the lagrangian relaxation method for solving integer programming problems this article originally appeared in management science january 1981 volume 27 number 1 pp 1 18 published by the institute of management sciences one of the most computationally useful ideas of the 1970s is the observation that many hard integer programming problems can be viewed as easy problems complicated by a relatively small set of side constraints dualizing the side constraints produces a lagrangian problem that is easy to solve and whose optimal value is a lower bound for minimization problems on the optimal value of the original problem the lagrangian problem can thus be used in place of a linear programming relaxation to provide bounds in a branch and bound algorithm this approach has led to dramatically improved algorithms for a number of important problems in the areas of routing location scheduling assignment and set covering this paper is a review of lagrangian relaxation based on what has been learned in the last decade

using extension sets to aggregate partial rankings in a flexible setting this paper deals with the rank aggregation problem in a general setting in particular we approach the problem for any kind of ranking complete or incomplete and with or without ties the underlying idea behind our approach is to take into account the so called extension set of a ranking that is the set of permutations that are compatible with the given ranking in this way we aim to manage the uncertainty inherent to this problem when not all the items are ranked and or when some items are equally preferred we develop two approaches a general one based on this idea and a constrained version in which the extension set is limited by not allowing non ranked items to be placed in an existent bucket of tied items we test our proposal by coupling it with two different algorithms we formalize our approaches mathematically and also carry out an extensive experimental evaluation by using 22 datasets the results show that the use of our extension sets based approaches to compute the precedence matrix clearly outperforms the standard way of computing the preference matrix by only using the information explicitly provided by the rankings in the sample

capacitated facility location separation algorithms and computational experience we consider the polyhedral approach to solving the capacitated facility location problem the valid inequalities considered are the knapsack cover flow cover effective capacity single depot and combinatorial inequalities the flow cover effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities the separation problem based on the family of submodular inequalities is np hard in general for the well known subclass of flow cover inequalities however we show that if the client set is fixed and if all capacities are equal then the separation problem can be solved in polynomial time for the flow cover inequalities based on an arbitrary client set and general capacities and for the effective capacity and single depot inequalities we develop separation heuristics an important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining the way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities the family of combinatorial inequalities was originally developed for the uncapacitated facility location problem but is also valid for the capacitated problem no computational experience using the combinatorial inequalities has been reported so far here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities we report on computational results from solving 60 medium size problems

facets and algorithms for capacitated lot sizing the dynamic economic lot sizing model which lies at the core of numerous production planning applications is one of the most highly studied models in all of operations research and yet capacitated multi item versions of this problem remain computationally elusive we study the polyhedral structure of an integer programming formulation of a single item capacitated version of this problem and use these results to develop solution methods for multi item applications in particular we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane branch and bound procedure moreover our results show that in many instances adding certain of these inequalities a priori to the problem formulation and avoiding the generation of cutting planes can be equally effective

optimal solution of the discrete cost multicommodity network design problem we investigate a multicommodity network design problem where a discrete set of technologies with step increasing cost and capacity functions should be installed on the edges this problem is a fundamental network design problem having many important applications in contemporary telecommunication networks we describe an exact constraint generation approach and we show that the conjunctive use of valid inequalities bipartition inequalities that are generated using max flow computations as well as an exact separation algorithm of metric inequalities makes it feasible to solve to optimality instances with up to 50 nodes and 100 edges

a multi period network design problem for cellular telecommunication systems abstract mathematical programming models for multi period network design problems which arise in cellular telecommunication systems are presented the underlying network topologies range from a simple star to complex multi layer steiner like networks linear programming lagrangian relaxation and branch and cut heuristics are proposed and a polynomial bounded heuristic based on an interior point linear programming implementation is described extensive computational results are presented on a number of randomly generated problem sets and the performance of the heuristic s are compared with an optimal branch and bound algorithm

minimum cost capacity installation for multicommodity network flows consider a directed graphg v a and a set of traffic demands to be shipped between pairs of nodes inv capacity has to be installed on the edges of this graph in integer multiples of a base unit so that traffic can be routed in this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs we study two different approaches for solving problems of this type the first one is based on the idea of metric inequalities see onaga and kakusho on feasibility conditions of multicommodity flows in networks ieee transactions on circuit theory ct 18 4 1971 425 429 and uses a formulation with only a variables the second uses an aggregated multicommodity flow formulation and has v a variables we first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on three nodes next we explain our solution methods for both of the approaches in detail and present computational results our computational experience shows that the two formulations are comparable and yield effective algorithms for solving real life problems 1998 the mathematical programming society inc published by elsevier science b v

approximating the single sink link installation problem in network design we initiate the algorithmic study of an important but np hard problem that arises commonly in network design the input consists of the following an undirected graph with one sink node and multiple source nodes a specified length for each edge and a specified demand demv for each source node v a small set of cable types where each cable type is specified by its capacity and its cost per unit length the cost per unit capacity per unit length of a high capacity cable may be significantly less than that of a low capacity cable reflecting an economy of scale i e the payoff for buying at bulk may be very high the goal is to design a minimum cost network that can simultaneously route all the demands at the sources to the sink by installing zero or more copies of each cable type on each edge of the graph an additional restriction is that the demand of each source must follow a single path the problem is to find a route from each source node to the sink and to assign capacity to each edge of the network such that the total costs of cables installed are minimized we call this problem the single sink link installation problem for the general problem we introduce a new moat type lower bound on the optimal value and we prove a useful structural property of near optimal solutions for every instance of our problem there is a near optimal solution whose graph is acyclic with a cost no more than twice the optimal cost we present efficient approximation algorithms for key special cases of the problem that arise in practice for points in the euclidean plane we give an approximation algorithm with performance guarantee o log d u1 where d is the total demand and u1 is the smallest cable capacity when the metric is arbitrary we consider the case where the network to be designed is restricted to be two level i e every source sink path has at most two edges for this problem we present an algorithm with performance guarantee o log n where n is the number of nodes in the input graph and also show that this performance guarantee is nearly best possible

diversification strategies in local search for a nonbifurcated network loading problem abstract this paper examines a variant of the network loading problem a network design problem found in the telecommunications industry in this problem facilities of fixed capacity must be installed on the edges of an undirected network to carry the flow from a central vertex to a set of demand vertices the objective is to minimize the total installation costs in this work the nonbifurcated version of the problem is considered where the demand at any given vertex must be satisfied through a single path the proposed heuristics alternate between a construction phase and a local search phase each new construction phase except the first one is part of a diversification strategy aimed at providing a new starting point for the following local search phase different diversification strategies are tested and compared on large scale instances with up to 500 vertices

a partitioning algorithm for the network loading problem this paper proposes a benders like partitioning algorithm to solve the network loading problem the approach is an iterative method in which the integer programming solver is not used to produce the best integer point in the polyhedral relaxation of the set of feasible capacities rather it selects an integer solution that is closest to the best known integer solution contrary to previous approaches the method does not exploit the original mixed integer programming formulation of the problem the effort of computing integer solutions is entirely left to a pure integer programming solver while valid inequalities are generated by solving standard nonlinear multicommodity flow problems the method is compared to alternative approaches proposed in the literature and appears to be efficient for computing good upper bounds

algorithms for generalized fractional programming a generalized fractional programming problem is specified as a nonlinear program where a nonlinear function defined as the maximum over several ratios of functions is to be minimized on a feasible domain of n the purpose of this paper is to outline basic approaches and basic types of algorithms available to deal with this problem and to review their convergence analysis the conclusion includes results and comments on the numerical efficiency of these algorithms

continuous knapsack sets with divisible capacities we study two continuous knapsack sets mathcal y ge y and mathcal y le y with n n integer one unbounded continuous and m m bounded continuous variables in either ge or le form when the coefficients of the integer variables are integer and divisible we show in both cases that the convex hull is the intersection of the bound constraints and 2 m 2m polyhedra arising as the convex hulls of continuous knapsack sets with a single unbounded continuous variable the latter convex hulls are completely described by an exponential family of partition inequalities and a polynomial size extended formulation is known in the ge case we also provide an extended formulation for the le case it follows that given a specific objective function optimization over both mathcal y ge y and mathcal y le y can be carried out by solving m m polynomial size linear programs a further consequence of these results is that the coefficients of the continuous variables all take the values 0 or 1 after scaling in any non trivial facet defining inequality of the convex hull of such sets

benders and cut algorithm for fixed charge capacitated network design problem abstract we develop a benders and cut algorithm for fixed charge capacitated network design problem which incorporates both benders cuts and polyhedral cuts into an implicit branch and bound algorithm the algorithm is tested over a wide range of problem instances with varying sizes and traffic conditions it is shown that this new algorithm s performance is competitive compared with existing algorithms and that benders cuts are more effective under heavy traffic load while cut set inequalities are more effective under light traffic load

projections of the capacitated network loading problem abstract consider an undirected network and a set of commodities with specified demands between various pairs of nodes of the network given two types of capacitated facilities that can be installed loaded for arc dependent costs we have to determine the integer number of facilities to load on each arc in order to send the required flow of all commodities at minimum total cost we present a natural mixed integer programming formulation of the problem and then consider its single commodity and multicommodity versions we develop equivalent formulations in a lower dimensional space by projecting out the flow variables and study the polyhedral properties of the corresponding projection cones our results strengthen an existing result for multicommodity flow problems we also characterize several classes of facet defining inequalities for this lower dimensional polyhedron and conclude by identifying some open problems and future research directions

on linear characterizations of combinatorial optimization problems we show that there can be no computationally tractable description by linear inequalities of the polyhedron associated with any np complete combinatorial optimization problem unless np co np a very unlikely event we also apply the ellipsoid method for linear programming to show that a combinatorial optimization problem is solvable in polynomial time if and only if it admits a small generator of violated inequalities

atm vp based network design abstract we consider the problem of designing an atm vp based leased line backbone network given point to point communication demands having predefined sizes in a network the problem is to find configurations of demand routes and link facilities installed on each edge satisfying all demands at minimum cost under some constraints one of the most important constraints is that a single demand cannot be split over multiple link facilities this is a sort of bin packing constraint we propose an integer programming formulation of the problem and an algorithm to solve it an efficient column generation technique to solve the linear programming relaxation is proposed and a valid inequality is used to strengthen the integer programming formulation the algorithm incorporates the column generation technique and the cutting plane approach into a branch and bound scheme we test the proposed algorithm on some real problems the results show that the algorithm can be used to solve the problems within reasonably small computing times

mixing mixed integer inequalities mixed integer rounding mir inequalities play a central role in the development of strong cutting planes for mixed integer programs in this paper we investigate how known mir inequalities can be combined in order to generate new strong valid inequalities given a mixed integer region s and a collection of valid base mixed integer inequalities we develop a procedure for generating new valid inequalities for s the starting point of our procedure is to consider the mir inequalities related with the base inequalities for any subset of these mir inequalities we generate two new inequalities by combining or mixing them we show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed integer region associated with the base inequalities we discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed integer programming problems in particular we present examples for production planning capacitated facility location capacitated network design and multiple knapsack problems we also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs finally we study some extensions of this mixing procedure

design of capacitated multicommodity networks with multiple facilities this paper addresses the problem of designing a multicommodity network using several facilities with different costs and capacities the problem is addressed in a special context of designing private telecommunications networks using fractional t1 services the algorithm starts from any given solution of the problem and gradually improves it by solving a series of subproblems arriving at a local minimum the subproblem is defined over a subset of links called the subnetwork by using one of the links as the base link it is shown that the subproblem can be formulated as a multiple choice knapsack problem that is solved by dynamic programming computational results and lower bounds are reported on problems of up to 20 nodes and up to 3 facilities on most problems the algorithm produces solutions within about 5 of lower bound on the average although it was not possible to compute lower bounds for larger problems heuristic solutions and running times are reported for problems of up to 99 nodes and four facilities

a dual ascent procedure for large scale uncapacitated network design the fixed charge network design problem arises in a variety of problem contexts including transportation communication and production scheduling we develop a family of dual ascent algorithms for this problem this approach generalizes known ascent procedures for solving shortest path plant location steiner network and directed spanning tree problems our computational results for several classes of test problems with up to 500 integer and 1 98 million continuous variables and constraints show that the dual ascent procedure and an associated drop add heuristic generate solutions that in almost all cases are guaranteed to be within 1 to 4 of optimality moreover the procedure requires no more than 150 seconds on an ibm 3083 computer the test problems correspond to dense and sparse networks including some models that arise in freight transport

a comparison of mixed integer programming models for nonconvex piecewise linear cost minimization problems we study a generic minimization problem with separable nonconvex piecewise linear costs showing that the linear programming lp relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope we also show a relationship between this result and classical lagrangian duality theory

on the posynomial fractional programming problems abstract the posynomial fractional programming pfp problem arises from the summation minimization of several quotient terms which are composed of posynomial terms appearing in the objective function subject to given posynomial constraints this paper proposes an approximate approach to solving a pfp problem a linear programming relaxation is derived for the problem based on piecewise linearization techniques which first convert a posynomial term into the sum of absolute terms these absolute terms are then linearized by some linearization techniques the proposed approach could reach a solution as close as possible to a global optimum

on capacitated network design cut set polyhedra this paper provides an analysis of capacitated network design cut set polyhedra we give a complete linear description of the cut set polyhedron of the single commodity single facility capacitated network design problem then we extend the analysis to single commodity multifacility and multicommodity multifacility capacitated network design problems valid inequalities described here are applicable to directed network design problems with any number of facility types and any level of capacities we report results from a computational study done for testing the effectiveness of the new inequalities

valid linear inequalities for fixed charge problems many problems in the operations research management science literature can be formulated with both zero one and continuous variables however the exact optimization of such mixed zero one models remains a computational challenge in this paper we propose to study mixed problems from a mathematical point of view that is similar in spirit to recent research on purely combinatorial problems that has investigated systems of defining linear inequalities or facets of the underlying polytope at least two numerical studies have validated this line of research computationally and the advances in the problem solving capabilities are considerable we expect that similar gains are possible for the mixed zero one problem more precisely we consider the mixed integer programs whose feasible region x is composed of i a simple additive constraint in the continuous variables xj for j 1 2 n and ii constraints 0 xj mjyj defined by binary variables yj for j 1 2 n this type of feasible region arises in a variety of mixed integer problems and particularly in network problems with fixed charges on the arcs we derive two classes of facet defining linear inequalities of the convex hull of x and show that the second of these classes gives a complete description of the convex hull when mj m for all j we also develop methods to detect violated inequalities from these classes so that these facets can be used as cutting planes to strengthen the formulations of certain mixed integer problems

piecewise linear multicriteria programs the continuous case and its discontinuous generalization in this paper we study piecewise linear multicriteria programs that is multicriteria programs with either a continuous or discontinuous piecewise linear objective function and a polyhedron set constraint we obtain an algebraic representation of a semi closed polyhedron and apply it to show that the image of a semi closed polyhedron under a continuous linear function is always one semi closed polyhedron we establish that the weak pareto solution point set of a piecewise linear multicriteria program is the union of finitely many semi closed polyhedra we propose an algorithm for finding the pareto point set of a continuous piecewise linear bi criteria program and generalize it to the discontinuous case we apply our algorithm to solve the discontinuous bi criteria portfolio selection problem with an l risk measure and transaction costs and show that this algorithm can be improved by using an ideal point strategy

variable disaggregation in network flow problems with piecewise linear costs we study mixed integer programming formulations based upon variable disaggregation for generic multicommodity network flow problems with nonconvex piecewise linear costs a problem class that arises frequently in many application domains in telecommunications transportation and logistics we present several structural results for these formulations and we analyze the results of extensive experiments on a large set of instances with various characteristics in particular we show that the linear programming relaxation of an extended disaggregated model approximates the objective function by its lower convex envelope in the space of commodity flows together the theoretical and computational results allow us to suggest which formulation might be the most appropriate depending on the characteristics of the problem instances

an efficient linearization approach for mixed integer problems abstract oral and kettani previously developed a linearization technique published in management science in 1990 and in operational research in 1992 for solving quadratic and cubic mixed integer problems for a quadratic problem with n 0 1 variables their method would introduce n additional continuous variables and n auxiliary constraints for a cubic problem with n 0 1 variables their method would introduce 3n additional continuous variables and 3n auxiliary constraints this linearization approach of oral and kettani has been accepted as the most efficient linearization technique published requiring the least number of additional continuous variables and auxiliary constraints however their method is difficult to extend for linearizing higher order polynomial terms that appear in mixed integer problems and in addition all constraints should be kept as linear this note proposes a new general model for linearizing various orders of mixed integer problems which cannot be solved by oral and kettani s model when the order is higher than three some computational results show that the proposed model is more efficient than oral kettani s method because it uses less additional variables and auxiliary constraints to linearize the same size of mixed integer problems in addition the proposed model can be easily applied to polynomial mixed integer terms that appear in the objective function and or constraints

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

solving mixed integer programming problems using automatic reformulation in this paper we describe computational experience in solving mixed 0 1 programming problems using strong valid inequalities as cutting planes in particular we report on the solution to optimality of 18 medium to large size problems including production planning problems with setup costs and capacity constraints multilevel distribution planning problems drainage and heating system design problems and electricity generator scheduling problems the solution approach uses the theory of strong valid inequalities that we developed in a series of earlier papers here we report specifically on the implementation of an experimental system mpsarx which consists of the sciconic mathematical programming system and an automatic reformulation executor arx that use this theory and on the results obtained with this system

a global approach for general 0 1 fractional programming abstract current methods of general 0 1 fractional programming g fp can only find the local optimum this paper proposes a new method of solving g fp problems by a mixed 0 1 linear program to obtain a global optimum given a mixed 0 1 polynomial term xy where x is a 0 1 variable and 0 y 1 we develop a theorem to transfer the xy term into a set of mixed 0 1 linear inequalities based on this theorem a g fp problem can be solved by a branch and bound method to obtain the global solution

design of survivable networks using three and four partition facets this paper considers the problem of designing a multicommodity network with single facility type subject to the requirement that under failure of any single edge the network should permit a feasible flow of all traffic we study the polyhedral structure of the problem by considering the multigraph obtained by shrinking the nodes but not the edges in a k partition of the original graph a key theorem is proved according to which a facet of the k node problem defined on the multigraph resulting from a k partition is also facet defining for the larger problem under a mild condition after reviewing the prior work on two partition inequalities we develop two classes of three partition inequalities and a large number of inequality classes based on four partitions proofs of facet defining status for some of these are provided while the rest are stated without proof computational results show that the addition of three and four partition inequalities results in substantial increase in the bound values compared to those possible with two partition inequalities alone problems of 35 nodes and 80 edges with fully dense traffic matrices have been solved optimally within a few minutes of computer time

modeling and solving the two facility capacitated network loading problem this paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry in this problem given point to point communication demand in a network must be met by installing loading capacitated facilities on the arcs loading a facility incurs an arc specific and facility dependent cost this paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost we consider two approaches for solving the underlying mixed integer program a lagrangian relaxation strategy and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem we show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the lagrangian relaxation bound our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation from an average of 25 to an average of 8 these results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry

a note on a global approach for general 0 1 fractional programming abstract based on an approach recently proposed for obtaining global optimal solution of general 0 1 fractional programming g fp problem a theorem is presented in this research through this theorem the nonlinear terms appearing in the formulation transformed from g fp can be directly replaced by a set of linear inequalities the g fp problem can hence be easily and more efficiently solved as a mixed integer program

on the polynomial mixed 0 1 fractional programming problems abstract this paper proposes a new method of solving polynomial mixed 0 1 fractional programming p01fp problems to obtain a global optimum given a polynomial 0 1 term x 1 x 2 x n y where x i is a 0 1 variable and y is a continuous variable we develop a linearization technique to transfer the x 1 x 2 x n y term into a set of mixed 0 1 linear inequalities based on this technique the p01fp can then be solved by a branch and bound method to obtain the global solution

the convex hull of two core capacitated network design problems the network loading problem nlp is a specialized capacitated network design problem in which prescribed point to point demand between various pairs of nodes of a network must be met by installing loading a capacitated facility we can load any number of units of the facility on each of the arcs at a specified arc dependent cost the problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost

discretized formulations for capacitated location problems with modular distribution costs abstract in this paper we study a discretization reformulation technique in the context of a facility location problem with modular link costs we present a so called traditional model and a straightforward discretized model with a general objective function whose variable coefficients are computed by solving a simple knapsack problem we show that the linear programming relaxation of the discretized model dominates the linear programming relaxation of the original model the discretized model suggests quite intuitive valid inequalities that considerably improve the linear programming relaxation of the original model computational results based on randomly generated data show that in the context of problems with modular costs the proposed discretized models perform significantly better than the traditional models an important outcome of our research is the result whose proof is also presented in this paper that a restricted version of the discretized model gives an extended description of the convex hull of the integer solutions of a subproblem that usually arises in network design problems with modular costs

strong inequalities for capacitated survivable network design problems we present several classes of facet defining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints these problems typically involve assigning capacities to a network with multicommodity demands such that after a vertex or edge deletion at least some prescribed fraction of each demand can be routed

polyhedral results for the edge capacity polytope network loading problems occur in the design of telecommunication networks in many different settings for instance bifurcated or non bifurcated routing also called splittable and unsplittable can be considered in most settings the same polyhedral structures return a better understanding of these structures therefore can have a major impact on the tractability of polyhedral guided solution methods in this paper we investigate the polytopes of the problem restricted to one arc edge of the network the undirected directed edge capacity problem for the non bifurcated routing case as an example one of the basic variants of network loading is described including an integer linear programming formulation as the edge capacity problems are relaxations of this network loading problem their polytopes are intimately related we give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope we describe classes of strong valid inequalities for the edge capacity polytopes and we derive conditions under which these constraints define facets for the diverse classes the complexity of lifting projected variables is stated the derived inequalities are tested on i the edge capacity problem itself and ii the described variant of the network loading problem the results show that the inequalities substantially reduce the number of nodes needed in a branch and cut approach moreover they show the importance of the edge subproblem for solving network loading problems

a strong cutting plane algorithm for production scheduling with changeover costs changeover costs and times are central to numerous manufacturing operations these costs arise whenever work centers capable of processing only one product at a time switch from the manufacture of one product to another although many researchers have contributed to the solution of scheduling problems that include changeover costs due to the problem s combinatorial explosiveness optimization based methods have met with limited success in this paper we develop and apply polyhedral methods from integer programming for a dynamic version of the problem computational tests with problems containing one to five products and up to 225 integer variables show that polyhedral methods based upon a set of facet inequalities developed in this paper can effectively reduce the gap between the value of an integer programming formulation of the problem and its linear programming relaxation by a factor of 94 to 100 these results suggest the use of a combined cutting plane branch and bound procedure as a solution approach in a test with a five product problem this procedure when compared with a standard linear programming based branch and bound approach reduced computation time by a factor of seven

a cutting plane algorithm for the linear ordering problem the linear ordering problem is an np hard combinatorial optimization problem with a large number of applications including triangulation of input output matrices archaeological senation minimizing total weighted completion time in one machine scheduling and aggregation of individual preferences in a former paper we have investigated the facet structure of the 0 1 polytope associated with the linear ordering problem here we report on a new algorithm that is based on these theoretical results the main part of the algorithm is a cutting plane procedure using facet defining inequalities this procedure is combined with various heuristics and branch and bound techniques our computational results compare favorably with the results of existing codes in particular we could triangulate all input output matrices of size up to 60 60 available to us within acceptable time bounds

the continuous knapsack set we study the convex hull of the continuous knapsack set which consists of a single inequality constraint with n n non negative integer and m m non negative bounded continuous variables when n 1 n 1 this set is a generalization of the single arc flow set studied by magnanti et al math program 60 233 250 1993 we first show that in any facet defining inequality the number of distinct non zero coefficients of the continuous variables is bounded by 2 n n 2n n our next result is to show that when n 2 n 2 this upper bound is actually 1 this implies that when n 2 n 2 the coefficients of the continuous variables in any facet defining inequality are either 0 or 1 after scaling and that all the facets can be obtained from facets of continuous knapsack sets with m 1 m 1 the convex hull of the sets with n 2 n 2 and m 1 m 1 is then shown to be given by facets of either two variable pure integer knapsack sets or continuous knapsack sets with n 2 n 2 and m 1 m 1 in which the continuous variable is unbounded the convex hull of these two sets has been completely described by agra and constantino discrete optim 3 95 110 2006 finally we show via an example that when n 3 n 3 the non zero coefficients of the continuous variables can take different values

survivable network design with shared protection routing abstract in this paper we study the problem of designing a survivable telecommunication network with shared protection routing we develop a heuristic algorithm to solve this problem recent results in the area of global re routing have been used to obtain very tight lower bounds for the problem our results indicate that in a majority of problem instances the average gap between the heuristic solutions and the lower bounds is within 5 computational experience is reported on randomly generated problem instances with up to 35 nodes 80 edges and 595 demand pairs and also on the instances available in sndlib database