A DERIVATIVE-FREE ALGORITHM FOR UNCONSTRAINED OPTIMIZATION

In this paper a hybrid algorithm which combines the pattern search method and the genetic algorithm for unconstrained optimization is presented. The algorithm is a deterministic pattern search algorithm,but in the search step of pattern search algorithm,the trial points are produced by a way like the genetic algorithm. At each iterate, by reduplication,crossover and mutation, a finite set of points can be used. In theory,the algorithm is globally convergent. The most stir is the numerical results showing that it can find the global minimizer for some problems ,which other pattern search algorithms don＇t bear.     

A derivative-free algorithm for linearly constrained optimization problems

Based on the NEWUOA algorithm, a new derivative-free algorithm is developed, named LCOBYQA. The main aim of the algorithm is to find a minimizer $x^{*} \in\mathbb{R}^{n}$ of a non-linear function, whose derivatives are unavailable, subject to linear inequality constraints. The algorithm is based on the model of the given function constructed from a set of interpolation points. LCOBYQA is iterative, at each iteration it constructs a quadratic approximation (model) of the objective function that satisfies interpolation conditions, and leaves some freedom in the model. The remaining freedom is resolved by minimizing the Frobenius norm of the change to the second derivative matrix of the model. The model is then minimized by a trust-region subproblem using the conjugate gradient method for a new iterate. At times the new iterate is found from a model iteration, designed to improve the geometry of the interpolation points. Numerical results are presented which show that LCOBYQA works well and is very competing against available model-based derivative-free algorithms.     

A stochastic and asymmetric-information framework for a dominant-manufacturer supply chain

Consider a dominant manufacturer wholesaling a product to a retailer, who in turn retails it to the consumers at $p/unit. The retail-market demand volume varies with p according to a given demand curve. This basic system is commonly modeled as a manufacturer-Stackelberg ([mS]) game under a “deterministic and symmetric-information” (“det-sym-i”) framework. We first explain the logical flaws of this framework, which are (i) the dominant manufacturer-leader will have a lower profit than the retailer under an iso-elastic demand curve; (ii) in some situations the system’s “correct solution” can be hyper-sensitive to minute changes in the demand curve; (iii) applying volume discounting while keeping the original [mS] profit-maximizing objective leads to an implausible degenerate solution in which the manufacturer has dictatorial power over the channel. We then present an extension of the “stochastic and asymmetric-information” (“sto-asy-i”) framework proposed in Lau and Lau [Lau, A., Lau, H.-S., 2005. Some two-echelon supply-chain games: Improving from deterministic–symmetric-information to stochastic-asymmetric-information models. European Journal of Operational Research 161 (1), 203–223], coupled with the notion that a profit-maximizing dominant manufacturer may implement not only [mS] but also “[pm]”—i.e., using a manufacturer-imposed maximum retail price. We show that this new framework resolves all the logical flaws stated above. Along the way, we also present a procedure for the dominant manufacturer to design a profit-maximizing volume-discount scheme using stochastic and asymmetric demand information.     

A derivative-free nonmonotone line-search technique for unconstrained optimization

A tolerant derivative–free nonmonotone line-search technique is proposed and analyzed. Several consecutive increases in the objective function and also nondescent directions are admitted for unconstrained minimization. To exemplify the power of this new line search we describe a direct search algorithm in which the directions are chosen randomly. The convergence properties of this random method rely exclusively on the line-search technique. We present numerical experiments, to illustrate the advantages of using a derivative-free nonmonotone globalization strategy, with approximated-gradient type methods and also with the inverse SR1 update that could produce nondescent directions. In all cases we use a local variation finite differences approximation to the gradient.     

A penalty derivative-free algorithm for nonlinear constrained optimization

In this paper, we modify a derivative-free line search algorithm (DFL) proposed in the Ref. (Liuzzi et al. SIAM J Optimiz 20(5):2614–2635, 2010) to minimize a continuously differentiable function of box constrained variables or unconstrained variables with nonlinear constraints. The first-order derivatives of the objective function and of the constraints are assumed to be neither calculated nor explicitly approximated. Different line-searches are used for box-constrained variables and unconstrained variables. Accordingly the convergence to stationary points is proved. The computational behavior of the method has been evaluated on a set of test problems. The performance and data profiles are used to compare with DFL.     

Knowledge-based framework for automated dynamic supply chain configuration

Supply chain management has gained renewed interest among researchers in recent years. This is primarily due to the availability of timely information across the various stages of the supply chain, and therefore the need to effectively utilize the information for improved performance. Although information plays a major role in effective functioning of supply chains, there is a paucity of studies that deal specifically with the dynamics of supply chains and how data collected in these systems can be used to improve their performance. In this paper I develop a framework, with machine learning, for automated supply chain configuration. Supply chain configuration used to be mostly a one-shot problem. Once a supply chain is configured, researchers and practitioners were more interested in means to improve performance given that initial configuration. However, recent developments in e-commerce applications and faster communication over the Internet in general necessitates dynamic (re)configuration of supply chains over time to take advantage of better configurations. Using examples, I show performance improvements of the proposed adaptive supply chain configuration framework over static configurations.     

A progressive barrier derivative-free trust-region algorithm for constrained optimization

We study derivative-free constrained optimization problems and propose a trust-region method that builds linear or quadratic models around the best feasible and around the best infeasible solutions found so far. These models are optimized within a trust region, and the progressive barrier methodology handles the constraints by progressively pushing the infeasible solutions toward the feasible domain. Computational experiments on 40 smooth constrained problems indicate that the proposed method is competitive with COBYLA, and experiments on two nonsmooth multidisciplinary optimization problems from mechanical engineering show that it can be competitive with the NOMAD software.     

A derivative-free approach for a simulation-based optimization problem in healthcare

Hospitals have been challenged in recent years to deliver high quality care with limited resources. Given the pressure to contain costs, developing procedures for optimal resource allocation becomes more and more critical in this context. Indeed, under/overutilization of emergency room and ward resources can either compromise a hospital’s ability to provide the best possible care, or result in precious funding going toward underutilized resources. Simulation-based optimization tools then help facilitating the planning and management of hospital services, by maximizing/minimizing some specific indices (e.g. net profit) subject to given clinical and economical constraints. In this work, we develop a simulation-based optimization approach for the resource planning of a specific hospital ward. At each step, we first consider a suitably chosen resource setting and evaluate both efficiency and satisfaction of the restrictions by means of a discrete-event simulation model. Then, taking into account the information obtained by the simulation process, we use a derivative-free optimization algorithm to modify the given setting. We report results for a real-world problem coming from the obstetrics ward of an Italian hospital showing both the effectiveness and the efficiency of the proposed approach.     

A derivative-free optimization algorithm based on conditional moments

In this paper we propose a derivative-free optimization algorithm based on conditional moments for finding the maximizer of an objective function. The proposed algorithm does not require calculation or approximation of any order derivative of the objective function. The step size in iteration is determined adaptively according to the local geometrical feature of the objective function and a pre-specified quantity representing the desired precision. The theoretical properties including convergence of the method are presented. Numerical experiments comparing with the Newton, Quasi-Newton and trust region methods are given to illustrate the effectiveness of the algorithm.     

Profit oriented supply chain network optimization

This paper proposes a novel mixed integer linear programming model to solve a supply chain network design problem. The proposed model deals with major issues for supply chains; product quality and cost. These issues are usually solved separately, but in this paper, we investigate effects of product quality on supply chain design and transportation flow. A trade-off between raw material quality, its purchasing and reprocessing costs was considered. Assuming decision maker (DM) wishes to work with a supplier which serves a low quality raw material; this raw material should be in need of reprocessing. To avoid the reprocessing costs, a supplier which serves a high quality raw material should be chosen but at this time the DM has to face a high purchasing cost. A supply chain network which consists of multiple suppliers, manufacturers, distribution centers and retailers is tried to be designed to accomplish aforementioned above trade-offs. The paper examines and discusses the relationship between product quality and supply chain design and offers several managerial insights.     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Scalable subspace methods for derivative-free nonlinear least-squares optimization

Mathematical Programming - We introduce a general framework for large-scale model-based derivative-free optimization based on iterative minimization within random subspaces. We present a...     

An optimization model for traceable closed-loop supply chain networks

In this paper, a new non-linear mixed-integer mathematical programming problem is proposed to model a stochastic multi-product closed-loop supply chain (CLSC). The radio frequency identification (RFID) system is implemented in the supply chain to decrease product losses and the overall lead time of transportation while computing the profit derived from internet and conventional sales. The resulting traceable CLSC improves upon the existing literature by allowing us to: (1) boost the incorporation of traceability assumptions in mathematical programming problems so as to enhance the efficiency and visibility of a supply chain, (2) analyze the strategic effects that different internet sale formats have on customers’ evaluations and acquisition choices, and (3) account for the environmental and socio-economical dimension by explicitly formalizing employment-based incomes as part of the profit function. Two meta-heuristic algorithms are introduced to solve the proposed optimization problem, namely, the greedy randomized adaptive search procedure (GRASP) and particle swarm optimization (PSO). Twelve test problems of different sizes are generated and solved using these algorithms. The computational results show that GRASP outperforms PSO in terms of both profit and CPU time values. Finally, a case study in the network marketing industry is presented and managerial implications outlined to show the validity of the proposed model and shed more light on its practical implications.     

A robust optimization model for agile and build-to-order supply chain planning under uncertainties

Supply chain planning as one of the most important processes within the supply chain management concept, has a great impact on firms’ success or failure. This paper considers a supply chain planning problem of an agile manufacturing company operating in a build-to-order environment under various kinds of uncertainty. An integrated optimization approach of procurement, production and distribution costs associated with the supply chain members has been taken into account. A robust optimization scenario-based approach is used to absorb the influence of uncertain parameters and variables. The formulation is a robust optimization model with the objective of minimizing the expected total supply chain cost while maintaining customer service level. The developed multi-product, multi-period, multi-echelon robust mixed-integer linear programming model is then solved using the CPLEX optimization studio and guidance related to future areas of research is given.     

Environmentally conscious optimization of supply chain networks

The optimization of supply chain structures considering both economic and environmental performances is nowadays an important research topic. However, enterprises are commonly faced with the competing issues of reduced cost, improved customer service and increased environmental factors as a multi-faceted trade-off problem when designing supply chains. Hence, this paper proposes an environmentally conscious optimization model of a supply chain network with a broader and more comprehensive objective function that considers not just the transportation costs, but also the costs for the amount of greenhouse gas emissions, fuel consumption, transportation times, noise and road roughness. The paper sheds light on the trade-offs between various parameters such as vehicle speed, fuel, time, emissions, noise and their total cost, and offers managerial insights on economies of environmentally conscious supply chain optimization. An integer non-linear programming model is developed to help decision makers find the optimal solution under mentioned considerations. The proposed model is validated through the solution of an example, where its applicability to supply chain problems is demonstrated for managerial insights.     

A modified multi-criterion optimization genetic algorithm for order distribution in collaborative supply chain

Supply chain collaboration is prevalent in today’s business model, and has been recognized to be one of the important issues in improving competition strength. Order distribution is to determine which order should be allocated to which supplier, it plays a very important role in a collaborative supply chain, because different order distribution infers different benefit under different criteria. However, there is not much research on the order distribution methodology. This paper adopted a framework of a central coordination system, which is equipped with a multi-criteria genetic optimization feature. In the previous multi-criterion optimization genetic algorithm (MCOGA), the analytic hierarchy process (AHP) is deployed to evaluate the fitness values. In this paper, a modified MCOGA is proposed based on the technique for order preference by similarity to ideal solution (TOPSIS). There are two main parts, namely, searching and evaluation, in genetic algorithms. Compared with the MCOGA, the proposed method takes the advantage of less complexity in the evaluation stage. The numerical example of order distribution is used to illustrate the efficiency of the proposed method.