Linear Programming Model Research Articles

Optimization of carbon emission in an integrated machine-piece scheduling and vehicle routing problem and its solution using MOPSO and NSGAII metaheuristic algorithms

The growth of the global economy is accompanied by significant energy consumption, and greenhouse gas emissions create various problems such as global warming and environmental degradation. To protect the environment, governments are seeking to reduce carbon emissions. Production systems that operate solely based on economic factors in the workshop only consider problems such as production speed, cost, and processing time. Two aspects can be effective in saving energy and reducing emissions at the production planning level: using routing to find the shortest path for collecting workpieces to the workshop, and turning off machines with long idle times and restarting them at the appropriate time. If the workshop production problem is combined with vehicle routing, a new problem arises. According to the research conducted so far, an integrated mathematical model for production routing has not been designed in a situation where the routing is before the production workshop. In this research, this bi-objective model is introduced, and it is solved using the augmented epsilon-constraint (AEC) method. The proposed mixed-integer linear programming model of this research includes three dimensions: environmental, social (customer satisfaction), and economic simultaneously. Given the high complexity of the mathematical model, MATLAB software and MOPSO and NSGA-II algorithms were used to solve it at higher dimensions. Seven evaluation criteria were used to compare the two proposed algorithms, and the results show that the MOPSO algorithm performs better. The findings suggest that minimizing pollution may involve sacrificing on-time delivery to customers. Consequently, decision-makers must carefully weigh the trade-off between reducing environmental impact and maintaining satisfactory delivery performance, ultimately deciding on an acceptable pollution level.

Optimizing Transportation Logistics for Cost Efficiency in a Garment Factory: A Linear Programming Approach Using LINDO

The garment industry in Bangladesh plays a pivotal role in the nation's economy, yet it faces significant challenges in optimizing transportation logistics to minimize costs and enhance efficiency. This paper uses linear programming techniques to negotiate the optimization of transportation logistics in a garment factory located in Gazipur, Bangladesh. The primary objective is to develop an optimized transportation plan that reduces costs while satisfying supply and demand constraints across multiple suppliers, production facilities, and distribution centers. The study formulates the transportation problem as a linear programming model and employs the simplex method to determine the optimal solution. LINDO software is utilized for model implementation and optimization. The model incorporates real data from the garment factory, including supply capacities, demand requirements, and transportation costs. Results from the LINDO optimization reveal a detailed transportation plan that minimizes total transportation costs, demonstrating significant cost savings compared to existing practices. The optimized plan improves operational efficiency and provides a robust decision-support tool for logistics managers. Sensitivity analysis is conducted to ensure the model's robustness under varying conditions. The methodologies and findings are scalable and adaptable to other industrial settings, offering valuable insights for enhancing logistical efficiency in the garment industry. The study also highlights potential areas for future research, including integrating environmental considerations and advanced optimization techniques. Through this research, the garment factory in Gazipur can achieve substantial cost reductions and operational improvements, enhancing its competitiveness in the global market.

Asynchronous decentralized traffic signal coordinated control in urban road network

AbstractThis study introduces an asynchronous decentralized coordinated signal control (ADCSC) framework for multi‐agent traffic signal control in the urban road network. The controller at each intersection in the network optimizes its signal control decisions based on a prediction of the future traffic demand as an independent agent. The asynchronous framework decouples the entangled interdependence between decision‐making and state prediction among different agents in decentralized coordinated decision‐making problems, enabling agents to proceed with collaborative decision‐making without waiting for other agents’ decisions. Within the proposed ADCSC framework, each controller dynamically optimizes its signal timing strategy with a unique rolling horizon scheme. The scheme's individualized parameters for each controller are determined based on the vehicle travel time between the adjacent intersections, ensuring that controllers can make informed control decisions with accurate arrival flow information from upstream intersections. The signal optimization problem is formulated as a mixed integer linear program model, which adopts a flexible signal scheme without a fixed phase structure and sequence. Simulation results demonstrate that the proposed ADCSC strategy significantly outperforms the benchmark signal coordination methods in terms of average delay, travel speed, stop numbers, and energy consumption. Experimental analysis on computation time validates the applicability of the proposed optimization model for real‐time implementation. Sensitivity analysis on key parameters in the framework is conducted, offering insights for parameter selection in practice. Furthermore, the ADCSC framework is extended to a road network in Qinzhou City, China, with 45 signalized intersections, demonstrating its effectiveness and scalability in the real‐world road network.

Value Assessment and Prediction of Regulating Ecosystem Services in Hainan Tropical Rainforest National Park, China

Ecosystem services serve as a bridge between the ecological environment and human society. The quantitative analysis and forecasting of ecosystem services can provide references for regional eco-environmental assessments and land-use planning for the future. In this study, taking Hainan Tropical Rainforest National Park (HTRNP) as an example, the value of regulating ecosystem services (RESs) in 2020 was assessed via ArcGIS 10.1 and the InVEST 3.5 model, and the per-unit value of RESs was calculated for different LULC types. In addition, in accordance with the Overall Planning for HTRNP and the objective of optimizing RESs, the value of RESs in short-term (to 2030) and long-term (to 2050) scenarios was forecast via a linear programming model. The results are as follows: (1) The RES value of HTRNP in 2020 was CNY 2090.67 × 108, with climate regulation accounting for the largest proportion; the spatial distribution of RESs in the eastern and central areas was higher than that in the western area, but different indicators of RESs differed in their spatial patterns in varied geographic units. (2) The natural forest ecosystem in HTRNP accounts for 76.94% of the total area but 84.82% of the total value of RESs. The per-unit value is ranked from highest to lowest as follows: montane rainforests > wetlands > lowland rainforests > lowland secondary rainforests > tropical coniferous forests > deciduous monsoon rainforests > tropical cloud forests > shrub forests > timber forests > economic forests > rubber forests > grasslands > farmlands > settlements. (3) In the short-term scenario, the value of RESs is CNY 2216.64 × 108, an increase of CNY 118.97 × 108 compared to 2020, with an increase rate of 5.67%. In the long-term scenario, the value of RESs is CNY 2472.48 × 108, an increase of CNY 374.81 × 108 compared to 2020, with an increase rate of 17.87%. The results reveal the significance of ecosystem services in the national park and can inform more targeted and scientifically sound decision-making in the future.

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

A Study on Energy Production and Consumption Based on Seasonal ARIMA and REAO

In the face of the global energy challenge, this study delves into the intricate landscape of energy dynamics within the rapidly evolving global economy and technological sphere. Recognizing energy as a pivotal driver of societal progress, we present a comprehensive evaluation and optimization model for China's energy sources, emphasizing economic, cost, electric energy, and environmental considerations. Employing time series models, specifically ARIMA and gray scale forecasting, we meticulously analyze wind and solar power generation, as well as overall power consumption. The results showcase nuclear power and hydropower as significant contributors to the energy landscape. Leveraging a linear programming model, we simulate and optimize the future energy development structure, considering constraints such as cost, energy, capacity, and carbon emissions. Our findings reveal the optimal quota contributions of thermal power, hydro power, nuclear power, wind power, and solar power as 13.7%, 14.2%, 17.6%, 54.5%, and 0%. Building upon these insights, we propose strategic recommendations for sustainable electric energy development. Emphasizing wind and solar energy's potential, we advocate for continued interest and increased support for technological advancements. Additionally, we call for the establishment of comprehensive power and energy development plans, integrating economic and environmental goals. To address imperfections in China's electricity tariff system, we recommend ongoing reforms aligned with market dynamics and environmental considerations.

Part Feeding and Internal Transportation Decision Making for a Machinery Manufacturer

This research validates a mixed integer linear programming model that minimizes total feeding and acquisition costs by integrating assembly line feeding and vehicle type decisions in a machinery manufacturing company. The results show that the optimal assignment reduces costs by 56% compared with the company's current assignment.

Research on cargo volume prediction and personnel scheduling optimization of sorting center based on BP neural network and improved genetic algorithm

With the development of Internet technology, e-commerce has become the primary choice for consumers to shop, and the huge order transaction volume has generated logistics problems. As an important part of the logistics network, the sorting platform is essential to improve its management efficiency. To enhance the efficiency of the sorting center, the topological relationship diagram is used to show the logistics network, and the cargo volume of the sorting center in the next 30 days and every hour is predicted based on the BP neural network model. Based on the mixed integer linear programming model and the improved genetic algorithm SCAGA model, the total man-days are minimized to ensure that the cargo volume is processed. At the same time, to ensure that the actual hourly efficiency is as balanced as possible, the optimization problem of personnel scheduling is studied. To maintain the efficient operation of the logistics industry, the accurate prediction of the number of goods and the reasonable scheduling of personnel, to provide a set of efficient and economical personnel scheduling programs for the logistics company, thus improving the overall operational efficiency and service quality.

Circular–Sustainable–Reliable Waste Management System Design: A Possibilistic Multi-Objective Mixed-Integer Linear Programming Model

Waste management involves the systematic collection, transportation, processing, and treatment of waste materials generated by human activities. It entails a variety of strategies and technologies to diminish environmental impacts, protect public health, and conserve resources. Consequently, providing an effective and comprehensive optimization approach plays a critical role in minimizing waste generation, maximizing recycling and reuse, and safely disposing of waste. This work develops a novel Possibilistic Multi-Objective Mixed-Integer Linear Programming (PMOMILP) model in order to formulate the problem and design a circular–sustainable–reliable waste management network, under uncertainty. The possibility of recycling and recovery are considered across incineration and disposal processes to address the main circular-economy principles. The objectives are to address sustainable development throughout minimizing the total cost, minimizing the environmental impact, and maximizing the reliability of the Waste Management System (WMS). The Lp-metric technique is then implemented into the model to tackle the multi-objectiveness. Several benchmarks are adapted from the literature in order to validate the efficacy of the proposed methodology, and are treated by CPLEX solver/GAMS software in less than 174.70 s, on average. Moreover, a set of sensitivity analyses is performed to appraise different scenarios and explore utilitarian managerial implications and decision aids. It is demonstrated that the configured WMS network is highly sensitive to the specific time period wherein the WMS does not fail.

Scheduling Splitable Jobs on Identical Parallel Machines to Minimize Makespan using Mixed Integer Linear

The scheduling of parallel machines with and without a job-splitting property, deterministic demand,and sequence-independent setup time with the goal of minimizing makespan is examined in this work.For simultaneous processing by multiple machines, single-stage splitable jobs are broken into random(job) sections. When a job starts to be processed on a machine, an operator has to setup the machinefor an hour. By creating two Mixed Integer Linear Programming models, this work proposes amathematical programming strategy (MILP). A MILP model takes the job-splitting property intoaccount. Another model, however, does not include the job-splitting property. This study investigatesthe performance of the proposed models using Gurobi solver. These programs' numerical calculationsare based on actual problems in the Indonesian city of Bandung's plastics industry. On four identicalparallel injection molding machines, 318 jobs must be finished in 22 periods. The real schedulingmethod is contrasted with these two MILP models. The maximum workload imbalance, the maximumrelative percentage of imbalance, and the makespan of these three scheduling systems are used toevaluate their effectiveness. Without the job-splitting property, MILP can handle the real issue ofscheduling identical parallel machines on injection molding machines to reduce makespan, resultingin a 36% average decrease. The MILP model's job-splitting property can reduce makespan by anadditional 2.40%. The order of relative ranking is MILP with job-splitting property, MILP withoutjob-splitting property, and actual scheduling based on the makespan minimization, workloadimbalance, and relative percentage of imbalance.

A mathematical model for potash supply chain management with a strategic logistics perspective

Abstract Accepted by: M. Zied Babai This paper introduces a novel Integer Linear Programming model designed to enhance the efficiency and sustainability of the potash supply chain, a crucial element supporting global agriculture. The developed mathematical optimization model focuses on fleet selection (private/outsource) and incorporates the concept of ‘inter-warehouse collaboration’, which addresses key logistics considerations. Integrating mining, processing, storage and transportation, the model encompasses decision variables like extracted carnallite amount, production, storage levels and shipped potash amount. Illustrated through a case study on the Arab Potash Company in Jordan, the results showcase the model’s proficiency in meeting local and international market demands. The model ensures resilient and sustainable supply chain performance by emphasizing logistics optimization, particularly in fleet selection. The study attains the highest ‘warehouse-to-warehouse’ support for Standard and Granular potash types in the international demand scenario, contributing to efficient production planning and fleet management. In conclusion, the presented mathematical model is a valuable tool for potash industry stakeholders, offering insights for strategic decision-makers involved in production planning and fleet management.

Optimizing Supply Chain Design under Demand Uncertainty with Quantity Discount Policy

In typical business situations, sellers usually offer discount schemes to buyers to increase overall profitability. This study aims to design a supply chain network under uncertainty of demand by integrating an all-unit quantity discount policy. The objective is to maximize the profit of the entire supply chain. The proposed model is formulated as a mixed integer nonlinear programming model, which is subsequently linearized into a mixed integer linear programming model and hence able to obtain a global solution. Numerical examples in the manufacturing supply chain where customer demand follows normal distributions are used to assess the effect of quantity discount policies. Key findings demonstrate that the integration of quantity discount policies significantly reduces total supply chain costs and improves inventory management under demand uncertainty, and decision makers need to decide a balance level between service levels and profits.

Deep reinforcement learning driven trajectory-based meta-heuristic for distributed heterogeneous flexible job shop scheduling problem

As the production environment evolves, distributed manufacturing exhibits heterogeneous characteristics, including diverse machines, workers, and production processes. This paper examines a distributed heterogeneous flexible job shop scheduling problem (DHFJSP) with varying processing times. A mixed integer linear programming (MILP) model of the DHFJSP is formulated with the objective of minimizing the makespan. To solve the DHFJSP, we propose a deep Q network-aided automatic design of a variable neighborhood search algorithm (DQN-VNS). By analyzing schedules, sixty-one types of scheduling features are extracted. These features, along with six shaking strategies, are used as states and actions. A DHFJSP environment simulator is developed to train the deep Q network. The well-trained DQN then generates the shaking procedure for VNS. Additionally, a greedy initialization method is proposed to enhance the quality of the initial solution. Seven efficient critical path-based neighborhood structures with three-vector encoding scheme are introduced to improve local search. Numerical experiments on various scales of instances validate the effectiveness of the MILP model and the DQN-VNS algorithm. The results show that the DQN-VNS algorithm achieves an average relative percentage deviation (ARPD) of 3.2%, which represents an approximately 88.45% reduction compared to the best-performing algorithm among the six compared, with an ARPD of 27.7%. This significant reduction in ARPD highlights the superior stability and performance of the proposed DQN-VNS algorithm.

Augmented ɛ‐constraint‐based matheuristic methodology for Bi‐objective production scheduling problems

AbstractMatheuristic is an optimisation methodology that integrates mathematical approaches and heuristics to address intractable combinatorial optimisation problems, where a common framework is to insert mixed integer linear programming (MILP) models as local search functions for evolutionary algorithms. However, since a mathematical programming formulation only tries to find the solution with the best objective value, matheuristics are rarely adopted to multi‐objective scenarios asking for a set of Pareto optimal solutions, for example, vehicle routing problems and production scheduling problems. In this situation, the ɛ‐constraint, which transforms multi‐objective problems into single‐objective formulations by considering selected objectives as constraints, seems to be a promising approach. First, an augmented ɛ‐constraint‐based matheuristic methodology (ɛ‐MH) is proposed to apply the idea of ɛ‐constraint to embedded MILP models, so that Pareto fronts obtained by meta‐heuristics can be further improved by solving a set of MILP models. Afterwards, four speed‐up strategies are developed to alleviate the computational burden resulting from repeatedly solving mathematical formulations, which also imply preferable scenarios for taking advantages of the ɛ‐MH. Finally, several real‐world bi‐objective scheduling problems are discussed to present potential applications for the proposed methodology.

Stochastic Risk-driven Bidding of a Solar and Storage Aggregator in Primary Frequency and Energy Markets: A Performance-based Capacity Allocation Approach

The ever-increasing impact of deploying renewable energy resources reducing power system inertia, requires distributed energy resource (DER) aggregators to secure high-performance, fast-responding primary frequency reserve (PFR) in ancillary service markets. However, enabling immediate and local regulation of the operating point upon detecting frequency deviations imposes the necessity of allocating specific headroom for DERs. The individual headroom susceptibility to uncertainty and response speed of a certain DER not only compromises aggregated headroom performance in the PFR market but also leads to incurring unnecessary opportunity costs in the energy market and inadequate frequency control in the power system. To address this challenge, this paper introduces an integrated probabilistic performance index for DERs that considers their individual interactive heterogeneous response speed and uncertainty. Using this index, aggregators overseeing photovoltaic and smart buildings equipped with energy storage can evaluate, allocate, and remunerate optimal headroom capacity for individual DERs based on their performance. This approach ensures maximum aggregated profits and minimum opportunity costs in the PFR and energy markets while improving the performance of the provided PFR capacity, respectively. These tasks are aimed by a novel optimal capacity allocation strategy for an aggregator in both markets. This strategy employs a mixed-integer linear programming model to find the optimal solution and the conditional value-at-risk measure to tackle the uncertainties faced with the problem. The efficiency of the proposed model is assessed within a system reflecting the ERCOT market structure, demonstrating improved aggregated headroom performance, reduced energy opportunity costs, diminished risk of capacity shortage, and improved load frequency control (LFC) from the power system perspective.

A sustainable network design for municipal solid waste management considering waste-to-energy conversion under uncertainty

This research proposes a sustainable municipal solid waste management (MSWM) system to address population growth, consumerism, and resource scarcity. It introduces a two-phase decision structure combining multi-attribute decision-making (MADM) tools. Phase one develops a hybrid risk-oriented fuzzy MADM tool for selecting optimal waste-to-energy technologies, considering environmental and technological factors. Phase two uses a fuzzy bi-objective multi-period mixed-integer linear programming model to design the MSWM supply chain efficiently under uncertainty. A case study in North Tehran demonstrates the framework’s practical applicability and effectiveness in real-world scenarios, highlighting that effective source separation boosts recycling, reduces costs, and provides social benefits. Sensitivity analyses offer insights to enhance waste segregation practices. The study emphasizes integrating economic, environmental, and social sustainability in waste management decision-making. By offering a novel, holistic approach, this research addresses existing gaps in systematic decision-making processes and provides a robust tool for municipalities to optimize strategies under uncertainty.

CMSA based on set covering models for packing and routing problems

AbstractMany packing, routing, and knapsack problems can be expressed in terms of integer linear programming models based on set covering. These models have been exploited in a range of successful heuristics and exact techniques for tackling such problems. In this paper, we show that integer linear programming models based on set covering can be very useful for their use within an algorithm called “Construct, Merge, Solve & Adapt”(CMSA), which is a recent hybrid metaheuristic for solving combinatorial optimization problems. This is because most existing applications of CMSA are characterized by the use of an integer programming solver for solving reduced problem instances at each iteration. We present applications of CMSA to the variable-sized bin packing problem and to the electric vehicle routing problem with time windows and simultaneous pickups and deliveries. In both applications, CMSA based on a set covering model strongly outperforms CMSA when using an assignment-type model. Moreover, state-of-the-art results are obtained for both considered optimization problems.

Sustainable optimization of balancing valve settings in urban heating systems with an enhanced Jaya algorithm

With the continuous growth of global energy demand, energy conservation in heating has become a research hotspot for urban sustainable development. However, the research on energy conservation by adjusting the balance valve in the secondary network of heating system is relatively few. To address this gap, this paper solves the balancing valve opening adjustment (BVOA) problem with the goal of minimizing heat consumption and the energy usage of circulation pumps. A mixed-integer linear programming model is designed for this problem, and an effective EJaya algorithm is proposed. Initially, a random initialization method with three scrambling strategies is employed to generate high-quality solutions. Moreover, a segmented operator is proposed to enhance the algorithm’s search efficiency. Subsequently, Q-learning is utilized to select from three neighborhood operators, enhancing the algorithm’s local search capabilities. Additionally, an adaptive search balance strategy is designed to balance the methods of updating solutions, aiding in finding better solutions. Finally, extensive experiments have validated the effectiveness and efficiency of the proposed algorithm in solving the BVOA problem, contributing to the promotion of urban energy sustainability and social welfare.

Efficient Algorithm for Generating Optimal Inequality Candidates for MILP Modeling of Boolean Functions

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds. It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature. In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

Optimizing fleet, staff configuration and operational strategies in one-way mixed fleet carsharing systems: a Lagrangian relaxation-based approach

ABSTRACT This study explores enhancing carsharing services by integrating gasoline and electric vehicles into a one-way mixed fleet carsharing system (OMFCS). The focus is on optimizing configurations (fleet and staff size, initial deployment) and operational strategies (vehicle relocation and staff rebalancing) while considering carbon emission costs. Employing a space-time-electricity network modeling approach, we developed an integer linear programming model to tackle the configurations and operational strategies optimization problem. For solving this model, we introduce a Lagrangian relaxation-branch bound approach, which integrates subgradient, dynamic programming and greedy-based heuristics algorithm. An illustrative case and a real-world case are conducted to demonstrate the efficiency of the proposed solution method and the analysis sheds light on the configurations and operational strategies of OMFCS. The sensitive analysis results suggest that OMFCS is more profitable and balances user service quality and carbon emissions better than carsharing systems using only one type of vehicle.