'F6 CAHIER DU LAMSADE. Laboratoire de Management Scientifique et Aide à la Décision. (Université Paris IX Dauphine). É'.;'OI?Ot`?

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1 F6 CAHIER DU LAMSADE Laboratoire de Management Scientifique et Aide à la Décision (Université Paris IX Dauphine) É;OI?Ot`?tllE ET GESTION C N A f,,1 2, Rue Conté 753 PARIS A conceptual framework for a normative theory of "decisionaid" Bernard ROY N février 1976 To be published in Management Science Special issue on multicriteria decisionmaking 1976

2 A CONCEPTUAL FRAMEWORK FOR A NORMATIVE THEORY OF "DECISIONAID" * B ROY Professeur à luniversité de Paris Conseiller Scientifique à la SEMA (France) Abstract The intent of this paper is to develop a conceptual framework for a normative theory of "decisionaid" Here the word normative does not apply to the decisionmaker, for whom aid is provided, but to the scientist and to his work of analysis and modelling This framework is appropriate to problems with multiple conflicting objectives The traditional optimization on a fixed set of alternatives (mutually exclusive actions) is treated as a particular case More generally,cases dealing with either fixed or evolutive sets of potential actions not necessarily pairwise incompatible are considered Amongst others, the concept of a consistent family of criteria together with those of truecriterion, precriterion, semicriterion, pseudocriterion are introduced Reasons for which multicriteria decisionaid may not fit in with the assessment of a unique truecriterion are briefly discussed Several situations calling for a modelling of gl,oba1 preferences so as tô "extract" good actions from a given set, otherwise than by optimizing a value function, are considered Lastly, a new interactive procedure called "evolutive target procedure" leading to compromises, in the presence of n conflicting criteria and flexible cons traints, is proposed Presented tothe XXII International Meeting of the Institute of Management, Sciences; 2426, 1975

3 INTRODUCTION Decisionaid refers to the activity of a scientist who tries, by means of more or less formalised models, to help a decisionmaker, so as to improve his contro 1 (this word having its cybernetica1 connotation) of the decisionmaking process To improve in this context signifies to increase the coherence between the evolution of the process and the different objectives intervening in it This presumes amongst others things to elicit the objectives, to clarify their antagonisms and to find implementable solutions which exceed them In this perspective, modelling has firstly a passive role in helping to comprehend, by mastering the various possible actionsand by the reflections it gives to preexisting preferences, and secondly an active role in the sense that the model contributes to the formation and evolution of the preferences of the different actors on stage so as to make acceptable or uncover possibilities which were previously refuted or not considered t In this vein I would like to analyse the "hinge" phase of modelling and propose a conceptual framework, useful to the scientists work This I intend to do by considering, successively in sections I; II,III, the three stages in modellinq shown in tabl é TABLE 1 THE THREE STAGES IN MODELLING SUBJECT MATTER OF THE DECISION Formal definition of the set A of POTENTIAL ACTIONS : STAGE I case globalised, fragmented, fixed, evolutive (cf table 2) Choice of PROBLEM FORMULATION : "one", "all", "some" (cf table 4) ELEMENTARY CONSEQUENCES Formal description of the cloud of conséquences (a) : SCALES, `VALUATION on each dimension, THRESHOLDS, (cf tables 5, STAGE II 6, 7) Choice of a CONSISTENT FAMILY OF CRITERIA, g, 9 adapted to the discriminating power and to mesurabinty or graduabih ty on each scale (cf table 8 and question QI) GLOBAL PREFERENCE Formal definition of BEST, WORST, GOOD and BAD :comparability, indépendance prcpertie,s,_ tradeoffs, suba?gregatiorr, STAGE III (cf table 9 and question Q2) Choice of an OPERATIONAL ATTITUDE adapted to the degrees of complexity, fuzziness and uncertainty of the "aggregation logic" and decision process (cf table 1) _ The retroaction shown on the left side of this table points out that, having reached stage III, the scientist is frequently led to modify work previous 1y accomplished in stage I and stage II, either because his first deductions incite him to do so or because he places himself at a different level of insertion into the decision process

4 I POTENTIAL ACTIONS AND PROBLEM FORMULATION I 1 Potential actions Before we can even begin to talk about an optimum we must first make reference to : alternatives conceived as mutually exclusive, each one representing a global action a including in a extensive way every aspect of the decision a set A embracing all imaginable global actions but only those which are implementable (usually called feasible alternatives), this a priori delimitation being based on the existence of a rigid objective frontier separating the admissible and inadmissible However, there may be problems for which it is futile or simply maladroit to use such a set as a starting basis for decisionaid Firstly, the frontier between the acceptable and the unacceptable is often fuzzy Sometimes this depends on the nature of the boundaries : a factorys maximum production capacity is affected by recourse to overtime or subcontactors, the possibility of loading certain machines beyond their normal capacity or increasing their potential by annexing other equipment In other cases (job planning with delivery date, portfolio com position, diet for an overweight person, ) it is the diagnosis of acceptability in its globality which will create the problems, due to the complex arrangement of the diverse fragments constituting the envisaged actions Secondly, insufficient performance brought to light by a preliminary calculation, the clash of ideas between the principal actors in the decisionmaking process, or simply the impossibility of imagining, a priori, all possible actions, are all circumstances which lead to the evolution of set A (cf table 2) Moreover, let us point out that analysis of the subject matter of the decision often brings out the artificial and uselessly complicated character of a conception which necessitates definition of mutually exclusive actions Many false problems are born from this conception Let us consider, for example, the decision regularly taken in a bank in relation to requests for credit)or in a private firm in relation to the remuneration of personnel, by a panel in connection with a diploma, by an individual with regard to his meal in the factory cafeteria, When the actions are not naturally mutually exclusive,we may seek to determine those configurations offragments which are Thus we are led to substitute for the natural set A of elementary compatible actions, a subset of 51(A) (set of all subsets of A), the elements of this subset appearing as global actions pairwise incompatible By doing this we are taking the risk of encounte ring difficulties (occasionally insurmontable)in delimiting the subset of 9(A) of acceptable configurations (feasible alternatives) Table 2 characterises the 4 cases which it seems natural to "add as a conclusion to the foregoing brief discussion Table 3 contains a list of examples illustrating each of these 4 cases and to which we will refer in preference The references cited will help the reader with his reflection on the interest of the 4 cases even though the models described in them may not be exactly the same as those referred to

5 , TABLE 2 FOUR CASES FOR THE MODELLING OF THE ET A F POTENTIAL ACTIONS The elements of A are mutually exclusive _ YES NO The set A is a priori / case globalised case ragmented d?ftnedjn a strict and? 1??d fixed and fixed exhaustive mannèr(by a rigid frontiey, a non case globalised case fragmented ambiguous test of NO and evolutive and évolutive Î embership, ) (or flexible) (or flexible) == The factor, in connection with the set A, is that the potential actions important ) within the context of a given stage in decisionàid are clearly identified This by no means signifies that the actions are mutual1y exclusive or independent, but that they may be considered separately from one another, without becoming devoid of meaning This does not mean either that they are all immediately acceptable, there is nothing binding and some may be considered unacceptable in a subsequent stage _ou,_ Depending on the problem studied, A may be defined (modelled) : by a list which very precisely identifies each potential action (eg 5, 11, 17, 23 in table 3) ; by a "generator" enabling systematic generation (at least in theory) of all potential actions (eg 3, 4, 19, 2 in table 3) ; as the solution set of a series of conditions or constraints, expressed mathematically, on the characteristics of the potential actions (eg 4, 12, 13, 15 in table 3) _ 12 Problem formulation In conjunction with this option as to the conception of the set A, the scientist must take another, just as fundamental It concerns the choice of the type of problem formulation, account being taken of the level of intervention of the model and its present stage of development It is often thought that the problem formulation o( in table 4 is the only natural one The unique quality of the final decision in the case of A globalised has come to reinforce this belief By experience we know how difficult it is for a scientist to cônvince the principal actors in the decisionmaking process that the optimal solution in keeping with his model is the one which should be adopted Moreover, this particular problem formulation oc ceases to be selfevident when A is evolutive and/or fragmented The scientist may then consider either problem formulations G or Y Let us give a brief presentation of each one (1) 1 feel that the terms "action" and "potential" are better respectively, than "alternative" and "feasible or admissible" which seemed to be too strongly related to case A globalised for the former and to case A fixed, for the latter

6 1 Mil L? m _ Cl ri # N Cr1 _ j d m > >> o G= ça : a n ; c Î S r i C: s ; hfl e rf Î ] O d Z o n OE S 5: m : b c N c n o à v â, ( 4 ra f), U u C c * ai <u ÉÎ e ; CD Ô 1 ) Ài M É À h É É M O É d 1 4J d ZO ô É V à,d U C3 Z Îr O <! =! cd m od? G$ Î ci s: r x nu " M 51 M O ca I 4e 4 ùi % o ci o ci w X V C) «J Q 4J ci d 8oe UO (1) at Ë L9 r1 Pi C ^ n m Q C 8 n OE j et, U 1 f) C La S M CMH P,,4 tû 8 4j M cn, s!e: ui ma! M (U Os zlà i 1 u ci O cd U r= C<3 S < a m r, 4q m m tf1 M à bà gu o : Cd d, G d? m j 4 OE Q J v1 V O r _ r v4 ce v4 Cu C\i <; T7 X d G7 IMI T i f t j t n I I r bd! 2f1 t ) r! 1 r tstj Le 11 bù M éli ii * v N rl ri mi Cu ri C\l C%J rl cn cy1 N ts^1 m, v4 N ` p C ) o! M j _ j t! fll : >,h n À S 3 CO oo ^! U a co* =) w p) S "l O M Ù <U ùi " *" C eo à t» o eu G &? > P5 À Ù 1 a U u r? ci : +7 G1 u ri O, " d P a 6, J 4j ^ v d v rf i v OK o r! f 1 C, ;j X " ^ 5 4, O 4j ri kj L) " VI E r_ u C :: C) 4J 4),t 44 C C n CI H o O 1 :: S M a C c) do M C?!= CUC! : sr À )) j f Lr\ c, ri rd ri r l! ; o G r d c?

7 5 TABLE 4 TYPE OF PROBLEM FORMULATION ON A,, one and only one action considered the "best" "one" the objective of 1 6 problem a those actions w ic seem goo amongs nose formulation Il = ; 1 studied is to r sélect select \ several actions amongst the "best" studied y some 1 / When the actions of A imply an examination with, for example, a view to entry into an éducationa1establishment, the awarding of a diploma, the allotment of credit, a grant or a subsidy, the acceptance of a rrinor or exploratory research and development project, etc, the scientist may envisage the problem in the following terms : accept all the "sufficiently" good actions, reject all those "far too" bad and ask for a complementary examination of the others He is then led for instance to use a procedure using a trichotomy of the set A : _ A for h É k an action a E A being : i in if it calls for A1 acceptance without the intervention of the decisionmaker, in if it cal1s for A3 rejection without the intervention of the decisionmaker, in if it calls for a A2 complementary examination ( requesting supplementary information, discussion, decisionmakers judgment, ) There are many cases in which the decisionmàker,without being constrained to accept only a single action of A, knows a priori that he must give up the idea of accepting all of the good ones This may be the case when the actions concern, for example, important research and development operations for a firm, régional development projects, equipment conceived to carry out certain functions, candidates destined to fill a series of similar posts, press supports before starting an advertising campaign, stocks or bonds entering into a portfolio, In these examples the idea of competition prevails The elements of A can :for instance be regrouped into equivaience classes, as small as possible, these classes being ordered so as to define a weak order on A It is this weak order which will be used to establish the final decision : the demarcation line (acceptance/rejection) may either be a subject for the decisionmakers judgement,

8 who will judge its level of acceptability (financial, physical, psychological, ) or else a subject for negotiation, or even aylocalstudy This double option (cf tables 2 and 4) leads to 4 x 3 = 12 cases, each corresponding to a real situation A superficial analysis may leave us with the impression that the globalised cases imply the X type problem formulation In actual fact, they do not, because decision aid, has for objective to present all of the efficient actions (optimal in the Pareto sense) with respect to n( 2) criteria (problem formulation A cf 4 and 16, table 3) or even the actions forming the kernel of an outranking relation (problem formulation Y, cf 6 and 18) : the fragmented cases do not exclude problem formulation c< since decisionaid can then proceed bysuccessive iterations, each consisting of a selection of a "best fragmenta (cf 7 and 19) These examples anticipate, to a slight extent, section III but in so doing they illustrate the retroaction from the 3rd to the 1st stage in the modelling : the choice of operational attitude is at the same time cause and consequence of the choice of problem formulation, in truth successive problem formulations adapted to the progressive development of the model In my view it is the quality of the insertion in the decisionmaking process (and not the facility in resolution) which is determinant in the fixing of : the nature of the potential actions backing up the reasoning and the structure and nature of the data ; the problem formulation, guideline for deduction and discussion, widely responsible for the adoption (or total rejection) of the model : problem formulation unacceptable, unrealistic, incomprehensible to the principal actors Now, having thus clarified the subject matter of the decision, the scientist has to bear in mind that a decision is very rarely the reflection of preferences of a single person or even of a wellidentified group of people The décision is an important momentin the evolution of a process involvinq man actors and provminc) decisionaid means TO iake part in this process This implies the identification of the one among those actors who play a determinant role in the achievement of_the process for whom or in whose name decisionaid is provided This means that decisionaid is very rarely conceivable without the scientists acceptance (provisionally) to "play the game" for a certain decisionmaker The scientist can do this by treating the global preference modelling problem not only for the decisionmaker, but successively, for several of the actors in thé decisionmaking process In fact, the notions of best, worst, good and bad have, only exceptionally, an absolute sense and it is unrealistic,i believe to talk about prererences without specifying the actor who expresses them and wants to have them accepted in the decisionmaking process Nevertheless,the analysis of the elementary consequences of the diverse potential actions can generally go ahead independently of thé chosen decisionmaker Then the scientist will have a scientific attitude since he will clearly dissociate : the formal description of all the elementary consequences that one at least of the actors may wish to be considered ; he will be able to try to synthesise them by using a consistent family of criteria which is acceptable and comprehensible to all (this will be the subject of the second section) ;

9 1 the modelling of global preference taking into account the decisionmakers personality ; he will be able to try to do this according to the operàtioni attitude which seems to him to be the most effective in the decisionmaking proces! (this will be the subject of the third section) II FROM CONSEQUENCES TO THE CONSISTENT FAMILY OF CRITERIA Even relative to a clearly identifiable decisionmaker (a manager, a selection committee, a community), the consequences of a potential action a, on which this action is supposed to be judged (with a view to eventually comparing it with others) will appear at first sight imprecise, badly differentiated, multiple and confused For this reason we call this complex reality the cloud of conse Quences of the action a, and denote it by J (a) II 1 Primary concepts The scientist must therefore devote himself to analysing and modelling in order to construct an abstract representation of v (a) integrating all the relevant consequences needed for the assessment of global preferences The élaboration of such a model is generally based on several primary concepts I have attempted to define these concepts in table 5 so that they underly a coherent methodology which is as general as possible (cf examples in table 3) Let us illustrate these definitions by an example In a problem ( manufacturing of windshields, printing of a magazine, ) involving the choice of priority rules designed _to establish sequencing of operations in a workshop and to determine the conditions under which recourse to exceptional means are necessary (overtime, subcontractors, ) three etementary conséquences can be identified : a) operatingcosts (energy, manpower, the fixed assets), b) customer satisfaction in relation to delivery dates, c) complexity of management in relation to planned adaptation in order to cope with habitual problems (breakdowns, illness, unforeseen jobs) With regard to the elementary consequence a) there is a financial aspect and a state indicator to appraise the average annual outlay using a certain priority rule A single point évaluation will be judged satisfactory even it it is approximate(it is important that it be unbiassed) A delay dimension is convenient to fix the possible states of elementary consequence b) Here the concept of average delay may be judged too rough to satisfactorily compare two priority rules on this single dimension If the sci:entist does not wish to lay himself open to prematurely preju49ing the way in which this consequence influencés global preferences, he need only introd ce single point state indicator completed by a moauiation indicator (cf table non The evaluation of the rule a on this dimension i can then be constituted by : the set ( i(a) of possible delays for an order, expressed by the number of working days, the distribution indicatïng the degreee of importance of each dealy e E yi(a), on the basis of the number of orders (per year) having a delay equal to e or of a theoretical probability of such a delay for an order

10 fls, z 2 c g p Z 4J S < S s r = d r 4J r rt3 _ < r À flj a # Ci,7 r CD 4* " cm 4à fl fl MS F E «Q OE N fl z Ln 4à g Z N É fly CL p r g N V "* o en CL g $* " = a ) g 7 N So tn 2 < r x ^ % GJ ai?r ^ CJ m C b4 S 4 r «2 t 4, tn " à CJ> É 2 ùj R Q flj Ç CA 5 rtf cn l *] OE E S x ùj Ç O E p 4) o tn r_ S CD É É <UtD <e " à à 4 S <u 4j r LLJ T (m c fl fl t Cu EU 4 a = Q % m aj «r c M s: o 4j ce: )< p p a ^ m 4j S tu <! e& S m O 4à <U (v z: 4) CD O à < G," = * Q G r V tu c O V > m > t t rr C7 p p 4 O 1 t z eu Xe: EO Z7 Cu a) 4 à à 5 V > > 3 H N c r =M N à " J LLJ r ç 2 z 4 N r <u 4à <u Mr 4) : G t= r i t i M SS fl? fl?? 4 <<omru ci C m? fls V E M <&C C 3 1 MU ci 4<D p &4 R o N C:) tu fa UJ té 2: 4J à MJ2 B & 5 2 < à flj, cy (A 43 3 tqe t/) 1 1 T!f, ce 2: C:4 e ùj W K m Ô rt3 C "UJ Cu r_ 43 c Cu S a " <j = g e M c o e, > ci g, W Q OMO""< flj &4 P fl < «! <E " L7 Co ki N t <u<ht tic Z E LLJ 4à % Ù tu t X s p r s J «4J Ù «4j Or C LLJ <" *fr r C G! a) fl c MrcE r= a î M i Q ù Ç C7 4j =! E VI =!_ ur) Q tu flj Cu S Cr_, C:) LL to ru 4 tu!> tu t7 ce _ > 4 4 cm <" (L) M? b 4 S R tt7 <UE Cu r a1 i r LLJ j â Qe e cm M E E<u CQ tz ai S Q Vl u Lii C) J L), g z W 2: O à = C C3 fc " tn r 4 N " c o c: z p U r p à r E in n3<l) % tt E r O r % ( 4 =S t < ~ C LLJ J <U 4à 4) 4< nccr M ( toforoi) LLJ (v E <D v tu y < r U C tu r % % Ô «tt5 tg rl5 r rcs a LLJ U Q 4 C)? 4 oc LLJ >e:

11 7 With regard to the last elementary consequence we associate a dimension "degree of flexibility" relecting the capacity of a priority rule to absorb incidents encountered in normal management (without provoking excessive tensions, damaging disorder, ) The associated scale can oniy be qualitative (without spending a great deal of time on its definition and corresponding valuations) It is not easy to code a priority rule on such a scale, although the head of the organisation and methods bureau often attaches a great deal of importance to the global and subjective idea he has of this degree of flexibility Therefore a realistic way to go about this might be to ask an expert to classify the different rules by means of this dimension, without seeking to code the states and their modulations ; ie only the relation established by the judge is important This is an example of a relational case (cf table 6) In certain cases the scientist may procede in a different way Let us suppose that an analysis of past activities of the shop reveals a small number of typical situations : normal situation,,, conjuncture producting a sudden increase in the workload, functioning below capacity because of the unavailability of a key machine or a specialised labour force, easily discriminant with respect to operating costs, customer satisfaction and the adaptation possibilities for each rule Lett denote the set of elements, called events, characterising each of the considered situations For each dimension i, the set of states, to which a specified event_e may lead, can generally bereduced to a proper subset of On certain dimensions,???(a) may be systematically reduced to a single element of this is E i (for a class of exclusive events) the case "single indexed event" of table 6 As for the dimensions for which this is not the case, a modulation indicator must specify (in a distributional or relational way) the relative importance of the states ofyf(a) : this is the case "complex indexed event" of table 6 We will leave it to the reader to reflect on this by reconsidering the preceding elementary consequences in this context Anyway, the act of clarifying the classa has, when the discriminant influence stands out clearly on at least two dimensions, the merit of bringing to light a causal liaison that global preference modelling can not ignore Within or outside of "indexed event" cases, there may exist other relations between the modelling indicators which can subsequently supply useful information and it is in the scientistinterest to diagnose them at this stage of the analysis These relations are laid down at the end of table 7 In order to illustrate the remainder of table 7, let us return to the three dimensions, finance, delay and flexibility, introduced above Note that for each of the three corresponding scales, the objective is to achieve (even if unobtainable) one of the two extreme grades (cost nil, no delays, maximum degree of flexibility) It could quite well be otherwise Let us suppose that deliveries are a source of problems for customers Each grade will then represent an algebraic difference between the actual and contractual delivery dates and the goal "no delay" will no longer be at the end of the scale If two grades of the same sign are still directly comparable in terms of preferences, a further step in the study of preferences must be taken so as to compare two grades with different signs

12 1 E r 1 N C1J 1 4 N C1J +1 CL U m m w J= r V1 r C1J U V 4 CJ fo c o C1J +1 C1J E r < V 4J r t N 4)!D 4) 4 U 4à r L E V1 «P««Co E ) (u à r 1 C1J O p E c l r C +1 r 4 fo OE O a > s V vf,n r )>a o N fa m VI C N C 2: < (/) Q) r V7 Vf (U O C1J C CU 3 N! ) +J U C1J +J E Q C1J, X 4à C1J C1J < 4( < o e: o 4 4) f OS Z +1 C1J ( éj 1 K " o OE Co +1 > S «LU d) J P N +1 r r > X > rc cn r K Q 3 C r +1 *o T Z r % OE <D r P o 1 > p r % " z o 4 S a > 4 Z z / <l) C " " N Q V (A r r E < Q r 4J C N? 4) ci 4)!l)r a CL r c: 3 C1J Q C3 4J 1 4J <C CL 4 t/) > «r N C 4 < u r= Q U _ < 5 C1J I CJ 4à «1 *< Q p J tn C1J O O ) O L Q Co ai 2: LLJ, 1I)r ) ro Qi U «C r= VI LU ic C1J 3: > s:: 3 r_ 1 S: Li " r (/)C1J C1J Q ci 4<a4 4 fo a 1 L C1J ro C rcf W N V LLJ C1J r C C Q1 r m (U VI M Cu CL C CL u tu m 1 K m a < f, UJ O J CQ J 4à! 4) < M _ C ( V) N = _ = ci à 6 É!= 2: 4à U C Q1 C1J 1, ci o > É _1 > r r C1J C1J Cr5 < m = c E C1J tn d) Il r? r r (1) x C) x L 4 4) r C1J Q ce Co < 1 % 4l O CI QD r a S 2: + Q OE n: t/ 7 (/)= U <_)= l W p L

13 c 1 C M <a C w c 3 4) <o rt3 Q r VI N V 4: 1/)"" ** u ui OU> M < QJQJ j= p N <U " Q) < (J E 3 C s v <n ce o a) Cu & N 1/) C QJ tc= C 4 4 p 113 cn ci ci 4j p 4) m L) i i < 4) N 4 p C) r <U r C u c c OC M 4 t S CL) fo O O 4 IJ p 4) CL " J= i i N Cu QJ M fa 4à <a tu CL +J E 4 > t j3 < M <) 4 <j >, E> <u e:* p p " >, i rtf o c Cu S 4* C CL > EM 4 N O 1/) QJ 113 >, Q) " o 1/) E C 3<i)fa p p i T Q) S! 43 <Ufa<j tn<o t <o O V v V 113 1/) tz u t 1/) Q) <u f p t <o& S 4 <U cc3 N ci V rj= < r V 4 M & S* 4 4 p < i T O i S E CL) Q) 4 " 4 ci T Cu <J e *t QJ C r w* r oi 4j S ajfa<u4<dfoo? u QJ 113 U w 43 CS p (U o 1/) u u () (U S j=: m M * ci t/)r3 II) " MU CL f <U> C T 4 ai VI CD Cu a) 4) (U " m ci a * p m O t &) tu " 1 r p r C c i *: Vf; ME II) == VI Co M p 1/) M r (a Z? ) II) 113 CL II) 4 QJ 1/) N vil IÓ p i LL r LLJ," LLJ, QJ ) LU C r r O Cu r S C:! LLJ C=,,, ocn cm << 4 <U m <U r = "S $ C 4à, r 3: f <Q tu VI C!J 11) II) E OJC: v1 V C1 <U) (ici Cu ru i Cu 4 ( 3 s <UE w c: Q) C C\) < <U <j p C *r CU rc5 C 113 C 3 C? c<u C p c <_< <u JE r f V a7 Q) V +r 4j E t C1 _ 3: 3 j: (v rll$ ci (J N tu CL) p S» M C =! M r j JE c 4à 4j m ci < IÓ va <U 4 <J i r6 s rc fa ru > r o"*o 3 f E >& r r > C) m " CfO EU p (U4 \/<U<U Q) V 4 &f ci f"" r C " C C 113 S r rr r C CL r <U r l C N 4) U 4 r +J U <1J r C) C: 4j E U r r tn N (u> > 4j V cn t a rp EUE U E C7 rc N rm f N C r fr IÓ 41 ci C x C) s IÓ 113 C nj <D C 1/) nj 113 Cr ci ai M Z s 2 E _1 <U ai C: V C 1 Ç r O p L II) r C Q) 4 cu E <U (U 4 v Q r <C 4 Q p i r C7 ici ^ 4 * < E Q) ro l7 y C*<4 s :::: YI) ClJ c LO " j= ;= = C) C) a) C) a) O V L ui C) LU w U w O 5 \1) r 1, o s r p r C C s a (li CM jf: s 4 CI C o to

14 The personality of the actors, the nature of the potential actions and scales, the mode of elaboration of the state and modulation indicators are the basis of different types of frequently intermingled thresholds To admit that the different actors are indifferent to two priori ty rules leading to the same valuations, except regarding average operating costs for which valuations are e and e fi may very well raise different explanations :? is a negligible sum in comparison with the sums in play elsewhere and with the sensitiveness of each actor in this dimension ; collection ; h is too low a sum in view of the techniques implemented for data calculation sets aside is a nonsignificant sum in view of the risks that the mode of By definition (cf table 7), is the maximum value ofq (on the financial dimension i considered) such that e h is not recognised as signi ficantly better than e the scientist may consider, either e %l as presumed preference to e, or (more simply) as indifferent from e = ) To avoid giving a discriminating role to differences of little significance, the scientist will sometimes be led to pay particular attention to such thresholds in the neighbourhood of the goal, and for example define an interval Es, sl (containing for which all the states will be oi) judged as "good" as they are sufficiently close to the goal II 2 The concept of a consistent family of criteria and the underlying natu re of the criteria the description obtained : or when the unique valuation is a non single point one, Ev merits a slight transformation so as to be more manageable as much for global preferences modelling as for decision aid This transformation consists in the elaboration of what I will cal! aconsistent family F of criteria (see tables 8 and 9) (1) Let us point out that, if f(x) is any increasing function, the subs titution in F of f Lgk(a)J for problem gk(a) gives a new consitent family for the It is important to note that the correspondence between criteria of the family and dimensions retained in the analysis is only simple (see in particular table 8 cingle point équivalent on the dimension i ") in the case where the criterion gk only brings into play indicators relative to a single dimension When this is the case the passage from thresholds relative to the scales, to those of the same nature relative to the criterion k (denoted by and q+(x) respectively) presents no major difficulties In the case of subaggrega tes, this passage may be a little more complex (1) The notation F designates the family of criteria as well as the set Ç 1 7 nl nf inriiiac

15 According to the discriminative power recognised for the criterion k, I suggest to call (see the end of table 8) : pseudocriterion (most general case) : a criterion for which an in difference threshold qk(x) and a threshold of presumed preference sk(x), s((x) >q((x) are defined ; + + semicriterion : a pseudocriterion for which = + qk(x) precriterion : a pseudocriterion for which qk(x) = or is not defined ; truecriterion : a pseudocriterion for which = = sk(x) (every difference is significant) These thresholds can not be any function of x, as is indicated in table 8 for a threshold of presumed preference, but remains true for an in difference threshold This is quite simply explained by the fact that, for y>x there can not exist (without inconsistencies) a value z of gk such that : y + sk(y) < x + sk(x) On studying the underlying structure of each one of these types of criterion, it is easy to deduce that we are concerned with :, a complete order for a truecriterion ; a semiorder for a semicriterion ; what appears to be an orientated semiorder for a precriterion ; a more complex structure for a pseudocriterion, to which I shall refer under the name of pseudoorder, In what follows a semiorder is characterised by the definition of two relations I and P such that : a) I is reflexive and symmetric ( which corresponds to indifférence) ; b) P is an antisymmetric "complement" of I in the sense that one and only one of the following three possibilities holds Vx, y : x I y, x P y, y P x (which correspond? toystrict preference) ; c) P I P C P (implies the transitivity of P) ; d) P2 ni2 (for further details, see FISHBURN (22) or JACQUETLAGREZE (34 ) In order to complete an elucidation of the use that the scientist may make of the criterion k, he must investigate the relations which connect any two intervals of the type : xk 1 defined by two orderedpairs (a,b) and (a`,b) of potential actions such that : 9j(a) = gj(b) = gj(al) = gj(b) k, 9k(a) =xk gk(b) gk(a) =wk gk( b )

16 Do such intervals sufficiently reveal an underlying reality to assess, in a significant and operational way, a comparison between the superiority of b over a and that of b over a? The objective of this comparison is to lead the scientist to opt (as in table 9) in favour of one of the following fundamental mutuall? exclusive situations : superiority identical : differs from (xk xk exactly as, (xik +«k) differs from superiority scrict1y greater : differs from (xk x? strictty more than differs from (xk superiority weakly greater differs from at 1east as :(xk + w k) xk much as differs from but it is xk impossible to say if it is strictly or exactly; superiority incomparable : neither of the three former situations dominate Q, These considerations lead me to formulate the question : _ How to discriminate the precedi?g four fundàmental_ mutually exclusive " situations soas to compare the "importance"ofany >o intervals of the type w wf? When one of the two intervals is null, the reply to this question has already been qiven through the concepts of thresholds (which may depend on the abscissa xk of the non null interval) Thus it is with the other cases that we are cunccrv2d It is rare that the scientist stops here in the sense that he implicitly admits that the reply is given(independent1y of xk and a simple examination of the sign of w k w k Frequently he goes as far as treating the ratio as a measure of the superiority of b over a when that of b over a is taken as unit It is clear that here very strong hypotheses make the criterion k appear as a high precision instrument Not only, gk is a truecriterion, but it is self imposing up to a positive linear transformation, ie gk is a measure (with reference to the set of intervals) : only the unit and origtrr are arbitrary If the scientist can define a function f(x) (non decreasing) for which f [gj (substituted for in gk F ) is a measure, then I will say gk rable criterion in F So that such a property holds, gk must satisfy 1 through 4 written below Before let us introduce a new definition is a measu the axioms 8y discriminating interval I will refer to an interval of the type w k = lxkl xk such that b is strictly preferred to a AXIOM 1 any non null interval is a discriminating interval ( gk is a truecriterion) AXIOM 2 : the reply to is Ql 1 independent of the values = gj(a) = gj(b) gj(b) j i k, whatever be the pairs a b and a b AXIOM 3 : the reply to Ql excludes superiority incomparable AXIOM 4 : on the subset of discriminating mtervals, the reply to question Ql excludes superiority Wéùkl>/ great and leads to transitive,ans,?4ers compatible with inclusion and union of the intervals, for superiority identical as well as for superiority strictly greater

17 1 E r VI Q i > OEk É 5 Q) r6 E C L L (U < r O <4 Z Q) VI U Q Q) «Q) < j= P w r r 3: Q) n:i G r 4J «r N < GJ t PO tu >< %W c c coj VI r àé 4 N r N U 5 n:i O r NC t Q)? " 4l C C1 MU r < r ai r r V1 VI 4 (U 4 E L C N " >, Q) VI r C f E rn:l cco4o N O > U,, c t 4 V1 <C u VI = c ) t E (U Q) f C) 4 E <U Cu ci en Q) E e: LU E r t (!) > r 4 4 4) OC ma Q) VI U O n:i m W fu<4 y b m tu f, o 4roC1 ftufc «ï 4) > Q) p 1 r r OE OEW n:i > Q) r 2: Cn tu a) *< C1?O? p C w, tn Z z: ^ * S 1 lm (U ) O a N tu (v C 2: rt rt1 O >> 4 y LLJ É ) i 3 i N p1 VI 1) 4 C r 4) N LL r r, Ln C 4 E Z w fo r O Q Ô U 1 n:i U, 4 <) 5 O! mr C 6i1 CC: Q) r N C <! r << r > LL U co (v c O i S N rtf Q) tjh 5 Q) r UO O C In <ur In crc: or < W p es a 4 c t/1 CtE 4l << Il eq U <D K m E f Lvl L y (UC QJ E O 2: 4 (u y OE %CJ j= r 3U a) Z m e:, tu O 5 c ( c < (n 5 S W OE In a 4l " tm (A O (V fa 4J 4) O n Il Il r OE Q) VI 1J un c (v Ln > e: te > «fa D, m eu d (m Dd) z: E y (D r r CL Q) CL n:i J OO tu > eo e: LLJ 4 1 i 1 1)fa X O O g <, N MIl Q) U Gl y m LU,1 J < e: «r L"7 ce Q OE = e: S, O tn l VI 2: c e: C _ O en L1 oj <4 N r l t < E fcr * ta a > a, r i 4j O E e: r QJ, OE< > " ^ M 4e: en 4 _ e:qj,, C1 r C y f!) U CCJ C CL O L H il II Q r r 4J y r<u] fl <1)1) r r c M m Q o n L C U U CL en) 8 C1 CD tn en fi 4J X

18 N aj C K 1 o V MU tts:3 U fo 3 K C 4< V +J en C L c WX aj _ o G)E m WX < C O É É <o 2 c < 4t t 4! " & %n " p C i P <U W Q a É r Q u Q E w É C O X ** E * <J C d m 2 * C L, i 2 QJX uo + 5 Ct fo C + 4, C7 r < Uà O ce > m v O m Q m " tts O m + C tts C tts 43 + U m U 9 X ce V tj>, P Và Q m m / m U :3 m R mà R m " 8 u E E 4 U j,j X X i p 47 X 4 Il?ttS _ 11 " QJ? m +J P"" * * <U4) 1 w x: WX X V + Il fl W b S à S Î C1J CC É E P"" f X p C1J 4Y p à v S C :3 p"" 1 m N? tts C1J C1J <U S U C1J S ro N t C Q) S C S t+ " P"" C1J S w Q P"" S C1J Q) +J p fl p C ro " Q) Q) :3 C1J V Q) U mi 4<U Îm :3 U 4 o I +J C 1 S X X " CIO " r 8# m V s I l c 1 rn 1 C g s m s, \ m p 7 T ce <U Q) ÀÀ rn Km rnq) +J C S P"", ro C C1J K 4 Q) 4 r:: s Co+ 1 a Co+ eg r V) r V Î, C ë O < Ô R C C +J s it ", r S b +J 1 V 1, m V «b + <J Z m rn t n3 ro ro rn «vr rn N r Y Y Y

19 These four axioms do not imply measurability as the reader may verify by reconsidering the example of II1 relative to the elementary consequence c) : adaptation possibilities Suppose that the scale Ei adopted is the following : e : rule very rigid, e 1 rule involving non negligible rigidity factors, e2 : rule normally flexible, e3 : rule exceptionally flexible, and to which corresponds a single point valuation rise to a truegiving criterion = index of the gk : gk(a) grade The scientist may then reply to question Q, (in agreeement with axioms 2, 3, 4) by defining on the set of non null intervals, a weak order (transitive and complete binary relation) such as the following : criterion classification gk (, 1)e,(l, 2)<(2 3) _ The table below shows that there are various transformations of the which render the lengths of the intervals compatible with the above but which lead to very different values for the ratios El eo el e2 e3 gk A set of actions necessary and sufficient (1) to imply measurability may be obtained by completing axioms 1, 2, 3, 4 by the following : of gk AXIOM 5 : for all three values of a gk unique value y such that the superiority of y over x is identical to that of y over x Past experience has shown that in a good many real world problem, it is impossible for the scientist to justify the exact values which axiom 5 postulates (either because there is more than one single value or because none is acceptable to all th actors who intervene in the decisionmaking process)the reliance on the idea of ap proximation to justify the arbitrary part included in the selected values (and in order to remain within the framework of the set of axioms) is not always a realistic attitude (1) The author thanks Ph VINCKE for his contribution to the verification of the exactnèss of this assertion

20 18 In fact this difficulty arises usually because axiom 1 is not opera tional It follows that axiom 4 has also to be reconsidered I suggest to call the criteria which satisfy axioms 2 and 3 graduable and to use the term of graduation to designate those (graduable) for which the values allow a direct comparison of the intervals (in the above table is gk graduable but only the last three criteria are graduations) The structure intoduced py an axiom similar to axiom 4) on the set of intervals of the type wk by replies to question is Q,, intimately related to the techniques used to assess these replies In the case of graduable criteria we can cha racterise the criteria : trulygraduable : weak order structure (see above) ; semigraduable : semiorder structure (see what follows) ; " pregraduable : oriented semiorder structure ; pseudograduable : pseudoorder structure In the example of II1 let us consider a semicriterion g associated with the dimension "delay" defined as follows : g(a) = x q+(x) = Min(qx, 1 x), 1, 2,, 1 ) expresses the percentage of orders for which the delay exceeds a week

1.- L a m e j o r o p c ió n e s c l o na r e l d i s co ( s e e x p li c a r á d es p u é s ).

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