ELECTRE and AHP MCDM methods versus CP method and the
official choice applied to high-speed railway layout alternative election
JOSE M. ANTONa, JUAN B. GRAUb, DIEGO ANGUINAC
Technical University of Madrid
E.T.S. de Ing. Agrónomos. Av. Complutense s/n, Ciudad Universitaria. 28040 Madrid. SPAIN
Abstract: - The application of different multicriteria methods for the election of alternatives for the layout of
high-speed railway in one specific corridor is considered in this paper using the outranking MCDM methods:
ELECTRE-I and AHP, in comparison with Compromise Programming (C.P.) methods. Starting from two
reported cost-effectiveness studies for Madrid-Valencia link, the feasibility may be assessed, considering only
a small set of criteria well elected and studied and containing relevant variables. Using outranking methods the
solution “North” was preferable, in coincidence with the official solution. C.P. methods for these applications
present several inconveniences. Thus, we propose outranking methods for such Decision-Making procedures.
Key-Words: Transportation systems, Multicriteria Methods, Decision Aiding.
TAV net was approved as a “Public-Works_20002007_Plan for TAV rail infrastructure”. The case in
this paper refers to layout of the link from Madrid to
Valencia at the Mediterranean coast at East, which is
an essential and decisive part of the approved TAV
SE Sub-net.
The officially adopted solution
communicates almost at best Valencia with Madrid,
and includes the northern city and region of Cuenca
in the area of the new facility. For that link MadridValencia the authors have taken for their academic
case-study for this paper three basic alternatives for
decision extracted from the six public official
alternatives for the whole TAV SE Sub-net that were
described in the document PS. These three
alternatives will be designed as (North, Centre and
South) in the following, and they were designed as
local optimums because of topography and use of
territory. There are, indexed by i or k:
a. The North solution (i = 1) is shorter for MadridValencia, at 350 km at first, through Cuenca,
which is also a main historical town. The cost is
higher than for the other solutions because the
terrain is a higher plateau.
b. The Centre or central solution (i = 2) follows in
the middle section the actual new MadridValencia expressway, is shorter with a length of
370 km roughly.
c. The South alternative (i = 3) follows roughly the
actual rail main line laid in the 19th century to
communicate the whole South and Southwest.
As a result of the flatter terrain and of the use of
older portions of the present line it is cheaper
1. Introduction.
The authors, involved in Decision Making in
U.P.M. and having some older experience in real
management, set first a prior soft academic casestudy using the methods ELECTRE and AHP
applied to the problem of high-speed railway layout
alternatives election. In that moment election of
alternatives for different corridors are part of a
general long term planning to upgrade the Spanish
rail network.. Then authors examined the relations of
different methods with the big and innovative real
project, and present in this paper case their revised
models and commentaries about their aptitude to
represent a real case.
2. Alternative layouts and criteria for
the TAV Madrid-Valencia.
The real decision was adopted (see
http://www.mfom.es) the 8th of January 2002 by the
administrations concerned of the sector SE of Spain
with centre in Madrid. The official decision
procedure involved the elaboration of a technical
study referred later as PS (from Public Survey, ref.
[7]), with plans at scale 1/25000) that was exposed
in a public information procedure. At present date
(winter of 2003) there is a trend for high-speed trains
in Spain coexistent with other diverse public work
plans. The future of trains require to upgrade totally
some main lines to passenger rail standards for
speeds of 300-350 km/h, designed here as TAV from
"Tren de Alta Velocidad" in Spanish, and a new
1
from Madrid to Valencia, although it has a much
bigger length of 470 km.
As said an ELECTRE method and an Expert
Choice AHP method will be used for the present
case-study, by experts using at best the previously
referred information, at least to make acceptable
trade-offs between variables that will take the form
of weights. Other methods could also be considered,
( [2] is a seminal reference, [5] contains a review of
MCDM methods). At the present academic level
four aggregated criteria that seem to be reasonably
independent and complete, are used and are
represented by j-criteria variables as follows:
1.
Cost (j = 1), is the value in million euros of the
initial costs, it is a "more is worse” variable, as
the total project budget from the survey PS at
state level to have the operating TAV new line
from present state.
2. Trip Duration (j = 2). That variable will be in
minutes, as travelling time between MadridValencia, and is a "more is worse” variable,
represents the important related benefits and is
roughly proportional to the distance.
3. Population index (j = 3). That aggregated
variable, of type "more is better", will include
the population served by the train, as disposed to
use it, that is habitants and their tendency to use
the train, and with a lower weight the population
which will have indirect benefits from the train
as said. At present level, as the effects are of
different nature, indirect, and not easily
predictable, a simple figure as population is
used. The North alternative will be given a
somehow higher value because it would
integrate a larger part of territory as said.
4. Environment index (j = 4). Some main
environment effects of that new train will be
positive and the proposed alternatives have
already avoided some critical sensitive areas, as
said. What remains are some unfavourable
environmental impacts to populations or to
natural areas, and what matters are the
differences between alternatives that are not
major. The “environmental impact” studies
include methods that are often ELECTRE
([9],[11],[12]), which result in indexes obtained
from a conventional Multicriteria aggregation,
that are “more is worse” index. The natural
environment is more sensitive at North to some
degree, and that can be saw in the PS study, and
that will result in growing indexes.
The cost and trip duration can now be well
predicted, and are of great relevance because the
costs are high and different, and because the trip
durations are short and also different. The trip
duration may be reduced to 90mn with the TAV new
project to compare to almost 4 hours actually by
train or bus, and the cost will be of the order of
4.000 million euros. The population index is itself an
aggregation, and could be decomposed, maybe in a
population variable and in a surface variable. As it
is, it must include how important is the region
beneficiated by the new transport system, direct or
indirectly, at long term. Note that the units for
criteria are different, they will be compared
specifically later and the intervention of experts will
matter then.
3. Method in an ELECTRE format.
The ELECTRE-I method from Roy, see Roy
in [12] or in [14], also in [9] and in [4], has been
applied using Microsoft Office-97 EXCEL, an
ELECTRE software made from Roy may be
purchased instead. The ELECTRE – IV, reported by
Roy in [13] with an academic model case “ex post”
for the enlargement of the METRO of Paris was
found not appropriate for the case herein, because
the goal is to choose one alternative instead of
putting in order different ones, and because the
variables are simple and because they do not
correspond to what he calls “pseudocriteria“”. To
start, for each i-alternative a value Iij has been
assessed to each variable valuating the j-criterion or
attribute, to get the Information Table or Initial
Decision Matrix, Table 1, with a row of  j being 1
if the j-criterion is “more is better” and –1 if it is
“more is worse”, and a row of the values wj of the
weights vector wT = ( wj ).
Table 1, Information Table or Initial Decision
Matrix I = ( Iij ).
TRIP POTEN- ENVIRONDURATIAL MENTAL
AlternaCOST
TION USERS IMPACT
tives:
NORTH 4207,0847
85
3654286
3
CENTRE 3606,0726
90
3000000
2
SOUTH 3005,0605 125 3000000
1
wj
0,3
0,4
0,2
0,1
j index
-1
-1
1
-1
The element Ii1 is the criteria variable Cost
in millions of Spanish pesetas (1 euro = 166.386 pts
exactly). The Ii2 is the Trip Duration in minutes. In
the context of these project and ranges of variation
of the corresponding variables, 1202.0242 and 40
respectively, the Trip Duration seems to the authors
somewhat more important than the Cost, and so has
2
where 
got a weight somehow higher, w2 = 0.4 versus
w1 = 0.3. That contains a comprehensive trade-off
between, estimating that an increase of 1202.242/0.3
in Cost is to be compensated by a decrease in Trip
Duration of 40/0.4 . The assignment of the relative
values of the weights is essential for the results and
it must be done by experts knowing the method and
the real case, that is available prior information,
documents as the PS [7] with “Demand prevision”
and “Profitability Studies”, and moreover some
subjective estimation about long term consequences,
because the Costs are considerable, but the benefits
of short Trip Duration are various ( e.g. [8] is a
survey of social local effects, etc... ) and will last
many years.
The element Ii3 of the Potential users column
corresponds to the “Population index” criteria as
defined above in a wide sense, and so a higher value
has been taken for the North solution, with about
20% increase, by subjective rational belief in fact.
The Centre and South alternatives got a similar
value 3000000 that represents the habitants in the
corridor and in the Valencia extremity beneficiated
by the solution. The weight w3 = 0.2 has been taken
lower than for the other variables but not too much,
because the variable includes long-term benefits for
organisation and development of territory, etc….
The element Ii4 of the Environment impact
column corresponds to a differential Environment
impact index, and it has got a lower weight, w4 = 0.1
because as said in that case the difference among
alternatives is not very relevant. The weight values
ought to be normalised so that they add 1 and are
somewhat imprecise, as a result the values in the
tableau are adopted. A sensibility analysis for the
weights wj values has proved that results do not
change for small changes in weights.
In this Information Table no alternative
dominates any other for the four criteria, in a
“Pareto-like” sense, and to go further ELECTRE
method outranking relationships R are obtained. By
definition the alternative Ei outranks the Ek , i. e.
Ei R E k ,as meaning that “Ei is at least so good as
Ek’”,
i. if the concordance C(Ei ,Ek ) of Ei to Ek is
greater than the concordance threshold, CT ,
ii. and if the discordance D(Ei ,Ek ) of Ei to Ek is
not greater than the discordance threshold DT.
First a Concordance Index Matrix C = (Cik) ,
Table 2, was calculated, as defined for i  k as
Cik = C(Ei ,Ek ) =   jik  w j ,
(1)
jik


is equal to 1 if  j  I ij  I kj  0 ,


½ if I ij  I kj , 0 if  j  I ij  I kj  0 .
Table 2, Concordance Index Matrix C = (Cik) .
NORTH CENTRE SOUTH
NORTH
0,6
0,6
CENTRE
0,4
0,5 0,49
SOUTH
0,4
0,5
For CT a first value is to be taken as the average
of these 6 elements, getting CT = 0.5. It is convenient
to change it slightly to CT = 0.49 after a posterior
sensitivity analysis, to get a better structure of
results.
Then a Standardized Decision Matrix S = (Sij)
is obtained as


S ij  I ij SupI rj  Inf I rj  ,
r  1,2,3 
 r  1,2,3
(2)
and to follow the columns of S , that have an equal
range of variation equal to 1, are weighted to get the
Standardized and Weighted Matrix T = (Tij), see
Table 3, in the form
Tij = Sij * wj .
(3)
Table 3. Standardized and Weighted Matrix
T = (Tij).
TRIP
DURACOST TION
NORTH 1,05
CENTRE 0,9
SOUTH 0,75
0,85
0,9
1,25
POTENTIAL
USERS
ENVIRONMENTAL
IMPACT
1,117030167
0,917030167
0,917030167
0,15
0,1
0,05
Let the Discordance Index Matrix D = (Dik) be
defined as, for (i,k = 1,2,3 ) and (i  k) ,
Max  SupTkj  Tij   j ,0 
Dik = D(E ,Ek) = j 1, 2,3, 4
, (4)
Max Tij  Tkj

j 1, 2 , 3, 4

giving the Table 4.
Table 4. Discordance Index Matrix D = ( Dik ).
NORTH CENTRE
NORTH
0,75
CENTRE
1
SOUTH
1
1
SOUTH
0,750
0,428571429
For DT a first value is to be taken as the average
of these 6 elements, getting DT = 0.8214328571.
That value was conserved after a sensibility analysis,
as the results were insensitive to small changes of it.
j1, 2 , 3, 4
3
To obtain the final outranking
relationships from precedent definition for
(i,k = 1,2,3 ) and (i  k):
i. An Outranking Concordance Matrix U was
obtained from the Concordance Index Matrix
C with the rule: if Cik > CT = 0.50  0.49
(sensitivity analysis) then Uik = 1, else Uik = 0.
ii. Then an Outranking Discordance Matrix W
was obtained from the Discordance Index
Matrix D with the rule: if Dik < DT =
0.8214328571 then Wik = 1, else Wik = 0.
iii. Finally the Aggregated Matrix A ,Table 5, was
obtained from U and D with the rule :
Aik = Uik * Wik .
(5)
That was considered appropriated for the case, in
view of the data and context, and the resultant graph
of Figure 1 was adopted as ELECTRE result.
It appears that the logic of judgement
between criteria was not the same as in a IRR-NPV
method, that many factors and data have intervened
and that the decision was in the same direction as
was taken by the group of real decision-makers.
4. Using Expert Choice for A.H.P. .
The “Analytic Hierarchy Process” was
originated as known by Saaty ( see [15] and [16]). It
appears as a close school of decision, proclaimed in
[17] as a distinct “theory of measurement” applied in
“decision making”, as a “theory of decision different
and independent from utility theory”. An Expert
Choice Inc. Company decision support AHP
software
(http://www.expertchoice.com/) for PC has been
used, obtained in a CD including sets of manuals
rich on managerial procedures but hiding the exact
mathematical formulas behind.
Following [9] for AHP, an assumed decision
problem, the goal, is structured as a Hierarchy by
experts including alternatives and criteria that are
called objectives; sub-criteria levels will be added to
criteria if convenient, but that has not been made for
the case studied herein. To rank the i-alternatives a
Table 5. Aggregated Matrix A = ( A i k ).
NORTH
NORTH
CENTER
SOUTH
CENTER
1
0
0
SOUTH
1
01
0
In ELECTRE-I as said the alternative Ei
outranks the alternative Ek if and only if Aik = 1, and
that means that “Ei is at least so good as Ek’”. Note
that before the sensitivity analysis CT was 0.5 and
hence A23 was 0 , and that afterwards the value 0.49
has been adopted CT obtaining the result A23 = 1 that
is preferred.
To have a graphical expression of the resulting
partial ordering of alternatives an ELECTRE graph
was drawn. In it an arc (i,k) was drawn if Ei outranks
Ek . The non-dominated alternatives form the kern, in
that case containing only the North alternative,
which is to be considered more favourable than the
others. The resulting graph for CT = 0.50 is in
Figure1.
set of m values vi adding 1,

m
i 1 i
v  1 , is to be
Figure 1.ELECTRE graph and kern
obtained through an “Analytic Hierarchy Process”
from successive analogous previous valuations
quantifying the opinion of experts in view of the real
problem, maybe involving successive meetings of
experts. Starting from the inferior level, for each jth
objective experts have to assess sets of weights wij
adding 1,
wij  1 , to valuate the ith alternative
North
respectively to the objective jth. Then to aggregate
the j-objectives they have to assess a set of n values

i
or weights uj adding 1,

n
j 1
u j  1 . If there is only
one level of objectives the final values are obtained
wij  u j or similarly, but if there were a
as vi 

Centre
j
South
“Hierarchy” of them, experts would have to made a
“Process” of similar assessments. For the case study
the weights wij were internally assessed for each jth
A sensibility analysis was then made to
check the results were sensitive to small changes of
the values CT and DT that are somehow conventional.
It was observed that a small decrease in CT from 0.5
to 0.49 changes U23 from 0 to 1, meaning that the
alternative Centre outranks the South alternative.
objective by means of pair-wise comparisons of the
three i-alternatives made by the authors as experts.
Then the weights uj were assessed as result of a
similar process of pair-wise comparisons of the four
j-objectives.
4
Table 7. Standardized and weighted distances from
From the wij and u j , and using internal
Ideal, pT 0,75
procedures in two modes, Ideal and Distributive, it
obtained the Resulting estimated weights v i that
follow in Table 6:
COST
NORTH
CENTER
SOUTH
Table 6. AHP global weights, v i .
Alternative
vi
NORTH CENTRE
0,484749 0,301688
SOUTH
0,213563
0,3
0,15
0
0,85
1,11703027
TRIP
DURA- POTENTIAL
TION
USERS
0
0,05
0,4
0
0,2
0,2
0,05
ENVIRONMENTAL
IMPACT
0,1
0,05
0
the ideal point, P = ( Pij ) .
For the ith alternative an “h-disutility” is defined
for h  1 as h-distance from the ideal point,
These weight values are indexes of “priority”
for the alternatives, the first one is clearly higher,
and thus they point that the North alternative is the
best, and that the Centre one is sensibly better than
the South.
The EC results contain several possible
additional presentations, including 4 charts of
sensitivity analysis, Performance, Dynamic,
Gradient, Head-to-head. The first chart Performance
of Figure 3 contains all the results.
1h

h
Dh,i    Pij   for 1  h < ,
 j

D,i  Inf Pij  for h = .
(6)
j
To conclude the alternatives are ranked in
opposite sense than the disutilities. For the present
case-study the disutilities were calculated for h = 1,
1.5, 2 and  are shown in the Table 8.
Figure 3. Chart with EC results.
Table 8. h-Distance (Minkowski) from the ideal
Altrntv. D1
D2
D
D1,5
North 0,4 0,31622777 0,3 0,33735053
Centre 0,45 0,25980762 0,2 0,30675555
South 0,6 0,4472136 0,4 0,48945216
point, P = ( Pij ) for distance Dh.
For h = 1 then D1,i 
P
ij
, and the obtained
j
ranking for this case the same as in ELECTRE, as in
EXPERT CHOICE and as in the official election, all
that being satisfactory. In fact there is only a
comparison of added rows, and it can be seen that
the result is independent on the ideal point that does
not represent a feasible solution, it depends on
values of variables, weights, in a more primitive
form that in ELECTRE.
But for the other higher values of h the Centre
solution is preferred without rational justification,
and that result is unsatisfactory. The ideal point has
then an influence that is not desirable in that case, it
has the lower cost of the South solution and that
enhances the influence of the cost criteria over trip
duration and population ones. The introduction of
other feasible non-optimal alternatives, as is the
South alternative in fact, may thus change the ideal
point and the results. In most cases in literature
about CP the feasible set of solutions in the criteria
variables space is convex and an intermediate
alternative is chosen. It is clear that in the present
5. Comparison with a Compromise
Programming method.
Starting from the table precedent in chapter III
“Standardised and weighted matrix” T and from the
 j , a Compromise Programming method CP (see
[9], [10] or [3], or [1] for a public utility case with
several methods) is applied now easily using the
precedent tables in EXCEL. First an “ideal point or
alternative” having the best rating for each criteria
was calculated, resulting in the row pT, that has been
subtracted from the rows of the matrix T so as to
obtain the matrix P = (Pij) of the Table 7.
5
Valencia-Madrid TAV case-study the feasible set is
not convex. That is at first because the number of
solutions is discrete, and that is probably because the
intermediate solutions are not feasible. For there is
not a feasible intermediate layout between the North
and the South solution, because between them are
the great Alarcón dam reservoir in a river Júcar deep
valley and a gentle rift that belongs to the separation
between Mediterranean and Atlantic rivers. This
non-convexity of sets of solutions happens often in
Public Works because of natural restrictions.
Selection in Madrid Metropolitan Area. Journal of the
Operations Research Society of Japan, 46, March 2003.
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(2nd edn). Wiley, New York, 1963.
[3] Ballestero E. and Romero C. Multiple Criteria
Decision Making and its Application to economic
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[4] Barba-Romero S. & Pomerol J.Ch. Decisiones
Multicriterio. Servicio de Publicaciones Universidad
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Decision Making, Advances in MCDM Models,
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[7] Ministerio de Fomento D. G. de Ferrocarriles
6. Comparison with the Official
Choice, Discussion and Conclusions.
The Official Choice, that in fact seems
appropriated to the real problem, contains the North
solution that was also preferred in the precedent
methods herein. That was because of lower Trip
Duration and higher long-term indirect benefits such
as better future Organisation of Territory, as
including the Cuenca corridor in the SE TAVSubnet, in spite of higher initial costs. That is
satisfactory for the authors, but what does it means
in view of practical use? One may think that the
academic case-study of this paper completed ex post
may be too strictly influenced by the official
decision. The authors think that their result is
honest, even having to incorporate the reality in a
very simple format.
It seems to the authors that Methods like
ELECTRE I and Expert-Choice are then of use to
take into account inhomogeneous factors, social
opinion and the knowledge of different kinds of
experts. But care must be taken to include the
relevant decision the pertinent relevant criteria and
to include at least as preliminary the results of some
quantitative studies such as the Profitability studies.
Only with good technical and decisional information
will experts made useful subjective assessments of
variables, weights or pair wise comparisons to use
methods like ELECTRE or AHP. In the case studied
herein the Cost and Trip Duration criteria were
relevant, but also were more imprecise factors about
the future of territory that were been incorporated
and aggregated in a Potential Users variable. The
differences on Environment Impact were here less
relevant, but they may be decisive in other cases,
and in fact the E.I. had a previous great influence on
the details of each alternative.
Estudio Informativo del Proyecto, línea de Alta Velocidad
“Madrid, Castilla La_Mancha, Comunidad Valenciana,
Region_de_Murcia”.
Official
Survey
made
in
collaboration with S.E.N.E.R., Madrid, September 2000.
[8] Ribalaygua C. B. et alt., Efectos Territoriales de
la alta velocidad ferroviaria. Estrategias para el
planeamiento supramunicipal. O.P. Ingeniería y
Territorio, 60, Madrid 2002.
[9] Romero C. Teoría de la decisión multicriteria:
Conceptos,técnicas
y
aplicaciones.
Alianza
Universidad Textos, Madrid, 1993.
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approach. Omega Int. Mgmt. Sci., 1999, 27, 341-347.
[11] Romero C. and González-Pachón. Aggregation
of partial ordinal ranking: an interval goal
programming approach. Computers & Operational
Research, 2001, 28, 827-834.
[12] Roy B., Bouyssou D. Aide Multicritère à la
Décision: Méthodes et cas. Economica, Paris, 1993.
[13] Roy B., Hugonnard J-Chr., Ranking of
suburban line extension projects on the Paris metro
system by multicriteria method. Transportation
Research 1982, 16, A, 301-312.
[14] Roy B. Méthodologie Multicritère d’Aide à la
Décision. Economica, Paris, 1985.
[15] Saaty T. The Analytic Hierarchy Process.
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[16] Saaty T. Multicriteria Decision Making: The
Analytic Hierarchy Process (extended edition Vol.
1, AHP Series), RWS Publications, 1996.
[17] Saaty T. Decision Making for Leaders
(extended edition Vol. 2, AHP Series), RWS
Publications, 1996.
References:
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Bielza C..
Compromise Based-Approach to Road Project
6