Robust Linear Programming Research Articles

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Optimisation of Ammonia Production and Supply Chain from Sugarcane Ethanol and Biomethane: A Robust Mixed-Integer Linear Programming Approach

Cross regional online food delivery: Service quality optimization and real-time order assignment

Online food delivery (OFD) represents a rapidly evolving e-business application that leverages cloud computing data centers, playing a crucial role in meeting the demands of urban lifestyles. With diverse order fulfillment features and increasing expectations for service quality, the task of effectively assigning riders for timely long-distance, cross-regional deliveries presents a significant engineering challenge. Previous studies often relied on traditional rider allocation methods that fail to account for varying capacities, or they utilized non-intelligent systems that did not adequately address fluctuating order demands and service delays. In this study, we introduce a robust Mixed Integer Linear Programming (MILP) optimization framework designed to minimize the total service time and delivery cost for cross-regional orders. This framework divides a large OFD area into multiple regions and utilizes both transfer vehicles and riders to optimize deliveries. To enhance the predictive accuracy of our model, we incorporate advanced machine learning techniques. Specifically, we employ the Long Short-Term Memory (LSTM) model to forecast regional order demands accurately, reflecting the dynamic nature of the marketplace. Additionally, Extreme Gradient Boosting (XGBoost) is tailored to dynamically predict travel times from restaurants to customer locations, facilitating more precise scheduling and resource allocation within the MILP framework. These machine learning techniques significantly bolster the MILP framework by providing detailed, accurate predictions that improve decision-making processes and adaptability to real-time conditions. Acknowledging the complexity of this optimization problem, we further enhance our approach by integrating a meta-heuristic algorithm, Adaptive Large Neighbor Search (ALNS), which efficiently assigns orders to the appropriate transfer vehicles and riders within polynomial time. Our Cross Regional Online Food Delivery (XROFD) system is meticulously designed to optimize both customer satisfaction and rider incentives. Simulation experiments confirm that the XROFD system not only reduces service times and delivery costs but also markedly enhances customer satisfaction and provides superior incentives for riders, outperforming existing state-of-the-art methods.

Accurate Flow Decomposition via Robust Integer Linear Programming.

Minimum flow decomposition (MFD) is a common problem across various fields of Computer Science, where a flow is decomposed into a minimum set of weighted paths. However, in Bioinformatics applications, such as RNA transcript or quasi-species assembly, the flow is erroneous since it is obtained from noisy read coverages. Typical generalizations of the MFD problem to handle errors are based on least-squares formulations or modelling the erroneous flow values as ranges. All of these are thus focused on error handling at the level of individual edges. In this paper, we interpret the flow decomposition problem as a robust optimization problem and lift error-handling from individual edges to solution paths. As such, we introduce a new minimum path-error flow decomposition problem, for which we give an Integer Linear Programming formulation. Our experimental results reveal that our formulation can account for errors significantly better, by lowering the inaccuracy rate by 30-50% compared to previous error-handling formulations, with computational requirements that remain practical.

Research on Vegetable Pricing and Replenishment Model Based on Greedy Algorithm

This article analyzes the pricing and replenishment decisions of supermarkets, exploring the distribution patterns and correlations between different vegetable categories and individual products. On the other hand, a robust linear programming model is constructed, which can iteratively optimize the corresponding replenishment plan, significantly improving the profitability of supermarkets. This provides important practical significance for the operation and sales of vegetable products in fresh food supermarkets. Through Spearman correlation coefficient analysis, it was found that there is a close relationship between cauliflower and leafy vegetables. The correlation between various vegetable items was processed using K-means clustering method, and the correlation between each item was discussed after classifying the groups. Finally, the grey correlation analysis method was used to further explore the ranking of the correlation between each category and individual product.

Reinforcement Learning-Based Multiobjective Evolutionary Algorithm for Mixed-Model Multimanned Assembly Line Balancing Under Uncertain Demand.

In practical assembly enterprises, customization and rush orders lead to an uncertain demand environment. This situation requires managers and researchers to configure an assembly line that increases production efficiency and robustness. Hence, this work addresses cost-oriented mixed-model multimanned assembly line balancing under uncertain demand, and presents a new robust mixed-integer linear programming model to minimize the production and penalty costs simultaneously. In addition, a reinforcement learning-based multiobjective evolutionary algorithm (MOEA) is designed to tackle the problem. The algorithm includes a priority-based solution representation and a new task-worker-sequence decoding that considers robustness processing and idle time reductions. Five crossover and three mutation operators are proposed. The Q -learning-based strategy determines the crossover and mutation operator at each iteration to effectively obtain Pareto sets of solutions. Finally, a time-based probability-adaptive strategy is designed to effectively coordinate the crossover and mutation operators. The experimental study, based on 269 benchmark instances, demonstrates that the proposal outperforms 11 competitive MOEAs and a previous single-objective approach to the problem. The managerial insights from the results as well as the limitations of the algorithm are also highlighted.

A robust LP-based approach for a dynamic surgical case scheduling problem with sterilisation constraints

The purpose of this article is to investigate a practical scheduling problem in which a group of elective surgical cases are scheduled over time, while considering their unpredictable durations and potential delays in the sterilisation of surgical instruments. The primary objectives were to schedule the maximum number of surgeries and decrease overtime for the surgical staff, as well as limit the number of instruments requiring emergency sterilisation. The study was conducted in collaboration with the University Hospital of Angers in France, which also contributed historical data for the experiments. We propose two robust mixed integer linear programming models, which are then solved iteratively through a rolling horizon approach, in which the objective functions are taken into account in lexicographic order. Experiments on randomly generated instances indicated which of the two approaches had better performance. Comparison of the results for a real-world scenario involving actual planning at the hospital indicated a greater than 69% decrease in overtime, and a minimum of 92% fewer stressful situations in the sterilising unit.

Vehicle-to-grid bidding for regulation and spinning reserve markets: A robust optimal coordinated charging approach

This paper handles the problem of optimal charging control for electric vehicle (EV) aggregators in energy and ancillary service markets under uncertainties. We propose a robust optimal coordinated charging (OCC) model that formulates a robust linear programming (RLP) for the EV aggregator formed on unidirectional vehicle-to-grid (V2G) and coordinates the provision of regulation and spinning reserves. Our model considers many uncertainties in the electricity markets, like ancillary service prices and the placement signals, as well as accurate parameters associated with EV behavior, like EV availability, time of trips, and trip duration. The objective of the optimization is to maximize the aggregator's income from V2G by contributing to the ancillary services markets. The proposed robust OCC model, a robust linear problem (RLP) model, is simulated using the CPLEX solver in GAMS software. We compare our model with the existing deterministic models and show that our model increases the aggregator's revenues and then reduces the deviation between predicted and realized revenues. We propose a new and effective method for EV aggregators to bid into energy and ancillary service markets under uncertainties. Our model shifts the charging time of EVs from peak to off-peak hours, resulting in an improved final state of charge.

Integrated supply chain network design and superstructure optimization problem: A case study of microalgae biofuel supply chain

In recent years, several strategies have been developed and adopted in a bid to manage the biofuel supply chain. In this paper, a two-stage optimization model is proposed for integrated microalgae biofuel supply chain network design and superstructure optimization problems. In the first stage, the design of the carbon capture, utilization, and storage (CCUS) network is taken into account. A robust mixed integer linear programming (RMILP) model is proposed to optimize the strategic CCUS decisions, including the simultaneous selection of emission sources, capture facilitates, Carbon Dioxide (CO2) pipelines, intermediate transportation sites and storage sites, or microalgae cultivation sites. The second stage is dedicated to biorefinery superstructure optimization in order to determine the optimal/promising biorefinery configurations. The presented model is able to handle the inherent uncertainty of critical input parameters. Moreover, the results show that biodiesel production cost cannot compete with current diesel price, but it can be reduced significantly by improving biomass productivity.

A Neurodynamic Approach for Solving Robust Linear Programming under the Polyhedral Uncertainty Set

A Neurodynamic Approach for Solving Robust Linear Programming under the Polyhedral Uncertainty Set

A robust optimization model for microgrid considering hybrid renewable energy sources under uncertainties.

Hybrid renewable energy sources and microgrids will determine future electricity generation and supply. Therefore, evaluating the uncertain intermittent output power is essential to building long-term sustainable and reliable microgrid operations to fulfill the growing energy demands. To address this, we proposed a robust mixed-integer linear programming model for the microgrid to minimize the day-ahead cost. To validate the proposed model piecewise linear curve is to deal with uncertainties of wind turbine, photovoltaic, and electrical load. The proposed solution is demonstrated through a case study compared under a robust worst-case scenario, deterministic model, and max-min robust optimization that aim to find optimal robustness. So, a piecewise linear curve is considered to obtain uncertain parameters in order to deal with uncertainties and predict the day-ahead cost. This study illustrates how the Uncertainty Budget Set selection used to integrate renewable energy sources into a microgrid, which manages the energy system. Therefore, the model complexity was slightly modified by adjusting the Uncertainty Budget Set to obtain the optimal decision and control the load demand and uncertainty of renewable energy sources. The comparative results demonstrate that the proposed robust optimization can achieve high solutions under microgrid's availability and is intended to confirm that the proposed method is more cost-effective than alternative optimization techniques. Additionally, the effectiveness and advantage of the proposed methodology in the IEEE 33-node system are validated in this case study by comparing it to the existing optimization. The comparison results show that the proposed robust optimization methods illustrate the model's efficiency, concluding remarks, and managerial insights of the research.

Pricing Strategy for a Virtual Power Plant Operator with Electric Vehicle Users Based on the Stackelberg Game

With the popularity and promotion of electric vehicles (EVs), virtual power plants (VPPs) provide a new means for the orderly charging management of decentralized EVs. How to set the price of electricity sales for VPP operators to achieve a win–win situation with EV users is a hot topic of current research. Based on this, this paper first proposes a Stackelberg game model in which the VPP participates in the orderly charging management of EVs as a power sales operator, where the operator guides the EV users to charge in an orderly manner by setting a reasonable power sales price and coordinates various distributed resources to jointly participate in the power market. Furthermore, taking into account the impact of wind power output uncertainty on VPP operation, a robust optimization method is used to extend the deterministic Stackelberg game pricing model into a robust optimization model, and a robust adjustment factor is introduced to flexibly adjust the conservativeness of the VPP operator’s bidding scheme in the energy market. The model is then transformed into a robust mixed-integer linear programming (RMILP) problem solved by Karush–Kuhn–Tucker (KKT) conditions and strong dyadic theory. Finally, the effectiveness of the solution method is verified in the calculation example, which gives the optimal pricing strategy for the VPP operator, the optimal charging scheme for EV users, and the remaining internal resources’ contribution plan, providing an important idea for the VPP to centrally manage the charging behavior of EVs and improve its own operating revenue.

A Robust Optimization Model for Multi-Period Railway Network Design Problem Considering Economic Aspects and Environmental Impact

The railway network design problem is one of the critical issues in the transportation sector due to its significance and variety of necessary applications. The major issue of this field relates to the decision of whether to increase the railways’ capacity or construct a new route to meet demand. Although the budget is a great concern of the managers for making such a decision, environmental factors should be necessarily included in the decision-making process. Therefore, this research proposes a novel robust bi-objective mixed-integer linear programming (MILP) model to simultaneously minimize the total cost and environmental impact under uncertain conditions and within a given time horizon. The proposed problem addresses strategic and operational decisions through railway project selection and product flow determination. To deal with the bi-objectiveness of the model and tackle the complexity of the problem, a nondominated sorting genetic algorithm (NSGA-II) is employed. The proposed NSGA-II could reach near-optimal Pareto solutions in a reasonable solution time and showed a reliable performance for being employed in large-sized instances. It also indicates that the proposed NSGA-II can be utilized for solving large-sized samples in a very short time.

Robust Optimization for the Self-Scheduling and Bidding Strategies of a Hydroproducer Considering the Impacts of Crossing Forbidden Zones

Giant hydrounits are usually accompanied by multiple head-dependent forbidden zones (FZs). FZs intensify hydropower systems’ head effects and affect the safe and economic operations in a deregulated market environment, especially in southwest China. This paper addresses the self-scheduling and bidding strategies of a typical hydroproducer (the Longtan hydroplant, the leading plant on the Hongshuihe River, one of the most important plants in the West-East Electricity Transmission Project in China) with irregular forbidden zones (IFZs) in a day-ahead market. Considering the impacts of crossing IFZs and the head effects, a robust optimization-based approach to maximize profits was developed to establish market-oriented strategies under asymmetric price intervals and is solved by an iterative algorithm.The complicated model with nonconvex and nonlinear features was recast into a robust mixed-integer linear programming (MILP) model. Finally, the optimal self-scheduling of a hydroproducer with IFZs in southwest China was used to show the effectiveness of the proposed model. The optimization results demonstrated that the proposed model can effectively incorporate the head effects and the impacts of crossing IFZs to maximize profits; furthermore, the model significantly reduced the phenomenon of crossing IFZs and provided more executable, stable, and safe generation scheduling. Additionally, the proposed model can obtain more profitable optimal bidding strategies than the model that ignores the impacts of crossing IFZs.

IoT Edge-Computing-Enabled Efficient Localization via Robust Optimal Estimation

Source localization within wireless sensor networks (WSNs) is one of the critical technologies in the Internet of Things (IoT). As the number of network nodes increases, so does the amount of data and computational requirement. It is imperative to introduce edge computing. However, there are still two issues when running existing wireless location algorithms on edge nodes: 1) conventional low-complexity approaches are easily affected by the bias generated in complex environments, leading to low locating accuracy and 2) the optimization algorithms considering the bias have good performances, but they are calculation-efficiency low on edge nodes. This study proposes a computationally efficient and high-precision location method to tackle the troubles. Precisely, we first introduce our previous research to construct a bias-considered nonconvex problem with a linear objective. Then, we propose an angle-assisted Taylor series with zero truncation error to linearize the second-order cone (SOC) constraint in the established problem. Next, we resort to the mini-max criterion to eliminate the angular uncertainty and get a robust linear programming (LP) problem with an optimal solution. So far, we have obtained a convex problem of low complexity. To ensure the calculated efficiency of the proposed problem on edge nodes, we proceed to give the solving process of the problem. Moreover, we provide a constraints tracking mechanism to reduce the number of iterations in the solution procedure, improving computational efficiency. Simulations and experiments demonstrate that the proposed method with similar locating accuracy to state-of-the-art optimization algorithms exhibits much higher computing efficiency on edge nodes.

Development of a robust multi-objective model for green capacitated location-routing under crisis conditions

Location-Routing Problem (LRP) is a strategic supply chain design problem aimed at meeting customer demands. LRPs involve selecting one or more depot sites from a set of potential locations and determining the best routes to connect them to demand points. With the rising awareness about the environmental impacts of transportation over the past years, the use of green logistics to mitigate these impacts has become increasingly important. To compensate for a gap in the literature, this paper presents a robust bi-objective mixed-integer linear programming (MILP) model for the green capacitated location-routing problem (G-CLRP) with demand uncertainty and the possibility of failure in depots and routes. The final result of this Robust Multi-Objective Model is to set up the depots and select the routes that offer the highest reliability (Maximizing network service) while imposing the lowest cost and environmental pollution. A Nondominated Sorting Genetic Algorithm (NSGA-II) is used to solve the large-sized instances of the modeled problem. The paper also provides a numerical analysis and a sensitivity analysis of the solutions of the model.

Strong duality in parametric robust semi-definite linear programming and exact relaxations

This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.

Closing the loop of a global supply chain through a robust optimal decentralized decision support system.

This paper presents a novel decentralized decision support system to optimally design a general global closed-loop supply chain. This is done through an original risk-based robust mixed-integer linear programming that is formulated based on an initial uncertain bi-level programming. Addressing the decision-maker's (DM's) attitude toward risk, a scenario-based conditional value-at-risk is used to deal with demand and return uncertainty. Also, the Karush-Kuhn-Tucker (KKT)conditions are employed to transform the model into its single-level counterpart. The results obtained from solving a numerical example through the proposed framework are compared with those of the corresponding centralized system, which is formulated through deterministic multi-objective programming and solved by the Lp-metric method. The results show that the use of the proposed framework improves the robustness of profit, income, and cost by about 28%, 34%, and 36% on average. However, a more conservative DM faces a larger cost of robustness than an optimistic DM while experiencing a more significant improvement in the system responsiveness. Using the proposed framework, the manager can measure the advantages, disadvantages, and consequences of their decisions before their actual implementation. This is because the model is capable of establishing fundamental trade-offs among risk, cost, profit, income, robustness, and responsiveness according to the DM's attitude toward risk.

Impact of the splitting of the German–Austrian electricity bidding zone on investment in a grid-scale battery energy storage system deployed for price arbitrage with gray and green power in Austrian and German day-ahead power markets

The splitting of the German–Austrian electricity bidding zone in October 2018 established limits for cross-border exchanges of power between Germany and Austria, which led to the decoupling of German and Austrian wholesale power prices. We analyze the impact of the split on the investment in lithium-ion grid-scale battery energy storage systems (BESS) used for intertemporal arbitrage in German and Austrian day-ahead power markets with hourly contracts operated by the Energy Exchange of Austria. The paper was written in response to the increasing share of green power in the national energy mix of the two countries, followed by a similar development in the volume of green power traded. Arbitrage with green power was analyzed and compared to arbitrage with power from conventional sources using a robust mixed-integer linear programming (MILP) model extended with a battery degradation process. The model was solved for different levels of depth of discharge (DOD) to identify the optimum between cycle life and calendar aging with respect to prevailing market conditions. Using predicted costs of battery, a break-even point was reached with 60% DOD identified as the optimal pattern for the dispatch of the BESS. A separate scenario was considered for the analysis of the impact of moving from the perfect knowledge of future prices to a situation where day-ahead prices are forecasted using the seasonal autoregressive integrated moving average (SARIMA) model.

Stochastic Decomposition Method for Two-Stage Distributionally Robust Linear Optimization

In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen [Math. Oper. Res., 16 (1991), pp. 650--669] for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm include (1) it works with data-driven approximations of ambiguity sets that are constructed using samples of increasing size and (2) efficient construction of approximations of the worst-case expectation function that solves only two second-stage subproblems in every iteration. We identify conditions under which the ambiguity set approximations converge to the true ambiguity sets and show that the DRSD method asymptotically identifies an optimal solution, with probability one. We also computationally evaluate the performance of the DRSD method for solving distributionally robust versions of instances considered in stochastic programming literature. The numerical results corroborate the analytical behavior of the DRSD method and illustrate the computational advantage over an external sampling-based decomposition approach (distributionally robust L-shaped method).

Equitable post-disaster relief distribution: a robust multi-objective multi-stage optimization approach

PurposeThis paper aims to address a location-distribution-routing problem for distributing relief commodities during a disaster under uncertainty by creating a multi-stage model that can consider information updates during the disaster. This model aims to create a relief network that chooses distribution centers with the highest value while maximizing equity and minimizing response time.Design/methodology/approachA hybrid algorithm of adaptive large neighborhood search (ALNS) and multi-dimensional local search (MDLS) is introduced to solve the problem. Its results are compared to ALNS and an augmented epsilon constraint (AUGMECON) method.FindingsThe results show that the hybrid algorithm can obtain high-quality solutions within reasonable computation time compared to the exact solution. However, while it yields better solutions compared to ALNS, the solution is obtained in a little longer amount of time.Research limitations/implicationsIn this paper, the uncertain nature of some key features of the relief operations problem is not discussed. Moreover, some assumptions assumed to simplify the proposed model should be verified in future studies.Practical implicationsIn order to verify the effectiveness of the designed model, a case study of the Sarpol Zahab earthquake in 2017 is illustrated and based on the results and the sensitivity analyses, some managerial insights are listed to help disaster managers make better decisions during disasters.Originality/valueA novel robust multi-stage linear programming model is designed to address the location-distribution-routing problem during a disaster and to solve this model an efficient hybrid meta-heuristic model is developed.