Chance-constrained Programming Research Articles

Bridging chance-constrained and stochastic optimization for risk estimation of virtual energy hubs dominated by hybrid vehicles under diverse uncertainties: To improve economic sustainability

Virtual energy hubs (VRHBs) combine dispersed energy resources to enhance the efficient use of various energy sources, grid flexibility, and renewable energy adoption. This paper investigates the adoption of VRHBs with power-to-gas (P2G) facilities and hybrid electric-compressed natural gas vehicles (HNGEVs) in the presence of renewable resources to effectively tackle carbon emissions and increase the flexibility of VRHBs. The proposed P2G facility is constructed from carbon capture to collect carbon pollution from fossil fuel resources and HNGEVs to recycle into synthesis gas in a methanation plant. Furthermore, due to severe uncertainties due to renewable resources and energy consumption intensity, the proposed framework utilizes stochastic-driven chance-constrained optimization, which enables decision-makers in VRHBs to account for the probabilistic nature of uncertainties. Chance-constrained optimization explicitly considers uncertainty through probabilistic constraints, providing a more robust and reliable solution compared to deterministic methods. By modeling uncertainties using chance-constrained programming, VRHBs can manage the economic risks under unexpected worst realizations in uncertainties. Based on the results, the P2G facility leads to a significant reduction in the daily emissions by 63.06% and 42.77% observed under no-risk and full-risk conditions, respectively. In addition, the suggested chance-constrained method resulted in a major decrease of 69% in overall operational costs.

A chance-constrained net revenue model for online dynamic predictive maintenance decision-making

Most models for predictive maintenance (PdM) decision-making focus on the expected value of a system performance metric (e.g., the expected cost rate). However, focusing solely on such expected values may overlook the significant risks of resulting PdM decisions. Indeed, it would be practically valuable to make the optimal PdM decision by considering the least probability of achieving a target system performance. Furthermore, emphasizing only maintenance costs without accounting for operational revenues can render the maintenance strategies unappealing, particularly when the revenues across system lifecycles are significant. This study introduces a method that employs a stochastic net revenue model and chance-constrained programming to address these issues in PdM decision-making, wherein the preventive maintenance cost is proportional to the system's current degradation state. By exemplifying system degradation with the Gamma process, we derive the probability density function of stochastic net revenue and present a Bayesian method to simultaneously update the drift and diffusion parameters of the Gamma process model. The effectiveness and practical applicability of our proposed chance-constrained net revenue model and parameter updating method are demonstrated through a numerical example and a case study.

Network-level pavement maintenance and rehabilitation planning considering uncertainties using chance-constrained programming

ABSTRACT In practice, given limited funds, to consider multiple strategic goals/objectives that different stakeholders concern, pavement network-level maintenance and rehabilitation (M&R) planning becomes a multi-objective optimisation (MOO) based project selection and budget allocation problem. In an attempt to solve this problem, most agencies established MOO models under the deterministic situation without appropriate consideration of uncertainties. However, ignoring performance uncertainties often leads to unreasonable decisions. To provide more convincing and reliable pavement M&R decisions, this paper proposes a Chance-Constrained Programming (CCP) based MOO method to incorporate performance uncertainties in network-level single period pavement M&R planning. First, a general deterministic MOO model with budget and network performance constraints is established. Then, three commonly-used statistical forms of network-level performance measures are introduced. To incorporate uncertainties, the probability distribution of each form of performance measure is derived. Based on the CCP method, the MOO model is transformed to an equivalent deterministic formulation as a mixed non-linear integer programming (MNLIP) problem. To demonstrate the proposed method, a case study using real data is conducted. The results show that the proposed method can effectively help decision-makers to appropriately incorporate performance uncertainties in conducting network-level pavement M&R planning.

The Stochastic Transportation Problem with Imprecise Data using Lomax Distribution

The stochastic transportation problem with imprecise data is a probabilistic chance-constrained programming (CCP) problem in which the objective function is fuzzy and the supply and demand are random. Three models for the STP with ID with mixed-type restrictions that follow the Lomax distribution (LD) are created in this research. Optimizing the transportation cost in FTP under probabilistic mixed constraints is the goal of the research project. To do this, the probabilistic mixed constraints are transformed into deterministic form using the LD, and the cost coefficient of the fuzzy objective function is changed with alpha cut representation. Numerical examples are presented to demonstrate the suggested models.

Chance-constrained programs with convex underlying functions: a bilevel convex optimization perspective

Chance-constrained programs with convex underlying functions: a bilevel convex optimization perspective

A CCP-based distributed cooperative operation strategy for multi-agent energy systems integrated with wind, solar, and buildings

To explore the bidirectional interaction between renewable energy and buildings in multi-agent energy systems, this paper proposes a distributed cooperative operation strategy for multi-agent energy systems integrated with wind, solar, and buildings based on chance-constrained programming (CCP). First, the multi-agent energy system integrated with wind, solar, and buildings is comprehensively modeled with detailed electric and thermal characteristics for flexibility enhancement. Then for maximizing the profits of the cooperative energy system and each engaged agent, a Nash bargaining model is presented and is divided into two subproblems: the coalition income and the power payment. To preserve the privacy of agents, the adaptive alternating direction method of multipliers (ADMM) is exploited to solve both subproblems. Meanwhile, the CCP method is applied to address diverse uncertainties from wind and solar power generation as well as outdoor temperature. Finally, the effectiveness of the proposed strategy is validated. The simulation results show that, besides the privacy of information among all agents being well preserved, our strategy enhances the profits not only for the energy system but also for all engaged agents.

Distributionally robust chance-constrained optimization with Gaussian mixture ambiguity set

Conventional chance-constrained programming methods suffer from the inexactness of the estimated probability distribution of the underlying uncertainty from data. To this end, a distributionally robust approach to the problem allows for a level of ambiguity considered around a reference distribution. In this work, we propose a novel formulation for the distributionally robust chance-constrained programming problem using an ambiguity set constructed from a variant of optimal transport distance that was developed for Gaussian Mixture Models. We show that for multimodal process uncertainty, our proposed method provides an effective way to incorporate statistical moment information into the ambiguity set construction step, thus leading to improved optimal solutions. We illustrate the performance of our method on a numerical example as well as a chemical process case study. We show that our proposed methodology leverages the multimodal characteristics from the uncertainty data to give superior performance over the traditional Wasserstein distance-based method.

A comparative study of fuzzy multi-objective investment project portfolio selection and optimization based on financial return and different risk measurements

Competitiveness in the global market is getting more intense. Due to resource and budget constraints, firms need to achieve their expected goals and satisfy all investment constraints under uncertainty. Selecting the set of projects among other candidates to get the most efficient portfolio requires a lot of attention from the Decision Makers (DMs) as this consideration no longer relies purely on the financial term. This problem becomes a multi-objective problem under uncertainty where the financial return and risk from uncertainty are required into the trading off consideration. Due to the financial uncertainty, the chance-constrained programming has been employed in this study for defuzzifying and solving uncertain optimization problems at a specified confidence level that is defined by the DMs. Then, various kinds of investment or financial risk measures, Lower-Semi Variance Index (LSVI), the absolute deviation with the expected FNPV, and the absolute mean-Conditional Value at Risk (CVaR) gap are provided in the selection of such risk measures to show their differences in characteristics and performances in the obtained results. Since, such problems can consist of many project candidates and complex constraints, which may grow beyond the application of the exact optimization approach, a meta-heuristic method, Genetic Algorithm (GA), is introduced to optimize this problem through designing and constructing a decision support tool for the investment portfolio selection and optimization. The applicability of the proposed comparative approach and the constructed tool are illustrated through examples.

Learning Fair Policies for Multi-Stage Selection Problems from Observational Data

We consider the problem of learning fair policies for multi-stage selection problems from observational data. This problem arises in several high-stakes domains such as company hiring, loan approval, or bail decisions where outcomes (e.g., career success, loan repayment, recidivism) are only observed for those selected. We propose a multi-stage framework that can be augmented with various fairness constraints, such as demographic parity or equal opportunity. This problem is a highly intractable infinite chance-constrained program involving the unknown joint distribution of covariates and outcomes. Motivated by the potential impact of selection decisions on people’s lives and livelihoods, we propose to focus on interpretable linear selection rules. Leveraging tools from causal inference and sample average approximation, we obtain an asymptotically consistent solution to this selection problem by solving a mixed binary conic optimization problem, which can be solved using standard off-the-shelf solvers. We conduct extensive computational experiments on a variety of datasets adapted from the UCI repository on which we show that our proposed approaches can achieve an 11.6% improvement in precision and a 38% reduction in the measure of unfairness compared to the existing selection policy.

A coordinated operation method for networked hydrogen-power-transportation system

Hydrogen fuel cell vehicles have been promoted as a complement to electric vehicles (EVs) to facilitate the decarbonization of transportation networks (TNs). The coordinated operation of a networked hydrogen-power-transportation system with distributed hydrogen supplies is proposed in this research. To maximize the synergistic effect, TN's couplings in hydrogen transport delays and refueling/charging demand are further integrated into the conventional hydrogen-power system. The objective is to maximize the total profits, subject to both the coupling and network constraints. To present the storage effect of hydrogen tube trailers with delays, an extended discrete user equilibrium (UE) is developed and the traditional UE is adopted to formulate refueling-charging demand couplings. Considering the hydrogen supply process, it especially involves the constraints of hydrogen production, storage, utilization, and dispensation. A data-driven robust chance-constrained programming is provided to account for multiple uncertainties from energy/traffic demand and renewable power. Simulation results of the 48-node system show that the proposed model improves the total profits.

Multi-objective optimization scheduling method considering flexible load integration for distribution network

In response to a series of issues in the distribution network, such as inadequate and inflexible utilization of flexible loads, delayed response to demand participation, and the uncertainty of new energy source output, a differentiated objective-based method for optimizing distribution network operations is proposed. Firstly, flexible loads are categorized, and corresponding mathematical models are established. Secondly, by employing chance-constrained programming to account for the uncertainty in new energy source output, a multi-objective optimization model is developed to reduce distribution network economic costs, decrease network losses, and enhance power supply reliability. Subsequently, an improved NSGA-III algorithm is introduced to address the multi-objective model. Finally, an 11-node distribution network is used as a case study, and three different algorithms are comprehensively compared. This confirms the rationality of the optimized scheduling scheme proposed in this paper.

Chance-Constrained Optimization for a Green Multimodal Routing Problem with Soft Time Window under Twofold Uncertainty

This study investigates a green multimodal routing problem with soft time window. The objective of routing is to minimize the total costs of accomplishing the multimodal transportation of a batch of goods. To improve the feasibility of optimization, this study formulates the routing problem in an uncertain environment where the capacities and carbon emission factors of the travel process and the transfer process in the multimodal network are considered fuzzy. Taking triangular fuzzy numbers to describe the uncertainty, this study proposes a fuzzy nonlinear programming model to deal with the specific routing problem. To make the problem solvable, this study adopts the fuzzy chance-constrained programming approach based on the possibility measure to remove the fuzziness of the proposed model. Furthermore, we use linear inequality constraints to reformulate the nonlinear equality constraints represented by the continuous piecewise linear functions and realize the linearization of the nonlinear programming model to improve the computational efficiency of problem solving. After model processing, we can utilize mathematical programming software to run exact solution algorithms to solve the specific routing problem. A numerical experiment is given to show the feasibility of the proposed model. The sensitivity analysis of the numerical experiment further clarifies how improving the confidence level of the chance constraints to enhance the possibility that the multimodal route planned in advance satisfies the real-time capacity constraint in the actual transportation, i.e., the reliability of the routing, increases both the total costs and carbon emissions of the route. The numerical experiment also finds that charging carbon emissions is not absolutely effective in emission reduction. In this condition, bi-objective analysis indicates the conflicting relationship between lowering transportation activity costs and reducing carbon emissions in routing optimization. The sensitivity of the Pareto solutions concerning the confidence level reveals that reliability, economy, and environmental sustainability are in conflict with each other. Based on the findings of this study, the customer and the multimodal transport operator can organize efficient multimodal transportation, balancing the above objectives using the proposed model.

Stochastic Modelling Frameworks for Dragon Fruit Supply Chains in Vietnam under Uncertain Factors

Managing uncertainties and risks is always a difficult but fascinating task in fresh fruit supply chains, especially when dealing with the strategy for the production and conveyance of fresh fruit in Vietnam. Following the COVID-19 outbreak, the confluence of economic recession and persistent adverse weather conditions has exacerbated challenges faced by dragon fruit cultivators. This research investigates a two-stage stochastic programming (TSSP) approach which is developed and served as a valuable tool for analyzing uncertainties, optimizing operations, and managing risks in the fresh fruit industry, ultimately contributing to the sustainability and resilience of supply chains in the agricultural sector. A prototype is provided to illustrate the complex and dynamic nature of dragon fruit cultivation and consumption in Vietnam. Data on the selling prices of dragon fruit were collected from several sources between 2013 and 2022 in Binh Thuan Province, Vietnam. The results were obtained from the model by using three different approaches in order of their versatility and efficacy: (1) Scenario tree generation; (2) Sample average approximation; (3) Chance-constrained programming.

The restaurant delivery problem with uncertain cooking time and travel time

Eliminating delays under uncertain delivery conditions presents a formidable challenge for meal delivery platforms. Nevertheless, by implementing effective measures, these platforms can successfully mitigate the impacts of uncertainties on delays. This paper introduces a scenario-based chance-constrained programming model tailored for meal delivery services with uncertain cooking time and travel time, ensuring a more accurate representation of real-world scenarios. The objective is to minimize the delivery cost while considering the decision preferences of a platform. To generate high-quality solutions, this work proposes a novel island harmony search algorithm that incorporates several key components, including greedy search, harmony selection, neighborhood reduction, island migration, and scenario-based simulation. Additionally, a matheuristic is introduced for the purpose of comparison. Comparative evaluation against a set of adopted benchmarks reveals the superior performance of both algorithms. A series of sensitivity analyses are conducted to explore the advantages of considering uncertainty, and investigate how decision preference values influence the feasibility of solutions for meal delivery platforms. The proposed formulation and algorithm yield high-quality solutions for delivery platforms with various preferences, facilitating decision-making that aligns with their operational context.

An ordered multi-objective fuzzy stochastic approach to sustainable water resources management: a case study from Taiyuan City, China

Abstract The uncertainty of socioeconomic development and climate change poses challenges to the sustainable management of water resources in Taiyuan, China. The study proposes a type-2 fuzzy chance-constrained ordered multi-objective fractional programming (T2FCC-ORMOFP) model to address the allocation of water under uncertainties. The model incorporates type-2 fuzzy programming and chance-constrained programming into an overall framework to handle multiple uncertainties under multi-level and multi-objective conflicts, while using fractional programming to reflect the marginal benefits of the system. It prioritizes the principles of optimal, fair, and stable distribution by the upper-level governing authorities, while considering the social and economic environmental benefits of lower-level decision-making. The optimal water allocation scenarios were obtained under different hydrological guarantee rates, violation levels of water supply constraints, and net economic benefits of type-2 fuzzy numbers for 2030. Additionally, groundwater is replaced by reclaimed water in varying proportions for flexible water supply. The results show that the water demand in Taiyuan cannot be ignored, and the risk of water shortage is more sensitive for the agricultural and industrial sectors. The recycled water substitution strategy can optimize the water supply structure and improve social and economic benefits (9–12%). Also T2FCC-ORMOFP can improve overall efficiency (20%) compared with a single-level model.

Two-stage chance-constrained programming based on Gaussian mixture model and piecewise linear decision rule for refinery optimization

The two-stage chance-constrained program (CCP) is studied for a refinery optimization problem. In stage-I, the refinery decision-makers determine the type and quantity of crude oil procurement under operational uncertainties to maximize the expected profit under all possibilities. In stage-II, process unit flowrates are adjusted based on the realized uncertainties and available crude oil, while introducing probabilistic constraints to manage the off-spec risk. To solve such a two-stage optimization problem, we propose a novel approach using Gaussian mixture model (GMM) to characterize uncertainties, and piecewise linear decision rule for stage-II operations. Comparing to the conventional scenario-based mixed-integer linear program (MILP), our new approach offers three advantages. First, it leverages a well-developed global optimization scheme for joint CCP to avoid scenario generation and potential bias. Second, the data-driven GMM enables CCP to handle uncertainties with general distributions. Third, the stage-II variables are parameterized via Gaussian component induced piecewise linear decision rule to strike an excellent trade-off between optimality and computational time. A simplified refinery plant, consisting of distillation, cracker, reformer, isomerization, and desulfurization units, is used as a test bed to demonstrate the superiority of the proposed optimization method in solution time, probabilistic feasibility, and optimality over the large-scale scenario-based MILP.

A stochastic simulation-based chance-constrained programming model for optimizing watershed best management practices for nonpoint source pollution control under uncertainty

In this study, a stochastic simulation-based chance-constrained programming (SSCP) model is developed by integrating soil and water assessment tool (SWAT), generalized likelihood uncertainty estimation (GLUE), chance-constrained programming (CCP) and genetic algorithm (GA) for managing non-point source (NPS) pollution under uncertainty. The SSCP model improves upon conventional simulation-based optimization methods by effectively (i) reflecting the propagation of stochastic uncertainties from the SWAT to the optimization framework by the joint GLUE and CCP (GLUE-CCP) approach, and (ii) identifying the optimal implementation levels of best management practices (BMPs) under different constraint-violation risk levels. The SSCP is solved using a GA with parallel sampling technique after conversion into the corresponding deterministic version. The model is applied to determine the optimal types, sizes and locations of BMPs for NPS pollution control in the Hanjiang River basin, China. The results indicate that, under low-risk conditions, grassed waterways and filter strips should be implemented in the northern part of the basin and larger detention ponds of about 637.8 ∼ 850.4 ha should be installed in the northern and eastern parts of the basin; whereas under high-risk conditions, filter strips should be installed in the northeastern part of the basin and larger detention ponds of about 562.1 ∼ 749.5 ha should be installed in the northern and southwestern parts of the basin. Moreover, as the risk level decreases, stricter control of constraint violations leads to higher BMP implementation costs in order to mitigate the risk of excessive NPS pollution loads. Thus, the proposed model provides valuable insights into the selection of BMP placement schemes at various risk levels, facilitating the reduction of NPS pollution at the outlet of the Hanjiang River basin to acceptable levels while considering the tradeoff between system cost and risk.

A Type-2 fuzzy hybrid preference optimization methodology for electric vehicle charging station location

This work focuses on a hybrid preference-based electric vehicle charging station location problem, which considers multiple optimization preferences of distribution network operators, charge station owners, and electric vehicle users. The problem is formulated by an uncertain mixed-integer programming model. Due to the multi-fold uncertainty of the charging process, the uncertain model parameters are expressed as Type-2 fuzzy variables (T2-FVs). The critical value-based type reduction method is adopted to handle the high computational complexity. The proposed uncertain model is converted to its equivalent deterministic chance-constrained programming model. The deterministic counterpart is solved by General Algebraic Modeling System (GAMS). At last, numerical simulations are performed to demonstrate the proposed location strategy as well as some sensitivity analyses. The results indicate that for any given parameters, the equivalent deterministic model follows the general form of mixed-integer programming one that can be easily solved by GAMS. The proposed methodology can effectively handle the multi-fold uncertainty of the charging process. Compared with crisp models, the proposed location strategy can provide more robust location decisions for electric vehicle charging stations (EVCSs). In addition, we also found that different interest groups have conflicting preferences for the locations of EVCSs, so considering multiple hybrid optimization preferences is necessary.

A novel two-phase approach for a sustainable p-hub median problem with multiple service levels and carbon cap-and trade policy under uncertainty

ABSTRACT To reduce the carbon emissions for society's sustainable development and satisfy customers' personalized delivery service, this paper addresses the cap-and-trade policy and multiple service levels into a p-hub median problem. In the hub network designing, single-assignment patterns between hubs and demand nodes, the choice of transportation mode for each O-D pair and the uncertainties of carbon emission and price are taken into account. Based on a fuzzy chance-constrained programming approach, a fuzzy mathematical model under an expected cost-minimization context with some fuzzy constraints is first presented. Then a two-phase approach is proposed to solve the proposed model. Finally, in view of the comparative analysis with small- and large-scale Turkish instances, serval managerial insights are finally obtained for government and logistics companies in the reduction of carbon emission and the hub network's total designing cost for the sustainable development.

Multi-objective migrating birds optimization for solving stochastic home health care routing and scheduling problems considering caregiver working time constraints

Currently, many countries are suffering from a heavy burden caused by population aging since the elderly occupy a large number of medical resources. Home health care (HHC) is a prospective approach to relieve and cope with this issue. This study addresses a multi-objective HHC routing and scheduling problem (HHCRSP) considering uncertainties and caregivers’ working time balance. First, a stochastic programming method is employed to formulate the uncertain service time and travel time, and then a chance-constrained programming model is developed to minimize the total operation cost and penalty cost of violating allowable time windows. Second, a multi-objective migrating birds optimization incorporating a stochastic simulation method is proposed. An ensemble of population iteration and external archive search is designed to obtain high-quality solutions. A stochastic simulation strategy is devised to evaluate the feasibility of obtained solutions and calculate their objective values. At last, numerical experiments are implemented for solving an extending benchmark set. The comparisons and discussions verify the better performance of the proposed algorithm compared with its peers.