A general approach for the location of transfer points on a network with a trip covering criterion and mixed distances

A gene al app oach o he loca ion o ans e poin s on a ne wo k wi h a ip co e ing c i e ion and mixed dis ances M. C. López-de-los-Mozos a, Juan A. Mesa b, Ani a Schöbel c a Depa amen o de Ma emá ica Aplicada I, E.T.S. de Ingenie ía In o má ica, A da. Reina Me cedes, s/n, 41012, Uni e sidad de Se illa, Spain b Depa amen o de Ma emá ica Aplicada II, E.T.S. de Ingenie ía, Camino de los Descub imien os, s/n, 41092, Uni e sidad de Se illa, Spain c Ins i u ü Nume ische and Angewand e Ma hema ik. Uni e si ä Gö ingen, Ge many Keywo ds: Loca ion Ne wo ks Co e ing p oblems Mixed dis ances a b s a c In his pape we conside a ip co e ing loca ion model in a mixed plana -ne wo k space. An embed- ded ne wo k in he plane ep esen s an al e na i e anspo a ion sys em in which a eling is as e han a eling wi hin he plane. We assume ha he demand o be co e ed is gi en by a se o o igin- des ina ion pai s in he plane, wi h some affic be ween hem. An o igin-des ina ion pai is co e ed by wo acili y poin s on he ne wo k (o ans e poin s), i he a el ime om he o igin o des ina ion by using he ne wo k h ough such poin s is no highe han a gi en accep ance le el ela ed o he a el ime wi hou using he ne wo k. The acili y loca ion p oblems s udied in his wo k consis o loca ing one o wo ans e poin s on he ne wo k such ha , unde se e al objec i e unc ions, he affic h ough he ne wo k is maximized. Due o he con inuous na u e o hese p oblems, a gene al app oach is p o- posed o disc e izing hem. Since he non-con exi y o he dis ance unc ion on cyclic ne wo ks also implies he absence o con exi y o he mixed dis ance unc ion, such an app oach is based on a decom- posi ion p ocess which leads o a collec ion o subp oblems whose solu ion se can be ound by adap ing he gene al s a egy o each p oblem conside ed. 1. In oduc ion Gi en a eal o i ual unde lying ne wo k, he gene al ne wo k design p oblem consis s o wo in e wined p oblems ( Con e as & Fe nández, 2012 ): o selec om he ne wo k a numbe o poin s o si ing acili ies, and o in e connec hese poin s by choosing links o he unde lying ne wo k. In ou p oblem we ha e a ne - wo k embedded in he plane and a se o demand poin s in he plane. The demand o affic be ween demand poin s is gi en by an OD-ma ix (O igin-Des ina ion ma ix) which is assumed o be s a ic, and i is sa isfied by a eling om one o he o he poin o each pai . We wan o selec poin s on he ne wo k o access/ exi o/ om i . These acili y poin s a e used o ans e om he plane o he embedded gi en ne wo k, hus allowing he connec ion o he pai s o demand poin s by using he ne wo k. A subp oblem o he gene al ne wo k design p oblem a ises in he design o a ailway ne wo k ( Lapo e & Mesa, 2015; Lapo e, Mesa, & O ega, 20 0 0) ha consis s in loca ing a numbe o s a ions and he acks o connec he s a ion- acili ies. The p oblem o loca ing s a ions in a ailway ne wo k was in- di ec ly ackled in Vuchic and Newell (1968) , in which he op imal in e s a ion spacing p oblem in a commu e line was esea ched. The objec i e o his p oblem is o minimize he o al ime o pas- senge s going o a ci y cen e along a ailway line. A commu e line compe ing wi h a eeway was conside ed in Vuchic (1969) in o de o maximize he numbe o passenge s on he basis o sho - es a el imes. The aim was also o de e mine he op imal in e - s a ion space be ween pai s o adjacen s a ions. Apa om he pape s ( Lapo e, Mesa, & O ega, 2002; Lapo e, Mesa, O ega, & Se illano, 2005; Repolho, An unes, & Chu ch, 2013 ) in which he easible solu ion space o loca ing s a ions was disc e e and he objec i es we e o maximize he passenge co e age, ip co e - age and sa ings in a el cos , espec i ely, se e al objec i e unc- ions ha e been conside ed in hose models in which s a ions can be loca ed along he edges o he ailway ne wo k: sa ing in pas- senge a el ime ( Hamache , Liebe s, Schöbel, Wagne , & Wag- ne , 2001 ), co e age/numbe o new s a ions ( Schöbel, 2005 ), ad- di ional a el ime ( Schöbel, Hamache , Liebe s, & Wagne , 2009 ), and o al a el ime ( Ca izosa, Ha be ing, & Schöbel, 2016 ). The maximal co e ing loca ion p oblem was in oduced in Chu ch and ReVelle (1974) in which a numbe o acili ies a e o be loca ed on a ne wo k so ha he popula ion wi hin a se ice dis ance is maximized. In Mu awski and Chu ch (2009) ime o access he a- cili ies is imp o ed by upg ading edges o he anspo a ion ne - wo k. The objec i e unc ion consis s in maximizing he numbe o people who a e co e ed by upg ading some pa s o he ne - wo k. In o de o loca e in e changes poin s in a highway ne wo k a cos -benefi objec i e unc ion is conside ed in Repolho, Chu ch, and An unes (2010) ha akes in o accoun a ou e choice model. In he p oblem deal wi h in his pape , he e a e exis ing de- mand poin s in he Euclidean plane whe eas new acili ies a e o be loca ed in a ne wo k. Ins ead o co e ing acili ies he objec- i e aims a co e ing ips be ween each pai o exis ing acili ies. These ips can be done ei he by using he plane wi h he Eu- clidean dis ance o by a plane-ne wo k combina ion in which he sec ion on he ne wo k is supposed o be a e sed as e han hose on he plane. The e o e, he e is compe i ion be ween he mode ha only uses he plana dis ance and he combined one. I he ime spen by he ip in he combined mode is lowe han ha o he plana mode hen he OD-pai is said o be co e ed by he ans e poin s used o access/exi he ne wo k. Each pai o exis ing acili ies has an associa ed demand and he objec i e is o maximize he numbe o ips o which he combined mode is p e e able o he plana one. In his pape we assume he decision space is he se o poin s on he edges o he ne wo k. The na u e o he decision space when loca ing new ans e poin s in long and medium dis ance ailway ne wo ks as well as in unde g ound u ban/subu ban apid ansi sys ems is con inuous. As was poin ed ou in Lapo e, Mesa, and Pe ea (2014) 30% o he s a ions o he Spanish high-speed ne wo k a e si ua ed ou side ci ies in he coun yside, hus al- lowing a con inuous decision space. Fu he mo e, he con inuous se ing o he decision space allows he use o con inuous op i- miza ion me hods hus p o iding insigh in o he beha io o he objec i e unc ion and he p oblem i sel . Mo eo e , new de el- opmen a eas, conges ion o en i onmen al and ene gy consump- ion a e easons o cons uc ing new s a ions. An example is he use o a sec ion o he ailway line Se ille-Huel a as a new com- mu e line (Line 5) o he me opoli an a ea o Se ille ( Lapo e e al., 2014 ). Fo his line new s a ions we e buil , some o hem be ween owns. Ano he example is he cons uc ion o a new s a- ion in he high-speed line Mad id-Se ille be ween Có doba and Pue ollano s a ions. The new s a ion: Villanue a de Có doba-Los Ped oches co e s an a ea o he no h o he p o ince o Có doba. Since he a eling demand is gi en by pai s o exis ing poin s i is cohe en o conside he loca ion o pai s o ans e poin s ac - ing as access/exi poin s on he ne wo k. Unde he assump ion o a ne wo k wi h a se o exis ing node-s a ions, he e a e wo p ob- lems pa icula ly ele an om an applied poin o iew, which deal wi h loca ing one, o wo, ans e poin s in o de o maxi- mize he amoun o OD-pai s addi ionally co e ed, ha is, hose OD-pai s which canno be co e ed by he exis ing s a ions. In he fi s p oblem only one ans e poin is loca ed since his poin in combina ion wi h each s a ion wo ks as an access/exi poin . In he second p oblem, which seeks o loca e wo ans e poin s, he addi ional co e age e e s he OD-pai s co e ed ei he by he wo ans e poin s o by a combina ion o each poin wi h some s a- ion, bu no ye co e ed by pai s o e ices. Bo h p oblems also exclude he OD-pai s p e iously co e ed by he exis ing s a ions. Mo eo e , he second p oblem p esen s a mo e gene al o mula- ion o he model, in he way ha i inco po a es he addi ional co e age p o ided bo h by he wo poin s and by each poin sepa- a ely. Fo sol ing hese addi ional co e ing p oblems, his pape p o- poses a me hodology which is based on conside ing a key p oblem, whose solu ion p ocedu e p o ides a gene al app oach which can be adap ed o he emaining p oblems. This key p oblem does no equi e any hypo hesis on he ne wo k (i is possible ha he e is no al eady loca ed s a ion), since i seeks o loca e wo ans- e poin s maximizing he amoun o OD-pai s co e ed. The ele- ance o his p oblem de i es om i allowing he de elopmen o a heo e ical amewo k which is sha ed by he abo e one o wo ans e addi ional co e ing p oblems, as well as by o he ela ed p oblems. Summa izing, he main con ibu ion o his pape is, in he fi s place, o p opose a comp ehensi e app oach o sol ing se e al p oblems dealing wi h loca ing ans e poin s on cyclic anspo a- ion ne wo ks unde se e al objec i e unc ions, all o hem o- cused on co e ing OD-pai s ins ead o single demand poin s. Mo e- o e , he p oposed app oach o sol e he p oblem o maximizing he co e age ob ained by 2- ans e poin s is used o he mo e eal p oblems o loca ing one o wo ans e poin s so ha he addi ional co e age (in p esence o al eady unc ioning s a ions) will be maximized. The p oblems s udied deal wi h a con inuous model, whose analysis enla ges he knowledge on bo h he geo- me ic s uc u e o he p oblems and he beha io and p ope ies o he objec i e unc ions. On he o he hand, such an app oach de- sc ibes a flexible me hodology which, by inco po a ing sligh mod- ifica ions, allows o disc e ize he solu ion se o all p oblems con- side ed. In a p e ious pape ( Kö ne , Mesa, Pe ea, Schöbel, & Scholz, 2014 ) he key p oblem was sol ed when he new acili ies a e o be loca ed in segmen s and ee ne wo ks. In his pape we ex end he app oach applied in Kö ne e al. (2014) o he case o gene al ne wo ks by aking in o accoun he loss o con exi y o he dis- ance unc ion be ween pai s o poin s h ough he ne wo k. The main aim o his esea ch is o sol e ha complex con inuous lo- ca ion p oblem by educing he candida e se o a fini e one om which an op imal solu ion is selec ed by means o a polynomial ime algo i hm. The pape is o ganized as ollows: a e his in oduc ion he elemen s o he model a e p esen ed in Sec ion 2 . The necessa y defini ions and esul s on he dis ance on ne wo ks a e summa- ized in Sec ion 3 . Sec ion 4 p o ides a decomposi ion o he fi s p oblem in o wo ypes o subp oblems, such ha o each sub- p oblem a solu ion me hod is p oposed in Sec ions 5 . Sec ions 6 and 7 s udy espec i ely wo new p oblems, bo h ela ed o he p e ious one, in which one and wo ans e poin s a e loca ed un- de di e en objec i e unc ions, and Sec ion 8 is de o ed o p e- sen ing he co esponding algo i hms and discussing i s compu a- ional complexi y. The pape ends wi h some conclusions and u - he esea ch. 2. Elemen s o he model In o de o o mula e he p oblems wi h a mixed mode o ans- po a ion, we conside a connec ed ne wo k N (V, E) ep esen ing a high-speed sys em, wi h | V | nodes and | E | edges (whe e | ·| deno es ca dinali y). We assume ha he ne wo k is embedded in he Eu- clidean plane and ha each undi ec ed edge e ∈ E can be modeled as a s aigh -line segmen . The embedding o N in he Euclidean plane as well as he coo dina es o he nodes in N will allow us o compu e he dis ances along he ne wo k. Mo e p ecisely, he leng h l(u, ) o an edge (o a subedge) [ u, ] is he Euclidean dis- ance || u − || be ween i s endpoin s, and by using sho es pa h algo i hms we can mo eo e compu e he dis ance be ween each pai o nodes. Mo eo e , he iangle inequali y holds in his model. Le N be he con inuum se o poin s o he edges. The edge leng hs induce a dis ance unc ion d such ha , o any wo poin s x, y ∈ N , d ( x , y ) is he leng h o any sho es pa h in N (V, E) con- nec ing x and y . I x and y a e on he same edge, hen d ( x , y ) coin- cides wi h he leng h o he subedge [ x , y ]. Fo x, y ∈ N , he a el dis ance be ween bo h poin s is gi en by αd ( x , y ), wi h α∈ (0, 1). Pa ame e αis a speed ac o , such ha Ai Aj X1 X2 Ai Aj X1 X2 Fig. 1. T a el pa hs o : h + ij (X 1 , X 2 ) (le ), and h − ij (X 1 , X 2 ) ( igh ). αd ( x , y ) ep esen s he a eling ime be ween x , y by using he high-speed ne wo k. I can immedia ely be seen ha , wi h hese dis ance, N is a me ic space. •Le A = { A i = (a i , b i ) , i = 1 , . . . , n } ⊂IR 2 be a se o exis ing de- mand poin s on he plane. We assume ha dis ances be ween wo poin s in he plane can be es ima ed by he Euclidean me - ic. •Le T = ( ij ) ∈ IR n ×n be an o igin-des ina ion ma ix in which ip pa e ns a e codified, i.e., ij is he weigh o he o de ed pai ( i , j ). This ma ix is known a p io i: o example, in a ans- po a ion con ex each ij can be iewed as he numbe o ips om an o igin A i o a des ina ion A j , and in a elecommunica- ion se ing i could ep esen he amoun o da a ans e ed om se e A i o se e A j . •Gi en an OD-pai ( i , j ), he a el dis ance by using he ne wo k h ough he poin s X 1 , X 2 ∈ N (o he 2- acili y poin ( X 1 , X 2 )) is ob ained om he wo possible a el pa hs: ( A i , X 1 , X 2 , A j ), and ( A i , X 2 , X 1 , A j ), linking o igin and des ina ion (see Fig. 1 ). Fo each o hem, he a el dis ance is gi en by h+ ij(X 1 , X 2 ) = || A i −X 1 || + αd(X 1 , X 2 ) + || X 2 −A j || h− ij(X 1 , X 2 ) = || A i −X 2 || + αd(X 2 , X 1 ) + || X 1 −A j || . The mixed a el dis ance be ween A i and A j by using he high- speed ne wo k h ough he ans e poin s X 1 , X 2 ∈ N is gi en by he sho es a el dis ance ob ained om bo h a el pa hs, and i is w i en as: ij (X 1 , X 2 ) = min { h+ ij (X1 ,X2 ),h− ij (X 1 , X 2 ) } . Symme y o such pa hs implies h + ij (X 1 , X 2 ) = h − ji (X 1 , X 2 ) , h− ij (X1 ,X2 )=h+ ji (X 1 , X 2 ) , and he e o e ij (X 1 , X 2 ) = ji (X 1 , X 2 ) . •Le  D = (  d ij ) ∈ IR n ×n be a symme ic ma ix, wi h 0 ≤ d ij < || A i −A j || , o i  = j , and  d ii = 0 , i = 1 , . . . , n . The alues o  D ep esen he accep ance le els o using he ne wo k, meaning ha he OD-pai ( i , j ) chooses he high-speed ne wo k i and only i he mixed a el dis ance by using i is less han o equal o  d ij . In o he wo ds, he OD-pai s always choose he as e op- ion. In he ollowing, we assume ha i  = j o a oid he i ial case. Defini ion 1. The OD-pai ( i , j ) is co e ed by X 1 , X 2 ∈ N i ij (X 1 , X 2 ) ≤ d ij . Le C ( X 1 , X 2 ) be he se o O/D pai s co e ed by X 1 , X 2 , gi en by C(X 1 , X 2 ) = { (i, j) , i  = j, 1 ≤i, j ≤n : ij (X 1 , X 2 ) ≤ d ij } . F om symme y bo h o he accep ance le el ma ix  D and he mixed a el dis ance we ha e (i, j) ∈ C(X 1 , X 2 ) i and only i (j, i ) ∈ C(X 1 , X 2 ) . As i has been poin ed ou in he p e ious sec ion, in he fi s place we o mula e he key p oblem, which p esen s s uc u al p ope ies sha ed by he emaining p oblems, and whose solu ion p ocedu e p o ides a gene al me hodology o analyzing and sol - ing he one and wo addi ional co e ing p oblems, which a e o - mula ed subsequen ly. Ai Aj X1 X2 Ak Fig. 2. Gi en he nonnega i e accep ance le els  d ij < || A i −A j || ,  d ik < || A i −A k || ,  d jk < || A j −A k || , and a sui able speed ac o α∈ (0, 1), we ha e ( i , j ), ( i , k ) ∈ C ( X 1 , X 2 ). Howe e ( j , k ) ∈ C ( X 1 , X 2 ). The objec i e alue a ( X 1 , X 2 ) is: F (X 1 , X 2 ) = ij + ji + ik + ki . The e may be se e al poin s ( X 1 , X 2 ) wi h he same objec i e alue. 1. The fi s objec i e unc ion measu es he o al weigh o OD- pai s cap u ed by each pai o ans e poin s X 1 , X 2 ∈ N , and i is gi en by: F (X 1 , X 2 ) =  (i,j) ∈ C(X 1 ,X 2 ) ij . In his unc ion whe he he node se V con ains al eady lo- ca ed s a ions o no is no ele an , since i seeks o com- pu e he weigh o pai s co e ed by he wo ans e poin s when hey assume he symme ic ole o access/exi poin s (see Fig. 2 ). The key p oblem, b iefly he 2- ans e co e ing p oblem (2- TC), is o find wo ans e poin s X 1 , X 2 ∈ N such ha he sum o weigh s o all OD-pai s co e ed by such poin s is maximized: max X 1 ,X 2 ∈N F (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) ij (2-TC) 2. Con a y o he abo e model, his p oblem and he ollow- ing one deal wi h addi ional co e age, and hey equi e he hypo hesis o ha ing a se o s a ions al eady loca ed, hese la e ones selec ed om he node se . Wi hou loss o gene ali y we can assume ha all nodes o V a e al eady loca ed ans e poin s. F om Defini ion 1 , he OD-pai ( i , j ) is co e ed by any X ∈ N i he e exis s ∈ V such ha ij (X, ) ≤ d ij . Thus, he se C ( X ) con aining he OD-pai s co e ed by X is C(X ) = { (i, j) , i  = j, 1 ≤i, j ≤n : ij (X, ) ≤ d ij o some ∈ V } Simila ly, he se C V o OD-pai s al eady co e ed by he nodes o V is gi en by C V = { (i, j) , i  = j, 1 ≤i, j ≤n : ij (w, ) ≤ d ij o some w, ∈ V } Thus, he second objec i e unc ion can be s a ed as F V (X ) :=  (i,j) ∈ C(X) C V ij And he 1- ans e addi ional co e ing p oblem (1-TAC) is: max X∈NF V (X ) :=  (i,j) ∈ C(X) ij (1-TAC) Finally, in his case wo ans e poin s ( X 1 , X 2 ) a e lo- ca ed by add essing he addi ional co e age. As abo e, we assume ha all nodes a e s a ions, and also he se C V is ex- cluded. The objec i e unc ion in eg a es he abo e o mu- la ions since i akes in o accoun bo h he amoun o OD- pai s co e ed by ( X 1 , X 2 ) and he OD-pai s co e ed by each poin in combina ion wi h he s a ions al eady loca ed a nodes, as ollows: F A (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) ∪ C(X 1 ) ∪ C(X 2 )  C V ij The 2- ans e addi ional co e ing p oblem (2-TAC), seeks he loca ion o wo ans e poin s maximizing he o al weigh o OD-pai s addi ionally co e ed: max X 1 ,X 2 ∈N F A (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) ∪ C(X 1 ) ∪ C(X 2 )  C V ij (2-TAC) Rema k 2. Each pai o associa ed p oblems ob ained by excluding, o no , C V om he co esponding o mula ion a e equi alen , bu no he same, in he ollowing way: hey sha e he same solu ion se (which is a s aigh o wa d consequence om he ac ha C V is a cons an se ), al hough hei op imal solu ions could be di e - en . Tha is, (2-TC) and max X 1 ,X 2 ∈N F (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) C V ij a e equi alen , and so on. Al hough p oblems (1-TAC) and (2-TAC) p esen special ele- ance ega ding he applica ions, he p ocedu e o sol ing p ob- lem (2-TC) p o ides a gene al me hodology which can be applied o all p oblems p esen ed in his pape . In ac , he na u e o such a me hodology allows he esolu ion o be ex ended o some gen- e aliza ions o hese p oblems, as he p - ans e addi ional co e - ing p oblem ( p > 2). This mo i a es ha p oblem (2-TC) is s udied in he fi s place. The s a egy o sol ing his p oblem lies in de- composing i in o a collec ion o subp oblems such ha , o each o hem, a pa i ion o he easible solu ion se is cons uc ed, and om such pa i ion, a fini e subse con aining some op imal solu- ion is selec ed. P oblem (2-TC) has been fi s sol ed o he pa icula case whe e N is a segmen o a s aigh line, and subsequen ly he me hod was ex ended o he case whe e N is a ee ne wo k T (see Kö ne e al., 2014 ). Thus, hence o h we will conside ha N con ains a leas one cycle. Unde his assump ion, he app oach applied o he ee ne wo k case canno be di ec ly ex ended o his case due o he absence o con exi y o he dis ance unc ion on a cyclic ne wo k. In o de o sol e his p oblem, we nex summa ize some con- cep s and esul s on dis ances in ne wo ks. 3. P e ious esul s on dis ances on ne wo ks Fo e alua ing he objec i e unc ion o p oblem (2-TC) , i is fi s necessa y o ob ain an analy ical exp ession o he dis ance be ween any wo poin s P , Q , o he ne wo k N . The a o emen- ioned non-con exi y leads o he ac ha he exp ession o he dis ance d ( P , Q ) could a y h ough he ne wo k. In his sec ion we pa i ion he ne wo k in o a collec ion o subedges such ha o each pai o subedges, he dis ance be ween hei poin s can be compu ed. To his end, we fi s need o e iew some concep s dealing wi h he dis ance on ne wo ks. He eina e we assume a ne wo k N wi h a dis ance unc ion d ( ·, ·) such ha he iangle inequali y holds. Le e = [ u, w ] ∈ Ebe an edge o N wi h leng h l e , and le P be a poin on edge e . Le x = l(u, P ) deno e he leng h o subedge [ u , P ], and le l(w, P ) = l e −x deno e he leng h o subedge [ P, w ] . I is well known ha o any node ∈ V, he dis ance d( , P ) = min { d( , u ) + x, d( , w ) + l e −x } is conca e and piecewise linea on x ∈ [0, l e ], wi h a mos wo pieces wi h slopes 1 and −1 . F om he iangle inequali y we ha e l e = d(u, w ) ≤| d( , u ) −d( , w ) | , he e o e he dis ance unc- ion d( , P ) on [ u, w ] is linea . Fo he sake o comple eness we in- clude he ollowing concep s and defini ions, al hough all o hem uq upwp wq uq wq wpup Fig. 3. An ipodal poin s. can be ound in Hooke , Ga finkel, and Chen. (1991) and e e ences he ein. Defini ion 3. Gi en a poin Q ∈ N and an edge [ u, w ] , he poin P ∈ [ u, w ] a which d ( Q , P ) is maximized is called an ipodal o Q in [ u, w ] . Defini ion 4. A poin ¯ ∈ [ u, w ] , o he han a node, is called an a c bo leneck poin i he e is a node o which ¯ is an ipodal o . In his case, we call i he a c bo leneck poin ¯ . Clea ly, he edge e = [ u, w ] con ains a mos | V | a c bo leneck poin s. I ¯ ∈ [ u, w ] is an ipodal o we ha e d( , u ) + l(u, ¯ ) = d( , w ) + l(w, ¯ ) , hence such a poin ¯ can be iden ified by he leng h o he subedge [ u, ¯ ] , gi en by: l(u, ¯ ) = d ( ,w ) −d ( ,u )+ l e 2 . Defini ion 5. Le B e be he se o a c bo leneck poin s o edge e = [ u, w ] , and le ¯ , ¯  be wo adjacen poin s o B e ∪ { u, w } . Then he closed subedge L = [ ¯ , ¯  ] is called a linea a c segmen (o a eelike segmen ( Hooke e al., 1991 )). We ema k ha B e has O (| V |) elemen s. On each linea a c segmen he dis ance d( , ·) is linea . I B e = ∅ he en i e edge e = [ u, w ] (including nodes) is a linea a c segmen . The se o all linea a c segmen s o an edge e is deno ed by L (e ) . Le [ u p , w p ] and [ u q , w q ] be wo edges o he ne wo k N , and le [ ¯ w p , ¯ u p ] ⊆[ u q , w q ] , [ ¯ w q , ¯ u q ] ⊆[ u p , w p ] be he subedges o which he ex eme poin s a e an ipodal poin s as ollows: ¯ u p , ¯ w p a e he an ipodal poin s o u p , w p , espec i ely, and ¯ u q , ¯ w q a e he an ipodal poin s o u q , w q , espec i ely (see Fig. 3 ). F om p e ious defini ions we ha e l( ¯ w q , ¯ u q ) = l( ¯ w p , ¯ u p ) and d( ¯ w q , ¯ w p ) = d( ¯ u q , ¯ u p ) (see Hooke e al., 1991 o a mo e de ailed explana ion). Defini ion 6. Le L p , L q be wo linea a c segmen s o di e en edges [ u p , w p ] and [ u q , w q ] , espec i ely, and le [ ¯ w q , ¯ u q ] ⊆[ u p , w p ] and [ ¯ w p , ¯ u p ] ⊆[ u q , w q ] be he subedges o which ¯ u p , ¯ w p a e he an ipodal poin s o u p , w p , espec i ely, and ¯ u q , ¯ w q a e he an ipo- dal poin s o u q , w q , espec i ely. I L p ⊆[ ¯ w q , ¯ u q ] and L q ⊆[ ¯ w p , ¯ u p ] , hen L p , L q a e called an ipodal segmen s o each o he . (This defini ion includes he case [ ¯ w q , ¯ u q ] = [ u p , w p ] and [ ¯ w p , ¯ u p ] = [ u q , w q ] ). The ollowing esul summa izes he beha io o dis ance d ( P , Q ) on linea a c segmen s o di e en edges (see Hooke e al., 1991 ). Lemma 7. Le P be es ic ed o linea a c segmen L p o edge [ u p , w p ] and Q o linea a c segmen L q o edge [ u q , w q ] , wi h [ u p , w p ]  = [ u q , w q ] . 1. I L p , L q a e an ipodal segmen s o each o he , d ( P , Q ) is con- ca e. 2. O he wise, d ( P , Q ) is linea . In case 1, he dis ance be ween a poin P on he segmen [ ¯ w q , ¯ u q ] and a poin Q on he segmen [ ¯ w p , ¯ u p ] beha es like he dis ance on he pa allelog am in Fig. 3 (a), and i is conca e. Dis- ance on any o he pai o segmen s beha es like dis ance on a line segmen and is he e o e linea . By deno ing x = l(u p , P ) , wi h x ∈ [0 , l(u p , w p )] and y = l(u q , Q) , wi h y ∈ [0 , l(u q , w q )] , we can compu e he dis ance d ( P , Q ) o he cases conside ed in his lemma. d(P, Q) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ min { x + d(u p , u q ) + y, l(u p , w p ) −x + d(w p , w q ) + l(u q , w q ) −y } , L p , L q an ipodal x + d (u p , u q ) + y, L p ⊆[ u p , ¯ w q ] , L q ⊆[ u q , ¯ w p ] o L q ⊆[ ¯ w p , ¯ u p ] x + d (u p , w q ) + l(u q , w q ) −y, L p ⊆[ u p , ¯ w q ] , L q ⊆[ ¯ u p , w q ] . No e ha i L p , L q a e an ipodal segmen s, wi h u p  = ¯ w q , ¯ u q  = w p and simila ly u q  = ¯ w p , ¯ u p  = w q , hen he dis ance d ( P , Q ) can also be compu ed by min { x + d(u p , w q ) + l(u q , w q ) −y, l(u p , w p ) −x + d(w p , u q ) + y } . Finally, he emaining cases ob ained by combin- ing L p , L q can be educed o one o hese by symme y ( o exam- ple, o he case L p ⊆[ u p , ¯ w q ] , L q ⊆[ ¯ u p , w q ] he dis ance can also be equi alen ly ob ained as l(u p , w p ) −x + d(w p , u q ) + y ). We now commen he case in which P , Q lie on he same edge e = [ u, w ] . Le L p , L q be wo linea a c segmen s o e such ha P ∈ L p , Q ∈ L q . The iangle inequali y implies ha L p and L q a e no an ipodal o each o he , consequen ly he dis ance d ( P , Q ) is ei he con ex (i L p = L q ), o linea (i L p  = L q ), and i is gi en by he leng h o subedge [ P , Q ]. By using he abo e no a ion x = l(u, P ) and y = l(u, Q) , we ha e d(P, Q) = | x −y | . 4. Decomposing he 2- ans e co e ing p oblem Summa izing he p e ious sec ion, o each pai o linea a c segmen s he dis ance d ( P , Q ) be ween hei espec i e poin s is ei he conca e o con ex, and i s analy ical exp ession can be com- pu ed. Fo his eason, he solu ion me hod is based on decompos- ing p oblem (2-TC) in o a collec ion o independen subp oblems (whe e each subp oblem is he es ic ion o (2-TC) o a gi en pai o linea a c segmen s), and sol ing each o hem ia disc e iza- ion o he solu ion se . To his end and o he sake o eadabili y, we will desc ibe he p ocess in h ee phases: he fi s one is de- o ed o bo h decomposing he p oblem (2-TC) and g ouping he subp oblems in o wo cases: he conca e and he con ex case. The second phase deals wi h he subp oblems o he fi s case, and fi- nally in he las s ep we will s udy he subp oblems o he second. These wo cases will be desc ibed as ollows. To decompose he p oblem we fi s compu e, in O (| V || E |) ime, he dis ance ma ix be ween all pai s o nodes o he ne wo k. Then, o each edge e ∈ E we ob ain, and so , he se B e (in O (| V |log | V |) ime). A he end o his p ocess we ha e, o each edge e o he ne wo k, he o de ed sequence L (e ) o all linea a c segmen s o he edge. Besides, gi en a pai o di e en edges e p = [ u p , w p ] and e q = [ u q , w q ] we know whe he o no wo linea a c segmen s L p ∈ L (e p ) and L q ∈ L (e q ) a e an ipodal. Le L =  e ∈ E L (e ) be he se o all linea a c segmen s o he o e all ne wo k. Thus p oblem (2-TC) is decomposed in o a se o subp oblems, whe e each o hem is ob ained by es ic ing he easible space N ×N o L p ×L q , wi h L p , L q ∈ L . By imposing ha X 1 ∈ L p , X 2 ∈ L q we ob ain he es ic ed p oblem: max X 1 ∈ L p , X 2 ∈ L q F (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) ij (2-TCR) A solu ion o p oblem (2-TC) is ound by sol ing he collec- ion o all O (|L| 2 ) es ic ed p oblems (2-TCR) (in ac , |L| (|L|−1) 2 es ic ed p oblems by symme y), and hen selec ing he bes so- lu ion. The classifica ion o he es ic ed p oblem (2-TCR) is made ac- co ding o he conca i y o he dis ance d ( X 1 , X 2 ), which is ela ed o he an ipodal cha ac e o L p , L q . Mo e specifically, p oblem (2- TCR) is conca e i he ollowing condi ion holds: Fig. 4. Ne wo k wi h V = { i , i = 1 , . . . , 4 } , and linea a c segmen s L p , p = 1 , . . . , 8 . L p ∈ L (e p ) , L q ∈ L (e q ) , wi h e p  = e q , and L p , L q a e an ipodal o each o he . Fo he emaining cases, he p oblem (2-TCR) is classified as con ex. 5. A p ocedu e o disc e izing he es ic ed p oblem (2-TCR) This sec ion is de o ed o s udying he es ic ed p oblem (2-TCR) o a gi en pai L p , L q ∈ L . Hence o h, we w i e ( X 1 , X 2 ) ∈ L p ×L q ins ead o X 1 ∈ L p , X 2 ∈ L q . The s a egy o sol ing (2-TCR) is based on iden i ying a Fi- ni e Domina ing Se (FDS), ha is, a fini e se o poin s ⊂L p × L q con aining an op imal solu ion. F om his se , (2-TCR) becomes he p oblem max (X 1 ,X 2 ) ∈ F (X 1 , X 2 ) . In o de o desc ibe he p o- cedu e o finding , he conca e and con ex cases a e analyzed sepa a ely. 5.1. The conca e case This p oblem can be o mula ed om (2-TCR) by adding he as- sump ion “L p , L q a e an ipodal o each o he ”, and i will be iden- ified as (2-TCR) (1). Each linea a c segmen is a ec ifiable subedge in which all dis- ance unc ions d( i , P ) a e linea . On he o he hand, we ha e: 1. Fo each A i ∈ A , he Euclidean dis ance || A i −X|| is con- ex when X a ies in any linea a c segmen . In e ec , his esul ollows s aigh o wa dly om con exi y heo y (see Rocka elle , 1970 ), since i is a con ex unc ion in an open se U hen he es ic ion o o any in e al (i.e., s aigh line segmen ) inside U is con ex. 2. When ( X 1 , X 2 ) ∈ L p ×L q , wi h L p , L q an ipodal o each o he , he dis ance αd ( X 1 , X 2 ) is conca e ( Lemma 7 ), whe e α∈ (0, 1) is he speed ac o . The e o e bo h he unc ion h + ij (X 1 , X 2 ) and h− ij (X 1 , X 2 ) a e no con ex on L p ×L q . To illus a e some concep s and p ope ies associa ed wi h his case, we will use he ne wo k wi h apezoidal shape shown in Fig. 4 . The leng hs o he basis edges o he apezoid a e 7 and 5, espec i ely, and he nodes { i , i = 1 , . . . , 4 } a e indica ed in he figu e wi h hei co esponding coo dina es. Likewise, each ¯ i ep- esen s he a c bo leneck poin opposi e o i , i = 1 , . . . , 4 . This figu e also shows he pa i ion o he ne wo k o igina ed by he se o linea a c segmen s L = { L p , p = 1 , . . . , 8 } . In his case, L 2 = [ ¯ 3 , ¯ 4 ] and L 6 = [ 4 , 3 ] a e an ipodal o each o he . Fig. 5. Su ace h + ij (X 1 , X 2 ) o α= 0 . 3 (le ), and se s H + ij (η) o such a su ace ( igh ). In o de o find an FDS o he p oblem, in he ollowing we in- oduce he necessa y defini ions on he suble el se s, om which he FDS is cons uc ed. 5.1.1. Cha ac e izing he suble el se s When ( X 1 , X 2 ) ∈ L p ×L q , and { L p , L q } a e an ipodal o each o he , he dis ance d ( X 1 , X 2 ) is a conca e unc ion ob ained om he lowe en elope o wo linea unc ions: d a ( X 1 , X 2 ) and d b ( X 1 , X 2 ). Tha is, d(X 1 , X 2 ) = min { d a (X 1 , X 2 ) , d b (X 1 , X 2 ) } , whe e his ex- p ession is ob ained om Lemma 7 by eplacing P and Q by X 1 and X 2 , espec i ely. Mo e specifically, i x = l(u p , X 1 ) and y = l(u q , X 2 ) , we ha e d a (X 1 , X 2 ) = x + d(u p , u q ) + y, and d b (X 1 , X 2 ) = l(u p , w p ) −x + d(w p , w q ) + l(u q , w q ) −y. I he e is no con usion we will iden i y X 1 wi h x and X 2 wi h y . The e o e, he a el dis ance h + ij (X 1 , X 2 ) o a el pa h ( A i , X 1 , X 2 , A j ) can be exp essed as h + ij (X 1 , X 2 ) = || A i −X 1 || + αmin { d a (X 1 , X 2 ) , d b (X 1 , X 2 ) } + || X 2 −A j || and consequen ly, h + ij (X 1 , X 2 ) = min { g + ,a ij (X 1 , X 2 ) , g + ,b ij (X 1 , X 2 ) } , whe e g + ,a ij (X 1 , X 2 ) = || A i −X 1 || + αd a (X 1 , X 2 ) + || X 2 −A j || g + ,b ij (X 1 , X 2 ) = || A i −X 1 || + αd b (X 1 , X 2 ) + || X 2 −A j || . Since d(X 2 , X 1 ) = d(X 1 , X 2 ) , in a simila manne we can define he a el dis ance h − ij (X 1 , X 2 ) o a el pa h ( A i , X 2 , X 1 , A j ): h − ij (X 1 , X 2 ) = min { g −,a ij (X 1 , X 2 ) , g −,b ij (X 1 , X 2 ) } , wi h g −, ij (X 1 , X 2 ) = || A i −X 2 || + αd (X 1 , X 2 ) + || X 1 −A j || , o ∈ { a , b }. Fo example, when X 1 ∈ L 6 , X 2 ∈ L 2 , Fig. 5 (le ) shows he unc ion h + ij (X 1 , X 2 ) on he ne wo k o Fig. 4 o α= 0 . 3 , A i (2.5, 6), A j (1 , −4) . No e he pagoda oo -shape o he su ace. Defini ion 8 (Suble el Se s) . Le { L p , L q } be wo linea a c seg- men s an ipodal o each o he such ha ( X 1 , X 2 ) ∈ L p ×L q . Fo η≥0, le us conside : 1. The ( η)-suble el se H + ij (η) o unc ion h + ij (X 1 , X 2 ) , gi en by: H + ij (η) = { (X 1 , X 2 ) ∈ L p ×L q : h + ij (X 1 , X 2 ) ≤η} . Analogously, H − ij (η) = { (X 1 , X 2 ) ∈ L p ×L q : h − ij (X 1 , X 2 ) ≤η} . 2. Fo ∈ { a , b }, he ( η)-suble el se s G + , ij (η) and G −, ij (η) , o unc ions g + , ij (X 1 , X 2 ) and g −, ij (X 1 , X 2 ) , espec i ely, gi en by: G + , ij (η) = { (X 1 , X 2 ) ∈ L p ×L q : g + , ij (X 1 , X 2 ) ≤η} G −, ij (η) = { (X 1 , X 2 ) ∈ L p ×L q : g −, ij (X 1 , X 2 ) ≤η} . Fo se e al η- alues, Fig. 5 ( igh ) displays he co esponding suble el se s H + ij (η) o he su ace ep esen ed on he le -hand side o he figu e. Fo ∈ { a , b }, he unc ion g + , ij (X 1 , X 2 ) is he sum o he con ex e m || A i −X 1 || 2 + || A j −X 2 || 2 and he linea unc ion αd ( X 1 , X 2 ). Simila ly, g −, ij (X 2 , X 1 ) is also he sum o a con ex and a linea e m. The e o e, he con exi y o he se s G + , ij (η) and G −, ij (η) , o ∈ { a , b }, is a s aigh o wa d consequence (see Boyd & Vande be ghe, 2004 ). F om hese defini ions, and aking in o accoun ha i h = min { , g} , he le el se o h is he union o he le el se s o and g , we ha e: Lemma 9. H + ij (η) = G + ,a ij (η) ∪ G + ,b ij (η) , and H − ij (η) = G −,a ij (η) ∪ G −,b ij (η) Consequen ly, bo h H + ij (η) and H − ij (η) a e he union o wo con- ex (bu possibili y no disjoin ) se s. The ollowing example shows ha G + ,a ij (η) ∩ G + ,b ij (η)  = ∅ , o some η- alues, wi h η< || A i −A j || . In Fig. 4 , we ha e || A i −A j || =√ 409 2 ≃ 10 . 11 . Fig. 6 (a) displays, o α= 0 . 4 and η= 10 , he bounda ies o he suble el se s G + ,a ij (η) and G + ,b ij (η) . I can be obse ed ha he in e sec ion o bo h se s is no emp y. P oposi ion 10. Fo any 0 ≤η< || A i −A j || , H+ ij (η) ∩ H − ij (η) = ∅ , i.e., i he e exis s a pa h om A i o A j wi h leng h sho e han he Euclidean (plana ) a el dis ance, hen he o de in which X 1 and X 2 a e passed is uniquely de e mined. ab c Fig. 6. (a): Fo  d ij = 10 , cu es G + ,a ij , G + ,b ij , and se Q + (i, j) = G + ,a ij ∩ G + ,b ij . (b): Se P((i, j) , (k, )) , o  d k = 10 . 5 . (c): Ve ex (X  1 , X  2 ) in he bounda y o a egion R . P oo . Le 0 < η<  A i −A j  and assume H + ij (η) ∩ H − ij (η)  = ∅ . Then he e exis s ( X 1 , X 2 ) such ha  A i −X 1  + αd(X 1 , X 2 ) +  A j −X 2  ≤η  A i −X 2  + αd(X 1 , X 2 ) +  A j −X 1  ≤η. Summing up and using ha d ( X 1 , X 2 ) ≥0 we ob ain ha  A i −X 1  +  A j −X 1  +  A i −X 2  +  A j −X 2  ≤2 η. Due o he iangle inequali y, we ha e ha  A i −A j  ≤ A i − X k  +  A j −X k  o k = 1 , 2 . This gi es us 2  A i −A j  ≤ A i −X 1  +  A j −X 1  +  A i −X 2  +  A j −X 2  ≤2 η, a con adic ion o he assump ion ha η<  A i −A j  .  Le { L p , L q } be an ipodal o each o he . Fo a gi en poin ( X 1 , X 2 ) ∈ L p ×L q , he objec i e alue F (X 1 , X 2 ) =  (i,j) ∈ C(X 1 ,X 2 ) ij quan i- fies he amoun o weigh s o OD-pai s cap u ed by such a poin . Taking in o accoun Defini ion 1 , le H ij be he se o poin s which co e he OD-pai ( i , j ), gi en by H ij = { (X 1 , X 2 ) ∈ L p ×L q : ij (X 1 , X 2 ) ≤ d ij } . The ollowing esul ollows di ec ly om he defini ion Co olla y 11. ( X 1 , X 2 ) ∈ H ij i and only i ( i , j ) ∈ C ( X 1 , X 2 ) . In o de o ob ain H ij om he abo e defined suble el se s, we eplace he η- alues by he specific accep ance le el  d ij associa ed wi h each OD-pai ( i , j ). Rema k 12 (No a ion) . Gi en an accep ance le el 0 ≤ d ij < || A i − A j || 2 , o ∈ { a , b } le us deno e G + , ij (  d ij ) and G −, ij (  d ij ) by G + , ij and G −, ij , espec i ely. This abb e ia ed no a ion is also applied o he se s H + ij (  d ij ) and H − ij (  d ij ) , which will be iden ified by H + ij and H − ij, espec i ely. Co olla y 13. Fo each OD-pai ( i , j ), H ij = H+ ij ∪ H − ij = G + ,a ij ∪ G + ,b ij ∪ G −,a ij ∪ G −,b ij . P oo . Since ij (X 1 , X 2 ) = min { h+ ij (X 1 , X 2 ) , h− ij (X 1 , X 2 ) } = ji (X 1 , X 2 ) , he esul ollows om Lemma 9 .  5.1.2. Iden i ying a fini e domina ing se We now b iefly commen on he ole o H ij in he cons uc- ion o a Fini e Domina ing Se (FDS). Recall ha C(X 1 , X 2 ) = { (i, j) , . . . , (k, ) } is he se o OD-pai s co e ed by ( X 1 , X 2 ). Co olla y 11 implies ha (X 1 , X 2 ) ∈ H ij ∩ . . . ∩ H k . Fo his eason, we nex analyze he in e sec ions o hese se s, since such in e - sec ions will p o ide he poin s o be inco po a ed in he FDS, as we will see in he ollowing. In ac , due o he absence o con ex- i y o bo h H + ij and H − ij, we will ocus he e o on selec ing poin s om he bounda ies o all hese se s (as well as om hei in e - sec ions), in o de o gua an ee ha he selec ed poin s belong o all se s in ol ed in he in e sec ion. Hence o h we will use he no a ion G (o H ), o iden i y he bounda y o le el cu e o a se G (o H ). Tha is, o ∈ { a , b }, we ha e G + , ij = { (X 1 , X 2 ) ∈ L p ×L q : g + , ij (X 1 , X 2 ) =  d ij } H + ij = { (X 1 , X 2 ) ∈ L p , L q : h+ ij (X 1 , X 2 ) =  d ij } , and analogously o G −, ij and H − ij . Clea ly, o τ∈ { + , −} , H τ ij ⊆ G τ,a ij ∪ G τ,b ij , and H τ ij = G τ,a ij ∪ G τ,b ij i he se G τ,a ij ∩ G τ,b ij con ains a mos one poin . Likewise H ij = { (X 1 , X 2 ) ∈ L p ×L q : ij (X 1 , X 2 ) =  d ij } . Since om P oposi ion 10 , H + ij ∩ H − ij = ∅ , we i ially conclude Co olla y 14. H ij = H + ij ∪ H − ij . As we ha e al eady seen, in o de o cons uc he FDS o p ob- lem (2-TCR) (1), we will successi ely selec se e al easible poin s om hese le el cu es and hei in e sec ions. Lemma 15. Gi en wo di e en OD-pai s ( i , j ), ( k , ), le P((i, j) , (k, )) ⊂L p ×L q be he se o in e sec ion poin s defined as ollows: P((i, j) , (k, )) =  ( G τ, ij ∩ G τ ,  k ) , τ, τ ∈ { + , −} , ,  ∈ { a, b}. Then, H ij ∩ H k ⊆P((i, j) , (k, )) . P oo . H ij = H + ij ∪ H − ij ⊆( G + ,a ij ∪ G + ,b ij ) ∪ ( G −,a ij ∪ G −,b ij ) , and a simi- la inclusion can be ob ained o H k . The e o e we can w i e H ij ∩ H k ⊆G + ,a ij ∪ G + ,b ij ∪ G −,a ij ∪ G −,b ij   G + ,a k ∪ G + ,b k ∪ G −,a k ∪ G −,b k . Clea ly, P((i, j) , (k, )) is he igh -hand side o his exp ession since P((i, j) , (k, )) is ob ained by in e sec ing each se o he fi s g oup wi h each se o he second g oup. This concludes he p oo .  Defini ion 16. Fo each OD-pai ( i , j ), le Q (i, j) = Q + (i, j) ∪ Q −(i, j) be a se o easible poin s o H ij = H + ij ∪ H − ij , whe e o each τ∈ { + , −} : Q τ(i, j) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ G τ,a ij ∩ G τ,b ij ,i his in e sec ion con ains a leas one easible poin . { X a , X b } , o he wise, whe e he poin s X a and X b a e a bi a ily selec ed om G τ,a ij , G τ,b ij , espec i ely. Clea ly Q τ(i, j) ⊂H τ ij , o τ∈ { + , −} . Fig. 6 (a) and (b) shows some o hese se s. Fig. 6 is based on he ne wo k o Fig. 4 , wi h α= 0 . 4 . We ha e conside ed he poin s A i (2.5, 6), A j (1 , −4) , A k (−2 , −4 . 5) and A (3, 5.5). Fo  d ij = 10 and  d k = 10 . 5 , Fig. 6 (a) shows he cu es G + ,a ij and G + ,b ij , as well as he co esponding se Q (i, j) = G + ,a ij ∩ G + ,b ij con aining wo in e sec ion poin s (in his case H − ij = ∅ ). Likewise, Fig. 6 (b) displays he se o in e sec ion poin s P((i, j) , (k, )) con- aining h ee poin s. In his figu e only a b anch o he cu e G −,a k is inside he easible domain. Theo em 17. Fo he conca e es ic ed p oblem (2-TCR) (1), le Q and Pbe he se s defined as ollows Q =  (i,j) Q (i, j) , P =  (i,j)  =(k, ) P((i, j) , (k, )) Then, o any a bi a y poin X pq ∈ L p ×L q , he se = Q ∪ P ∪ { X pq } is an FDS o he p oblem (2-TCR) (1) P oo . Le G =  (i,j) { G τ, ij , τ∈ { + , −} , ∈ { a, b}} ⊂L p ×L q be he collec ion o all le el cu es o he es ic ed p oblem (2-TCR) (1). F om bo h Defini ion 8 and subsequen esul s, he collec ion Gin- duces a pa i ion o he easible domain L p ×L q in o a se o egions { R s , s ∈ I } (whe e I is an index se ), such ha bo h F ( X 1 , X 2 ) and C ( X 1 , X 2 ) a e cons an in he in e io poin s o each egion (see Fig. 6 (c)). The possible changes in he objec i e unc ion may only occu a he poin s on he bounda y o each egion. The e- o e an FDS o p oblem (2-TCR) (1) can be cons uc ed by selec ing poin s om he se o bounda ies { R s , s ∈ I} . Gi en a egion R ∈ , we analyze he cases R = L p ×L q and R ⊂L p ×L q . 1. I R = L p ×L q , hen G = ∅ . Hence, F ( X 1 , X 2 ), C ( X 1 , X 2 ) a e cons an , o all ( X 1 , X 2 ) ∈ L p ×L q . Equi alen ly, any poin X pq = (X 1 , X 2 ) ∈ L p ×L q is op imal. 2. O he wise, R ⊂L p ×L q , and i s bounda y R con ains a se o edges: pieces o le el cu es, and possibly some e ex which is a poin sha ed by (a leas ) wo di e en le el cu es ( Fig. 6 (c)). All poin s o an edge ob ained om he le el cu e G τ, ij ∈ Gco e he OD-pai ( i , j ). The e o e, gi en a se o edges inciden o a e ex, such a e ex co e s all OD-pai s associa ed wi h hese edges. In o he wo ds: i (X  1 , X  2 ) is a e ex, hen F (X  1 , X  2 ) ≥F (X 1 , X 2 ) , o any poin ( X 1 , X 2 ) o each edge inciden o (X  1 , X  2 ) . F om his a gu- men , all e ices o he pa i ion a e selec ed o be in- co po a ed o he FDS. Addi ionally, i he e is some R s wi h none e ex, hen R s con ains a single le el cu e, in which case an a bi a y poin o such a cu e is also added o he FDS. In he ollowing we p o e ha he FDS hus cons uc ed is he se . In e ec , a e ex (X  1 , X  2 ) o R is he in e sec ion poin o (a leas ) wo le el cu es, and hese cu es can be ob ained ei he om a single OD-pai ( i , j ) o om (a leas ) wo OD-pai s ( i , j ), ( k , ). (a) In he fi s case, Lemma 9 and P oposi ion 10 imply ha (X  1 , X  2 ) ∈ G τ,a ij ∩ G τ,b ij . F om Defini ion 16 , his in- e sec ion is he se Q τ(i, j) . (b) Le assume wo edges associa ed wi h he OD-pai s ( i , j ) and ( k , ) a e inciden o (X  1 , X  2 ) . This means ha (X  1 , X  2 ) is ob ained om he in e sec ion G τ, ij ∩ G τ , k , o some τ, τ ∈ { + , −} and ,  ∈ { a , b }, which implies ha i belongs o he se P((i, j) , (k, )) . S eps (a) y (b) a e epea ed wi h each e ex o he pa i ion . A he end o his p ocess, all e ices o he case (b) a e he se P. On he o he hand, an a bi a y poin is selec ed om each R s wi hou e ices: hese a bi a y poin s, o- ge he wi h all e ices o he case (a), a e he se Q . Finally, he a bi a y poin X pq may be selec ed as one o poin s p e- iously selec ed (in ac , X pq is necessa y only i G = ∅ ). This concludes he p oo .  5.2. The con ex case This p oblem, iden ified by (2-TCR) (2), is o mula ed by adding he assump ion ha { L p , L q } a e no an ipodal o each o he o p ob- lem (2-TCR) . In his case, when ( X 1 , X 2 ) ∈ L p ×L q , om Lemma 7 and he subsequen easoning, he dis ance d ( X 1 , X 2 ) is con ex. Mo e specifically, we ha e d a ((X 1 , X 2 ) = d b (X 1 , X 2 ) = d(X 1 , X 2 ) . Consequen ly, all esul s o he p e ious sec ion a e alid o (2- TCR) (2) aking in o accoun ha we now ha e: H + ij = G + ,a ij = G + ,b ij , and H − ij = G −,a ij = G −,b ij . The e o e, om P oposi ion 10 and Co olla y 13 , bo h H + ij and H − ij a e con ex and disjoin se s such ha H ij = H+ ij ∪ H − ij. Likewise, o his p oblem he se Q (i, j) o Defini ion 16 becomes he se Q (i, j) = { X + , X −} , whe e X + = (X + 1 , X + 2 ) and X −= (X − 1 , X − 2 ) a e poin s a bi a ily selec ed om H + ij and H − ij , espec i ely. And o each wo di e en OD-pai s ( i , j ), ( k , ), he se P((i, j) , (k, )) o Lemma 15 is now gi en by P((i, j) , (k, )) =  (i,j)  =(k, ) ( H ij ∩ H k ) = { H + ij ∩ H + k } ∪ { H + ij ∩ H − k } ∪{ H − ij ∩ H + k } ∪ { H − ij ∩ H − k } . Wi h hese se s, Theo em 17 emains alid and es ablishes ha = Q ∪ P ∪ { X pq } is a FDS o his p oblem. Finally, since C V is a cons an se , i ollows s aigh o wa dly: Co olla y 18. Le us conside ha he e exis s a se o s a ions loca ed on he nodes o he ne wo k. Then, is also a FDS o he p ob- lem in which he se C V is excluded: max (X 1 ,X 2 ) ∈ L p ×L q F (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) C V ij . 5.3. Example This example, based on he ne wo k o Fig. 4 , illus a es he abo e p ocedu e. The exis ing demand is loca ed a he poin s: A 1 (2.5, 6), A 2 (3, 5.5), A 3 (5 . 5 , √ 6 / 10) , A 4 (2 , −4 . 5) , and A 5 (1 , −4) . The pa i- ion o he ne wo k o igina ed by he se o linea a c segmen s L = { L p , p = 1 , . . . , 8 } p o ides 36 di e en es ic ed p oblems on he pai s {{ L p , L q }, p = 1 , . . . , 7 , q = p + 1 , . . . , 8 } ∪ {{ L p , L p } , p = 1 , . . . , 8 } , ( he pai s { L p , L q } and { L q , L p } gi e ise o symme ic e- s ic ed p oblems wi h he same solu ion). Among all hese pai s, { L 2 , L 6 }, { L 1 , L 5 } and { L 3 , L 7 } a e an ipodal o each o he , and he emaining pai s a e no . Fig. 7. Le : Collec ion Gand pa i ion  o he conca e es ic ed p oblem on { L 2 , L 6 }, and poin s o he FDS . Righ : FDS o he subp oblem (1) on { L 2 , V ( L 6 )}. We conside α= 0 . 4 . The ma ices T = ( ij ) and  D = (  d ij ) con- aining he weigh s and he accep ance le els o all OD-pai s, e- spec i ely, a e gi en by T = ⎛ ⎜ ⎜ ⎝ 0 30 15 28 14 16 0 32 45 23 40 25 0 20 20 28 45 26 0 30 14 20 23 30 0 ⎞ ⎟ ⎟ ⎠  D =⎛ ⎜ ⎜ ⎝ 0 0 . 5 6 10 . 2 10 0 . 5 0 5 9 . 2 8 . 5 6 5 0 5 . 5 5 10 . 2 9 . 2 5 . 5 0 1 10 8 . 5 5 1 0 ⎞ ⎟ ⎟ ⎠ Wi h his scena io we fi s ha e conside ed he conca e es ic ed p oblem: max (X 1 ,X 2 ) ∈ L 2 ×L 6 F (X 1 , X 2 ) :=  (i,j) ∈ C(X 1 ,X 2 ) ij No e ha o ( X 1 , X 2 ) ∈ L 2 ×L 6 , X 1 = (x 1 , 0) ∈ L 2 and X 2 = (x 2 , 2 √ 6 ) ∈ L 6 , he e o e each poin ( X 1 , X 2 ) ∈ is iden ified as x = (x 1 , x 2 ) . Wi h his no a ion, x k = (x k 1 , x k 2 ) deno es he k h local solu ion (X k 1 , X k 2 ) o his subp oblem (i any). Fig. 7 (le ) shows he FDS , cons uc ed om all e ices o he pa i ion . The local solu ion o his subp oblem is eached a bo h x 1 = (4 . 556013 , 2 . 90379) and x 2 = (4 . 776812 , 2 . 950328) , wi h x 1 ∈ P((2 , 3) , (2 , 4)) and x 2 ∈ P((1 , 5) , (2 , 4)) . Fo hese poin s, he pai s co e ed and he objec i e alue a e (1, 3), (1, 4), (1, 5), (2, 3), (2, 4) and 286, espec i ely. The ollowing objec i e alues (in dec easing o de ) a e ob- ained om he con ex es ic ed p oblems on { L 3 , L 6 } and { L 1 , L 6 }. Le F ∗ p,q be he op imum objec i e alue o he es ic ed p ob- lem (2-TCR) . Table 1 summa izes he esul s o he abo e h ee subp oblems. The local solu ions o he emaining es ic ed p ob- lems p o ide subs an ially lowe objec i e alues, and hey ha e no been included in his able. The solu ions ob ained om he es ic ed p oblem on { L 2 , L 6 } a e also global solu ions o he 2- ans e p oblem on he o e - Table 1 Bes op imum alues o he 2- ans e p oblem. Res ic ed p oblem Local solu ions C ( X 1 , X 2 )F ∗ p,q { L 2 , L 6 }x 1 = (4 . 556013 , 2 . 90379) (1, 3), (1, 4), (1, 5), (2, 3), (2, 4) 286 x 2 = (4 . 776812 , 2 . 950328) { L 3 , L 6 }x 3 = (5 . 1 , 3 . 075407) (1, 3), (1, 4), (2, 3), (2, 4) 258 { L 1 , L 6 }x 4 = (0 , 2 . 2126927) (1,4), (1,5), (2,4), (2,5) 217 x 5 = (0 , 3 . 1439062) all ne wo k. The global op imum is F ∗= F ∗ 2 , 6 = F (X ∗ 1 , X ∗ 2 ) = 286 , o (X ∗ 1 , X ∗ 2 ) ∈ { (X k 1 , X k 2 ) , k = 1 , 2 } . 6. The case o loca ing a single ans e poin The abo e p ocedu e also sol es he case o loca ing one ans- e poin . In e ec , as we ha e a gued a he beginning o his pape , he o mula ion o he 1- ans e addi ional co e ing p ob- lem (1-TAC) is made unde he hypo hesis ha all nodes o V (o a subse o hem) a e al eady loca ed ans e poin s. F om his as- sump ion, p oblem (1-TAC) can also be sol ed by adap ing he p o- cedu e desc ibed o he 2- ans e es ic ed p oblem (2-TCR) o a se o subp oblems dealing wi h he es ic ion o p oblem (1- TAC) o each linea a c segmen L p ∈ L . The 1- ans e p oblem (1-TAC) can be e o mula ed as ol- lows: max L p ∈L max X∈ L p F V (X ) :=  (i,j) ∈ C(X) C V ij =  (i,j) ∈ X 2 ∈ V C(X,X 2 )  C V ij whe e we ha e applied ha C(X) =  X 2 ∈ V C(X, X 2 ) . The es ic ion o (1-TAC) o L p ∈ L is: max X∈ L p F V (X ) :=  (i,j) ∈ X 2 ∈ V C(X,X 2 )  C V ij (1-TACR)