Mixed-integer Programming Method Research Articles

A fuzzy bi-objective mixed-integer programming method for solving supply chain network design problems under ambiguous and vague conditions

Supply chain (SC) network design problems are complex problems with multi-layer levels and dynamic relationships which involve a considerable amount of uncertainty concerning customer demand, facility capacity, or lead times, among others. A large number of optimization methods (i.e., fuzzy mathematical programming, stochastic programming, and interval mathematical programming) have been proposed to cope with the uncertainties in SC network design problems. We propose a fuzzy bi-objective mixed-integer linear programming (MILP) model to enhance the material flow in dual-channel, multi-item, and multi-objective SCs with multiple echelons under both ambiguous and vague conditions, concurrently. We use a computationally efficient ranking method to resolve the ambiguity of the parameters and propose two methods for resolving the vagueness of the objective functions in the proposed fuzzy MILP model. The preferences of the decision makers (DMs) on the priority of the fuzzy goals are represented with crisp importance weights in the first method and fuzzy preference relations in the second method. The fuzzy preference relations in the second method present a unique practical application of type-II fuzzy sets. The performance of the two methods is compared using comprehensive statistical analysis. The results show the perspicuous dominance of the method which uses fuzzy preference relations (i.e., type-II fuzzy sets). We present a case study in the food industry to demonstrate the applicability of the proposed model and exhibit the efficacy of the procedures and algorithms. To the best of our knowledge, a concurrent interpretation of both ambiguous and vague uncertainties, which is applicable to many real-life problems, is novel and has not been reported in the literature.

MEIGO: an open-source software suite based on metaheuristics for global optimization in systems biology and bioinformatics.

BackgroundOptimization is the key to solving many problems in computational biology. Global optimization methods, which provide a robust methodology, and metaheuristics in particular have proven to be the most efficient methods for many applications. Despite their utility, there is a limited availability of metaheuristic tools.ResultsWe present MEIGO, an R and Matlab optimization toolbox (also available in Python via a wrapper of the R version), that implements metaheuristics capable of solving diverse problems arising in systems biology and bioinformatics. The toolbox includes the enhanced scatter search method (eSS) for continuous nonlinear programming (cNLP) and mixed-integer programming (MINLP) problems, and variable neighborhood search (VNS) for Integer Programming (IP) problems. Additionally, the R version includes BayesFit for parameter estimation by Bayesian inference. The eSS and VNS methods can be run on a single-thread or in parallel using a cooperative strategy. The code is supplied under GPLv3 and is available at http://www.iim.csic.es/~gingproc/meigo.html. Documentation and examples are included. The R package has been submitted to BioConductor. We evaluate MEIGO against optimization benchmarks, and illustrate its applicability to a series of case studies in bioinformatics and systems biology where it outperforms other state-of-the-art methods.ConclusionsMEIGO provides a free, open-source platform for optimization that can be applied to multiple domains of systems biology and bioinformatics. It includes efficient state of the art metaheuristics, and its open and modular structure allows the addition of further methods.

Comparison of linear, mixed integer and non-linear programming methods in energy system dispatch modelling

In the paper, three frequently used operation optimisation methods are examined with respect to their impact on operation management of the combined utility technologies for electric power and DH (district heating) of eastern Denmark. The investigation focusses on individual plant operation differences and differences between the solution found by each optimisation method. One of the investigated approaches utilises LP (linear programming) for optimisation, one uses LP with binary operation constraints, while the third approach uses NLP (non-linear programming). The LP model is used as a benchmark, as this type is frequently used, and has the lowest amount of constraints of the three. A comparison of the optimised operation of a number of units shows significant differences between the three methods. Compared to the reference, the use of binary integer variables, increases operation of selected units by 23%, while for a non-linear approach the increase can be higher than 39%. The results indicate a higher coherence between the two latter approaches, and that the MLP (mixed integer programming) optimisation is most appropriate from a viewpoint of accuracy and runtime.

Integrated modeling approach for sustainable municipal energy system planning and management – A case study of Shenzhen, China

Abstract In this study, a fractile-based interval mixed-integer programming (FIMP) method is advanced for sustainable municipal-scale energy system planning and management. FIMP can handle uncertainties presented in terms of fuzzy boundary intervals that represent interval coefficients with independently fuzzy lower and upper bounds with possibility distributions. A FIMP-based municipal energy model (FIMP-MEM) is then formulated for managing various energy activities in the City of Shenzhen, China. Solutions for energy supply, electricity generation, oil-product production, air-pollutant mitigation, carbon dioxide control, capacity expansion, and electricity import/export are obtained. Results can be used to help the city's managers to identify desired system designs and to determine which of these designs can most efficiently accomplish optimizing the system objective under diverse p-necessity fractiles. The generated decision alternatives are beneficial for the city's energy system planning and management through (a) generating desired energy resources allocation, (b) identifying electricity generation and capacity-expansion scheme, (c) providing air pollution control plan, (d) analyzing the tradeoff among system cost, environmental impact, and system-failure risk.

A sampling and classification item selection approach with content balancing

Existing automated test assembly methods typically employ constrained combinatorial optimization. Constructing forms sequentially based on an optimization approach usually results in unparallel forms and requires heuristic modifications. Methods based on a random search approach have the major advantage of producing parallel forms sequentially without further adjustment. This study incorporated a flexible content-balancing element into the statistical perspective item selection method of the cell-only method (Chen et al. in Educational and Psychological Measurement, 72(6), 933-953, 2012). The new method was compared with a sequential interitem distance weighted deviation model (IID WDM) (Swanson & Stocking in Applied Psychological Measurement, 17(2), 151-166, 1993), a simultaneous IID WDM, and a big-shadow-test mixed integer programming (BST MIP) method to construct multiple parallel forms based on matching a reference form item-by-item. The results showed that the cell-only method with content balancing and the sequential and simultaneous versions of IID WDM yielded results comparable to those obtained using the BST MIP method. The cell-only method with content balancing is computationally less intensive than the sequential and simultaneous versions of IID WDM.

Comparison of Mixed-Integer Programming and Genetic Algorithm Methods for Distributed Generation Planning

This paper applies recently developed mixed-integer programming (MIP) tools to the problem of optimal siting and sizing of distributed generators in a distribution network. We investigate the merits of three MIP approaches for finding good installation plans: a full AC power flow approach, a linear DC power flow approximation, and a nonlinear DC power flow approximation with quadratic loss terms, each augmented with integer generator placement variables. A genetic algorithm-based approach serves as a baseline for the comparison. A simple knapsack problem method involving generator selection is presented for determining lower bounds on the optimal design objective. Solution methods are outlined, and computational results show that the MIP methods, while lacking the speed of the genetic algorithm, can find improved solutions within conservative time requirements and provide useful information on optimality.

Supplier evaluation and demand allocation among suppliers in a supply chain

This paper presents a hybrid algorithm that prioritizes the suppliers and then allocates the demand among the suppliers. The objective here is to maximize the total purchase value of the items taking into consideration budget constraint, demand condition, delivery lead-time and supplier capacity. Since the problem is multi-criteria decision making, we solve this problem by integrating the supplier rating with mixed linear integer programming method. The customer demand is allocated by using a hybrid algorithm based on the technique for order preference by similarity to ideal solution (TOPSIS) and the mixed linear integer programming (MILP) approaches. The effectiveness of the proposed algorithm is validated with computational results. Drawing to a case, a supplier S3 is identified as the best supplier by using the TOPSIS method for demand allocation under no restrictions. On the contrary, under constrained scenario, supplier S2 is selected as the best supplier by using the hybrid algorithm for demand allocation and maximum units are allocated to S2.

Measuring world governance: revisiting the institutions hypothesis

We consider the weighting scheme that results in a best-case scenario in the construction of the world governance indicators (WGIs), a proxy of institutional quality. To perform that, we use an approach that relies on consistent tests for stochastic dominance efficiency of a given index with respect to all possible indices constructed from a set of individual components. The test statistics and the estimators are computed using mixed integer programming methods. The results show that the equally weighted (fixed weights) composite WGI index is not the best-case scenario and that governance indicators at different years should be weighted differently. Furthermore, we revisit the institutions hypothesis in the empirical growth literature, where institutional quality is the main determinant of long-term development. We find that not only do institutions matter for economic development but also geography and macroeconomic policies do affect economic development directly.

A Constraint Programming Method for Advanced Planning and Scheduling System with Multilevel Structured Products

This paper deals with the advanced planning and scheduling (APS) problem with multilevel structured products. A constraint programming model is constructed for the problem with the consideration of precedence constraints, capacity constraints, release time and due date. A new constraint programming (CP) method is proposed to minimize the total cost. This method is based on iterative solving via branch and bound. And, at each node, the constraint propagation technique is adapted for domain filtering and consistency check. Three branching strategies are compared to improve the search speed. The results of computational study show that the proposed CP method performs better than the traditional mixed integer programming (MIP) method. And the binary constraint heuristic branching strategy is more effective than the other two branching strategies.

Multi-piece mould design based on a mixed-integer programming method

Multi-piece mould design is a moulding technology that involves three-dimensional spatial construction of two or more mould pieces in a manner similar to assembling/dissembling a three-dimensional puzzle to build production parts. Using such a moulding technology, complex parts with intricate geometries can be made for limited run productions. Compared to traditional two-piece moulds and rapid prototyping, the multi-piece mould approach has many advantages with respect to part complexity and production speed, etc.; however, the technology has challenges in designing the actual multi-piece moulds. Previous methodologies address this problem primarily using heuristics. We present a multi-piece mould design (MPMD) framework that is based on a mixed-integer programming approach. The method constructs the MPMD by minimising the number of mould pieces that is required for a given Computer-Aided Design (CAD) model. The solution strategy for the formulated linear mixed-integer optimisation problem is presented. The algorithmic strategy for solving the resulting mixed-integer programming problem is also provided with examples that illustrate the effectiveness and efficiency of the approach.

Locating multiple types of recharging stations for battery-powered electric vehicle transport

This study used the concepts of set- and maximum-coverage to formulate capacitated multiple-recharging-station-location models, using a mixed integer programming method, based on a vehicle-refueling logic. The results of the case study demonstrate that the use of mixed stations can achieve the optimal deployment for the planning area, with results that are better than those achieved with a single type of recharging stations. While in some paths the use of slow-recharging stations means that tours are not feasible, the deployment of mixed stations can provide an economical approach which ensures the completion of overall tours on each path.

Integrated lot-sizing and scheduling with overlapping for multi-level capacitated production system

In this paper, multi-product multi-period integrated lot-sizing and scheduling problem with a new practical condition, overlapping in operations, is discussed in a capacitated multi-stage production system. In many real situations, products are carried using pallets or conveyers, so there is no constraint to complete the entire process of products lots and then transfer them to the next stage. To consider this issue in the problem an approach, named overlapping in operations, is developed in this paper. For each product, embedded operations on the two consecutive machines can be performed due to overlapping consideration in which each operation of certain product may be overlapped with the another operation of the same product on the next machine. A novel mixed-integer programming (MIP) model is proposed for the problem to minimise the maximum makespan over the planning horizon for the corresponding production plan. The mathematical model is impractical to solve large-scale problems in reasonable time. So, two MIP-based heuristics using reduced size MIP models and simulated annealing algorithm is proposed to solve non-small instances of the problem. Moreover, a MIP method is used to validate the proposed solution approaches. The presented algorithms explore the solution space for both lot-sizing and scheduling problems. A combination of production plan and sequence that is feasible and close to optimum can be found using these algorithms. Computational experiences show that the proposed algorithms can find good quality solution for the problem in a reasonable time. The computational experiences also confirm that the makespan measure and machines utilisation rate can be improved by overlapping consideration.

An interval joint-probabilistic programming method for solid waste management: a case study for the city of Tianjin, China

Currently, environmental protection and resources conservation continue to be challenges faced by solid-waste managers in China. These challenges are being further compounded by rapid socioeconomic development and population growth associated with increased waste generation rates and decreased waste disposal capacities. In response to these challenges, an interval joint-probabilistic mixed-integer programming (IJMP) method is developed for supporting long-term planning of waste management activities in the city of Tianjin, which is one of the largest municipalities in the northern part of China. In the IJMP, joint probabilistic constraints are introduced into an interval-parameter mixed-integer programming framework, such that uncertainties presented in terms of interval values and random variables can be reflected. Moreover, a number of violation levels for the waste-management-capacity constraints are examined, which can facilitate in-depth analyses of tradeoffs among economic objective and system-failure risk. The results indicate that reasonable solutions have been generated. They are valuable for supporting the adjustment of the city’s existing waste-management practices and the long-term planning of the city’s waste-management facilities.

Efficient scheduling method of complex batch processes with general network structure via agent‐based modeling

A novel efficient agent‐based method for scheduling network batch processes in the process industry is proposed. The agent‐based model is based on the resource‐task network. To overcome the drawback of localized solutions found in conventional agent‐based methods, a new scheduling algorithm is proposed. The algorithm predicts the objective function value by simulating another cloned agent‐based model. Global information is obtained, and the solution quality is improved. The solution quality of this approach is validated by detailed comparisons with the mixed‐integer programming (MIP) methods. A solution close to the optimal one can be found by the agent‐based method with a much shorter computational time than the MIP methods. As a scheduling problem becomes increasingly complicated with increased scale, more specifications, and uncertainties, the advantages of the agent‐based method become more evident. The proposed method is applied to simulated industrial problems where the MIP methods require excessive computational resources. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2884–2906, 2013

Solving siphons with the minimal cardinality in Petri nets and its applications to deadlock control

Siphons can be used to characterise deadlock states and solve deadlock problems in Petri nets that model flexible manufacturing systems. By modifying the objective function and adding new constraints to the mixed-integer programming (MIP) method proposed by Park and Reveliotis, this paper presents a revised MIP (RMIP) to directly solve siphons, called smart siphons, with the minimal cardinality as well as the minimal number of resource places. Accordingly, an iterative siphon-based control (ISC) method adds a proper control place (CP) to make each smart siphon max-controlled until the controlled system is live. Since any siphon that has more resource places can be composed of those containing less resource places, a small number of CPs are required during the iterative control process due to the solved smart siphons containing the minimal number of resource places. Compared with the existing methods in the literature, the proposed ISC method using the RMIP avoids computing a maximal deadly marked siphon from which a minimal siphon is then derived, adds a small number of proper CPs, and leads to a liveness-enforcing supervisor with a simple structure. Finally, a case study demonstrates the effectiveness of the proposed ISC method.

A new country risk index for emerging markets: A stochastic dominance approach

An optimal weighting scheme is proposed to construct economic, political and financial risk indices in emerging markets using an approach that relies on consistent tests for stochastic dominance efficiency. These tests are considered for a given risk index with respect to all possible indices constructed from a set of individual risk factors. The test statistics and the estimators are computed using mixed integer programming methods. We derive an economic, political and financial risk ranking of emerging countries. Finally, an overall risk index is constructed. One main result is that the financial risk is the leading contributor to sovereign risk in emerging markets followed by the economic and political risks.

Item Selection for the Development of Parallel Forms From an IRT-Based Seed Test Using a Sampling and Classification Approach

Two sampling-and-classification–based procedures were developed for automated test assembly: the Cell Only and the Cell and Cube methods. A simulation study based on a 540-item bank was conducted to compare the performance of the procedures with the performance of a mixed-integer programming (MIP) method for assembling multiple parallel test forms. The study investigated the statistical equivalence of the forms generated by the three test assembly methods (Cell Only, Cell and Cube, and MIP) in terms of test information functions, test characteristic curves, mean square deviations, and practical constraints, such as content balancing and nonoverlap among forms. The results indicated that the 13-point MIP method outperformed the other two methods in terms of the “closeness” test information functions between the reference form and the assembled parallel tests. Regarding test characteristic curves, the Cell Only and Cell and Cube methods yielded more similar test characteristic curves than the MIP method. Constraining test information functions apparently does not guarantee that the assembled forms will yield similar test characteristic curves. Overall, the Cell Only and Cell and Cube methods have the potential to provide results similar to the optimization approach.

Solving the spatial scheduling problem: a two-stage approach

The spatial scheduling problem that arises in hull block assembly shops occurs when scheduling and spatial allocation of the blocks must be considered simultaneously. We present a two-stage approach to this type of problem. The first stage aims to reduce the number of blocks. The second stage optimises the scheduling and spatial allocation of blocks using nonlinear mixed integer programming (NMIP) methods. The procedure proposed in this paper uses an agglomeration algorithm (AA) for the blocks. This procedure is based on the space–time coupling mechanism. The AA is a three-dimensional classification used to cluster blocks linked closely in time and space into virtual blocks. Extensive computational results from real cases are presented to demonstrate the effectiveness of the proposed approach, demonstrating a significant improvement over results obtained from existing methods.

CP and MIP Methods for Ship Scheduling with Time-Varying Draft

Existing ship scheduling approaches either ignore constraints on ship draft (distance between the waterline and the keel), or model these in very simple ways, such as a constant draft limit that does not change with time. However, in most ports the draft restriction changes over time due to variation in environmental conditions. More accurate consideration of draft constraints would allow more cargo to be scheduled for transport on the same set of ships. We present constraint programming (CP) and mixed integer programming (MIP) models for the problem of scheduling ships at a port with time-varying draft constraints so as to optimise cargo throughput at the port. We also investigate the effect of several variations to the CP model, including a model containing sequence variables, and a model with ordered inputs. Our model allows us to solve realistic instances of the problem to optimality in a very short time, and produces better schedules than both scheduling with constant draft, and manual scheduling approaches used in practice at ports.

Hybrid Inexact Optimization Approach with Data Envelopment Analysis for Environment Management and Planning in the City of Beijing, China

Abstract In this study, a two-stage interval-stochastic mixed integer programming method was developed for supporting long-term planning of solid waste management in the city of Beijing, China. The developed method reflects uncertainties expressed as probability density functions and intervals, as well as offers a linkage between predefined environmental policies and associated economic implication. The method has advantages in tackling dynamic, interactive, and uncertain characteristics of solid waste management system in the city, and addressing issues regarding waste diversion and landfill prolongation. Reasonable solutions were generated for waste flow allocation and system capacity expansion. Data envelopment analysis was then utilized for analyzing these solutions under different policy scenarios. Obtained results can provide useful information and decision-support for the city's solid waste management planning. Results are valuable for adjustment of the existing waste management practice and identi...