Mixed Integer Linear Programming Research Articles

Study on Optimizing Inventory to Minimize Waste in the Fresh Produce Supply Chain

Abstract: The increasing demand for fresh produce, particularly greens, presents unique challenges in inventory management due to their short shelf life and susceptibility to spoilage. Inefficient management often results in significant waste, leading to financial losses for retailers and increased environmental burdens from the disposal of spoiled goods. Despite various strategies being employed in the agri-food sector, gaps remain in understanding how to optimize inventory levels to minimize waste at the retails level while maintaining high service levels for customers. This research aims to develop an inventory management model tailored to fresh produce that can mitigate these issues by balancing stock availability and minimizing wastage, particularly in large retail shops. To address the research problem, a mixed-integer linear programming (MILP) model was developed, focusing on three service levels—90%, 92%, and 95%—to evaluate the trade-offs between customer satisfaction and total inventory costs. The model incorporates key constraints, including stock balance, shelf-life limitations, and demand satisfaction. The model was validated using actual data, including stock, sales, and wastage records for the month of August, 2024. The objective function aimed to minimize total costs, encompassing holding costs, wastage, and lost revenue. The findings indicate that higher service levels, while reducing stockouts, significantly increase costs due to excess inventory and waste. The most cost-effective solution was observed at the 90% service level, which balanced waste reduction and customer satisfaction. The study also emphasizes the importance of integrating total cost reduction and minimize wastage. Future work should focus on incorporating consumer behavior patterns and extending the model to multi-echelon inventory systems, which include both distribution centers and retail outlets, to capture a more holistic view of the supply chain.

Re-Supplying Autonomous Mobile Parcel Lockers in Last-Mile Distribution

This paper investigates a practical last-mile delivery scenario where a fleet of trucks replenishes autonomous mobile parcel lockers (AMPLs) in an urban setting. The lockers move along specified paths within restricted zones to reach customers’ locations. Ensuring seamless coordination between trucks and AMPLs requires the identification of suitable locations to exchange empty or loaded modular lockers. We first introduce a mixed-integer linear programming (MILP) formulation for the investigated problem. The proposed formulation establishes the basis for optimizing meeting point selection and routing decisions. Additionally, the study introduces a cluster-based simulated annealing (CSA) algorithm tailored for addressing larger-scale instances of the studied problem. The CSA algorithm incorporates the K-means clustering method with specialized operators rooted in an extensive neighborhood search, aiming to improve the effectiveness of solution discovery. We generated a new set of benchmark instances to assess the MILP formulation’s efficiency and the proposed metaheuristic algorithm and conducted comprehensive numerical experiments.

Reallocation of chemotherapy appointments in a large health system using a mixed integer linear programming approach

ABSTRACT Outpatient chemotherapy scheduling has significant implications for both patients and health systems. Consideration of treatment location preference is important for patient satisfaction and outcomes, and it is a complex decision impacted by travel distance. In health systems with one treatment site that stands out from the rest as a destination medical center (the primary site), there are financial and resource utilization incentives to free up as much space as possible for appointments at that site. In this study, we demonstrate that leveraging the underutilized health system sites allows decompression of appointment volume at the primary site, and it takes full advantage of valuable resources such as oncology nurses and chair availability. A Mixed Integer Linear Programming approach was used to develop a model under four scenarios which reallocates appointments from the primary site to other health system sites based on patient travel distance to the sites. This approach was applied to data from the Mayo Clinic Health System Minnesota region, which demonstrated that the health system has the potential to move approximately 50% of eligible appointments out of the primary site, resulting in an overall volume change of approximately 30%. Implications for scheduling policies and infrastructure are discussed.

Precision game engineering through reshaping strategic payoffs

Nash equilibrium is a key concept in game theory fundamental for elucidating the equilibrium state of strategic interactions, with applications in diverse fields such as economics, political science, and biology. However, the Nash equilibrium may not always align with desired outcomes within the broader system. This article introduces a novel game engineering framework that tweaks strategic payoffs within a game to achieve a pre-defined desired Nash equilibrium while averting undesired ones. Leveraging mixed-integer linear programming, this framework identifies intricate combinations of players and strategies and optimal perturbations to their payoffs that enable the shift from undesirable Nash equilibria to more favorable ones. We demonstrate the effectiveness and scalability of our approach on games of varying complexity, ranging from simple prototype games such as the Prisoner’s Dilemma and Snowdrift games with two or more players to complex game configurations with up to 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${10}^{6}$$\\end{document} entries in the payoff matrix. These studies showcase the capability of this framework in efficiently identifying the alternative ways of reshaping strategic payoffs to secure desired Nash equilibria and preclude undesired equilibrium states. Our game engineering framework offers a versatile toolkit for precision strategic decision-making with far-reaching implications across diverse domains.

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.

Tscluster: A python package for the optimal temporal clustering framework

Temporal clustering extends the conventional task of data clustering by grouping time series data according to shared temporal trends across sociospatial units, with diverse applications in the social sciences, especially urban science. The two dominant methods are as follows: Time Series Clustering (TSC), with dynamic cluster centres but static labels for each entity, and Sequence Label Analysis (SLA), with static cluster centres but dynamic labels. To implement the universe of models spanning the design space between TSC and SLA, we present tscluster, an open-source Python framework. tscluster offers: (1) several innovative techniques, such as Bounded Dynamic Clustering (BDC), that are not available in existing libraries, allowing users to set an upper bound on the number of label changes and identify the most dynamically evolving time series; (2) a user-friendly interface for applying and comparing these methods; (3) globally optimal solutions for the clustering objective by employing a mixed-integer linear programming formulation, enhancing the reproducibility and robustness of the results in contrast to existing methods based on initialization-sensitive local optimization; and (4) a suite of visualization tools for interpretability and comparison of clustering results. We present our framework using a case study of neighbourhood change in Toronto, comparing two methods available in tscluster. Supplemental materials provide an additional case study of local business development in Chicago and a detailed mathematical exposition of our framework. tscluster can be installed via PyPI (pypi.org/project/tscluster), and the source code is accessible on Github (github.com/tscluster-project/tscluster). Documentation is available online at the tscluster website (tscluster.readthedocs.io).

Including Lifetime Hydraulic Turbine Cost into Short-Term Hybrid Scheduling of Hydro and Solar

In traditional short-term hydropower scheduling problems, which usually determine the optimal power generation schedules within one week, the off-design zone of a hydraulic turbine is modeled as a forbidden zone due to the significantly increased risk of turbine damage when operating within this zone. However, it is still plausible to occasionally operate within this zone for short durations under real-world circumstances. With the integration of Variable Renewable Energy (VRE) into the power system, hydropower, as a dispatchable energy resource, operates complementarily with VRE to smooth overall power generation and enhance system performance. The rapid and frequent adjustments in output power make it inevitable for the hydraulic turbine to operate in the off-design zone. This paper introduces the operating zones associated with various production costs derived from fatigue analysis of the hydraulic turbine and calculated based on the turbine replacement cost. These costs are incorporated into a short-term hybrid scheduling tool based on Mixed Integer Linear Programming (MILP). Including production costs in the optimization problem shifts the turbine’s working point from a high-cost zone to a low-cost zone. The resulting production schedule for a Hydro-Solar hybrid power system considers not only short-term economic factors such as day-ahead market prices and water value but also lifetime hydraulic turbine cost, leading to a more comprehensive calculation of the production plan. This research provides valuable insights into the sustainable operation of hydropower plants, balancing short-term profits with lifetime hydraulic turbine costs.

Contemporary approaches in matheuristics an updated survey

AbstractMatheuristics are problem independent frameworks that use mathematical programming tools to obtain high quality heuristic solutions. They are structurally general enough to be applied to different problems with little adaptation to their abstract structure, so they can be considered as new or hybrid metaheuristics based on components derived from the mathematical model of the problems of interest. In this survey, we emphasize the mathematical tools and describe how they can be used to design heuristics. We focus on mixed-integer linear programming and report representative examples from the literature of how it has been used for effective heuristic optimization. References to contributions to matheuristics deriving from neighboring research areas such as Artificial Intelligence or Quantum Computing are also included. We conclude with some ideas for possible future developments. This paper extends an original version published in 4OR with new sections on CMSA, Incremental Core, AI hybrids and Quantum Heuristics, and includes references to several recent publications.

A multi-level rescheduling approach for a dynamic remote operations scheduling problem

ABSTRACT In this paper, we tackle a dynamic scheduling problem faced by a large international company. The problem involves assigning installation projects arriving over time to specialised technicians who execute them remotely. Each project consists of several tasks having processing times, release dates, and execution deadlines. The company needs to assign projects to technicians and schedule tasks complying with technicians' skills, precedence constraints between tasks, and tasks requiring multiple technicians simultaneously. The problem is dynamic as new projects and tasks become available over time, requiring their allocation to technicians. We formulate the offline problem as a mixed integer linear program that minimises the makespan, and we address the dynamic version solving restricted problems within a rolling horizon framework. The approach systematically implements different levels of schedule adjustment to incorporate new information. To study scalability, we validate our algorithm by using both real-world and synthetic simulations demonstrating its efficiency and effectiveness. Additionally, we provide interesting managerial insights for the company.

Operational optimization of a rural multi-energy system supported by a joint biomass-solid-waste-energy conversion system and supply chain

Biomass, as one of the most accessible renewable resources, could reduce dependency on fossil fuels and mitigate associated environmental impacts. With economic development and improved living standards, biomass in rural areas is converted into various forms of energy to meet high energy demands. However, existing studies on rural multi-energy system operations assume energy production from biomass is a given input parameter and consider it to be a fixed constant or subject to upper and lower bounds, independent of the biomass supply chain. Indeed, biomass energy production is significantly influenced by the collection, transportation, storage, and other processes within the supply chain. This work proposes a mixed integer linear program (MILP) for the joint optimization of an integrated rural multi-biomass-solid waste energy conversion system with a source-grid-demand-based biomass-solid waste supply chain (IRMBS-BSC system). The IRMBS-BSC system encompasses three sub-systems: the biomass-solid waste source sub-system, the biomass-solid waste grid sub-system, and the biomass-solid waste demand sub-system. Various biomass-solid wastes are classified, and a source model is established to evaluate their availability. Subsequently, a supply chain model is developed within the biomass-solid waste grid sub-system. Based on the available and transported quantity from the biomass-solid waste source sub-system and grid sub-system, a multi-energy system operation model, considering refined biomass conversion processes, is proposed. A case study involving a multi-energy system that integrates multiple rural community biomass-solid waste supplies is conducted. The results demonstrate that the proposed MILP, which considers the biomass-solid waste supply chain, can reduce energy imports during the load peak periods and can also decrease the overall operational costs of the multi-energy system.

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.

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.

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.

Exact methods for the Selective Assessment Routing Problem

AbstractThe Selective Assessment Routing Problem (SARP) is a problem in humanitarian logistics addressing the site selection and routing decisions of rapid needs assessment teams which aim to evaluate the post-disaster conditions of different community groups, each carrying a distinct characteristic. The aim is to construct an assessment plan that maximizes the covering of different characteristics in a balanced way. We explore exact approaches based on mixed integer linear programming. Different mathematical formulations are presented, and theoretical results regarding their strengths are derived. The models are experimentally evaluated on a set of test instances including a real-world scenario.

Comparative Analysis of Conventional (Financial, Network) and Optimized Mixed Integer Linear Programming Scenarios for Telecom Network Investment Decision Making

The telecommunications industry in Indonesia faces numerous challenges in allocating resources for network development, including network complexity, technological changes, market competition, and customer expectations. To achieve optimal financial outcomes and customer satisfaction, investments must be well-managed, well-placed, and well-executed. This study analyses network investment portfolio selection outcomes at Telkomsel; Indonesia's largest cellular operator, comparing conventional scenarios (financial and network) with an optimized scenario using Mixed Integer Linear Programming (MILP). The research evaluates these scenarios based on total portfolio score, incremental revenue, and Internal Rate of Return (IRR). Financial indicators such as net present value (NPV), IRR, EBIT margin, incremental revenue, and network indicators such as capacity and customer satisfaction (download/upload throughput, latency, packet loss, jitter) assess investment feasibility. The MILP optimization scenario strongly correlates with incremental revenue, NPV, and IRR, indicating higher financial performance. Sites selected in the MILP optimization outperscenario formed others in total score (22% better than the financial scenario, 63% better than the network scenario), incremental revenue (3.5% better than the financial scenario, 90.2% better than the network scenario), and portfolio IRR (4% better than the financial scenario, 70% better than the network scenario).

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.

A hybrid local search algorithm for the continuous energy-constrained scheduling problem

AbstractWe consider the continuous energy-constrained scheduling problem (CECSP). A set of jobs has to be processed on a continuous, shared resource. A schedule for a job consists of a start time, completion time, and a resource consumption profile. The goal is to find a schedule such that each job does not start before its release time, is completed before its deadline, satisfies its full resource requirement, and respects its lower and upper bounds on resource consumption during processing. The objective is to minimize the total weighted completion time. We present a hybrid local search approach, using simulated annealing and linear programming, and compare it to a mixed-integer linear programming (MILP) formulation. We show that the hybrid local search approach matches the MILP formulation in solution quality for small instances and is able to find a feasible solution for larger instances in reasonable time.

Optimal Placement and Sizing of Distributed Generation in Electrical DC Distribution Networks Using a Stochastic Mixed-Integer LP Model

AbstractThis paper presents a stochastic mixed-integer linear mathematical model for finding the optimal placement and sizing of distributed generation in a DC distribution network, considering the uncertainty of electrical demand and distributed renewable sources. The proposed model accurately represents the original mixed-integer nonlinear model, obtaining a globally optimal solution in less computational time with low errors. The mathematical model allows for considering constraints related to the maximum limits for the penetration of distributed generation, such as those specified by Resolution CREG 174 of 2021. Furthermore, the uncertainties of the electrical demand, wind energy-based distributed generation (DG), and solar energy-based DG are considered in the mathematical models using a two-stage stochastic programming approach. The accuracy and efficiency of the proposed model were tested and validated on a 21-node DC test system from the specialized literature, and the effectiveness and robustness were assessed on a 69-node DC test system. The obtained results show that the proposed stochastic mixed-integer linear mathematical model performs well.

Minimizing the earliness–tardiness for the customer order scheduling problem in a dedicated machine environment

AbstractThe customer order scheduling problem has garnered considerable attention in the recent scheduling literature. It is assumed that each of several customer orders consists of several jobs, and each customer order is completed only if each job of the order is completed. In this paper, we consider the customer order scheduling problem in a machine environment where each customer places exactly one job on each machine. The objective is to minimize the earliness–tardiness, where tardiness is defined as the time an order is finished past its due date, and earliness is the time a job is finished before its due date or the completion time of the corresponding order, whichever is later. Even though the earliness–tardiness criterion is an important objective for just-in-time production, this problem has not been studied in the context of the customer order scheduling problem. We provide a mixed-integer linear programming (MILP) formulation for this problem and derive multiple problem properties. Furthermore, we develop six different heuristics for this problem configuration. They follow the structure of the iterated greedy algorithm and additionally use a refinement function in which they differ. In a computational experiment, the algorithms were compared with each other and outperformed a solver solution of the MILP, which proves their ability to efficiently solve the problem configuration.