Winner Determination in Combinatorial Reverse Auctions

M. Ali et al. (Eds.): Contemporary Challenges & Solutions in Applied AI, SCI 489, pp. 35–40.
DOI: 10.1007/978-3-319-00651-2_5 © Springer International Publishing Switzerland 2013
Winner Determination in Combinatorial
Reverse Auctions
Shubhashis Kumar Shil, Malek Mouhoub, and Samira Sadaoui
Abstract. Since commercially efficient, combinatorial auctions are getting more
interest than traditional auctions. However, winner determination problem is still
one of the main challenges of combinatorial auctions. In this paper, we propose a
new method based on genetic algorithms to address two important issues in the
context of combinatorial reverse auctions: determining the winner(s) in a reasona-
ble processing time and reducing the procurement cost. Indeed, not much work
has been done using genetic algorithms to determine the winner(s) specifically for
combinatorial reverse auctions. To evaluate the performance of our method, we
conducted several experiments comparing our proposed method with another me-
thod related to determining winner(s) in combinatorial reverse auctions. The expe-
riment results clearly demonstrate the superiority of our method in terms of
processing time and procurement cost.
1 Introduction
An auction is a market scenario in which bidders compete for item(s). In tradition-
al auctions, an individual item is auctioned separately, which leads to an ineffi-
cient allocation and processing time [7, 10]. Combinatorial auctions have been
proposed to improve the efficiency of bid allocation by allowing bidders to bid on
multiple items [7, 10]. These auctions provide a combinatorial allocation that mi-
nimizes the procurement cost and running time [5, 7, 10]. They have been used in
various real-world situations [1] such as resource allocation with real-time con-
straints [10], sensor management [9, 11], supply chain management [12] and
computer grids [2]. A combinatorial auction problem is actually a winner determi-
nation problem [4]. Winner determination is still one of the main challenges of
combinatorial auctions [10]. Indeed, determining the winner(s) in combinatorial
Shubhashis Kumar Shil ⋅ Malek Mouhoub ⋅ Samira Sadaoui
Department of Computer Science, University of Regina, Regina, SK, Canada
e-mail: {shil200s,mouhoubm,sadaouis}@uregina.ca

36 S.K. Shil, M. Mouhoub, and S. Sadaoui
auctions is a complex problem and it has been shown to be NP-complete [5, 7].
However, applying combinatorial auctions to procurement scenario [13] is cost–
saving [7]. Many algorithms have been developed to solve combinatorial auction
problems, e.g. Hsieh and Tsai presented a Langrangian heuristic method [7], and
Sitarz introduced Ant algorithms and simulated annealing [7]. Furthermore, many
research works have been carried out to figure out the efficient way to solve win-
ner determination in combinatorial auctions. Most of the proposed algorithms
restrict the bundles on which bids can be submitted in order to solve the problem
optimally but these restrictions introduce economic inefficiencies [7]. Some algo-
rithms find optimal solutions but are very slow; others avoid restrictions but allow
bidding on a small number of items [7]. We are interested in combinatorial re-
verse auctions in which we consider the procurement of a single unit of
multi-items. In our auction, there is one buyer and several sellers who compete
according to the buyer’s requirements. First the buyer announces his demand
(multiple items) in the auction system. Then the interested sellers register for that
auction and bid on a combination of items. Genetic Algorithms (GAs) are success-
ful to solve many combinatorial optimization problems [7]. GAs are powerful
search techniques consisting of selection, crossover and mutation methods [3].
Sometimes simple crossover and mutation operators produce inappropriate chro-
mosomes. To avoid this problem, special or modified crossover and mutation
operators are defined [1]. GAs can terminate anytime as required and the current
best chromosome can be the best solution. Nevertheless, not much work has been
done by using GAs to solve winner determination problem in the context of com-
binatorial reverse auctions. To the best of our knowledge, only one research paper
[7] employed GAs to tackle this problem. However, the method proposed in [7]
needs comparatively many generations and a considerable amount of time to pro-
duce good solutions. In this paper, our research goal is twofold: (1) solve the
winner determination problem in combinatorial reverse auctions in a reasonable
processing time, and (2) reduce the procurement cost with fewer generations. For
this purpose, we define a new GA-based method that uses two repairing tech-
niques to repair infeasible chromosomes as well as a modified two-point crossover
operator that is capable of distributing the solutions and preventing a premature
convergence. Furthermore, we conduct several experiments by comparing our
proposed method with the one defined in [7]. The experimental results clearly
demonstrate the superiority of our method in terms of processing time and pro-
curement cost.
2 Proposed GA-Based Method
In Fig. 1, we define our GA-based method that we name GACRA (Genetic Algo-
rithms for Combinatorial Reverse Auctions). Assume there are m items and n
sellers. So the number of bid items combination is 2m
-1 and we use m×n bits to
represent each chromosome. In case of m=2 and n=3, a chromosome represented
by 100100 means that seller 1 bids for only item A (first two bits 10), seller 2 bids

Winner Determination in Combinatorial Reverse Auctions 37
for only item B (next two bits 01) and seller 3 bids for no items (last two bits 00).
To generate bid prices, we consider random values between 200 and 500 for each
item for each seller. In Step 2, the initial chromosomes are generated randomly.
To avoid redundancy, the RemoveRedundancy function ensures exactly one selec-
tion of a particular item from all sellers in each chromosome. To avoid emptiness,
the RemoveEmptiness function guarantees at least one selection of every item in
each chromosome. So, we repair infeasible chromosomes using these two func-
tions. Our RemoveRedundancy function works with the following steps.
1. For each chromosome, selects bits for each seller.
2. Tests bits for one seller to verify if item(s) are selected and stores this
information.
3. Continue testing bits of next sellers; if item(s) are already selected by the pre-
vious seller then converts the current bit value to 0.
Our RemoveEmptiness function works with the following steps.
1. For each chromosome, selects bits for each seller.
2. Tests bits for all sellers and stores the information of the non- selected item(s).
3. Continue converting bit value to 1 until all item(s) are selected.
Fig. 1 Pseudo code of GACRA

In step 4, fitness value of chromosome is generated. We propose the following
fitness function to calculate the fitness value of every chromosome. Since the
motivation of this research work is to minimize the procurement cost for the buy-
er, our fitness function for Chromosome Xi is defined as follows.
=
−
=
×
= n
s C
ss
i m
CxCb
XF
1
12
1
)()(
1
)(
}1,0{)(/ ∈Cxs
(1)
where )(Cbs represents a bid for the item combination C submitted from the th
s sel-
ler; )(Cxs is 1 when the item combination C is selected for the th
s seller and 0
otherwise.
We use gambling-wheel disk selection method [4] to select chromosomes for
cross-over operation. In step 7, the crossover operation is performed. A child
chromosome takes two portions from one parent and one portion from another
parent in two-point crossover. In our modified two-point crossover operation, the
basic idea is same but the direction of taking portions from parents is different.
The first child takes portions in forward direction but the second child takes it in
reverse order. It creates positive effect to increase diversity in the solution spaces
to allow all bidders to get more chances to be selected. In some cases, after remov-
ing redundancy and emptiness, some bidders get deprived but this modification of
crossover gives them a chance again. This will prevent the procedure to converge
prematurely. Then, the procedure will move to mutation operation as indicated in
our algorithm. In step 9, RemoveRedundancy and RemoveEmptiness functions
remove redundancy and emptiness respectively. In step 11, the procedure selects
the better chromosomes among the initial and new chromosomes of the generation
based on fitness values. Since genetic algorithm is called anytime algorithm, our
procedure can be stopped anytime and it produces the best solution. The entire
process is repeated until the termination condition is fulfilled, which is here the
number of generations. In step 12, the procedure returns the winner(s). The solu-
tion is not improving in all generations but in our procedure we always preserve
the current winner. So there is no chance to produce worse solution than the
previous generation.
3 Experiment
We have conducted several experiments to determine the winner in combinatorial
reverse auctions by using our method GACRA. We also compare GACRA with
the technique presented in [7] that we call CRA (Combinatorial Reverse Auc-
tions). We have implemented GACRA as well as CRA as described in [7] in Java.
These two methods are both executed on an AMD Athlon (tm) 64 X2 Dual Core
Processor 4400+ with 3.43 GB of RAM and 2.30 GHz of processor speed. We
have used the following parameters and settings for the experiments: Chromosome
Encoding: Binary String; Selection: Gambling-Wheel Disk; Crossover: Modified

Winner Determination in Combinatorial Reverse Auctions 39
Two-point; Crossover Rate: 0.6; Mutation Rate: 0.01 and Termination Condition:
Generation Number.
In the first experiment, we measure the required time of our proposed method
and compare it with CRA. In Fig. 2, we show the required time (in milliseconds)
versus the number of generations for both GACRA and CRA. This is the average
required time of 20 runs. From the comparison we can see that our proposed me-
thod needs less processing time. This happens because of two reasons: (1) we
represent the chromosome with less number of bits, and (2) we keep the calcula-
tion of fitness value simple.
We have also done some comparative experiments on the procurement cost and
report the results in Fig. 2. Since our procedure always maintains feasible solu-
tions and never accepts redundant bid item, it is able to produce good solutions
from the very first generations. Moreover, it keeps producing better solutions in
consecutive generations.
Fig. 2 Left figure - required time (in milliseconds) vs number of generation; right figure –
bid price vs number of generation
In addition, we assess the processing time for GACRA by varying the number
of sellers and keeping the number of items fixed to 2, 4, 6 and 8. Finally we assess
the processing time for GACRA by varying the number of items and keeping the
number of sellers fixed to 20, 40, 60 and 80. As the number of bits required to
represent the chromosomes of our algorithm directly depends on both the number
of items and the number of sellers, so with increasing any of these, the required
running time also increases. Due to using less number of bits to represent chromo-
some, required running time of GACRA is less than that of CRA. In CRA fitness
function tries to mitigate the effect of allowing redundancy but in GACRA fitness
function is simple and has no additional task. In CRA repair function only re-
moves emptiness but in GACRA RemoveRedundancy, RemoveEmptiness func-
tions and Modified two-point crossover operator remove redundancy, emptiness
and early convergence respectively. Due to not allowing redundancy, GACRA is
able to minimize procurement costs from very early generations.

4 Conclusion and Future Work
Motivated to save significant time and reduce the procurement cost, this paper
solves the problem of winner determination in combinatorial reverse auctions.
With the help of the two repairing method, the modified two-point crossover and a
careful selection of operator of GAs, it is notable that our method can produce
optimal solutions in a reduced processing time. As it is obvious that parallel GAs
are capable of providing the solution more efficiently [6, 8], the future target of
this research is to determine the winner by using the most efficient methods of
parallel GAs. Another future direction is to determine the winner(s) according to
multiple attributes of items.
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