Mathematical Programming Algorithms Research Articles

Combining Exact and Heuristic Approaches for the Capacitated Fixed-Charge Network Flow Problem

We develop a solution approach for the fixed-charge network flow (FCNF) problem that produces provably high-quality solutions quickly. The solution approach combines mathematical programming algorithms with heuristic search techniques. To obtain high-quality solutions, it relies on neighborhood search with neighborhoods that involve solving carefully chosen integer programs derived from the arc-based formulation of FCNF. To obtain lower bounds, the linear programming relaxation of the path-based formulation of FCNF is used and strengthened with cuts discovered during the neighborhood search. The solution approach incorporates randomization to diversify the search and learning to intensify the search. Computational experiments demonstrate the efficacy of the proposed approach.

Balancing fuzzy multi-objective two-sided assembly lines via Bees Algorithm

Many real life problems contain imprecise variables, constraints and objectives. Fuzzy set theory gives an opportunity to handle imprecise terms in such situations. Two-sided assembly line balancing (2sALB) problem which is a generalization of the well known simple assembly line balancing problem can also be modeled more realistically by employing fuzzy approaches. Such an approach is presented in this study to model and solve 2sALB problem by employing fuzzy mathematical programming and Bees Algorithm (BA). 2sALB problem is a combinatorial complex problem. For this reason BA is employed as a search mechanism for obtaining good solutions to it. BA is a relatively new member of swarm intelligence based meta-heuristics that tries to mimic natural behavior of real honey bees in food foraging in solving complex optimization problems. BA is generally applied to continuous optimization in the literature. Its application to combinatorial problems is rare. This study also presents one of the first application of BA to an assembly line balancing problem which is member of combinatorial optimization.

Cost Sharing under Uncertainty: An Algorithmic Approach to Cooperative Interval-Valued Games

AbstractRecently, Branzei, Dimitrov, and Tijs (2003) introduced cooperative interval-valued games. Among other insights, the notion of an interval core has been coined and proposed as a solution concept for interval-valued games. In this paper we will present a general mathematical programming algorithm which can be applied to find an element in the interval core. As an example, we discuss lot sizing with uncertain demand to provide an application for interval-valued games and to demonstrate how interval core elements can be computed. Also, we reveal that pitfalls exist if interval core elements are computed in a straightforward manner by considering the interval borders separately.

Selection of dispatching rules in FMS: ANP model based on BOCR with choquet integral

Scheduling policy which depends on selection of dispatching rules is one of the reasons which affect FMS’ productivity. Selection of a dispatching rule can only be based on one or two criteria such as processing time, due date, or system components at traditional methods such as mathematical programming, simulation, and heuristic algorithms. However, traditional methods have some weakness; the first one is about selection criteria based on only one dispatching rule or a few combination of the rules, the second one is related with not considering most of the manufacturing system information, and the last one is to deal with selection decision is not a dynamic structure. In this study, an analytical network process (ANP) model has been developed in order to eradicate these weaknesses. The model based on multiple criteria decision-making process. In addition, most of manufacturing system information have been employed, and it consists of a dynamic decision-making process. The proposed model has a comprehensive structure which consists of a company’s manufacturing criteria and macro-strategies. Although interdependence is accepted among criteria in ANP, non-interaction is assumed among them (i.e., additive measure). Because of existence of interactivities among criteria, a choquet integral has been adapted in an ANP model.

A genetic algorithm for bi-level linear programming dynamic network design problem

Solving the Bi-level Linear Programming Network Design Problem (BLPNDP), which is typically regarded as a strongly NP-complete problem, with traditional mathematical programming algorithms is computationally intractable. A majority of the research efforts have been focused on tackling this problem using meta-heuristics. However, the evaluation functions in most meta-heuristic approaches involve dynamic traffic simulation when evaluating feasible capacity expansion policies. This commonly used evaluation function becomes a major bottleneck constraining the applicability of meta-heuristics as it is computationally expensive and analytically complicated. To address this issue, we present a modified Genetic Algorithm (GA) to solve the BLPNDP with a novel evaluation function which reduces the number of dynamic traffic simulations needed. The heuristic exploits the underlying linear programming structure of the BLPNDP and employs dual variable approximation techniques to estimate the evaluation function instead of dynamic traffic simulation. Empirical analyses are conducted on three networks of varying sizes to examine the performance of the GA based heuristic. From the experiments conducted, the proposed GA based heuristic outperforms the traditional GA. Also the proposed method of estimating the evaluation function is flexible and can be used within the framework of many meta-heuristic approaches.

Dynamic Programming Algorithms for Solving Single Machine Scheduling Problems with a Fixed Complexity Time

We solve in this paper some machine scheduling problems with certain conditions; the solution had been made by using dynamic programming procedures with a fixed time for all the algorithms. Some of the algorithms described in the paper represent a modification to some mathematical programming algorithms and some of them are new or a generalization to others. The optimal solution of all these algorithms will be done with a fixed complexity time, which may lead us to solve them easier and faster.

An extended mathematical programming framework

Extended mathematical programs are collections of functions and variables joined together using specific optimization and complementarity primitives. This paper outlines a mechanism to describe such an extended mathematical program by means of annotating the existing relationships within a model to facilitate higher level structure identification. The structures, which often involve constraints on the solution sets of other models or complementarity relationships, can be exploited by modern large scale mathematical programming algorithms for efficient solution. A specific implementation of this framework is outlined that communicates structure from the GAMS modeling system to appropriate solvers in a computationally beneficial manner. Example applications are taken from chemical engineering.

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

In this paper, we describe a successive approximation and smooth sequential quadratic programming (SQP) method for mathematical programs with nonlinear complementarity constraints (MPCC). We introduce a class of smooth programs to approximate the MPCC. Using an l1 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints (MPEC) when the algorithm terminates finitely.

Designing cutouts for optimum residual strength in plane structural elements

The paper presents a new approach for shape optimisation of structures with residual strength as the design objective. It must be emphasized that flaws are inevitably present in most structures, and hence the influence of cracks on optimised shapes needs to be investigated. Numerical simulation of cracks using the finite element method requires a very fine mesh to model the singularity at crack tip. This makes fracture calculations computationally intensive. Furthermore, for a damage tolerance based optimisation numerous cracks are to be considered along the structural boundary, and fracture analysis needs to be repeated for each crack at every iteration, thus making the whole process extremely computationally expensive for practical purpose. Moreover, the lack of information concerning crack size, orientation, and location makes the formulation of the optimisation problem difficult. As a result, little attention has been paid to date to consider fracture parameters in the optimisation objective. To address this, the paper presents a methodology for the shape optimisation of structures with strength and durability as the design objectives. In particular, the damage tolerance optimisation is illustrated via the problem of optimal design of a ‘cutout in a rectangular block under biaxial loading’. A parametric shape representation has been used to describe the problem geometry. Damage tolerance based optimisation was performed using nonlinear programming algorithms, and NE-NASTRAN was used for finite element analysis. The first order mathematical programming algorithms, viz: the Broydon–Fletcher–Goldfarb–Shanno (BFGS) and the Fletcher–Reeves (Conjugate Direction) Methods were evaluated as the potential algorithms for optimising the residual strength in the presence of flaws. Another recently developed Sequential Unconstrained Minimisation Technique (SUMT), based on an exterior penalty function method and especially suited for large problems, was also investigated for fracture based optimisation. The effects of the orientation and the number of boundary cracks on the optimal solutions were also studied. It has been shown that the residual strength optimised shapes can be different from the corresponding and commonly adopted stress optimised solution. This emphasises the need to explicitly consider residual strength as the design objective. In all cases a significant reduction in the maximum stress intensity factor was achieved with the generation of a ‘near’ uniform fracture critical surface. The design space near the optimal region was found to be relatively flat. This is beneficial as a significant structural performance enhancement is important rather than precise identification of the local/global optimum solution. The optimal solutions obtained using the nonlinear programming algorithms were compared against those obtained in the literature using a heuristic optimisation method (Biological algorithm). The results obtained using the two methods, employing inherently different (gradient-based and gradient-less) algorithms, were found to agree very well.

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.

Structural analysis and optimization in the framework of reduced-basis method

This work focus on the implementation and application of Reduced Basis (RB) techniques for structural and sensitivities analysis of trusses, 2D elasticity, heat conduction and coupled thermal–elastic problems. This techniques has been implemented in order to economically and accurately compute outputs and their gradients, as they are calculated several times due to the changes of the parameters in an optimization process. The tools developed here are combined with mathematical programming algorithms and multiobjective strategies to form a robust and reliable optimization procedure to achieve better performance on structural designs. Some simple academic applications have been considered for the validation of all proposed methodologies. Further studies will addressed more complex problems.

Bidding Decision in Reverse Auction Solved by Fuzzy Mathematical Programming and Genetic Algorithm

Abstract Abstract This study proposes a mixed-integer programming with fuzzy constraints for solving the bidding problem in a reverse auction environment. Suppliers’ optimum bid price is determined through the compromise between profitability of the bid and possibility to win the contract. The proposed model embeds the advanced available-to-promise (AATP) concept in order to support accurate computation of profit and customer order promising. Moreover, the feasible bid price is constrained by a fuzzy upper bound to model the decision-maker's subjective judgment. A genetic algorithm is formulated to solve the problem. Performance of the proposed approach is evaluated through computer simulated experiments.

Mathematical Programming Algorithms for Two-Path Routing Problems with Reliability Considerations

Most traditional routing problems assume perfect operability of all arcs and nodes. However, when independent arc failure probabilities exist, a secondary objective must be present to retain some measure of expected functionality, introducing nonlinear, nonconvex constraints. We examine the robust two-path problem, which seeks to establish two paths between a source and destination node wherein at least one path must remain fully operable with some threshold probability. We consider the case where both paths must be arc disjoint and the case where arcs can be shared between the paths. We examine various strategies for solving the resulting nonlinear integer program, including pruning, coefficient tightening, lifting, and branch-and-bound partitioning schemes. We discuss the advantages and disadvantages of these methods and conclude with computational results.

Topology optimization for nano‐scale heat transfer

AbstractWe consider the problem of optimal design of nano‐scale heat conducting systems using topology optimization techniques. At such small scales the empirical Fourier's law of heat conduction no longer captures the underlying physical phenomena because the mean‐free path of the heat carriers, phonons in our case, becomes comparable with, or even larger than, the feature sizes of considered material distributions. A more accurate model at nano‐scales is given by kinetic theory, which provides a compromise between the inaccurate Fourier's law and precise, but too computationally expensive, atomistic simulations.We analyze the resulting optimal control problem in a continuous setting, briefly describing the computational approach to the problem based on discontinuous Galerkin methods, algebraic multigrid preconditioned generalized minimal residual method, and a gradient‐based mathematical programming algorithm. Numerical experiments with our implementation of the proposed numerical scheme are reported. Copyright © 2008 John Wiley & Sons, Ltd.

Optimal design and operation of irrigation pumping stations using mathematical programming and Genetic Algorithm (GA)

For many water authorities worldwide, one of the greatest potential areas for energy savings is in pump selection and in the related effective scheduling of daily pump operations. The optimal control and operation of an irrigation pumping station is achieved here by first solving the nonlinear governing model using Lagrange Multipliers (LM) and then through Genetic Algorithm (GA) approach. Computation in both methods is driven by an objective function that includes operating and capital costs subject to various performance and hydraulic constraints. The LM approach first specifies the annual energy costs and minimizes the total cost for all sets of pumping stations; the method then selects the least-cost pumps from among the feasible sets. The GA model simultaneously determines the least total annual cost of the pump station and its operation. The solution includes the selection of pump type, capacity, and the number of units, as well as scheduling the operation of irrigation pumps that results in minimum design and operating cost for a set of water demand curves. Application of the two models to a real-world project shows not only considerable savings in cost and energy but also highlights the efficiency and ease of the GA approach for solving complex problems of this type.

Merit-Function Piecewise SQP Algorithm for Mathematical Programs with Equilibrium Constraints

We propose a merit-function piecewise SQP algorithm for mathematical programs with equilibrium constraints (MPEC) formulated as mathematical programs with complementarity constraints. Under mild conditions, the new algorithm is globally convergent to a piecewise stationary point. Moreover, if the partial MPEC linear independence constraint qualification (LICQ) is satisfied at the accumulation point, then the accumulation point is an S-stationary point.

Design of problem-specific evolutionary algorithm/mixed-integer programming hybrids: two-stage stochastic integer programming applied to chemical batch scheduling

Engineering optimization often deals with large, mixed-integer search spaces with a rigid structure due to the presence of a large number of constraints. Metaheuristics, such as evolutionary algorithms (EAs), are frequently suggested as solution algorithms in such cases. In order to exploit the full potential of these algorithms, it is important to choose an adequate representation of the search space and to integrate expert-knowledge into the stochastic search operators, without adding unnecessary bias to the search. Moreover, hybridisation with mathematical programming techniques such as mixed-integer programming (MIP) based on a problem decomposition can be considered for improving algorithmic performance. In order to design problem-specific EAs it is desirable to have a set of design guidelines that specify properties of search operators and representations. Recently, a set of guidelines has been proposed that gives rise to so-called Metric-based EAs (MBEAs). Extended by the minimal moves mutation they allow for a generalization of EA with self-adaptive mutation strength in discrete search spaces. In this article, a problem-specific EA for process engineering task is designed, following the MBEA guidelines and minimal moves mutation. On the background of the application, the usefulness of the design framework is discussed, and further extensions and corrections proposed. As a case-study, a two-stage stochastic programming problem in chemical batch process scheduling is considered. The algorithm design problem can be viewed as the choice of a hierarchical decision structure, where on different layers of the decision process symmetries and similarities can be exploited for the design of minimal moves. After a discussion of the design approach and its instantiation for the case-study, the resulting problem-specific EA/MIP is compared to a straightforward application of a canonical EA/MIP and to a monolithic mathematical programming algorithm. In view of the results the benefits of customising the EA are discussed.

Optimization of control strategy for a single-shaft parallel hybrid electric vehicle

An optimal control strategy is obtained for a parallel hybrid electric vehicle (HEV). Scale factors in weighting the motor usage are selected as the control variables on how to distribute the demanded vehicle power into the electric motor and the internal combustion engine. Dynamic models of the HEV powertrain are represented by state space equations. In order to find the optimal control strategy, an optimization problem is defined, which finds the optimal control variables to minimize the fuel consumption while satisfying the constraint equations. The optimization problem is solved using the HEV performance simulator developed in MATLAB Simulink and the mathematical optimization sequential quadratic programming algorithm which is based on FORTRAN for multi-variables design optimization. It is expected that the optimization method proposed in this study provides the control strategy of the HEV systematically, which minimizes the fuel consumption.

Application of optimisation technique to tyre design

A new tyre design procedure, which is capable of determining the optimum tyre contour, optimum compound characteristics in each tyre material, optimum tyre construction and optimum pitch sequence, has been applied to the actual tyre design. This technology is developed by combining the finite element method or analytical method with an optimisation technique, such as mathematical programming or genetic algorithm, and is applied not only to the shape optimisation, such as tyre crown shape, but also to the topological optimisation, such as composite construction. The new design procedure is verified to improve the manoeuvrability, rolling resistance, durability, tyre noise and other tyre performances through indoor drum tests and field tests, if the appropriate design variable, the constraints and the objective function are selected. In this paper, studies on the application of the optimisation technique to design tyres are reviewed.

Mathematical programming models and algorithms for a class–faculty assignment problem

This paper presents mathematical programming models for assigning faculty members to classes including, among typical academic class scheduling issues, certain specialized central policies at Kuwait University. The time-slots for classes are initially assumed to be given and an integer programming model (CFAM) is constructed to solve the resulting problem, which aims to minimize the individual and collective dissatisfaction of faculty members in a fair fashion, where dissatisfaction is measured by a function of the assignment of faculty members to time-slots and specific classes. In order to enhance the quality of results obtained in practice, the model is modified (ECFAM) so that the time-slots for the classes can be changed, however, with restrictions related to efficient facility utilization and permitting an administratively regulated maximum number of changes. Gender-based modeling considerations are also introduced in order to maintain desirable class offering patterns. Computational results are presented based on solving the models directly by the CPLEX-MIP (version 7.5) package and also using a specialized LP-based heuristic. The faculty schedules generated via the proposed approach based on a number of case studies related to the Department of Mathematics and Computer Science at Kuwait University reveal that this approach yields improved schedules in terms of fairness and enhanced satisfaction levels among faculty members.