Robust Mixed-integer Linear Programming Research Articles

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.

Robust coordinated optimization of the thickening-dewatering process under uncertain feeding conditions

The thickening-dewatering process is extensively used in mineral processing process, but there is usually strong uncertainty in the feeding condition of the thickener. In this work, we present a scenario tree-based mixed-integer linear programming (MILP) method for addressing the coordinated optimization of the thickening-dewatering process under uncertain feeding conditions. First, we developed a prediction model using partial least squares (PLS) to estimate the underflow concentration and operation time of each piece of equipment. Subsequently, we formulated a robust MILP model to minimize the energy consumption of the process. This model incorporates several constraints, including starting time, operation time, shutdown time, safety limits, and ladder electricity prices for each piece of equipment. Finally, we divided the uncertain feeding condition into three scenarios to solve the robust coordinated optimization problem. Case studies show that the proposed method can effectively mitigate constraint violations, and reduce safety risks.

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 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.

Multi-Objective Bus Timetable Coordination Considering Travel Time Uncertainty

This paper proposes a timetable coordination method for transfer problems in a bus transit system. With a given bus network, a stochastic mixed-integer linear programming (MILP) model has been formulated to obtain coordinated bus timetables with the objective of minimizing a weighted sum of the average value of total waiting time and its average absolute deviation value, allowing for random bus travel time. The vital decision variable is the terminal departure offset time of each target bus trip within a certain off-peak period. The robust MILP model can also be used to solve the first-bus transfer problem with the introduction of several new linear constraints. A solution method based on the Monte Carlo simulation has been developed to solve the MILP model. Numerical experiments have been conducted for different scenarios. The results indicate that bus timetables coordinated by the developed model are capable of substantially reducing waiting time for transfer and non-transfer passengers. In addition, the feasibility of simplifying a common sub-route into a single transfer stop in a timetable coordination problem has been explored based on numerical experiments.

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.

Robust trajectory-constrained frequency control for microgrids considering model linearization error

Grid supportive modes integrated within inverter-based resources can improve the frequency response of renewable-rich microgrids. The synthesis of grid supportive modes to guarantee frequency trajectory constraints under a predefined disturbance set is challenging but essential. To tackle this challenge, a numerical optimal control (NOC)-based control synthesis methodology is proposed. Without loss of generality, a wind-diesel fed microgrid is studied, where we aim to design grid supportive functions in the wind turbine. In the control design, linearized models are used, and the linearization-induced errors are quantitatively analyzed by reachability and interval arithmetics and represented in the form of interval uncertainties. Then, the NOC problem can be formulated into a robust mixed-integer linear program. The control structure is strategically configured into two levels to realize online deployment. The proposed control is verified on the modified 33-node microgrid with a full-order three-phase nonlinear model in Simulink. The simulation results show the effectiveness of the proposed control paradigm and the necessity of considering linearization-induced uncertainty.

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.

Optimal configuration of polygeneration plants under process failure, supply chain uncertainties, and emissions policies

The COVID-19 pandemic exacerbated the erratic demand, supply, and prices of energy. It is unlikely that these effects would subside post-pandemic, especially with the pre-existing climate change crisis that also needs to be addressed. Emissions policies aimed at mitigating climate change place economic pressures on already disrupted energy systems, which could worsen energy insecurity. Configuring disrupted energy systems to build robustness to supply chain-related uncertainties and economic pressures of emissions policies are desired to simultaneously address these problems. To this end, this study introduces a robust mixed-integer linear program that simultaneously incorporates the abovementioned needs for configuring energy production systems. The proposed model is tested through a demonstrative case study that deals with a biomass-based polygeneration plant design problem. The scenario analysis and sensitivity test on the model concerning the case under consideration yields the following results: (1) setting ambitious target profits reduces the probability of the resulting plant configuration to achieving the set targets in the presence of supply chain-related uncertainties, while conservative targets promote the opposite; (2) the inoperability of the plant’s process units reduces the robustness of optimal process configurations, and drastic configurations may be required to achieve targets despite the inoperability of process units; (3) a hybrid cap-and-trade and emissions tax policy yields approximately similar implications to the robustness of the resulting optimal plant configurations compared to a pure cap-and-trade policy, but the rate of decrease in robustness with respect to the initial emissions cap is lesser in the hybrid policy than in the pure cap-and-trade policy.

Robust optimization of sustainable closed-loop supply chain considering carbon emission schemes

In the context of the circular economy, this paper focuses on developing an optimal sustainable Closed-Loop Supply Chain (CLSC) by considering such Carbon Emission Schemes (CESs) as carbon cap, carbon cap-and-trade, and carbon tax. A risk-based robust mixed-integer linear programming is formulated using a scenario-based Conditional Value-at-Risk (CVaR) to deal with demand uncertainty. The results reveal that the model performance is affected by both the Decision-Maker's (DM's) risk-aversion and the CESs. However, in all the CESs, the more conservative the DM, the lower the cost performance, and the greater the cost/solution robustness. The carbon cap-and-trade policy not only has the best cost-performance but, ignoring the very small amount of cost robustness deficit compared to the carbon cap model, can also be considered acceptable in the case of robustness. The cost-saving that the carbon cap-and-trade policy brings to the optimistic DM is greater than that brings to the pessimistic one. However, different levels of the DM's risk aversion have a negligible effect on the cost increase of the carbon tax model compared to the carbon cap one.

A sustainable robust optimization model to design a sugarcane-based bioenergy supply network: A case study

Reasons such as volatile fossil fuel prices, environmental limitations, and swift economic growth have increased the demand for clean renewable energy. Bioenergy has been received much attention from researchers due to its various benefits compared to other renewable energy alternatives. In this paper, a three-stage approach is developed to design a sustainable sugarcane-based bioenergy supply chain network under an uncertain environment. The fuzzy data envelopment analysis method is employed in the first stage to determine the appropriate regions for opening the sugarcane fields according to climatic, ecological, and social criteria. The nominated regions are then integrated into a mathematical optimization model as candidate locations for sugarcane fields to configure the proposed supply chain design. This stage helps reduce the computational complexity and provides a reasonable solution by excluding the inappropriate regions. In the second stage, a robust mixed-integer linear programming model (MILP) is formulated to optimize a set of strategic and tactical decision variables. The enviromental impacts from CO2 equivalent emissions (CO2-eq emissions), water, and energy consumption in all echelons of the supply chain are incorporated in the developed model. This model is capable of providing a steady design against different expected scenarios. In the last stage, an experimental analysis is conducted, to examine the performance of the developed model in comparison with an expected scenario model. A real case study in Iraq is applied to check the applicability of the developed model. The results demonstrated the superiority of the proposed robust model over the expected scenario model in terms of the mean and standard deviation of their objective functions by 18% and 51% respectively.

A Rolling-Horizon Approach for a Surgery Case Scheduling Problem with Sterilizing Constraints

In this paper, we study a real scheduling problem which consists in scheduling a set of elective surgical cases that arrive over the time taking into account the uncertainties on their duration, and the delay to sterilize surgical instruments. The objectives are to schedule as many surgeries as possible, and to minimize the overtime of the surgical block staff and the number of instruments processed in emergency in the sterilizing unit. This research was performed in collaboration with the University Hospital of Angers in France, which has also provided historical data for the experiments. We propose a rolling horizon approach based on the iterative resolution of a robust mixed integer linear programming model, in which the objective functions are taken into account in a lexicographic order. The experiments on real data show a reduction of more than 50% of overtime, and around 87% less stress situations in the sterilizing unit.

Convalescent plasma bank facility location-allocation problem for COVID-19

With convalescent plasma being recognized as an eminent treatment option for COVID-19, this paper addresses the location-allocation problem for convalescent plasma bank facilities. This is a critical topic, since limited supply and overtly increasing cases demand a well-established supply chain. We present a novel plasma supply chain model considering stochastic parameters affecting plasma demand and the unique features of the plasma supply chain. The primary objective is to first determine the optimal location of the plasma banks and to then allocate the plasma collection facilities so as to maintain proper plasma flow within the network. In addition, recognizing the perishable nature of plasma, we integrate a deteriorating rate with the objective that as little plasma as possible is lost. We formulate a robust mixed-integer linear programming (MILP) model by considering two conflicting objective functions, namely the minimization of overall plasma transportation time and total plasma supply chain network cost, with the latter also capturing inventory costs to reduce wastage. We then propose a CPLEX-based optimization approach for solving the MILP functions. The feasibility of our results is validated by a comparison study using the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) and a proposed modified NSGA-III. The application of the proposed model is evaluated by implementing it in a real-world case study within the context of India. The optimized numerical results, together with their sensitivity analysis, provide valuable decision support for policymakers.

Decentralized Decision System for Closed-Loop Supply Chain: A Bi-Level Multi-Objective Risk-Based Robust Optimization Approach

This paper proposes a novel risk-based robust mixed-integer linear programming to design a decentralized closed-loop supply chain. The model is formulated as an uncertain bi-level multi-objective programming with multiple suppliers, manufacturers, and distributors, as the leader, and recovery, recycling, and disposal centers as the follower. A Scenario-based Conditional Value-at-Risk is employed to capture the demand uncertainty. The Karush-Kuhn-Tucker approach, ε−constraint, and LP-metric are leveraged to deal with the complexity of the bi-level coordination, and the multi-objectivity of the leader and follower. The performance of the model is compared with the performance of the deterministic decentralized model and the corresponding multi-objective model designed for the centralized system in both the robust and deterministic modes. Results indicate better performance of robust approaches compared to deterministic approaches. The decentralized approach provides better performance for the cost-sensitive decision-maker, especially the optimistic one, and those who are sensitive to social parameters prefer the centralized approach.

A robust MILP and gene expression programming based on heuristic rules for mixed-model multi-manned assembly line balancing

Current dynamic markets require manufacturing industries to organize a robust plan to cope with uncertain demand planning. This work addresses the mixed-model multi-manned assembly line balancing under uncertain demand and aims to optimize the assembly line configuration by a robust mixed-integer linear programming (MILP) model and a robust solution generation mechanism embedded with dispatching rules. The proposed model relaxes the cycle time constraint and designs robust sequencing constraints and objective functions to ensure the line configuration can meet all the demand plans. Furthermore, two solution generation mechanisms, including a task-operator-sequence and an operator-task-sequence, are designed. To quickly find a suitable line configuration, a gene expression programming (GEP) approach with multi-attribute representation is proposed to obtain efficient dispatching rules which are ultimately embedded into the solution generation mechanisms. Experimental results show that solving the proposed MILP model mathematically is effective when tackling small and medium-scale instances. However, for large instances, the dispatching rules generated by the GEP have significant superiority over traditional heuristic rules and those rules mined by a genetic programming algorithm.

Robust self-scheduling of parking lot microgrids leveraging responsive electric vehicles

The penetration of plug-in electric vehicles is increasing in power systems all over the world. Due to imperfect data about the availability of electric vehicles, the uncertainty of the demand-side increases considerably. To overcome the problem, the power storage capacity of electric vehicles is addressed to provide demand flexibility for the supply-side. This paper proposes a novel structure for parking lot microgrids to provide day-ahead peak-shaving and valley-filling for power systems with distinct peak hours in the daily operation. The microgrid is comprised of electrical demands, e.g. heat ventilation and air conditioning, lighting, escalators, as well as self-generation facilities, including roof-top photovoltaic sites and gas-fired engines. Besides, the parking lot has smart charging stations to charge/discharge the electric vehicles based on the flexibility requirements of power systems and/or owners’ preferences. The microgrid is supplied by the wholesale electricity market and bilateral contracts. To leverage the flexibility potentials of electric vehicles, a data-driven approach is suggested that classifies the electric vehicles based on distinct characteristics, e.g. dwell time and preferred state of charge. The robust optimization approach is adopted to determine the optimum procurement strategies for the microgrid in the worst-case realization of the wholesale market price uncertainty. Finally, the robust-mixed integer linear programming is examined in Iran Power Grid not only to provide flexibility for the power network but also to make a profit for the microgrid.

Robust multi-objective thermal and electrical energy hub management integrating hybrid battery-compressed air energy storage systems and plug-in-electric-vehicle-based demand response

A compressed air energy storage (CAES) can operate together with a battery energy storage system (BESS) to enhance the economic and environmental features of the energy hubs (EH). In this regard, this paper investigates their mutual cooperation in a multi-objective thermal and electrical residential EH optimization problem, which aims to diminish the total operational cost and emission. The proposed formulation also includes a PEV-based demand response program (DRP), renewable energy production units, solar heat collectors (SHE), thermal energy storage (TES), and hot water storage. Additionally, the CAES is operated as a combined heat and power unit in discharging and simple cycle modes. A linearized battery degradation cost model is integrated into the cost objective function, which ensures global optimization. Moreover, the inherent uncertainties of wind speed, solar irradiation, residential loads, electricity market price and PEVs’ load demand are handled by the computationally effective robust mixed-integer linear programming (RMILP) method. The PEVs’ uncertain load demand is obtained from their corresponding arrival–departure time, daily traveled miles, and vehicle type. The optimal Pareto front of the multi-objective problem is obtained by ε-constrained method for different robustness adjustments. Different deterministic and robust case studies are designed to evaluate the functionality of the proposed formulation.