Comparison Training for a Rescheduling 
Problem in Neural Networks 
Didier Keymeulen 
Artificial Intelligence Laboratory 
Vrije Universiteit Brussel 
Pleinlaan 2, 1050 Brussels 
Belgium 
Martine de Gerlache 
Prog Laboratory 
Vrije Universiteit Brussel 
Pleinlaan 2, 1050 Brnssels 
Belgium 
Abstract 
Airline companies usually schedule their flights and crews well in 
advance to optimize their crew pools activities. Many events such 
as flight delays or the absence of a member require the crew pool 
rescheduling team to change the initial schedule (rescheduling). In 
this paper, we show that the neural network comparison paradigm 
applied to the backgammon game by Tesauro (Tesatlro and Se- 
jnowski, 1989) can also be applied to the rescheduling problem of 
an aircrew pool. Indeed both problems correspond to choosing 
the best solution from a set of possible ones without ranking them 
(called here best choice problem). The paper explains from a math- 
ematical point of view the architecture and the learning strategy of 
the backpropagation neural network used for the best choice prob- 
lem. We also show how the learning phase of the network can be 
accelerated. Finally we apply the neural network model to some 
real rescheduling problems for the Belgian Airline (Sabena). 
I Introduction 
Due to merges, reorganizations and the need for cost reduction, airline companies 
need to improve the efficiency of their manpower by optimizing the activities of 
their crew pools as much as possible. A standard scheduling of flights and crews is 
usually made well in advance but many events, such as flight delays or the absence 
of a crew member make many schedule changes (rescheduling) necessary. 
801 
802 Keymeulen and de Gerlache 
Each day, the CPR 1 team of an airline company has to deal with these pertur- 
bations. The probleln is to provide the best answer to these regularly occurring 
perturbations and to limit their impact on the general schedule. Its solution is hard 
to find and usually the CPR team calls on hall reserve crews. An efficient reschedul- 
ing tool taking into account the experiences of the CPR team could substantially 
reduce the costs involved in rescheduling notably by limiting the use of a reserve 
crew. 
The paper is organized as follow. In the second section we describe the rescheduling 
task. In the third section we argue for the use of a neural network for the reschedul- 
ing task and we apply an adequate architecture for such a network. Finally in 
the last section, we present results of experiments with schedules based on actual 
schedtales used by Sabena. 
2 Rescheduling for an Airline Crew Pool 
When a pilot is unavailable for a flight it becomes necessary to replace him, e.g. 
to reschedule the crew. The rescheduling starts froin a list of potential substitute 
pilots (PSP) given by a schedtaling prograan based generally on operation research or 
expert system technology (Steels, 1990). The PSP list obtained respects legislation 
and security rules fixing for example the mareher of flying hours per month, the 
maximum number of consecutive working hour and the number of training hours 
per year and their schedule. From the PSP list, the CPR team selects the best 
candidates taking into account the schedule stability and equity. The schedule 
stability requires that possible perturbations of the schedule can be dealt with with 
only a minimal rescheduling effort. This criterion ensures work stability to the crew 
members and has an important influence on their social behavior. The schedule 
equity ensures the equal distribution of the work and payment among the crew 
members during the schedule period. 
One may think to solve this rescheduling problem in the same way as the scheduling 
problem itself using software tools based on operational research or expert system 
approach. But this is inefficient for two reasons, first, the scheduling issued from a 
scheduling system and its adaptation to obtain an acceptable schedule takes days. 
Second this system does not take into accotint the previous schedule. It follows 
that the lapdated one anay (lifter significantly fi'om the previous one after each 
perturbation. This is unacceptable fa'oan a pilot's point of view. ttence a specific 
procedure for rescheduling is necessary. 
3 Neural Network Approach 
The problem of reassigning a new crew member to replace a missing member can 
be seen as the problem of finding the best pilot in a pool of potential substittate 
pilots (PSP), called the bcst choicc problem. 
To solve the best choice problem, we choose the neural network approach for two 
reasons. First the rules used by the expert are not well defined: to find the best PSP, 
Crew Pool Rescheduler 
Comparison Training for a Rescheduling Problem in Neural Networks 803 
the expert associates implicitly a score vahle to each profile. The learning approach 
is precisely well suited to integrate, ill a short period of time, the expert knowledge 
given in an implicit form. Second, the neural network approach was applied with 
success to board-games e.g. the Backgammon game described by Tesauro (Tesanro 
and Sejnowski, 1989) and the Nine Men's Morris game described by Bratin (Braun 
and al., 1991). These two games are also examples of best choice problem where 
the player chooses the best move from a set of possible ones. 
3.1 Profile of a Potential Substitute Pilot 
To be able to use the neural network approach we have to identify the main fea- 
tures of the potential substitute pilot and to codify them in terms of rating values 
(de Gerlache and Keymeulen, 1993). We based our coding scheme on the way the 
expert solves a rescheduling problem. He identifies the relevant parameters associ- 
ated with the PSP and the perturbed schedule. These parameters give three types 
of information. A first type describes the previous, present and furtire occupation 
of the PSP. The second type represents information not in the schedule such as 
the human relationship factors. The associated vaines of these two types of pa- 
rameters differ for each PSP. The last type of pa.rameters describes the context 
of the reschednling, namely the characteristics of the schedule. This last type of 
parameters are the same for all the PSP. All these paranleters form the profile of 
a PSP associated to a perturbed schedule. At each rescheduling problem corre- 
sponds one perturbed schedule j and a group of, PSP i to which we associate a 
Profile _= (PSP i, Perturbed_Schedule j). Implicitly, the expert associates a rat- 
ing value between 0 and 1 to each parameter of the Profi! based on respectively 
its little or important impact on the resulting schedule if the PSP i was chosen. 
The rating value reflects the relative ilnportance of the parameters on the stability 
and the equity of the resulting schedule obtained after the pilots substitution. 
3.2 Dual Neural Network 
It would have been possible to get more information froill the expert than only the 
best profile. One of the possibilities is to ask him to score every profile associated 
with a perturbed planning. From this association we could immediately construct 
a scoring function which couples each profile with a specific vahle, namely its score. 
Another possibility is to ask the expert to rank all profiles associated with a per- 
turbed schedule. The corresponding ranki,.g function couples each profile with a 
value snch that the vahles associated with the profiles of the same perturbed sched- 
ule order the profiles according to their rank. The decision making process used by 
the rescheduler team for the aircrew rescheduling problem does not consist in the 
evaluation of a scoring or ranking function. Indeed only the knowledge of the best 
profile is useful for the rescheduling process. 
From a neural network architectural point of view, because the ranking problem is a 
generalization of the best choice problem, a same neural network architecture can be 
used. But the difference between the best choice problem and the scoring problem 
is such that two different neural network architectures are associated to them. As 
we show in this section, although a backpropagation network is sufficient to learn a 
scoring function, its architecture, its learning and its retrieval procedures must be 
804 Keyrneulen and de Gerlache 
adapted to learn the best profile. Through a mathematical formulation of the best 
choice problem, we show that the comparison paradigm of Tesauro (Tesauro, 1989) 
is suited to the best choice problem and we suggest how to improve the learning 
convergence. 
3.2.1 Comparing Function 
For the best choice problem the expert gives the best profile Profile .Bet associated 
$ 
with the perturbed schedule j and that for m perturbed schedules. The problem 
consists then to learn the mapping of the m. n profiles associated with the m 
perturbed schedules into the rn best profiles, one for each perturbed schedule. One 
way to represent this association is through a corn, paring function. This function 
has as input a profile, represented by a vector Xj, and returns a single value. When 
a set of profiles associated with a perturbed schedule are evaluated by the function, 
it returns the lowest value for the best profile. This comparing timetlon integrates 
the information given by the expert and is sufficient to reschedule any perturbed 
schedule solved in the past by the expert. Formally it is defined by: 
Co,,pa,'e. = (Profile) (1) 
Compareff *t <Com. parc. { Vj with j = 1,...,m 
ViBest with i=l,...,n 
The value of Compare are not known a priori and have only a meaning when they 
are compared to the value Compare * of the comparing filnction for the best 
profile. 
3.2.2 Geometrical Interpretation 
To illustrate the difference between the neural network learning of a scoring function 
and a comparing function, we propose a geometrical interpretation in the case of 
a linear network having as input vectors (profiles) ,..., X,..., associated 
with a perturbed schedule j. 
The learning of a scoring filnction which associates a score Score with each input 
vector X consists in finding a hyperplane in the input vector space which is tangent 
to the circles of center ./ and ra(lius 8co,' (Fig. 1). On the contrary the learning 
of a comparing function consists to obtain the equation of an hyperplane such that 
the end-point of the vector Sff* is nearer the hyperplane than the end-points of 
the other input vectors  associated with the same perturbed schedule j (Fig. 1). 
3.2.3 Learning 
We use a neural network approach to build the comparing filnction and the mean 
squared error as a measure of the quality of the approximation. The comparing 
function is approximated by a non-linear function' C(Profile) = 
where I is the weight vector of the neural network (e.g backpropagation network). 
The problem of finding C which has the property of (1) is equivalent to finding the 
function C that minimizes the following error filnction (Braun and al., 1991) where 
 is the sigmoid timetlon: 
Comparison Training for a Rescheduling Problem in Neural Networks 805 
Figtare 1' Geometrical Interpretation of the learning of a Scoring Function 
(Rigth) and a Comparing Function (Left) 
j=l 
i y Best 
(q [C( rrofile '') - (rrofilej )])2 
(2) 
To obtain the weight vector which minimizes the error filnction (2), we use the 
property that the -grM,(ff/) points in the direction in which the error function 
will decrease at the fastest possible rate. To update the weight we have thus to 
calculate the partial derivative of (2) with each components of the weight vector 
1' it is made of a prodtact of three factors. The evaluation of the first two factors 
(the sigmoid and the derivative of the sigmoid) is immediate. The third factor is 
the partial derivative of the non-linear function A/' which is generally calculated 
by using the generalized delta rule learning law (Rumelhart and McClelland, 1986). 
Unlike the linear associator network, for the backpropagation network, the error 
function (2) is not equivalent to the error function where the difference ..Bet _ )j 
is associated with the input vector of the backpropagation network becatse: 
5r(, - 27 ~' 
_.) # - 
(3) 
By consequence to calculate the three factors of the partial derivative of (2), we 
have to introduce separately at the bottom of the network the input vector of the 
best profile ,.,t and the input vector of a less good profile X-'j. Then we have to 
memorize their partial contribution at each node of the network and multiply their 
contributions before tipdating the weight. Using this way to evaluate the derivative 
of (2) and to update the weight, the simplicity of the generalized delta rule learning 
law has disappeared. 
806 Keymeulen and de Gerlache 
3.2.4 Architecture 
Wesauro (Tesauro and Sejnowski, 1989) proposes an architecture, that we call dual 
neural network, and a learning procedure such that the simplicity of the generalized 
delta rule learning law can still be used (Fig. 2). The same kind of architecture, 
called siamese network, was recently used by Bromley for the signature verification 
(Bromley and al., 1994). The dual neural network architecture and the learning 
strategy are justified mathematically at one hand by the decomposition of the partial 
derivative of the error function (2) in a slim of two terms and at the other hand by 
the asymmetry property of the sigmoid and its derivative. 
The architecture of the dual neural network consists to duplicate the multi-layer 
network approximating the comparing function (1) and to connect the output of 
both to a unique output node through a positive unit weight for the left network and 
negative unit weight, for the right network. Dnring the learning a couple of profiles 
is presented to the dual neural network: a best profile e.t and a less good profile 
. The desired value at the output node of the dual neural network is 0 when the 
left network has for input the best profile and the right network has for input a less 
good profile and 1 when these profiles are permuted. During the recall we work only 
with one of the two multi-layer networks, suppose the left one (the choice is of no 
importance because they are exactly the same). The profiles Xj associated with a 
perturbed schedule j are presented at the input of the left, multi-layer network. The 
best profile is the one having the lowest value at the output of the left multi-layer 
network. 
Through this mathelnatical formulation we can use the suggestion of Braun to 
improve the learning convergence (Bratin and al., 1991). They propose to replace 
the positive and negative unit weight between the output node of the multi-layer 
networks and the outplat node of the dtlal neural network by respectively a weight 
value eqnal to 2 for the left network and -2 for the right network. They modify the 
value of 2 by applying the generalized delta rule which has no significant impact on 
the learning convergence. By manually increasing the factor 2 during the learning 
procedure, we improve considerably the learning convergence due to its asymmetric 
impact on the derivative of ,v(k) with I: the modification of the weight vector 
is greater for couples not yet learned than for couples already learned. 
4 Results 
The experiments show the ability of onr model to help the CPR team of the Sabena 
Belgian Airline company to choose the best profile in a group of PSPs based on 
the learned expertise of the team. To codify the profile we identify 15 relevant 
parameters. They constitute the input of our neural network. The training data 
set was obtained by analyzing the CPR team at work during 15 days from which 
we retain our training and test perturbed schedules. 
We consider that the network has learned when the coinparing value of the best 
profile is less than the comparing value of the other profiles and that for all training 
perturbed schedules. At that time (I/7) is less than .5 for every couple of profiles. 
The left graph of Figure 3 shows the evohltion of the mean error over the couples 
Comparison Training for a Rescheduling Problem in Neural Networks 807 
Dual Neural Network 
Figure 2: The training of a dual neural network. 
during the training. The right graph shows the improvement of the convergence 
when the weight 12 is increased regularly during the training process. 
 Dual Neural Network hror 
0.4 0.4 
0,2 02 
Incnsing V 
of the Dual Neural l',ktwork 
Numb of 
a 
Figtire 3: Convergence of the dual neural network architecture. 
The network does not converge when we introduce contradictory decisions in our 
training set. It is possible to resolve them by adding new context parameters in the 
coding scheme of the profile. 
After learning, our network shows generalization capacity by retrieving the best 
profile for a new perturbed schedule that is similar to one which has already been 
learned. The degree of similarity required for the generalization remains a topic for 
further study. 
808 Keyrneulen and de Gerlache 
15 Conclusion 
In conclusion, we have shown that the rescheduling problem of an airline crew pool 
can be stated as a decision making problem, namely the identification of the best 
potential substitute pilot. We have stressed the importance of the codification of the 
information used by the expert to evaluate the best candidate. We have applied the 
neural network learning approach to help the rescheduler team in the rescheduling 
process by using the experience of already solved rescheduling problems. By a 
mathematical analysis we have proven the efficiency of the dual neural network 
architecture. The mathematical analysis permits also to improve the convergence 
of the network. Finally we have illustrated the method on rescheduling problems 
for the Sabena Belgian Airline company. 
Acknowledgments 
We thank the Scheduling and Rescheduling team of Mr. Verworst at Sabena for 
their valuable information given all along this study; Professors Steels and D'Hondt 
from the VUB and Professors Pasttin, Leysen and Declerck from the Military Royal 
Academy who supported this research; Mr. Horner and Mr. Pau from the Digital 
Europe organization for their funding. We specially thank Mr. Decuyper and Mr. 
de Gerlache for their advices and attentive reading. 
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