EVOLVING EFFECTIVE BIDDING FUNCTIONS FOR AUCTION

BASED RESOURCE ALLOCATION FRAMEWORK

Mohamed Bader-El-Den and Shaheen Fatima

Department of Computer Science, Loughborough University, Leicestershire LE11 3TU, U.K.

Keywords:

Genetic programming, Multi-agent systems, Resource allocation, Timetabling, Auctions based systems,

Heuristics.

Abstract:

In this paper, we present an auction based resource allocation framework. This framework, called GPAuc,

uses genetic programming for evolving bidding functions. We describe GPAuc in the context of the exam

timetabling problem (ETTP). In the ETTP, there is a set of exams, which must be assigned to a predeﬁned

set of slots. Here, the exam time tabling system is the seller that auctions a set of slots. The exams are

viewed as the bidding agents in need of slots. The problem is then to ﬁnd a schedule (i.e., a slot for each

exam) such that the total cost of conducting the exams as per the schedule is minimised. In order to arrive

at such a schedule, we need to ﬁnd the bidders’ optimal bids. This is done using genetic programming. The

effectiveness of GPAuc is demonstrated experimentally by comparing it with some existing benchmarks for

exam time-tabling.

1 INTRODUCTION

Decentralised scheduling is the problem of allocat-

ing resources to alternative possible uses over time,

where competing uses are represented by autonomous

agents. This scheduling can be done using differ-

ent methods such as such as ﬁrst-come ﬁrst-served,

priority-ﬁrst, and combinations thereof. But, these

methods do not generally possess globally efﬁcient

solutions. Due to this limitation, considerable re-

search is now focussing on the use of market mech-

anisms for distributed resource allocation problems

(Krishna, 2002). Market mechanisms use prices de-

rived through distributing bidding protocols, such as

auctions, to determine schedules.

In an auction, there are two types of agents: the

auctioneer and the bidders. The auctioneer could be

a seller of a resource and the bidders are buyers that

are in need of the resource. The bidders bid for the re-

source being auctioned and one of them is selected as

the winner. An agent’s bid, in general, indicates the

price it is willing to pay to buy the resource. On the

basis of the agent’s bids, the resource is allocated to

the winning agent. The auction protocol determines

the rules for bidding and also for selecting a winner.

There are several protocols such as the English auc-

tion, the Dutch auction, and the Vickrey auction pro-

tocol (Krishna, 2002).

Given an auction protocol, a key problem for the

bidders is to ﬁnd an optimal bidding function for the

protocol (Krishna, 2002). An agent’s bidding func-

tion is a mapping from its valuation or or utility or

preference (for the resource being auctioned) to a bid.

An agent’s valuation is a real number and so is its bid.

Since there are several agents bidding for a single re-

source, an agent must decide how much to bid so that

its chance of winning is maximized and the price at

which it wins is minimised. Such a bid is called the

agent’s optimal bid. An agent’s optimal bidding func-

tion is then the function that takes the valuation as

input and returns its optimal bid.

For a single auction, ﬁnding an agent’s optimal

bidding function is easy. But in the context of the

distributed scheduling problem we focus on, there are

several auctions that are held sequentially one after

another. Furthermore, an agent may need more than

one resource and must therefore bid in several auc-

tions. In such cases, an agent’s bidding function de-

pends on several parameters such as how many auc-

tions will be held, how many bidders will bid in each

of these auctions, and how much the agent and the

other bidders value the different resources. This com-

plicates the problem of ﬁnding optimal bids. In order

to overcome this problem, our objective is to use GP

to evolve bidding functions.

We study the distributed scheduling problem in

310

Bader-El-Den M. and Fatima S. (2009).

EVOLVING EFFECTIVE BIDDING FUNCTIONS FOR AUCTION BASED RESOURCE ALLOCATION FRAMEWORK.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 310-313

DOI: 10.5220/0002324203100313

 SciTePress

the context of the famous exam time tabling prob-

lem (ETTP) (Carter et al., 1996). The ETTP can be

viewed as a decentralised scheduling problem where

the exams represent independent entities (users) in

need of resources (slots) with possibly conﬂicting and

competing schedule requirements. The problem is

then to assign exams to slots (i.e., ﬁnd a schedule)

such that the total cost of conducting the exams as per

the schedule is minimised.

2 EXAM TIMETABLING

PROBLEM

The exam timetabling problem is a common problem

in most educational institutions. Although the prob-

lem’s details tend to vary from one institution to an-

other, the core of the problem is the same. There is a

set of exams (tasks), which have to be assigned to a

predeﬁned set of slots and rooms (resources).

In our research we will be using on the follow-

ing formulation for the exam timetabling problem.

The problem consists of the a set of n exams E =

, . . . e

}, a set of m students S = {s

, . . . s

}, a set

of q time slots P = {p

, p

, . . . p

} and a registration

function R : S → E, indicating which student is attend-

ing which exam. Seen as a set R = {(s

, e

) : 1 ≤ i ≥

m, 1 ≤ j ≥ n}, where student s

is attending exam e

A scheduling algorithm assigns each exam to a cer-

tain slot. A solution then has the form O : E → P or,

as a set, O = {(e

, p

) : 1 ≤ k ≥ n, 1 ≤ l ≥ q}.

The problem is similar to the graph colouring

problem but it includes extra constraints, as shown by

Welsh and Powell (Welsh and Powell, 1967). These

constraints are categorised into two main types: (a)

Hard Constraints, violating any of these constraints

is not permitted since it would lead to an unfeasible

solution, and (b) Soft Constraints, which are desir-

able but not crucial requirements. Violating any of the

soft constraints will only affect the solution’s quality.

All hard constraints are equally important, while soft

constraints are not. The importance of soft constraints

vary. Usually a cost function is designed to calculate

the cost of violating each of the soft constraints. So-

lutions with lower cost have better quality.

In general, there are two phases in solving

scheduling problems, construction phase for generat-

ing initial solution, the second phase is to improve the

quality of the initially constructed solutions, method

we present here is for the ﬁrst phase.

3 AUCTION BASED

TIMETABLING

We call our exam time tabling system GPAuc. In

GPAuc, the seller is the exam time tabling system

(ETTS) and it auctions the slots one at a time. The

exams are the bidders. Every slot could be sold more

than once (because one slot could contain more than

one conﬂicting exam), but in each auction the slot

could be sold only for one exam. For an auction, the

winning bid is determined as follows. If the highest

bid does not increase the solution cost beyond a cer-

tain limit (Accepted-Cost), the highest bid becomes

the winning bid. Otherwise, the same rule is applied

to the second highest bid. If the second highest bid

causes the cost to increase beyond the Accepted-Cost,

the slot is left unsold and the next auction is initiated

for the followingslot. If no slots havebeen sold in full

round on all available slot, in this case the Accepted-

Cost are increased. this process is repeated till all ex-

ams are scheduled, or reaching a deadlock, where no

more exams could be scheduled without violating a

hard constraint.

The cost for a schedule is calculated using the fol-

lowing function (Carter et al., 1996):

Cost =

N−1

∑

i=1

∑

j=i+1

[w(|p

− p

|)a

] (1)

where N is the total number of exams in the problem,

S the total number of students, a

is the number of

students attending both exams i and j, p

is the time

slot where exam i is scheduled, w(|p

− p

|) returns

5−|p

−p

if |p

− p

| ≤ 5, and 0 otherwise.

3.1 Optimisation via Genetic

Programming

GP (Koza, 1992; Langdon and Poli, 2002; Poli et al.,

2008) is an evolutionary algorithm which is inspired

by biological evolution. The target of a GP system

is to ﬁnd computer programs that perform a user-

deﬁned task. It is a specialisation of genetic algo-

rithms where each individual is a computer program.

GP is a machine learning technique used to optimise

a population of computer programs depending on a

ﬁtness function that measures the program’s perfor-

mance on a given task. Tree presentation of the indi-

viduals is the most common presentation which also

we will be using here. The function and terminal set

used is shown in table 1, the terminal set are inspired

from some standard graph coloring heuristics.

The GP’s ﬁtness function we used is the follow-

ing:

EVOLVING EFFECTIVE BIDDING FUNCTIONS FOR AUCTION BASED RESOURCE ALLOCATION

FRAMEWORK

311

Table 1: GP function and terminal sets.

Function Set

add(d

, d

) : returns the sum of d

and d

sub(d

, d

) : subtracts d

from d

mul(d

, d

) : returns the multiplication of d

by d

div(d

, d

) : protected division of d

by d

abs(d

) : returns the absolute value of d

neg(d

) : multiplies d

by −1

fneg(d

) : abs(d

) multiplied by −1, to

force

negative value

sqrt(d

) : returns the a protected square

root of d

Terminal Set

slt : total number of available slots

for the currently bidding exam e

std : the total number of students at-

tending the bidding exam e

conf : the total number of all exams

(scheduled and not scheduled)

in conﬂict with exam e

cSched : number of already scheduled

exams that are in conﬂict with e

cPendd : number of exams(not scheduled

yet) that are in conﬂict with e

cost : current increase in the cost if e

is allocated the current slot

f = [

M−1

∑

i=1

∑

j=i+1

w(|p

− p

|)a

] + (N − M) ×C (2)

where: N is the total number of exams in the prob-

lem, M is the total number of exams that have been

successfully scheduled, (N − M) ≥ 0 is the number of

unscheduled exams, C is constant. The objective is to

minimise this equation, so the lower the ﬁtness value

the better the individual is.

The ﬁrst part of the ﬁtness function in Equation (2)

is almost the same as the cost function in Equation 1.

The second part adds extra penalty for each unsched-

uled exam. Even though solutions with unscheduled

exams are considered to be invalid solutions, this ex-

tra penalty for unscheduled exams is introduced to

give GP better ability to differentiate between indi-

viduals.

3.2 Experimental Results

We tested our method for timetabling by applying it

to one of the most widely used benchmarks in exam

timetabling, against which many state-of-the-art algo-

rithms have been compared in the past. the bench-

mark’s details could be found in (Carter et al., 1996)

where it was ﬁrst introduced.

We ran a number of experiments using differ-

ent GP parameters, with population size varies be-

tween 50 to 1000, number of generations range be-

tween 50 and 100, the production and mutation rate is

5% or 10% of the total population, and 80% for the

crossover rate. The selection is done using tourna-

ment selection of 5.

Table 2 shows the cost (using equation number

(2)) of the GPAuc compared to a number of other

construction techniques, as it could be noticed the

GPAuc is very competitive with other methods tak-

ing in consideration that GPAuc does not use back-

tracking. Moreover, GPAuc is as distributed methods

and that all biding functions are automatically evolved

without human interaction.

Figures 1, 2, 3 and 4 provide some analysis

into the behaviour of the best performing individ-

uals throughout the generations. These graphs are

drawn from evolving biding functions on the York83

instance, with population size 500, number of gener-

ations 100, mutation and reproduction rate 10% and

crossover rate of 90%.

Figure 1: The average value of the ﬁnal Accepted cost of

the best evolved functions.

Figure 2: Percentage of individuals that have been able to

schedule all exams in the best evolved function

IJCCI 2009 - International Joint Conference on Computational Intelligence

312

Table 2: Results from the GP-HH for time tabling among with other results reported in literature on benchmark exam

timetabling problems, the table shows the cost for each case, the cost is calculated using equation 1

car91 car92 ear83 hec92 kfu93 lse91 sta83 tre92 uta93

GPAuc 7.03 5.80 41.2 13,01 15.90 13.01 157.3 9.32 3.82

(Burke et al., 2007) 5.41 4.84 38.19 12.72 15.76 13.15 141.08 8.85 3.88

(Asmuni et al., 2004) 5.20 4.52 37.02 11.78 15.81 12.09 160.42 8.67 3.57

(Carter et al., 1996) 7.10 6.20 36.40 10.80 14.00 10.50 161.50 9.60 3.50

(Burke et al., 2007) 5.41 4.84 38.84 13.11 15.99 13.43 142.19 9.2 4.04

Figure 3: Average number of remaining exams in all best

behaving individuals.

Figure 4: Average number of all auction taking place in best

performing heuristics through out all the generations.

4 CONCLUSIONS

In this paper we introduced GPAuc, a genetic pro-

gramming framework for evolving agent’s bidding

function for resource allocation problems. The frame-

work is described in the context of the exam time

tabling problem.

Results shows that the framework is competitive

with other existing methods for constructing exam

timetables, taking in consideration that the GPAuc has

no backtracking and uses a distributed approach.

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FRAMEWORK

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