a heuristic approach for the travelling purchaser problem abstract the travelling purchaser problem tpp is a known generalization of the travelling salesman problem and is defined as follows let us consider a set of products and a set of markets each market is provided with a limited amount of each product at a known price the tpp consists in selecting a subset of markets such that a given demand of each product can be purchased minimizing the routing cost and the purchasing cost this problem arises in several applications mainly in routing and scheduling contexts and it is np hard in the strong sense a new heuristic approach based on a local search scheme exploring a new neighbourhood structure is proposed the key idea is to perform a k exchange of markets instead of the classical 1 exchanges a neighbour of a given tpp solution is another tpp solution obtained by removing a path of consecutive markets and by inserting other markets so as to restore the feasibility this proposal is evaluated on a broad class of instances from literature where the routing costs are euclidean distances computational results show that our proposal is favourably compared to previous heuristic algorithms in literature

on the approximate tradeoff for bicriteria batching and parallel machine scheduling problems we consider multiobjective scheduling problems i e scheduling problems that are evaluated with respect to many cost criteria and we are interested in determining a trade off pareto curve among these criteria we study two types of bicriteria scheduling problems single machine batching problems and parallel machine scheduling problems instead of proceeding in a problem by problem basis we identify a class of multiobjective optimization problems possessing a fully polynomial time approximation scheme fptas for computing an approximate pareto curve this class contains a set of problems whose pareto curve can be computed via a simple pseudo polynomial dynamic program for which the objective and transition functions satisfy some easy to verify arithmetical conditions our study is based on a recent work of woeginger electronic colloquium on computational complexity report 84 short version appeared in soda 99 pp 820 829 for the single criteria optimization ex benevolent problems we show how our general result can be applied to the considered scheduling problems

tradeoff based decomposition and decision making in multiobjective programming to facilitate the evaluation of tradeoffs and the articulation of preferences in multiple criteria decision making a multiobjective decomposition scheme is proposed that restructures the original problem as a collection of smaller sized subproblems with only subsets of the original criteria a priori preferences on objective tradeoffs are integrated into this process by modifying the ordinary pareto order by more general domination cones and decision makers are supported by an interactive decision making procedure to coordinate any remaining tradeoffs using concepts of approximate efficiency a theoretical foundation for this method is provided and an illustrative application to multiobjective portfolio optimization is described in detail

fast evaluation of gaunt coefficients recursive approach solution of electromagnetic multisphere scattering problems involves the determination of a large number of gaunt coefficients that are connected with definite integrals of products of three associated legendre functions this paper presents general recursion formulae cycling a single index for the gaunt coefficient with the use of the recursive scheme gaunt coefficients of either low or high degree and order can be evaluated accurately and expeditiously

optimality conditions via scalarization for a new e efficiency concept in vector optimization problems abstract in this work necessary and sufficient conditions for approximate solutions of vector optimization problems are obtained via scalarization i e by considering approximate solutions of associated scalar optimization problems these conditions are proved through a new e efficiency concept and two very general assumptions on the scalarization that extend the usual order representing and monotonicity properties moreover neither solidness hypothesis on the order cone nor monotonicity property on the scalarization are assumed

the hd and hd methods for accelerating the convergence of three center nuclear attraction and four center two electron coulomb integrals over b functions and their convergence properties abstract three center nuclear attraction and four center two electron coulomb integrals over slater type orbitals are required for ab initio and density functional theory dft molecular structure calculations they occur in many millions of terms even for small molecules and require rapid and accurate evaluation the b functions are used as a basis set of atomic orbitals these functions are well adapted to the fourier transform method that allowed analytical expressions for the integrals of interest to be developed rapid and accurate evaluation of these analytical expressions is now made possible by applying the hd and hd methods for accelerating the convergence of the semi infinite oscillatory integrals the convergence properties of the new methods are analysed the numerical results section shows the high predetermined accuracy and the substantial gain in the calculation times obtained using the new methods

epsilon approximate solutions for multiobjective programming problems abstract in this paper approximate solutions for multiobjective programming problems up to 0 are investigated the concepts of approximate solutions were introduced by white and loridan the connections with these concepts are studied also the sufficient and necessary conditions to obtain the elements of the smallest approximate solution set are verified

algorithms to solve the orienteering problem a comparison abstract the multiobjective vending problem mvp keller 1985 is a generalization of the traveling salesman problem tsp where it is not necessary to visit all nodes in the problem definition a special case of the mvp is the orienteering problem op tsiligirodes 1984 et al the problems are np hard a heuristic was designed to solve the general mvp keller 1985 three other heuristics were designed to solve the op tsiligirides 1984 golden et al 1985 the algorithms underlying these heuristics are outlined their performances are compared against three op problems

ϵ pareto optimality for nondifferentiable multiobjective programming via penalty function abstract necessary and sufficient conditions without a constraint qualification for pareto optimality of multiobjective programming are derived the necessary kuhn tucker condition suggests the establishment of a wolf type duality theorem for nondifferentiable convex multiobjective minimization problems the generalized saddle point for pareto optimality of the vector lagrangian is studied

an extremely efficient approach for accurate and rapid evaluation of three centre two electron coulomb and hybrid integrals over b functions analytic expressions of three centre two electron coulomb and hybrid integrals over b functions are obtained using the fourier transform method thoroughly explored by steinborn s group these analytic expressions involve semi infinite integrals which are slowly convergent due to the presence of hypergeometric and spherical bessel functions in the integrands we have proven that these hypergeometric functions can be expressed as finite expansions and the integrands involving these series satisfy all the conditions required to apply the h approach which greatly simplifies the application of the nonlinear transformation this work presents a rapid and accurate evaluation of these integrals obtained by using a new approach which we called s this new method is based on the h and methods and some practical properties of spherical bessel reduced bessel and sine functions the s method has greatly simplified the calculations avoiding the long and difficult implementation of the successive zeros of the spherical bessel function and a method for solving linear systems which are required by h and

the biobjective travelling purchaser problem the purpose of this article is to present and solve the biobjective travelling purchaser problem which consists in determining a route through a subset of markets in order to collect a set of products minimizing the travel distance and the purchasing cost simultaneously the most convenient purchase of the product in the visited markets is easily computed once the route has been determined therefore this problem contains a finite set of solutions one for each route and the problem belongs to the field of the biobjective combinatorial optimization it is here formulated as a biobjective mixed integer linear programming model with an exponential number of valid inequalities and this model is used within a cutting plane algorithm to generate the set of all supported and non supported efficient points in the objective space a variant of the algorithm computes only supported efficient points for each efficient point in the objective space exactly one pareto optimal solution in the decision space is computed by solving a single objective problem each of these single objective problems in turn is solved by a specific branch and cut approach a heuristic improvement based on saving previously generated cuts in a common cut pool structure has also been developed with the aim of speeding up the algorithm performance results based on benchmark instances from literature show that the common cut pool heuristic is very useful and that the proposed algorithm manages to solve instances containing up to 100 markets and 200 different products the general procedure can be extended to address other biobjective combinatorial optimization problems whenever a branch and cut algorithm is available to solve a single objective linear combination of these criteria

a branch and cut algorithm for the undirected traveling purchaser problem the purpose of this paper is to present a branch and cut algorithm for the undirectedtraveling purchaser problem which consists of determining a minimum cost route through a subset of markets where the cost is the sum of travel and purchase costs the problem is formulated as an integer linear program and several families of valid inequalities are derived to strengthen the linear relaxation the polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facet defining a branch and cut procedure is developed and tested over four classes of randomly generated instances results show that the proposed algorithm outperforms all previous approaches and can optimally solve instances containing up to 200 markets

epsilon dominating solutions in mean variance portfolio analysis abstract in this paper the problem of finding the effcient set of portfolios in a general constraint set is replaced by finding a set of epsilon dominating protfolios the number of which is determined by the size of epsilon algorithms and associated theory are given together with some possible modifications

generating epsilon efficient solutions in multiobjective programming scalarization approaches to purposely generating e efficient solutions of multiobjective programs are investigated and a generic procedure for computing these solutions is proposed and illustrated with an example real life decision making situations in which the solutions are of significance are described

efficient evaluation of vector translation coefficients in multiparticle light scattering theories vector addition theorems are a necessary ingredient in the analytical solution of electromagnetic multiparticle scattering problems these theorems include a large number of vector addition coefficients there exist three basic types of analytical expressions for vector translation coefficients stein s quart appl math 19 15 1961 cruzan s quart appl math 20 33 1962 and xu s j comput phys 127 285 1996 stein s formulation relates vector translation coefficients with scalar translation coefficients cruzan s formulas use the wigner 3jm symbol xu s expressions are based on the gaunt coefficient since the scalar translation coefficient can also be expressed in terms of the gaunt coefficient the key to the expeditious and reliable calculation of vector translation coefficients is the fast and accurate evaluation of the wigner 3jm symbol or the gaunt coefficient we present highly efficient recursive approaches to accurately evaluating wigner 3jm symbols and gaunt coefficients armed with these recursive approaches we discuss several schemes for the calculation of the vector translation coefficients using the three general types of formulation respectively our systematic test calculations show that the three types of formulas produce generally the same numerical results except that the algorithm of stein s type is less accurate in some particular cases these extensive test calculations also show that the scheme using the formulation based on the gaunt coefficient is the most efficient in practical computations

necessary and sufficient karush kuhn tucker conditions for multiobjective markov chains optimality the solution concepts proposed in this paper follow the karush kuhn tucker kkt conditions for a pareto optimal solution in finite time ergodic and controllable markov chains multi objective programming problems in order to solve the problem we introduce the tikhonov s regularizator for ensuring the objective function is strict convex then we consider the c variable method for introducing equality constraints that guarantee the result belongs to the simplex and satisfies ergodicity constraints lastly we restrict the cost functions allowing points in the pareto front to have a small distance from one another the computed image points give a continuous approximation of the whole pareto surface the constraints imposed by the c variable method make the problem computationally tractable and the restriction imposed by the small distance change ensures the continuation of the pareto front we transform the multi objective nonlinear problem into an equivalent nonlinear programming problem by introducing the lagrange function multipliers as a result we obtain that the objective function is strict convex the inequality constraints are continuously differentiable and the equality constraint is an affine function under these settings the kkt optimality necessary and sufficient conditions are elicited naturally a numerical example is solved for providing the basic techniques to compute the pareto optimal solutions by resorting to kkt conditions

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

strong linear programming relaxations for the orienteering problem abstract consider a set of nodes each with an associated profit and a set of arcs each with an associated length the objective of the orienteering problem is to find the path beginning at a specified origin and terminating at a specified destination that maximizes total profit subject to 1 a constraint on the length of the path and 2 the condition that no node is visited more than once the problem may be formulated as a 0 1 integer program and it has been shown to be np hard prior research has focused on heuristics that obtain lower bounds on the optimal objective function value some recent research has proposed a branch and bound algorithm that solves a linear programming lp relaxation of the 0 1 model at each node and obtains an optimal orienteering path our research is concerned with tightening the lp relaxation by adding constraints and valid inequalities we propose a procedure to obtain upper bounds by solving three successive linear programs we test our procedure on datasets in existing literature and demonstrate that the average deviation between our upper bounds and the best known lower bounds feasible solutions is less than five percent the quality of upper bound obtained is significantly enhanced over the bound obtained from solving the basic lp relaxation

minimum directed 1 subtree relaxation for score orienteering problem score orienteering is a variant of the orienteering game where competitors start at a specified control point visit as many control points as possible and thereby collect prizes and return to the starting point within a prescribed amount of time the competitor with the highest amount of prizes wins the game in this paper we propose the minimum directed 1 subtree problem as a new relaxation to this problem and develop two methods to improve its lower bound we describe a cut and dual simplex method and a lagrangian relaxation method and construct an algorithm that combines these two methods in an appropriate way computational experiments are carried out to analyse the behavior of the proposed relaxation with respect to the characteristics of the test problems especially for large scale instances we find that the proposed relaxation is superior to the ordinary assignment relaxation

transgenetic algorithm for the traveling purchaser problem in this paper an evolutionary algorithm is presented for the traveling purchaser problem an important variation of the traveling salesman problem the evolutionary approach proposed in this paper is called transgenetic algorithm it is inspired on two significant evolutionary driving forces horizontal gene transfer and endosymbiosis the performance of the algorithm proposed for the investigated problem is compared with other recent works presented in the literature computational experiments show that the proposed approach is very effective for the investigated problem with 17 and 9 new best solutions reported for capacitated and uncapacitated instances respectively

theory and methodology abstract the hierarchial network design problem with transshipment facilites was first formulated in current j 1988 the design of a hierarchical transportation network with transshipment facilities transportation science 22 270 277 the objective of the fc hndp is to identify a least cost two level network the network must include a primary path from a prespecified origin node to a prespecified terminus node all nodes not on the primary path must be connected to that path via secondary arcs in addition transshipment facilities are required at the intersections of the two network levels in this paper we present two efficient heuristics for solving the fc hndp these heuristics were based upon a lagrangian relaxation of the problem to facilitate this relaxation we have reformulated the original problem results for 240 randomly generated problems are summarized the heuristics yielded the optimal solution in 72 of these problems the mean gap between the lagrangian lower bound and the feasible solution value was 0 38 and the maximum gap was 4 33 of the lower bound

fast evaluation of the gaunt coefficients addition theorems for vector spherical harmonics require the determination of the gaunt coefficients that appear in a linearization expansion of the product of two associated legendre functions this paper presents an algorithm for the efficient calculation of these coefficients through solving the most appropriate lower triangular linear system and derives all relevant recurrence relations needed in the calculation this algorithm is also applicable to the calculation of the clebsch gordan coefficients that are closely related to the gaunt coefficients and are frequently encountered in the quantum theory of angular momentum the new method described in this paper reduces the computing time to 1 compared to the existing formulation that is widely used this new method can be applied to the calculation of both low and high degree coefficients while the existing formulation works well only for low degrees

a combined scalarizing method for multiobjective programming problems abstract in this paper a new general scalarization technique for solving multiobjective optimization problems is presented after studying the properties of this formulation two problems as special cases of this general formula are considered it is shown that some well known methods such as the weighted sum method the constraint method the benson method the hybrid method and the elastic constraint method can be subsumed under these two problems then considering approximate solutions some relationships between e weakly properly efficient points of a general without any convexity assumption multiobjective optimization problem and optimal solutions of the introduced scalarized problem are achieved

the properties of sine spherical bessel and reduced bessel functions for improving convergence of semi infinite very oscillatory integrals the evaluation of three centre nuclear attraction integrals over b functions this paper focuses on the use of useful properties of sine spherical bessel and reduced bessel functions to simplify the application of the nonlinear d and transformations for accelerating the convergence of semi infinite very oscillatory integrals and to reduce the calculation times keeping a high predetermined accuracy three centre nuclear attraction integrals which are one of the most difficult type involved in density functional theory methods when using a basis set of b functions are evaluated using the new approach the numerical results show the efficiency of the new method compared with other alternatives

a numerical method for constructing the pareto front of multi objective optimization problems in this paper a new numerical method is presented for constructing an approximation of the pareto front of multi objective optimization problems this method is based on the well known scalarization approach by pascoletti and serafini the proposed method is applied to four test problems that illustrate specific difficulties encountered in multi objective optimization problems such as nonconvex disjoint and local pareto fronts the effectiveness of the proposed method is demonstrated by comparing it with the nsga ii algorithm and the normal constraint method

three centre two electron coulomb and hybrid integrals evaluated using nonlinear d and bard transformations of the two electron integrals occuring in the molecular context the three centre coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work accelerates the convergence of these integrals by first manipulating the indices of the hypergeometric function and exploiting relationships to express this function as a finite expansion and exploiting the properties of bessel functions which satisfy second order linear differential equations this is a suitable form of the integrand to apply the nonlinear d due to d levin and a sidi and due to a sidi transformations the extensive numerical results section illustrates the accuracy and unprecedented efficiency of evaluation of these integrals

efficient and rapid numerical evaluation of the two electron four center coulomb integrals using nonlinear transformations and useful properties of sine and bessel functions two electron four center coulomb integrals are undoubtedly the most difficult type involved in ab initio and density functional theory molecular structure calculations millions of such integrals are required for molecules of interest therefore rapidity is the primordial criterion when the precision has been reached this work presents an extremely efficient approach for improving convergence of semi infinite very oscillatory integrals based on the nonlinear d transformation and some useful properties of spherical bessel reduced bessel and sine functions the new method is now shown to be applicable to evaluating the two electron four center coulomb integrals over b functions the section with numerical results illustrates the unprecedented efficiency of the new approach in evaluating the integrals of interest

regular article efficient and rapid evaluation of three center two electron coulomb and hybrid integrals using nonlinear transformations among the two electron integrals occurring in the molecular context the three center coulomb and hybrid integrals are numerous and difficult to evaluate to high accuracy the analytical and numerical difficulties arise mainly from the presence of the spherical bessel function and hypergeometric series in these integrals the present work pursues the acceleration of convergence for three center two electron coulomb and hybrid integrals we have proven that the hypergeometric function can be expressed as a finite expansion and that the integrand involving this series and a product of bessel functions satisfies a linear differential equation with coefficients having a power series expansion in the reciprocal of the variable suitable for application of the nonlinear d and d transformations these transformations depend strongly on the order of the differential equation that the integrand of interest satisfies this work concentrates on reduction of this order to two exploiting properties of spherical and reduced bessel functions leading to greatly simplified calculations to evaluate the integrals precisely by reducing the order of linear systems to be solved it also avoids the long and difficult implementations of successive derivatives of the integrands the numerical results section illustrates clearly the reduction of the calculation time we obtained for a high predetermined accuracy

a branch and bound algorithm for the traveling purchaser problem an important generalization of the traveling salesman problem called the traveling purchaser problem is considered a branch and bound algorithm which solves a related simple plant location problem for calculating the bounds is proposed for this problem computational experiments with this algorithm provide the evidence that moderate size problems of upto 25 cities and 100 commodities can be solved in reasonable computation time the problem is shown to have applications in the areas of scheduling warehousing and routing