.
Cellular algori
Fig. 4. Algorithm run time comparison - experiment with 20 shops
m = 5 Forecas
(including the optimal solution).
as the best Cel
Table 3
ucts. On the other hand, all heuristics are very fast so the ideamore
is than m >
theitsother
hand,
heuristics
very fast issues
so
A comparison of algorithm results’ dispersion for the coefficient of products.
to furtherOn
test
quality
andallrun
time forare
scalability
by
among the test
the
idea is to further
test itsofquality
and runThe
timeresults
for scalability
variation (CV) normalized measure
increasing
the number
products m.
are presented
the quality of s
issues
bynext
increasing
the number
of products
m. The
results are
in
the
subsection
after
the
dispersion
analysis.
products m (co
products Greedy Forecasting Cellular MinMin
BB
presented in the next subsection after the dispersion analysis.
The last heu
12
2.62%
1.02%
0.78%
6.33%
0.00%
6.3. Scalability: a set of experiments for heuristic algolutions for a lo
6.3.
Scalability:
a set ofgroup
experiments
for heuristic
algo- abebigger numb
rithms.
In the second
of experiments
a comparison
13
3.96%
2.11%
4.23%
7.67%
0.00%
rithms. In the second group of experiments a comparison be- the Forecasting
tween the set of algorithms was done regarding scalability. In
tween the set of algorithms was done regarding scalability. In provides soluti
14
3.63%
2.53%
1.68%
7.21%
0.00%
these examples we increase the size of the instances as follow:
these examples we increase the size of the instances as follow: than best Cellu
n 2 f20, 40g, m 2 f5, 10,. .15,
…, 100g,
and piecewise
discount15
3.72%
3.61%
2.33%
6.55%
0.00%
n∈
{20, 40}, m ∈ {5, 10, 15,
. , 100},
and piecewise
discountFigure 6 co
ing
functions
follow
definition
from
subsection
6.1.
ing functions follow definition from subsection 6.1.
16
3.27%
3.31%
2.17%
5.69%
0.00%
sults dispersio
Foreach
each
instances
generated.
Instances
For
pairpair
(n, (n, m),
m), 100 100
instances
were were
generated.
Instances
value (deviatio
weregenerated
generated
using
procedure
described
17
3.62%
3.70%
2.71%
6.32%
0.00%
were
using
the the
samesame
procedure
described
in Sec-in Secrithms) from 1
tion 6.2.
tion
6.2.
value for each
18
3.78%
3.81%
2.54%
5.93%
0.00%
to compute
the optimal
solution
valuesult
in and the be
As As
we we
are are
not not
ableable
to compute
the optimal
solution
value in
19
2.67%
2.93%
1.82%
4.84%
0.00%
a reasonable
computational
timetime
givengiven
the size
the of
instances
a reasonable
computational
theofsize
the instances
Forecasting Cell
, be
best
weweevaluate
error
γ γofofeach
evaluatethetherelative
relative
error
eachstrategy
strategyunder
undereach
each met10
3.64%
3.28%
1.84%
4.48%
0.00%
tion value is pr
F(X)
metric.
The
relativeerror
errorisisformally
formally defined
defined as
= F(X) , where
ric. The
relative
as γγ = products for 10
best
F(X)F(X)
represents
the solution
found
by by
a heuristic
and
where
represents
the solution
found
a heuristic
andF(X)ful
bestinformation
thesolution
best solution
among
all heuristics.
test from a given instance (sometimes optimum or close to, F(X)
denotes
the best
among
all heuristics.
best denotes
the algorithm r
sometimes quite far from it). We used the results of the BB
is represented
measure (perce
set of of
experiments
withwith
20 shops.
FigureFigure
5 presents
algorithm as a reference value. In each cell the CV value is 6.3.1.
6.3.1.AA set
experiments
20 shops.
5 presents
solution
values
to the
error error
for thefor100
presented for the same instance of n shops and m products for average
average
solution
values
to relative
the relative
thein-100 in-Figure 7 exp
stances
over
n =n = 20
20 shops,
which
are obtained
by theby
evaluated
algorithm run t
100 computational tests.
stances
over
shops,
which
are obtained
the evaluated
heuristics.
from 100 com
An important factor to consider in the context of online heuristics.
0.001
0.0001
1
2
3
4
5
6
7
8
9
10
Products
shopping is the run time needed by any algorithm to compute J. Musial et al.
a solution. Figure 4 displays the results regarding the compari- 8
20 Shops
20 Shops
Table 3
1.181.18
We
can observe that Cellular outperforms the rest of
son of algorithmsArun
time
in
microseconds
[ms]
(including
the
Greedy
Greedy
comparison of algorithm results’ dispersion for the coefficient of variation
Forecasting
Forecasting
the
algorithms.
For all instances n, m where m =
Cellular
Cellular
(CV) normalized
measure value from
optimal solution). Each cell represents
the average
MinMin
1.161.16
{5,MinMin
10,
15, . . . , 100}, it provides the best quality solution. The
Forecasting Cellular
MinMin
BB
100 computational products
tests forGreedy
20 shops.
behavior of Cellular is the same than in the first set of experi2 products
2.62% (m ∙ 5)
1.02% MinMin
0.78% is the
6.33%
0.00% 1.141.14 ments.
For a low number of
fastest
3
3.96%
2.11%
4.23% of 7.67%
0.00%
algorithm. It can be observed
that when
the number
products
1.121.12
4
3.63%
2.53%
1.68%
7.21%
0.00%
is bigger than five (m > 5),
Greedy
outperforms
the
rest
of
5
3.72%
3.61%
2.33%
6.55%
0.00%
1.1 1.1
algorithms. Differences
between
algorithms
6
3.27% all3.31%
2.17%are very
5.69% sig0.00%
nificant.
7
3.62%
3.70%
2.71%
6.32%
0.00% 1.081.08
8 for BB
3.78%grows
3.81%
2.54%
5.93%
0.00%
Run time execution
exponentially
and
it was
2.67%
0.00% 1.061.06
impossible to prepare 9experiments
for2.93%
a bigger1.82%
number4.84%
of prod20 Shops
Relative error
Relative error
1.18
1.16
Greedy
Forecasting
Cellular
MinMin
1.14
10
3.64%
3.28%
1.84%
4.48%
0.00%
1.041.04
1.1
1.08
1.06
20 Shops
20 Shops
1.021.02
20 Shops
1000 1000
1.04
1000
Greedy
Greedy
Forecasting
Forecasting
Cellular
Cellular
MinMin
MinMin
BB BB
100 100
Relative error
1.12
Greedy
Forecasting
Cellular
MinMin
BB
100
1 1
1.02
10 10
20 20
30 30
40 40
1
10
10
Time [ms]
Time [ms]
1
1
0.1
0.1
30
40
50 50
60 60
Products
50Products
60
70
70 70
80
80
90
80
90
90
100 100
100
Products
Fig.
5. Algorithm
resultscomparison
comparison -–experiment
withwith
20 shops
Fig. 5.
Algorithm
results
experiment
20 shops
F(X)
(without
solution).Relative
Relativeerror
error is
= F(X) , where
where F(X)
(without
BB BB
solution).
is γγ = F(X) repbest
denotes the
represents
the solution
by a heuristicand
andF(X)
F(X)best
resents
the solution
foundfound
by a heuristic
the best
best denotes
best solution among
all heuristics.
solution
among all heuristics
10
Time [ms]
10
20
1
0.1
0.01
Greedy presents better scalability than MinMin and Forecasting by providing the second best quality of solutions. For
We
canthan
observe
Cellular
outperforms
the restofofthethe almore
m > 15that
products
it stabilized
within 10-11%
gorithms.
Foralgorithm
all instances
n, mFor
where
m 2 f5, …, 100g,
Cellular
solution.
a low
number10, of 15, products
Fig. 4. Algorithm run time comparison - experiment with 20 shops
0.001 0.001
m = 5 Forecasting
propose almost
the same
of solution
(including the optimal solution).
it provides
the best quality
solution.
The quality
behavior
of Cellular
the best
Cellular
more expensive). From
is theassame
than
in thealgorithm
first set(0.48%
of experiments.
more than m > 15 products it becomes the worst algorithm
0.00010.0001
are
very
1
1
2
2
3products.
3
4
4On the
5
5other
6 hand,
6
7all 7 heuristics
8
8
9
9
10fast
10 so Greedy presents better scalability than MinMin and Foreamong the tested ones. Moreover, it is easily noticeable that
Products
Products
the idea is to further
test its quality and run time for scalability
casting
by providing
second
quality number
of solutions.
the quality
of solutionthe
degrades
withbest
the increasing
of
issues by increasing the number of products m. The results are
For more
than
m > 15 products
it stabilized
within 10‒11%
Fig. 4. Algorithm run time comparison – experiment with 20 shops
products
m (compared
to the best Cellular
algorithm).
presented in the next subsection after the dispersion analysis.
last heuristic
algorithm
- MinMin provides
the worst
so(including the optimal solution)
of the The
Cellular
algorithm
solution.
For a low
number
of
lutions for a lower number of products (m < 15). However, for
6.3. Scalability: a set of experiments for heuristic algo- a bigger number of products it provides better solutions than
rithms. In the second group of experiments a comparison be- the Forecasting algorithm (but worse than Greedy). MinMin
512
Tech.bigger
64(3) 2016
tween the set of algorithms was done regarding scalability. In provides solutions between 6% andBull.
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these examples we increase the size of the instances as follow: than best Cellular solutions. - 10.1515/bpasts-2016-0056
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via point
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sults dispersion. Each
represents
the of
standard
deviation
For each pair (n, m), 100 instances were generated. Instances value (deviation around the best value calculated from all algo0.001
0.01 0.01
0.0001
1
2
3
4
5
6
Products
7
8
9
10
Greedy presents better scalability than MinMin and ForeGreedy presents better scalability than MinMin and Fore- casting by providing the second best quality of solutions. For
casting by providing the second best quality of solutions. For more than m > 15 products it stabilized within 10-11% of the
Fig.more
4. Algorithm
comparison
experiment
with
20optimization
shops
than m Algorithms
>run
15time
products
it stabilized
within
10-11%
of the Cellular
solving
the-Internet
shopping
problemalgorithm
with pricesolution.
discounts For a low number of products
m
=
5
Forecasting
propose
almost the same quality of solution
(including
the
optimal
solution).
Cellular
algorithm
solution.
For
a
low
number
of
products
xperiment with 20 shops
m = 5 Forecasting propose almost the same quality of solution as the best Cellular algorithm (0.48% more expensive). From
more than m > 15 products it becomes the worst algorithm
Shops
2020
Shops
as the best
algorithm
moreare
expensive).
From
easily
products.
OnCellular
the other
hand, all(0.48%
heuristics
very fast
so noticeable that the quality of solution degrades with
0.08
0.08
among the tested ones. Moreover, it is easily noticeable that
Greedy
Greedy
more
than
m
>
15
products
it
becomes
the
worst
algorithm
the increasing
number of products m (compared to the best
Forecasting the idea is to further test its quality and run time for scalability
Forecasting
ristics are very fast
so
Cellular
Cellular
the quality of solution degrades with the increasing number of
among
the tested the
ones.
Moreover,
it is easily
MinMin
MinMin
0.07
issues
by increasing
number
of products
m. Thenoticeable
resultsCellular
arethat algorithm).
0.07
run time for
scalability
m (compared to the best Cellular algorithm).
the quality of solution degrades with the increasing number of products
heuristic algorithm – MinMin provides the worst
ucts m. The results are presented in the next subsection after the dispersion analysis. The last
The
last
heuristic algorithm - MinMin provides the worst soproducts
m
(compared
to
the
best
Cellular
algorithm).
0.06
0.06
solutions for a lower number of products (m < 15). However,
e dispersion
analysis.
The last heuristic algorithm - MinMin provides the worst so- lutions for a lower number of products (m < 15). However, for
6.3. Scalability: a set of experiments for heuristic algofor a bigger number of products it provides better solutions than
0.05
0.05
lutions for a lower number of products (m < 15). However, for a bigger number of products it provides better solutions than
In the second group of experiments a comparisonthe
be-Forecasting algorithm (but worse than Greedy). MinMin
ts for heuristic algo- rithms.
a bigger number of products it provides better solutions than the Forecasting algorithm (but worse than Greedy). MinMin
the set of algorithms was done regarding scalability.
In
provides
solutions
between
6% and
bigger
ments a comparison
be- tween
0.04
0.04
solutions
between
6%16%
and (on
16%average)
(on average)
bigger
the Forecasting algorithm (but worse than Greedy). MinMin provides
these
examples
we
increase
the
size
of
the
instances
as
follow:
best
Cellular
solutions.
egarding scalability. In provides solutions between 6% and 16% (on average)than
than
best
Cellular
solutions.
bigger
0.03
0.03
{20, 40}, m ∈ {5, 10, 15, . . . , 100}, and piecewise discount-Figure Figure
he instances
as follow: n ∈than
6 contains
a comparison
of heuristic
algorithm
re- re6 contains
a comparison
of heuristic
algorithm
best Cellular solutions.
follow definition from subsection 6.1.
nd piecewise discount- ing functions
sults
dispersion.
Each
point
represents
the
standard
deviation
sults
dispersion.
Each
point
represents
the
standard
deviation
Figure
6
contains
a
comparison
of
heuristic
algorithm
re0.02
For each pair (n, m), 100 instances were generated. Instances
bsection 6.1.0.02
value (deviation
aroundaround
the best
calculated
from
allall
al-algovalue (deviation
the value
best value
calculated
from
sults dispersion. Each point represents the standard deviation
were generated using the same procedure described in gorithms)
Sece generated.
Instances
computationaltests
testsfor
for 20
20 shops
fromfrom
100100
computational
shopswhere
whereevery
0.01
value (deviation around the best value calculated from all algo- rithms)
0.01
6.2.
dure described in Sec- tionrithms)
valuefor
foreach
each test
test is presented
correlation between
the reevery
presentedasasa a correlation
between
from 100 computational tests for 20 shops where
everyvalue
Greedy
As
we are not able to compute the optimal solution value in sult and the best algorithm result for this computation ( Greedy
0 0
the
each
presented
as70 a correlation
theresult
re- and the best algorithm result for this computationbest
best ,
1010
20value
30for
4040 test50is
6060
70
8080
9090 between
100
20
30
50
100
a
reasonable
computational
time
given
the
size
of
the
instances
Forecasting
Forecasting
Greedy
Cellular
Cellular MinMin
MinMin
timal solution value in sult and the best Products
Products
, , best
deviaalgorithm result for this computation ( ( best ,
). In each
eachpoint
pointthe
thestandard
standard
best
best
best , best
best ).
the
relative
error γ of each strategy under each
he size of the
instances we evaluate
Cellular chart
MinMin
tion
valueisispresented
presented for
for the same
instance
ofofn nshops
and m
Fig. 6. AlgorithmForecasting
standard
deviation
–
experiment
with
20
shops
deviation
value
same
instance
shops
,
,
).
In
each
point
the
standard
deviaF(X)
F(X)
best error
bestis formally defined as γ =
, products for 100 computational tests. It can provide very usech strategy under each metric.bestThe relative
F(X)
F(X)
best
best
tion value is presented for the same instance of n shopsand
andmmproducts for 100 computational tests. It can provide very
F(X)
F(X)for
represents
the solutiontests.
found
by aprovide
heuristic
and
defined as γ = F(X)
, where
ful information
concerning
quality
(based
stability) of
products
100 computational
It can
very
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thethe
quality
(based
ononstability)
best
F(X)
denotes the
best solution
among
all
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best
best
the
algorithm
results.
Table
4
contains
the
information
which
und by a heuristic
and
ful
information
concerning
the
quality
(based
on
stability)
of
products m = 5 Forecasting propose almost the same qual- of the algorithm results. Table 4 contains the information which
is
represented
by
the
coefficient
of
variation
(CV)
normalized
ng all heuristics.
the
algorithm
results.
Table
4
contains
the
information
which
ity of solution as the best Cellular algorithm (0.48% more is represented by the coefficient of variation (CV) normalized
(percentage
Amore
set of than
experiments
with
20ofshops.
Figure
presents
is represented
by the
coefficient
variation
(CV)5 normalized
expensive).6.3.1.
From
m > 15
products
it becomes
measuremeasure
(percentage
value).value).
average
solution
valuesvalue).
to the relative error for the 100 inFigure 7 exposes information regarding the comparison of
measure
(percentage
hops. Figure
5
presents
the worst algorithm among the tested ones. Moreover, it is
Figure 7 exposes information regarding the comparison
over n7 =
20 shops,
which areregarding
obtained by
evaluated of algorithm run time [ms]. Each cell represents the average value
error for the 100 in- stances
Figure
exposes
information
thethe
comparison
of algorithm run time [ms]. Each cell represents the average
from 100 computational tests for 20 shops. We can observe
tained by the evaluated heuristics.
algorithm run time
Each cell represents the average
valuefrom
Table[ms].
4
value
100 computational tests for 20 shops. We can obfrom
100
computational
tests
for
20
shops.
We
can
observe
Heuristic algorithm results’ dispersion for the coefficient
serve that for a low number of products m = 5 MinMin is the
8 of variation (CV) normalized measure
Bull.
Bull. Pol.
Pol. Ac.:
Ac.: Tech.
Tech. XX(Y)
XX(Y) 2016
2016
0.001
0.001
0.0001
0.0001
11
22
33
44
55
66
77
88
99
10
10
Products
Products
8
9
10
Standard Deviation
Standard Deviation
fastest algorithm. For other instances of the ISOPwD problem
(number of products m > 5), algorithm Greedy is the fastest.
Differences between all algorithms are very significant. The
following example illustrates these differences. A few observations are worth noting.
Bull. Pol. Ac.: Tech. XX(Y) 2016
products
shops
Greedy
Forecasting
Cellular MinMin
15
20
4.45%
2.92%
2.62%
6.58%
10
20
3.75%
3.73%
0.94%
5.24%
15
20
2.79%
3.17%
0.00%
3.86%
20
20
3.18%
3.19%
0.00%
3.93%
25
20
2.65%
2.65%
0.00%
3.74%
30
20
2.71%
3.09%
0.00%
3.26%
35
20
2.36%
2.47%
0.00%
3.50%
40
20
3.42%
2.36%
0.00%
3.82%
45
20
3.04%
2.38%
0.00%
3.62%
50
20
3.09%
2.40%
0.00%
3.61%
55
20
2.69%
2.06%
0.00%
3.29%
60
20
2.51%
2.27%
0.00%
3.40%
65
20
2.10%
2.17%
0.00%
3.72%
70
20
3.77%
2.12%
0.00%
3.62%
75
20
2.57%
2.35%
0.00%
3.37%
80
20
2.73%
1.98%
0.00%
3.23%
85
20
3.48%
2.36%
0.00%
3.56%
90
20
2.06%
1.99%
0.00%
3.05%
95
20
3.47%
1.98%
0.00%
3.49%
100
20
2.71%
3.09%
0.00%
3.26%
Shops
20 20
Shops
2000
2000
Greedy
Greedy
Forecasting
Forecasting
Cellular
Cellular
MinMin
MinMin
1500
1500
Time [ms]
Time [ms]
7
1000
1000
500
500
00
1010
2020
3030
40 40
50 50
Products
Products
60 60
70 70
80 80
90 90
100100
Fig. 7. Algorithm processing time comparison – experiment with
20 shops (without BB solution)
6.3.2. A set of experiments with 40 shops. There were no
changes regarding the behavior of Cellular, it provides the best
quality of solutions. That is, for all instances n, m it provided
the best solution.
513
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J. Musial et al.
4040
Shops
Shops
1.16
1.16
1.14
1.14
Greedy
Greedy
Forecasting
Forecasting
Cellular
Cellular
MinMin
MinMin
1.12
1.12
Relative error
Relative error
1.11.1
1.08
1.08
1.06
1.06
1.04
1.04
1.02
1.02
1
1
10
10
20
20
30
30
40
40
50
50
Products
Products
60
60
70
70
80
80
90
90
100
100
Fig. 8. Algorithm results comparison (Relative error measure) – experiment with 40 shops (without BB solution)
Regarding 40 shops experiment general observations are
similar to the ones from the 20 web-stores experiment. However, some differences are significant and worth to notice. All
standard deviation and CV values are lower for the experiment
with 40 shops. Algorithm Greedy provides a lower dispersion
level than Forecasting (slightly but still). Observations for the
Cellular and MinMin algorithms are similar to the experiment
with 20 web-stores.
Regarding the computation time (see Fig. 10), for all instances with n = 40 web-stores, algorithm Greedy is the fastest. Differences between all algorithms are very significant.
Calculation times are even more important here, since goal application will work as an online web-site. There is plenty of
research showing that users do not want to wait to see a webpage content, because they quickly lose interest. An up-to-date
survey is given in [32].
40 Shops
40 Shops
4000
4000
3000
3000
2000
2000
1000
1000
0 0
4040Shops
Shops
Greedy
Greedy
Forecasting
Forecasting
Cellular
Cellular
MinMin
MinMin
Standard Deviation
Standard Deviation
0.06
0.06
0.05
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
00
1010
2020
3030
4040
5050
Products
Products
6060
7070
8080
9090
100
100
Fig. 9. Algorithm standard deviation chart – experiment with 40 shops
514
10 10
20 20
30 30
40 40
50 50
Products
Products
60 60
70 70
80 80
90 90
100 100
Fig. 10. Algorithm processing time comparison – experiment with
40 shops (without BB solution)
0.08
0.08
0.07
0.07
Greedy
Greedy
Forecasting
Forecasting
Cellular
Cellular
MinMin
MinMin
Time [ms]
Time [ms]
Figure 8 displays relative error average results. We can notice that characteristics of the results are not far away from
experiment with m = 20 shops.
Algorithm Greedy provides the second best quality of
solutions. For more than m > 25 products it stabilized within
9.6‒10.8% of the Cellular algorithm solutions. For a low number
of products m = 5 Forecasting computes almost the same quality
of solution as the best Cellular algorithm (1.55% more expensive). From more than m > 10 products it becomes the worst algorithm among the tested ones. Moreover, it is easily noticeable
that the quality of solution degrade with the increasing number
of products m (compared to the best {Cellular algorithm}) from
1.55% to 15.83% percent on average greater than Cellular. MinMin provides the worst solutions for a lower number of products
(m = 5). However, for a bigger number of products it provides
better solutions than the Forecasting algorithm (but worse than
Greedy). The algorithm also provides better scalability than the
previous experiment for smaller number of web-stores.
Figure 9 contains a comparison of heuristic algorithm results dispersion.
5000
5000
Concerning all results collected in the computational experiment it can be stated that the Cellular algorithm provides the
best solutions among all heuristic algorithms. The algorithm also
computes more stable solutions than the rest of compared algorithms. On the other hand, it is the slowest one. What is worth
noting is that it could be much more efficient when parallelized
on five machines (each cell can process on a different machine).
Another important consideration is that in a real e-commerce
web site the instances of the algorithms will be executed in the
server machines (i.e. in a web-server of a data center), which
usually are more powerful than customers’ machines.
Algorithm Greedy can result in a lot of interest due to very
fast processing times in which it can provide a good quality
of solutions. Both algorithms Greedy and Forecasting can be
parallelized for two machines (both algorithms make two separate executions and pick the best of these two at the final step).
The performance of MinMin can be improved by using
a local search algorithm [33]. To improve the run time execution MinMin can be easily parallelized. There are different
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Algorithms solving the Internet shopping optimization problem with price discounts
parallel implementations of the min-min scheduling algorithm
reported in literature [33, 34], both CPU and GPU parallel
implementations.
Given the fast run time execution and the good quality of
solutions that Greedy, Forecasting and MinMin can compute
in average their solutions can be used as initial seeds for Cellular. It could help to improve the run time execution of the
algorithm without degrading its performance. Special database
system optimization could be performed to decrease all-around
computational times [35].
7. Conclusions
In this paper, we addressed the Internet shopping optimization
problem including price discounts. For the practical application,
we created a working model as close to real Internet shopping
conditions as possible. The main reason was their wide choice
in Internet web-stores and frequency purchase. A new set of
algorithms is presented, three heuristics and a new cellular processing optimization algorithm. Computer experiments demonstrated their good performance. The results generated by the
algorithms were compared to the optimal solutions computed
by a proposed branch and bound algorithm.
As for current version of the problem (doesn’t count yet
with a linear model) it is not possible at the moment to perform
a relaxation technique on the restrictions or to implement it on
a MIP solver such as CPLEX or Gurobi. Nonetheless, presentation of a valid MILP model is certainly worth considering and
interesting future step – this will be one of our next challenges.
In the future, we plan to extend the Internet shopping model to
include additional constraints such as: minimum delivery times,
incomplete shopping lists realization, and maximum budget.
Moreover, we plan to propose new quality algorithms for dual
discounting function ISOP [36]. The experimental results demonstrated the potential by the new Cellular processing optimization
algorithm. However, one of the main concerns for its applicability to ISOP is the time needed to find a solution. To alleviate the
problem and also deal with scalability we consider investigating
a parallel version of the algorithm on a GPU infrastructure. Furthermore, there are some interesting similarities between ISOP
and exciting Cloud Brokering [37]. Linking it with ISOP may
result in the possibility of using algorithms prepared for ISOP.
Acknowledgments. This study was partially supported by the
FNR (Luxembourg) and NCBiR (Poland), through IShOP project, INTER/POLLUX/13/6466384.
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