Robust Counterpart Model Research Articles

A distributionally robust optimization approach of multi-park integrated energy systems considering shared energy storage and Uncertainty of Demand Response

To enhance the economic efficiency and renewable energy integration capacity of multi-park integrated energy systems (MPIES) and address the issue of insufficient consideration of demand response uncertainty in existing studies, this paper proposes a distributionally robust optimization approach for multi-park integrated energy systems, considering shared energy storage and the uncertainty of demand response. First, models of the shared energy system and demand response are established. Based on these models, a deterministic optimization scheduling model for MPIES is developed, aiming to minimize system costs while considering constraints such as grid power balance. To address uncertainty in demand response, this paper employs Interval Type-2 Fuzzy Theory to construct uncertainty sets and considers system reliability constraints. The original optimization problem is then transformed into an equivalent robust counterpart model, and the optimal distributionally robust solution is obtained through parameter domain decomposition methods. Finally, the proposed method is validated using the IEEE 33-bus system. The results show that considering shared energy storage and demand response individually can reduce total system costs by 4.86 % and 26.46 %, respectively. After accounting for the uncertainty in demand response, the total system cost increases only slightly by 4.36 %, but this improves the system's robustness.

Robust pricing and inventory decisions in ship‐from‐store omnichannel operations

AbstractThis work studies the deployment of the ship‐from‐store omnichannel strategy and the pricing and inventory decisions for an online retailer. Robust optimization models are constructed for the online‐only and the ship‐from‐store modes under a budgeted uncertainty set. The ARIMA model is used to predict the parameter values of the budgeted uncertainty set using historical demand data. The closed‐form optimal solution for the online‐only mode is obtained. The robust counterpart model for the ship‐from‐store mode is converted to a mixed integer quadratic programming model. Numerical studies are conducted to validate the theoretical results and to verify the effectiveness and practicality of the developed robust optimization solution approach. The results show that adopting a ship‐from‐store strategy may hurt the retailer's profit if a significant proportion of consumers are time‐sensitive with high travel cost. The ship‐from‐store strategy is optimal if it significantly boosts market growth.

A matheuristic-oriented iterated greedy algorithm for multi-mode resource-constrained project scheduling problem under uncertainty

Multi-mode resource-constrained project scheduling problem is one of the most important branches of combinatorial optimization problems. The aim is to find a feasible solution consisting of activity start time and execution mode so that the project makespan is minimized. However, there are a lot of uncertain events, that bring great challenges to project scheduling. We first propose a mixed integer linear programming model where the activity duration is uncertain. Then two uncertain parameters are introduced to describe the disturbance degree of uncertain activity duration and the allowable violation degree of constraints respectively, and the MILP model with uncertain activity duration is converted to its deterministic robust counterpart model. A matheuristic local optimization approach is proposed to balance computational time with the optimization of the computational result. A matheuristic-oriented iterated greedy algorithm that combines a matheuristic local optimization approach and an iterated greedy algorithm is proposed to solve this problem in this paper. Experimental results indicate that the robust counterpart model can obtain robust optimal solutions for small-scale instances in accepted computational time. Meanwhile, the proposed matheuristic-oriented iterated greedy algorithm can obtain robust near-optimal solutions for large-scale instances and is superior to other compared algorithms.

Robust Optimization of Vaccine Distribution Problem with Demand Uncertainty

This study proposes a multi objective optimization model for vaccine distribution problems using the Maximum Covering Location Problem (MCLP) model. The objective function of the MCLP model in this study is to maximize the fulfillment of vaccine demand for each priority group at each demand point. In practice, the MCLP model requires data on the amount of demand at each demand point, which in reality can be influenced by many factors so that the value is uncertain. This problem makes the optimization model to be uncertain linear problem (ULP). The robust optimization approach converts ULP into a single deterministic problem called Robust Counterpart (RC) by assuming the demand quantity parameter in the constraint function is in the set of uncertainty boxes, so that a robust counterpart to the model is obtained. Numerical simulations are carried out using available data. It is found that the optimal value in the robust counterpart model is not better than the deterministic model but is more resistant to changes in parameter values. This causes the robust counterpart model to be more reliable in overcoming uncertain vaccine distribution problems in real life. This research is limited to solving the problem of vaccine distribution at a certain time and only assumes that the uncertainty of the number of requests is within a specified range so that it can be developed by assuming that the number of demand is dynamic.

En-route charge scheduling for an electric bus network: Stochasticity and real-world practice

Bus electrification, with its high energy conversion efficiency and zero tailpipe emissions, has a significant impact on addressing the petroleum crisis and lowering carbon emissions in urban transportation. However, long charging times and uncertain operational risks hinder their wider adoption. This paper explores the robust en-route charge scheduling problem for electric buses to overcome uncertain energy consumption while ensuring charging accessibility. The problem is first described as a deterministic mixed-integer linear programming model that employs a time-of-use electricity price to balance the charging demand. Additionally, a robust counterpart model is formulated to account for stochastic energy. Numerical studies are designed to evaluate the proposed model in a real network, followed by a sensitivity analysis to investigate the impact of battery type, fleet composition, and depth of discharge on the charging schedule. The results show that the robust optimization model provides system feasibility against uncertainty with a comparable price and emissions.

Robust optimization method of emergency resource allocation for risk management in inland waterways

This study proposes a robust optimization method for waterborne emergency resource allocation in inland waterways that addresses the uncertainties and mismatches between supply and demand. To accomplish this, we integrate the risk evaluation of maritime with a robust optimization model and employ the Entropy Weighted Method (EWM)-Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)-Analytic Hierarchy Process (AHP) method to evaluate the risk of various areas. The approach enables exploration of the relationship between maritime risk and emergency resource allocation strategy. The robust optimization method is used to deal with uncertainty and derive the robust counterpart of the proposed model. We establish an emergency resource allocation model that considers both the economy and timeliness of emergency resource allocation. We construct an optimization model and transform it into an easily solvable robust counterpart model. The results demonstrate that the proposed method can adapt to real-world scenarios, and effectively optimize the configuration effect while improving rescue efficiency under reasonable resource allocation. Specifically, the proportion of rescue time saved ranges from 28.52% to 92.60%, and the proportion of total cost saved is 95.82%. Our approach has significant potential to provide a valuable reference for decision-making related to emergency resource allocation in maritime management.

Data-driven robust optimization for a multi-trip truck-drone routing problem

Using drones along with conventional vehicles, such as trucks, can potentially improve the cost-effectiveness and speed of delivery operations. This research considers a multi-trip truck-drone routing problem under uncertainty. Each truck is allowed to take multiple trips from the depot,while each drone is allowed to perform a single trip from the launch locations. Based on customer preferences, deliveries are grouped into deliveries by trucks to distribution centers and deliveries by drones from distribution centers to customers. A mixed-integer linear programming model is formulated to minimize the sum of the waiting times of all customers. To deal with uncertainty, a new two-stage clustering algorithm that uses multiple-kernel learning-based and single-kernel learning-based methods is introduced to construct uncertainty sets. In the case of column-wised uncertainty, a two-stage clustering with a dimensional separation algorithm is developedto avoid over-conservatism.A two-step solution method is provided for solving the proposed robust model to reduce the CPU time. The performance of the proposed algorithms is evaluated on several test problems. The results indicate that the developed algorithms can effectively avoid over-conservatism, reduce variables and constraints of the data-driven robust counterpart model, and reduce the CPU time required to solve the problem.

Robust optimization for a dynamic emergency materials supply chain network under major infectious disease epidemics

ABSTRACT The COVID-19 outbreak has posed serious challenges to logistics and supply chain management. This study examines how living materials can be provided to a population under quarantine control due to the spread of infectious diseases. First, we present a modified SEIR model for predicting the number of people and materials demanded. Considering the dynamic evolution and other characteristics of infectious diseases, and the resulting demand uncertainty, we develop a robust optimization model for a multi-period dynamic emergency materials supply chain network with the objective of minimising total cost. We then transform it into a solvable robust counterpart model, propose a dynamic adjustment strategy, and cope with it using the Rolling Horizon approach to obtain dynamic location and materials distribution decisions. Finally, we conduct a case study of the Shanghai epidemic in 2022 to verify the feasibility and effectiveness of our model and method.

Location and inventory pre-positioning problem under uncertainty

In humanitarian logistics, the location and inventory pre-positioning problem (LIPP), making decisions on facility location, inventory pre-positioning, and allocation, is critical to post-disaster rescue efficiency. Since post-disaster conditions are complex and uncertain, it is challenging for planners to make LIPP decisions and perform rescue activities. To reduce operational costs while also guaranteeing service levels to customers, it is necessary to take uncertainties into account when making decisions. This study explores a service-oriented LIPP under two types of uncertainties, i.e., uncertain customer demand and third-party supply. We propose a two-stage robust optimization framework to solve the problem, considering the service-level constraints and minimizing the total cost. We develop two algorithms to solve the problem. Specifically, a column-and-constraint generation algorithm is utilized to solve the original two-stage robust model, and an improved row generation algorithm is developed to tackle the affinely adjustable robust counterpart model. Simulation results show that both algorithms can provide satisfying solutions. A case study based on a power grid company in China is conducted to perform sensitivity analysis and compare models, producing several managerial insights.

Robust scheduling of EMU first-level maintenance in a stub-end depot under stochastic uncertainties

Frequently occurring and unavoidable perturbations in railway systems cause a reduction in performance and even the infeasibility of the original schedule. In addition, existing related studies assume that the maintenance routes are fixed in a deterministic situation. Therefore, it is necessary to focus on robust scheduling of electric multiple units first-level maintenance with flexible maintenance routes to defend against stochastic uncertainties. Firstly, a mixed-integer linear programming model with uncertain parameters in objective functions and constraints is built for the first time. Multiple constraints are formulated to consider flexible maintenance routes, train shunting conflicts, and track occupation conflicts in a stub-end depot. A robust optimization approach is adopted to obtain a deterministic robust counterpart model with uncertainties in processing/arrival/departure times. In this model, uncertainty and reliability levels are introduced to describe and quantify the disturbance degree of processing/arrival/departure times and the allowable violation degree of resource constraints respectively. Moreover, an adaptive iterative local search is proposed by embedding heuristic rules for the initialization. The proposed algorithm includes problem-specific neighborhood structures to improve the evolutionary ability, a variable neighborhood descent method to guide and drive the search toward promising areas of the search space, and an adaptive perturbation mechanism to enable the algorithm to jump out of local optimum. Numerical results from China’s railway system validate the proposed model and quantitatively demonstrate the merit of the proposed algorithm. Further analysis shows that our approach can achieve an appropriate trade-off between robustness and efficiency for various uncertainty and reliability levels.

Bus Bridging for Rail Disruptions: A Distributionally Robust Fuzzy Optimization Approach

Dealing with uncertain rail disruptions effectively raises a significant challenge for computational intelligence research. This article studies the bus bridging problem under demand uncertainty, where the passenger demand is represented as parametric interval-valued fuzzy variables and their associated uncertainty distribution sets. A distributionally robust fuzzy optimization model is proposed to minimize the maximum travel time and to search for the optimal scheme for vehicle allocation, route selection, and frequency determination. To solve the proposed robust model, we discuss the computational issues concerning credibilistic constraints, turning the robust counterpart model into computationally tractable equivalent formulations. The proposed approach is verified, and the resulting method is validated with a report on uncertain parameters in a real-world disrupted event of Shanghai Rail Line 1. Experimental results show that the distributionally robust fuzzy optimization approach can provide a better uncertainty-immunized solution.

Robust goal programming approach for healthcare network management for perishable products under disruption

In this paper, a robust goal programming model for a healthcare network design problem under disruption is developed. The objective of the proposed multi-period, multi-echelon model is to simultaneously optimize the total deviation from the considered three goals, including minimizing network cost, maximizing network coverage, and maximizing network reliability. Distribution uncertainty of healthcare demand and facility capacity is captured. The perishability of healthcare products is regarded with limited shelf-life. Using an option contract ensures appropriate coordination with suppliers during pre and post-disaster periods. Via handling the tractable robust budget-based counterpart model, suitable strategies are proposed to present the designed network's efficiency and effectiveness. Risk mitigation strategies deal with disrupted demand and supply in the healthcare network with specific impacts on coverage, cost, and reliability of the services. The proposed cooperate coverage mechanism provides appropriate service coverage under disruption propagation. Finally, the option contract has achieved cost-efficient coordination under demand uncertainty. Furthermore, practical managerial insights are demonstrated by investigating a real-life case study and conducting extensive sensitivity analyses.

Robust Counterpart Models for Fresh Agricultural Product Routing Planning Considering Carbon Emissions and Uncertainty

Cold chain transportation guarantees the quality of fresh agricultural products in people’s lives, but it comes with huge environmental costs. In order to improve transportation efficiency and reduce environmental impact, it is crucial to quantify the routing planning problem under the impact of carbon emissions. Considering fixed costs, transportation costs, and carbon emission costs, we propose a mixed integer linear programming model with the aim of minimizing costs. However, in real conditions, uncertainty poses a great challenge to the rationality of routing planning. The uncertainty is described through robust optimization theory and several robust counterpart models are proposed. We take the actual transportation enterprises as the research object and verify the validity of the model by constructing a Benders decomposition algorithm. The results reveal that the increase in uncertainty parameter volatility forces enterprises to increase uncontrollable transportation costs and reduce logistics service levels. An increase in the level of security parameters could undermine the downward trend and reduce 1.4% of service level losses.

Worst-case analysis of Omega-VaR ratio optimization model

The Omega ratio, a performance measure that separately considers upside and downside deviations from a fixed threshold, improves the Sharpe ratio by incorporating the higher-order moments. In this paper, we analyse the performance of a robust optimization model based on maximizing the worst-case of Omega ratio by taking its threshold point as the robust value of a defined percentile of the underlying loss distribution. To this aim, the threshold point is computed as the worst-case of the value-at-risk at a particular confidence level. We formulate robust model of the proposed strategy under two cases of uncertainty sets, the mixture distribution uncertainty and the box uncertainty. We show that, in the first case, the problem reduces to a second-order cone program (SOCP) and, in the second one, to a semi-definite program (SDP), hence tractable in both the cases. We conduct a comprehensive empirical investigation of the proposed models over six data sets across the globe, namely BSE 100 (India), FTSE 100 (UK), Hang Seng (Hong Kong), S&P Asia 50 (Asia), Dow Jones Industrial Average (USA), and IBEX (Spain). We compare our models with the three variants of the Omega ratio model, one its robust variant taking worst-case conditional value-at-risk as threshold point, and two of its nominal variants using value-at-risk and conditional value-at-risk as threshold points, respectively. We find that the proposed model under mixture distribution uncertainty exhibits a better performance over most of the data sets and scenarios than its CVaR-based robust counterpart. Under the box set, the proposed model performs similar or generates mixed results compared to its CVaR-based robust counterpart model. We also note that both the proposed robust models save investors against the risk of losses over the bearish phase of market in comparison to their nominal counterparts. Finally, on comparing the proposed model under mixture distribution uncertainty with the box uncertainty, the former model is found to be more suitable for optimistic investors, whereas the later strategy is more ideal for pessimistic investors.

Robust collaborative optimization for train timetabling and short-turning strategy in urban rail transit systems

This paper investigates the robust collaborative train timetabling optimization problem by integrating short-turning strategy and train circulation plan while considering the uncertainty in passenger demand. The passenger dynamics equations are proposed to represent the relationship among headways, short-turning services, and passenger load. A mixed-integer nonlinear programming (MINLP) model is constructed to balance train utilization and stranded passengers. The extra variables are introduced to linearize the nonlinear constraints and convert the original model into a robust counterpart model according to the strong duality theory. Finally, two numerical experiments are carried out to verify the validity of the train timetabling model integrating with short-turning strategies. The study shows that the proposed strategy can better support the balance between stranded passengers and train utilization compared to other regular strategies. Moreover, the results indicate that robust strategies perform well in the trade-off between the optimality and the level of conservatism of the solutions.

Contribution of robust optimization on handling agricultural processed products supply chain problem during Covid-19 pandemic

This research aims to show how decision sciences can make a significant contribution on handling the supply chain problem during Covid-19 Pandemic. The paper discusses how robust optimization handles uncertain demand in agricultural processed products supply chain problems within two scenarios during the pandemic situation, i.e., the large-scale social distancing and partial social distancing. The study assumes that demand and production capacity are uncertain during a pandemic situation. Robust counterpart methodology is employed to obtain the robust optimal solution. To this end, the uncertain data is assumed to lie within a polyhedral uncertainty set. The result shows that the robust counterpart model is a computationally tractable through linear programming problem. Numerical experiment is presented for the Bandung area with a case on sugar and cooking oil that is the most influential agricultural processed products besides the main staple food of the Indonesian people, rice.

A Mixed Integer Linear Programming Support Vector Machine for Cost-Effective Group Feature Selection: Branch-Cut-and-Price Approach

Recently, cost-based feature selection has received significant attention due to its great ability to achieve promising prediction accuracy at a minimum feature acquisition cost. To further improve its predictive and economic performances, this research proposes a cost-effective 1-norm support vector machine with group feature selection as GFS-CESVM1. Its robust counterpart model, GFS-RCESVM1, is also introduced to address the cost uncertainty of features and feature groups because cost variation commonly exists in real-world problems. The proposed models are formulated as Mixed Integer Linear Programming (MILP). To efficiently solve the proposed SVM MILP models, we develop a Branch-Cut-and-Price (BCP) algorithm that considers only a limited number of variables and/or constraints, which thereby leads to rapid convergence to an optimal solution. Various experimental results on benchmark and synthetic datasets demonstrate that GFS-CESVM1 can achieve competitive outcomes by considering not only individual feature evaluation but also group structural information among features. The GFS-RCESVM1 can identify the subset of features that is immune to cost uncertainty and therefore provide feasible and optimal solutions. Furthermore, our BCP algorithm can dominantly outperform the ordinary BB algorithm for finding better objective value and integrality gap within a short period of time.

Robust strategic planning of dynamic wireless charging infrastructure for electric buses

Electromobility in public bus systems is growing rapidly and experiencing a fundamental transformation in their infrastructure and operations. The dilemma of limited driving range and charging time of battery electric buses (BEBs) hinders their adoption. A novel approach to address BEB limitations is to utilize dynamic wireless charging (DWC) technology that allows buses to charge while in motion. This paper aims to analyze robust strategic planning of DWC and BEB fleet scheduling based on a real bus network at Binghamton University. The problem is first formulated as a new deterministic mixed-integer linear programming model to simultaneously optimize both the charging planning problem and fleet scheduling problem in an integrated fashion. To address the uncertainty of energy demand and charging time, a robust counterpart model (RCM) has been derived. To increase RCM flexibility, the battery status variable is formulated in a cumulative form. Dependent and independent budget uncertainty sets have been developed to control the robustness. A sensitivity analysis has been conducted to study the system behavior in response to different charging types, auxiliary energy demand, depth of discharge, charging options at terminals, battery degradation, and electricity cost. The deterministic model shows that eight homogeneous BEBs are required to operate on the selected routes with a battery capacity of 54.01 kWh and a total cost of $3,636,347. The results show that joint planning of charging infrastructure and fleet scheduling can save 19.2% of total cost compared to disjoint planning. The RCM results in 10 BEBs to ensure feasiblility against uncertainty.

Designing a food supply chain strategy during COVID-19 pandemic using an integrated Agent-Based Modelling and Robust Optimization

Coronavirus disease (COVID-19) has spread for over a year and affected many aspects, including the food supply chain. One of the ways COVID-19 has impacted the food supply chain is the food production capacity reduction. It is necessary to develop the optimum food supply chain strategy by determining the optimum food hub location and food network to maintain food security which robust against disruptions and uncertainties. In this study, Robust Optimization (RO) is applied to handle the uncertainties. Nevertheless, the actual uncertain data might be hard to be collected or even unavailable at the moment. Therefore, an innovative framework is proposed to integrate RO with Agent-Based Modelling (ABM). ABM is used to simulate the upstream actor of the food supply chain and predict the uncertain food production capacity, which RO later handles. Particularly, this study focused on rice supply chain. The result shows that the framework is able to handle the uncertain rice supply chain problem, in which the actual uncertain data might be unavailable, and give the robust optimum food hub location and food network. The food hub location and food network are obtained by solving the Robust Counterpart (RC) model with respect to the uncertainty set obtained from the ABM simulation result.

Two-Stage Robust Counterpart Model for Humanitarian Logistics Management

In the early stages of a major public emergency, decision-makers were troubled by the timely distribution of a large number of donations. In order to distribute caring materials reasonably and efficiently, considering the transportation cost and time delay cost, this paper takes the humanitarian logistics management as an example to study the scheduling problem. Based on the actual situation of insufficient supply during the humanitarian logistics management, this paper using optimization theory establishes a two-stage stochastic chance constrained (TS-SCC) model. In addition, due to the randomness of emergency occurrence and uncertainty of demand, the TS-SCC model is further transformed into the two-stage robust counterpart (TS-RC) model. At the same time, the validity of the model and the efficiency of the algorithm are verified by simulations. The result shows that the model and algorithm constructed are capable to obtain the distribution scheme of caring materials even in worst case. In the TS-BRC (with box set) model, the logistics service level increased from 89.83% to 93.21%, while in the TS-BPRC (with mixed box and polyhedron set) model, it increases from 90.32% to 94.96%. Besides, the model built in this paper can provide a more reasonable dispatching plan according to the actual situation of caring material supply.