the equilibrium generalized assignment problem and genetic algorithm abstract the well known generalized assignment problem gap is to minimize the costs of assigning n jobs to m capacity constrained agents or machines such that each job is assigned to exactly one agent this problem is known to be np hard and it is hard from a computational point of view as well in this paper follows from practical point of view in real systems the gap is extended to the equilibrium generalized assignment problem egap and the equilibrium constrained generalized assignment problem ecgap a heuristic equilibrium strategy based genetic algorithm ga is designed for solving the proposed egap finally to verify the computational efficiency of the designed ga some numerical experiments are performed on some known benchmarks the test results show that the designed ga is very valid for solving egap

branch and price algorithms for the dual bin packing and maximum cardinality bin packing problem there appear to be two versions of the dual bin packing problem in the literature in addition one of the versions has a counterpart in the cutting stock literature known as the skiving stock problem this paper outlines branch and price algorithms for both we introduce combinatorial upper bounds and well performing heuristics from the literature in the branch and price framework extensive computational tests indicate that the branch and price approach is superior to the existing branch and bound procedures based on combinatorial bounds the tests illustrate the influence of different problem characteristics on the computation time and the limits of the branch and price approach c 2004 elsevier b v all rights reserved

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

classification of dantzig wolfe reformulations for binary mixed integer programming problems abstract in this note we provide a classification of dantzig wolfe reformulations for binary mixed integer programming problems we specifically focus on modeling the binary conditions in the convexification approach to the dantzig wolfe decomposition for a general binary mixed integer programming problem an extreme point of the overall problem does not necessarily correspond to an extreme point of the subproblem therefore the binary conditions cannot in general be imposed on the new master problem variables but must be imposed on the original binary variables in some cases however it is possible to impose the binary restrictions directly on the new master problem variables the issue of imposing binary conditions on the original variables versus the master problem variables has not been discussed systematically for mip problems in general in the literature and most of the research has been focused on the pure binary case the classification indicates in which cases you can and cannot impose binary conditions on the new master problem variables

a set partitioning heuristic for the generalized assignment problem this paper discusses a heuristic for the generalized assignment problem gap the objective of gap is to minimize the costs of assigning j jobs to m capacity constrained machines such that each job is assigned to exactly one machine the problem is known to be np hard and it is hard from a computational point of view as well the heuristic proposed here is based on column generation techniques and yields both upper and lower bounds on a set of relatively hard test problems the heuristic is able to find solutions that are on average within 0 13 from optimality

on dantzig wolfe decomposition in integer programming and ways to perform branching in a branch and price algorithm dantzig wolfe decomposition as applied to an integer program is a specific form of problem reformulation that aims at providing a tighter linear programming relaxation bound the reformulation gives rise to an integer master problem whose typically large number of variables is dealt with implicitly by using an integer programming column generation procedure also known as branch and price algorithm there is a large class of integer programs that are well suited for this solution technique in this paper we propose to base the dantzig wolfe decomposition of an integer program on the discretization of the integer polyhedron associated with a subsystem of constraints as opposed to its convexification this allows us to formulate the integrality restriction directly on the master variables and sets a theoretical framework for dealing with specific issues such as branching or the introduction of cutting planes in the master we discuss specific branching schemes and their effect on the structure of the column generation subproblem we give theoretical bounds on the complexity of the separation process and the extent of the modifications to the column generation subproblem our computational tests on the cutting stock problem and a generalisation the cutting strip problem show that in practice all fractional solutions can be eliminated using branching rules that preserve the tractability of the subproblem but there is a trade off between branching efficiency and subproblem tractability

a new optimization algorithm for the vehicle routing problem with time windows the vehicle routing problem with time windows vrptw is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service in this paper we present the development of a new optimization algorithm for its solution the lp relaxation of the set partitioning formulation of the vrptw is solved by column generation feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming the lp solution obtained generally provides an excellent lower bound that is used in a branch and bound algorithm to solve the integer set partitioning formulation our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark vrptw test problems the algorithm was capable of optimally solving 100 customer problems this problem size is six times larger than any reported to date by other published research

selected topics in column generation dantzig wolfe decomposition and column generation devised for linear programs is a success story in large scale integer programming we outline and relate the approaches and survey mainly recent contributions not yet found in textbooks we emphasize the growing understanding of the dual point of view which has brought considerable progress to the column generation theory and practice it stimulated careful initializations sophisticated solution techniques for the restricted master problem and subproblem as well as better overall performance thus the dual perspective is an ever recurring concept in our selected topics

a path relinking approach with ejection chains for the generalized assignment problem the generalized assignment problem is a classical combinatorial optimization problem known to be np hard it can model a variety of real world applications in location allocation machine assignment and supply chains the problem has been studied since the late 1960s and computer codes for practical applications emerged in the early 1970s we propose a new algorithm for this problem that proves to be more effective than previously existing methods the algorithm features a path relinking approach which is a mechanism for generating new solutions by combining two or more reference solutions it also features an ejection chain approach which is embedded in a neighborhood construction to create more complex and powerful moves computational comparisons on benchmark instances show that the method is not only effective in general but is especially effective for types d and e instances which are known to be very difficult

combinatorial optimization current successes and directions for the future our ability to solve large important combinatorial optimization problems has improved dramatically in the past decade the availability of reliable software extremely fast and inexpensive hardware and high level languages that make the modeling of complex problems much faster have led to a much greater demand for optimization tools this paper highlights the major breakthroughs and then describes some very exciting future opportunities previously large research projects required major data collection efforts expensive mainframes and substantial analyst manpower now we can solve much larger problems on personal computers much of the necessary data is routinely collected and tools exist to speed up both the modeling and the post optimality analysis with the information technology revolution taking place currently we now have the opportunity to have our tools embedded into supply chain systems that determine production and distribution schedules process design and location allocation decisions these tools can be used industry wide with only minor modifications being done by each user

optimization models for targeted offers in direct marketing exact and heuristic algorithms this paper presents an optimization model for the selection of sets of clients that will receive an offer for one or more products during a promotion campaign we show that the problem is strongly np hard and that it is unlikely that a constant factor approximation algorithm can be proposed for solving this problem we propose an alternative set covering formulation and develop a branch and price algorithm to solve it we also describe eight heuristics to approximate an optimal solution including a depth first branch and price heuristic and a tabu search algorithm we perform extensive computational experiments both with the exact as well as with the heuristic algorithms based on our experiments we suggest the use of optimal algorithms for small and medium size instances while heuristics especially tabu search and branch and price based routines are preferable for large instances and when time is an important factor

a branch and price algorithm for the multilevel generalized assignment problem the multilevel generalized assignment problem mgap is a variation of the generalized assignment problem in which agents can execute tasks at different efficiency levels with different costs we present a branch and price algorithm that is the first exact algorithm for the mgap it is based on a decomposition into a master problem with set partitioning constraints and a pricing subproblem that is a multiple choice knapsack problem we report on our computational experience with randomly generated instances with different numbers of agents tasks and levels and with different correlations between cost and resource consumption for each agent task level assignment experimental results show that our algorithm is able to solve instances larger than those of the maximum size considered in the literature to proven optimality

a hybrid tabu search branch bound approach to solving the generalized assignment problem a new approach for solving the generalized assignment problem gap is proposed that combines the exact branch bound approach with the heuristic strategy of tabu search ts to produce a hybrid algorithm for solving gap the algorithm described uses commercial software to solve sub problems generated by the ts guiding strategy the ts approach makes use of the concept of referent domain optimisation and introduces novel add drop strategies in addition the linear programming relaxation of gap that forms part of the branch bound approach is itself helpful in suggesting which variables might take binary values computational results on benchmark test instances are presented and compared with results obtained by the standard branch bound approach and also several other heuristic approaches from the literature the results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances

a tabu search heuristic for the generalized assignment problem abstract this paper considers the generalized assignment problem gap it is a well known np hard combinatorial optimization problem that is interesting in itself and also appears as a subproblem in other problems of practical importance a tabu search heuristic for the gap is proposed the algorithm uses recent and medium term memory to dynamically adjust the weight of the penalty incurred for violating feasibility the most distinctive features of the proposed algorithm are its simplicity and its flexibility these two characteristics result in an algorithm that compared to other well known heuristic procedures provides good quality solutions in competitive computational times computational experiments have been performed in order to evaluate the behavior of the proposed algorithm

a new dantzig wolfe reformulation and branch and price algorithm for the capacitated lot sizing problem with setup times although the textbook dantzig wolfe decomposition reformulation for the capacitated lot sizing problem as already proposed by manne manne a s 1958 programming of economic lot sizes management sci 4 2 115 135 provides a strong lower bound it also has an important structural deficiency imposing integrality constraints on the columns in the master program will not necessarily give the optimal integer programming solution manne s model contains only production plans that satisfy the wagner whitin property and it is well known that the optimal solution to a capacitated lot sizing problem will not necessarily satisfy this property the first contribution of this paper answers the following question unsolved for almost 50 years if manne s formulation is not equivalent to the original problem what is then a correct reformulation we develop an equivalent mixed integer programming mip formulation to the original problem and show how this results from applying the dantzig wolfe decomposition to the original mip formulation the set of extreme points of the lot size polytope that are needed for this mip dantzig wolfe reformulation is much larger than the set of dominant plans used by manne we further show how the integrality restrictions on the original setup variables translate into integrality restrictions on the new master variables by separating the setup and production decisions our new formulation gives the same lower bound as manne s reformulation second we develop a branch and price algorithm for the problem computational experiments are presented on data sets available from the literature column generation is accelerated by a combination of simplex and subgradient optimization for finding the dual prices the results show that branch and price is computationally tractable and competitive with other state of the art approaches found in the literature

on the cauchy problem for axi symmetric vortex rings we consider the classical cauchy problem for the three dimensional navier stokes equation with the initial vorticity 0 concentrated on a circle or more generally a linear combination of such data for circles with common axis of symmetry we show that natural approximations of the problem obtained by smoothing the initial data satisfy good uniform estimates which enable us to conclude that the original problem with the singular initial distribution of vorticity has a solution we impose no restriction on the size of the initial data

a survey of algorithms for the generalized assignment problem abstract this paper surveys algorithms for the well known problem of finding the minimum cost assignment of jobs to agents so that each job is assigned exactly once and agents are not overloaded all approaches seem to be based on branch and bound with bound supplied through heuristics and through relaxations of the primal problem formulation from the survey one can select building blocks for the design of one s own tailor made algorithm the survey also reveals that although just about every mathematical programming technique was tried on this problem there is still a lack of a representative set of test problems on which competing enumeration algorithms can be compared as well as a shortage of effective heuristics

global time varying stabilization of underactuated surface vessel this note considers the stabilization problem of an underactuated surface vessel three global smooth time varying control laws are proposed with the aid of different techniques the first proposed control law makes the state of the closed loop system asymptotically converge to zero while the second and the third control laws make the state of the closed loop system globally exponentially converge to zero moreover the exponential convergence rate of the state of the closed loop system can be arbitrarily assigned with the third control law simulation results show that the proposed control laws are effective

generating experimental data for the generalized assignment problem the generalized assignment problem gap is the problem of finding the minimal cost assignment of jobs to machines such that each job is assigned to exactly one machine subject to capacity restrictions on the machines we propose a new stochastic model for the gap a tight condition on this stochastic model under which the gap is feasible with probability one when the number of jobs goes to infinity is derived this new stochastic model enables us to analyze the adequacy of most of the random generators given for the gap in the literature we demonstrate that the random generators commonly used to test solution procedures for the gap tend to create easier problem instances when the number of machines m increases we describe a greedy heuristic for the gap and use it to illustrate the results from the paper

technical note an improved dual based algorithm for the generalized assignment problem the generalized assignment problem gap determines the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent subject to an agent s capacity existing solution algorithms have not solved problems with more than 100 decision variables this paper designs an optimization algorithm for the gap that effectively solves problems with up to 500 variables compared with existing procedures this algorithm requires fewer enumeration nodes and shorter running times improved performance stems from an enhanced lagrangian dual ascent procedure that solves a lagrangian dual at each enumeration node adding a surrogate constraint to the lagrangian relaxed model and an elaborate branch and bound scheme an empirical investigation of various problem structures not considered in existing literature is also presented

a branch and price algorithm for the generalized assignment problem the generalized assignment problem examines the maximum profit assignment of jobs to agents such that each job is assigned to precisely one agent subject to capacity restrictions on the agents a new algorithm for the generalized assignment problem is presented that employs both column generation and branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem

robust optimization approach for a chance constrained binary knapsack problem we consider a certain class of chance constrained binary knapsack problem where each item has a normally distributed random weight that is independent of the other items for this problem we propose an efficient pseudo polynomial time algorithm based on the robust optimization approach for finding a solution with a theoretical bound on the probability of satisfying the knapsack constraint our algorithm is tested on a wide range of random instances and the results demonstrate that it provides qualified solutions quickly in contrast a state of the art mip solver is only applicable for instances of the problem with a restricted number of items

effective algorithm and heuristic for the generalized assignment problem we present new branch and bound algorithm for the generalized assignment problem a standard subgradient method sm used at each node of the decision tree to solve the lagrangian dual provides an upper bound our key contribution in this paper is a new heuristic applied at each iteration of the sm which tries to exploit the solution of the relaxed problem by solving a smaller generalized assignment problem the feasible solution found is then subjected to a solution improvement heuristic we consider processing the root node as a lagrangian heuristic computational comparisons are made with new existing methods

exact solutions to a class of stochastic generalized assignment problems abstract this paper deals with a stochastic generalized assignment problem with recourse only a random subset of the given set of jobs will require to be actually processed an assignment of each job to an agent is decided a priori and once the subset of jobs which have to be executed is known reassignments can be performed if there are overloaded agents we construct a convex approximation of the objective function that is sharp at all feasible solutions we then present three versions of an exact algorithm to solve this problem based on branch and bound techniques optimality cuts and a special purpose lower bound numerical results are reported

harvest scheduling subject to maximum area restrictions exploring exact approaches we consider a spatial problem arising in forest harvesting for regulatory reasons blocks harvested should not exceed a certain total area typically 49 hectares traditionally this problem called the adjacency problem has been approached by forming a priori blocks from basic cells of 5 to 25 hectares and solving the resulting mixed integer program superior solutions can be obtained by including the construction of blocks in the decision process the resulting problem is far more complex combinatorially we present an exact algorithmic approach that has yielded good results in computational tests this solution approach is based on determining a strong formulation of the linear programming problem through a clique representation of a projected problem

column generation in the integral simplex method the integral simplex method for set partitioning problems allows only pivots on one to be made which results in a primal all integer method in this technical note we outline how to tailor the column generation principle to this method because of the restriction to pivots on one only local optimality can be guaranteed and to ensure global optimality we consider the use of implicit enumeration

logic cuts for multilevel generalized assignment problems abstract in the multilevel generalized assignment problem mgap agents can perform tasks at more than one efficiency level important manufacturing problems such as lot sizing can be easily formulated as mgaps however the large number of variables in the related 0 1 integer program makes it hard to find optimal solutions to these problems even when using powerful commercial optimization packages the mgap includes a set of knapsack constraints one per agent that can be useful for generating simple logical constraints or logic cuts we exploit the fact that logic cuts can be generated in linear time and can be easily added to the model before solving it with classical branch and bound methodology we generate all contiguous 1 cuts for every knapsack in large mgap s problems and report the effect of adding these cuts in our experimental results

a new lagrangian relaxation approach to the generalized assignment problem abstract in this paper we present a new lagrangian relaxation approach to the generalized assignment problem gap the new approach is based on a reformulation of gap into an equivalent problem which is then relaxed by traditional lagrangian relaxation techniques the reformulation is created by introducing a set of auxiliary variables and a number of coupling constraints by relaxing the coupling constraints we get subproblems where both types of constraint structures present in the gap are active the lower bound from the lagrangian dual of this approach is shown to be at least as strong as that from the best of the traditional lagrangian relaxation approaches an important feature is also the possibility of generating upper bounds quite easily the new approach has been tested on 10 problems of size m 4 n 25 i e 100 0 1 variables and the results have been compared with the traditional approaches

algorithms for the generalized quadratic assignment problem combining lagrangean decomposition and the reformulation linearization technique in this paper we propose two exact algorithms for the gqap generalized quadratic assignment problem in this problem given m facilities and n locations the facility space requirements the location available space the facility installation costs the flows between facilities and the distance costs between locations one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized this problem generalizes the well known quadratic assignment problem qap both exact algorithms combine a previously proposed branch and bound scheme with a new lagrangean relaxation procedure over a known rlt reformulation linearization technique formulation we also apply transformational lower bounding techniques to improve the performance of the new procedure we report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time six of these instances were open

a multiplier adjustment method for the generalized assignment problem we describe a branch and bound algorithm for the generalized assignment problem in which bounds are obtained from a lagrangian relaxation with the multipliers set by a heuristic adjustment method the algorithm was tested on a large sample of small random problems and a number of large problems derived from a vehicle routing application computation times were reasonable in all cases and the branch and bound trees generated had nearly two orders of magnitude fewer nodes than for competing algorithms although comparison of running times on different machines is difficult the multiplier adjustment method appears to be about one order of magnitude faster than the best previously existing algorithms for this problem