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Abstract

Internet shopping optimization problem A high number of Internet shops makes it difficult for a customer to review manually all the available offers and select optimal outlets for shopping. A partial solution to the problem is brought by price comparators which produce price rankings from collected offers. However, their possibilities are limited to a comparison of offers for a single product requested by the customer. The issue we investigate in this paper is a multiple-item multiple-shop optimization problem, in which total expenses of a customer to buy a given set of items should be minimized over all available offers. In this paper, the Internet Shopping Optimization Problem (ISOP) is defined in a formal way and a proof of its strong NP-hardness is provided. We also describe polynomial time algorithms for special cases of the problem.
Int. J. Appl. Math. Comput. Sci., 2010, Vol. 20, No. 2, 385–390
DOI: 10.2478/v10006-010-0028-0
INTERNET SHOPPING OPTIMIZATION PROBLEM
JACEK BŁA
˙
ZEWICZ
,∗∗∗
,MIKHAIL Y. KOVALYOV
∗∗
,J˛EDRZEJ MUSIAŁ
,
A
NDRZEJ P. URBA
´
NSKI
,ADAM WOJCIECHOWSKI
Institute of Computing Science
Pozna
´
n University of Technology, ul. Piotrowo 2, 60–965 Pozna
´
n, Poland
e-mail: {Jacek.Blazewicz,edrzej.Musial}@cs.put.poznan.pl
{Andrzej.Urbanski,Adam.Wojciechowski}@cs.put.poznan.pl
∗∗
United Institute of Informatics Problems
National Academy of Sciences of Belarus, Surganova 6, 220012 Minsk, Belarus
e-mail: kovalyov_my@newman.bas-net.by
∗∗∗
Institute of Bioorganic Chemistry
Polish Academy of Sciences, Z. Noskowskiego 12/14, 61–704 Pozna
´
n, Poland
A high number of Internet shops makes it difficult for a customer to review manually all the available offers and select
optimal outlets for shopping. A partial solution to the problem is brought by price comparators which produce price rankings
from collected offers. However, their possibilities are limited to a comparison of offers for a single product requested by
the customer. The issue we investigate in this paper is a multiple-item multiple-shop optimization problem, in which total
expenses of a customer to buy a given set of items should be minimized over all available offers. In this paper, the Internet
Shopping Optimization Problem (ISOP) is defined in a formal way and a proof of its strong NP-hardness is provided. We
also describe polynomial time algorithms for special cases of the problem.
Keywords: algorithms, computational complexity, combinatorial algorithms, optimization, Internet shopping.
1. Introduction
On-line shopping is one of key business activities offered
over the Internet. A survey concerning American Inter-
net users behavior published by Pew Internet & Ameri-
can Life Project in February 2008 (Horrigan, 2008) shows
that the population of on-line customers grows rapidly and
systematically from year to year. The number of on-line
users either buying or searching for products on-line since
2000 has roughly doubled. While in 2000 22% of Ame-
ricans (46% of on-line users) had some experience with
buying products in virtual shops, the ratio grew to 39%
in 2003 and reached 49% (66% of on-line users) in 2007.
The developmentof on-line shopping is also stimulated by
the increasing number of Internet users. E-commerce re-
venue has grown from $7.4 billion in the middle of 2000 to
$34.7 billion in the third quarter of 2007. A survey concer-
ning the behavior of customers in Poland (Gemius, 2008)
confirms the tendency observed in America. Among va-
rious business instruments commonly available on-line
since 1990s, like auctions (Lee, 1998; Klein, 2000) and
(Vulkan, 2003, pp. 149-178), banking and secure pay-
ment (Langdon et al., 2000), shopping (Liang and Hu-
ang, 1998), electronic libraries (Lesk, 1997), etc., on-line
retail remains the service offered by the highest number of
providers. A growing number of on-line shops and incre-
ased accessibility to customers world-wide due to the use
of credit card payment in on-line transactions (Langdon
et al., 2000) are key attributes that force strong compe-
tition on the market, keep prices low (when compared to
off-line shopping) and make more and more customers in-
terested in on-line purchasing (Vulkan, 2003, pp. 22-27).
However, a wide choice of on-line shops makes it diffi-
cult to manually compare all the offers and choose optimal
providers for the required set of products.
A solution of this problem has been supported by
software agents (Tolle and Chen, 2000), so-called pri-
ce comparison sites. The concept of a price compara-
tor is built on the idea of collecting offers of many
on-line shops and building a price ranking on a custo-
386
J. Bła
˙
zewicz et al.
mer’s request. This approach is commonly accepted by
customers and, according to Alexa Rank, popular price
comparison services belong to the group of 1000 most
viewed sites world-wide: shopping.com: 518-th pla-
ce, nextag.com: 533-th place, bizrate.com: 600-
th place, shoplocal.com: 932-th place (site popularity
results registered in October 2008, www.alexa.com).
It is worth noting that price ranking built on-line on a
customer’s request expressed in a text query (product de-
scription) is a solution to a specific case of shopping, in
which a customer wants to buy a single product. Multi-
ple item shopping is not supported by price comparators
available nowadays. Furthermore, price comparison sites,
being commercial projects, tend to optimize their incomes
from directing customers to particular on-line shops. As a
result, price comparison sites play the role of recommen-
der systems (Satzger et al., 2006) which tend to detect a
customer’s preferences and interests in order to suggest
products to buy. A side effect of the problems mentioned
above is the loss of customer confidence. To illustrate the
optimization process, we would like to consider and assess
its benefits. To this end, let us consider an example below.
A customer wants to buy five books. The prices of the bo-
oks and delivery costs in six shops which the customer
considers as potential shopping locations are collected in
Table 1.
The customer’s goal is to buy all five books at mini-
mum expense. The support from currently available price
comparators allows building the customer’s basket based
on optimal offers for each individual book. The result of
the selection process is presented in Table 2.
The shopping performed upon price comparator sug-
gestions would not be optimal (total cost: 210) because
simple price comparison does not include delivery cost,
which grows as the number of shopping locations grows.
In order to find the cheapest solution for the ISOP illu-
strated above, one can perform a complete search of all
possible realizations of shopping. The simple example we
analyze shows that the optimization process may bring so-
me savings. In our example, the optimal cost of purcha-
se (in Shop 1 and Shop 4) equals 189 (see Table 3). The
problem addressed in this paper is to manage a multiple-
item shopping list over several shopping locations. The
objective is to have all the shopping done at minimum to-
tal expense. One should notice that dividing the original
shopping list into several sub-lists whose items will be
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 opti-
mization process.
In the sequel, we consider the above mentioned Inter-
net shopping optimization problem in a more formal way.
In Section 2, a formal definition of the problem is given.
In Section 3, we prove that the ISOP is NP-hard in the
strong sense and that it is not approximable in polynomial
time. In Section 4, we demonstrate that the ISOP is po-
lynomially solvable if the number of products to buy, n,
or the number of shops, m, is a given constant. The paper
concludes with a summary of the results and suggestions
for future research.
2. Problem definition
The notation used throughout this paper is given in Ta-
ble 4. We study the following problem of Internet shop-
ping. A single buyer looks for a multiset of products
N = {1,...,n} to buy in m shops. A multiset of ava-
ilable products N
l
,acostc
jl
of each product j N
l
,
and a delivery cost d
l
of any subset of the products from
the shop to the buyer are associated with each shop l,
l =1,...,m. It is assumed that c
jl
= if j ∈ N
l
.
The problem is to find a sequence of disjoint selections
(or carts) of products X =(X
1
,...,X
m
),whichwe
call a cart sequence, such that X
l
N
l
, l =1,...,m,
m
l=1
X
l
= N , and the total product and delivery cost, de-
noted by F (X):=
m
l=1
δ(|X
l
|)d
l
+
jX
l
c
jl
,is
minimized. Here |X
l
| denotes the cardinality of the mul-
tiset X
l
,andδ(x)=0if x =0and δ(x)=1if x>0.
We denote this problem as the ISOP, its optimal solution
as X
, and its optimal solution value as F
.
3. Strong NP-hardness and
inapproximability
In this section we will analyze the computational com-
plexity of the ISOP. We will demonstrate its strong NP-
hardness by proving strong NP-completeness of its deci-
sion counterpart—Problem P1. The latter has the same in-
put as the ISOP plus an additional parameter y,andthe
question is to determine whether there exists a selection
of products with the total cost F (X) y.
Theorem 1. The ISOP is NP-hard in the strong sense even
if all costs of the available products are equal to zero and
all the delivery costs are equal to one.
Proof. We construct a pseudo-polynomial transformation
of Problem P1 from the strongly NP-complete problem
E
XACT COVER BY 3-SETS (X3C), see (Garey and John-
son, 1979).
E
XACT COVER BY 3-SETS (X3C) can be defined
as follows: Given a family E = {E
1
,...,E
L
} of three-
element subsets of the set K = {1,...,3k}, does E con-
tain an exact cover of K, i.e., a subfamily Y E such that
each j K belongs to exactly one three-element set in Y ?
It is clear that if Y is a solution to X3C, then |Y | = k.
Given an instance of X3C, we construct the following
instance of Problem P1. There are m = L shops with
available products of the sets N
l
= E
l
, l =1,...,L.The
Internet shopping optimization problem
387
Table 1. Prices of books and delivery costs offered by six internet shops.
Cost Book a Book b Book c Book d Book e Delivery Total
Shop 1 18 39 29 48 59 10 203
Shop 2 24 45 23 54 44 15 205
Shop 3 22 45 23 53 53 15 211
Shop 4 28 47 17 57 47 10 206
Shop 5 24 42 24 47 59 10 206
Shop 6 27 48 20 55 53 15 218
Table 2. Price comparator solution—the result of the selection process.
Book a Book b Book c Book d Book e Delivery Total
Price 18 39 17 47 44 45 210
Shop Shop 1 Shop 1 Shop 4 Shop 5 Shop 2
Table 3. Optimal purchase cost in selected shops.
Book a Book b Book c Book d Book e Delivery Total
Cost 18 39 17 48 47 20 189
Shop Shop 1 Shop 1 Shop 4 Shop 1 Shop 4
buyer would like to purchase the set of products N = K.
The cost of any product available in any shop is equal to
zero (c
jl
=0, j N
l
, l =1,...,m), the delivery cost
from any shop to the buyer is equal to one (d
l
=1, l =
1,...,m), and the threshold value of the criteria is y = k.
We show that X3C has a solution if and only if there exists
a solution X for the constructed instance of Problem P1
with F (X) k. It is easy to see that our transformation
is polynomial and pseudo-polynomial at the same time.
Therefore, Problem P1 belongs to the class NP.
Let Y be a solution to X3C. Construct a solution to
Problem P1, in which the required products are purchased
in k shops determined by their sets of products N
l
= E
l
Y , i.e., X
l
= N
l
if N
l
Y and X
l
= if N
l
∈ Y. Since
Y is an exact cover of K, all the required products are
purchased, and the cost of the corresponding solution is
F (X)=k.
Now assume that there exists a solution X to Pro-
blem P1 with the cost value F (X) k. On the one hand,
for this solution the number of shops with X
l
= φ should
not exceed k because otherwise F (X) >k. On the other
hand, the number of these shops should not be less than
k because otherwise at least one product j N will not
be purchased. Therefore, there are exactly k shops with
X
l
= . Since the purchased products form the set K,the
collection of the shops with X
l
= represents a solution
to X3C.
We now discuss the approximability of the ISOP. Let
us consider its special case, in which the cost of any pro-
duct in any shop is equal to zero, and the delivery cost
from any shop to the buyer is equal to one. This special
case is equivalent to the following M
INIMUM SET CO-
VER problem, see (Crescenzi and Kann, 2008).
M
INIMUM SET COVER: Given a collection C of subsets
of a finite set S, find a set cover for S, i.e., a subset C
C such that every element in S belongs to at least one
member of C
, which minimizes the cardinality of the set
cover, i.e., |C
|.
Due to (Raz and Safra, 1997), the problem M
INI-
MUM SET COVER is polynomially non-approximable wi-
thin the ratio c·ln |S|, for some constant c>0. Therefore,
the following statement can be formed.
Statement 1. There exists no polynomial (c · ln n)-
approximation algorithm for the ISOP, unless P = NP.
4. Polynomial algorithms for special cases
Notice that the intractability of the ISOP is established un-
der the assumption that both the number of products to
buy, n, and the number of shops, m, are variables. In this
section we present two solution approaches for the ISOP,
which are polynomial if either n or m is a constant.
The idea of our first algorithm SHOP-ENUM is to
enumerate all possible selections of shops containing all
the required products, to choose the best cart sequence
X
(M)
for each selection of shops M , M ⊆{1,...,m},
to calculate the total product and delivery cost for each M ,
and, finally, to find a cart sequence X
with the minimum
total cost, F
.
Algorithm SHOP-ENUM
Step 1. Set F
= and X
l
= , l =1,...,m.
Step 2. Consider selections of shops to buy all the requ-
ired products. Each shop l, l ∈{1,...,m}, such that
388
J. Bła
˙
zewicz et al.
Table 4. Problem definition—table of notation.
Symbol Explanation
n number of products
m number of shops
N
l
multiset of products available in shop l
c
jl
cost of each product j N
l
d
l
delivery cost for shop l, l =1,...,m
X =(X
1
,...,X
m
) sequence of selections of products in shops 1,...,m
F ( X) sum of product and delivery costs
δ(x) 0-1 indicator function for x =0and x>0
X
optimal sequence of selections of products
F
optimal (minimum) total cost
N
l
is non-empty can be selected or not; therefore, the-
re are at most 2
m
selections. For each selection of shops
M perform the following computations. If M does not
contain all the required products, then skip considering
this selection. Otherwise, do the following: Determine a
cart sequence X
(M)
=(X
(M)
1
,...,X
(M)
m
), X
(M)
l
N
l
,
l =1,...,m, as follows. For each product i N ,se-
lect a shop l M in which the cost of product i is lo-
west. This can be done in constant time if costs c
il
are
stored for each product i in a heap. Assign product i to
the corresponding multiset X
(M)
l
. Notice that the assign-
ment strategy can eliminate some of the selected shops.
This strategy is optimal for a given selection of shops due
to the fact that the total product costs are minimized and
the total delivery cost does not exceed the sum of deli-
very costs for the selected shops. Calculate the total cost,
F (X
(M)
).IfF (X
(M)
) <F
, re-set F
:= F (X
(M)
)
and X
:= X
(M)
.
Step 3. Output optimal solution X
with the minimum
cost F
.
The time complexity of the SHOP-ENUM algorithm
is O(n2
m
), which is polynomial (linear) if the number of
shops m is a constant.
The idea of our second algorithm, PRODUCT-
ENUM, is to enumerate all possible shop choices for each
product. Let S =(S
1
,...,S
n
) be a shop sequence such
that S
i
∈{1,...,m} is a shop in which product i will
be purchased, i =1,...,n. The algorithm determines a
shop sequence S
with the minimum total cost, F
,ofthe
corresponding cart sequence.
Algorithm PRODUCT-ENUM
Step 1. Set F
= and S
i
= , i =1,...,n.
Step 2. Consider shop sequences S =(S
1
,...,S
n
) such
that product i N will be purchased in shop S
i
, i =
1,...,n. There are at most m
n
such sequences. For each
sequence S, calculate the cost of the corresponding cart
sequence, F (S).IfF (S) <F
, re-set F
= F (S), S
:=
S and pass on to considering the next shop sequence.
Step 3. Output optimal shop sequence S
with the mini-
mum total cost F
.
The time complexity of the algorithm PRODUCT-
ENUM is O(nm
n
), which is polynomial if the number
of products n is a constant.
The algorithm PRODUCT-ENUM can be applied in
practice if the number of products to buy, n,issmalland
the number of shops having these products, m,islarge.
Alternatively, if the number of shops m is small and the
number of the required products n is large, the algorithm
SHOP-ENUM can be efficient.
5. Conclusions
Internet shopping is often attributed with prices lower
than in traditional shops. Another strong advantage of an
on-line purchase is a wide choice of alternative retailers
which, in general, remain at the same distance from the
customer—at least one day for shopping delivery. Posta-
ge cost is often non-zero, which makes it reasonable to
group purchased products so that each group is ordered in
the same shop, and the total purchase and delivery cost is
minimized. Changing retail location for the same product
can be possible, provided that the customer is guarante-
ed that the product in a new location is identical to that
in an old one. In the case of changing the retail location,
the quantity and quality of the product must be identical.
However, the total price can change because of different
times of delivery, different profit margins, etc. Customers
are interested in minimizing the total product and delive-
ry cost. Currently available price comparators can be used
for these purposes only occasionally, because they do not
provide multiple-item basket optimization.
We introduced the Internet shopping optimization
problem and provided a proof of its strong NP-hardness.
Furthermore, two polynomial time algorithms for special
cases of the ISOP were described. In the future work, we
Internet shopping optimization problem
389
intend to derive and experimentally test heuristic appro-
aches for the ISOP to make the suggested approach ap-
plicable for solving complex shopping cart optimization
problems in on-line applications. The ideas and algorith-
mic results given in (Musiał and Wojciechowski, 2009)
for a simplified version of the ISOP can be generalized
and extended for these purposes.
Acknowledgment
The work was partially supported by the grant
from the Ministry of Science and Higher Education
no. N519188933. Also, Jacek Bła
˙
zewicz acknowledges
the support of an INRIA Rhone-Alpes grant.
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Satzger, B. , Endres, M. and Kielssing, W. (2006). A preference-
based recommender system, in K. Bauknecht, B. Pröll and
H. Werthner (Eds.), E-Commerce and Web Technologies,
Lecture Notes in Computer Science, Vol. 4082, Springer-
Verlag, Berlin/Heidelberg, pp. 31–40.
Tolle, K. and Chen, H. (2000). Intelligent software agents
for electronic commerce, in M. Shaw, R. Blanning,
T. Strader and A. Whinston (Eds.), Handbook on
Electronic Commerce, Springer-Verlag, Berlin/Heidelberg,
pp. 265–382.
Vulkan, N. (2003). The Economics of E-Commerce. A Strategic
Guide to Understanding and Designing the On-line Mar-
ketplace, Princeton University Press, Princeton, NJ.
Jacek Bła
˙
zewicz, born in 1951 (M.Sc. in control engineering: 1974,
Ph.D. and D.Sc. in computer science: 1977 and 1980, respectively), is
a professor of computer science at the Pozna
´
n University of Technology.
Presently he is a deputy director of the Institute of Computing Science.
His research interests include algorithm design and complexity analysis
of algorithms, especially in bioinformatics as well as in scheduling the-
ory. He has published widely in the above fields (over 300 papers) in
many outstanding journals. He is the author and co-author of 14 mono-
graphs, an editor of the International Series of Handbooks in Information
Systems (Springer-Verlag) as well as a member of the editorial boards of
several scientific journals. His science citation index reaches 2300 (ac-
cording to the ISI database). In 1991 he was awarded an EURO Gold
Medal for his scientific achievements in the area of operations research.
In 2002 he was elected a corresponding member of the Polish Academy
of Sciences. In 2006 he was awarded an honorary doctoral degree by the
University of Siegen.
Mikhail Y. Kovalyov, born in 1959 (Ph.D. and D.Sc. degrees in ma-
thematical cybernetics in 1986 and 1999, respectively, professorial title
in computer science: 2004), is a deputy general director of the United
Institute of Informatics Problems, National Academy of Sciences of Be-
larus, and a professor of Belarusian State University. His research inte-
rests include combinatorial optimization, scheduling, logistics and bio-
informatics. He has authored over 100 publications in refereed scientific
journals. His paper co-authored with C.N. Potts is included in the list of
30 best papers published by the European Journal of Operational Rese-
arch in 1975–2005. His citation h-index is 14 according to Scopus and
16 according to Google Scholar. He is a co-author of one monograph
and a chapter in a monograph. Mikhail Y. Kovalyov is an associate edi-
tor for Omega and the Asia-Pacific Journal of Operational Research,a
member of the editorial boards of the European Journal of Operatio-
nal Research and Computers and Operations Research, a member of the
EURO working group on production management and scheduling, and
the vice-president of the Byelorussian Operational Research Society.
edrzej Musiał, born in 1982 (M.Sc. in computer science from the Po-
zna
´
n University of Technology: 2006) is a Ph.D. student, researcher and
lecturer at the Institute of Computer Science of the Pozna
´
n University of
Technology. His research interests include electronic commerce, algo-
rithm design and combinatorial optimization. He has published several
papers. Since 2008 has been a board member of the Polish Information
Processing Society (Major Poland Branch).
Andrzej P. Urba
´
nski graduated in computer science from the Pozna
´
n
University of Technology, Poland, and obtained his PhD in the same field
from the Polish Academy of Sciences. Presently he is a researcher and
lecturer at the Institute of Computer Science of the Pozna
´
n University of
Technology. His research includes the fields of computer-aided design,
artificial intelligence, and electronic commerce. He has been publishing
and presenting papers at many European scientific conferences and has
also published articles and books popularizing computer science, espe-
cially for children, with some literary setting. Some of his computer pro-
grams have been marketed and reviewed in popular media, among others
390
J. Bła
˙
zewicz et al.
by Polish TV networks, Deutsche Rundfunk, BBC, Die Welt, The New
York Times, and Canadian CRBC.
Adam Wojciechowski graduated from the Pozna
´
n University of Tech-
nology (PUT) in 1995 with an M.Sc. in computer science. In 1997/98 he
worked at Dublin City University, Ireland. In 2002 he received a Ph.D.
degree in computer science/software engineering at the PUT, where he
currently works as an assistant professor. His research interests include
software engineering, electronic business, optimization issues on nan-
cial markets and distant education. He is an author or co-author of over
12 papers published in international scientific journals and internatio-
nal conference proceedings. He has also prepared Polish handbooks for
ECDL applicants and published a distant education course for webma-
sters.
Received: 19 June 2009
Revised: 24 October 2009
... [13] has been formally dened as Internet Shopping Optimization Problem (ISOP). The task is to make a purchase consisting of many products paying the minimum possible amount at the end. ...
... This chapter provides basic information on computational complexity, algorithms, combinatorial optimization, and task scheduling. In Internet Shopping Optimization Problem (Chapter 4) Internet Shopping Optimization Problem (ISOP), where the customer wants to complete the shopping list using the online shops oer, was originally proposed by Blazewicz et al. [13]. The aim is to make purchases with possibly the lowest nal invoice. ...
... We denote this problem as ISOP (Internet Shopping Optimization Problem), its optimal solution as X * , and its optimal solution value as F * . It has been proven that the ISOP problem belongs to the class of strongly NP-hard problems [13]. ...
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Computer science is constantly developing in a very dynamic way. At the same time, the scope of its applications, solutions and learning necessary to explore the presented problems, is constantly growing. Complicated mathematical calculations, using specially designed sets of instructions (algorithms) can lead to effective solution of problems from various areas of life. Computer science is definitely applicable science - one whose results, often very complex scientific analyses, can and should be used in practice. It is a science that is ultimately supposed to change the everyday life of every person. In recent years, e-commerce has become a very important aspect not only of life, but also of information technology.\\ This monograph touches upon the fascinating subject of online shopping, paying particular attention to the innovative, yet unexploited possibilities of applying science in this sphere of life. One of such unresolved problems / aspects of online shopping is the situation in which we intend to buy more than one product at a time (a situation that occurs extremely often). Of course, we want to achieve everything by minimizing the final costs, while being in line with the list of additional requirements and limitations. The paper contains a number of mathematical formulations for the next presented variants of the problem of online shopping optimization, solutions are discussed and comprehensive computational experimental tests are widely commented on. The work is a collection of results of research conducted by the habilitatant on the problem of ISOP in the last 8-9 years. The monograph provides an extensive overview of the current state of knowledge concerning new technologies enabling optimal online shopping. The current e-commerce market still lacks effective analysis tools supporting online shopping. The key problem is the lack of any analysis of purchases consisting of several / many products. On the other hand, such purchases are by far the most frequent. The book contains a collection of the latest knowledge in the field of online shopping optimization, divided according to the complexity of the problem and possible solutions to be applied. The problem of online shopping optimization \cite{Blazewicz2010} has been formally defined as Internet Shopping Optimization Problem (ISOP). The task is to make a purchase consisting of many products paying the minimum possible amount at the end. The choice can be made from a number of online store offers. It is also possible to define many additional requirements or limitations. The problem is evolving into new, more interesting and more specialized models (more and more complicated and more difficult to solve). Subsequent versions presented with each step reflect the market situation even better and respond to the customer's expectations. The following versions of the problem concern in particular discounts offered for the purchase of all products, very diverse (often complex) tables of shipping costs, where the price depends on different forms of payment or purchase amounts. Trust as one of the most important purchasing factors could not be omitted in the analysis of the whole problem. It is natural that sometimes complex mathematical analysis should not be presented without the inclusion and presentation of appropriate basics. They are provided on the front pages of the book so that the reader can confidently turn the pages without having to read other supportive literature items to understand the main content. It is obvious that some excerpts are briefly presented in a way that is necessary and sufficient to move smoothly within the framework of this publication, although not exhaustive in a given section of the subject matter. The whole material is supplemented with a wide list of references to which the reader can refer at any time. The results of the conducted research may be used by students, doctoral students, as well as research workers in the field of computer science, or more general fields of study related to computer science. They may also form the basis of a separate multidisciplinary subject or part of already prepared classes on e.g. algorithmics or optimization. They can also be simply a collection of materials presented in the form of an interesting publication that describes (and solves) the real problem of almost every Internet user. They are an ideal material of knowledge, which have wide possibilities of application and implementation. They provide a comprehensive overview of the most recent knowledge of the ISOP problem. The main scientific objectives and the results achieved are presented below.
... It is worth mentioning that the Internet Shopping Optimization Problem (ISOP) [7] has some similarities with Cloud Brokering problem. Basic version of the ISOP could be defined as follows. ...
... The idea was to propose different possibilities to customers to find shops in a geographically defined area that they can go and realize their shopping list at different total price. It could be perceived as a basic prototype of the ISOP, which was presented and formally modeled as an optimization problem [7]. It was proved that the problem is NPhard in the strong sense. ...
... Therefore, a new Forecasting algorithm was proposed [5]. These heuristics are used to validate the correctness and performance of the proposed solution methods presented in the original article about ISOP [7]. Recently different types of the ISOP have been examined. ...
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Cloud computing has become one of the major computing paradigms. Not only the number of offered cloud services has grown exponentially but also many different providers compete and propose very similar services. This situation should eventually be beneficial for the customers, but considering that these services slightly differ functionally and non-functionally -wise (e.g., performance, reliability, security), consumers may be confused and unable to make an optimal choice. The emergence of cloud service brokers addresses these issues. A broker gathers information about services from providers and about the needs and requirements of the customers, with the final goal of finding the best match.
... The main contributions of this paper are as follows: (i) the relationship between improvised exploitation and each parameter under asymmetric interval is established; (ii) sufficient condition on iterative convergence is proven theoretically; (iii) the WCA has been successfully applied for solving computationally expensive CEC 2015 benchmark problems (Chen et al., 2014); (iv) Complexity analysis regarding computational time required by the WCA has also been investigated in this paper; (v) a new discretisation strategy is proposed to make WCA able to solve combinatorial optimisation problems raised in Internet shopping optimisation problem (ISOP) (Blazewicz et al., 2010). ...
... Feature selection is another application of discrete optimisation problem widely studied in the literature Mafarja et al., 2018). Internet shopping optimisation problem is a new benchmark and challenge in combinatorial optimisation problem first proposed by Blazewicz et al. (2010) studied in this research. Furthermore, CEC'15 benchmarks, widely studied in the literature (Heidari & Pahlavani, 2017), along with Internet shopping optimisation problem are considered in this research. ...
... The objective in the Internet shopping optimisation problem (ISOP) is to find a selection of products (including all product set) purchased from different Internet shops, X = (x 1 , x 2 , x 3 ,. . ., x m ), which minimises the total cost (i.e., products costs + delivery costs) as given follows (Blazewicz, Cheriere, Dutot, Musial, & Trystram, 2014;Blazewicz et al., 2010): ...
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Water cycle algorithm (WCA) is a population-based metaheuristic algorithm, inspired by the water cycle process and movement of rivers and streams towards sea. The WCA shows good performance in both exploration and exploitation phases. Further, the relationship between improvised exploitation and each parameter under asymmetric interval is derived and an iterative convergence of WCA is proved theoretically. In this paper, CEC’15 computationally expensive benchmark problems (i.e., 15 problems) have been considered for efficiency measurement of WCA accompanied with other optimisers. Also, a new discretisation strategy for the WCA has been proposed and applied along with other optimisers for solving combinatorial Internet shopping optimisation problem. By applying complexity analysis, it shows that using the WCA intricacy from dimension 10–30 is increased for almost three times. Proposing a unique discretisation approach along with providing iterative convergence proof can be considered as novelty of this research. By observing the attained numerical results, the WCA could find the minimum average error of CEC’15 in 12 and 8 out of 15 cases for dimensions 10 and 30, respectively. Experimental optimisation results for a wide range computationally expensive problems reveal the effectiveness and advantage of WCA for solving both continuous and discrete optimisation problems.
... This is a variant of Internet shopping optimization problem [13]. The problem assumes buying some set of items. ...
... The objective is to select a seller for each item to minimize the total cost of purchases and shipping costs. What differs this problem from the classic definition [13] is an extension of an objective function, taking into account a component depending on the number of parcels that will be shipped (we also want to consider minimization of the number of parcels). ...
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This paper concludes the Brilliant Challenges contest. Participants had to design interesting optimization problems and publish them using the Optil.io platform. It was the first widely-advertised contest in the area of operational research where the objective was to submit the problem definition instead of the algorithmic solutions. Thus, it is a crucial contribution to Open Science and the application of crowdsourcing methodology to solve discrete optimization problems. The paper briefly describes submitted problems, presents the winners, and discusses the contest's achievements and shortcomings. Finally, we define guidelines supporting the organization of contests of similar type in the future.
... The problem of managing multiple-item complex shopping lists over several shopping locations, called the Internet Shopping Optimization Problem (ISOP), in its very general form, aims to have all the shopping done at the minimum total expense subject to the satisfaction of all necessary requirements and conditions. The ISOP problem was defined and proved to be NP-hard [4]. Therefore, achieving results as close to optimum as possible within a reasonable time requires efficient heuristic approaches. ...
Chapter
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... The problem of managing multiple-item complex shopping lists over several shopping locations, called the Internet Shopping Optimization Problem (ISOP), in its very general form, aims to have all the shopping done at the minimum total expense subject to the satisfaction of all necessary requirements and conditions. The ISOP problem was defined and proved to be NP-hard [4]. Therefore, achieving results as close to optimum as possible within a reasonable time requires efficient heuristic approaches. ...
Chapter
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... The problem of managing multiple-item complex shopping lists over several shopping locations, called the Internet Shopping Optimization Problem (ISOP), in its very general form, aims to have all the shopping done at the minimum total expense subject to the satisfaction of all necessary requirements and conditions. The ISOP problem was defined and proved to be NP-hard [4]. Therefore, achieving results as close to optimum as possible within a reasonable time requires efficient heuristic approaches. ...
Chapter
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