existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in banach spaces in this paper we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a banach space we prove that ifsis an amenable semigroup and s tt t s is a nonexpansive semigroup on a closed convex subsetcin a uniformly convex banach spaceesuch that the setf s of common fixed points of s is nonempty then there exists a nonexpansive retractionpfromcontof s such thatptt ttp pfor eacht sandpx co ttx t s for eachx c in this case there exists a net a of finite averages of s such that for eacht sand for each bounded subsetbofc lim a ttx a x 0 and lim tta x a x 0 uniformly forx b also if the norm ofeis frechet differentiable then for eachx c pxis the unique common fixed point in s s co ttsx t s furthermore if is an asymptotically invariant net of means then for eachx c t x converges weakly topx

two agent scheduling in a flowshop abstract in this paper we study two agent scheduling in a two machine flowshop the cost function is the weighted sum of some common regular functions including the makespan and the total completion time specifically we consider two problems namely the problem to minimize the weighted sum of both agents makespan and the problem to minimize the weighted sum of one agent s total completion time and the other agent s makespan for the first problem we give an ordinary np hardness proof and a pseudo polynomial time algorithm we also analyze the performance of treating the problem using johnson s rule and propose an approximation algorithm based on johnson s rule for the second problem we propose an approximation algorithm based on linear programming relaxation of the problem finally we show that some simple algorithms can be used to solve special cases of the two problems

weighted flow time bounds for scheduling identical processors abstract we present two lower bounds for the identical parallel processor weighted flow time problem and identify theoretical properties regarding relationships among these and other bounds for the problem the new bounds along with other bounds from the literature are evaluated in a computational study and are found to be among the top three in performance conditions favorable to each of the top three performing bounds are identified

scheduling to minimize average completion time off line and on line approximation algorithms in this paper we introduce two general techniques for the design and analysis of approximation algorithms for np hard scheduling problems in which the objective is to minimize the weighted sum of the job completion times for a variety of scheduling models these techniques yield the first algorithms that are guaranteed to find schedules that have objective function value within a constant factor of the optimum in the first approach we use an optimal solution to a linear programming relaxation in order to guide a simple list scheduling rule consequently we also obtain results about the strength of the relaxation our second approach yields on line algorithms for these problems in this setting we are scheduling jobs that continually arrive to be processed and for each time t we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards our on line technique yields constant performance guarantees for a variety of scheduling environments and in some cases essentially matches the performance of our off line lp based algorithms

scheduling interfering job sets on parallel machines abstract we consider bicriteria scheduling on identical parallel machines in a nontraditional context jobs belong to two disjoint sets and each set has a different criterion to be minimized the jobs are all available at time zero and have to be scheduled non preemptively on m parallel machines the goal is to generate the set of all non dominated solutions so the decision maker can evaluate the tradeoffs and choose the schedule to be implemented we consider the case where for one of the two sets the criterion to be minimized is makespan while for the other the total completion time needs to be minimized given that the problem is np hard we propose an iterative spt lpt spt heuristic and a bicriteria genetic algorithm for the problem both approaches are designed to exploit the problem structure and generate a set of non dominated solutions in the genetic algorithm we use a special encoding scheme and also a unique strategy based on the properties of a non dominated solution to ensure that all parts of the non dominated front are explored the heuristic and the genetic algorithm are compared with a time indexed integer programming formulation for small and large instances results indicate that the both the heuristic and the genetic algorithm provide high solution quality and are computationally efficient the heuristics proposed also have the potential to be generalized for the problem of interfering job sets involving other bicriteria pairs

fixed point properties of semigroups of non expansive mappings abstract in recent years there have been considerable interests in the study of when a closed convex subset k of a banach space has the fixed point property i e whenever t is a non expansive mapping from k into k then k contains a fixed point for t in this paper we shall study fixed point properties of semigroups of non expansive mappings on weakly compact convex subsets of a banach space or more generally a locally convex space by considering the classes of bicyclic semigroups we answer two open questions one posted earlier by the first author in 1976 dalhousie and the other posted by t mitchell in 1984 virginia we also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non expansive mappings related to another open problem raised by the first author in 1976

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

lower bounds for the earliness tardiness scheduling problem on parallel machines with distinct due dates this paper addresses the parallel machine scheduling problem in which the jobs have distinct due dates with earliness and tardiness costs new lower bounds are proposed for the problem they can be classed into two families first two assignment based lower bounds for the one machine problem are generalized for the parallel machine case second a time indexed formulation of the problem is investigated in order to derive efficient lower bounds throught column generation or lagrangean relaxation a simple local search algorithm is also presented in order to derive an upper bound computational experiments compare these bounds for both the one machine and parallel machine problems and show that the gap between upper and lower bounds is about 1 5

minimizing weighted completion times with deadlines we examine a one machine scheduling problem whose objective is to minimize the weighted sum of completion times subject to deadline constraints we synthesize and extend current known theory and derive precedence relations between jobs these relationships along with tight new bounding criteria are incorporated in a branch and bound solution procedure

iterative algorithms for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of families and semigroups of nonexpansive mappings abstract we introduce iterative algorithms for finding a common element of the set of solutions of a system of equilibrium problems and of the set of fixed points of a finite family and a left amenable semigroup of nonexpansive mappings in a hilbert space we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality which is the optimality condition for a minimization problem our results extend for example the recent result of v colao g marino h k xu an iterative method for finding common solutions of equilibrium and fixed point problems j math anal appl 344 2008 340 352 to systems of equilibrium problems

a branch and cut algorithm for the resolution of large scale symmetric traveling salesman problems an algorithm is described for solving large scale instances of the symmetric traveling salesman problem stsp to optimality the core of the algorithm is a polyhedral cutting plane procedure that exploits a subset of the system of linear inequalities defining the convex hull of the incidence vectors of the hamiltonian cycles of a complete graph the cuts are generated by several identification procedures that have been described in a companion paper whenever the cutting plane procedure does not terminate with an optimal solution the algorithm uses a tree search strategy that as opposed to branch and bound keeps on producing cuts after branching the algorithm has been implemented in fortran two different linear programming lp packages have been used as the lp solver the implementation of the algorithm and the interface with one of the lp solvers is described in sufficient detail to permit the replication of our experiments computational results are reported with up to 42 stsps with sizes rangin

single machine scheduling with precedence constraints we discuss the problem of sequencing precedence constrained jobs on a single machine to minimize the average weighted completion time this problem has attracted much attention in the mathematical programming community since sidney s pioneering work in 1975 sidney j b 1975 decomposition algorithms for single machine scheduling with precedence relations and deferral costs operations research23 283 298 we look at the problem from a polyhedral perspective and uncover a relation between sidney s decomposition theorem and different linear programming relaxations more specifically we present a generalization of sidney s result which particularly allows us to reason that virtually all known 2 approximation algorithms are consistent with his decomposition moreover we establish a connection between the single machine scheduling problem and the vertex cover problem indeed in the special case of series parallel precedence constraints we prove that the sequencing problem can be seen as a special case of the vertex cover problem we also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem finally we give a 3 2 approximation algorithm for two dimensional partial orders and we also provide a characterization of the active inequalities of a linear programming relaxation in completion time variables

parallel machine scheduling linear programming and parameter list scheduling heuristics in this paper we consider a class of parallel machine scheduling problems and their associated set partitioning formulations we show that the tightness of the linear programming relaxation of these formulations is directly related to the performance of a class of heuristics called parameter list scheduling heuristics this makes it possible to characterize the worst possible gap between optimal solutions for the scheduling problems and the corresponding linear programming relaxations in the case of the classical parallel machine weighted completion time model we also show that the solution to the linear programming relaxation of the set partitioning formulation is asymptotically optimal under mild assumptions on the distribution of job weights and processing times finally we extend most of the results to the time discretized formulation of machine scheduling problems

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

extension and generalization properties of the weighted minkowski inverse in a minkowski space for an arbitrary matrix the weighted minkowski inverse a m n m n m related to the positive definite matrices m m m and n m n of an arbitrary matrix a m m n including singular and rectangular is one of the important generalized inverses for solving matrix equations in minkowski space in this paper the results are introduced in the following three ways first we establish some new and attractive properties of the weighted minkowski inverse a m n in a minkowski space second new representations and conditions for the continuity of the weighted minkowski inverse a m n in a minkowski space are discussed finally some illustrated counterexamples are also studied to show that some well known properties of the weighted moore penrose inverse a m n in a hilbert space h are not valid in the minkowski inverse a m n in a minkowski space

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

the relation of time indexed formulations of single machine scheduling problems to the node packing problem the relation of time indexed formulations of nonpreemptive single machine scheduling problems to the node packing problem is established and then used to provide simple and intuitive alternate proofs of validity and maximality for previously known results on the facial structure of the scheduling problem previous work on the facial structure has focused on describing the convex hull of the set of feasible partial schedules schedules in which not all jobs have to be started the equivalence between the characteristic vectors of this set and those of the set of feasible node packings in a graph whose structure is determined by the parameters of the scheduling problem is established the main contribution of this paper is to show that the facet inducing inequalities for the convex hull of the set of feasible partial schedules that have integral coefficients and right hand side 1 or 2 are the maximal clique inequalities and the maximally and sequentially lifted 5 hole inequalities of the convex hull of the set of feasible node packings in this graph respectively

hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings abstract recently takahashi and takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 2006 doi 10 1016 j jmaa 2006 08 036 suggested and analyzed an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space in this paper we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a hilbert space then we prove a strong convergence theorem which is the improvements and extension of takahashi and takahashi s 2006 corresponding result using this theorem we obtain two corollaries which improve and extend their corresponding results

optimal mechanism design for a sequencing problem with two dimensional types we study the design of mechanisms for a sequencing problem where the types of job agents consist of processing times and waiting costs that are private to the jobs in the bayes nash setting we seek to find a sequencing rule and incentive compatible payments that minimize the total expected payments that have to be made to the agents it is known that the problem can be efficiently solved when jobs have single dimensional types here we address the problem with two dimensional types we show that the problem can be solved in polynomial time by linear programming techniques answering an open problem formulated by heydenreich et al our implementation is randomized and truthful in expectation remarkably it also works when types are correlated across jobs the main steps are a compactification of an exponential size linear programming formulation and a convex decomposition algorithm that allows us to implement the optimal linear programming solution in addition by means of computational experiments we generate some new insights into the implementability in different equilibria

project scheduling with resource constraints a branch and bound approach abstract this paper describes a branch and bound algorithm for project scheduling with resource constraints the algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods four lower bounds are examined the first is a simple lower bound based on longest path computations the second and third bounds are derived from a relaxed integer programming formulation of the problem the second bound is based on the linear programming relaxation with the addition of cutting planes and the third bound is based on a lagrangean relaxation of the formulation this last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient though not polynomial algorithm is given the fourth bound is based on the disjunctive arcs used to model the problem as a graph we report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources

the achievable region approach to the optimal control of stochastic systems summary the achievable region approach seeks solutions to stochastic optimization problems by characterizing the space of all possible performances the achievable region of the system of interest and optimizing the overall system wide performance objective over this space this is radically different from conventional formulations based on dynamic programming the approach is explained with reference to a simple two class queuing system powerful new methodologies due to the authors and co workers are deployed to analyse a general multiclass queuing system with parallel servers and then to develop an approach to optimal load distribution across a network of interconnected stations finally the approach is used for the first time to analyse a class of intensity control problems

a time indexed formulation of non preemptive single machine scheduling problems we consider the formulation of non preemptive single machine scheduling problems using time indexed variables this approach leads to very large models but gives better lower bounds than other mixed integer programming formulations we derive a variety of valid inequalities and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities a cutting plane branch and bound algorithm based on these inequalities has been implemented computational experience on small problems with 20 30 jobs and various constraints and objective functions is presented

on the minimization of total weighted flow time with identical and uniform parallel machines we consider the np hard problem of scheduling jobs on identical parallel machines to minimize total weighted flow time we discuss the properties that characterize the structure of an optimal solution present a lower bound and propose a branch and bound algorithm the algorithm is superior to prior methods presented in the literature we also extend the algorithm to uniform parallel machines and solve medium sized problem instances

submodular function minimization submodular functions often arise in various fields of operations research including discrete optimization game theory queueing theory and information theory in this survey paper we give overview on the fundamental properties of submodular functions and recent algorithmic devolopments of their minimization

hybrid optimization methods for time dependent sequencing problems abstract in this paper we introduce novel optimization methods for sequencing problems in which the setup times between a pair of tasks depend on the relative position of the tasks in the ordering our proposed methods rely on a hybrid approach where a constraint programming model is enhanced with two distinct relaxations one discrete relaxation based on multivalued decision diagrams and one continuous relaxation based on linear programming both relaxations are used to generate bounds and enhance constraint propagation experiments conducted on three variants of the time dependent traveling salesman problem indicate that our techniques substantially outperform general purpose methods such as mixed integer linear programming and constraint programming models

worst case bound of an lrf schedule for the mean weighted flow time problem this paper studies the problem of scheduling a set of n independent tasks on m identical processors so as to minimize mean weighted flow time the problem is known to be np complete for m geqq 2 and to be np complete in the strong sense for m arbitrary the worst case behavior of a heuristic algorithm which requires time o n log n is investigated and it is shown that the mean weighted flow time obtained by the algorithm does not exceed sqrt 2 1 2 cong 1 0207 times that of an optimal schedule moreover the bound sqrt 2 1 2 is best possible

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

dissipative matrices and invariant maximal semidefinite subspaces abstract an explicit construction of invariant maximal semidefinite subspaces for a dissipative matrix in an indefinite scalar product is given necessary and sufficient conditions for uniqueness of such a subspace are given these subspaces play an important role in the explicit construction of pseudo canonical factorizations for certain rational matrix valued functions

a general lower bound for the makespan problem abstract the problem of scheduling jobs on identical parallel processors to minimize makespan is np hard we introduce a general lower bound which includes as special cases bounds that either dominate or are equivalent to lower bounds previously proposed in the literature the general lower bound unifies earlier results and can be used to identify new bounds improve previously proposed bounds and extend bounds for the problem with identical processor ready times to the problem with variable processor ready times

convex quadratic and semidefinite programming relaxations in scheduling we consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs the main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem the best previously known approximation algorithms are based on lp relaxations in time or interval indexed variables those lp relaxations however suffer from a huge number of variables as a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2 approximation algorithm which can be further improved to performance guarantee 3 2 in the absence of release dates we also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings finally for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by goemans and williamson for the maxcut problem this leads to an improved 1 2752 approximation

the constrained minimum weighted sum of job completion times problem we consider the problem of minimizing the weighted sum of job completion times on a single machine subject to certain job weights with an additional side constraint on the weighted sum of job completion times with respect to different job weights this problem is np hard and we provide a polynomial time approximation scheme for this problem our method is based on lagrangian relaxation mixed with carefully guessing the positions of certain jobs in the schedule

single machine scheduling polyhedra with precedence constraints we consider nonpreemptive single machine scheduling subject to precedence constraints we define feasible schedules by the vector of the job completion times and study the structure of the convex hull of all feasible schedules called the scheduling polyhedron p we derive classes of valid inequalities for p and necessary and sufficient conditions under which they are facet inducing our main result is a complete description of the minimal linear system defining p when the precedence constraints are series parallel moreover this system consists of two classes of facet inducing inequalities associated with series and parallel compositions respectively if the precedence constraints are not series parallel we present another class of facet inducing inequalities for p associated with induced z subgraphs we also show that the convex hull of all preemptive feasible schedules is identical to p if and only if the precedence constraints form an out forest

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

approximation of fixed points for amenable semigroups of nonexpansive mappings in banach spaces abstract the purpose of this paper is to study iterative schemes of browder and halpern types for a semigroup of nonexpansive mappings on a compact convex subset of a smooth and strictly convex banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented

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

on character amenability of banach algebras abstract we continue our work e kaniuth a t lau j pym on amenability of banach algebras math proc cambridge philos soc 144 2008 85 96 in the study of amenability of a banach algebra a defined with respect to a character of a various necessary and sufficient conditions of a global and a pointwise nature are found for a banach algebra to possess a mean of norm 1 we also completely determine the size of the set of means for a separable weakly sequentially complete banach algebra a with no mean in a itself a number of illustrative examples are discussed

minimizing average completion time in the presence of release dates a natural and basic problem in scheduling theory is to provide good average quality of service to a stream of jobs that arrive over time in this paper we consider the problem of schedulingn jobs that are released over time in order to minimize the average completion time of the set of jobs in contrast to the problem of minimizing average completion time when all jobs are available at time 0 all the problems that we consider are np hard and essentially nothing was known about constructing good approximations in polynomial time we give the first constant factor approximation algorithms for several variants of the single and parallel machine models many of the algorithms are based on interesting algorithmic and structural relationships between preemptive and nonpreemptive schedules and linear programming relaxations of both many of the algorithms generalize to the minimization of averageweighted completion time as well 1998 the mathematical programming society inc published by elsevier science b v

convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in banach spaces abstract the purpose of this paper is to study hybrid iterative schemes of halpern types for a semigroup t s s s of relatively nonexpansive mappings on a closed and convex subset c of a banach space with respect to a sequence n of asymptotically left invariant means defined on an appropriate invariant subspace of l s we prove that given a certain sequence n in 0 1 x c we can generate an iterative sequence x n which converges strongly to f x where f x is the generalized projection from c onto the fixed point set f our main result is even new for the case of a hilbert space

parallel machine scheduling by column generation parallel machine scheduling problems concern the scheduling of n jobs on m machines to minimize some function of the job completion times if preemption is not allowed then most problems are not only np hard but also very hard from a practical point of view in this paper we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule to that end we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables n covering constraints and a single side constraint we show that the linear programming relaxation can be solved efficiently by column generation because the pricing problem is solvable in pseudo polynomial time we display this approach on the problem of minimizing total weighted completion time on m identical machines our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral moreover they show that our branch and bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm

a polyhedral approach to single machine scheduling problems we report new results for a time indexed formulation of nonpreemptive single machine scheduling problems we give complete characterizations of all facet inducing inequalities with integral coefficients and right hand side 1 or 2 for the convex hull of the set of feasible partial schedules i e schedules in which not all jobs have to be started furthermore we identify conditions under which these facet inducing inequalities are also facet inducing for the original polytope which is the convex hull of the set of feasible complete schedules i e schedules in which all jobs have to be started to obtain insight in the effectiveness of these classes of facet inducing inequalities we develop a branch and cut algorithm based on them we evaluate its performance on the strongly np hard single machine scheduling problem of minimizing the weighted sum of the job completion times subject to release dates

minimizing mean flow time with release time constraint abstract we consider the problem of preemptively scheduling a set of n independent tasks with release times on m identical processors with the objective of minimizing the mean flow time for one processor baker gives an o n log n time algorithm to find an optimal schedule lawler asks the question whether the problem can be shown to be solvable in polynomial time or shown to be np hard for m 2 in this paper we answer this question by showing that it is np hard for each fixed m 2

lagrangian relaxation and column generation based lower bounds for the p m h j 1 σ w i c i scheduling problem we consider an identical parallel machine scheduling problem to minimize the sum of weighted completion times of jobs however instead of allowing machines to be continuously available as it is generally assumed we consider that each machine is subject to a deterministic and finite maintenance period this condition increases the problem complexity we provide for the problem an adapted heuristic lower bounds based on lagrangian relaxation and column generation methods computational study shows satisfactory results

on the relationship between combinatorial and lp based lower bounds for np hard scheduling problems enumerative approaches to solving optimization problems such as branch and bound require a subroutine that produces a lower bound on the value of the optimal solution in the domain of scheduling problems the requisite lower bound has typically been derived from either the solution to a linear programming lp relaxation of the problem or the solution to a combinatorial relaxation in this paper we investigate from a theoretical perspective the relationship between several lp based lower bounds and combinatorial lower bounds for three scheduling problems in which the goal is to minimize the average weighted completion time of the jobs scheduled we establish a number of facts about the relationship between these different sorts of lower bounds including the equivalence of certain lp based lower bounds for these problems to combinatorial lower bounds used in successful branch and bound algorithms as a result we obtain the first worst case analysis of the quality of the lower bounds delivered by these combinatorial relaxations

approximation techniques for average completion time scheduling we consider the problem of nonpreemptive scheduling to minimize average weighted completion time allowing for release dates parallel machines and precedence constraints recent work has led to constant factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs we introduce several new techniques which generalize this basic paradigm we use these ideas to obtain improved approximation algorithms for one machine scheduling to minimize average completion time with release dates in the process we obtain an optimal randomized on line algorithm for the same problem that beats a lower bound for deterministic on line algorithms we consider extensions to the case of parallel machine scheduling and for this we introduce two new ideas first we show that a preemptive one machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times second we show that a nongreedy rounding of the relaxation yields better approximations than a greedy one we also prove a general theorem relating the value of one machine relaxations to that of the schedules obtained for the original m machine problems this theorem applies even when there are precedence constraints on the jobs we apply this result to obtain improved approximation ratios for precedence graphs such as in trees out trees and series parallel graphs

solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings in this paper we introduce a hybrid viscosity approximation scheme for finding a common element of the solution set for a system of equilibrium problems the solution set for a system of variational inequality problems and the common fixed point set for an infinite family of nonexpansive mappings in hilbert spaces then we prove the strong convergence of the proposed iterative scheme our results improve and extend the results announced by ceng and yao 2008 4

a cutting plane algorithm for mv portfolio selection model this paper deals with a portfolio selection problem with fuzzy return rates a possibilistic mean variance fmvc portfolio selection model was proposed the possibilistic programming problem can be transformed into a linear optimal problem with an additional quadratic constraint by possibilistic theory for such problems there are no special standard algorithms we propose a cutting plane algorithm to solve fmvc the nonlinear programming problem can be solved by sequence linear programming problem a numerical example is given to illustrate the behavior of the proposed model and algorithm

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

a comparison of branch and bound algorithms for a family scheduling problem with identical parallel machines this paper investigates branch and bound algorithms for the problem of scheduling jobs with family setups on identical parallel machines to minimize the weighted sum of completion times in particular we propose a new branching scheme that appears to substantially outperform current procedures in terms of computation time and search tree size

improved approximation algorithms for shop scheduling problems in the job shop scheduling problem there are m machines and n jobs a job consists of a sequence of operations each of which must be processed on a specified machine and the aim is to complete all jobs as quickly as possible this problem is strongly cal np hard even for very restrictive special cases the authors give the first randomized and deterministic polynomial time algorithms that yield polylogarithmic approximations to the optimal length schedule these algorithms also extend to the more general case where a job is given not by a linear ordering of the machines on which it must be processed but by an arbitrary partial order comparable bounds can also be obtained when there are m types of machines a specified number of machines of each type and each operation must be processed on one of the machines of a specified type as well as for the problem of scheduling unrelated parallel machines subject to chain precedence constraints

scheduling parallel machines with inclusive processing set restrictions and job release times we consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule the machines have the same processing speed but each job is compatible with only a subset of those machines the machines can be linearly ordered such that a higher indexed machine can process all those jobs that a lower indexed machine can process we present an efficient algorithm for this problem with a worst case performance ratio of 2 we also develop a polynomial time approximation scheme ptas for the problem as well as a fully polynomial time approximation scheme fptas for the case in which the number of machines is fixed

lp models for bin packing and cutting stock problems abstract we review several linear programming lp formulations for the one dimensional cutting stock and bin packing problems namely the models of kantorovich gilmore gomory onecut models as in the dyckhoff stadtler approach position indexed models and a model derived from the vehicle routing literature we analyse some relations between the corresponding lp relaxations and their relative strengths and refer how to derive branching schemes that can be used in the exact solution of these problems using branch and price

hybrid pseudo viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non expansive mappings abstract in this paper we introduce hybrid pseudo viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems the set of fixed points of an infinite family and left amenable semigroup of non expansive mappings in the frame work of hilbert spaces our goal is to prove a result of strong convergence for hybrid pseudo viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non expansive mappings the results presented in this paper can be treated as an extension and improvement of the corresponding results announced by ceng et al l c ceng q h ansari and j c yao hybrid pseudo viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non expansive mappings nonlinear analysis 4 2010 743 754 and many others

the one machine sequencing problem let m 1 and m 3 be non bottleneck machines and m 2 a bottleneck machine processing only one job at a time suppose that n jobs have to be processed on m 1 m 2 and m 1 in that order and job i has to spend a time a on m 1 d 1 on m 2 and q 1 on m 3 we want to minimize the makespan this problem is important since its resolution provides a bound on the makespan of complicated systems such as job shops it is np hard in the strong sense however efficient branch and bound methods exist and we describe one of them our bound for the tree search is very close to the bound used by florian et al but the principle of branching is quite different at every node we construct by an o n log n algorithm a schrage schedule then we define a critical job c a critical set j and consider two subsets of schedules the schedules when c procedes every job of j and the schedules when c follows every job of j we give the results of this method and prove that the difference between the optimum and the schrage schedule is less than d 1

a polyhedral study of the cardinality constrained knapsack problem a cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive this structure occurs for example in areas such as finance location and scheduling traditionally cardinality constraints are modeled by introducing auxiliary 0 1 variables and additional constraints that relate the continuous and the 0 1 variables we use an alternative approach in which we keep in the model only the continuous variables and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables to derive the valid inequalities we extend the concepts of cover and cover inequality commonly used in 0 1 programming to this class of problems and we show how cover inequalities can be lifted to derive facet defining inequalities we present three families of non trivial facet defining inequalities that are lifted cover inequalities finally we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0 1 variables over the traditional mip approach for this class of problems

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

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

lagrangian domain reductions for the single machine earliness tardiness problem with release dates this paper presents new elimination rules for the single machine problem with general earliness and tardiness penalties subject to release dates these rules based on a lagrangian decomposition allow to drastically reduce the execution windows of the jobs we measure the efficiency of these properties by integrating them in a branch and bound tests show that instances with up to 70 jobs without release dates and up to 40 jobs with release dates can be optimally solved within 1000 seconds

scheduling jobs of equal length complexity facets and computational results the following problem was originally motivated by a question arising in the automated assembly of printed circuit boards given are n jobs which have to be performed on a single machine within a fixed timespan 0 t subdivided into t unit length subperiods the processing time or length of each job equals p p n the processing cost of each job is an arbitrary function of its start time the problem is to schedule all jobs so as to minimize the sum of the processing costs this problem is proved to be np hard already for p 2 and 0 1 processing costs on the other hand when t np c with c constant the problem can be solved in polynomial time a partial polyhedral description of the set of feasible solutions is presented in particular two classes of facet defining inequalities are described for which the separation problem is polynomially solvable also we exhibit a class of objective functions for which the inequalities in the lp relaxation guarantee integral solutions finally we present a simple cutting plane algorithm and report on its performance on randomly generated problem instances

decompositions network flows and a precedence constrained single machine scheduling problem we present an in depth theoretical algorithmic and computational study of a linear programming lp relaxation to the precedence constrained single machine scheduling problem 1 prec s jw jc jto minimize a weighted sum of job completion times on the theoretical side we study the structure of tight parallel inequalities in thelp relaxation and show thatevery permutation schedule that is consistent with sidney s decomposition has total cost no more than twice the optimum on the algorithmic side we provide a parametric extension to sidney s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum cut problem finally we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2 000 jobs

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

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 deadline constrained weighted completion time problem analysis of a heuristic a greedy heuristic that schedules jobs on one machine is examined the problem considered is to minimize the sum of weighted completion times subject to deadline constraints both theoretical and empirical evidence are presented to suggest that this method finds solutions that are near optimal optimality conditions are developed also an analysis of the relative error is presented worst case bounds are found that are based on the problem parameters in addition we show that the relative error goes to zero as the problem size goes to infinity

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

the single machine weighted mean squared deviation problem abstract this paper studies a single machine problem related to the just in time jit production objective in which the goal is to minimize the sum of weighted mean squared deviation of the completion times with respect to a common due date in order to solve the problem several structural and dominance properties of the optimal solution are investigated these properties are then integrated within a branch and cut approach to solve a time indexed formulation of the problem the results of a computational experiment with the proposed algorithm show that the method is able to optimally solve instances with up to 300 jobs within reduced running times improving other integer programming approaches

approximation bounds for a general class of precedence constrained parallel machine scheduling problems an important class of scheduling problems concerns parallel machines and precedence constraints we consider precedence delays which associate with each precedence constraint a certain amount of time that must elapse between the completion and start times of the corresponding jobs together with ordinary precedence constraints release dates and delivery times can be modeled in this manner we present a 4 approximation algorithm for the total weighted completion time objective for this general class of problems the algorithm is a rather simple form of list scheduling the list is in order of job midpoints derived from a linear programming relaxation our analysis unifies and simplifies that of a number of special cases heretofore separately studied while actually improving many of the former approximation results

technical note an improved lower bound for minimizing weighted completion times with deadlines we consider the np hard one machine scheduling problem whose objective is to minimize the weighted sum of completion times subject to meeting deadlines we propose a new lower bound that dominates the bounds found in the literature

two agent scheduling to minimize the total cost abstract two agents each having his own set of jobs compete to perform their own jobs on a common processing resource each job of the agents has a weight that specifies its importance the cost of the first agent is the maximum weighted completion time of his jobs while the cost of the second agent is the total weighted completion time of his jobs we consider the scheduling problem of determining the sequence of the jobs such that the total cost of the two agents is minimized we provide a 2 approximation algorithm for the problem show that the case where the number of jobs of the first agent is fixed is np hard and devise a polynomial time approximation scheme for this case

a ptas for minimizing the total weighted completion time on identical parallel machines we consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times this problem is np hard in the strong sense the best approximation result known so far was a documentclass aastex usepackage amsbsy usepackage amsfonts usepackage amssymb usepackage bm usepackage mathrsfs usepackage pifont usepackage stmaryrd usepackage textcomp usepackage portland xspace usepackage amsmath amsxtra pagestyle empty declaremathsizes 10 9 7 6 begin document 1 over 2 1 sqrt 2 end document approximation algorithm that has been derived by kawaguchi and kyan back in 1986 the contribution of this paper is a polynomial time approximation scheme for this setting which settles a problem that was open for a long time moreover our result constitutes the first known approximation scheme for a strongly np hard scheduling problem with minsum objective

strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings in this paper we introduce a suitable mann type algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings in hilbert spaces then we prove the strong convergence of the proposed iterative scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings our results extend and improve the recent result of colao and marino v colao and g marino strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings nonlinear anal 73 2010 3513 3524 and many others

positive real matrices in indefinite inner product spaces and invariant maximal semidefinite subspaces in this paper positive real matrices in indefinite inner product spaces are studied this class of matrices is intimately connected to dissipative matrices in the complex case first the complex case and then the real case are treated an explicit construction is given for invariant semidefinite subspaces

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 new average case analysis for completion time scheduling we present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized our goal is to use the concept of competitive ratio which is a typical worst case notion also within an average case analysis we show that the classic sept scheduling strategy with n worst case competitive ratio achieves an average of o 1 under several natural distributions among them the exponential distribution our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio thus our result bridges the gap between worst case and average case performance guarantee

tight bounds for permutation flow shop scheduling in flow shop scheduling there are m machines and n jobs such that every job has to be processed on the machines in the fixed order 1 m in the permutation flow shop problem it is also required that each machine process the set of all jobs in the same order formally given n jobs along with their processing times on each machine the goal is to compute a single permutation of the jobs n n that minimizes the maximum job completion time makespan of the schedule resulting from the previously best known approximation guarantee for this problem was o m log m 1 2 sviridenko m 2004 a note on permutation flow shop problem ann oper res 129 247 252 in this paper we obtain an improved o min m1 2 n1 2 approximation algorithm for the permutation flow shop scheduling problem by finding a connection between the scheduling problem and the longest increasing subsequence problem our approximation ratio is relative to the lower bounds of maximum job length and maximum machine load and is the best possible such result this also resolves an open question from potts et al potts c d shmoys d williamson 1991 permutation vs nonpermutation flow shop schedules oper res lett 10 281 284 by algorithmically matching the gap between permutation and nonpermutation schedules we also consider the weighted completion time objective for the permutation flow shop scheduling problem using a natural linear programming relaxation and our algorithm for the makespan objective we obtain an o min m1 2 n1 2 approximation algorithm for minimizing the total weighted completion time improving on the previously best known guarantee of em for any constant e 0 smutnicki c 1998 some results of the worst case analysis for flow shop scheduling eur j oper res 109 66 87 we give a matching lower bound on the integrality gap of our linear programming relaxation

simple forms and invariant subspaces of h expansive matrices in this paper we consider a simple form for pairs of matrices a h where h is a real symmetric invertible matrix and a is a real h expansive matrix that is atha h is positive semidefinite for such pairs we use the simple form to give an explicit construction of real a invariant maximal h semidefinite subspaces

the asymptotic performance ratio of an on line algorithm for uniform parallel machine scheduling with release dates jobs arriving over time must be non preemptively processed on one of m parallel machines each running at its own speed so as to minimize a weighted sum of the job completion times in this on line environment the processing requirement and weight of a job are not known before the job arrives the weighted shortest processing requirement wspr heuristic is a simple extension of the well known wspt heuristic which is optimal for the single machine problem without release dates according to wspr whenever a machine completes a job the next job assigned to it is the one with the least ratio of processing requirement to weight among all jobs available for processing at this point in time we analyze the performance of this heuristic and prove that its asymptotic competitive ratio is one for all instances with bounded job processing requirements and weights this implies that the wspr algorithm generates a solution whose relative error approaches zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies extensively on properties of optimal solutions to a single machine relaxation of the problem

sharing supermodular costs we study cooperative games with supermodular costs we show that supermodular costs arise in a variety of situations in particular we show that the problem of minimizing a linear function over a supermodular polyhedron a problem that often arises in combinatorial optimization has supermodular optimal costs in addition we examine the computational complexity of the least core and least core value of supermodular cost cooperative games we show that the problem of computing the least core value of these games is strongly np hard and in fact is inapproximable within a factor strictly less than 17 16 unless p np for a particular class of supermodular cost cooperative games that arises from a scheduling problem we show that the shapley value which in this case is computable in polynomial time is in the least core while computing the least core value is np hard

asymptotic performance ratio of an online algorithm for the single machine scheduling with release dates we analyze the single machine weighted completion time problem with release dates and prove that the asymptotic performance ratio of a simple online algorithm is one this implies that this algorithm generates a solution whose relative error decreases to zero as the number of jobs increases our proof does not require any probabilistic assumption on the job parameters and relies heavily on properties of optimal solutions to a linear programming relaxation of the problem

new bounds for the identical parallel processor weighted flow time problem in 1964 eastman even and isaacs eei presented a polynomial time lower bound for the np hard problem of scheduling n jobs on m available and identical processors to minimize weighted flow time a general bound of which the eei bound is a special case is proposed four new bounds for the identical processor problem are given all of the new bounds can be applied to problems with variable processor ready times the eei bound is limited to problems where all processors are initially available at the same time two of the four new bounds are shown to dominate the eei bound the two other bounds are found to be effective for a particular problem class two polynomial time lower bounds for problems with nonidentical processors are introduced one bound is applicable to the uniform processor problem the other bound can be applied to the general processor problem no bounds have previously been proposed for these problems