How to Efﬁciently Solve Internet Shopping Optimization Problem

with Price Sensitive Discounts?

Jedrzej Musial

, Johnatan E. Pecero

, Mario C. Lopez

, Hector J. Fraire

, Pascal Bouvry

and Jacek Blazewicz

Institute of Computing Science, Poznan University of Technology, ul. Piotrowo 2, 60-965 Poznan, Poland

Computer Science and Communications Research Unit, University of Luxembourg,

6 rue Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg

Instituto Tecnol´ogico de Ciudad Madero, Ciudad Madero, Mexico

Keywords:

Optimization, e-Commerce, Internet Shopping, Applications, Algorithms, Computational Complexity.

Abstract:

In this paper we deal with the Internet Shopping Optimization Problem. An extended model that includes price

sensitive discounts is considered. A set of algorithms to solve the Internet Shopping Optimization Problem

with Price Sensitivity Discounts (ISOPwD) is introduced. The algorithms are designed to consider a different

solution quality regarding computational time and results close to the optimum solution. Simulations based on

real world data assess the new set of heuristics. The results of the proposed algorithms were compared with

the optimal solutions, computed by a branch and bound algorithm. The scalability is evaluated by increasing

the problem sizes. Computational experiments are performed and their results are carefully analyzed and

discussed. The paper should be perceived as a work in progress - position paper.

1 INTRODUCTION

The Internet gives access to a huge marketplace. By

simply clicking or by touching a screen customers

can buy any product online. Electronic commerce (e-

commerce) enables consumers to shop at thousands

of online stores and to pay for their purchases with-

out leaving the comfort of their homes (Timmers,

1998). Technology (e.g., Cloud Computing, mo-

bile devices - smartphones, tablets) is present into

daily business and administrative operations. Al-

ready since 2005 one third of European companies

used e-commerce in business procurement, logis-

tics, ﬁnance and product development (Luxembourg-

Embassy-Copenhagen, 2005).

Internet shopping, ﬁtting into a business-to-

consumer subcategory, becomes more and more pop-

ular. Products available in online 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 (Lee, 1998). Based on out-

standing logistics, the delivery can usually be oper-

ated within 48 hours or less. A crucial aspect of on-

line shopping is the time spent on comparing offers.

Customers often need to take into account that ship-

ping 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 where all the products

from the customer’s shopping list could be bought at

the lowest price (Musial, 2012).

We investigate an extended version of the Inter-

net Shopping Optimization Problem (Blazewicz et al.,

2010) which additionally considers price sensitive

discounts (Blazewicz et al., 2014). A new set of

heuristic approaches to solve the problem was intro-

duced. The set of heuristics is composed of a new

lightweight metaheuristic based on a cellular opti-

mization process, a new greedy algorithm and two

state-of-the-art greedy algorithms. We have designed

different heuristics to consider a different solution

quality regarding computational time and results close

to the optimum solution. Moreover, the performance

of the proposed heuristics to the optimal values was

tested. The optimal solutions for small problem in-

stances are computed using a branch and bound algo-

rithm. To evaluate the scalability of the heuristics the

problem size was increased.

209

Musial J., Pecero J., Lopez M., Fraire H., Bouvry P. and Blazewicz J..

How to Efﬁciently Solve Internet Shopping Optimization Problem with Price Sensitive Discounts?.

DOI: 10.5220/0005112602090215

In Proceedings of the 11th International Conference on e-Business (ICE-B-2014), pages 209-215

ISBN: 978-989-758-043-7

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

2 RELATED WORK

Motivated by the problem of buying multiple prod-

ucts from different e-commerce web sites, Blazewicz

et al. (Blazewicz et al., 2010) modeled Internet shop-

ping as an optimization problem, where a customer

wants to buy a list of products from online stores at

the minimum ﬁnal price. The authors showed that the

problem is NP-hard in the strong sense and designed a

set of polynomial time algorithms for special cases of

the problem. During previous research different ver-

sions (specializations) of the Internet Shopping Op-

timization Problem were examined (Wojciechowski

and Musial, 2010; Blazewicz et al., 2014). For exam-

ple, due to NP-hardness of the optimization problem,

Wojciechowski and Musial (Wojciechowski and Mu-

sial, 2010) designed a heuristic solution to optimize

the shopping basket and evaluate it for the customer

basket optimization problem to make it applicable for

solving complex shopping cart optimization in online

applications. Moreover, it is proven that the problem

is not approximable in polynomial time (Blazewicz

et al., 2010). The archetype of the presented prob-

lem was a web-based customer assistance system ded-

icated to pharmacyshopping that helps customers ﬁnd

shops in a geographically deﬁned range where the en-

tire shopping list could be realized at the best total

price (Wojciechowski and Musial, 2009).

It is worth noticing that there are some similarities

between the Internet Shopping Optimization Prob-

lem and the well-known Facility Location Problem

(FLP) (Revelle et al., 2008). The main characteris-

tics of the FLP are space, the metric, given customer

locations and given or not given positions for facility

locations. A traditional FLP is to open a number of

facilities in arbitrary positions of the space (continu-

ous problem) or in a subset of given positions (dis-

crete problem), and to assign customers to the opened

facilities so that the sum of opening costs and costs re-

lated to the distances between customer locations and

their corresponding facility locations is minimized.

Discussions of FLPs can be found in (Krarup

et al., 2002; Eiselt and Sandblom, 2004; Melo et al.,

2009; Iyigun and Ben-Israel, 2010). The traditional

discrete FLP is NP-hard (Garey and Johnson, 1979)

in the strong sense. Note, however, that the general

problem ISOP with price sensitive discounts cannot

be treated as a traditional discrete FLP since there is

no evident motivation for a discount on the cumula-

tive cost in the sense of distances. It can be noticed

that the FLP problem and problem ISOP are not each

other’s sub-cases, while the traditional discrete FLP is

a special case of any of these problems.

The problem of the Internet Shopping with Price

Sensitive Discounts (let us call it ISOPwD) has been

introduced in (Blazewicz et al., 2014). Computa-

tional complexity of various special cases is estab-

lished. Properties of optimal solutions are proved and

polynomial time and exponential time solution algo-

rithms based on these properties are designed. The

authors presented two greedy heuristics. In this paper,

we consider the same problem. We provide a working

model for the experimental results, which is close to

real Internet shopping conditions. For that, we pre-

pared a model on the basis of data from the online

book industry reported in (Clay et al., 2001) (see sub-

section 4.1).

3 PROPOSED ALGORITHMS

The ISOPwD is strongly NP-hard. Moreover, to our

best knowledge it cannot be a goal of reduction from

any one of the known problems. Therefore, it is ap-

parent right to propose heuristic solution, simple efﬁ-

cient greedy algorithms (Blazewicz et al., 2014; Cor-

men et al., 2001) that use local knowledge and do

not allow any backtracking for efﬁciency purpose. It

is worth noticing that the greedy algorithm does not

always yield optimal solutions. However, it could

provide an optimal or close to optimal solution us-

ing much less resources and time than other optimal

working algorithms (i.e., full scan, branch&bound).

More sophisticated algorithms should provide bet-

ter quality solutions. Therefore, we propose very

efﬁcient pseudo-parallel optimization approach al-

gorithm. State-of-the-art minimization scheduling

driven algorithm was used as a competitor to the pre-

vious ones.

3.1 Greedy Algorithm - GREEDY

In the ﬁrst heuristic for the ISOPWD, denoted as

GREEDY, products are considered in a certain order.

The algorithm is run for various product orders and

the best solution found is presented to the customer.

Let us consider that the products are sorted in an as-

cending order 1, . . . , n. Values of the total delivery

and the standard price for all shops are initially set

as T

= d

, i = 1, . . . , m. In iteration j of algorithm

GREEDY, product j is selected in its eligible shop

i ∈ M

with minimum value f

+ p

), and the cor-

responding T

-value is re-set: T

:= T

+ p

Discounting function f

, j) returns value of product

j after applying discount for shop i.

We observed that algorithm GREEDY demon-

strates very good performance on the experimental

data. The ﬁrst experimental evaluation of algorithm

ICE-B2014-InternationalConferenceone-Business

210

GREEDY can be found in (Wojciechowski and Mu-

sial, 2010) and (Blazewicz et al., 2014).

3.2 Algorithm with

Forecasting - Forecasting

Observed weak points of algorithm GREEDY (poor

results for a very speciﬁc situation) led to the cre-

ation of a new, upgraded version. The local step

choice analysis is more complicated than in the

basic algorithm GREEDY. The forecasting method

is looking for a step ahead (Błazewicz and Musiał,

2011). Therefore, technically this algorithm is not

a strict greedy algorithm. Sometimes it proposes a

current solution which is not optimal for the current

step (local solution), but for a better overall solution

in hope of providing an optimal global solution. The

main idea is to check the situation one step ahead

(forecasting bad situations).

From the ﬁrst step to the penultimate, the al-

gorithm calculates the ”choosing factor” to pick an

eligible shop i for product j. Instead for looking

for a local optimum it looks one step ahead (which

prevents occurring a bad case) by calculating a

choosing factor as f

ij+1

) for every shop

i, and actual product j, as well as following product

j+ 1. Then it pick a shop i with the lowest calculated

value. In each following step next product j is taken

into account, j = j + 1. The last step of the algorithm

works in a different way (forecast could not work

beyond the set of products j). The last product is

selected in its eligible shop i ∈ M

with minimum

value f

+ p

Discounting function f

, j) returns value of

product j after applying the discount for shop i.

3.3 Cellular Processing

Algorithm - Cellular

The cellular processing based algorithm is a

new pseudo-parallel optimization approach (Ter´an-

Villanueva et al., 2013). It includes multiple process-

ing cells that explore different regions of the search

space. Each processing cell can be implemented us-

ing population or search based heuristics or a hy-

bridization of them. The main idea and the principle

of the algorithm are to split a sequential algorithm into

several pseudo-parallel processing (i.e. cell) modules,

so that each cell can explore different regions of the

search space. The main feature of the new approach

is that the iterative veriﬁcation of the stagnation con-

ditions prevents wasting time on unnecessary tasks.

The components of the algorithm are a pool of

candidate solutions, generated either by a construc-

tive or a random algorithm and a cell set that is sim-

ple, independent, self contained and applied to work

with the subset of candidate solutions that were given

to solve. This process continues until the cells stall

all the solutions in their local optimum. After that,

the solutions return to the pool, and the cells share in-

formation with each other in order to escape from the

local optimum and continue the search for the global

optimum.

In this work we prepared a special designed and

implementation of the cellular processing algorithm

idea. The candidate solutions were generated at ran-

dom. An Iterated Local Search algorithm (ILS) was

designed as the core of the cells. This choice was

made due to the simplicity and high conﬁgurability

of this structure, which allows it to be highly scal-

able and to run in a variety of hardware conﬁgura-

tions. Moreover, the ILS algorithm is a trajectory-

based metaheuristic that can be seen as a straightfor-

ward, yet powerful technique for extending simple lo-

cal search algorithms.

The algorithm starts off by generating an initial

solution. Then, a local search process is applied to

the candidate solution. After that, wallowing iteration

based approach, it seeks to improvethe solutions from

one iteration to the next. At each iteration, a perturba-

tion 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 ap-

plied to the perturbed solution. The new solution is

then evaluated and accepted as the new current solu-

tion under some conditions. The algorithm ﬁnishes

when the termination condition is met.

Algorithm is still under upgrade. Its complicated

nature results in updates after completion of each set

of experiments.

Cellular Processing Algorithm should be per-

ceived as a new, strong contribution to this paper. The

idea is to use a new pseudo-parallel optimization ap-

proach. However, all the algorithm development pro-

cess was created from the basis. All steps in devel-

opment process were carefully analyzed according to

the problem nature.

3.4 Minimum-Minimum

Algorithm - MinMin

The min-min algorithm is a heuristic that is used pri-

marily in the scheduling of tasks or processes (Ibarra

and Kim, 1977). As noted in (Wu et al., 2000), the

min-min algorithm is a fast method that offers good

HowtoEfficientlySolveInternetShoppingOptimizationProblemwithPriceSensitiveDiscounts?

211

performance, but as it schedules those tasks or pro-

cesses with minimum cost ﬁrst, it may result in an

imbalanced solution.

In (Braun et al., 2001) the process of the heuristic

min-min is described in the context of the scheduling

tasks onto heterogeneous distributed computing sys-

tems. This method was adapted for the Internet Shop-

ping Optimization Problem for the selection of a list

of products in a set of stores.

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 dif-

ferent stores M, given the cost of the product, p

plus

delivery cost, d

. In the case of a tie, the product with

the lower delivery cost is selected, and if both stores

havethe same delivery cost, the product is chosen ran-

domly.

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 con-

tinues until the unassigned product list is empty.

Minimum-Minimum algorithm should be per-

ceived as partly new contribution to this paper. Al-

gorithm was developed according to the literature and

then tuned to the ISOPwD.

3.5 Branch and Bound Algorithm - BB

To calculate the optimal solution, we designed a

Branch and Bound Algorithm (Land and Doig, 1960).

The algorithm starts off by calculating an upper

bound (UB) employing the solution given by the Cell

Processing Algorithm and proceeds to branch the ﬁrst

level of the search tree in the stack.

Subsequently, the algorithm pops the top element of

the stack and evaluates the objective value it would

have if it were part of the current solution. If the

partial solution exceeds the limit given by the upper

bound (UB), the current branch is fathomed. Other-

wise, if it were not a leaf of the tree, the algorithm

would pile up the following elements within the stack.

If it were a leaf, it would mean that the current solu-

tion is better than the best global solution found so

far. Consequently it would update the upper bound

with the new value found.

This process continues as long as there are ele-

ments in the stack, which means that the whole search

tree has been explored. Founded upper bound is now

considered as the optimal solution of the instance be-

ing evaluated.

Branch and Bound algorithm should be perceived

as partly new contribution to this paper. Algorithm

was developed according to the literature and then

tuned to the ISOPwD.

4 COMPUTATIONAL

EXPERIMENTS RESULTS

Computational experiments were performed and di-

vided into two groups (due to the computational com-

plexity time) - a set of experiments including Branch

and Bound exact algorithm, and a set of experi-

ments without optimal solution as a comparison of all

heuristics to evaluate scalability issues by increasing

instances’ size. Very ﬁrst experimental results from

the ﬁrst group will be presented within this section.

Full results will be presented in the ﬁnal version of

the article.

Algorithms were compared using three metrics:

an approximation factor, the running time spent by

each heuristic to compute a solution, and the disper-

sion analysis based on the standard deviation. The

approximation factor of a heuristic is deﬁned as ρ =

F(X)

∗

, where F(X) represents the solution found by a

heuristic and F(X)

∗

denotes the optimal solution.

4.1 World Working Model and

Instances Generator

A challenging step in experimental research was to

create a model, which should be as close to real Inter-

net shopping conditions as possible. We studied the

relationship between the competitive structure, adver-

tising, prices and price dispersion over Internet stores.

As a group of representative products to be taken into

account in our computational experiment we chose

books, because of their wide choice in virtual (In-

ternet) stores and frequency of purchase through this

kind of shopping channel. We adopted some informa-

tion and computational results from Clay et al. (Clay

et al., 2001) for our model.

In the computational experiments we as-

sume that number of stores m ∈ {20, 40},

and number of products in our shopping cart

n ∈ {2, 3, . . . , 10, 15, . . . , 100}. It is assumed that

each bookstore has all the required books. For

each pair (n, m), 100 instances were generated. In

each instance, the following values were randomly

generated for all i (shop index) and j (product

index) in the corresponding ranges. Delivery price:

∈ {5, 10, 15, 20, 25, 30}, publisher’s recommended

price of book j: r

∈ {5, 10, 15, 20, 25}, and price

of book j in bookstore i: p

∈ [a

, b

], where

≥ 0.69r

, b

≤ 1.47r

, and the structure of

intervals [a

, b

] follow from information from

Clay et al. (Clay et al., 2001) and observations on

the biggest Polish Internet bookstores. Discounting

functions were prepared for every shop.

ICE-B2014-InternationalConferenceone-Business

212

During full paper submission it is planed to prepare

open access online instance generator that was used

during computation experiments.

4.2 A Set of Experiments including

Branch and Bound Algorithm

The ﬁrst group of experiments is the one in which the

optimal solutions obtained by the exact BB algorithm

were compared to two state-of-the-art heuristic algo-

rithms: GREEDY, FORECASTING, and two newly de-

veloped algorithms: CELLULAR and MINMIN (for

the ISOPWD). In these examples, m ∈ {20}, n ∈

{2, 3, 4, 5, 6, 7, 8, 9, 10}, and discounts follow the pro-

posed discounting function. For each pair (n, m), 100

instances using the information in subsection 4.1.

0.98

1.02

1.04

1.06

1.08

1.1

1.12

2 4 6 8 10

Approximation Ratio

Products

20 Shops

Greedy

Forecasting

Cellular

MinMin

Figure 1: Algorithm results comparison - experiment with

20 shops (including the optimal solution).

After some very ﬁrst computational experiment

we one can notice that among all heuristics, CELLU-

LAR provides the best quality of solutions (closest to

the optimum). In the worst case, the solution pro-

posed by this algorithm was merely 1.47% more ex-

pensive than the cheapest one. For many instances

CELLULAR algorithm computes the optimal solution.

It is worth noticing that the algorithm is very stable

as regards the quality of solutions (for most cases the

solution is between 1.24% - 1.47% worse than opti-

mum).

Both GREEDY and FORECASTING algorithms pro-

vide similar quality of solutions (for a lower number

of products n, the latter was better and for a higher

number of products n > 5 the former was better).

Moreover, it is easily noticeable that the quality of so-

lution degrades with the increasing number of prod-

ucts n.

The last heuristic algorithm - MINMIN provides the

worst solutions for a lower number of products. How-

ever, the algorithm is quite stable in quality, therefore

for a bigger number of products it provides better so-

lutions than GREEDY and FORECASTING.

If one is looking solely for the quality of solutions

the undisputed leader among heuristic algorithms is

CELLULAR. The following paragraph will address

running time, which is also a very important factor.

For a low number of n products algorithm MIN-

MIN is the fastest. For a bigger number of prod-

ucts, algorithm GREEDY is the fastest. Differences

between all algorithms are very signiﬁcant. The fol-

lowing example illustrates these differences. Compu-

tation time for an instance of n = 5 products (from

m = 20 shops) vary from 0.0047 ms for MINMIN)

across 0.0085 ms for GREEDY, 0.0205 ms for FORE-

CASTING, 0.3795 for CELLULAR to the 2.5998 ms

for BB (computes 556 times longer than MINMIN).

For a maximum number of products (n = 10) compu-

tation times are as follows: 0.0072 ms for GREEDY

across 0.0083 ms for MINMIN, 0.0132 ms for FORE-

CASTING, 0.2183 for CELLULAR to the 707.4174 ms

for BB (98k times longer than GREEDY).

Computation times for BB algorithm grow exponen-

tially and it was impossible to prepare experiments for

a bigger number of products. On the other hand, all

heuristics are very fast so the idea is to further test its

quality and computational times for scalability issues

by increasing the number of products n - let us follow

the next subsection.

5 CONCLUSIONS AND

CONTRIBUTIONS

In this position paper, the Internet Shopping Opti-

mization Problem including delivery discounts was

addressed. For the practical application, a working

model as close to real Internet shopping conditions as

possible was created.

The main contributions of current research are as

follows. A new set of algorithms is presented. Cel-

lular Processing Algorithm should be perceived as a

new, strong contribution to this paper. The idea is

to use a new pseudo-parallel optimization approach.

However, all the algorithm development process was

created from the basis. All steps in development pro-

cess were carefully analyzed according to the prob-

lem nature. Another Minimum-Minimum algorithm

should be perceived as partly new contribution to this

paper. Algorithm was developed according to the lit-

erature and then tuned to the ISOPwD. Subsequent

Branch and Bound algorithm should be perceived as

partly new contribution to this paper. Algorithm was

developed according to the literature and then tuned

to the ISOPwD. Another two presented algorithms

HowtoEfficientlySolveInternetShoppingOptimizationProblemwithPriceSensitiveDiscounts?

213

Greedy and Forecasting were known from the liter-

ature. They were perfectly background and com-

petitors for a new breaking through Cellular Process-

ing Algorithm. Experimental data model was pre-

sented in previous research, however, for the ﬁrst

time an instance generator was created to addition-

ally contribute the paper. It will be available online as

open access. First computational experimentsdemon-

strated results obtained by all the algorithms. Results

were carefully analyzed and widely commented.

Our current work is focused on computational ex-

periment run. It is planed to solve a vast number of

instances using all the algorithms. Moreover, the idea

is to provide scalability analysis - with a set of exper-

iments for heuristic algorithms. This type of exper-

iment will be performed for much bigger instances

of the problem (with a signiﬁcantly bigger number

of shops and products) to compare all heuristic algo-

rithms. Important part of a discussion will be disper-

sion analysis. In the full version of this position pa-

per a very detailed description of the computational

experiment will follow a vast number of exhaustive

tests that will be performed. Moreover, it is planed

to describe each algorithm in a very detailed way (in-

cluding algorithm pseudo-code) to enable repetition

and evaluation of these algorithms. Furthermore, an

instance generator described within this paper will be

available online for open usage. An interesting ex-

tension can be made when the model (with instance

generator) will be enhanced to complete to analysis

several kinds of products and compare the different

scenarios.

To alleviate the problem, and also deal with scalabil-

ity we consider investigating a parallel version of the

algorithm on a GPU infrastructure. Moreover, very

interesting topic could be to combine a method of

selecting column widths for a given set of advertise-

ment on a web page (Marszałkowskiand Drozdowski,

2013) with ISOP when preparing online working ap-

plication.

ACKNOWLEDGEMENTS

This study was partially supported by the FNR

(Luxembourg) and NCBiR (Poland), through IShOP

project, INTER/POLLUX/13/6466384.

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