混合整数双层线性规划的全局优化算法

本文讨论了上层决策变量为整数变量、下层决策变量为连续变量的混合整数双层线性规划问题,利用其可行解均落在约束域边界上的性质,提出了一种求解混合整数双层线性规划全局最优解的算法,并举例说明了算法的执行过程。     

An exact penalty function method for optimising QAP formulation in facility layout problem

A quadratic assignment problem (QAP), which is a combinatorial optimisation problem, is developed to model the problem of locating facilities with material flows between them. The aim of solving the QAP formulation for a facility layout problem (FLP) is to increase a system’s operating efficiency by reducing material handling costs, which can be measured by interdepartmental distances and flows. The QAP-formulated FLP can be viewed as a discrete optimisation problem, where the quadratic objective function is optimised with respect to discrete decision variables subject to linear equality constraints. The conventional approach for solving this discrete optimisation problem is to use the linearisation of the quadratic objective function whereby additional discrete variables and constraints are introduced. The adoption of the linearisation process can result in a significantly increased number of variables and constraints; solving the resulting problem can therefore be challenging. In this paper, a new approach is introduced to solve this discrete optimisation problem. First, the discrete optimisation problem is transformed into an equivalent nonlinear optimisation problem involving only continuous decision variables by introducing quadratic inequality constraints. The number of variables, however, remains the same as the original problem. Then, an exact penalty function method is applied to convert this transformed continuous optimisation problem into an unconstrained continuous optimisation problem. An improved backtracking search algorithm is then developed to solve the unconstrained optimisation problem. Numerical computation results demonstrate the effectiveness of the proposed new approach.     

遗传算法与惩罚函数法在机械优化设计中的应用

提出了应用于机械优化设计的"遗传算法+惩罚函数法"的通用算法.它非常适合求解复杂的非线性约束优化问题.本通用算法既克服了传统优化方法的缺点,得到了一个较为理想的全域最优解;同时也改善了遗传算法的局限性.     

Industrial waste recycling strategies optimization problem: mixed integer programming model and heuristics

Manufacturers have a legal accountability to deal with industrial waste generated from their production processes in order to avoid pollution. Along with advances in waste recovery techniques, manufacturers may adopt various recycling strategies in dealing with industrial waste. With reuse strategies and technologies, byproducts or wastes will be returned to production processes in the iron and steel industry, and some waste can be recycled back to base material for reuse in other industries. This article focuses on a recovery strategies optimization problem for a typical class of industrial waste recycling process in order to maximize profit. There are multiple strategies for waste recycling available to generate multiple byproducts; these byproducts are then further transformed into several types of chemical products via different production patterns. A mixed integer programming model is developed to determine which recycling strategy and which production pattern should be selected with what quantity of chemical products corresponding to this strategy and pattern in order to yield maximum marginal profits. The sales profits of chemical products and the set-up costs of these strategies, patterns and operation costs of production are considered. A simulated annealing (SA) based heuristic algorithm is developed to solve the problem. Finally, an experiment is designed to verify the effectiveness and feasibility of the proposed method. By comparing a single strategy to multiple strategies in an example, it is shown that the total sales profit of chemical products can be increased by around 25% through the simultaneous use of multiple strategies. This illustrates the superiority of combinatorial multiple strategies. Furthermore, the effects of the model parameters on profit are discussed to help manufacturers organize their waste recycling network.     

A generalized interval fuzzy mixed integer programming model for a multimodal transportation problem under uncertainty

A generalized interval fuzzy mixed integer programming model is proposed for the multimodal freight transportation problem under uncertainty, in which the optimal mode of transport and the optimal amount of each type of freight transported through each path need to be decided. For practical purposes, three mathematical methods, i.e. the interval ranking method, fuzzy linear programming method and linear weighted summation method, are applied to obtain equivalents of constraints and parameters, and then a fuzzy expected value model is presented. A heuristic algorithm based on a greedy criterion and the linear relaxation algorithm are designed to solve the model.     

A 0-1 integer programming model and solving strategies for the slab storage problem

We consider the slab storage problem (SSP) in slab yard operations. A set of slabs enter a slab yard in a specific order. A proper stack needs to be selected for each inbound slab, so that the number of relocations in the subsequent retrieval stage is minimised. We present a 0-1 integer programming model of the SSP that minimises the lower bound of the number of relocations. Four solving strategies are derived from several interesting properties of the mathematical model to speed up the solving process of the model. Making use of randomly generated instances and practical instances, we testify the effectiveness of the solving strategies and study the influence of problem parameters on the computational time of the model. Computational results show that the solving strategies can effectively reduce the computational time of the model and is applicable in medium-sized practical instances.     

A consistent penalty method for contact between a deforming viscoplastic workpiece and a rigid tool

A methodology is presented for enforcing a contact constraint between a viscoplastic workpiece and the tooling which deforms it. The algorithm is based on a consistent penalty approach wherein an elemental discretization permits elimination of contact tractions prior to solution for the velocity field. When written in rate form, the new algorithm allows contact constraints to be embedded in a viscoplastic formulation without altering the structure of the global matrix equation. Examples are presented to illustrate its use in metal forming simulations.     

Production scheduling with outsourcing scenarios: a mixed integer programming and efficient solution procedure

When market demand exceeds the company's capacity to manufacture, outsourcing is commonly considered as an effective alternative option. In traditional scheduling problems, processing of received orders is just possible via in-house resources, while in practice, outsourcing is frequently found in various manufacturing industries, especially in electronics, motor and printing companies. This paper deals with the scheduling problem, minimising the cost of outsourcing and a scheduling measure represented by weighted mean flow time, in which outsourcing of manufacturing operations is allowed through subcontracts. Each order can be either scheduled for in-house production or outsourced to an outside supplier in order to meet customer due dates. In this problem, not only should the sequence of orders be determined, but also decision on picking the jobs for outsourcing, selecting the appropriate subcontractor, and scheduling of the outsourced orders are considered as new variables. To formulate the given problem, four different outsourcing scenarios are derived and mixed integer programming models are developed for each one separately. Furthermore, to solve the suggested problem, a computationally effective team process algorithm is devised and then a constraint handling technique is embedded into the main algorithm in order to ensure satisfaction of customer due dates. Numerical results show that the suggested approach possesses high global solution rates as well as fast convergence.     

Explicit model of dual programming and solving method for a class of separable convex programming problems

An objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.     

A 3D shell-like approach using a natural neighbour meshless method: Isotropic and orthotropic thin structures

A three-dimensional shell-like approach for the analysis of composite thin plates and shells using a meshless method, the natural neighbour radial point interpolation method (NNRPIM), is presented. In the NNRPIM the nodal connectivity is enforced using the natural neighbour concept. The node-depending background mesh used in the numerical integration of the NNRPIM interpolation functions is entirely created from the unstructured nodal arrangement. The radial point interpolators are used to construct the NNRPIM interpolation functions, which possesses the delta Kronecker property, used in the Galerkin weak form. The novelty of this work lays on the development of a unique NNRPIM approach when 3D thin structures are considered. This new approach leads to remarkable results and it is extremely suitable to the composite structure problem. In order to demonstrate the effectiveness of the method the 3D shell-like NNRPIM analysis is used to solve several isotropic and orthotropic thin plates and shells problems.     

Multiple failures of API 5L X42 natural gas pipe: Experimental and computational analysis

Multiple failures of API 5L X42 (X42) coal–tar coated natural gas feeder line due to vertical jetting of high pressure erosive slurry was studied experimentally and computationally. Computational Fluid Dynamic (CFD) was used as the simulation tools to study the flow pattern, velocity distribution and strain rates on pipe surface. Experimental work was performed to determine the erosion pattern. Three different diameters of jetting sources produced three distinct impact patterns were clearly identified. These patterns were frequently referred in the experimental study. A CFD simulation result shows that the highest shear strain rate area coincides with the leakage point. These results thus support the hypothesis that it is very likely that the failure of the gas pipeline was caused by the high pressure water jet gusting from the failed welded joint of the water pipeline.     

A parallel machine lot-sizing and scheduling problem with a secondary resource and cumulative demand

We investigate a parallel machine multi-item lot-sizing and scheduling problem with a secondary resource, in which demands are given for the entire planning horizon rather than for every single period. All-or-nothing assumption of the discrete lot-sizing and scheduling problem is valid so that a machine is either idle or works at full capacity in a period. The objective is to minimise the number of setups and teardowns. We prove that the problem is NP-hard and present two equivalent formulations. We show some properties of the optimal objective value, give optimality conditions and suggest a heuristic algorithm. We discuss and formulate two possible extensions related to real-life applications. Finally, we carry out computational experiments to compare the two formulations, to determine the effect of our proposed modeling improvements on solution performance, and to test the quality of our heuristic.     

Numerical analysis of conjugate natural and mixed convection heat transfer of nanofluids in a square cavity using the two-phase method

In the present study, the problem of conjugate natural and mixed convection of nanofluid in a square cavity containing several pairs of hot and cold cylinders is visualized using non-homogenous two-phase Buongiorno's model. Such configuration is considered as a model of heat exchangers in order to prevent the fluids contained in the pipelines from freezing or condensing. Water-based nanofluids with Cu, Al₂O₃, and TiO₂ nanoparticles at different diameters ($25\text{nm}?d_p?145\text{nm}$) are chosen for investigation. The governing equations together with the specified boundary conditions are solved numerically using the finite volume method based on the SIMPLE algorithm over a wide range of Rayleigh number (${10}^4?\mathit{Ra}?{10}^7$), Richardson number (${10}^{-2}?\mathit{Ri}?{10}^2$) and nanoparticle volume fractions ($0?\phi ?5\%$). Furthermore, the effects of three types of influential factors such as: orientation of conductive wall, thermal conductivity ratio ($0.2?K_r?25$) and conductive obstacles on the fluid flow and heat transfer rate are also investigated. It is found that the heat transfer rate is significantly enhanced by incrementing Rayleigh number and thermal conductivity ratio. It is also observed that at all Rayleigh numbers, the total Nusselt number rises and then reduces with increasing the nanoparticle volume fractions so that there is an optimal volume fraction of the nanoparticles where the heat transfer rate within the enclosure has a maximum value. Finally, the results reveal that by increasing the thermal conductivity of the nanoparticles and Rayleigh number, distribution of solid particles becomes uniform.     

A decision support framework for global supply chain modelling: an assessment of the impact of demand,supply and lead-time uncertainties on performance

We developed a decision support framework for a global manufacturer of specialty chemicals to study the relative impact of demand, supply and lead-time uncertainties on cost and customer service performance. Our approach combines optimisation and simulation methodologies as follows: mathematical models provide optimal plans via a novel approach to the supply chain planning mechanism of the Company. Simulation models execute the supply chain plans so as to allow the examination of the outcomes under the various sources of uncertainty. The iterative use of optimisation and simulation methodologies allows the user the benefit of obtaining optimal solutions while revealing the impact of uncertainties on system performance. Our results indicate that demand uncertainty has the greatest negative impact on performance for the supply chain that we modelled in this study, emphasising the importance of effective forecasting. The relative importance of supply and lead-time uncertainties varies according to the performance measures. While our results are valid for the specific supply chain and the operating environment we modelled, our study emphasises the importance of the ability to model supply chains realistically to obtain valid and useful results.     

Field services design and management of natural gas distribution networks: a class of vehicle routing problem with time windows approach

Service operations management of metropolitan gas networks at operational level implies the optimisation of decisions related to logistic activities, taking into account multi-objectives and operational constraints. This paper proposes a metaheuristic approach for the operational planning of the daily logistic activities based on vehicle routing with time window model. Experimental results for a real planning case in a gas distribution network demonstrate the approach effectiveness.     

Cloud manufacturing service composition and optimal selection with sustainability considerations: a multi-objective integer bi-level multi-follower programming approach

The process of service composition and optimal selection (SCOS) is an important issue in cloud manufacturing (CMfg). However, the current studies on CMfg and SCOS have generally focused on optimising the allocation of resources against quality of service (QoS), in terms of e.g. cost, quality, and time. They have seldom taken the perspective of sustainability into discussion, although sustainability is indispensable in the CMfg environment. Addressing this gap, we aim to (1) propose a comprehensive method to assess the sustainability of cloud manufacturing (SoM) in terms of the economic, environmental, and social aspects; (2) establish a multi-objective integer bi-level multi-follower programming (MOIBMFP) model to simultaneously maximise SoM and QoS from the perspectives of both platform operator and multiple service demanders; and (3) design a hybrid particle swarm optimisation algorithm to solve the proposed MOIBMFP model. The experimental results show that the proposed algorithm is more feasible and effective than the typical multi-objective particle swarm optimisation algorithm when solving the proposed model. In other words, the proposed model and algorithm suggest better alternatives to meet the needs of the platform operator and service demanders in the CMfg environment.     

The extended resource task network: a framework for the combined scheduling of batch processes and CHP plants

The issue of energy has emerged as one of the greatest challenges facing mankind. In an industrial perspective, the development of site utility systems (generally combined heat and power (CHP) systems) for the generation and management of utilities provides a great potential source for energy savings. However, in most industrial sites, a master–slave relationship usually governs this kind of system and limits the potential operating capacity of CHP. To improve the decision-making process, Agha et al. (2010. Integrated production and utility system approach for optimising industrial unit operation. Energy, 35, 611–627) have proposed an integrated approach that carries out simultaneous and consistent scheduling of batch production plants and site utility systems. The modelling of the problem relies on a mixed integer linear programming (MILP) formulation. Nevertheless, although it is a powerful mathematical tool, it still remains difficult to use for non-expert engineers. In this framework, a graphical formalism based on existing representations (STN, RTN) has been developed: the extended resource task network (ERTN). Combined with an efficient and generic MILP formulation, it permits various kinds of industrial problems, including production and consumption of utility flows to be modelled homogenously. This paper focuses on the semantic elements of the ERTN formalism and illustrates their use through representative examples.     

The dielectric function of a GaAs/AlGaAs single quantum well: calculation and comparison with experiment

The dielectric function (DF) of a modulation doped (0 0 1) GaAs/AlGaAs (Al content of x=0.33) single quantum well structure containing a two-dimensional electron gas (2DEG) was calculated as a function of 2DEG density and temperature, taking into account the nonparabolicity of the valence band and many-body effects. The DF was used to simulate electroreflectance and photoluminescence excitation spectra. The results were found to be in a good agreement with experimental data at T>80 K and 2DEG concentrations of more than 2×10¹¹ cm⁻².     

A note on the divergence problem arising in calculation of Green’s function for a 2D piezoelectric thin film strip containing straight line defects and free boundaries: Fourier approach for bodies of unrestricted anisotropy

In this paper we derive a set of novel formulas for computation of the Green’s function and the coupled electro-elastic fields in a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects. The strip is assumed to be of unrestricted anisotropy, but allowing piezoelectricity, and in this sense situation is more general than in the available literature where only cubic symmetry was investigated. We employ a set of already known analytic formulas for the Fourier amplitude of the Green’s function and the corresponding electro-elastic fields. The key novelty of this paper is solution for the divergence problem occurring during integration of the Fourier amplitude. This problem is caused by poles at k = 0 in various matrix components of the amplitude. From purely mathematical point of view such poles lead to quantities which do not tend to zero at infinity, and this situation is clearly unphysical. To resolve this issue it is demonstrated by means of rigorous analysis that when some additional physical conditions are imposed, physical fields exhibit regular behavior at infinity - the poles do not contribute. Nevertheless, they lead to irremovable numerical ∞ − ∞ uncertainties spreading over the whole domain of integration. This motivates us to compute exact formulas for all these poles to enable engineering calculations involving the system in question.     

A note on the divergence problem arising in calculation of Green’s function for a 2D piezoelectric thin film strip with free boundaries and containing straight line defects: Fourier approach for bodies of cubic symmetry

Analytic formulas for the Green’s function and the coupled electro-elastic fields for a 2D piezoelectric strip with free boundaries and containing a distribution of straight line defects have already been found some years ago. These formulas exploit the well-known Stroh formalism and the Fourier approach, so the result is given as the Fourier integral and therefore its numerical implementation should pose no problem. However, in this note we show that for the case of cubic symmetry this form of the Green’s function contains strong divergences, excluding possibilities of direct application of well-known numerical schemes. It is also shown that these divergences translate to divergences of the corresponding electro-elastic fields of a single defect. By means of a rigorous analysis it is demonstrated that imposing physical conditions implied by the nature of the problem all of these divergences cancel and the final, physical result exhibits expected, regular behavior at infinity. Unfortunately, although the nature of this problem is purely mathematical, it leads to irremovable numerical ∞ − ∞ uncertainties which tend to spread over the whole Fourier domain and severely impede engineering applications of the Green’s function. This motivates us to compute the exact form of all divergent terms. These novel formulas will serve as a guide when establishing numerically stable algorithms for engineering computations involving the system in question.