Formulation and solution of the non-linear,damped eigenvalue problem for skeletal systems

This paper presents and discusses an Arnoldi-based eigensolution technique for evaluating the complex natural frequencies and mode shapes from frequency dependent quadratic eigenproblems associated with vibration analysis of damped structures. The new solution technique is used in conjunction with a mixed finite element modelling procedure which utilizes both the polynomial and frequency dependent displacement fields in formulating the system matrices. This modelling provides the ability to represent a frequency dependent damping matrix in vibration analysis of skeletal systems. The eigensolution methodology presented here is based upon the ability to evaluate a specific set of parametrized curves for the non-linear eigenvalue problem at given values of the parameter. Numerical examples illustrate that this method, used in conjunction with a secant interpolation, accurately evaluates the complex natural frequencies and modes of the quadratic non-linear eigenproblem and verifies that the new eigensolution technique coupled with the mixed finite element modelling procedure is more accurate than the conventional finite element models.     

Application of ELSP solution techniques to the deterministic robotic scheduling problem

This paper studies the flexible deterministic robotic scheduling problem (FDRSP) where one tool changer (robot) is in charge of several machining operations and overlapping tool changes are not allowed. The relationship between the FDRSP and the economic lot scheduling problem (ELSP) is explored. The most promising solution techniques for the ELSP guaranteeing feasibility in advance are modified and applied to the FDRSP. It is concluded that the basic period approach performs the best in small problems, while the rounding to powers of two approach is superior for medium to large problems.     

A comparison of solution approaches for the fixed-sequence economic lot scheduling problem

The economic lot scheduling problem (ELSP) has received much attention recently. The general version of the problem has a non-convex objective function, so it is difficult to find truly optimal solutions. We examine the three most popular heuristic approaches to the fixed-sequence ELSP. Each approach imposes one or both of these simplifying constraints: the zero-switch constraint (production of a part is started only when its inventory is depleted) and the equal-lot constraint (the lot size of a given part is constant through time). We provide a formulation that clarifies the relationships between the general problem and the three constrained versions, and compare their performances in a computational study.     

On the efficient modeling and solution of the multi-mode resource-constrained project scheduling problem with generalized precedence relations

For variants of the single-mode resource-constrained project scheduling problem, state-of-the-art exact algorithms combine a Branch and Bound algorithm with principles from Constraint Programming and Boolean Satisfiability Solving. In our paper, we propose new exact approaches extending the above principles to the multi-mode RCPSP (MRCPSP) with generalized precedence relations (GPRs). More precisely, we implemented two constraint handlers cumulativemm and gprecedencemm for the optimization framework SCIP. With the latter, one can model renewable resource constraints and GPRs in the context of multi-mode activities, respectively. Moreover, they integrate domain propagation and explanation generation techniques for the above problem characteristics. We formulate three SCIP-models for the MRCPSP with GPRs, two without and one with our constraint handler gprecedencemm. Our computational results on instances from the literature with 30, 50 and 100 activities show that the addition of this constraint handler significantly strengthens the SCIP-model. Moreover, we outperform the state-of-the-art exact approach on instances with 50 activities when imposing time limits of 27 s. In addition, we close (find the optimal solution and prove its optimality for) 289 open instances and improve the best known makespan for 271 instances from the literature.     

Production scheduling with uncertain supply: a new solution to the open pit mining problem

The annual production scheduling of open pit mines determines an optimal sequence for annually extracting the mineralized material from the ground. The objective of the optimization process is usually to maximize the total Net Present Value (NPV) of the operation. Production scheduling is typically a Mixed Integer Programming (MIP) type problem containing uncertainty in the geologic input data and economic parameters involved. Major uncertainty affecting optimization is uncertainty in the mineralized materials (resource) available in the ground which constitutes an uncertain supply for mine production scheduling. A new optimization model is developed herein based on two-stage Stochastic Integer Programming (SIP) to integrate uncertain supply to optimization; past optimization methods assume certainty in the supply from the mineral resource. As input, the SIP model utilizes a set of multiple, stochastically simulated scenarios of the mineralized materials in the ground. This set of multiple, equally probable scenarios describes the uncertainty in the mineral resource available in the ground, and allows the proposed model to generate a single optimum production schedule. The method is applied for optimizing the annual production scheduling at a gold mine in Australia and benchmarked against a traditional scheduling method using the traditional single “average type” assessment of the mineral resource in the ground. In the case study presented herein, the schedule generated using the proposed SIP model resulted in approximately 10% higher NPV than the schedule derived from the traditional approach.     

A correction to the analytical solution of the mixed-mode bending (MMB) problem

This paper addresses the analytical solution of the mixed-mode bending (MMB) problem. The first published solutions used a load separation in pure mode I and mode II and were applied for a crack length less than the beam half-span, a ⩽ L. In later publications, the same mode separation was used in deriving the analytical solution for crack lengths bigger than the beam half-span, a > L. In this paper it is shown that this mode separation is not valid when a > L and in some cases may lead to very erroneous results. The correct mode separation and the corresponding analytical solutions, when a > L, are presented. Results, of force vs. displacement and force vs. crack length graphs, obtained using the existing formulation and the corrected formulation are compared. A finite element solution, which does not use mode separation, is also presented.     

The general lotsizing and scheduling problem

The GLSP (GeneralLotsizing andSchedulingProblem) addresses the problem of integrating lotsizing and scheduling of several products on a single, capacitated machine. Continuous lotsizes, meeting deterministic, dynamic demands, are determined and scheduled with the objective of minimizing inventory holding costs and sequence-dependent setup costs. As the schedule is independent of predefined time periods, the GLSP generalizes known models using restricted time structures. Three variants of a local search algorithm, based onthreshold accepting, are presented. Computational tests show the effectiveness of these heuristic approaches and are encouraging for further extensions of the basic model.     

Formulation and solution of hierarchical finite element equations

The paper presents a general hierarchical formulation applicable to both elliptic and hyperbolic problems. Static and eigenvalue linear elastic problems as well as convection–diffusion problems are studied. The hierarchical formulation is well suited for adaptive procedures. For the convection-diffusion problem the hierarchical approximation is made in time only. Different hierarchical functions are proposed for different types of problems. Both weighted residual and least-squares formulations are applied. A combination of these two gives a penalty method with a constraint equation corresponding to the least-squares method. A whole class of time integration formulae is obtained. These are all suitable for adaptive procedures owing to the hierarchical approximation in the time domain. If a linear discontinuous hierarchical base function is used in the Galerkin weak formulation, the method so obtained corresponds to the discontinuous Galerkin method in time and is especially suited for convection dominated problems. The streamline-diffusion method is found to be the aforementioned penalty method. This paper also examines the sequence of nested equation systems that results from a hierarchical finite element formulation. Properties of these systems arising from static problems are investigated. The paper presents some new possibilities for iterative solution of hierarchic element equations, and different procedures are compared in a numerical example. Finally, a simple ID convection-diffusion problem clearly shows that the proposed hierarchical formulation in time gives a stable and accurate solution even for convection dominated flow.     

On the solution of a minimum compliance topology optimisation problem by optimality criteria without a priori volume constraint specification

The paper presents a topology optimisation method based on optimality criteria for total potential energy maximisation with a volume constraint. The final volume of the optimal structural configuration has not to be specified a priori and is directly controlled by the stress, displacement or stiffness constraints defined at the design problem layout phase. The proposed method leads to the identification of well defined structures characterised by a small number of discrete elements with intermediate material properties within a limited number of iterations. The results obtained by solving several two dimensional benchmark problems are shown.     

An optimal solution of energy scheduling problem based on chance-constraint programming model using Interval-valued neutrosophic constraints

Optimization and Engineering - Neutrosophic sets have been commenced as a generalization of crisp, fuzzy and intuitionistic fuzzy sets to depict vague, incompatible and deficient information about...     

A tabu search algorithm with solution space partition and repairing procedure for cyclic robotic cell scheduling problem

This paper proposes a tabu search (TS) algorithm to solve an NP-hard cyclic robotic scheduling problem. The objective is to find a cyclic robot schedule that maximises the throughput. We first formulate the problem as a linear program, provided that the robot move sequence is given, and reduce the problem to searching for an optimal robot move sequence. We find that the solution space can be divided into some specific subspaces by the maximal number of works-in-process. Then, we propose a TS algorithm to synchronously perform local searches in each subspace. To speed up our algorithm, dominated subspaces are eliminated by lower and upper bounds of the cycle time during the iterations. In the TS, a constructive heuristic is developed to generate initial solutions for each subspace and a repairing procedure is proposed to maintain the feasibility of the solutions generated in the initialisation stage and the neighbours search process. Computational comparison both on benchmark instances and randomly generated instances indicates that our algorithm is efficient for the cyclic robotic scheduling problem.     

An industrial extension of the discrete lot-sizing and scheduling problem

We propose a model and solution algorithm for an industrial production planning problem at Solideal, an international tire manufacturer. The tires are built in molds and are produced in heaters in large series production runs. Some tires can be cured in two different types of heaters with different efficiencies. Preparing a heater and mold for a specific tire type requires a start up. The resulting lot-sizing problem is an extension of the standard Discrete Lot-Sizing and Scheduling Problem. The specific extensions which complicate the problem are: (i) general start-up times, which can be a fraction of the time bucket; (ii) multiple alternative machines with different efficiencies; (iii) multiple capacitated resources, namely the molds and heaters; and (iv) backlogging. These issues are directly motivated by our real life production planning problem. We propose a column-generation-based algorithm for this problem. The dynamic programming recursion for the subproblem is substantially improved by using valid bounds on the state space and cost function. Further, Lagrange relaxation is used to reduce the degeneracy of the master problem. We test our algorithm on real life data sets with up to 30 products and 30 periods and find good quality solutions and lower bounds within a reasonable computation time. Our best implementation has an overall average gap of 0.25% on these test problems.     

A survey on the resource-constrained project scheduling problem

In this paper, research on the resource-constrained project scheduling problem is classified according to specified objectives and constraints. Each classified area is extensively surveyed, and special emphasis is given to trends in recent research. Specific papers involving nonrenewable resource constraints and time/cost-based objectives are discussed in detail because they present models that are close representations of real-world problems. The difficulty of solving such complex models by optimization techniques is noted. For the purposes of this survey, a set of 78 optimally solved test problems from the literature and a second set of 110 benchmark problems have been subjected to analysis with some well-known dispatching rules and a scheduling algorithm that consists of a decision-making process utilizing the problem constraints as a base of selection. The computational results are reported and discussed in the text. Constructive scheduling algorithms that are directly based on the problem constraints and whose performances are independent of problem characteristics are identified as a promising area for future research.     

Modeling the satellite placement problem as a network flow problem with one side constraint

Summary The placement of telecommunication satellites in the geostationary orbit (GSO) gives rise to NP-hard optimization problems usually approached with iterative neighborhood (possibly tabu) search schemes. A typical iteration thereof consists in fixing an order for the satellites and determining their actual locations by linear programming. In such procedures it is crucial to efficiently solve the very large number of arising special linear programs. We describe those linear programs, characterize their duals as special network flow problems with one side constraint and then present an efficient network simplex method to solve them. Since these problems can be highly degenerate, we generalize Cunningham's concept of strongly feasible bases to our case and present a procedure based thereupon which prevents cycling. Computational experience with our algorithms substantiates our efficiency claims.     

The warehouse scheduling problem: formulation and algorithms

The Warehouse Scheduling Problem is a deterministic multi-item inventory problem with a restriction on warehouse floor space available. We formulate a mixed integer nonlinear programming problem for the objective of minimizing long run inventory holding and order costs per unit of time. We integrate algorithms for staggering orders, described in companion papers, with a heuristic to choose the order sequences. The result is called Sequenced Staggering. We describe a new algorithm to generate order frequencies, called the powers-of-two-factor-of-three technique, as a generalization of Roundy's roundoff technique for powers-of-two policies. We report on a computational study of four hybrid algorithms for solving the warehouse scheduling problem, including the competing algorithm of Gallego, Queyranne, and Simchi-Levi. Based on these results, we recommend the combination of powers-of-two frequencies with Sequenced Staggering.     

Method of bilateral approximations in the solution of certain nonlinear constraint problems

The solution of nonlinear constraint problems in the theory of heterogeneous ignition and heat conduction reduces to the solution of a system of nonlinear Volterra integral equations, followed by a numerical evaluation the algorithm for which is based on the use of sums analogous to Darbu sums.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 2, pp. 356–363, August, 1972.