Multi-Path Approach for Reliability-Redundancy Allocation Using a Scaling Method

The reliability-redundancy allocation problem is an optimization problem that achieves better system reliability by determining levels of component redundancies and reliabilities simultaneously. The problem is classified with the hardest problems in the reliability optimization field because the decision variables are mixed-integer and the system reliability function is nonlinear, non-separable, and non-convex. Thus, iterative heuristics are highly recommended for solving the problem due to their reasonable solution quality and relatively short computation time. At present, most iterative heuristics use sensitivity factors to select an appropriate variable which significantly improves the system reliability. The sensitivity factor represents the impact amount of each variable to the system reliability at a designated iteration. However, these heuristics are inefficient in terms of solution quality and computation time because the sensitivity factor calculations are performed only at integer variables. It results in degradation of the exploration and growth in the number of subsequent continuous nonlinear programming (NLP) subproblems. To overcome the drawbacks of existing iterative heuristics, we propose a new scaling method based on the multi-path iterative heuristics introduced by Ha (2004). The scaling method is able to compute sensitivity factors for all decision variables and results in a decreased number of NLP subproblems. In addition, the approximation heuristic for NLP subproblems helps to avoid redundant computation of NLP subproblems caused by outlined solution candidates. Numerical experimental results show that the proposed heuristic is superior to the best existing heuristic in terms of solution quality and computation time.     

Ant colony approach to constrained redundancy optimization in binary systems

This paper presents an ant colony optimization algorithm to address the constrained redundancy allocation problem in order to maximize system reliability for complex binary systems. The constraints involved, though separable, are both linear and non-linear. We couple an adaptive penalty function with the basic ant colony approach to handle highly constrained problems and embed a local search technique to find still better locally optimal solutions. The proposed algorithm has been tested on a large number of problems, containing even up to 500 subsystems, with both fixed and randomly generated parameters. Experimental results demonstrate the advantage of the proposed algorithm to solve similar types of problems.     

A Hybrid Particle Swarm Optimization Algorithm for the Redundancy Allocation Problem

The Redundancy Allocation Problem generally involves the selection of components with multiple choices and redundancy levels that produce maximum system reliability given various system level constraints as cost and weight. In this paper we investigate the series–parallel redundant reliability problems, when a mixing of components was considered. In this type of problem both the number of redundancy components and the corresponding component reliability in each subsystem are to be decided simultaneously so as to maximise the reliability of system. A hybrid algorithm is based on particle swarm optimization and local search algorithm. In addition, we propose an adaptive penalty function which encourages our algorithm to explore within the feasible region and near feasible region, and discourage search beyond that threshold. The effectiveness of our proposed hybrid PSO algorithm is proved on numerous variations of three different problems and compared to Tabu Search and Multiple Weighted Objectives solutions.     

Pruned Pareto-optimal sets for the system redundancy allocation problem based on multiple prioritized objectives

In this paper, a new methodology is presented to solve different versions of multi-objective system redundancy allocation problems with prioritized objectives. Multi-objective problems are often solved by modifying them into equivalent single objective problems using pre-defined weights or utility functions. Then, a multi-objective problem is solved similar to a single objective problem returning a single solution. These methods can be problematic because assigning appropriate numerical values (i.e., weights) to an objective function can be challenging for many practitioners. On the other hand, methods such as genetic algorithms and tabu search often yield numerous non-dominated Pareto optimal solutions, which makes the selection of one single best solution very difficult. In this research, a tabu search meta-heuristic approach is used to initially find the entire Pareto-optimal front, and then, Monte-Carlo simulation provides a decision maker with a pruned and prioritized set of Pareto-optimal solutions based on user-defined objective function preferences. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

A note of using effective immune based approach for the flow shop scheduling with buffers

This note considers flow shop scheduling with buffers. The objective is to minimize the makespan. As we know, such a problem has several applications in manufacturing and has gained wide attention both in academic and engineering fields. In this note, we propose an immune based approach (IA) to solve this NP-hard problem. Numerical results of 29 benchmark problems from the OR-Library are reported and compared with those of a hybrid genetic algorithm (HGA). As shown, the proposed IA is effective and robust. The solutions by proposed IA are superior to those of HGA reported in the literature. In addition, the average of the relative errors of the proposed IA is only 0.85% for 29 instances with infinite buffers. Limited numerical results show that, as expected, buffer size has a great impact on the scheduling objective especially for larger-scale instances, and there are also significant differences with or without buffers for all instances. But the effect of increasing buffer size from 1 up to 2, 4 or ∞ decreases drastically for all instances.     

基于遗传算法的多目标可靠性优化

在元件的体积、重量和造价的共同约束下的多级串并联系统的可靠性优化问题是一个具有多局部极值的、非线性的、同时具有整数和实数变量的混合优化问题.将遗传算法和多目标可靠性分配问题相结合,对可靠性分配问题进行求解,得到较好效果,从而得出结论,遗传算法在求解多目标可靠性优化问题中是一种行之有效的方法.     

Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.     

A mixed integer linear programming model for reliability optimisation in the component deployment problem

Component deployment is a combinatorial optimisation problem in software engineering that aims at finding the best allocation of software components to hardware resources in order to optimise quality attributes, such as reliability. The problem is often constrained because of the limited hardware resources, and the communication network, which may connect only certain resources. Owing to the non-linear nature of the reliability function, current optimisation methods have focused mainly on heuristic or metaheuristic algorithms. These are approximate methods, which find near-optimal solutions in a reasonable amount of time. In this paper, we present a mixed integer linear programming (MILP) formulation of the component deployment problem. We design a set of experiments where we compare the MILP solver to methods previously used to solve this problem. Results show that the MILP solver is efficient in finding feasible solutions even where other methods fail, or prove infeasibility where feasible solutions do not exist.     

Generalized Surrogate Problem Methodology for Online Stochastic Discrete Optimization

We consider stochastic discrete optimization problems where the decision variables are nonnegative integers and propose a generalized surrogate problem methodology that modifies and extends previous work in Ref. 1. Our approach is based on an online control scheme which transforms the problem into a surrogate continuous optimization problem and proceeds to solve the latter using standard gradient-based approaches while simultaneously updating both the actual and surrogate system states. In contrast to Ref. 1, the proposed methodology applies to arbitrary constraint sets. It is shown that, under certain conditions, the solution of the original problem is recovered from the optimal surrogate state. Applications of this approach include solutions to multicommodity resource allocation problems; in these problems, exploiting the convergence speed of the method, one can overcome the obstacle posed by the presence of local optima.     

A branch-and-price algorithm to solve the integrated berth allocation and yard assignment problem in bulk ports

In this research, two crucial optimization problems of berth allocation and yard assignment in the context of bulk ports are studied. We discuss how these problems are interrelated and can be combined and solved as a single large scale optimization problem. More importantly we highlight the differences in operations between bulk ports and container terminals which highlights the need to devise specific solutions for bulk ports. The objective is to minimize the total service time of vessels berthing at the port. We propose an exact solution algorithm based on a branch and price framework to solve the integrated problem. In the proposed model, the master problem is formulated as a set-partitioning problem, and subproblems to identify columns with negative reduced costs are solved using mixed integer programming. To obtain sub-optimal solutions quickly, a metaheuristic approach based on critical-shaking neighborhood search is presented. The proposed algorithms are tested and validated through numerical experiments based on instances inspired from real bulk port data. The results indicate that the algorithms can be successfully used to solve instances containing up to 40 vessels within reasonable computational time.     

MIP approaches for the integrated berth allocation and quay crane assignment and scheduling problem

In this paper we consider an integrated berth allocation and quay crane assignment and scheduling problem motivated by a real case where a heterogeneous set of cranes is considered. A first mathematical model based on the relative position formulation (RPF) for the berth allocation aspects is presented. Then, a new model is introduced to avoid the big-M constraints included in the RPF. This model results from a discretization of the time and space variables. For the new discretized model several enhancements, such as valid inequalities, are introduced. In order to derive good feasible solutions, a rolling horizon heuristic (RHH) is presented. A branch and cut approach that uses the enhanced discretized model and incorporates the upper bounds provided by the RHH solution is proposed. Computational tests are reported to show (i) the quality of the linear relaxation of the enhanced models; (ii) the effectiveness of the exact approach to solve to optimality a set of real instances; and (iii) the scalability of the RHH based on the enhanced mathematical model which is able to provide good feasible solutions for large size instances.     

A finite element approach to the position problems in open-loop variable geometry trusses

The present paper looks at some kinematic and static-equilibrium problems that arise with variable-geometry trusses (VGTs). The first part of the paper looks at the use of active controls in the correction of static deformations, the second part at the position problems. The separation between deformable- and rigid-body displacements makes it possible to consider separately the corrections in each component of the structure. VGTs are considered as open-loop linkages with redundant rigid-body degrees of freedom. Owing to this redundancy, possible solutions to the inverse problem are in general infinite, for which reason it is necessary to use some optimization criteria. To tackle the problem an optimization procedure with constraints is developed for the purpose of minimizing the displacements of the actuators. Suitable use of the constraints allows us to solve the direct position problem using the same optimization procedure.     

A benchmarking approach to optimal asset allocation for insurers and pension funds

We solve the optimal asset allocation problem for an insurer or pension fund by using a benchmarking approach. Under this approach the objective is an increasing function of the relative performance of the asset portfolio compared to a benchmark. The benchmark can be, for example, a function of an insurer’s liability payments, or the (either contractual or target) payments of a pension fund. The benchmarking approach tolerates but progressively penalizes shortfalls, while at the same time progressively rewards outperformance. Working in a general, possibly non-Markovian setting, a solution to the optimization problem is presented, providing insights into the impact of benchmarking on the resulting optimal portfolio. We further illustrate the results with a detailed example involving an option based benchmark of particular interest to insurers and pension funds, and present closed form solutions.     

Advances in Interval Methods for Deterministic Global Optimization in Chemical Engineering

In recent years, it has been shown that strategies based on an interval-Newton approach can be used to reliably solve a variety of nonlinear equation solving and optimization problems in chemical process engineering, including problems in parameter estimation and in the computation of phase behavior. These strategies provide a mathematical and computational guarantee either that all solutions have been located in an equation solving problem or that the global optimum has been found in an optimization problem. The primary drawback to this approach is the potentially high computational cost. In this paper, we consider strategies for bounding the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method. Recent preconditioning techniques for this purpose are reviewed, and a new bounding approach based on the use of linear programming (LP) techniques is presented. Using this approach it is possible to determine the desired bounds exactly (within round out), leading to significant overall improvements in computational efficiency. These techniques will be demonstrated using several global optimization problems, with focus on problems arising in chemical engineering, including parameter estimation and molecular modeling. These problems range in size from under ten variables to over two hundred, and are solved deterministically using the interval methodology.     

A heuristic method for non-homogeneous redundancy optimization of series-parallel multi-state systems

This paper develops an efficient heuristic to solve the non-homogeneous redundancy allocation problem for multi-state series-parallel systems. Non identical components can be used in parallel to improve the system availability by providing redundancy in subsystems. Multiple component choices are available for each subsystem. The components are binary and chosen from a list of products available on the market, and are characterized in terms of their cost, performance and availability. The objective is to determine the minimal-cost series-parallel system structure subject to a multi-state availability constraint. System availability is represented by a multi-state availability function, which extends the binary-state availability. This function is defined as the ability to satisfy consumer demand that is represented as a piecewise cumulative load curve. A fast procedure is used, based on universal generating function, to evaluate the multi-state system availability. The proposed heuristic approach is based on a combination of space partitioning, genetic algorithms (GA) and tabu search (TS). After dividing the search space into a set of disjoint subsets, this approach uses GA to select the subspaces, and applies TS to each selected subspace. The design problem, solved in this study, has been previously analyzed using GA. Numerical results for the test problems from previous research are reported, and larger test problems are randomly generated. These results show that the proposed approach is efficient both in terms of both of solution quality and computational time, as compared to existing approaches.     

Auxiliary tool allocation in flexible manufacturing systems

This paper analyses a new approach to the machine loading problem arising in flexible manufacturing systems (FMSs). This approach allows the operations to be assigned to machines assuming that machines have access to all the tools required for their operations. This exploits the flexibility of the FMS completely. Next an allocation of tools to machines is determined which satisfies the tool requirements for each machine and minimizes the total number of tools. Thus this approach minimizes the unnecessary tool duplications in the system and maximizes the tool utilization. The problem is modeled as an integer linear program (ILP). We notice that the main problem has a block diagonal structure which is decomposable by relaxing a set of linking constraints. Each separated sub-problem represents a problem of allocation of a single type of tools. We develop a branch-and-bound based exact solution procedure and three heuristic procedures to solve the sub-problems. Our lower bounding approach uses Lanrangean relaxation. The solutions to the Lagrangean relaxation are further used to determine the branching sequences and to develop heuristic approaches. Since finding even a feasible solution to the main problem is NP-hard, we develop only enumerative procedures to solve the main problem. Finally, these solution procedures are tested on randomly generated test problems.     

A GA based penalty function technique for solving constrained redundancy allocation problem of series system with interval valued reliability of components

In this paper, we have considered the problem of constrained redundancy allocation of series system with interval valued reliability of components. For maximizing the overall system reliability under limited resource constraints, the problem is formulated as an unconstrained integer programming problem with interval coefficients by penalty function technique and solved by an advanced GA for integer variables with interval fitness function, tournament selection, uniform crossover, uniform mutation and elitism. As a special case, considering the lower and upper bounds of the interval valued reliabilities of the components to be the same, the corresponding problem has been solved. The model has been illustrated with some numerical examples and the results of the series redundancy allocation problem with fixed value of reliability of the components have been compared with the existing results available in the literature. Finally, sensitivity analyses have been shown graphically to study the stability of our developed GA with respect to the different GA parameters.     

Time-variant reliability-based structural optimization using sorm

Structural optimization under time-invariante reliability constraints is sufficiently well known. The same problem under time-dependent loads and resistances has not yet found satisfying solutions. Recently, a new attempt has been made where structural reliability is determined by the outcrossing approach in the context of first-order reliability methodology (FORM). In the paper an algorithm is designed with which outcrossing rates determined by asymptotic second-order reliability methods (SORM) can be used as constraints in structural optimization. The method is developed for two different types of stationary load models, rectangular wave renewal processes and Gaussian processes, respectively. An example application demonstrates the new methodology     

Importance measures in reliability and mathematical programming

The importance measures have been a sensitivity analysis for a probabilistic system and are applied in diverse fields along with other design tools. This paper provides a comprehensive view on modeling the importance measures to solve the reliability problems such as component assignment problems, redundancy allocation, system upgrading, and fault diagnosis and maintenance. It also investigates importance measures in broad applications such as networks, mathematical programming, sensitivity and uncertainty analysis, and probabilistic risk analysis and probabilistic safety assessment. The importance-measure based methods are among the most practical decision tools.