Practical solution approaches to solve a hierarchical stochastic production planning problem in a flexible automated workshop in China

This paper deals with a Hierarchical Stochastic Production Planning (HSPP) problem of Flexible Automated Workshops (FAWs), each with a number of Flexible Manufacturing Systems (FMSs). The problem includes not only the standard (demand, capacity and material supply) uncertainties but also uncertainties in processing times, necessity for rework and scrap. In contrast to most work that only considers either single period or infinite horizons, we also considers multiple time periods and multiple products. One objective of this paper is to determine a cost minimizing production plan for each FMS taking into consideration work-inprocess inventory, work centers overload and underload, cumulative over- and under-production of finished products over a finite time horizon. The HSPP problem is formulated as a stochastic nonlinear programming model whose constraints are linear but whose objective function is piecewise linear. To facilitate the solution procedure, the model is first transformed into a deterministic nonlinear programming model and then into a linear programming model. For medium- or small-scale problems, Karmarkar's algorithm is applied to obtain the solution. For large-scale problem, an interaction/prediction algorithm is used. The effectiveness of these approaches is benchmarked against the linear programming method in Matlab 5.0 in various HSPP settings.     

The Replication of Multi-year Solutions Using Single Period Models of Electric Utility Capacity Expansion Planning

When fuel prices are projected to change over the planning horizon, a standard approach is to formulate a multiperiod capacity expansion model for a utility. However, multiperiod models are considerably larger and more cumbersome than a single period model. An alternative strategy is to develop a single period model whose optimal solution replicates the first period part of the multiperiod optimal solution. This allows the model builder to trade-off a more aggregate multiperiod model for a more detailed single period, rolling horizon model without increasing computational complexity. When and how this can be accomplished in a model of electric utilities is the subject of this paper.     

REPLACEMENT OF TECHNOLOGY WHEN A NEW TECHNOLOGICAL BREAKTHROUGH IS EXPECTED

A replacement problem is presented in which two technologies are involved. One technology is immediately available while the second, considered a technological breakthrough, is expected at some uncertain time in the future. The technological environment is dynamic. Therefore, special attention is given to the availability and reliability of data. Decision horizon stopping rules are suggested when long time horizons are considered. The algorithm is different from currently known standard procedures. It does not rely on the monotonicity properties of the optimal solution that occur when the horizon is extended, nor on action elimination considerations based on non-homogeneous Markov decision analysis. Rather, a special procedure is suggested in which an auxiliary problem binds the solution of the original problem. Interrelationships between the two problems lead to the desired horizon results. The procedure detects much shorter forecast horizons than the Bess and Sethi (1988) procedure without increasing the computational effort needed.     

Multiperiod competitive supply chain networks with inventorying and a transportation network equilibrium reformulation

In this paper, we present a multitiered dynamic supply chain network equilibrium modeling framework in which the decision-makers have sufficient information about the future and seek to determine their optimal plans that maximize their profits over the multiperiod planning horizon. We construct the finite-dimensional variational inequality governing the equilibrium of the multiperiod competitive supply chain network. The model allows us to investigate the interplay of the heterogeneous decision-makers in the supply chain in a dynamic setting, and to compute the resultant equilibrium pattern of product outputs, transactions, inventories, and product prices. We then establish the supernetwork equivalence of the multiperiod supply chain model with a properly configured transportation network, which provides a new interpretation of the equilibrium conditions of the former in terms of paths and path flows. This framework offers great modeling flexibility so that, for example, transportation delay and/or perishable products can be easily handled, as we also demonstrate. Numerical examples are provided to illustrate how such multiperiod supply chain problems can be reformulated and solved as transportation network equilibrium problems in practice.     

A multiperiod stochastic production planning and sourcing problem with service level constraints

We study a stochastic multiperiod production planning and sourcing problem of a manufacturer with a number of plants and/or subcontractors. Each source, i.e. each plant and subcontractor, has a different production cost, capacity, and lead time. The manufacturer has to meet the demand for different products according to the service level requirements set by its customers. The demand for each product in each period is random. We present a methodology that a manufacturer can utilize to make its production and sourcing decisions, i.e., to decide how much to produce, when to produce, where to produce, how much inventory to carry, etc. This methodology is based on a mathematical programming approach. The randomness in demand and related probabilistic service level constraints are integrated in a deterministic mathematical program by adding a number of additional linear constraints. Using a rolling horizon approach that solves the deterministic equivalent problem based on the available data at each time period yields an approximate solution to the original dynamic problem. We show that this approach yields the same result as the base stock policy for a single plant with stationary demand. For a system with dual sources, we show that the results obtained from solving the deterministic equivalent model on a rolling horizon gives similar results to a threshold subcontracting policy. Correspondence to: Fikri KaraesmenThe authors are grateful to Yves Dallery for his ideas, comments and suggestions on the earlier versions of this paper.     

Rolling-horizon lot-sizing when set-up times are sequence-dependent

The challenging problem of efficient lot sizing on parallel machines with sequence-dependent set-up times is modelled using a new mixed integer programming (MIP) formulation that permits multiple set-ups per planning period. The resulting model is generally too large to solve optimally and, given that it will be used on a rolling horizon basis with imperfect demand forecasts, approximate models that only generate exact schedules for the immediate periods are developed. Both static and rolling horizon snapshot tests are carried out. The approximate models are tested and found to be practical rolling horizon proxies for the exact model, reducing the dimensionality of the problem and allowing for faster solution by MIP and metaheuristic methods. However, for large problems the approximate models can also consume an impractical amount of computing time and so a rapid solution approach is presented to generate schedules by solving a succession of fast MIP models. Tests show that this approach is able to produce good solutions quickly.     

A Two-Stage Planning Model for Power Scheduling in a Hydro-Thermal System Under Uncertainty

A two-stage stochastic programming model for the short- or mid-term cost-optimal electric power production planning is developed. We consider the power generation in a hydro-thermal generation system under uncertainty in demand (or load) and prices for fuel and delivery contracts. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. A stochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints that couple power units. It is assumed that the stochastic load and price processes are given (or approximated) by a finite number of realizations (scenarios). Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single unit subproblems. The stochastic thermal and hydro subproblems are solved by a stochastic dynamic programming technique and by a specific descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the primal problem is developed. Numerical results are presented for realistic data from a German power utility and for numbers of scenarios ranging from 5 to 100 and a time horizon of 168 hours. The sizes of the corresponding optimization problems go up to 400.000 binary and 650.000 continuous variables, and more than 1.300.000 constraints.     

基于Benders分解的租赁车队规划

以汽车租赁业的日常车辆调配为背景,研究租赁车队的战术规划问题。将车辆调配情况抽象到时空网络结构中,并根据车辆需求的供应策略和时空节点的流量平衡得到约束条件,以企业运营成本最小为目标建立优化模型。针对模型特点采用Benders分解算法将原问题分解为两类子问题,给出对应的算法步骤。以一周为战术规划期设计算例,对模型和算法的有效性进行检验,结果表明能够为优化车队调配提供较好的辅助决策支持。     

Multi-objective short-term production scheduling for open-pit mines: a hierarchical decomposition-based algorithm

This article presents a novel algorithm for solving a short-term open-pit production-scheduling problem in which several objectives, of varying priority, characterize the quality of each solution. A popular approach employs receding horizon control, dividing the horizon into N period-aggregates of increasing size (number of periods or span). An N-period mixed integer program (MIP) is solved for each period in the original horizon to incrementally construct a production schedule one period at a time. This article presents a new algorithm that, in contrast, decomposes the horizon into N period-aggregates of equal size. Given a schedule for these N periods, obtained by solving an N-period MIP, the first of these aggregates is itself decomposed into an N-period scheduling problem with guidance provided on what regions of the mine should be extracted. The performance of this hierarchical decomposition-based approach is compared with that of receding horizon control on a suite of data sets generated from an operating mine producing millions of tons of ore annually. As the number of objectives being optimized increases, the hierarchical decomposition-based algorithm outperforms receding horizon control, in a majority of instances.     

Multi-product continuous plant scheduling: combination of decomposition,genetic algorithm,and constructive heuristic

We propose a polylithic method for medium-term scheduling of a large-scale industrial plant operating in a continuous mode. The method combines a decomposition approach, a genetic algorithm (GA) and a constructive MILP-based heuristic. In the decomposition, decisions are made at two levels, using the rolling horizon approach. At the upper level, a reduced set of products and the time period is chosen to be considered in the lower level. At the lower level, a short-term scheduling MILP-model with event-based representation is used. A heuristic solution to the lower level problem is found using a constructive Moving Window heuristic guided by a genetic algorithm. The GA is applied for finding efficient utilisation of critical units in the lower level problem. For solving the one unit scheduling problem, a parallel dynamic programming algorithm is proposed. Implementation of the dynamic programming algorithm for a graphics processing unit (GPU) is incorporated in the GA for improving its performance. The experimental study of the proposed method on a real case of a large-scale plant shows a significant improvement of the solution quality and the solving time comparing to the pure decomposition algorithm proposed in the earlier study, and confirmed suitability of the proposed approach for the real-life production scheduling. In particular, the reduction of the number of changeovers and their duration in the obtained solution as well as the CPU time of solving the problem was about 60% using the new approach.     

SOLUTION OF SOME COMBINATORIAL OPTIMIZATION PROBLEMS ENCOUNTERED IN WATER RESOURCES DEVELOPMENT†

The solution of some constrained combinatorial optimization problems encountered in the preinvestment planning of large-scale water resources systems is discussed. Mathematical structures are formulated for several problems involving the optimal selection, sequencing and timing of a set of water resources development projects which, in the aggregate, must satisfy a number of continuous time demand projections at every point in a finite planning horizon. An overview of four different solution techniques is also given. Specifically, myopic decision rules, integer programming formulations, and both implicit enumeration by branch-and-bound algorithms and dynamic programming algorithms are discussed. A computational comparison of these solution techniques on a number of real-world water resources problems of various sizes is also reported.     

A study of convergence in decentralized design processes

The decomposition and coordination of decisions in the design of complex engineering systems is a great challenge. Companies who design these systems routinely allocate design responsibility of the various subsystems and components to different people, teams or even suppliers. The mechanisms behind this network of decentralized design decisions create difficult management and coordination issues. However, developing efficient design processes is paramount, especially with market pressures and customer expectations. Standard techniques to modeling and solving decentralized design problems typically fail to understand the underlying dynamics of the decentralized processes and therefore result in suboptimal solutions. This paper aims to model and understand the mechanisms and dynamics behind a decentralized set of decisions within a complex design process. By using concepts from the fields of mathematics and economics, including Game Theory and the Cobweb model, we model a simple decentralized design problem and provide efficient solutions. This new approach uses matrix series and linear algebra as tools to determine conditions for convergence of such decentralized design problems. The goal of this paper is to establish the first steps towards understanding the mechanisms of decentralized decision processes. This includes two major steps: studying the convergence characteristics and finding the final equilibrium solution of a decentralized problem. Illustrations of the developments are provided in the form of two decentralized design problems with different underlying behavior.     

Optimizing the performance of a repairable system under a maintenance and repair contract

In this article a number of maintenance models for finite horizons are reviewed. These methods have not been applied as frequently as their infinite horizon counterparts, and yet are very much applicable to systems under maintenance and repair contracts. The emphasis in this paper is on repairable systems under maintenance and repair contracts, where the decision has to be made whether and when to repair or replace the system. Based on a case study, a new finite horizon model will be constructed and new approaches to analyze and improve repair/replacement decisions under a finite horizon introduced. Finally, the meaning of ‘risk’ and ‘criticality’ in the context of maintenance contracts will be discussed and quantified. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic-demand capacitated facility location problems with and without relocation

We consider two multi-period dynamic-demand capacitated location problems. In the first problem, the facilities are allowed to be relocated in each period, whereas in the second they are kept at a fixed location determined at the beginning of the planning horizon. We provide Lagrangian Relaxation and Benders Decomposition algorithms, including an ?-optimal BD algorithm, for the solution for the first model and a Benders Decomposition algorithm for the second. For detailed analysis, we generate a wide variety of instances to test the performance of the algorithms by taking into account varying number of customer locations and time periods in the planning horizon as well as fixed cost structures and facility capacities. We observe that the efficiency of the solution algorithms depends on the input data structure, specifically the cost structures, the facility capacities (which, in turn, dictate the expected number of open facilities), and the variation in the total customer demand from period to period.     

On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical methods for interactive multiple‐class image segmentation problems

In this article, we consider a bilaterally constrained optimization model arising from the semisupervised multiple‐class image segmentation problem. We prove that the solution of the corresponding unconstrained problem satisfies a discrete maximum principle. This implies that the bilateral constraints are satisfied automatically and that the solution is unique. Although the structure of the coefficient matrices arising from the optimality conditions of the segmentation problem is different for different input images, we show that they are M‐matrices in general. Therefore, we study several numerical methods for solving such linear systems and demonstrate that domain decomposition with block relaxation methods are quite effective and outperform other tested methods. We also carry out a numerical study of condition numbers on the effect of boundary conditions on the optimization problems, which provides some insights into the specification of boundary conditions as an input knowledge in the learning context. © 2010 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 20, 191–201, 2010     

Quadratic Posylognomials: An Extension of Posynomial Geometric Programming

The theory of geometric programming is extended to include a new function with logarithmic exponents. The function, defined as a Quadratic Posylognomial (QPL), is a series of nonlinear product terms with positive coefficients and positive variables. A QPL may be created by adding a linear function of the logarithms of the variables to the constant exponents of a posynomial. The logarithm of each nonlinear term is a quadratic form in the logarithms of the primal variables, whereas the logarithm of a general term of a posynomial is linear in the logarithms of the variables. The primal-dual relationships and necessary conditions are developed, and special cases where sufficient conditions exist are derived. The theory is the basis for solution of a machining economics problem using a more accurate QPL tool life equation and has the potential for solution of other engineering problems where a posynomial formulation is inadequate. The extended theory may also prove useful for condensing posynomial geometric programming problems by reducing the “degree of difficulty,” a measure of the number of decision variables in the dual problem.     

Modelling of coal trade process for the logistics enterprise and its optimisation with stochastic predictive control

In the paper, a typical coal trade process is described and modelled, where one logistics enterprise with blending equipments lies in the core and two types of common contracts are elucidated to define constraints. A mixed-integer model is built and featured by addressing contract violation, blending operation, real-time price information and arbitrarily distributed stochastic demands. To deal with the stochastic demands, probabilistic constraints are formed. Accordingly, stochastic model predictive control strategy with both receding horizon and decreasing horizon formulations is developed to handle the probabilistic constraints and exploit the value of newest price information. By solving a series of mixed-integer linear programmes, optimal coal trade decisions for the logistics enterprise can be obtained, including procurement decision, selling decision and operational decision of the blending equipments. Thorough simulation experiments are carried out and compared with three different strategies, which interpret the effectiveness of the proposed strategy.     

Integrating production-transportation planning decision with fuzzy multiple goals in supply chains

This work presents a novel fuzzy multi-objective linear programming (f-MOLP) model for solving integrated production-transportation planning decision (PTPD) problems in supply chains in a fuzzy environment. The proposed model attempts to simultaneously minimise total production and transportation costs, total number of rejected items, and total delivery time with reference to available capacities, labor level, quota flexibility, and budget constraints at each source, as well as forecast demand and warehouse space at each destination. An industrial case demonstrates that the proposed f-MOLP model achieves an efficient compromise solution and overall decision maker satisfaction with determined goal values. Additionally, the proposed model provides a systematic framework that facilitates decision makers to interactively modify the fuzzy data and parameters until a satisfactory solution is obtained. Overall, the f-MOLP model offers a practical method for solving PTPD problems with fuzzy multiple goals, and can effectively improve producer–distributor relationships within a supply chain.     

Network Application in Industry and Government

The primary purpose of this paper is to provide the practitioner tools to use in modeling decision making situations as network flow problems. These tools are presented as part of the discussion of recent industrial and governmental applications. The intent is not to enumerate all applications of networks, but rather to give the reader a flavor of the versatility and usability of networks. An additional objective is to acquaint the reader with the process of visualizing a problem by means of network diagrams, thereby making it possible to capture important interrelationships in an easily understood “pictorial” framework. A secondary purpose is to familiarize the reader with recent computational advances in the development of computer codes to solve these problems. For example, recent breakthroughs in the solution and human engineering aspects of minimum cost flow transshipment problems have made it possible to solve problems that require many hours of computing time with state-of-the-art commercial LP packages in only a few minutes.