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

dynamic recomputation cannot extend the optimality range of priority indices sequencing rules that rely on priority indices are particularly attractive because they are simple to calculate and easy to implement it has been established that there are exactly two classes of delay cost functions for which policies that are determined by time invariant priority indices are capable of producing optimal sequences linear delay costs and discounted linear delay costs we consider index based policies that allow dynamic recalculation of priority indices and show that this class does not enlarge the set of delay cost functions for which index based rules can produce optimal policies our analysis relies on the argument that index based rules must induce a transitive ranking of the tasks we show that the classes of linear and discounted linear functions are the only ones that can be associated with such rankings

single machine scheduling to minimize total weighted earliness subject to minimal number of tardy jobs abstract motivated by just in time manufacturing we consider a single machine scheduling problem with dual criteria i e the minimization of the total weighted earliness subject to minimum number of tardy jobs we discuss several dominance properties of optimal solutions we then develop a heuristic algorithm with time complexity o n 3 and a branch and bound algorithm to solve the problem the computational experiments show that the heuristic algorithm is effective in terms of solution quality in many instances while the branch and bound algorithm is efficient for medium size problems

dominance and decomposition heuristics for single machine scheduling new heuristic dominance rules and a flexible decomposition heuristic are developed for the problem of minimizing weighted tardiness on a single processor extensive computational experience demonstrates that when our new heuristic dominance rules were incorporated into an optimal algorithm optimal or nearly optimal solutions were obtained quickly in fact solution times were orders of magnitude faster than those using the optimal algorithm alone on larger problems our decomposition heuristic obtained better solutions than previous heuristics furthermore on 50 job problems our decomposition heuristic obtained an optimal solution over ten times more often on the average than the best competing heuristic 22 versus 2 of the time since both our approaches are basically relaxations of optimal solution algorithms they could easily be adapted for use in the solution of other scheduling problems

on the equivalence of the max min transportation lower bound and the time indexed lower bound for single machine scheduling problems new observations are made about two lower bound schemes for single machine min sum scheduling problems we find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program we show the equivalence of this strongest bound and the bound provided by the lp relaxation of the time indexed integer programming formulation these observations lead to a new lower bound scheme that yields fast approximation of the time indexed bound several techniques are developed to facilitate the effective use of the new lower bound in branch and bound numerical experiments are conducted on 375 benchmark problems of the total weighted tardiness problem from or library results obtained with our new method are spectacular we are able to solve all 125 open problems to optimality

comparing efficiencies of genetic crossover operators for one machine total weighted tardiness problem in this study the well known one machine problem with the performance criterion of minimizing total weighted tardiness is considered this problem is known to be np hard and consists of one machine and n independent jobs each of these jobs has a distinct integer processing time a distinct integer weighting factor and a distinct integer due date the purpose of this problem is to find a sequence of these jobs minimizing the sum of the weighted tardiness using benchmarking problems this study compares performances of eleven genetic crossover operators which have been widely used to solve other types of hard scheduling problems

a multi start dynasearch algorithm for the time dependent single machine total weighted tardiness scheduling problem we extend the dynasearch technique recently proposed by congram et al in the context of time dependent combinatorial optimization problems as an application we consider a general time dependent idleness version of the well known single machine total weighted tardiness scheduling problem in which the processing time of a job depends on its starting time of execution we develop a multi start local search algorithm and present experimental results on several types of instances showing the superiority of the dynasearch neighborhood over the traditional one

single machine earliness and tardiness scheduling abstract we examine the problem of scheduling a given set of jobs on a single machine to minimize total early and tardy costs without considering machine idle time we decompose the problem into two subproblems with a simpler structure then the lower bound of the problem is the sum of the lower bounds of two subproblems a lower bound of each subproblem is obtained by lagrangian relaxation rather than using the well known subgradient optimization approach we develop two efficient multiplier adjustment procedures with complexity o nlog n to solve two lagrangian dual subproblems a branch and bound algorithm based on the two efficient procedures is presented and is used to solve problems with up to 50 jobs hence doubling the size of problems that can be solved by existing branch and bound algorithms we also propose a heuristic procedure based on the neighborhood search approach the computational results for problems with up to 3 000 jobs show that the heuristic procedure performs much better than known heuristics for this problem in terms of both solution efficiency and quality in addition the results establish the effectiveness of the heuristic procedure in solving realistic problems to optimality or near optimality

the two machine flowshop scheduling problem with sequence independent setup times new lower bounding strategies abstract the two machine flowshop environment with sequence independent setup times has been intensely investigated both from theoretical and practical perspectives in the scheduling literature nevertheless very scant attention has been devoted to deriving effective lower bounding strategies in this paper we propose new lower bounds for the total completion time minimization criterion these bounds are based on three relaxation schemes namely the waiting time based relaxation scheme the single machine based relaxation scheme and the lagrangian relaxation scheme extensive computational study carried on instances with up to 500 jobs reveals that embedding the waiting time based bounding strategy within the lagrangian relaxation framework yields the best performance while requiring negligible cpu time

new precedence theorems for one machine weighted tardiness in an earlier paper by emmons emmons h 1969 one machine sequencing to minimize certain functions of job tardiness oper res 17 701 715 the problem of sequencing jobs on a single machine in order to minimize total tardiness was analyzed emmons provided three theorems for specifying precedence relations for pairs of jobs his theorems apply when the tardiness penalty for each job grows at the same rate rinnooy kan et al rinnooy kan a h g b j lageweg j k lenstra 1975 minimizing total costs in one machine scheduling oper res 23 908 927 later extended emmons s theorems to the case when job tardiness penalties can grow at different rates for different jobs provided here is a set of theorems stronger than those of rinnooy kan et al that more fully exploits the special properties of the weighted tardiness function allowing for greater reduction of the solution space

stronger lagrangian bounds by use of slack variables applications to machine scheduling problems lagrangian relaxation is a powerful bounding technique that has been applied successfully to manynp hard combinatorial optimization problems the basic idea is to see annp hard problem as an easy to solve problem complicated by a number of nasty side constraints we show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds the trick is widely applicable but we focus on a broad class of machine scheduling problems for which it is particularly useful we provide promising computational results for three problems belonging to this class for which lagrangian bounds have appeared in the literature the single machine problem of minimizing total weighted completion time subject to precedence constraints the two machine flow shop problem of minimizing total completion time and the single machine problem of minimizing total weighted tardiness

an exact algorithm for the precedence constrained single machine scheduling problem abstract this study proposes an efficient exact algorithm for the precedence constrained single machine scheduling problem to minimize total job completion cost where machine idle time is forbidden the proposed algorithm is based on the ssdp successive sublimation dynamic programming method and is an extension of the authors previous algorithms for the problem without precedence constraints in this method a lower bound is computed by solving a lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence constrained total weighted tardiness and total weighted earliness tardiness problems and most instances with 100 jobs of the former problem

scheduling on parallel identical machines to minimize total tardiness this paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs we develop dominance properties and lower bounds and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

an algorithm for single machine sequencing with deadlines to minimize total weighted completion time abstract each of n jobs is to be processed without interruption on a single machine each job becomes available for processing at time zero has a deadline by which it must be completed and has a positive weight the objective is to find a processing order of the jobs which minimizes the sum of weighted completion times in this paper a branch and bound algorithm for the problem is presented which incorporates lower bounds that are obtained using a new technique called the multiplier adjustment method firstly several dominance conditions are derived then a heuristic is described and sufficient conditions for its optimality are given the lower bound is obtained by performing a lagrangean relaxation of the deadline constraints the lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimal solution of the relaxed problem thus yielding a lower bound the algorithm is tested on problems with up to fifty jobs

an irregular filter model in this paper we introduce a new filter model which is of a kind that has escaped investigation up to now it is induced by an intersection type theory generated in a non standard way by a preorder which puts into relation an atom with an arrow type without equating them we study the domain theoretic implications of this choice that are not trivial in order to describe this filter model a new category is introduced and a special purpose functor defined the filter model is then characterized as the initial algebra of the functor

a tabu search algorithm for the single machine total weighted tardiness problem abstract in this study a tabu search ts approach to the single machine total weighted tardiness problem smtwt is presented the problem consists of a set of independent jobs with distinct processing times weights and due dates to be scheduled on a single machine to minimize total weighted tardiness the theoretical foundation of single machine scheduling with due date related objectives reveal that the problem is np hard rendering it a challenging area for meta heuristic approaches this paper presents a totally deterministic ts algorithm with a hybrid neighborhood and dynamic tenure structure and investigates the strength of several candidate list strategies based on problem specific characteristics in increasing the efficiency of the search the proposed ts approach yields very high quality results for a set of benchmark problems obtained from the literature

existence and non existence of global solutions to initial boundary value problems for nonlinear evolution equations with strong dissipation abstract the main purpose in this paper is to investigate existence and non existence of global solutions of the initial dirichlet boundary value problem for evolution equations with strong dissipation many authors studied classes consisting of such types of equations for which initial boundary value problems possess global solutions for this purpose we consider a related problem and seek global solutions and blow up solutions of it depending on whether it belongs to such classes or not

single machine scheduling of unit time jobs with earliness and tardiness penalties the problem of determining a schedule of jobs with unit time lengths on a single machine that minimizes the total weighted earliness and tardiness penalties with respect to arbitrary rational due dates is formulated as an integer programming problem we show that if the penalties meet a certain criterion called the dominance condition then there exists an extremal optimal solution to the lp relaxation that is integral leading to a polynomial time solution procedure the general weighted symmetric penalty structure is one cost structure that satisfies the dominance condition we point out other commonly found penalty structures that also fall into this category

a branch and bound algorithm for the two stage assembly scheduling problem this paper develops a branch and bound algorithm for the two stage assembly scheduling problem in this problem there are m machines at the first stage each of which produces a component of a job when all m components are available a single assembly machine at the second stage completes the job the objective is to schedule the jobs on the machines so that the maximum completion time or makespan is minimized a lower bound based on solving an artificial two machine flow shop problem is derived also several dominance theorems are established and incorporated into the branch and bound algorithm computational experience with the algorithm is reported for problems with up to 8000 jobs and 10 first stage machines

dual decomposition of a single machine scheduling problem we design a fast ascent direction algorithm for the lagrangian dual problem of the single machine scheduling problem of minimizing total weighted completion time subject to precedence constraints we show that designing such an algorithm is relatively simple if a scheduling problem is formulated in terms of the job completion times rather than as an 0 1 linear program also we show that upon termination of such an ascent direction algorithm we get a dual decomposition of the original problem which can be exploited to develop approximative and enumerative approaches for it computational results exhibit that in our application the ascent direction leads to good lagrangian lower and upper bounds

minimizing total tardiness on a single machine with unequal release dates this study addresses the problem of minimizing total tardiness on a single machine with unequal release dates dominance properties established in previous literatures and herein are adopted to develop branch and bound and heuristic procedures computational experiments were conducted to evaluate the approaches the results revealed that the branch and bound algorithm is efficient in solving hard problems and easy problems that involve up to 50 and 500 jobs respectively the computational effectiveness of the heuristic is also reported

one machine sequencing to minimize total tardiness a fourth theorem for emmons in 1969 h emmons provided three theorems emmons 1 3 for determining precedence relations between pairs of jobs for the single machine tardiness problem we show here a fourth straightforward theorem that uses the information when the jobs in the pair are both known to precede a third job in an optimum sequence the new theorem augments the three emmons theorems and is shown to be a generalization of a theorem by elmaghraby

hybrid genetic algorithm for permutation flowshop scheduling problems with total flowtime minimization in this paper a hga hybrid genetic algorithm is proposed for permutation flowshop scheduling problems pfsp with total flowtime minimization which are known to be np hard one of the chromosomes in the initial population is constructed by a suitable heuristic and the others are yielded randomly an artificial chromosome is generated by a weighted simple mining gene structure with which a new crossover operator is presented additionally two effective heuristics are adopted as local search to improve all generated chromosomes in each generation the hga is compared with one of the most effective heuristics and a recent meta heuristic on 120 benchmark instances experimental results show that the hga outperforms the other two algorithms for all cases furthermore hga obtains 115 best solutions for the benchmark instances 92 of which are newly discovered

a dynamic programming method for single machine scheduling abstract this paper treats the single machine scheduling problem that minimizes the sum of general cost functions defined for jobs e g weighted sums of earliness and tardiness since the straightforward dynamic programming recursion requires 2 n states where n is the number of jobs an ssdp successive sublimation dynamic programming method is developed to keep the number of generated states within manageable level by utilizing several types of dynamic programming recursions representing varied degrees of state space relaxations computational results on problem instances of up to n 35 exhibit a clear superiority of this ssdp approach over the original dynamic programming recursion

a fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date we develop a fully polynomial time approximation scheme fptas for minimizing the weighted total tardiness on a single machine provided that all due dates are equal the fptas is obtained by converting an especially designed pseudopolynomial dynamic programming algorithm