Mathematical Mixed Integer Linear Programming Research Articles

A Mathematical Programming Approach for Multi-Objective Optimization in Designing Flexible Manufacturing Cells

The study aims to design flexible manufacturing cells with routing flexibility. A weighted mixed-integer linear mathematical programming model that aims to find optimal routing of parts in flexible manufacturing cells under constraints such as minimum machine utilization rate, maximum machine utilization rate, tool capacities, the utilization rate of workers, and labor-system unbalance is developed. The mathematical programming model aims to minimize the weighted sum of five objective functions: (1) the total number of intracellular movements; (2) the total number of intercellular movements; (3) the total workload unbalance of the machine system; (4) the total number of tools in all machines in the cells; and (5) the total labor-system workload unbalance. The main contribution of this study is to obtain these five objectives simultaneously, which have not been encountered to handle together before. By integrating these factors, the study presents a comprehensive approach to optimizing the design of flexible manufacturing cells. This study also has the potential to enhance system performance by addressing these factors. An illustrative problem tests the developed model, and the LINGO 17.0 optimization program is used to solve the generated mathematical programming model. Moreover, the related sensitivity analysis is performed with some parameters to examine the obtained results.

Partial coverage optimization under network connectivity constraints in heterogeneous sensor networks

The maximum network lifetime problem (MLP) in heterogeneous wireless sensor networks under connectivity and coverage constraints is addressed in this paper. Two main variants of the studied problem are considered. The first variant is α-coverage where a portion ((1 - α) percent) of the targets are allowed to be left uncovered. The second one is called β-coverage or β-constraint where each target has a minimum coverage rate β during the network’s lifetime service. When the two variants are considered at the same time, the problem is called αβ−Connected Maximum Lifetime Problem (αβ−CMLP) where we consider both global (whole targets) resp. local (individual target) monitoring level thresholds to improve the coverage quality of the deployed WSN. Unlike earlier works devoted to only α coverage, we deal with both local (α) and global (β) coverage leveling thresholds under network connectivity constraint. One approach to optimize the network’s lifetime is to divide the sensor nodes into Non-Disjoint subsets of sensors, or cover sets, and to schedule these covers with variable activation time periods, so that the global time lifespan of the network is optimized. To this end, we provide both exact and heuristic approaches in this study. First, a novel mathematical Mixed Integer Linear Programming (MILP) is presented to solve the αβ-coverage with network connectivity requirement to optimality. Unfortunately, due to the NP-Completeness of the addressed problem, the MILP’s resolution becomes impracticable for large optimization problems. To remedy this and to cope with large instances, we propose a new exact approach based on column generation able to achieve optimal solutions in reasonable time. In addition, since the CG’s subproblem resolution is also an NP-Hard problem, a new dedicated Heuristic (DH) is designed to solve the CG’s subproblem in polynomial time complexity. Moreover, we propose an exact ILP formulation for the CG’s subproblem if the DH heuristic fails to compute an attractive solution in each iteration of the CG computation process. Finally, several experiments are performed to evaluate the performances of our proposals.

RETRACTED ARTICLE: A multi-objective mixed integer linear programming model proposed to optimize a supply chain network for microalgae-based biofuels and co-products: a case study in Iran.

Recently, with efficient methods of cultivating microalgae at hand, researchers and investors have paid considerable attention to the production of different environmentally friendly biofuel products. In this study, a two-stage deterministic model is proposed to design a microalgae-based biofuel and co-product supply chain network (MBCSCN). In the first stage, appropriate locations to cultivate microalgae are identified through the analytical hierarchy process (AHP). In the second stage, a deterministic mathematical mixed integer linear programming (MILP) model is developed for a period of 5 years based on economic and environmental impacts as two criteria. The economic objective function maximizes the overall profit, while the objective function of the environmental impacts seeks to minimize fossil fuel consumption throughout the supply chain. Then, a multi-objective MILP optimization problem is solved using the ε-constraint method. The proposed model is evaluated through a case study in Iran. It has helped to identify appropriate locations for the cultivation of microalgae and to specify the required quantity of feedstock, the species of microalgae, the required technology, and the transportation modes in each step of the supply chain.

An Energy-Efficient Internet of Things Relaying System for Delay-Constrained Applications

The emerging Internet-of-things (IoT) systems contain a large number of small wireless devices with limited energy, communication, and computational capabilities. In such systems, a helping station located between the IoT devices and backhaul servers can be deployed to broadcast the IoT devices to the backhaul networks. This paper investigates a hybrid energy-efficient framework using multiple energy harvested relays with data buffering capabilities. The relays are powered by a hybrid energy supply consisting of a traditional electric grid and renewable energy grid. We propose an energy efficient novel approach aiming to support the wireless uplink transmission from IoT devices to backhaul servers with an acceptable delay threshold or transmission deadline. A mathematical mixed-integer linear programming (MILP) optimization problem is formulated to optimize the relays’ placement and energy consumption considering the association between relays and devices, instantaneous relays’ battery level, and transmit power budget. Due to the non-convex nature of the formulated optimization problem, we propose two heuristic low-complexity solutions to solve this problem. Finally, we compare the performance of the proposed algorithms with exhaustive search solutions as a benchmark.

Exact optimization and decomposition approaches for shelf space allocation

Shelf space is one of the scarcest resources, and its effective management to maximize profits has become essential to gain a competitive advantage for retailers. We consider the shelf space allocation problem with additional features (e.g., integer facings, rectangular arrangement restrictions) motivated by literature and our interactions with a local bookstore. We determine optimal number of facings of all products in two aspects (width and height of a rectangular arrangement space for each product), and allocate them as contiguous rectangles to maximize profit. We first develop a mixed-integer linear mathematical programming model (MIP) for our problem and propose a solution method based on logic-based Benders decomposition (LBBD). Next, we construct an exact 2-stage algorithm (IP1/IP2), inspired by LBBD, which can handle larger and real-world size instances. To compare performances of our methods, we generate 100 test instances inspired by real-world applications and benchmarks from the literature. We observe that IP1/IP2 finds optimal solutions for real-world instances efficiently and can increase the local bookstore’s profit up to 16.56%. IP1/IP2 can provide optimal solutions for instances with 100 products in minutes and optimally solve up to 250 products (assigned to 8 rows x 160 columns) within a time limit of 1800 s. This exact 2-stage IP1/IP2 solution approach can be effective in solving similar problems such as display problem of webpage design, allocation of product families in grocery stores, and flyer advertising.

Exact Optimization and Decomposition Approaches for 2D Shelf Space Allocation

Shelf space is one of the scarcest resources, and its effective management to maximize profits has become essential to gain a competitive advantage for retailers. We consider the two-dimensional shelf space allocation problem (2DSSAP) with additional features motivated by literature and our interactions with a local bookstore. Two dimensions represent the width and height of rectangular arrangement space of a product. We determine optimal number of facings of all products in both dimensions and allocate them as contiguous rectangles to maximize profit. We first develop a mixed-integer linear mathematical programming model (MIP) for our problem and propose a solution method based on logic-based Benders decomposition (LBBD). Next, we construct an exact 2-stage algorithm (IP1/IP2), inspired by LBBD, which can handle larger and real-world size instances. To compare performances of our methods, we generate 100 test instances inspired by real-world applications and benchmarks from the literature. We observe that IP1/IP2 finds optimal solutions for real-world instances efficiently and can increase the local bookstore's profit up to 16.56\%. IP1/IP2 can provide optimal solutions for instances with 100 products in minutes and optimally solve up to 250 products (assigned to 8 rows x 160 columns) within a time limit of 1800 seconds. This exact 2-stage IP1/IP2 solution approach can be effective in solving similar problems such as display problem of webpage design, allocation of product families in grocery stores, and flyer advertising.

Waste-to-Resource value chain optimisation: Combining spatial, chemical and technoeconomic aspects

Due to complex composition of carbohydrates, lipid, protein, cellulose, hemicellulose and lignin, wastewater (WW) and organic fraction municipal solid waste (OFMSW) represent nutrient and carbon rich resources. Conventionally, value chains in the waste sector have considered OFMSW and WW as unwanted by-products as opposed to potential valuable resources. Full exploitation of these resources calls for a value chain transformation towards proactive resource recovery. This study focuses on the waste supply chain optimisation to recover value added products from OFMSW.The research leads to a systems-modelling approach, which integrates spatial data analyses, mathematical mixed integer linear programming (MILP) optimisation and technology performance evaluation to inform the design of waste-to-resource value chains. A UK based study on OFMSW is presented to demonstrate the efficacy of the approach. The study captures variation in OFMSW quantity and composition, incorporating over 600 existing anaerobic digestion (AD) operational plants in the UK, while potential sites for new waste-recovery facilities are identified, accounting for transportation and logistics, using a GIS-based analysis. Key outcomes are analysed (technology type, size, location, logistical connections), placing emphasis on the need to consider the value of the resource recovery potential over the lifetime of an AD or thermochemical treatment facility in the design process. Such an approach offers a promising pathway for tackling the open challenges currently hindering the waste-to-resource transformation.

Multi-objective optimization of complex scheduling problems in low-volume low-variety production systems

In this paper, a new approach for solving scheduling problems in low-volume low-variety production systems is proposed. Products assembled in such production systems follow a pre-defined processing order through a series of unique work centers, each budgeted with multiple classifications of resources, responsible to complete a pre-defined statement of work, over the span of an imposed takt-time. Aircraft, heavy aero-structures, and heavy mining and military equipment are examples of products assembled in such production systems. Despite prominent scholarly advancements in sequencing and scheduling optimization of a wide range of production systems, limited research has been reported on mathematical programming approaches for scheduling optimization of activities in low-volume low-variety production systems. This paper fills the gap in the current literature, through the formulation of a set of multi-objective mixed-integer linear mathematical programming models, developed for solving discrete-time work center scheduling problems in low-volume low-variety production systems. Three mathematical models are proposed in this paper, two of which are formulated for scheduling optimization of activities within a work center, differentiated by their objectives and underlying assumptions, reflective of two distinct industrial approaches to scheduling. Additionally, an alternative optimization model is proposed for evaluating a work center's maximum capacity given the complete saturation of resources, recommended for capacity studies and early detection of bottlenecks. The models proposed in this paper are validated and verified for compatibility and reliability through a real-world case study with a global leader in the aerospace industry.

A novel model for the integrated planning of part quality inspection and preventive maintenance in a linear-deteriorating serial multi-stage manufacturing system

This paper presents a mixed-integer linear mathematical programming model for the integrated planning problem of the part quality inspection and preventive maintenance activities in serial multi-stage manufacturing system. The model concurrently determines the right time and place for performing the above-mentioned activities while the production stages are linearly deteriorating. These two decisions are made while the model is to minimize the total cost including the production, maintenance, inspection, scrap, replacement, and the penalty of shipped defective items to customer(s). A numerical example and a real case study are investigated to validate and verify the proposed model. The results show that the determination of inspection locations along a manufacturing line in different periods of time regarding the impact of preventive maintenance activities on defective production probability results in a more efficient manufacturing system.

Synthesis and optimization of manufacturing systems configuration using co-platforming

A mathematical mixed integer linear programming momdel to synthesize the manufacturing system configuration based on co-platforming strategy by mapping product platform to a corresponding machines platform is introduced. The mathematical model is verified through mathematical example and implemented in case study from automotive cylinder block manufacturer. The proposed model is beneficial in synthesizing manufacturing system to reduce investment costs by maintaining a group of platform machines that do not change with the change in product variants in different production periods. The synthesized manufacturing system platform does not change with the introduction of new product variants with new features belonging to the same product family which supports economic sustainability of manufacturing systems.

An MILP Optimization Problem for Sizing Port Rail Networks and Planning Shunting Operations in Container Terminals

This paper proposes an optimization approach for sizing port rail networks and planning railway shunting operations by adopting a discrete-time model of the overall system. First, a mixed-integer linear mathematical programming problem is defined in order to optimize shunting operations to be performed on the considered network by satisfying certain arrivals and departures of import and export flows. Moreover, the proposed procedure can be used to evaluate the capacity of a port rail network, in terms of maximum number of trains that can be managed over a certain time horizon, and to carry out what-if analyses aimed at testing different scenarios. The effectiveness of the proposed approach is shown by applying the optimization problem to a real case study referred to the port rail network of La Spezia Container Terminal located in Northern Italy. A computational analysis realized by varying the dimension and complexity of the problem instances is also reported in this paper to discuss the computational performance of the proposed model.

An Event-Triggered Receding-Horizon Scheme for Planning Rail Operations in Maritime Terminals

This paper proposes a planning approach to optimize railway operations in seaport terminals by adopting a queue-based discrete-time model of the considered system. First, a mixed-integer linear mathematical programming problem is defined in order to optimize the timing of import trains and the use of the handling resources devoted to rail port operations. Second, in order to deal with unexpected situations or uncertainty in estimating some data necessary to the planning, an event-triggered receding-horizon planning approach is proposed, in which the finite horizon optimization problem is solved whenever a critical event happens or the real values of some problem data significantly differ from the predicted ones. Both these planning approaches are tested on data referred to a real terminal and deeply discussed in this paper.

Modeling and solving mixed-model assembly line balancing problem with setups. Part I: A mixed integer linear programming model

This paper is the first one of the two papers entitled “modeling and solving mixed-model assembly line balancing problem with setups”, which has the aim of developing the mathematical programming formulation of the problem and solving it with a hybrid meta-heuristic approach. In this current part, a mixed-integer linear mathematical programming (MILP) model for mixed-model assembly line balancing problem with setups is developed. The proposed MILP model considers some particular features of the real world problems such as parallel workstations, zoning constraints, and sequence dependent setup times between tasks, which is an actual framework in assembly line balancing problems. The main endeavor of Part-I is to formulate the sequence dependent setup times between tasks in type-I mixed-model assembly line balancing problem. The proposed model considers the setups between the tasks of the same model and the setups because of the model switches in any workstation. The capability of our MILP is tested through a set of computational experiments. Part-II tackles the problem with a multiple colony hybrid bees algorithm. A set of computational experiments is also carried out for the proposed approach in Part-II.

Two-stage assembly flow-shop scheduling problem with non-identical assembly machines considering setup times

In this paper, we address the two stage assembly flow-shop problem with multiple non-identical assembly machines in stage two to minimise weighted sum of makespan and mean completion time. Also, sequence dependent setup times are considered for the first stage. This problem is a generalisation of previously proposed two stage assembly flow-shop problems (TSAFSP). In many real world industrial and production systems, there is more than one assembly machine to assemble job components. After extending a mathematical mixed-integer linear programming model to solve the problem, we use GAMS software. The TSAFSP has been known as NP-hard. Therefore, our more general problem is NP-hard too and so for large sized problems the right way to proceed is with the use of heuristic algorithms. So in this paper a hybrid VNS heuristic, which is a combination of the variable neighbourhood search (VNS) algorithm and a novel heuristic is developed and its solutions compared with solutions obtained by GAMS. Computational experiments reveal that the hybrid VNS heuristic performs much better than GAMS with respect to the percentage errors and run times.

Transportation cost analysis of the Hellenic system for alternative management of Waste Electrical and Electronic Equipment

This work presents an ex post analysis of the reverse logistics system for the alternative management of WEEE (Waste Electrical and Electronic Equipment) in Greece, in view of the system’s initiation in 2004. The aim was to predict how this system would provide guidelines concerning facility locations for WEEE collection and to calculate the transportation costs for WEEE management. A mathematical mixed integer linear programming model of the system was developed on the basis of both extensive survey of international cases and careful considerations of both the national EEE market, as well as the national and local waste management conditions, practices and possibilities.

Optimum loading of machines in a flexible manufacturing system using a mixed-integer linear mathematical programming model and genetic algorithm

Machine loading problem in a flexible manufacturing system (FMS) encompasses various types of flexibility aspects pertaining to part selection and operation assignments. The evolution of flexible manufacturing systems offers great potential for increasing flexibility by ensuring both cost-effectiveness and customized manufacturing at the same time. This paper proposes a linear mathematical programming model with both continuous and zero-one variables for job selection and operation allocation problems in an FMS to maximize profitability and utilization of system. The proposed model assigns operations to different machines considering capacity of machines, batch-sizes, processing time of operations, machine costs, tool requirements, and capacity of tool magazine. A genetic algorithm (GA) is then proposed to solve the formulated problem. Performance of the proposed GA is evaluated based on some benchmark problems adopted from the literature. A statistical test is conducted which implies that the proposed algorithm is robust in finding near-optimal solutions. Comparison of the results with those published in the literature indicates supremacy of the solutions obtained by the proposed algorithm for attempted model.

Modelling of hybrid energy system—Part I: Problem formulation and model development

A well designed hybrid energy system can be cost effective, has a high reliability and can improve the quality of life in remote rural areas. The economic constraints can be met, if these systems are fundamentally well designed, use appropriate technology and make use effective dispatch control techniques. The first paper of this tri-series paper, presents the analysis and design of a mixed integer linear mathematical programming model (time series) to determine the optimal operation and cost optimization for a hybrid energy generation system consisting of a photovoltaic array, biomass (fuelwood), biogas, small/micro-hydro, a battery bank and a fossil fuel generator. The optimization is aimed at minimizing the cost function based on demand and potential constraints. Further, mathematical models of all other components of hybrid energy system are also developed. This is the generation mix of the remote rural of India; it may be applied to other rural areas also.

Solving the response time variability problem by means of the cross-entropy method

The response time variability problem (RTVP) is an NP-hard combinatorial scheduling problem that has recently appeared in the literature. The RTVP has a wide range of production line systems applications such as sequencing the models to be produced on a mixed-model assembly line in a just-in-time context. This problem occurs whenever several units of different models need to be sequenced so as to minimise the variability of the distance between any two consecutive units of the same model. A mathematical mixed integer linear programming (MILP) model has been presented by another study, but the practical limit for obtaining optimal solutions is around 40 units to be scheduled. Another study has developed five heuristic algorithms to solve non-small RTVP instances. We propose to solve the RTVP by means of the metaheuristic cross-entropy (CE) method, which has been developed recently. We report on the computational experiments in which the CE method is compared with the five heuristic algorithms proposed in the literature.

Steady-state modelling of hybrid energy system for off grid electrification of cluster of villages

Electrification of villages from the main grid leads to large investments and losses, and this forms the basis of decentralized Hybrid Energy System. In order to evaluate the techno-economic performance of hybrid energy system for remote rural area electrification, a mixed integer linear mathematical programming model (time-series) has been developed to determine the optimal operation, optimal configuration including the assessment of the economic penetration levels of photovoltaic array area, and cost optimization for a hybrid energy generation system consisting of small/micro hydro based power generation, biogas based power generation, biomass (fuelwood) based power generation, photovoltaic array, a battery bank and a fossil fuel generator. An optimum control algorithm written in C++, based on combined dispatch strategy, allowing easy handling of the models and data of hybrid energy system components is presented. A special feature of the proposed model is that a cost constant (cost/unit) for each of the proposed resource is introduced in the cost objective function in such a way that resources with lesser unit cost share the greater of the total energy demand in an attempt to optimize the objective function. To demonstrate the use of model and algorithm, a case study for a rural remote area is also presented.

Variations in the efficiency of a mathematical programming solver according to the order of the constraints in the model

It is well-known that the efficiency of mixed integer linear mathematical programming depends on the model (formulation) used. With the same mathematical programming solver, a given problem can be solved in a brief calculation time using one model but requires a long calculation time using another. In this paper a new, unexpected feature to be taken into account is presented: the order of the constraints in the model can change the calculation time of the solver considerably. For a test problem, the Response Time Variability Problem (RTVP), it is shown that the ILOG CPLEX 9.0 optimizer returns a ratio of 17.47 between the maximum and the minimum calculations time necessary to solve optimally 20 instances of the RTVP, according to the order of the constraints in the model. It is shown that the efficiency of the mixed integer linear mathematical programming depends not only on the model (formulation) used, but also on how the information is introduced into the solver.