Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

By further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.     

Competitive production scheduling: A two-firm,noncooperative finite dynamic game

In this paper, we study the production scheduling problem in a competitive environment. Two firms produce the same product and compete in a market. The demand is random and so is the production capacity of each firm, due to random breakdowns. We consider a finite planning horizon. The scheduling problem is formulated as a finite dynamic game. Algorithms are developed to determine the security, hazard, and Nash policies. Numerical examples are discussed. A single-firm optimization model is also analyzed and it is observed that the production control policy from the single-firm optimization model may not perform well in a competitive environment.     

A mixed integer programming model for advanced planning and scheduling (APS)

This paper presents a mixed integer programming (MIP) model which succeeds in a system integration of the production planning and shop floor scheduling problems. The proposed advanced planning and scheduling (APS) model explicitly considers capacity constraints, operation sequences, lead times and due dates in a multi-order environment. The objective of the model is to seek the minimum cost of both production idle time and tardiness or earliness penalty of an order. The output of the model is operation schedules with order starting time and finish time. Numerical result shows that the suggested APS model can favorably produce optimal schedules.     

A profit-maximizing economic lot scheduling problem with price optimization

The economic lot scheduling problem (ELSP) is a well known problem that focuses on scheduling the production of multiple items on a single machine such that inventory and setup costs are minimized. In this paper, we extend the ELSP to include price optimization with the objective to maximize profits. A solution approach based on column generation is provided and shown to produce very close to optimal results with short solution times on a set of test problems. The results are discussed and recommendations for further research are provided.     

Relaxation of Product Markov Chains on Product Spaces

We study the relaxation time of product-type Markov chains approaching a product distribution. We bound the approach to stationarity for such Markov chains in terms of the mixing times of the component Markov chains. In cases where the component mixing times differ considerably we propose an optimized visiting scheme which makes such product-type Markov chains comparable to Gibbs-type samplers. We conclude the paper by discussing the relaxation of Metropolis-type samplers for separable energy functions.     

A coupling cutting stock-lot sizing problem in the paper industry

An important production programming problem arises in paper industries coupling multiple machine scheduling with cutting stocks. Concerning machine scheduling: how can the production of the quantity of large rolls of paper of different types be determined. These rolls are cut to meet demand of items. Scheduling that minimizes setups and production costs may produce rolls which may increase waste in the cutting process. On the other hand, the best number of rolls in the point of view of minimizing waste may lead to high setup costs. In this paper, coupled modeling and heuristic methods are proposed. Computational experiments are presented.     

The Scheduling of Magistrates to Courts

The problem of scheduling Magistrates to Courts may be formulated generally as an integer programming problem. However, when a particular problem is considered it is found that heuristic scheduling techniques will produce an acceptable solution to this particular problem without the need to use complex algorithms. The use of a computer program to implement the heuristic scheduling was found to produce a considerable saving in clerical effort.     

Zero-Inventory Conditions for a Two-Part-Type Make-to-Stock Production System

We consider the dynamic scheduling of a two-part-type make-to-stock production system using the model of Wein [12]. Exogenous demand for each part type is met from finished goods inventory; unmet demand is backordered. The control policy determines which part type, if any, to produce at each moment; complete flexibility is assumed. The objective is to minimize average holding and backorder costs. For exponentially distributed interarrival and production times, necessary and sufficient conditions are found for a zero-inventory policy to be optimal. This result indicates the economic and production conditions under which a simple make-to-order control is optimal. Weaker results are given for the case of general production times.     

Production under periodic demand update prior to a single selling season: A decomposition approach

This paper focuses on dynamic, continuous-time production control problems in the fashion industry. Similar to the classical news-vendor problem, we consider a single product-type and the cumulative demand for items is not known until the end of the production horizon and therefore must be forecasted. Since there are periodic updates before a single selling season, actual demand during a period of time can only be determined by the end of the period. If the overall demand is overestimated, excessive inventory holding and production costs are paid and surpluses are sold at low prices at the end of the production horizon. If it is under-estimated, then sales are lost. The objective is to dynamically determine production orders which minimize overall expected costs. Since the optimal feedback for such a problem is characterized by thresholds evolving with time and system states, there is a significant computational burden in determining them. With the aid of the variational analysis and a decomposition, we derive a closed-form solution for the thresholds. A numerical study carried out to compare the decomposition and straightforward simulation-based solutions indicates the high accuracy of the suggested approach while the computational burden is dramatically reduced.     

On constraint preconditioners for generalized saddle point matrices

For the iterative solution of large sparse generalized saddle point problems, a class of new constraint preconditioners is presented, and the spectral properties and parameter choices are discussed. Numerical experiments are used to demonstrate the feasibility and effectiveness of the new preconditioners, as well as their advantages over the modified product-type skew-Hermitian triangular splitting (MPSTS) preconditioners.     

Economic lot scheduling problem with imperfect production processes

In this paper, we model the effects of imperfect production processes on the economic lot scheduling problem (ELSP). It is assumed that the production facility starts in the in-control state producing items of high or perfect quality. However the facility may deteriorate with time and shifts at a random time to an out of control state and begins to produce nonconforming items. A mathematical model is developed for ELSP taking into account the effect of imperfect quality and process restoration. Numerical examples are presented to illustrate important issues related to the developed model.     

Solving a chemical batch scheduling problem by local search

In this paper the following chemical batch scheduling problem is considered: a set of orders has to be processed on a set of facilities. For each order a given amount of a product must be produced by means of chemical reactions before a given deadline. The production consists of a sequence of processes whereby each process has to be performed by one facility out of a given subset of facilities allowed for this process. The processing times depend on the choice of the facility and the processing is done in batch mode with given minimum and maximum sizes. The problem is to assign the processes to the facilities, splitting them into batches, and scheduling these batches in order to produce the demands within the given deadlines. For the scheduling part of the problem we present an approach based on the following steps. First, a procedure to calculate the minimum number of batches needed to satisfy the demands is presented. Based on this, the given problem is modeled in two different ways: as a general shop scheduling problem with set-up times or as scheduling problem with positive time-lags. Finally, a two-phase tabu search method is presented which is based on the two different formulations of the problem. The method is tested on some real world data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal mining of a stratiform phosphate field

An optimal control model is developed to determine optimal surface mining operations of a phosphate field. This field1806 1040 contains both a bottom layer and a top layer of phosphates. The deeper the digging, the more phosphates will be exploited but the higher will be the cost per ton. If the bottom layer is not exploited, it will be lost to future production. The methods of Optimal Control Theory are applied and produce expressions which can be evaluated for the optimal scheduling around the phosphate field.     

Multi-item lot size determination and scheduling under capacity constraints

This paper deals with the planning of a production group, which has to produce several products. For each product there is a delivery plan covering several periods. Moreover, there are capacity constraints. Such a situation requires integrated optimization of lot sizes and lot scheduling. Since exact solution of the problem is in general not feasible, we will present a non exact approach which gives quite good results in some practical cases and might be a good starting point in other cases.     

Solving a concrete sleepers production scheduling by genetic algorithms

PRECON S.A. is a manufacturing company devoted to produce prefabricated concrete parts for several industries as railway transportation and agricultural industries. Recently, PRECON S.A. signed a contract with RENFE, the Spanish National Railway Company, to manufacture pre-stressed concrete sleepers for the sidings of the new railways of the high speed train (AVE). The scheduling problem associated with the manufacturing process of the sleepers is very complex, since this involves several constraints and objectives. These constraints are related to production capacity, the quantity of available moulds, demand satisfaction and other operational constraints. The two main objectives are related to the way to maximize the utilization of manufacturing resources and minimize mould movements. We developed a deterministic crowding genetic algorithm for this multiobjective problem. The algorithm has proved to be a powerful and flexible tool to solve large-scale instances of this real and complex scheduling problem.     

A fuzzy logic-based hybrid estimation of distribution algorithm for distributed permutation flowshop scheduling problems under machine breakdown

As the research interest in distributed scheduling is growing, distributed permutation flowshop scheduling problems (DPFSPs) have recently attracted an increasing attention. This paper presents a fuzzy logic-based hybrid estimation of distribution algorithm (FL-HEDA) to address DPFSPs under machine breakdown with makespan criterion. In order to explore more promising search space, FL-HEDA hybridises the probabilistic model of estimation of distribution algorithm with crossover and mutation operators of genetic algorithm to produce new offspring. In the FL-HEDA, a novel fuzzy logic-based adaptive evolution strategy (FL-AES) is adopted to preserve the population diversity by dynamically adjusting the ratio of offspring generated by the probabilistic model. Moreover, a discrete-event simulator that models the production process under machine breakdown is applied to evaluate expected makespan of offspring individuals. The simulation results show the effectiveness of FL-HEDA in solving DPFSPs under machine breakdown.     

Scheduling production of multiple part-types in a system with pre-known demands and deterministic inactive time intervals

We introduce and formalise a scheduling problem that consists in determining an optimum policy (i.e. one that minimises total inventory holding costs) to produce n part-types using a system that is able to share its capacity at all times among these part-types and that switches between an active and an inactive state for pre-known periods of time. Consequently, when active the system must produce enough reserves to meet the demand during the inactive interval. We show that there is always a simple optimum policy in which the production of the part-types is prioritised and, provided the units are properly defined, the optimum priority ordering corresponds to a non-decreasing sequence of the unit holding costs of the part-types.     

Hilbert-type inequalities with a product-type homogeneous kernel and Schur polynomials

The main objective of this paper is to prove Hilbert-type and Hardy-Hilbert-type inequalities with a product-type homogeneous kernel, thus generalizing a result obtained in [Z. Xie, Z. Zheng, A Hilbert-type integral inequality whose kernel is a homogeneous form of degree −3, J. Math. Anal. Appl. 339 (2008) 324-331]. In some cases the best possible constants obtained in these inequalities are expressed using the Schur polynomials.     

Block GPBi-CG method for solving nonsymmetric linear systems with multiple right-hand sides and its convergence analysis

Numerical Algorithms - In this paper, the block generalized product-type bi-conjugate gradient (GPBi-CG) method for solving large, sparse nonsymmetric linear systems of equations with multiple...     

Multilevel Scheduling of Multistage Production with Limited In-Process Inventory

A multistage, multiproduct production scheduling with limited in-process buffers between the successive stages is considered. Each stage is made up of identical parallel machines. The problem objective is to determine an assignment of products to machines over a scheduling horizon, which minimizes the completion time of the production order, with the in-process inventory holding costs as low as possible. The problem is modelled by multilevel integer programmes, each of which is concerned with scheduling a different stage of production. The interactions among the stages are modelled by additional constraints, which take into account the semi-finished product's availability and the limited storage capacity of the intermediate buffers. An hierarchical scheduling algorithm is presented, and an illustrative example for a three-stage production is included.