Multi-criteria analysis with partial information about the weighting coefficients

ELSEVIER Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theo y and Me hodology Mul i-c i e ia analysis wi h pa ial in o ma ion abou he weigh ing coe icien s E. Ca izosa, E. Conde, F.R. Fe nandez, J. Pue o * Depa amen o de Es ad s ica e In es igaci&n Ope a i a, Uni e sidad de Se illa, 41012 Se illa, Spain Recei ed Oc obe 1992; e ised Ap il 1993 Abs ac In his pape we add ess he p oblem o anking a se o al e na i es wi h pa ial in o ma ion abou he weigh ing coe icien s. We in oduce a amily o quasio de s ha a e easily in e p e able and manageable, which includes among o he s, he na u al quasio de in R n and o he well known p e e ence s uc u es in he li e a u e. The en ichmen o he p e e ence s uc u e wi h espec o he na u al quasio de is measu ed by means o an absolu e measu e we in oduce. Keywo ds: Mul iple c i e ia decision making; Pa ial in o ma ion; Weigh s 1. In oduc ion Nea ly all he eal wo ld decision p oblems in ol e mo e han one objec i e and can be o - mula ed in a na u al way using he mul i-c i e ia app oach: 'Max' (x) - ( ,(x),..., n(X)) (1) whe e i:X--* ~ is he e alua ion o he i- h objec i e (i = 1 .... , n) and X is he se o al e - na i es o be anked. In his o mula ion, and wi hou addi ional in- o ma ion abou he objec i es, one can ob ain a pa ial o de R I in X: Gi en wo al e na i es x, y ~ X, we say ha x is as p e e ed as y ollow- ing R, (xRzy) i * Co esponding au ho . i(x)> i(y) Vi=l,...,n, o equi alen ly, xRiy i ~_, Will(X)> ~_, wi i(y ) l <i <_n l <_i <_n Vw>O, w~ ~. AS he pa ial o de Rz is ypically oo ague (and he se o nondomina ed al e na i es is oo la ge) a numbe o p ocedu es has been p oposed in o de o en ich he p e e ence s uc u e abo e (P ome hee [5], Elec e [19], in e ac i e me hods [11], he u ili y app oach [9], e c.). The in e es ed eade is e e ed o he seminal pape o Roy [20] o a syn hesis o he main app oaches o his p oblem which ha e been s udied. In he u ili y app oach, one assumes he exis- ence o a unc ion U: ~ ~ R, in such a way ha an al e na i e x is conside ed as p e e ed o y i U( i(x) ..... n(X)) >-- U( l(Y) ..... n(Y))" 0377-2217/95/$09.50 © 1995 Else ie Science B.V. All igh s ese ed SSDI 0377-2217(93)E0270-8 292 E. Ca izosa e al. ~Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 Fu he mo e, unde mild egula i y condi ions [12], U is a linea unc ion, i.e.: one assumes he exis ence o a ec o w* = (w*,...,w*)_> 0 such ha he Decision Make (D-M) p e e ences a e gi en by he o de R: xRy i ~,, wTk(x ) ~ ~_, w~k(y ). l~i~n l~i~n Howe e , one o he main d awbacks o his app oach is ha he D-M would ha e o es ima e nume ically he weigh ( he ec o w*) ha each c i e ion b ings o he inal sco e o an al e na i e and he is no always willing o do so [6]. This choice is a c i ical s ep, and, al hough some me hodologies ha e been de eloped o a ain his goal, such as he En opy me hod [11], he Saa y me hod [21], he Solymosi and Dombi echnique [23,15], e c., hey equi e, in ou opinion, oo much specialized in o ma ion om he D-M. Ins ead o p o iding he exac w*, he D-M migh ha e some knowledge abou w* and his in o ma ion can be used o ob ain a (pa ial) anking in X which en iches he o iginal p e e - ence s uc u e R I. In his pape we show ha exac nume ical weigh s a e no always necessa y. Ins ead o his exac es ima ion, he D-M gi es only ce ain lin- ea ela ions which exp ess pa ial in o ma ion abou he ma ginal subs i u ion a es be ween he c i e ia. This app oach is no new and some wo k in his ield can be ound in he li e a u e; Ki - wood and Sa in [13], Hazen [10], o Eisel and Lapo e [8], de i e condi ions o de e mine o di- nal ankings be ween al e na i es using pa ial in o ma ion abou weigh ing cons an s. Fo ins ance, in he quali a i e app oach [17] he D-M is eques ed o es ima e only he ank o de o he c i e ia. I he c i e ia a e a anged in dec easing o de o p e e ence, wi h w 1 > • • • > wn, hen xny i w( (x)- (y))>O Vw>_O, WI~_ ... ~_Wn o , equi alen ly (see [17]) xRy i g j(x)> ~] j(y) Vi=l,...,n. j<_i j<i In a mo e gene al se ing, he D-M is e- ques ed o es ima e a linea ope a o which mixes he weigh s, i.e. he D-M asse s ha he ec o w* belongs o a ce ain polyhed on. Thus, gi en a linea ope a o A, we de ine he p e e ence be- ween he al e na i es by means o he bina y ela ion R A as xRay i w( (x)- (y))>_O Vw>O, Aw > O. (2) All hese bina y ela ions a e quasi-o de s (i.e. e lexi e and ansi i e) [26] and can be used o pa ially ank he al e na i es, al hough hei use equi es he solu ion o a linea p oblem o com- pa e e e y pai o al e na i es. Howe e , i he ex eme poin s wl,...,w m o n he poly ype {w : Aw > O, w > O, F~i= lwi - 1} a e known, he ela ion R a is xRAy i wk( (x)-- (y)) >O Vk = 1,...,m, (3) which simpli ies i s use. The ollowing example iUus a es he com- men s. Example 1.1. The s a manage o a consul ing i m mus ank ou di e en execu i es o a bank acco ding o he budge hey es ima e o nex yea based on h ee inancial c i e ia. These c i- e ia a e: loans gi en o clien s (C1), clien s' sa - ings deposi s (C 2) and edemp ions achie ed (C3). Each manage 's budge mus adhe e o he inancial policy o he bank. This policy is gi en in o m o h ee cons ain s: 1. The bank wishes ha he c edi s a e g ea e han he sum o he edemp ions plus 1.5 imes he sa ings. 2. The edemp ions mus be less han 0.1 imes he sa ings deposi s. 3. The edemp ions mus be non-nega i e. Conside he ollowing ma ix whe e he ow i ep esen s he ac ion p oposed by he manage i, i= 1,...,4: C1 C2 C3 a 1 11 12.2 0 a 2 5 4 5 a3 11 11 13 a 4 11 12 2.3 E. Ca izosa e al. / Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 293 Le F deno e he abo e ma ix. The con- s ain s imposed by he bank can be w i en as Wl>I.5w2+w 3, W2>10W 3, W3>__0. Thus, he linea ope a o A and he ex eme poin s ma ix E, o he poly ope {w : Aw > O, w > 0, Ei~__ lW~ = 1} a e espec i ely (1.-1.5-1) 1527 A= 0 1 -10 and E= 2 10 5 27 1 0 0 1 0 and he qnasio de gi en by he abo e cons ain s in he ac ion's se is ob ained om he ma ix F × E b means o he na u al quasio de . Hence, -11 287 298 ] 25 27 5 2...~3 125 5 27 F×E= 11 11 22- ~ 11 zs~ 298.3. 25 27 gene a es a ela ion which can be ep esen ed by he g aph o Fig. 1. Howe e , wi h a di e en se o cons ain s imposed by he bank gi en by he ope a o A , and whose ma ix ep esen a ion and ex emes a e 1 -2.5 -2.5) A1 = 0 1 -13 , 0 0 1 1 5 35 7 49 E 1 0 2 13 7 49 , 0 0 +9 i is easy o see ha he quasio de gene a ed is ac ually an o de whose g aph is gi en in Fig. 2. Thus, i is possible in many cases o iden i y he Fig. 1. The g aph o R A. [K]-- [?N-- [] Fig. 2. The g aph o RA p e e ed al e na i es wi hou knowing he nu- me ical es ima es o he c i e ion weigh s. Howe e , he p e e ence s uc u e de e mina- ion in i s gene al o mula ion ( he de e mina ion o he se o non-domina ed al e na i es) is equi - alen o he e ex polyhed on enume a ion, which is no a polynomially sol able p oblem in he size o he inequali y sys em de ining he polyhed on [7,22]. The pape is s uc u ed as ollows: In Sec ion 2, we in oduce a class o ma ices (Q-ope a o s) and s udy some p ope ies o he quasio de s hey induce. In Sec ion 3, we cha ac e ize some Q-ope a o s ha a e easily unde s andable and manageable. In Sec ion 4, we ex end he esul s ob ained in p e ious sec ions o a b oade class. The pape inishes wi h some conclusions and possible ex ensions. 2. The class o Q-ope a o s E e y linea ope a o A belonging o he se o eal ma ices o dimension k×n, de ines a quasi-o de R n on he se X o al e na i es: Gi en a ma ix A ~ R kxn we de ine he poly- opes C~ and CA: C~=(w:Aw>O,w>O, i=l~Wi=l} (4) and CA = w:Aw>_O, wi=l , (5) i= and he quasio de R a on X: xRaY i ~wi i(x)>__ ~wi i(y ) i=1 i=1 Vw ~ C + . (6) In his sec ion we in oduce a class o qua- 294 E. Ca izosa e aL / Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 sio de s induced by linea ope a o s (Q-ope - a o s) ha ha e e y in e es ing p ope ies: he ex eme poin s o he co esponding C~ can be e y easily ob ained. De ini ion 2.1. A linea ope a o A ~R.x. is said o be a Q-ope a o i de (A) ~ 0 and A -1 _> 0 (componen wise). Fo a Q-ope a o A, he in e se ope a o A-1 exis s; we deno e i s elemen s by aii, and by /x he ec o o sums o he columns o A-1: The ma ix abo e is easily shown o be a Q-ope - a o , hus Paelinck's heo em appea s as a di ec consequence o ou Theo em 2.1. Indeed, A -a can be eadily ob ained, and he ex eme poin s o C~ a e he columns o he ollowing ma ix: 1 1/n 1 ~ ... 1 1/n 0 -~ ... 0 0 ... 1/n 1/n 0 0 ... 1/n A -l= (aij) and /xj= E aij" (7) l <_i <_n Theo em 2.1. I A E ~Xn is a Q-ope a o , hen C] is he con ex hull o he columns o A- 1, each one no malized in o de o add 1. P oo . As A-l>-0, i ollows ha /~i > 0, Vj = 1,...,n. Le D be he diagonal ma ix such ha dii = 1/Izi, Vi, and le e e R ~ deno e he ec o o ones. Gi en w e E', one has w~C~ i 3z such ha w=A-aDz, Dz>_O, A-aDz >_ O, eA-1Dz = 1 i.e. ( ecall ha D >_ 0, A -1 _> 0, eA-1D = e)" weC~ i 3z such ha w=A-1Dz, z>O, ez=l. In o he wo ds, w ~ C~ i w can be ep esen ed as a con ex combina ion o he columns o A-aD, as asse ed. [] Rema k 2.1. The well-known Paelinck heo em [17] p o en, among o he s, in [17,2,6,13], educes o he calculus o he ex eme poin s o he poly- hed on {w e R ~ : w a > w 2 > • • • >_ w~ >_ O, ~i= aWi = 1}, which is o he o m C~ o he ollowing ma ix A: (010 0 1 ~1 ... 0 0 0 ... 1 Rema k 2.2. In he cen oid me hod o Solymosi and Dombi [23,15], gi en he polyhed on C~, one p oposes as w* he a e age ec o o he ex eme poin s o C]. By he heo em abo e, his w* is easy o ob ain when A is a Q-ope a o : w*= (1/n)A-ae. Rema k 2.3. I should be no ed ha hese kinds o quasi-o de s do no necessa ily need n linea ela ions. I he D-M is only able o supply k < n, hese ope a o s can be ans o med wi hou in- co po a ing any addi ional in o ma ion. This is possible by adding he na u al ela ions w i > 0 and emo ing he edundan ones. As an illus a- ion, conside a p oblem wi h h ee objec i es, whe e he D-M s a es ha w a > w 2 bu is unable o p o ide mo e in o ma ion abou he weigh s. Then he will ha e he ollowing ope a o A and i s in e se A-a: A= 1 , A -1= 1 0 , 0 0 1 and he quasi-o de would be ( a(x) > l(Y), XRAY i ~ a(x) + 2(x) > x(Y) + 2(Y), I 3(x) > 3(Y)" Rema k 2.4. Ano he impo an p ope y o Q- ope a o s is he ac ha hey induce in e al weigh s, which a e easy o ob ain, allowing a ce ain deg ee o sensi i i y analysis in he nu- E. Ca izosa e al. / Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 295 me ical es ima ion o weigh s. The exac in e al o w i ~ C~, Vi = 1 ..... n, is gi en by WiG [Inj!n(olij//~j), max(o ij/IXj)l, i= 1,...,n, whe e aij and /xy we e de ined in (7). Indeed, o i = 1,...,n, le z i be he op imal alue o he linea p og am max{w i : w ~ C~ }. This z~ is a ained a an ex eme poin o C~ and, hus, by Theo em 2.1, z~ = maxj(aiJixj). Simi- la ly, one concludes wi h he minimum. Fo ins ance, o he quasio de R A desc ibed in Rema k 2.1, i is easily seen ha wl~[1/n, 11, w2~[0,½1,..., w,~[O, 1/n]. As a inal consequence, obse e ha one can also ob ain he maximum and minimum alue associa ed wi h each al e na i e x E X when he weigh w a ies in C~, which is he basis o some decision-making me hods (see, e.g. [3]). Indeed, as inding he maximum ( espec . he minimum) alue o w (x) when w a ies in C~ educes o sol ing he linea p og am max{w (x):w ~ C~} ( espec i ely min{w (x) : w ~ C~}), Theo em 2.1 implies ha min ~ ai---k i(x) <_w (x) < max ~ ° ik i(x ) k i=1 Zk k i=1 ~k Vw~C~, x~X. The amily o ope a o s p oposed in he p e i- ous heo em is maximal in he sense ha he unique se o weigh s in ~n wi h n ex eme poin s whe eby all he weigh s gene a ed a e non-nega- i e a e hose wi h A- ~ _> 0. This is s a ed in he ollowing heo em: Fi s , we show ha D A = D~. Indeed, i is e iden ha D~ c D A. Now, le w ~ D A, and we will show ha w ~ D~. Ob iously, i w = 0, hen w ~ D~, so we can assume ha w ~ 0; in o he wo ds, we only ha e o conside he cases (ew ~ O) and (ew < O, w 4= 0). Case 1. ew>O. Le w l=(1/(ew))w~C A= C~. Hence, w 1 > 0, hus w >_ 0, which ( ecall ha w E D A) implies ha w ~ D~. Case 2. ew < O, w 4= O. As by assump ion C A ~ ¢, he e exis s w°~ C A = CJ. De ine w 1 as ol- lows: 1 -- ew w 1 -- --w d- w °. 1 - ew 1 - ew Such w 1 e i ies ha w 1 ~ D A and ew I = 0. Fu - he mo e, a leas one componen o w 1 is nega- i e. Indeed, i w 1 > 0, hen, as ew 1 = O, i would ollow ha w I = 0, hus (ew)w ° = w; as w 4= 0, ew < O, one would ob ain ew < 0, hus 0 <Aw = (ew)Aw°; as 0<Aw ° and ew <0, his would imply ha Aw °= O, i.e.: ( ecall ha A -x exis s) w ° = 0, which is a con adic ion. Hence, w a has a leas a nega i e componen , hus he e exis s some A, 0 < h < 1 such ha he ec o w 2 = hw* + (1 - h)w ° e i ies ha Aw 2 > O, ew 2 > 0, w 2 has a leas a nega i e componen . By case 1, w e ~D~, which is a con adic ion. Hence, D A = D~, as we claimed. Wi h his, we a e in posi ion o show ha A-l> 0. Indeed, le be a column o A-l; as AA -1 gi es he iden i y ma ix, i ollows ha A >_ O, i.e., ~ D A, hus > 0. Then, we ha e shown ha all he columns o A-1 e i y ha >_ 0, hus A-1 >_ 0, as asse ed. [] Theo em 2.2. Le A ~ ~n×n be a linea ope a o such ha de (A) ~ 0 and C A -4= ~J. Then C + = C A i A-l >O. P oo . I is e iden ha , i A-l> 0 hen C~ = C A. We now show he con e se. Le A be an n × n ma ix wi h de (A) ~ 0 such ha C~ = C A. De ine he se s D n and D~: D4={wE~n:Aw >>. O}, D~={w~n:Aw >O, w>O}. The p ocess o supplying in o ma ion o he ini ial mul i-c i e ia p oblem ans o ms he p e - e ence scheme. Thus in he beginning, i.e. when no in o ma ion is a ailable, one al e na i e x ~ X is p e e ed o ano he y~X i (x)>_ (y) (componen -wise), Tha is, wi h no in o ma ion, he p e e ence scheme coincides wi h he Pa e o quasi-o de . So, i seems na u al ha in he p o- cess o supplying in o ma ion, he mo e p ecise in o ma ion he D-M gi es, he mo e accu a e quasi-o de will be gene a ed. I is e iden ha 296 E. Ca izosa e al. / Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 e e y quasi-o de R A wi h A- 1 >_ 0, imp o es he no-in o ma ion- ela ion because i educes he se o weigh s. Bu gi en wo ela ions R A and R B i is no clea how o de e mine which o hem is he mos accu a e. Howe e , his is e y impo an because i al- lows us o know he deg ee o knowledge shown by he D-M abou his own p oblem. The mo e accu a e he quasi-o de , he be e he knowl- edge o he p oblem. Mo eo e , i seems na u al ha he accu acy o a ela ion is in e sely p opo - ional o he magni ude o i s se o weigh s as is p oposed by Rios [18], so we de ine he accu acy o R A in he ollowing way. De ini ion 2.2. Gi en a Q-ope a o A, he accu- acy o RA, AC(RA), is de ined as IXn_l(C~) / /xn_l(C~), whe e /x,_ 1 ep esen s he Lebesgue measu e in Nn-~. The limi alues o he accu acy a e gi en in he ollowing p oposi ion• P oposi ion 2.1. I A is a Q-ope a o , hen 1 < AC(RA) < + oo. P oo . Fi s o all, ix~_l(C~)<lx,_~(C -), VA E N~xn. Second, as A -1 > 0, C~ is a simplex in he hype plane El<_i<,wi = 1 and /X,_x(C~)>0. Then 1 _< AC(R A) < +m. [] The alue 1 co esponds o he i s no-in o - ma ion case and, hence, we can see he accu acy as a educ ion measu e o he se o weigh s. Mo e p ecisely e e y ope a o ha gene a es an o de on R n has an accu acy +~. An o de ela ion o his kind could be seen as he limi case o a con e gen sequence o quasi-o de s wi h inc easing and mo e p ecise in o ma ion. Finally we shall gi e he exp ession o he accu acy: Theo em 2.3. Le A be a Q-ope a o . Then, AC(R A) = [de (A)lFIl<j_<~/xj, whe e z k was de- ined in (7). P oo . Fi s , ecall ha he olume o he simplex gene a ed by he poin s xX,..., x n in ~-1 is [1] 1 1 ... 1 1 x~ x 2 ... x~' (n- 1)~ de ". X 1 X 2 X n n--1 n--1 • " " n--1 Le e i ~ ~" be he ec o whose i- h componen is one and ze o he es . Conside he sys em o e e ence ~ in he hype plane H = {w:ew = 1} ha has e 1 as o igin and he ec o s e i-e 1, i = 2 ..... n, as gene a o s. Then, any poin w = (wa,. •., w n) ~ H has coo dina es (w2,..., w n) in ~9~. By Theo em 2.1, CJ is gene a ed by he ec- o s Olli// ]Z i I azz/ Zi. ], i= l,...,n. I Olni/~i ] Hence, AC(RA) 0/21/],-£1 0/22///£ 2 ... 012n/1.£ n = de . . O/nl//l-L 1 ang///L~ 2 ... Olnn//l~n Thus ( ecall ha /-~i = E]=laij, Vi), AC(RA) [ a11//]£ 1 O/12///'£2 ... Olln//~n --1. = de " -. O/nl//~L~ 1 Oln2/~ 2 ... ann/]£n In o he wo ds, 1 AC(RA) = [de (A-1)[FI]=l(1//xj) n = [de (A) I 1-I/xj, j=l as asse ed. [] Example 2.1. Fo he quasio de R A desc ibed in Rema k 2.1, one has ha de (A)= 1 and /xj =j, Vj', hus AC(R A) = n! E. Ca izosa e al. / Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 297 Fo he quasio de s R A and RA~ in oduced in Example 1.1, he imp o emen in he o de ela- ion om R A o RA1 can be measu ed by means o he accu acy: AC(R A) = 67.5 and AC(Rq)= 171.5. 3. Some amilies o Q-ope a o s The wides class o ope a o s we can deal wi h is cha ac e ized in Theo em 2.2. Howe e , he condi ion shown in he p e ious pa ag aph is di icul o check be o ehand. So in o de o enable he D-M o apply hese ela ions, we p opose wo sub-classes belonging o he o iginal one wi h h ee impo an p ope ies: 1. o know be o ehand ha hey belong o he b oad class; 2. o be easy o he D-M o unde s and and accep ; 3. o be su e ha he se o weigh s i gene a es is no emp y. A p ocedu e o ob ain a class o hese ope a o s consis s in o e ing he D-M he compa ison o each c i e ion i(-) wi h a mos a coali ion o he emaining c i e ia. Usually, he exac de e mina ion o he weigh s is made by he ade-o be ween a c i e ion and he emainde s. Bu his me hodology canno be used i he D-M does no gi e i s p e e ences so p ecisely. Al e na i ely, ou app oach p oposes o e- place his equi alence by inequali ies which a e qui e accep able o he D-M. In his p ocess, he mo e p ecise in o ma ion he D-M supplies, he mo e accu a e he quasi-o de i gene a es, and hence i is close o he ca dinal u ili y. The e o e, he in o ma ion equi ed om he D-M abou he c i e ion i(.) which he is willing o gi e, will ha e he ollowing o m: Wi ~--- E aijwj, aij ~ O, E aij ~ 1, j4=i j~i whe e aij ep esen s he minimum ma ginal sub- s i u ion a e o i o j. We should no ice ha when no in o ma ion is a ailable, his kind o ela ion will be wi > 0. Bu , e en by supplying small alues o ais one imp o es he accu acy o he quasi-o de gi en by he D-M and a oids he p oblem o he exac es ima ion o he weigh s (ca dinal u ili y). Example 3.1. We shall deal wi h he ollowing example in which he D-M has h ee objec i es, a se X o easible al e na i es, and he is able o gi e he equi ed in o ma ion in he o m o inequali ies. W 1 ~ 0.5W2, W 2 ~ 0.5W 1 q- 0.5W3, w 3 > 0.5w 2 . Then he ope a o is de ined by he ma ix A, whose ex eme poin s a e gi en by he columns o he second ma ix and he in e al weigh s a e 1 -0.5 0 A = -0.5 1 -0.5 0 -0.5 1 1 1 1 2 4 ~ W1 ~ [61 ' 1] 1 1 1 E = 3 2 X W2 ~ [31-' 1]. : : : w.~[L½] 6 4 g Hence i s accu acy AC(R A) = 18. This example sugges s he possibili y ha hese kinds o ope a o s ha e he p ope y o in e se posi i e. In o de o cla i y he e ms used in he ollowing esul s we in oduce wo classical con- cep s [16]. De ini ion 3.1. A linea ope a o A ~ R nxn is diagonally dominan i ~_, laijl < laiil, i= l,...,n, ]~i and s ic ly diagonally dominan i s ic inequali- ies hold o all i = 1 ..... n. The esul sugges ed by he example abo e is s a ed in he ollowing heo em. Theo em 3.1. Le A ~ ~nXn be a diagonally domi- nan linea ope a o such ha aij <_ 0, i ~ j, aii > 0, V i = 1 ..... n, and de (A) ~ 0. Then A -: > O. 298 E. Ca izosa e aL ~Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 P oo . Fo simplici y and wi hou loss o gene al- i y, we conside aii = 1, Vi. As A is diagonally dominan , he inequali ies aii >~ --Y'.j~iaij, Vi = 1 .... ,n, hold. Then e e y co ac o Aij o he ma ix A is nonnega i e [4]. Le A -1 be he in e se o A, whose elemen s a e a u =Aii/de (A). Then he sign o aij., Vi, j, coincides wi h he sign o de (A) and de (A) 1 a12 ... aln 0 1 -- a12a21 ... a2n -- alna21 = de 0 an2 -- al2anl ... 1 -- alnanl 1 a12 "'" aln ~ = de O A 1 J =dee(A1). Besides, he ma ix A~ is diagonally dominan because o all i = 2 .... , n hei elemen s e i y 1 - aliail > 0 and 1+ ~ aij+ail (- ~au)>l+ Zaij>_O. j¢i,j~a j4=l / j#i Thus, we can epea his easoning n imes and we ob ain ha de (A) > 0. [] Co olla y 3.1. Le A ~ ~x~ be a s ic ly diago- nally dominan linea ope a o such ha aii ~ 0, i =~ j, aii > O, Vi = 1 ..... n. Then A- 1 ~ O. he can compa e he impo ance o se e al c i e ia be ween hem. So he exp ession ob ained o he c i e ion ~ anked in i- h posi ion is wi>- ~,auw j, ai~>-O, j>i whe e ai ep esen s he minimum ma ginal sub- s i u ion a e o i o p In his sub-class he h ee p ope ies enume - a ed a he beginning o Sec ion 3 also hold and hey belong o he class cha ac e ized by Theo- em 2.2. Theo em 3.2. Le A ~ W '×n be a diagonal posi i e iangula linea ope a o such ha a u < O, Vj > 1. Then A- 1 > O. P oo . In his si ua ion de (A)= ~[l<i<naii > 0 and he co ac o s a e Aij= ]'-I akk>0 Vi < j, k~i,k~j Aii = YI akk > O, i = l,...,n, k~i Aq=0 Vi>j. Hence, easons analogous o he ones we used in Theo em 3.1 p o e his heo em. [] 4. Non-homogeneous Q-ope a o s This class o ope a o s is closely ela ed o a well known amily o linea ope a o s called M- ope a o s [16,25], bu he i s one exhibi s in i s a o he easy in e p e abili y and manipula ion because he M-ope a o s equi e p ope ies o i educibili y o s ic diagonal dominance, no needed in he p oposed class. I mus also be no ed ha he well known o dinal ela ion men ioned in Rema k 2.1 is an example o such an ope a o . Ano he impo an sub-class o ope a o s be- longing o he class cha ac e ized by Theo em 2.2 a e he iangula and nonposi i e o -diagonal elemen s. This ype o ope a o co esponds o he si ua ion in which he D-M is able o do a ce ain ank o de in he c i e ia and in addi ion, On many occasions, he D-M is willing o ob ain a leas ce ain le els in his weigh s [14], so he quasi-o de is gi en by a linea ope a o A and a le el ec o A _> 0. In his case, he se o easible weigh s is gi en by he poly ope (w ~ ~ :w >__ O, Aw >_, , C~,~) = w i = 1 / . (8) E i=l,...,n I Hence he pai (A, A) induces he ela ion R(A,a ) gi en by XR(A,;~)y i w( (x)-- (y)) >O Vw e C(5,, ). (9) E. Ca izosa e al. /Eu opean Jou nal o Ope a ional Resea ch 81 (1995) 291-301 299 An in e es ing ype o hese ela ions a e hose de ined by a ma ix A wi h non-nega i e in e se (componen wise) and eA-1A < 1. In wha ollows we say hese ela ions a e induced by a non-ho- mogeneous Q-ope a o (A, A). To his kind o e- la ions we can ex end all he p e ious esul s wi h minimum e o as can be seen om he ollowing heo em. Theo em 4.1. Le (A, A) be a non-homogeneous Q-ope a o . Then w -A-1A w~C~,A) i l_eA_lh ~CJ. P oo . w~C A,h ) i 3 >OIAw=h + , w>0, ew = 1; i.e. ( ecall ha A -1 >__ 0): w~C~,~) i 3 >OIw-A-~h=A-l , ew = 1. As e(w -A-1A) = 1 - eA-1h > 0, i ollows ha 1 w~C~,x) i :l >Ol l_eA_lh (W-A-1h) =A -1 i 3 >__0[W =A-i , 1 - eA-1h e~ = 1, w -A-1A whe e - 1-eA-1A i ~C~. Which concludes he p oo . [] Co olla y 4.1. I (A, A) is a non-homogeneous Q-ope a o , hen he se C(~,x) is he con ex hull o he columns o he ollowing ma ix: /x11(1 - Eke_l ZkAk) A2 A_I[ 1 - mogeneous case, we wish o pay some a en ion o he ollowing p oblem. We know ( ecall Re- ma k 2.3) ha he e exis many quasi-o de s ha p oduce he same in e als o hei weigh s; bu , a e he e any o hem which mus be empha- sized? The answe is a i ma i e and among hese ope a o s one is pa icula ly in e es ing; i is known in he li e a u e as E-cone [24] which is one o ou non-homogeneous Q-ope a o s. This kind o ope a o ( he E-cone) is based on ela- ions o he o m w i>_k i, i = 1,...,n (one pe objec i e)• Thei wo main p ope ies a e ha he se o weigh s i de ines includes he in e al weigh s conside ed be o ehand and also is mini- mal (in he inclusion sense among he ope a o s o his kind). We s a e and p o e his in he ollowing heo em• Le k = (k 1 .... , k n) >_ 0. Conside he se C+(l,k)---- ( ~n Wi__ Wi= } w ~ : >ki; ~ 1 l <_i <_n and o each a,/3 ~ ~n such ha 0 _< ag _< ig _< 1, Vi, and F,~=la i < 1 < E~=I/3g, he se (see [2]) X(a,13)=(w~n:O NWN/3; ~ wi=l}. l <_i <_n Theo em 4.2. The ec o k = (kl,..., k n) wi h k i = max(a i, 1 - Ej~//3j), i = 1 ..... n, de e mines he minimum se C Lk) con aining he se X(~, ~). A 1 1 -/xKl(1 - Ek,2/ZkAk) An whe e IX j, j = 1 ..... n, we e de ined in (7). ... A 1 ''• /~2 ... 1--/X21(1--Ek.n~k; k) Al hough all he p ope ies holding in he ho- mogeneous case can be ex ended o he non-ho- P oo . Fo all (kl,..., k n) >_ O, El <_i<_nki __< 1, one can ob ain, using Co olla y 4,1, ha C(~,k ) is he