Considering Multiple Instances of Items in Combinatorial Reverse Auctions

A. Moonis et al. (Eds.): IEA/AIE 2014, Part II, LNAI 8482, pp. 487–496, 2014.
© Springer International Publishing Switzerland 2014
Considering Multiple Instances of Items
in Combinatorial Reverse Auctions
Shubhashis Kumar Shil and Malek Mouhoub
Department of Computer Science, University of Regina, Regina, SK, Canada
{shil200s,mouhoubm}@uregina.ca
Abstract. Winner determination in combinatorial reverse auctions is very im-
portant in e-commerce especially when multiple instances of items are consid-
ered. However, this is a very challenging problem in terms of processing time
and quality of the solution returned. In this paper, we tackle this problem using
genetic algorithms. Using a modification of the crossover operator as well as
two routines for consistency checking, our proposed method is capable of
finding the winner(s) with a minimum procurement cost and in an efficient
processing time. In order to assess the performance of our GA-based method,
we conducted several experiments on generated instances. The results clearly
demonstrate the good time performance of our method as well as the quality of
the solution returned.
Keywords: Winner Determination, Combinatorial Reverse Auctions, Genetic
Algorithms.
1 Introduction
The main purpose in combinatorial auctions is to increase the efficiency of bid alloca-
tion especially when biding on multiple items [3], [13]. More precisely, the goal here
is to minimize the procurement cost in an efficient running time [3], [13], [21] when
determining the winner [7]. Winner determination is still one of the prime challenges
in combinatorial auctions [3] and is identified as an NP-complete problem [13], [21].
Nevertheless, applying combinatorial auctions to a procurement scenario [11], [14],
such as transportation services, travel packages and sales of airport time slots, is cost
saving [13]. Many algorithms have been introduced to tackle combinatorial auctions,
e.g. Sitarz defined Ant algorithms and Hsieh and Tsai developed a Langrangian heu-
ristic method [13]. While Genetic Algorithms (GAs) have been applied successfully
to solve many combinatorial optimization problems [13], such as quadratic assign-
ment problem [18], job scheduling [15] and travelling salesman problem [20], not
much work has been done to solve the problem of winner determination in the context
of Combinatorial Reverse Auctions (CRAs) with GAs.
In [17], we proposed a GA based method, called GACRA, to solve winner deter-
mination specifically for CRAs by considering multiple items. In [17], we introduced
two repairing methods, RemoveRedundancy and RemoveEmptiness, along with a

488 S.K. Shil and M. Mouhoub
modified two-point crossover operator. By using these two methods and an efficient
crossover operator, we solved the problem of winner determination successfully and
efficiently. Indeed, the goal of these two methods is to repair the infeasible chromo-
somes and the main task of the crossover operator is to ensure a diversity of solutions.
Indeed, increasing diversity will allow bidders to get more chances to be selected. In
[17], we have demonstrated that GACRA is better in terms of processing time and
procurement cost when compared to another GA-based winner determination method
[13]. Subsequently in [16], we conducted several statistical experiments and showed
that GACRA is a consistent method since it was able to reduce the solution variations
in different runs. Moreover, we described the any-time behaviour of GACRA in [16].
In this paper, we consider the procurement of a single unit of multiple instances of
items and propose an improved GA-based method, that we call GAMICRA (Genetic
Algorithms for Multiple Instances of Items in Combinatorial Reverse Auctions), by
modifying the methods and operators of [16] and [17] as follows.
1. We change the configuration of chromosomes to represent the bids in the case of
multiple instances of items. We define the configuration of the chromosomes as
well as the length of the chromosomes based on the number of sellers, items and
instances of items.
2. We modify the fitness function to consider multiple instances of items. The fitness
function is kept simple enough in order not to perform a lot of calculation. This
will help the processing time to remain reasonable.
3. We also improve RemoveRedundancy and RemoveEmptiness methods to solve the
winner determination problem associated with multiple instances of items. Re-
moveRundancy guarantees that the selected number of instances for each item does
not exceed the buyer’s requirements. On the other hand, RemoveEmptiness ensures
that the selected number of instances for each item is not less than that of the
buyer’s requirements.
With these modifications and using the modified two-point crossover defined in
[17], we solve efficiently the problem of winner determination for multiple instances
of items in CRAs. In order to evaluate the performance of our method in terms of
procurement cost and processing time, we perform several experiments. The results
we report in this paper clearly demonstrate that our method is prominent in terms of
time and cost and also show that it does not suffer from the inconsistency issue.
In many e-commerce and resource allocation problems, real-time response to a
very large problem is expected [8]. When solutions are needed quickly and problem
instances are large, exact algorithms that produce solutions at the expense of consum-
ing extended required time are not only inadequate but also become infeasible as
instances become larger. In real life, certain domains may require high-quality and
approximate solutions within a suitable running time [8]. Sometimes it is unnecessary
to spend a lot of time to slightly improve the quality of the solution. For these reasons,
in this research we consider GAs to solve the winner determination problem as they
have the ability to produce good solutions from the very first generations.
The rest of this paper is organized as follows. In Section 2, our GAMICRA method
is presented and then explained through an example. In Section 3, the experimental

Considering Multiple Instances of Items in Combinatorial Reverse Auctions 489
study we conducted to evaluate the processing time, procurement cost and the consis-
tency of our method is reported. In Section 4, concluding remarks with some future
research directions are given.
2 Proposed Method (GAMICRA)
In a CRA, there is one buyer and several sellers who compete according to the
buyer’s requirements. First the buyer announces his demand (multiple instances of
items) in the auction system. Then, the interested sellers register for that auction and
bid on a combination of items. In this paper, our research goal is twofold: (1) to solve
the winner determination problem for multiple instances of items in the context of
CRAs, and (2) to find the winners in a reasonable processing time and reduced pro-
curement cost. For this purpose, we propose the method GAMICRA that utilizes two
techniques, namely RemoveRedundancy and RemoveEmptiness, to repair infeasible
chromosomes. In addition, we utilize the modified two-point crossover operator de-
fined in [17] that provides a good distribution of the solutions and preventing a pre-
mature convergence. Figure 1 illustrates a possible scenario in CRAs. Here the buyer
wants to purchase three instances of item A and five instances of item B. There are
three sellers (S1, S2 and S3) who want to bid on items A and B. Therefore, the maxi-
mum instance between these two instances of items A (here 3) and B (here 5) is equal
to 5. Hence, 3 bits are required to represent each item for each seller considering the
following equation (1).
max , 2 (1)
With 3 bits, we can generate 23
= 8 combinations {000, ... , 111}. We assume for
this example that the number of chromosomes = 4, and the length of chromosomes =
2 × 3 × 3 = 18 according to the following equation (2).
(2)
Figure 2 presents the pseudo code of our GAMICRA method (Algorithm 1) while
RemoveRedundancy and RemoveEmptiness procedures are reported respectively in
Figures 3 and 4. In Algorithm 1, the bid price of each item for each seller is generated
randomly from the interval [100, 1000] (see Table 1). Table 2 shows the initial chro-
mosomes that are produced randomly.
The next steps of Algorithm 1 are to convert infeasible chromosomes into feasible ones
with the help of RemoveRedundancy and RemoveEmptiness methods. For our example,
the conversion of chromosome X1 from infeasible to feasible one is described below.
Table 1. Bid Price Generation
S1 S2 S3
Bid Price of Item A 951 868 525
Bid Price of Item B 826 920 696

Table 2. Initial Chromosomes
S1 S2 S3
Chromosome Item A Item B Item A Item B Item A Item B
X1 010(2) 011(3) 110(6) 001(1) 001(1) 101(5)
X2 100(4) 100(4) 110(6) 011(3) 011(3) 101(5)
X3 010(2) 010(2) 010(2) 101(5) 000(0) 010(2)
X4 010(2) 101(5) 101(5) 010(2) 101(5) 110(6)
Fig. 1. An example in CRAs
Initially, chromosome X1 is 010 011 110 001 001 101. As shown above, the num-
ber of instances for Item A is 9 which is more than the buyer’s requirement. The
number of instances of Item B also exceeds the initial requirement. Hence, this chro-
mosome is infeasible and needs to be repaired.
At first we will remove redundancy by using RemoveRedundancy method accord-
ing to Algorithm 2 shown in Figure 3. After removing the redundancy, X1 is now
equal to 000 000 000 000 000 101. Although the number of instances of Item B is the
same as the buyer’s requirement, chromosome X1 is still infeasible as the number of
instances of Item A is less than that of the buyer’s requirement. In this case, we will
apply the RemoveEmptiness method on this chromosome according to Algorithm 3
given in Figure 4. After removing the emptiness, X1 becomes 000 000 000 000 011
101. Now, chromosome X1 is a feasible one. By applying the same process to all the

chromosomes, the initial chromosomes after remove redundancy and remove empti-
ness are as follows.
X1: 000 000 000 000 011 101
X2: 000 000 000 101 011 000
X3: 000 000 010 101 001 000
X4: 000 101 011 000 000 000
Fig. 2. Algorithm for GAMICRA

Fig. 3. RemoveRedundancy pseudo code
Fig. 4. RemoveEmptiness pseudo code
For evaluating the fitness value of each chromosome, the following fitness function
is applied.
1
∑ ∑
/ 0,1
(3)
Where F(Xi) is the fitness value of chromosome Xi; bNM is the bid price of item M
submitted by seller N; lNM is the number of instances of item M submitted by seller N;
xNM(C) is 1 when the item combination C is selected for item M of seller N, and 0
otherwise.
Gambling Wheel Disk [7] is used in the selection operation. After selection, the
following chromosomes are produced based on the fitness values calculated by using
equation (3).

X1: 000 000 000 101 011 000 (Previous X2)
X2: 000 000 000 101 011 000 (Previous X2)
X3: 000 101 011 000 000 000 (Previous X4)
X4: 000 101 011 000 000 000 (Previous X4)
Then the modified two-point crossover is applied as shown in Figure 5(a). In our
modified two-point crossover operation, the first child inherits the parent chromo-
somes between the crossover points in the forward direction as the basic two-point
crossover, but the second child inherits it in reverse order to create a positive effect.
After crossover, the following chromosomes are generated.
X1: 000 000 000 101 011 000
X2: 011 000 000 101 000 000
X3: 000 101 011 000 000 000
X4: 000 101 011 000 000 000
Then the procedure will enter into the mutation operation as illustrated in Figure
5(b). For each chromosome, two unique random numbers are selected between 1 and
n. Assume for this example, 1 and 3 are chosen, so the bit configurations of sellers S1
and S3 are interchanged. Algorithm 1 is repeated until the specified number of gen-
erations is completed. Thus, the winner(s) are determined based on the minimum bid
price in an optimal processing time.
Fig. 5. (a) Modified two-point crossover (b) Mutation Operation
3 Experimentation
We have fully implemented GAMICRA in Java. This method is 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 use the following parameters and settings for all the experiments. The popula-
tion size is 100 and we use the binary string for chromosomes encoding. The selection
method is based on the Gambling Wheel Disk technique [7]. The crossover and
mutation rates are 0.6 and 0.01 respectively. The running time in milliseconds and
procurement cost (bid price) are averaged over 20 runs. The maximum number of
generations is 50. The total number of sellers, items and instances are 500, 8 and 5
respectively unless indicated otherwise.

Figures 6(a), 6(b), 6(c), 6(d), 6(e) and 6(f) report the results of these experiments.
Figure 6(a) shows how the running time varies with the increase of the number of
generations. As we can notice our method can still run in a reasonable time (under the
minute) even for a large number of iterations. Figure 6(b) shows how does the
bid price improve with the increase of the number of generations, thanks to our modi-
fied two-point crossover initially defined in [14]. As we can easily see, this cost sig-
nificantly improves in the first 30 iterations and stabilized afterwards. Figures 6(c)
and 6(d) show how the running time increases when we augment the number of
Fig. 6. Running time and procurement cost performances

sellers as well as the number of items. In this regard our method is capable of produc-
ing the solutions in less than 30 seconds even for large number of sellers and a good
number of items. Figure 6(e) shows the time performance of our method when vary-
ing the number of instances from 1 to 128. Here too, GAMICRA successfully returns
the solutions in a reasonable running time (under than 2 minutes in the case of 128
instances). Finally Figure 6(f) demonstrates the consistency of GAMICRA. The bid
price is indicated by a solid line with the maximum and minimum values represented
by + and - respectively. The error bars are depicted with a confidence level of 95%.
From the figure, it is clearly notable that the variations of bid price in a particular
generation over different runs are quite small. This means that GAMICRA is able to
reduce the solution variations. It is also notable that the solution quality increases
steadily over the generations. After 5 and 10 generations, our method produces an
average bid price of 6267.55 and 5731.3 respectively. It then continues producing
better solutions generation after generation. Furthermore, the variability of the solu-
tion quality over multiple runs improves over generations. The variability of the solu-
tions fluctuates till the 25th generation and stabilizes afterwards. In addition, the
minimum bid price remains constant after 25 generations. This suggests that the best
solution found by GAMICRA might be the optimal solution.
4 Conclusion and Future Work
We propose a GA based method for winner determination with multiple instances of
items in CRAs. Thanks to the RemoveRedundancy and RemoveEmptiness functions
as well as the modified two-point crossover operator, GAMICRA can produce opti-
mal solutions in a reasonable processing time and optimal procurement cost. More-
over, GAMICRA is a consistent method as it is able to reduce the solution variations
over different runs.
As it is obvious that parallel GAs are capable of providing the solutions in a better
running time [1], the future target of this research is to design our proposed method in
parallel based on Parallel GAs (PGAs). Another future direction is to consider other
bidding features besides ’Price’, such as ’Seller Reputation’, ’Delivery Time’, ’Feed-
back from Other Buyers’ and ’Warranty’. The winner will then be determined accord-
ing to these criteria.
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