the convergence of stochastic algorithms solving flow shop scheduling abstract in the paper we apply logarithmic cooling schedules of simulated annealing based algorithms to flow shop scheduling in our problem setting the objective to minimize the overall completion time which is called the makespan we prove a lower bound for the number of steps that are sufficient to approach an optimum solution with a certain probability the result is related to the maximum escape depth from local minima of the underlying energy landscape in our approach we need n o log o 1 1 steps to be in an optimum solution with probability 1 where n denotes the total number of tasks the auxiliary computations are of polynomial complexity since the model cannot be approximated arbitrarily closely in the general case unless p np the approach can be used to obtain approximation algorithms that work well in the average case

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

robust inventory management an optimal control approach we formulate and solve static and dynamic models of inventory management that lie at the intersection of robust optimization and optimal control theory our objective is to minimize cumulative ordering holding and shortage costs over a horizon 0 t where the variable is a nonnegative ordering rate function q t 2 0 t the demand rate function d t is unknown and is only assumed to belong to an uncertainty set d t l2 0 t a 1 t 0td t dt b a d t b t 0 t this set is motivated by the strong law of large numbers for stochastic processes limt 1 t 0td t dt where is the mean drift we analyze a static model where the ordering rate function must be fully specified at time zero and three dynamic variants where re optimizations are allowed during the planning horizon 0 t at prespecified review epochs in the dynamic models at review epoch 0 t the past demand on 0 is observable in the first dynamic model we ignore this information and define a variant of that i

stability of adaptive and nonadaptive packet routing policies in adversarial queueing networks we investigate the stability of packet routing policies in adversarial queueing networks we provide a simple classification of networks which are stable under any greedy scheduling policy we show that a network is stable if and only if the underlying undirected connected graph contains at most two edges we also propose a simple and distributed policy which is stable in an arbitrary adversarial queueing network even for the critical value of the arrival rate r 1 finally a simple and checkable network flow type load condition is formulated for adaptive adversarial queueing networks and a policy is proposed which achieves stability under this new load condition this load condition is a relaxation of the integral network flow type condition considered previously in the literature

adversarial queuing theory we consider packet routing when packets are injected continuously into a network we develop an adversarial theory of queuing aimed at addressing some of the restrictions inherent in probabilistic analysis and queuing theory based on time invariant stochastic generation we examine the stability of queuing networks and policies when the arrival process is adversarial and provide some preliminary results in this direction our approach sheds light on various queuing policies in simple networks and paves the way for a systematic study of queuing with few or no probabilistic assumptions

tractable stochastic analysis in high dimensions via robust optimization modern probability theory whose foundation is based on the axioms set forth by kolmogorov is currently the major tool for performance analysis in stochastic systems while it offers insights in understanding such systems probability theory in contrast to optimization has not been developed with computational tractability as an objective when the dimension increases correspondingly some of its major areas of application remain unsolved when the underlying systems become multidimensional queueing networks auction design in multi item multi bidder auctions network information theory pricing multi dimensional options among others we propose a new approach to analyze stochastic systems based on robust optimization the key idea is to replace the kolmogorov axioms and the concept of random variables as primitives of probability theory with uncertainty sets that are derived from some of the asymptotic implications of probability theory like the central limit theorem in addition we observe that several desired system properties such as incentive compatibility and individual rationality in auction design are naturally expressed in the language of robust optimization in this way the performance analysis questions become highly structured optimization problems linear semidefinite mixed integer for which there exist efficient practical algorithms that are capable of solving problems in high dimensions we demonstrate that the proposed approach achieves computationally tractable methods for a analyzing queueing networks b designing multi item multi bidder auctions with budget constraints and c pricing multi dimensional options

an algorithm for solving the job shop problem in this paper we propose a branch and bound method for solving the job shop problem it is based on one machine scheduling problems and is made more efficient by several propositions which limit the search tree by using immediate selections it solved for the first time the famous 10 10 job shop problem proposed by muth and thompson in 1963

new search spaces for sequencing problems with application to job shop scheduling in this paper search heuristics are developed for generic sequencing problems with emphasis on job shop scheduling the proposed methods integrate problem specific heuristics common to operations research and local search approaches from artificial intelligence in order to obtain desirable properties from both the applicability of local search to a wide range of problems and the incorporation of problem specific information are both properties of the proposed algorithms two methods are proposed both of which are based on novel definitions of solution spaces and of neighborhoods in these spaces applications of the proposed methodology are developed for job shop scheduling problems and can be easily applied with any scheduling objective to demonstrate effectiveness the method is tested on the job shop scheduling problem with the minimum makespan objective encouraging results are obtained

an analytic approach to a general class of g g s queueing systems we solve the queueing system ck cm s where ck is the class of coxian probability density functions pdfs of order k which is a subset of the pdfs that have a rational laplace transform we formulate the model as a continuous time infinite space markov chain by generalizing the method of stages by using a generating function technique we solve an infinite system of partial difference equations and find closed form expressions for the system size general time prearrival post departure probability distributions and the usual performance measures in particular we prove that the probability of n customers being in the system when it is saturated is a linear combination of geometric terms the closed form expressions involve a solution of a system of nonlinear equations that involves only the laplace transforms of the interarrival and service time distributions we conjecture that this result holds for a more general model following these theoretical results we propose an exact algorithm for finding the system size distribution and the system s performance measures we examine special cases and apply this method for numerically solving the c2 c2 s and ek c2 s queueing systems

two simulated annealing based heuristics for the job shop scheduling problem abstract in this paper we present two simulated annealing based algorithms for the classical general job shop scheduling problem where the objective is to minimize the makespan we consider sets of jobs consisting of tasks and sets of machines which can handle at most one task at a time to represent the problem we employ the model of disjunctive graphs simulated annealing has been applied to this problem earlier e g by van laarhoven et al where the neighborhood function is based on reversing a single arc of a longest path of the underlying graph in our approach we analyze a neighborhood function which involves a non uniform generation probability to obtain the neighbors of a schedule we reverse more than a single arc of longest paths and perform a control on the increase of the makespan the selection of the arcs depends on the number of longest paths to which a particular arc belongs furthermore we have designed two cooling schedules which employ a detailed analysis of the objective function depending on the specified neighborhood relation the expected run times can be either o n2 e or o n3 e m for the first cooling schedule and o n5 2 e m1 2 or o n7 2 e m1 2 for the second cooling schedule where n is the number of tasks m the number of machines and e represents o ln ln n ln n our computational experiments on small to large benchmark problems have shown that within short series of consecutive trials relatively stable results equal or close to optimal solutions are repeatedly obtained including the well known benchmark problems ft10 and la38 we could improve five upper bounds for the yn1 yn4 swv12 swv13 and swv15 benchmark problems e g for swv13 the gap between the lower and the former upper bound has been shortened by about 57 in our approach we rely only on basic information about the given problem instance

variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem abstract practical constraint satisfaction problems csps such as design of integrated circuits or scheduling generally entail large search spaces with hundreds or even thousands of variables each with hundreds or thousands of possible values often only a very tiny fraction of all these possible assignments participates in a satisfactory solution this article discusses techniques that aim at reducing the effective size of the search space to be explored in order to find a satisfactory solution by judiciously selecting the order in which variables are instantiated and the sequence in which possible values are tried for each variable in the csp literature these techniques are commonly referred to as variable and value ordering heuristics our investigation is conducted in the job shop scheduling domain we show that in contrast with problems studied earlier in the csp literature generic variable and value heuristics do not perform well in this domain this is attributed to the difficulty of these heuristics to properly account for the tightness of constraints and or the connectivity of the constraint graphs induced by job shop scheduling csps a new probabilistic framework is introduced that better captures these key aspects of the job shop scheduling search space empirical results show that variable and value ordering heuristics derived within this probabilistic framework often yield significant improvements in search efficiency and significant reductions in the search time required to obtain a satisfactory solution the research reported in this article was the first one along with the work of keng and yun 1989 to use the csp problem solving paradigm to solve job shop scheduling problems the suite of benchmark problems it introduced has been used since then by a number of other researchers to evaluate alternative techniques for the job shop scheduling csp the article briefly reviews some of these more recent efforts and shows that our variable and value ordering heuristics remain quite competitive

a hybrid method for solving stochastic job shop scheduling problems this paper presents a nonlinear mathematical programming model for a stochastic job shop scheduling problem due to the complexity of the proposed model traditional algorithms have low capability in producing a feasible solution therefore a hybrid method is proposed to obtain a near optimal solution within a reasonable amount of time this method uses a neural network approach to generate initial feasible solutions and then a simulated annealing algorithm to improve the quality and performance of the initial solutions in order to produce the optimal near optimal solution a number of test problems are randomly generated to verify and validate the proposed hybrid method the computational results obtained by this method are compared with lower bound solutions reported by the lingo 6 optimization software the compared results of these two methods show that the proposed hybrid method is more effective when the problem size increases

optimal placements of flexible objects i analytical results for the unbounded case the authors consider optimal placements of two dimensional flexible elastic deformable objects the objects are discs of equal size placed within a rigid boundary the paper is divided into two parts in the first part analytical results for three types of regular periodic arrangements the hexagonal square and triangular placements are presented the regular arrangements are analyzed for rectangular boundaries and radii of discs that are small compared to the area of the placement region because in this case the influence of boundary conditions can be neglected this situation is called the unbounded case they show that for the unbounded case among the three regular placements the type of hexagonal arrangements provides the largest number of placed units for the same deformation depth furthermore it can be proved that these regular placements are not too far from the truly optimal arrangements for example hexagonal placements differ at most by the factor of 1 1 from the largest possible number of generally shaped units in arbitrary arrangements these analytical results are used as guidances for testing stochastic algorithms optimizing placements of flexible objects in the second part mainly two problems are considered the underlying physical model and a simulated annealing algorithm maximizing the number of flexible discs in equilibrium placements along with the physical model an approximate formula is derived reflecting the deformation force relationship for a large range of deformations

a fast taboo search algorithm for the job shop problem a fast and easily implementable approximation algorithm for the problem of finding a minimum makespan in a job shop is presented the algorithm is based on a taboo search technique with a specific neighborhood definition which employs a critical path and blocks of operations notions computational experiments up to 2 000 operations show that the algorithm not only finds shorter makespans than the best approximation approaches but also runs in shorter time it solves the well known 10 10 hard benchmark problem within 30 seconds on a personal computer

strict comparison and z absorption of nuclear c algebras for any unital separable simple infinite dimensional nuclear c algebra with finitely many extremal traces we prove that mathcal z absorption strict comparison and property si are equivalent we also show that any unital separable simple nuclear c algebra with tracial rank zero is approximately divisible and hence is mathcal z absorbing

on unapproximable versions of np complete problems we prove that all of karp s 21 original np complete problems have a version that is hard to approximate these versions are obtained from the original problems by adding essentially the same simple constraint we further show that these problems are absurdly hard to approximate in fact no polynomial time algorithm can even approximate log k of the magnitude of these problems to within any constant factor where logk denotes the logarithm iterated k times unless np is recognized by slightly superpolynomial randomized machines we use the same technique to improve the constant epsilon such that max clique is hard to approximate to within a factor of n epsilon finally we show that it is even harder to approximate two counting problems counting the number of satisfying assignments to a monotone 2sat formula and computing the permanent of 1 0 1 matrices

robust queueing theory we propose an alternative approach for studying queues based on robust optimization we model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability using the generalized central limit theorem this framework allows us to model heavy tailed behavior characterized by bursts of rapidly occurring arrivals and long service times we take a worst case approach and obtain closed form upper bounds on the system time in a multi server queue these expressions provide qualitative insights that mirror the conclusions obtained in the probabilistic setting for light tailed arrivals and services and generalize them to the case of heavy tailed behavior we also develop a calculus for analyzing a network of queues based on the following key principles a the departure from a queue b the superposition and c the thinning of arrival processes have the same uncertainty set representation as the original arrival processes the proposed appr

optimal stochastic allocation of machines under waiting time constraints the nonpreemptive scheduling of n geqslant 1 stochastic jobs is considered to minimize the expected number of parallel machines needed to meet given waiting time constraints the number of machines available is unlimited the running times of the jobs are denoted t 1 ldots t n and are taken to be independent samples of an exponentially distributed random variable t with mean 1 job waiting times are to be bounded stochastically by a nonnegative random variable w independent of t 1 ldots t n at time zero a timer is started with an initial value w and job scheduling begins when the timer expires all jobs still waiting for a machine are assigned to available machines only the distributions of w and the job durations t 1 ldots t n are known to the scheduler in advance a scheduling policy is defined which is proven to be optimal when w has an exponential distribution and is asymptotically optimal as n to infty when w is a constant hard deadline in the exponential case

closed form solutions for robust inventory management we propose and analyze robust optimization models of an inventory management problem where cumulative purchase inventory and shortage costs over n periods are minimized for correlated nonidentically distributed demand we assume that the means and covariance matrix of stochastic demand are known the distributions are not needed we derive closed form ordering quantities for both symmetric and asymmetric uncertainty sets under capacitated inventory constraints in both static and dynamic settings the behaviors of our robust strategies differ qualitatively depending on the symmetry of the uncertainty set for instance considering our simplest static problem 1 if the uncertainty set is symmetric then there is positive ordering in all periods whereas 2 if the set is asymmetric then there is a set of periods in the middle of the planning horizon with zero orders we also link the symmetry of the uncertainty set to the symmetry of the demand distribution finally we present encouraging computati

stochastic scheduling with variable profile and precedence constraints in this paper we consider the stochastic profile scheduling problem of a partially ordered set of tasks on uniform processors the set of available processors varies in time the running times of the tasks are independent random variables with exponential distributions we obtain a sufficient condition under which a list policy stochastically minimizes the makespan within the class of preemptive policies this result allows us to obtain a simple optimal policy when the partial order is an interval order an in forest or an out forest

optimal design for multi item auctions a robust optimization approach in this paper we revisit the auction design problem for multi item auctions with budget constrained buyers by introducing a robust optimization approach to model a concepts such as incentive compatibility and individual rationality that are naturally expressed in the language of robust optimization and b the auctioneer s beliefs on the buyers valuations of the items rather than using probability distributions the classical probabilistic approach or an adversarial model to model valuations we introduce an uncertainty set based model for these valuations we construct these uncertainty sets to incorporate historical information available to the auctioneer in a way that is consistent with limit theorems of probability theory or knowledge of the probability distribution in this setting we formulate the auction design problem as a robust optimization problem and provide a characterization of the optimal solution as an auction with reservation prices thus extending the work of myerson myerson rb

job shop scheduling by simulated annealing we describe an approximation algorithm for the problem of finding the minimum makespan in a job shop the algorithm is based on simulated annealing a generalization of the well known iterative improvement approach to combinatorial optimization problems the generalization involves the acceptance of cost increasing transitions with a nonzero probability to avoid getting stuck in local minima we prove that our algorithm asymptotically converges in probability to a globally minimal solution despite the fact that the markov chains generated by the algorithm are generally not irreducible computational experiments show that our algorithm can find shorter makespans than two recent approximation approaches that are more tailored to the job shop scheduling problem this is however at the cost of large running times

performance analysis of queueing networks via robust optimization performance analysis of queueing networks is one of the most challenging areas of queueing theory barring very specialized models such as product form type queueing networks there exist very few results that provide provable nonasymptotic upper and lower bounds on key performance measures in this paper we propose a new performance analysis method which is based on the robust optimization the basic premise of our approach is as follows rather than assuming that the stochastic primitives of a queueing model satisfy certain probability laws such as i i d interarrival and service times distributions we assume that the underlying primitives are deterministic and satisfy the implications of such probability laws these implications take the form of simple linear constraints namely those motivated by the law of the iterated logarithm lil using this approach we are able to obtain performance bounds on some key performance measures furthermore these performance bounds imply similar bounds in the underlying stochastic queueing models we demonstrate our approach on two types of queueing networks a tandem single class queueing network and b multiclass single server queueing network in both cases using the proposed robust optimization approach we are able to obtain explicit upper bounds on some steady state performance measures for example for the case of tsc system we obtain a bound of the form c 1 1 ln ln 1 1 on the expected steady state sojourn time where c is an explicit constant and is the bottleneck traffic intensity this qualitatively agrees with the correct heavy traffic scaling of this performance measure up to the ln ln 1 1 correction factor