single variable term semi latticized fuzzy relation geometric programming with max product operator considering the practical application background in peer to peer network system we investigate the single variable term semi latticized fuzzy relation geometric programming with max product operator in this paper based on the characteristics of the objective function and feasible domain we develop a matrix approach to deal with the proposed problem a step by step algorithm is proposed to find an optimal solution without solving all the minimal solutions of the constraint two examples are given to illustrate the feasibility and efficiency of the algorithm

the more for less paradox in fuzzy posynomial geometric programming the more for less mfl problem in fuzzy posynomial geometric programming fpgp is advanced in this paper the research results presented here focus primarily on the nonconvex fpgp in both objective functions and constraint functions convexification quasiconvex or pseudoconvex is extended in the sense of an mfl paradox by consolidating the necessary and sufficient conditions since the fpgp is equivalent to fuzzy linear programming correspondingly there exists a solution to the fpgp furthermore the duality or strong duality theorem the equivalent condition of the mfl paradox and its condition under expansion are examined in detail it is well known that the fundamental understanding of problems on mfl paradox is of paramount importance to applications of resource allotments and optimal resource management and correspondingly that the information science and technology advancement play a rule to resource allotments and resource option in management problems in fact they are dependent and interwinded

strong sums of projections in von neumann factors abstract this paper presents necessary and sufficient conditions for a positive bounded operator on a separable hilbert space to be the sum of a finite or infinite collection of projections not necessarily mutually orthogonal with the sum converging in the strong operator topology if the collection is infinite a similar necessary condition is given when the operator and the projections are taken in a type ii von neumann factor and the condition is proven to be also sufficient if the operator is diagonalizable a simpler necessary and sufficient condition is given in the type iii factor case

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

operator valued frames we develop a natural generalization of vector valued frame theory we term operator valued frame theory using operator algebraic methods this extends work of the second author and d han which can be viewed as the mul tiplicity one case and extends to higher multiplicity their dilation approach we prove several results for operator valued frames concerning duality dis jointeness complementarity and composition of operator valued frames and the relationship between the two types of similarity left and right of such frames a key technical tool is the parametrization of parseval operator val ued frames in terms of a class of partial isometries in the hilbert space of the analysis operator we apply these notions to an analysis of multiframe gener ators for the action of a discrete group g on a hilbert space one of the main results of the han larson work was the parametrization of the parseval frame generators in terms of the unitary operators in the von neumann algebra gen erated by the group representation and the resulting norm path connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von neumann algebra in this paper we general ize this multiplicity one result to operator valued frames however both the parameterization and the proof of norm path connectedness turn out to be necessarily more complicated and this is at least in part the rationale for this paper our parameterization involves a class of partial isometries of a different von neumann algebra these partial isometries are not path connected in the norm topology but only in the strong operator topology we prove that the set of operator frame generators is norm pathwise connected precisely when the von neumann algebra generated by the right representation of the group has no minimal projections as in the multiplicity one theory there are analogous results for general non parseval frames

geometric programming with a single term exponent subject to bipolar max product fuzzy relation equation constraints abstract we study geometric programming with a single term exponent subject to bipolar max product fuzzy relation equation constraints in the area of economics and the covering problem the structure of its feasible domain is characterized and the lower and upper bound vectors on its solution set are determined it is shown that each component of one of its optimal solutions is the corresponding component of either the lower bound or the upper bound vector this interesting property helps us to create a value matrix and present some necessary and sufficient conditions for its consistency checking moreover some sufficient conditions are proposed to detect one of its optimal solutions without solving the problem a modified branch and bound method is extended to solve the problem in a general case with use of the value matrix an efficient algorithm is then designed to solve the problem using the sufficient conditions and the modified branch and bound method its computational complexity is also analyzed finally some examples are provided to illustrate its importance and the steps of the algorithm

asymptotics of eigenvalues of large symmetric toeplitz matrices with smooth simple loop symbols abstract this paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric in general non hermitian toeplitz matrix the symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane the main result describes the asymptotic structure of all eigenvalues the asymptotic expansions constructed are uniform with respect to the location of the eigenvalues in the bulk of the spectrum

solutions of fuzzy relation equations based on continuous t norms this study is concerned with fuzzy relation equations with continuous t norms in the form atr b where a and b are the fuzzy subsets of x and y respectively email protected xxy is a fuzzy relation and t is a continuous t norm the problem is how to determine a from r and b first an if and only if condition of being solvable is presented novel algorithms are then presented for determining minimal solutions when x and y are finite the proposed algorithms generate all minimal solutions for the equations making them efficient solving procedures

optimal reconstruction systems for erasures and for the q potential abstract in this paper we introduce the q potential as an extension of the benedetto fickus frame potential defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions we extend recent results of b g bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst normalized reconstruction error is computed with respect to a compatible unitarily invariant norm we finally describe necessary and sufficient spectral conditions that we call q fundamental inequalities for the existence of protocols with prescribed properties by relating this problem to klyachko s and fulton s theory on sums of hermitian operators

deriving minimal solutions for fuzzy relation equations with max product composition this work considers fuzzy relation equations with max product composition the critical problem in solving such equations is to determine the minimal solutions when an equation is solvable however this problem is np hard and difficult to solve a v markovskii on the relation between equations with max product composition and the covering problem fuzzy sets and systems 153 2005 261 273 this work first examines the attributes of a solvable equation and characteristics of minimal solutions then reduces the equation to an irreducible form and converts the problem into a covering problem for which minimal solutions are correspondingly determined furthermore for theoretical and practical applications this work presents a novel method for obtaining minimal solutions the proposed method easily derives a minimal solution and obtains other minimal solutions from this predecessor using a back tracking step the proposed method is compared with an existing algorithm and some applications are described in detail

solving a linear programming problem with the convex combination of the max min and the max average fuzzy relation equations abstract in this paper we introduce a fuzzy operator constructed by the convex combination of two known operators max min and max average compositions h j zimmermann fuzzy set theory and it s application kluwer academic publishers boston dordrecht london 1999 this operator contains some properties of the two known compositions when it generates the feasible region for linear optimization problems we investigate linear optimization problems whose feasible region is the fuzzy sets defined with this operator thus firstly the structure of these fuzzy regions is considered and then a method to solve the linear optimization problems with fuzzy equation constraints regarding this operator is presented

linear optimization problem constrained by fuzzy max min relation equations fang and li introduced the optimization model with a linear objective function and constrained by fuzzy max min relation equations they converted this problem into a 0 1 integer programming problem and solved it using the jump tracking branch and bound method subsequently wu et al improved this method by providing an upper bound on the optimal objective value and presented three rules for simplifying the computation of an optimal solution this work presents new theoretical results concerning this optimization problem they include an improved upper bound on the optimal objective value improved rules for simplifying the problem and a rule for reducing the solution tree accordingly an accelerated approach for finding the optimal objective value is presented and represents an improvement on earlier approaches its potential applications are discussed

eigenvalues of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper presents higher order asymptotic formulas for the eigenvalues of large hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the formulas are established not only for the extreme eigenvalues but also for the inner eigenvalues the results extend and make more precise existing results which so far pertain to banded matrices or to matrices with infinitely differentiable symbols also given is a fixed point equation for the eigenvalues which may be solved numerically by an iteration method

a contractive version of a schur horn theorem in ii1 factors we prove a contractive version of the schur horn theorem for submajorization in ii1 factors that complements some previous results on the schur horn theorem within this context we obtain a reformulation of a conjecture of arveson and kadison regarding a strong version of the schur horn theorem in ii1 factors in terms of submajorization and contractive orbits of positive operators

resolution of fuzzy relational equations method algorithm and software with applications analytical methods and algorithms for inverse problem resolution of fuzzy linear systems of equations in some bl algebras godel algebra in case of max min and min max compositions and goguen algebra in case of max product composition are presented algorithms with software realization for solving fuzzy linear systems of equations are proposed applications include fuzzy optimization with fuzzy linear systems of equation constraint fuzzy machines and covering problem

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

pinchings and positive linear maps abstract we employ the pinching theorem ensuring that some operators a admit any sequence of contractions as an operator diagonal of a to deduce improve two recent theorems of kennedy skoufranis and loreaux weiss for conditional expectations onto a masa in the algebra of operators on a hilbert space similarly we obtain a proof of a theorem of akeman and anderson showing that positive contractions in a continuous masa can be lifted to a projection we also discuss a few corollaries for sums of two operators in the same unitary orbit

minimizing a linear function under a fuzzy max min relational equation constraint in this paper we investigate the problem of minimizing a linear objective function subject to a fuzzy relational equation constraint a necessary condition for optimal solution is proposed based on this necessary condition we propose three rules to simplify the work of computing an optimal solution numerical examples are provided to illustrate the procedure experimental results are reported showing that our new procedure systematically outperforms our previous work

an algorithm for solving optimization problems with fuzzy relational inequality constraints an algorithm for solving a kind of optimization problems with fuzzy relational inequalities was proposed by guo and xia 6 however it is too expensive to verify the optimal condition in this paper some rules for reducing these problems are proposed and the relationship between minimal solutions and fri paths is also given these lead to a new algorithm for solving this kind of problems numerical experiments are presented for illustrating the efficiency of the proposed algorithm

diagonals of normal operators with finite spectrum let x lambda1 lambdan be a finite set of complex numbers and let a be a normal operator with spectrum x that acts on a separable hilbert space h relative to a fixed orthonormal basis e1 e2 for h a gives rise to a matrix whose diagonal is a sequence d d1 d2 with the property that each of its terms dn belongs to the convex hull of x not all sequences with that property can arise as the diagonal of a normal operator with spectrum x the case where x is a set of real numbers has received a great deal of attention over the years and is reasonably well though incompletely understood in this work we take up the case in which x is the set of vertices of a convex polygon in the critical sequences d turn out to be those that accumulate rapidly in x in the sense that sigmainfinityn 1 dist dn x s d e gammax with the following property if s d not equal 0 then d is not the diagonal of any such operator a we also show that while this is the only obstruction when n 2 there are other as yet unknown obstructions when n 3

asymptotic distribution of the spectra of a class of generalized kac murdock szego matrices we consider symmetric toeplitz matrices tn t r s r s 1n with tr r r where and are real and 0 1 we give formulas for det tn and tn 1 and show that if 1 and 0 then tn has eigenvalues 1n 2n nn such that limn 1n limn nn and 2n n 1 n are equally distributed as n with values of f 1 21 2 cos 2 at n 2 equally spaced points in 0

on the relation between fuzzy max archimedean t norm relational equations and the covering problem the problem of solving a system of fuzzy relational equations with max archimedean t norm composition is studied it is shown that this problem is closely related to the covering problem which belongs to the class of np hard problems it is proved that there is a one to one correspondence between the minimal solutions of the equations and the irredundant coverings as previously discovered by markovskii on the relation between equations with max product composition and the covering problem fuzzy sets and systems 153 2005 261 273 for fuzzy relational equations with max product composition since max product composition is a special case of max archimedean t norm composition this work extends markovskii s work to fuzzy relational equations with max archimedean t norm composition an extension of markovskii s algorithm is implemented yielding a processing time linearly proportional to the square of the number of minimal solutions

inside the eigenvalues of certain hermitian toeplitz band matrices while extreme eigenvalues of large hermitian toeplitz matrices have been studied in detail for a long time much less is known about individual inner eigenvalues this paper explores the behavior of the jth eigenvalue of an n by n banded hermitian toeplitz matrix as n tends to infinity and provides asymptotic formulas that are uniform in j for email protected email protected n the real valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum and then to decrease strictly back from the maximum to the minimum having nonzero second derivatives at the minimum and the maximum the results which are of interest in numerical analysis probability theory or statistical physics for example are illustrated and underpinned by numerical examples

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

solving fuzzy relation equations with a linear objective function abstract an optimization model with a linear objective function subject to a system of fuzzy relation equations is presented due to the non convexity of its feasible domain defined by fuzzy relation equations designing an efficient solution procedure for solving such problems is not a trivial job in this paper we first characterize the feasible domain and then convert the problem to an equivalent problem involving 0 1 integer programming with a branch and bound solution technique after presenting our solution procedure a concrete example is included for illustration purpose

min max programming problem subject to addition min fuzzy relation inequalities a bittorrent like peer to peer file sharing system can be reduced into a system of addition min fuzzy relation inequalities addition min is a new composition and the solution set of addition min fuzzy relation inequalities differs much from that of general max t norm fuzzy relation inequalities or equations in order to avoid network congestion and improve the stability of data transmission a min max programming problem is proposed subject to addition min fuzzy relation inequalities in this paper based on some relevant theorems on resolution a novel algorithm is developed step by step to find an optimal solution of the proposed problem moreover an application example is given to illustrate the feasibility and efficiency of the algorithm finally some further discussions are made concerning the optimal solution of the proposed problem

multivariable schur horn theorems fil massey pedro gustavo universidad nacional de la plata facultad de ciencias exactas departamento de matematicas argentina consejo nacional de investigaciones cientificas y tecnicas oficina de coordinacion administrativa saavedra 15 instituto argentino de matematica alberto calderon argentina

an infinite dimensional schur horn theorem and majorization theory the main result of this paper is the extension of the schur horn theorem to infinite sequences for two nonincreasing nonsummable sequences and that converge to 0 there exists a positive compact operator a with eigenvalue list and diagonal sequence if and only if j 1n j j 1n j for every n if and only if q for some orthostochastic matrix q when and are summable requiring additionally equality of their infinite series obtains the same conclusion extending a theorem by arveson and kadison our proof depends on the construction and analysis of an infinite product of t transform matrices

an algorithm for solving fuzzy relation equations with max t composition operator this paper studies the problem of solving a max t composite finite fuzzy relation equation where t is a special class of pseudo t norms if the equation is solvable then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value under these conditions some minimal solutions of the system can be obtained easily some procedures are also proposed in order to simplify the original system the simplified system is then decomposed if possible into several subsystems with smaller dimensions which are very easy to solve furthermore a method is presented to solve each subsystem by combining the method and those procedures an efficient algorithm is proposed to obtain the set of feasible solutions of the original system two examples are also given to illustrate the algorithm

the fisher hartwig conjecture and toeplitz eigenvalues abstract the conjecture of fisher and hartwig published in 1968 describes the asymptotic expansion of toeplitz determinants with singular generating functions for more than twenty years progress was made in extending the validity of the conjecture but recent computer experiments led to counterexamples that show the limits of the original conjecture and pointed the way to a revised conjecture this paper describes the history of the problem several numerical examples and the revised conjecture

optimization of fuzzy relation equations with max product composition abstract an optimization problem with a linear objective function subject to a system of fuzzy relation equations using max product composition is considered since the feasible domain is non convex traditional linear programming methods cannot be applied we study this problem and capture some special characteristics of its feasible domain and the optimal solutions some procedures for reducing the original problem are presented the problem is transformed into a 0 1 integer program which is then solved by the branch and bound method for illustration purpose an example of the procedures is provided

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

an accelerated approach for solving fuzzy relation equations with a linear objective function in literature the optimization model with a linear objective function subject to fuzzy relation equations has been converted into a 0 1 integer programming problem by fang and li 1999 they proposed a jump tracking branch and bound method to solve this 0 1 integer programming problem in this paper we propose an upper bound for the optimal objective value based on this upper bound and rearranging the structure of the problem we present a backward jump tracking branch and bound scheme for solving this optimization problem a numerical example is provided to illustrate our scheme furthermore testing examples show that the performance of our scheme is superior to the procedure in the paper by fang and li several testing examples show that our initial upper bound is sharp

the pythagorean theorem ii the infinite discrete case the study of the pythagorean theorem and variants of it as the basic result of noncommutative metric euclidean geometry is continued the emphasis in the present article is the case of infinite discrete dimensionality

method of solution to fuzzy equations in a complete brouwerian lattice abstract up to now how to solve a fuzzy relation equation in a complete brouwerian lattice is still an open problem as di nola et al point out to this problem the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable in this paper we first show that there is a minimal element in the solution set of a fuzzy relation equation a x b where a a 1 a 2 a n and b are known and x x 1 x 2 x n t is unknown when its solution set is nonempty and b has an irredundant finite join decomposition further we give the method to solve a x b in a complete brouwerian lattice under the same conditions finally a method to solve a more general fuzzy relation equation in a complete brouwerian lattice when its solution set is nonempty is also given under similar conditions

an algorithm for minimizing a linear objective function subject to the fuzzy relation inequalities with addition min composition abstract in this paper we study the optimal solution of minimizing a linear objective function subject to the fuzzy relation inequalities with addition min composition we first discuss some properties about the minimal solutions of fuzzy relation inequalities with addition min composition and define the pseudo minimal indexes of this system next we give an algorithm to get the set of the pseudo minimal indexes which is called pmi algorithm finally we obtain an algorithm for this optimization system by utilizing these concepts and results the example is provided to show that our algorithm is simple and convenient

addition min fuzzy relation inequalities with application in bittorrent like peer to peer file sharing system abstract the data transmission mechanism in bittorrent like peer to peer p2p file sharing systems may be reduced to some addition min fuzzy relation inequalities the solution set of addition min fuzzy relation inequalities plays an important role in the corresponding optimization problem in this paper we study some properties of the solutions to such system convexity of the solution set and number of minimal solutions are discussed with comparison to those of the classical max t fuzzy relation equations or inequalities besides vertex solution and variable ordering minimal solution are also investigated with application in bittorrent like p2p file sharing system two numerical examples are given to illustrate the feasibility and efficiency of the algorithm for solving the variable ordering solution

leximax minimum solution of addition min fuzzy relation inequalities abstract addition min fuzzy relation inequalities are introduced to characterize the peer to peer p2p network system in order to decrease the network congestion considering both efficiency and fairness we define the leximax minimum solution for system of addition min fuzzy relation inequalities the leximax optimality refines both pareto optimality and max optimality for solving the leximax minimum solution concepts of minimum zero point and minimax value are defined with corresponding resolution algorithms i e algorithms i and ii based on these two algorithms algorithm iii is further developed to find the unique leximax minimum solution numerical examples illustrate the validity of our proposed algorithms

on fuzzy relational equations and the covering problem previous studies have shown that fuzzy relational equations fres based on either the max continuous archimedean t norm or the max arithmetic mean composition can be transformed into the covering problem which is an np hard problem exploiting the properties common to the continuous archimedean t norm and the arithmetic mean this study proposes a generalization of them as the u norm enabling fres that are based on the max continuous u norm composition also to be transformed into the covering problem this study also proposes a procedure for transforming the covering problem into max product fres consequently max continuous u norm fres can be solved by extending any procedure for solving either the covering problem or max product fres

toeplitz eigenvalues for radon measures abstractit is well known that for toeplitz matrices generated by a suf ciently smooth real valuedsymbol the eigenvalues behave asymptotically as the values of the symbol on uniform mesheswhile the singular values even for complex valued functions do as those values in modulus these facts are expressed analytically by the szego and szego like formulas and as is provedrecently the smoothness assumptions are as mild as those of l 1 in this paper it is shownthat the szego like formulas hold true even for toeplitz matrices generated by the so calledradon measures 2002 elsevier science inc all rights reserved ams classi cation 15a12 65f10 65f15 65t10keywords toeplitz matrices eigenvalues singular values szego formulas radon measures 1 introductionwe consider a sequence of toeplitz matrices cf 3 a n a kl a kl a k l 0 k l n 1 1 constructed from the coef cients of a formal fourier series f x k a k e i kx 2 supported by the russian fund of basic research under grant 99 01 00017 and completed while the rst author was a guest professor at the university of saarland germany under support of daad

eigenvectors of hermitian toeplitz matrices with smooth simple loop symbols abstract the paper is devoted to the structure and the asymptotics of the eigenvector matrix of hermitian toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line the results extend existing results on banded toeplitz matrices to full toeplitz matrices with temperate decay of the entries in the first row and column we establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors these formulas are in terms of the wiener hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue the paper does not aim at providing numerical algorithms its main purpose is rather to reveal the structure underneath the eigenvectors which are shown to be the sum of two harmonics and certain edge corrections

the schur horn theorem for operators with three point spectrum abstract we characterize the set of diagonals of the unitary orbit of a self adjoint operator with three points in the spectrum our result gives a schur horn theorem for operators with three point spectrum analogous to kadison s pythagorean theorem and carpenter s theorem which characterize the diagonals of orthogonal projections

cyclic displacements and decompositions of inverse matrices for cupl toeplitz matrices abstract in this paper we deal mainly with a class of cupl toeplitz matrices 32 without toeplitz structure which are close to the toeplitz matrices in the sense that their 1 1 cyclic displacements coincide with cyclic displacement of some toeplitz matrices by constructing the corresponding displacement of the matrices we derive the formulas on representation of the inverses of the cupl toeplitz matrices in the form of sums of products of factor row first plus last right circulants furthermore the inverses of the cupl hankel matrices can be also obtained which benefits from the relation between the cupl toeplitz matrices and the cupl hankel matrices

fuzzy relation equations and inequalities abstract this paper generalizes a method of solution of fuzzy relation equations introduced by sanchez 1976 and developed by pedrycz 1982 we find the greatest or the least solution of a composite l fuzzy relation equation with arbitrary monotone composition

an efficient solution procedure for fuzzy relation equations with max product composition we study a system of fuzzy relation equations with max product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions instead of solving the problem combinatorially the procedure identifies the nonminimal solutions and eliminates them from the set of minimal solutions

fuzzy relation equations on a finite set abstract the family of all the solutions of a fuzzy relation equation on a finite set is considered it is characterized by the set of all the lower solutions which can be obtained by a combinatorial algorithm

posynomial geometric programming problem subject to max min fuzzy relation equations we discuss a class of posynomial geometric programming problem pgpf aimed at minimizing a posynomial subject to fuzzy relational equations with max min composition by introducing auxiliary variables we convert the pgpf into an equivalent programming problem whose objective function is a non decreasing function with an auxiliary variable we show that an optimal solution consists of a maximum feasible solution and one of the minimal feasible solutions by an equivalent programming problem in addition we introduce some rules for simplifying the problem then by using a branch and bound method and fuzzy relational equations fre path we present an algorithm to obtain an optimal solution to the pgpf finally numerical examples are provided to illustrate the steps of the procedure