Polyhedral Uncertainty Sets Research Articles

Robust Optimization Model for Internet Shopping Online Problems with Endorsement Costs in the Fashion Industry

Online business is a business activity carried out via the internet or digitally. Buying, selling, and advertising are done online through e-commerce, social media, or online shops. The products offered vary, including services, food, household needs, and fashion. Selling online is not limited by time and distance, and consumers can obtain information about products and services that can influence their decisions. At the same time, sellers also have the opportunity to advertise their products in a broader range by making endorsements. An endorsement is a form of advertising using well-known figures who are recognized, trusted, and respected by people. In this thesis, a model for optimizing the problem of online internet shopping with endorsement fees is formulated. This optimization model aims to maximize the profits gained by sellers in marketing their products online. In marketing products, there is uncertainty in the number of requests. To overcome this uncertainty, an approach is needed that can handle this uncertainty, namely Robust optimization. The Robust optimization model is solved using the polyhedral uncertainty set approach, resulting in a computationally tractable optimal solution. Keywords: internet shopping online; endorsement costs; robust optimization.

Robust Optimization Model for Internet Shopping Online Problems with Endorsement Costs in the Fashion Industry

Online business is a business activity carried out via the internet or digitally. Buying, selling, and advertising are done online through e-commerce, social media, or online shops. The products offered vary, including services, food, household needs, and fashion. Selling online is not limited by time and distance, and consumers can obtain information about products and services that can influence their decisions. At the same time, sellers also have the opportunity to advertise their products in a broader range by making endorsements. An endorsement is a form of advertising using well-known figures who are recognized, trusted, and respected by people. In this thesis, a model for optimizing the problem of online internet shopping with endorsement fees is formulated. This optimization model aims to maximize the profits gained by sellers in marketing their products online. In marketing products, there is uncertainty in the number of requests. To overcome this uncertainty, an approach is needed that can handle this uncertainty, namely Robust optimization. The Robust optimization model is solved using the polyhedral uncertainty set approach, resulting in a computationally tractable optimal solution. Keywords: internet shopping online; endorsement costs; robust optimization.

Adaptive robust scheduling of a hydro/photovoltaic/pumped-storage hybrid system in day-ahead electricity and hydrogen markets

A hybrid energy generation system (HEGS) exploits the synergies of various energy to improve the utilization rate of renewable energy while enhancing economic efficiency. In deregulated energy markets, this study proposes a scheduling model for HEGS trading in both day-ahead electricity and hydrogen markets to maximize total profits based on price arbitrage. This HEGS is based on a real case in Southwest China and comprises three hydro units, two photovoltaic (PV) plants a pumped-storage facility, and an electrolyzer. The uncertainty of available PV power generation is modelled using polyhedral uncertainty set and then the proposed model is cast as an adaptive robust optimization (ARO) model in which the conservativeness can be controlled by the PV uncertainty budget (ΓPV). Then the column-and-constraint generation algorithm is employed to solve this model. The numerical results reveal that the total profit of HEGS trading in two markets can be increased compared to the case of only trading in the electricity market. For instance, the increase in the total profit is 1.01 % (i.e. 5091 CNY) when ΓPV = 2. Besides, the total profit is considerably affected by ΓPV. As ΓPV increases from 2 to 8 (i.e. the robustness increases), the total profit decreases from 507,045.5 CNY to 503,036.1 CNY. Moreover, sensitivity analysis of the total profit related to the pumped-storage capacity and hydrogen price are also presented.

A robust optimization framework for forest biorefineries design considering uncertainties on biomass growth and product selling prices

The dependence of biomass growth on uncontrolled environmental factors and the lack of confidence in product selling price estimation imposes challenges for the efficient design of biorefineries, especially for forest systems, which present complex and long-termed growth behavior. The present work proposes the expansion of an optimization framework for forest biorefineries design to handle uncertainties on both biomass productivity and product selling prices. A robust formulation is proposed under a box and polyhedral uncertainty set formulation allowing its conservatism degree to be controlled. A case study of a eucalyptus biorefinery in Brazil illustrates the model's capabilities. The canonical worst-case approach to uncertainties on selling prices leads to a null optimal Net Present Value (NPV) and, on biomass growth, leads to a design that uses a 70% excess of lands. Scenarios of a controlled degree of conservatism lead to designs closer to the uncertainty-free optimal NPV of 136 bi BRL.

A double-oracle, logic-based Benders decomposition approach to solve the [formula omitted]-adaptability problem

The K-adaptability problem is a special case of adaptive robust optimization with discrete recourse that aims to prepare K solutions under uncertainty, and select among them upon full knowledge of the realized scenario. We propose a novel approach to solve K-adaptability problems with linear objective and constraints, binary first-stage decision variables, second-stage objective uncertainty, and a polyhedral uncertainty set. A logic-based Benders decomposition is applied to handle the first-stage decisions in a master problem, thus the Benders subproblem becomes a min–max–min robust combinatorial optimization problem. To solve the subproblem, a double-oracle algorithm that iteratively generates adverse scenarios and recourse decisions and assigns scenarios to K-subsets of the decisions by solving p-center problems is devised. Extensions of the proposed approach to handle parameter uncertainty in both the first-stage objective and the second-stage constraints, and for integer first-stage decision variables and nonlinear functions, are also provided. We show that, under mild conditions, the proposed algorithm converges to an optimal solution and terminates in a finite number of iterations. Numerical results obtained from experiments on benchmark instances of the adaptive shortest path problem, the regular knapsack problem, the asset liability-management problem, and a generic K-adaptability problem demonstrate the performance advantage of the proposed approach when compared to state-of-the-art methods in the literature.

Systematic Literature Review on Robust Optimization in Solving Sustainable Development Goals (SDGs) Problems during the COVID-19 Pandemic

Handling uncertainty is important in decision making, especially for SDGs problems. Robust Optimization (RO) is an applied optimization method that can be employed to handle optimization under uncertain data. With SDGs problems, many uncertain data have been considered in decision making. With RO, the data uncertainties are assumed to lay within a compact, convex continuous set. There are three special sets that can be used to represent the data, i.e., box, ellipsoidal, or polyhedral uncertainty sets. These special sets lead the SDGs problems to a computationally tractable optimization model, such that the global optimal solution is attained. However, literature reviews on the application of RO in SDGs decision-making is sparse, especially during the COVID-19 pandemic period. This paper examines the following topics: (1) the purposes of studies of RO and SDGs during the COVID-19 pandemic, (2) the state-of-the-art in RO-SDGs to determine the research objectives, and (3) the SDGs type of problems that have been modeled using RO. A systematic literature review is conducted in this paper, wherein discussion is based on a PRISMA (Preferred Reporting Items for Systematic Review and Meta-Analyses) flowchart. To this end, the database reference searching conducted on the Scopus, Science Direct, and SAGE databases, is completed using the help RStudio software. The analysis was carried out on two datasets, assisted by the output visualization using RStudio software with the “bibliometrix” package, and using the ‘biblioshiny()’ command to create a link to the “shiny web interface”. In this paper, the research gap on application of RO to SDGs problems is analyzed in order to identify the research objectives, methods, and specific RO-SDGs problems. As a result, the application of RO to SDGs problems is rare; this finding provides a motivation to conduct a further study of RO and SDGs during the COVID-19 pandemic. An expansion is presented using the key phrase “Operations Research and Optimization Modeling”, or “OROM”. SDGs in Indonesia may be referenced as an example of the capacity building available through RO/OROM.

Digital twin-driven robust bi-level optimisation model for COVID-19 medical waste location-transport under circular economy

COVID-19 medical waste collection-transport system, including the location of the related facilities, transport, and disposal, is a critical component in the circular economy. To promote the circularity of the waste management system and mitigate the spread of novel coronavirus, how to optimise COVID-19 medical waste location-transport strategies remains an open but urgent issue. In this paper, a novel digital twin-driven conceptual framework is proposed to improve the strategic decision on the location of temporary disposal centres and, subsequently, the operational decision on the transport of COVID-19 medical waste in the presence of hierarchical relationships amongst stakeholders, circular economy, environmental regulations, service level, and uncertainty in infection probability. The polyhedral uncertainty set is introduced to characterise stochastic infection probability. Digital twin technology is further used to estimate the upper and lower bound of the uncertainty set. Such a problem is formulated as a digital twin-driven robust bi-level mixed-integer programming model to minimise total infection risks on the upper level and total costs on the lower level. A hybrid solution strategy is designed to combine dual theory, Karush-Kuhn-Tucker (KKT) conditions, and a branch-and-bound approach. Finally, a real case study from Maharashtra in India is presented to evaluate the proposed model. Results demonstrate that the solution strategy performs well for such a complex problem because the CPU time required to conduct all experiments is less than one hour. Under a given uncertainty level of 36 and perturbation ratio of 20%, a regional transport strategy is preferred from generation points to transfer points, while a cross-regional one is usually implemented from transfer points to disposal centres. It is of significance to determine the bound of available temporary disposal centres. Using digital technology (e.g., digital twin) to accurately estimate the amount of COVID-19 medical waste is beneficial for controlling the pandemic. Reducing infection risks relative to cost is the prioritised goal in cleaning up COVID-19 medical waste within a relatively long period.

Robust Optimal Control With Inexact State Measurements and Adjustable Uncertainty Sets

The efficacy of robust optimal control with adjustable uncertainty sets is verified in several domains under the perfect state information setting. This paper investigates constrained robust optimal control for linear systems with linear cost functions subject to uncertain disturbances and state measurement errors that are both residing in adjustable uncertainty sets. We first show that the class of affine feedback policies of state measurements are equivalent to the class of affine feedback policies of estimated disturbances in terms of their conservativeness. Then, we formulate and solve a robust optimal control problem with adjustable uncertainty sets by considering the disturbance feedback policies. In contrast to the conventional robust optimal control, where uncertainty sets are fixed and known a priori, the uncertainty sets themselves are regarded as decision variables in our design. In particular, given the metrics for evaluating the optimal size/shape of the polyhedral uncertainty sets, a bilinear optimization problem is formulated to decide the optimal size/shape of uncertainty sets and a corresponding optimal control policy to robustly guarantee that the system will respect its constraints for all admissible uncertainties. In addition, we introduce a convex approximation for the proposed scheme to provide a computationally efficient inner approximation of the original problem. The proposed scheme is illustrated by numerical simulation of a building temperature control problem to demonstrate its effectiveness.

Robust Inventory Management: A Cycle-Based Approach

Problem definition: We study the robust formulation of an inventory model with positive fixed ordering costs, where the unfulfilled demand is either backlogged or lost, the lead time is allowed to be positive, the demand is potentially intertemporally correlated, and the information about the demand distribution is limited. Methodology/results: We propose a robust cycle-based policy that manages inventory by dividing the planning horizon into nonoverlapping inventory cycles, where an order is placed at the beginning of each cycle. Our policy selects the lengths and order quantities for all inventory cycles to minimize the worst-case total cost incurred over the planning horizon. When the uncertain demand belongs to a general polyhedral uncertainty set, the decisions in our policy can be computed by solving linear programs (LPs) for the backlogging model with any lead time and the lost-sales model with zero lead time; however, the number of LPs that need to be solved grows exponentially in the length of the planning horizon. In the special case where the uncertain demand belongs to a box uncertainty set, the decisions in our policy can be computed using a dynamic programming (DP) recursion whose complexity grows polynomially in the length of the planning horizon. We also propose a one-cycle look-ahead heuristic to handle large problem instances with a general polyhedral uncertainty set. This heuristic can be applied for both the backlogging and lost-sales models with any lead time, and it only requires solving LPs whose number grows quadratically in the length of the planning horizon. Results from extensive computational experiments clearly show that both a rolling-cycle implementation of our policy and the one-cycle look-ahead heuristic have very strong empirical performance. Managerial implications: Our robust cycle-based policy and the one-cycle look-ahead heuristic are conceptually simple and can accommodate multiple realistic features in inventory management problems. They provide a very effective approach to robust inventory management, especially in the lost-sales setting. Funding: Y. Chen was supported by a start-up grant from Nanyang Technological University. C. Wang was supported by the National Natural Science Foundation of China [Grant 71802115] and the Tsinghua University Initiative Scientific Research Program. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.1168 .

A novel and efficient exact technique for integrated staffing, assignment, routing, and scheduling of home care services under uncertainty

We model and solve integrated multi-period staffing, assignment, routing, and scheduling of caregivers for home care services and obtain insights for the case under uncertainty. The goal is to construct a weekly schedule that adheres to related operational considerations and determines optimal staffing of caregivers by minimizing caregivers’ fixed- and overtime costs. For tractability, we incorporate a priori-generated visit pattern—an existing practical approach that deals effectively with hard assignment decisions. First, we propose a novel mixed-integer program (MIP) for the nominal (deterministic) problem. We then incorporate uncertainty in service and travel times and develop a robust counterpart by hybridizing interval and polyhedral uncertainty sets. Second, we show that there is a special mathematical structure within the model that allows us to develop a novel logic-based Benders branching-decomposition algorithm that systematically delays the resolution of difficult routing/ scheduling problems and efficiently solves both the nominal and robust MIP models, i.e., our solution approach for the deterministic case allowed us to solve the robust model. Using a benchmark and newly generated instances from the literature, we show that CPLEX can solve our nominal model with average optimality gap of 56%. On the other hand, our new exact technique can solve our nominal model to an average optimality gap of 5%. Third, we provide practical insights into (i) the price of robustness and (ii) the impacts of nurse flexibility and overtime. The average total cost does not increase beyond 18% than the nominal solution and the cost-savings of nurse flexibility is about 19% on average and higher than overtime’s. We also demonstrate that nurse flexibility impact is high and comparable in both cases, deterministic and stochastic.

Conic Duality for Multi-Objective Robust Optimization Problem

Duality theory is important in finding solutions to optimization problems. For example, in linear programming problems, the primal and dual problem pairs are closely related, i.e., if the optimal solution of one problem is known, then the optimal solution for the other problem can be obtained easily. In order for an optimization problem to be solved through the dual, the first step is to formulate its dual problem and analyze its characteristics. In this paper, we construct the dual model of an uncertain linear multi-objective optimization problem as well as its weak and strong duality criteria via conic duality. The multi-objective form of the problem is solved using the utility function method. In addition, the uncertainty is handled using robust optimization with ellipsoidal and polyhedral uncertainty sets. The robust counterpart formulation for the two uncertainty sets belongs to the conic optimization problem class; therefore, the dual problem can be built through conic duality. The results of the analysis show that the dual model obtained meets the weak duality, while the criteria for strong duality are identified based on the strict feasibility, boundedness, and solvability of the primal and dual problems.

Robust aircraft conflict resolution under trajectory prediction uncertainty

In air traffic control, aircraft velocity may be perturbed due to weather effects or measurements errors and affect trajectory prediction. We address the aircraft conflict resolution problem in the presence of such perturbations. We consider polyhedral uncertainty sets on aircraft velocity, develop a budgeted robust optimization approach and show that it is amenable to mixed-integer optimization. Numerical experiments reveal that perturbations of the order of 5% on aircraft velocities can be accounted for without significantly impacting performance.

A Systematic Review on Integer Multi-objective Adjustable Robust Counterpart Optimization Model Using Benders Decomposition

Multi-objective integer optimization model that contain uncertain parameter can be handled using the Adjustable Robust Counterpart (ARC) methodology with Polyhedral Uncertainty Set. The ARC method has two stages of completion, so completing the second stage can be assisted by the Benders Decomposition. This paper discusses the systematic review on this topic using the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA). PRISMA presents a database mining algorithm for previous articles and related topics sourced from Scopus, Science Direct, Dimensions, and Google Scholar. Four stages of the algorithm are used, namely Identification, Screening, Eligibility, and Included. In the Eligibility stage, 16 articles obtained and called Dataset 1, used for bibliometric mapping and evolutionary analysis. Moreover, in the Included stage, six final databases obtained and called Dataset 2, which was used to analyze research gaps and novelty. The analysis was carried out on two datasets, assisted by the output visualisation using RStudio software with the " bibliometrix" package, then we use the command 'biblioshiny()' to create a link to the “shiny web interface”. At the final stage of the article using six articles analysis, it is concluded that there is no research on the ARC multi-objective integer optimization model with Polyhedral Uncertainty Sets using the Benders Decomposition Method, which focuses on discussing the general model and its mathematical analysis. Moreover, this research topic is open and becomes the primary references for further research in connection.

Robust Optimization Model for Twitter Sentiment Analysis of PeduliLindungi Application

Technological advances during the COVID-19 pandemic in Indonesia gave rise to the PeduliLindungi application which is developed by the government to prevent the spread of COVID-19. The advantages and disadvantages of developing PeduliLindungi can be seen from the responses and opinions from users, one of which is through the Twitter. A person's opinion about PeduliLindungi based on the tweet can be classified into positive, negative, or neutral categories using a Machine Learning approach with the Support Vector Machine (SVM) algorithm. In this paper, multiobjective optimization modeling is used to maximize the performance metrics, which are the value of Accuracy, Precision, Recall, and F1-Score. The value of the performance metrics is considered to contain uncertainty factors. Therefore, the optimization problem is solved by using Robust Optimization to handle the uncertainty factor. The data uncertainty is assumed to be belongs to polyhedral uncertainty set thus the resulted robust is computationally tractable. Numerical experiment is presented to complete the discussion.

Block-Coordinate-Descent Adaptive Robust Operation of Industrial Multi-layout Energy hubs under Uncertainty

This paper presents an adaptive robust optimization approach to optimal operation of multi-layout energy hubs under uncertainty. In the first step, the multi-layout energy hub concept is presented and discussed comprehensively followed by its required energy management model, but in the deterministic form. In the next step, an adaptive robust optimization approach is developed for the energy management model of multi-layout energy hubs. The uncertainties of energy hub load as well as upstream energy market prices are considered through bounded intervals using polyhedral uncertainty sets. The proposed adaptive-robust multi-layout EHS optimizer (ARMEO) is developed as a tri-level min-max-min optimization problem which cannot be solved directly. To do so, column-and-constraint (C&C) technique is used to recast the tri-level model into a "min" master problem and a "max-min" sub-problem. However, the "max-min" sub-problem is still a bi-level model and cannot be solved directly. To cope, block coordinate descent (BCD) methodology is applied to the sub-problem to iteratively solve the "max-min" sub-problem. An industrial-based case study is conducted to show the effectiveness of the proposed model in 1) managing multi-layout energy hubs, and 2) provide immunized operational solutions against uncertainties. Based on the results, it is observed that the ARMEO model is subject to a higher operation cost (compared to deterministic model), however, the obtained operating solutions are immunized against the uncertainties. Moreover, it has been shown that the proposed multi-layout EHS model can provide reasonable operating solutions for all layouts of the system as a whole.

Robust alternative fuel refueling station location problem with routing under decision-dependent flow uncertainty

The refueling station location problem with routing (RSLP-R) is defined as a maximal coverage problem that locates alternative fuel refueling stations (AFSs) on a road network to maximize the refueled alternative fuel vehicle flows by considering the limited range of vehicles and the willingness of drivers to deviate from their paths for refueling. In this study, we introduce the robust counterpart of RSLP-R using a decision-dependent polyhedral uncertainty set. We model the flow uncertainty set using a hybrid model that comprises a hose model and individual flow bounds. To take into account the fact that vehicle flows are affected by AFS deployment decisions in their neighborhoods, we incorporate the decision-dependency notion into the flow uncertainty set. We propose two linear mixed integer programming formulations and a Benders reformulation. Our computational experiments on instances based on the road network of Belgium confirm the effectiveness of the reformulation in solving larger instances. We also report the results of experiments to assess the value of incorporating uncertainty and decision-dependency into the problem.

Two-Stage Robust Optimization Under Decision Dependent Uncertainty

In the conventional robust optimization (RO) context, the uncertainty is regarded as residing in a predeter-mined and fixed uncertainty set. In many applications, however, uncertainties are affected by decisions, making the current RO framework inapplicable. This paper investigates a class of two-stage RO problems that involve decision-dependent uncertainties. We introduce a class of polyhedral uncertainty sets whose right-hand-side vector has a dependency on the here-and-now decisions and seek to derive the exact optimal wait-and-see decisions for the second-stage problem. A novel iterative algorithm based on the Benders dual decomposition is proposed where advanced optimality cuts and feasibility cuts are designed to incorporate the uncertainty-decision coupling. The computational tractability, robust feasibility and optimality, and convergence performance of the proposed algorithm are guaranteed with theoretical proof. Four motivating application examples that feature the decision-dependent uncertainties are provided. Finally, the proposed solution methodology is verified by conducting case studies on the pre-disaster highway investment problem.

Covid-19 PPE distribution planning with demand priorities and supply uncertainties

The recent Covid-19 outbreak put healthcare resources under enormous pressure. Governments and healthcare authorities faced major challenges in securing and delivering critical supplies such as personal protective equipment (PPE) and test kits. As timely distribution of critical supplies exceeded government resources, certain sectors, negatively impacted by the pandemic, offered their storage and distribution capabilities; both helping with the crisis and creating economic revenue. We investigate the problem of optimally leveraging the capacity and efficiency of underutilized distribution networks to enhance the capability of government supply networks to meet healthcare needs for critical supplies. We model the problem as a dynamic distribution planning problem that decides on the re-purposing of storage facilities, the allocation of demand, and the timely distribution of limited PPE supplies to different jurisdictions. From a resource provider’s perspective, the goal is to maximize demand fulfillment based on priorities set out by the government, as well as maximize economic value to participating networks. As uncertainty is a prevalent feature of the problem, we adopt a robust framework due to the lack of historical data on such supply uncertainties. We provide a mixed integer programming formulation for the adversarial problem and present a cutting plane algorithm to solve the robust model efficiently under both polyhedral and ellipsoidal uncertainty sets. We build a case study for the province of Ontario, Canada, and run extensive analysis of the service and economic value trade-off, and the effects of modeling demand priorities and supply uncertainties.

A New Robust Bidding Approach for Wind Power Producers Participating in Competitive Power Markets with Correlated Market Prices

In this research, a bidding problem for a wind-power plant participating in a day-ahead power market with uncertain correlated market prices is studied. A new robust optimization approach considering correlation among uncertainty on the hourly prices in a day-ahead market is developed. This results in solutions with lower level of over-conservatism. For this purpose a new correlated polyhedral uncertainty set is introduced. To consider the uncertainty of market clearing prices and the value of power produced by wind power producer a bidding algorithm is developed. Results of the study using a robust modelling the bidding problem reveal that the appliance of the proposed model on the bidding problem for a price-taker wind power plant in a day-ahead market with uncertain correlated data leads to solutions with superior performance than that of the conventional polyhedral uncertainty sets.

Robust Multi-Period Maximum Coverage Drone Facility Location Problem Considering Coverage Reliability

This study proposes a multi-period facility location formulation to maximize coverage while meeting a coverage reliability constraint. The coverage reliability constraint is a chance constraint limiting the probability of failure to maintain the desired service standard, commonly followed by emergency medical services and fire departments. Further, uncertainties in the failure probabilities are incorporated by utilizing robust optimization using polyhedral uncertainty sets, which results in a compact mixed-integer linear program. A case study in the Portland, OR metropolitan area is analyzed for employing unmanned aerial vehicles (UAVs) or drones to deliver defibrillators in the region to combat out-of-hospital cardiac arrests. In the context of this study, multiple periods represent periods with different wind speed and direction distributions. The results show that extending to a multi-period formulation, rather than using average information in a single period, is particularly beneficial when either response time is short or uncertainty in failure probabilities is not accounted for. Accounting for uncertainty in decision-making improves coverage significantly while also reducing variability in simulated coverage, especially when response times are longer. Going from a single-period deterministic formulation to a multi-period robust formulation boosts the simulated coverage values by 57%, on average. The effect of considering a distance-based equity metric in decision-making is also explored.