Error analysis of penalty function techniques for constraint definition in linear algebraic systems

The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.     

Liner and nonlinear elastic analysis of cracked plate: Application of a penalty function and superposition method

Cracked plates of power hardening material under plane strain and incompressibility conditions are investigated in this work by using a penalty function and superposition method. The present numerical method is found to be quite efficient for the relevant problems if the hardening exponent of material is not so large.     

Genetic algorithm with adaptive and dynamic penalty functions for the selection of cleaner production measures: a constrained optimization problem

This paper presents a new approach of genetic algorithm (GA) to solve the constrained optimization problem. In a constrained optimization problem, feasible and infeasible regions occupy the search space. The infeasible regions consist of the solutions that violate the constraint. Oftentimes classical genetic operators generate infeasible or invalid chromosomes. This situation takes a turn for the worse when infeasible chromosomes alone occupy the whole population. To address this problem, dynamic and adaptive penalty functions are proposed for the GA search process. This is a novel strategy because it will attempt to transform the constrained problem into an unconstrained problem by penalizing the GA fitness function dynamically and adaptively. New equations describing these functions are presented and tested. The effects of the proposed functions developed have been investigated and tested using different GA parameters such as mutation and crossover. Comparisons of the performance of the proposed adaptive and dynamic penalty functions with traditional static penalty functions are presented. The result from the experiments show that the proposed functions developed are more accurate, efficient, robust and easy to implement. The algorithms developed in this research can be applied to evaluate environmental impacts from process operations.     

Closed form models for determining the optimal dwell point location in automated storage and retrieval systems

This paper examines the problem of where a storage/retrieval machine should reside, or dwell, when an automated storage and retrieval system (AS/RS) becomes idle to minimize the expected value of the next transaction time. After a review of the relevant literature on AS/RS dwell point strategies, this paper proposes several analytical models of these expected response times of the AS/RS based on the relative locations of the input and output ports of the AS/RS. It uses a continuous rack approximation to provide analytical models of the dwell point location problem. These models provide closed form solutions for the dwell point location in an AS/RS. Extensions are made to consider AS/RS with a variety of configurations including multiple input and output ports. These models not only provide solutions to the dwell point location problem, but they provide considerable insight into the nature of this problem, which is particularly valuable when the requirements facing the AS/RS are uncertain.     

Mesoscopic fabric models using a discrete mass-spring approach: Yarn-yarn interactions analysis

A meso/macro discrete model of fabric has been developed, accounting for yarn-yarn interactions occurring at the crossing points. The fabric yarns, described initially by a Fourier series development, are discretized into elastic straight bars represented by stretching springs and connected at frictionless hinges by rotational springs. The motion of each node is described by a lateral displacement and a rotation. The compressibility of both yarns is expressed as a kinematic relationship, considering frictionless motions of the yarns. The expression of the reaction force exerted by the transverse yarns at the contact points is then assessed, from which the work of the reaction forces is established. The equilibrium shape of the yarn is obtained as the minimum of its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. Simulations of a traction curve of a single yarn are performed, that evidence the effect of the yarn interactions and the main deformation mechanisms.     

Algorithm for a class of constrained weight least square problems and applications

An algorithm, based on ordinal optimisation (OO) and sensitive theories, is presented to solve a class of constrained weight least square problems with continuous and discrete variables. the proposed algorithm can cope with an enormous amount of computational complexity problems and has a high probability of obtaining a good enough solution according to the oo theory. this method has some advantages, such as computational efficiency, numerical stability and the superiority of the good enough solution. the proposed algorithm is explicit, compact and easy to program. test results demonstrate that the proposed approach is more computational-efficient than other existing approaches for solving constrained-state estimation problems with continuous and discrete variables on the ieee 30-bus and the ieee 118-bus systems.     

On a branch-and-bound approach for a Huff-like Stackelberg location problem

Modelling the location decision of two competing firms that intend to build a new facility in a planar market can be done by a Huff-like Stackelberg location problem. In a Huff-like model, the market share captured by a firm is given by a gravity model determined by distance calculations to facilities. In a Stackelberg model, the leader is the firm that locates first and takes into account the actions of the competing chain (follower) locating a new facility after the leader. The follower problem is known to be a hard global optimisation problem. The leader problem is even harder, since the leader has to decide on location given the optimal action of the follower. So far, in literature only heuristic approaches have been tested to solve the leader problem. Our research question is to solve the leader problem rigorously in the sense of having a guarantee on the reached accuracy. To answer this question, we develop a branch-and-bound approach. Essentially, the bounding is based on the zero sum concept: what is gain for one chain is loss for the other. We also discuss several ways of creating bounds for the underlying (follower) sub-problems, and show their performance for numerical cases. This work has been supported by the Ministry of Education and Science of Spain through grant SEJ2005/06273/ECON. M. Elena Sáz was supported by a junior research grant of Mansholt Graduate School (Wageningen Universiteit).     

Iterative procedures for improving penalty function solutions of algebraic systems

The standard implementation of the penalty function approach for the treatment of general constraint conditions in discrete systems of equations often leads to computational difficulties as the penalty weights are increased to meet constraint satisfaction tolerances. A family of iterative procedures that converges to the constrained solution for fixed weights is presented. For a discrete mechanical system, these procedures can be physically interpreted as an equilibrium iteration resulting from the appearance of corrective force patterns at the nodes of ‘constraint members’ of constant stiffness. Three forms of the iteration algorithm are studied in detail. Convergence conditions are established and the computational error propagation behaviour of the three forms is analysed. The conclusions are verified by numerical experiments on a model problem. Finally, practical guidelines concerning the implementation of the corrective process in large-scale finite element codes are offered.     

A constrained EPSAC approach to inventory control for a benchmark supply chain system

The design of an appropriate inventory control policy for a supply chain (SC) plays an essential role in tempering inventory instability and bullwhip effect. Several constraints are commonly encountered in actual operations so managers are required to take these physical restrictions into account when designing the inventory control policy. Model predictive control (MPC) appears as a promising solution to this issue, due to its capability of finding optimal control actions for a constrained SC system. Therefore, the inventory control problem for a benchmark SC is solved using the extended prediction self-adaptive control approach to MPC. To extend methodologies in our previous work, the control framework relies on generic process model and incorporates the physical constraints arising from practical operations to form the general constrained optimisation problems. The managers can choose from decentralised and centralised control structures according to specific informational and organisational factors of their SCs. The proposed control schemes in this study may be appropriate for industrial practice because the designed policy can bring a reduction of over 30% in operating cost and a significant increase of customer satisfaction level compared with that of the conventional policy.     

Demand Driven MRP: assessment of a new approach to materials management

Demand driven material requirements planning or DDMRP is a recent and promising material management method that has been developed and implemented in the practitioner world. Essentially, DDMRP represents a rethinking of the basic MRP logic. By incorporating elements drawn from Lean Systems and the Theory of Constraints and by introducing new features such as dynamic buffers, DDMRP modifies the basic MRP logic so that it is better able to satisfy customer demands in an increasingly demanding, turbulent and dynamic environment. Claims have been made by firms that DDMRP represents a superior planning approach. In this paper, we introduce and explore DDMRP. In addition, we evaluate its effectiveness relative to two other widely accepted approaches – MRP II and Kanban/Lean production – through a series of structured computer simulation experiments. The results strongly indicate that DDMRP does represent a superior approach – one that warrants further academic study.     

A class of universal relations for constrained,isotropic elastic materials

Summary A class of universal relations for all kinematically constrained, isotropic, elastic materials is described by the equationSB=BS relating the symmetric extra stress and the Cauchy-Green deformation tensors. This rule generates easily at most three universal relations for all kinematically admissible deformations of any constrained, isotropic body for which these tensors are nondiagonal. New universal formulae for homogeneous, compressible and incompressible materials reinforced by inextensible fibers in a variety of arrangements are presented for several kinds of homogeneous and nonhomogeneous, controllable universal deformations.     

A natural gas cash-out problem: A bilevel programming framework and a penalty function method

One of the many complex problems that arise from the transmission and marketing of natural gas is when a shipper draws a contract with a pipeline company to deliver a certain amount of gas among several points. What is actually delivered is often different from the amount that had been originally agreed upon. This phenomenon is called an imbalance. When an imbalance occurs, the pipeline penalizes the shipper by imposing a cash-out penalty policy. Since this penalty is a function of the operating daily imbalances, an important decision-making problem for the shippers is how to carry out their daily imbalances so as to minimize their incurred penalty. In this paper, we introduce the problem of minimizing the cash-out penalty costs from the point of view of a natural gas shipping party. We present a mixed integer bilevel linear programming model and discuss its underlying assumptions. To solve it efficiently, we reformulate it as a standard mathematical program and describe a penalty-function algorithm functions for its solution. The algorithm is well-founded and its convergence is proved. Results of numerical experiments support the algorithm’s robustness providing a valuable solution technique for this very important and complex problem in the natural gas market.     

Human reliability analysis: a human point of view

Present human reliability analysis (HRA) is like a Centaur in that it is only half human. Error psychology is expected to provide a path leading us in the right direction. AI technology can be utilized to better understand errors. It is pointed out that the next generation HRA will require two additional components in which an analysis is made to identify two sets of error prone situations (EPSs): one in which human operators are likely to initiate critical events, and another in which they are likely to override engineered safeguard features. Carefully designed empirical studies are believed to be useful in obtaining general knowledge about critical EPSs. It is suggested that any future effort should be made based on sound psychological concepts.     

A constrained non‐linear system approach for the solution of an extended limit analysis problem

A number of recent papers (see, e.g. (Int. J. Mech. Sci. 2007; 49 :454–465; Eur. J. Mech. A/Solids 2008; 27 :859–881; Eng. Struct. 2008; 30 :664–674; Int. J. Mech. Sci. 2009; 51 :179–191)) have shown that classical limit analysis can be extended to incorporate such important features as geometric non‐linearity, softening and various so‐called ductility constraints. The generic formulation takes the form of a challenging (nonconvex and nonsmooth) optimization problem referred to in the mathematical programming literature as a mathematical program with equilibrium constraints (MPEC). Similar to a classical limit analysis, the aim is to compute in a single step a bound (upper bound, in the case of the extended problem) to the maximum load. The solution algorithm so far proposed to solve the MPEC is to convert it into an iterative non‐linear programming problem and attempts to process this using a standard non‐linear optimizer. Motivated by the fact that no method is guaranteed to solve such MPECs and by the need to avoid the use of an optimization approach, which is unfamiliar to most practising engineers, we propose, in the present paper, a novel numerical scheme to solve the MPEC as a constrained non‐linear system of equations. We illustrate the application of this approach using the simple class of elastoplastic softening skeletal structures for which certain ductility conditions are prescribed. Copyright © 2009 John Wiley & Sons, Ltd.     

Hybrid-stress models for elastic–plastic analysis by the initial-stress approach

Two alternative numerical methods are presented, based on the assumed-stress hybrid finite element model and the initial-stress approach, for the elastic–plastic small-deflection analysis of structures under static loading. The use of the initial-stress approach results in a set of simultaneous linear algebraic incremental equations to be solved at each loading step, with the elastic stiffness matrix remaining unchanged throughout the loading process and the effects of plasticity included as equivalent element loads. The derivation of these alternative methods differs in the assigning of the incremental stress which satisfies equilibrium; in one case it is the actual stress increment while in the other it is a fictitious stress increment. An equilibrium imbalance correction is included in each of these methods to prevent drifting of the solution during the incremental process. Example solutions are presented which demonstrate the accuracy of the two methods and permit comparisons of the relative efficiencies of the two methods.     

Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage

This paper deals with a class of rate-independent metal plasticity models which exhibit non-linear isotropic hardening, non-linear kinematic hardening (Chaboche-Marquis model) and ductile damage (Lemaitre-Chaboche model). The backward Euler scheme is used to integrate the rate constitutive relations. The non-linear equations obtained are solved by the Newton method. The consistent tangent operator is obtained by exact linearization of the algorithm. Despite the complexity of the constitutive equations, closed-form expressions are derived, without any approximations. Analytical, numerical and experimental results are presented and discussed.