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.
1
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 case-study 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 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 anew TAV net was approved as a “Public-Works_2000-2007_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 Madrid-Valencia 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:
- The North solution (i = 1) is shorter for Madrid-Valencia, 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.
- The Centre or central solution (i = 2) follows in the middle section the actual new Madrid-Valencia expressway, is shorter with a length of 370 km roughly.
- 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 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.
- Trip Duration (j = 2). That variable will be in minutes, as travelling time between Madrid-Valencia, and is a "more is worse” variable, represents the important related benefits and is roughly proportional to the distance.
- 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.
- 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 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 ).
Alterna-tives: / COST / TRIP DURA-TION / POTEN-TIAL USERS / ENVIRON-MENTAL IMPACTNORTH / 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 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 relationshipsR are obtained. By definition the alternative Eioutranks the Ek , i. e. ,as meaning that “Ei is at least so good as Ek’”,
- if the concordance C(Ei ,Ek ) of Ei to Ek is greater than the concordance threshold, CT ,
- and if the discordanceD(Ei ,Ek ) of Eito 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 ) =, (1)
where is equal to 1 if , ½ if , 0 if .
Table 2, Concordance Index Matrix C = (Cik) .
NORTH / CENTRE / SOUTHNORTH / 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
, (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). / COST / TRIP DURA-TION / POTENTIAL USERS / ENVIRON-MENTAL IMPACTNORTH / 1,05 / 0,85 / 1,117030167 / 0,15
CENTRE / 0,9 / 0,9 / 0,917030167 / 0,1
SOUTH / 0,75 / 1,25 / 0,917030167 / 0,05
Let the Discordance Index Matrix D = (Dik) be defined as, for (i,k = 1,2,3 ) and (i k) ,
Dik = D(E ,Ek) = ,(4)
giving the Table 4.
Table 4. Discordance Index Matrix D = ( Dik ).
NORTH / CENTRE / SOUTHNORTH / 0,75 / 0,750
CENTRE / 1 / 0,428571429
SOUTH / 1 / 1
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.
To obtain the final outranking relationships from precedent definition for (i,k = 1,2,3 ) and (i k):
- An Outranking Concordance Matrix U was obtained from the Concordance Index Matrix C with the rule: if CikCT = 0.50 0.49 (sensitivity analysis) then Uik = 1, else Uik = 0.
- Then an Outranking Discordance Matrix W was obtained from the Discordance Index Matrix D with the rule: if DikDT =0.8214328571 then Wik = 1, else Wik = 0.
- Finally the Aggregated Matrix A ,Table 5, was obtained from U and D with the rule:
Aik= Uik * Wik .(5)
Table 5. Aggregated Matrix A = ( A i k ).
NORTH / CENTER / SOUTHNORTH / 1 / 1
CENTER / 0 / 0 1
SOUTH / 0 / 0
In ELECTRE-I as said the alternative Eioutranks the alternative Ek if and only if Aik = 1, and that means that “Eiis 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 Eioutranks 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.
Figure 1.ELECTRE graph and kern
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. 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
( 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 set of m values vi adding 1, , is to be 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, , to valuate the ith alternative respectively to the objective jth. Then to aggregate the j-objectives they have to assess a set of n values or weights uj adding 1, . If there is only one level of objectives the final values are obtained as or similarly, but if there were a “Hierarchy” of them, experts would have to made a “Process” of similar assessments. For the case study the weights were internally assessed for each jth 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.
From the and , and using internal procedures in two modes, Ideal and Distributive, it obtained the Resulting estimated weights that follow in Table 6:
Alternative / NORTH / CENTRE / SOUTH/ 0,484749 / 0,301688 / 0,213563
Table 6. AHP global weights,.
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.
Figure 3. Chart with EC results.
- Comparison with a Compromise Programming method.
Starting from the table precedent in chapter III “Standardised and weighted matrix” T and from the , 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.
Table 7. Standardized and weighted distances from the ideal point, P = ( Pij ) .
Ideal, pT / 0,75 / 0,85 / 1,11703027 / 0,05COST / TRIP DURA-TION / POTENTIAL USERS / ENVIRON-MENTAL IMPACT
NORTH / 0,3 / 0 / 0 / 0,1
CENTER / 0,15 / 0,05 / 0,2 / 0,05
SOUTH / 0 / 0,4 / 0,2 / 0
For the ith alternative an “h-disutility” is defined
for h 1as h-distance from the ideal point,