Algebraic Solutions to Multidimensional Minimax Location Problems with Chebyshev Distance

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1 Recent Researches n Appled and Computatonal Mathematcs Algebrac Solutons to Multdmensonal Mnmax Locaton Problems wth Chebyshev Dstance NIKOLAI KRIVULIN Faculty of Mathematcs and Mechancs St Petersburg State Unversty 28 Unverstetsky Ave, St Petersburg, RUSSIA nkk@mathspburu Abstract: Multdmensonal mnmax sngle faclty locaton problems wth Chebyshev dstance are examned wthn the framework of dempotent algebra A new algebrac soluton based on an extremal property of the egenvalues of rreducble matrces s gven The soluton reduces both unconstraned and constraned locaton problems to evaluaton of the egenvalue and egenvectors of an approprate matrx Key-Words: Sngle faclty locaton problem, Chebyshev dstance, Idempotent semfeld, Egenvalue, Egenvector 1 Introducton Locaton problems form one of the classcal research domans n optmzaton that has ts orgn datng back to XVIIth century Over many years a large body of research on ths topc contrbuted to the development n varous areas ncludng nteger programmng, combnatoral and graph optmzaton Among other soluton approaches to locaton problems are models and methods of dempotent algebra [1 6], whch fnd expandng applcatons n the analyss of actual problems n engneerng, manufacturng, nformaton technology, and other felds Specfcally, an algebrac soluton to a one-dmensonal locaton problem on a graph s proposed n [2, 7] A constraned locaton problem and ts representaton n terms of dempotent algebra are examned n [8, 9] In ths paper, we consder a multdmensonal mnmax sngle faclty locaton problem wth Chebyshev dstance, and show how the problem can be solved based on new results of the spectral theory n dempotent algebra The am of the paper s twofold: frst, to gve a new algebrac soluton to the locaton problem, and second, to extend the area of applcaton of dempotent algebra The rest of the paper s as follows We begn wth an overvew of prelmnary algebrac defntons and results Specfcally, we present an extremal property of the egenvalue of rreducble matrces Furthermore, we examne an unconstraned mnmax locaton problem and represent t n the terms of dempotent algebra A new soluton s gven that reduces the problem to evaluaton of the egenvalue and egenvectors of an rreducble matrx Fnally, the soluton s extended to solve a constraned locaton problem 2 Prelmnary Results We start wth a bref overvew of defntons, notaton and prelmnary results of dempotent algebra that underle the soluton approach developed n subsequent sectons Further detals can be found n [1 6] 21 Idempotent Semfeld Let X be a set wth two operatons, addton and multplcaton, and ther respectve neutral elements, 0 and 1 We suppose that X, 0, 1,, s a commutatve semrng where the addton s dempotent and the The work was partally supported by Russan Foundaton for Basc Research Grant ISBN:

2 Recent Researches n Appled and Computatonal Mathematcs multplcaton s nvertble Snce the nonzero elements of the semrng form a group under multplcaton, the semrng s usually referred to as dempotent semfeld The nteger power s defned n the ordnary way Let us put X + = X \ {0} For any x X + and nteger p > 0, we have x 0 = 1, x p = x p 1 x = x x p 1, x p = x 1 p, and 0 p = 0 We assume that the nteger power can naturally be extended to the case of ratonal and real exponents In what follows, we drop, as s customary, the multplcaton sgn The power notaton s used n the sense of dempotent algebra The dempotent addton allows one to defne a relaton of partal order such that x y f and only f x y = y From the defnton t follows that x x y and y x y, as well as that the addton and multplcaton are both sotonc Below the relaton symbols are thought of as referrng to ths partal order It s easy to verfy that the bnomal dentty now takes the form x y α = x α y α for all α 0 As an example, one can consder the dempotent semfeld of real numbers R max,+ = R { },, 0, max, + In the semfeld R max,+, there are the null and dentty elements defned as 0 = and 1 = 0 For each x R, there exsts ts nverse x 1 equal to the opposte number x n conventonal arthmetc For any x, y R, the power x y concdes wth the arthmetc product xy 22 Vectors and Matrces Vector and matrx operatons are routnely ntroduced based on the scalar addton and multplcaton defned on X Consder the Cartesan power X n wth ts elements represented as column vectors For any two vectors x = x and y = y, and a scalar c X, vector addton and multplcaton by scalars follow the rules {x y} = x y, {cx} = cx The set X n wth these operatons s a vector semmodule over the dempotent semfeld X As t usually s, a vector y X n s lnearly dependent on vectors x 1,, x m X n, f there are scalars c 1,, c m X such that y = c 1 x 1 c m x m In partcular, the vector y s collnear wth x, f y = cx For any column vector x = x X n +, we defne a row vector x = x wth ts elements x = x 1 For all x, y X n +, the componentwse nequalty x y mples x y For any conformng matrces A = a j, B = b j, and C = c j, matrx addton and multplcaton together wth multplcaton by a scalar c X are performed accordng to the formulas {A B} j = a j b j, {BC} j = k b k c kj, {ca} j = ca j A matrx wth all zero entres s referred to as zero matrx and denoted by 0 Consder the set of square matrces X n n The matrx that has all dagonal entres equal to 1 and off-dagonal entres equal to 0 s the dentty matrx denoted by I Wth respect to matrx addton and multplcaton, the set X n n forms dempotent semrng wth dentty For any matrx A 0 and an nteger p > 0, we have A 0 = I and A p = A p 1 A = AA p 1 The trace of the matrx A = a j s defned as tr A = a 11 a nn A matrx s rreducble f and only f t cannot be put n a block trangular form by smultaneous permutatons of rows and columns Otherwse the matrx s reducble 23 Egenvalues and Egenvectors of Matrces A scalar λ s an egenvalue of a square matrx A X n n f there exsts a vector x X n \ {0} such that Ax = λx Any vector x 0 that satsfes the equaton s an egenvector of A, correspondng to λ If the matrx A s rreducble, then t has only one egenvalue gven by λ = tr 1/m A m 1 m=1 ISBN:

3 Recent Researches n Appled and Computatonal Mathematcs The correspondng egenvectors of A are found as follows Frst we evaluate the matrx A = λ 1 A λ 1 A n Let a and a be respectve column and dagonal entry, of A Consder the subset of columns a such that a = 1, = 1,, n In the subset, fnd those columns that are lnearly ndependent of the others, and take them to form a matrx A + The set of all egenvectors of A correspondng to λ together wth zero vector concdes wth the lnear span of the columns of A, whereas each vector takes the form where v s a nonzero vector of approprate sze x = A + v, 3 An Extremal Property of Egenvalues Suppose A X n n s an rreducble matrx wth an egenvalue λ Consder a functon ϕx = x Ax defned on X n + It has been shown n [10, 11] that ϕx has λ as ts mnmum, whch s attaned at any egenvector of A Now we mprove the above result by extendng the set of vectors that provde for the mnmum of ϕx Lemma 1 Let A = a j X n n be an rreducble matrx wth an egenvalue λ Suppose u = u and v = v are egenvectors of the respectve matrces A and A T Then t holds that mn x Ax = λ, x X n + where the mnmum s attaned at any vector x = u α 1 vα 1 1,, u α nv α 1 n T for all α such that 0 α 1 Proof It s easy to verfy as n [11] that any vector x wth nonzero elements satsfes the nequalty x Ax λ Indeed, snce xu x u 1 I, we have x Ax = x Axu u x Aux u 1 = λx ux u 1 = λ It remans to present a vector x that turns the nequalty nto an equalty Wth x = u we mmedately have x Ax = λu u = λ Smlarly, f x = v T, then x Ax = x T A T x T = v A T v = λv v = λ Let us now take any vector x = x wth elements x = u α vα 1, where α s a real number such that 0 α 1 Wth the followng calculatons λ = u Au α v A T v 1 α = =1 j=1 u α a α ju α j k=1 l=1 =1 j=1 u α v α 1 k a 1 α lk v 1 α l v 1 α a j u α j vj α 1 = =1 j=1 x 1 a j x j = x Ax, we arrve at the nequalty x Ax λ Snce the opposte nequalty s always vald, we conclude that the vector x satsfes the condton x Ax = λ 4 The Unconstraned Locaton Problem In ths secton we consder a mnmax sngle faclty locaton problem wth Chebyshev dstance For any two vectors r = r 1,, r n T and s = s 1,, s n T n R n, the Chebyshev dstance L or maxmum metrc s calculated as ρr, s = max r s 2 1 n Gven m 2 vectors r = r 1,, r n T R n and constants w R, = 1,, m, the problem under consderaton s to fnd a vector x = x 1,, x n T so as to provde mn max ρr, x + w 3 1 m x R n ISBN:

4 Recent Researches n Appled and Computatonal Mathematcs It s not dffcult to solve the problem by usng geometrc arguments see, eg, [12, 13] Below we gve a new algebrac soluton that s based on representaton of the problem n terms of the dempotent semfeld R max,+ and applcaton of the result from the prevous secton Frst we rewrte 2 as follows ρr, s = s r r s Denote the objectve functon of the problem by ϕx Wth the vectors we can wrte p = w 1 r 1 w m r m, q = w 1 r 1 w mr m, ϕx = and then represent problem 3 as m w ρr, x = =1 m w x r r x = x p q x, =1 mn ϕx = mn x Rn x R nx p q x 4 Furthermore, we ntroduce a vector y and a matrx A of order n + 1 as follows 1 0 q y =, A = x Snce we now have ϕx = x p q x = y Ay, problem 4 reduces to that of the form mn y R y Ay 5 n+1 Note that the vectors y R n+1 that solve 5 do not always have an approprate form to gve a solutons to 4 Specfcally, to be consstent to 4, the vector y must have the frst element equal to 1 5 Algebrac Soluton of the Unconstraned Problem Consder problem 5, and note that the matrx A s rreducble It follows from Lemma 1 that the mnmum n 5 s equal to the egenvalue λ of the matrx A For all k = 1, 2, we have 0 A 2k 1 = q p k 1 q, A 2k = q p k 1 q 0 pq, and therefore, tra 2k 1 = 0, tra 2k = q p k Fnally, applcaton of 1 gves λ = q p 1/2 To fnd vectors that produce the mnmum n 5, we need to derve the egenvectors of the matrces A and A T Note that A T s obtaned from A by replacement of p wth q T and q wth p T Therefore, t wll suffce to fnd the egenvectors for A, and then turn them nto the egenvectors for A T by the above replacement To get the egenvectors of A, we frst fnd the matrx A Snce for any k = 1, 2, t holds that λ 1 A 2k 1 = q p 1/2 0 q, λ 1 A 2k = q p 1 q 0 pq, we arrve at the matrx A n the form A = λ 1 A λ 1 A n+1 = 1 q p 1/2 q q p 1/2 p q p 1 pq It s not dffcult to verfy that n the matrx A, any column that has 1 on the dagonal s collnear wth the frst column Indeed, suppose that the submatrx q p 1 pq has a dagonal element equal to 1, say the element ISBN:

5 Recent Researches n Appled and Computatonal Mathematcs n ts frst column that corresponds to the second column of A In ths case, we have q p = q1 1 p 1, whereas the matrx A takes the form A = 1 q 1/2 1 q q 1/2 1 p q 1 p 1 1 pq = 1 q 1/2 1 q 1/2 1 p p 1 1 p where the second column proves to be collnear wth the frst Let us construct a matrx A + that ncludes such columns of A that have the dagonal element equal to 1 Snce all these columns are collnear wth the frst one, they can be omtted Wth the matrx A + formed from the frst column of A, we fnally represent any egenvector of A as u = 1 q p 1/2 p s, s R By replacng p wth q T and q wth p T, we get the egenvectors of A T y = v = 1 q p 1/2 q T It follows from Lemma 1 that the soluton of 4 takes the form u α 0 vα u α 1 vα 1 1 q p 1/2 α p α 1 q1 α 1 u α n+1 vα 1 n+1 = q p 1/2 α p α nq 1 α n t, t R sα t α 1, s, t R Wth the condton that the frst element of y must be equal to 1, we have to set s α t α 1 = 1 Gong back to problem 4, we arrve at the followng result Lemma 2 The mnmum n problem 4 s gven by λ = q p 1/2, and t s attaned at any vector x = λ 1 2α p α 1 q1 α 1 p α nq 1 α n, 0 α 1 Wth the usual notaton, we can reformulate the statement of Lemma 2 as follows Corollary 1 The mnmum n 3 s gven by λ = maxp 1 q 1,, p n q n /2, and t s attaned at any vector p 1 λ q 1 + λ x = α + 1 α, 0 α 1, p n λ q n + λ where p = maxr 1 + w 1,, r m + w m, q = mnr 1 w 1,, r m w m for each = 1,, n, 6 A Constraned Locaton Problem Suppose that there s a set S R n gven to specfy feasble locaton ponts Consder a constraned locaton problem that s represented n terms of the semfeld R max,+ n the form mn x S m w ρr, x 6 =1 ISBN:

6 Recent Researches n Appled and Computatonal Mathematcs To solve the problem we put t n the form of 4 by ncludng the area constrants nto the objectve functon of an unconstraned problem Frst we slghtly transform problem 4 to enable accommodaton of the constrants n a natural way Wth the notaton = w 1 r 1 w m r m, q 0 = w 1r 1 w mr m, λ 0 = q 0 1/2, ϕ 0 x = λ 1 0 x q 0 x, we turn to the problem mn x R ϕ 0x n It follows from Lemma 2 that the last problem has ts mnmum equal to 1 = 0, whereas ts soluton set obvously concdes wth that of 4 Suppose that there are constrants on the maxmum dstance from the faclty locaton pont to each gven ponts that determne the feasble locaton set n the form S = {x R n ρr, x d, = 1,, m} For each = 1,, m, the nequalty x r r x = ρr, x d j can be rewrtten n an equvalent form as d 1 x r d 1 x 1, whereas all constrants can be replaced wth one nequalty r We ntroduce the notaton x d 1 1 r 1 d 1 m r m d 1 1 r 1 d 1 m r mx 1 p 1 = d 1 1 r 1 d 1 m r m, q 1 = d 1 1 r 1 d 1 m r m, ϕ 1 x = x p 1 q 1 x, and note that ϕ 1 x 1 when and only when the above constrants are satsfed Furthermore, we put p = λ 1 0 p 1, q = λ 1 0 q 0 q 1, ϕx = ϕ 0x ϕ 1 x = x p q x Now we can replace the orgnal problem 6 by an unconstraned problem wth the objectve functon ϕx that has the form of problem 4 wth the soluton gven by Lemma 2 It s clear that both problems gve the same soluton set provded that a soluton exsts At the same tme, the new problem allows one to get approxmate solutons n the case when the orgnal problem does not have a soluton References: 1 F L Baccell, G Cohen, G J Olsder, J-P Quadrat, Synchronzaton and Lnearty, Wley, Chchester, R A Cunnghame-Green, Mnmax Algebra and Applcatons, Advances n Imagng and Electron Physcs, Vol 90, Academc Press, San Dego, 1994, pp V N Kolokoltsov, V P Maslov, Idempotent Analyss and ts Applcatons, Kluwer, Dordrecht, J S Golan, Semrngs and Affne Equatons Over Them: Theory and Applcatons, Sprnger, New York, B Hedergott, G J Olsder, J van der Woude, Max-Plus at Work: Modelng and Analyss of Synchronzed Systems, Prnceton Unversty Press, Prnceton, P Butkovč, Max-Lnear Systems: Theory and Algorthms, Sprnger, London, R A Cunnghame-Green, Mnmax Algebra and Applcatons, Fuzzy Sets and Systems, Vol 41, No 3, 1991, pp K Zmmermann, Dsjunctve Optmzaton, Max-Separable Problems and Extremal Algebras, Theoretcal Computer Scence, Vol 293, No 1, 2003, pp A Tharwat, K Zmmermann, One Class of Separable Optmzaton Problems: Soluton Method, Applcaton, Optmzaton, Vol 59, No 5, 2008, pp N K Krvuln, On Evaluaton of Bounds on the Mean Growth Rate of the State Vector for the Lnear Stochastc Dynamcal System n Idempotent Algebra, Vestnk St Petersburg Unversty: Mathematcs, Vol 38, No 2, 2005, pp N K Krvuln, Egenvalues and Egenvectors of Matrces n Idempotent Algebra, Vestnk St Petersburg Unversty: Mathematcs, Vol 39, No 2, 2006, pp D R Sule, Logstcs of Faclty Locaton and Allocaton, Marcel Dekker, New York, E Morad, M Bdkhor, Sngle Faclty Locaton Problem, Faclty Locaton, Physca-Verlag, Hedelberg, 2009, pp ISBN: