Chance-constrained Programming Model Research Articles

A distribution-free-based approach for stochastic food closed-loop supply chain

ABSTRACT Resource scarcity has driven growing interest in circular economy (CE). Closed-loop supply chain (CLSC) with returnable transport items (RTIs) in the food industry is an important component of CE. However, existing works on food CLSC with RTIs have not simultaneously considered the perishability, facility location, and uncertain demand under limited information. Therefore, this work addresses a new food CLSC optimisation problem. We first propose a non-linear chance-constrained programming model. It is then transformed into a mixed-integer linear programming model via using the distribution-free (DF) method and sample average approximation (SAA) method, respectively. An illustrative example reveals that the DF method needs only 10.50% of the computation time of the SAA method. To address large-scale problems, an improved Lagrangian relaxation (LR) method is developed. To address the computational challenge in large-scale problems, an improved Lagrangian relaxation (LR) algorithm is developed. Results show that CPLEX achieves a gap of 75.57%, while the LR surpasses it by finding near-optimal solutions with a gap of 1.22%, using only 31.82% of the computation time required by CPLEX. For this work, the main insights are summarised: (1) extending product shelf life can reduce the total cost; and (2) to alleviate uncertain demand and production risks, production capacity and product inventory capacity can be appropriately expanded, but excessive investment may not improve returns.

Joint grey time windows and two-dimensional loading design for modelling and solving multi-objective vehicle routing problems

This study defines grey time windows as periods during which customers expect deliveries at a specific time, with an allowance for a certain amount of advance or delay. For model solving, the grey time window is first whitened using the whitening triangle function based on different satisfaction thresholds and corresponding delivery costs. A tightness function is then designed for two-dimensional loading, which measures the effectiveness of different key points and loading postures. Finally, the Iori dataset was utilised to verify the effectiveness of the proposed solution algorithm. To the best of our knowledge, this is the first time the joint design concept of grey time windows and the whitening method have been introduced. Additionally, a grey opportunity-constrained programming model was constructed, with the model transformation and solution methods provided. Simulation results demonstrate that the proposed two-dimensional loading algorithm is both simple and efficient, and that the constructed grey chance-constrained programming model is solvable and has practical application value. Ultimately, this study provides insights for logistics companies on optimising delivery plans under various time window concepts.

Chance-constrained programming for drinking water quality management under fuzziness and randomness

ABSTRACT In this paper, an inexact fuzzy random chance-constrained programming (IFRCCP) model was proposed to deal with the randomness and fuzziness in the optimization of booster cost for the water distribution system (WDS). The IFRCCP model was applied to a WDS to analyze the effects of randomness and fuzziness on the booster costs. The booster cost intervals were obtained under various violation levels and confidence levels. The results indicated that the lower and upper booster costs increased with the confidence levels of the lower limits and decreased with the violation levels of the lower limits for both normal and log-normal distribution fuzzy random variables (FRVs). Moreover, the increased magnitude of lower and upper booster costs under normal distribution FRV is a little less than that under log-normal distribution FRV, while the decreased magnitude of lower and upper booster costs under normal distribution FRV is larger than that under log-normal distribution FRV. The same regulation can be found for the FVR with both triangular and trapezoidal distribution FRVs. With the increase of booster number, the increased magnitude of the lower and upper booster costs decreased for both normal and log-normal distribution FRVs.

Designing a multi-level reverse logistics network for waste batteries of electric vehicles under uncertainty—A case study in the Yangtze River Delta Urban Agglomerations of China

As a breakthrough for alleviating the impact of the energy crisis and improving the environment, the electric vehicle (EV) industry is entering a period of rapid development. The growth of this industry is expected to lead to a dramatic increase in the quantity of end-of-life EV traction batteries in the future, posing significant challenges to environmental pollution and the sustainable development of the EV industry. This study addresses the cascade utilization and uncertainties of waste batteries, by incorporating the development weight of the match degree between urban commercial environments and facility development, proposes a fuzzy chance constrained programming model for the design of multi-level reverse logistics network (RLN) for retired EV batteries. The objective is to minimize network costs and carbon emissions, determining optimal facility locations and transportation distribution schemes. The mathematical model is validated through a real case study in the Yangtze River Delta Urban Agglomerations in China, and sensitivity analysis is conducted to explore the impact of various influencing factors on network design. The results indicate that the development weight significantly influence facility location decisions, while the quantity and quality of battery recycling have a substantial impact on the operational quantities of network facilities at different levels.

Integrated multi-objective chance-constrained fuzzy interval linear programming model with principal component analysis for optimizing agricultural water resource management under uncertainties

ABSTRACT This study addresses the pivotal challenge of water resource allocation in urban environments by introducing a novel approach – a multi-objective chance-constrained fuzzy interval linear programming model integrated with principal component analysis (PCA). This innovative model aims to alleviate subjectivity in urban water management processes, particularly in adjusting water demands across various sectors. The proposed model incorporates correlation analysis to identify dimensionality-reducing factors of multitarget components, determining the proportion of each target component relative to the total components. Fuzzy sets are applied to irrigation water resource allocation quantity, segmented into six levels of fuzzy membership to analyze the stochasticity of water supply. Results demonstrate the model's efficacy, revealing that variations in risk probabilities impact water supply, necessitating positive water management strategies to enhance agricultural efficiency and negative strategies to mitigate the risk of inadequate water supply. Key findings emphasize the significance of agricultural water availability and the structure of irrigation water use in optimal resource allocation. Importantly, the study showcases the enhanced precision achieved through the proposed multi-objective chance-constrained fuzzy interval linear programming with PCA, thereby refining the optimization outcomes for water management under multifaceted objectives.

Wasserstein distributionally robust optimization for train operation and freight assignment in a metro-based underground logistics system

As a shared passenger-and-freight transportation mode, the metro-based underground logistics system (M-ULS) can significantly alleviate negative effects caused by ground transportation vehicles and improve urban sustainability. Against this background, this paper addresses dedicated freight train operation and freight assignment problems in the M-ULS without affecting passenger service during the off-peak period. More importantly, this paper proposes a flexible skip-stop strategy to reduce the increased operational cost of freight trains. To hedge against uncertainty, this paper formulates a distributionally robust chance-constrained programming model by limiting uncertain freight demands to a Wasserstein ambiguity set. Moreover, the distributionally robust optimization model for a Wasserstein ball with the 1-norm can be transformed into a mixed-integer linear program, which allows for an efficient solution via CPLEX, thereby enabling metro operators to use it. To implement this study in real-life applications, a case study is conducted based on the Beijing Subway Yizhuang Line. The computational results verify the superiority of the skip-stop strategy and distributionally robust optimization approach proposed in this study. Finally, some managerial suggestions are provided for the metro operators.

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.

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.

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.

Integrated remanufacturing scheduling of disassembly, reprocessing and reassembly considering energy efficiency and stochasticity through group teaching optimization and simulation approaches

The energy crisis and environmental pollution are receiving increasing attention from governments and communities. This study researches energy-aware remanufacturing systems. Remanufacturing aims to reuse valuable resources from end-of-life products and produce as-new products. Since remanufacturing systems involve a series of disassembly, processing and assembly operations, remanufacturing schedule integrates disassembly, processing and assembly shops. A multi-objective scheduling of remanufacturing systems is proposed, considering workstation use, energy consumption and customer satisfaction simultaneously. A chance-constrained programming model is established to minimize makespan and energy consumption while satisfying total tardiness requirements. A hybrid method is developed, using group teaching optimization and a discrete event simulation system, which can seek and evaluate potentially favourable solutions. The approach is validated on a group of test instances using well-known methods. The results reveal that this method can find non-dominated solutions with well-converged and well-diversified performance, verifying its advantages in providing informed decisions for managers and engineers.

Fourth-party logistics network design with demand surge: A greedy scenario-reduction and scenario-price based decomposition algorithm

The sharing economy has offered new viable approaches for firms to tackle demand surges. In this paper, we study a novel network design problem for the fourth-party logistics network (4PLN) to cope with unpredictable demand surges, in which temporary renting of logistics resources from a sharing market is available to supplement network capacity. A chance-constrained stochastic programming model is established to minimize the overall cost for 4PLN under service level targets. To deal with the difficulties brought by the evaluation of service level, we reformulate a mixed-integer linear programming (MILP) model, that can be solved straightly, based on the sample average approximation method. Then, to address the enormous challenge posed by the coupling of the basic NP-hard network design problem and the large number of demand scenarios in the MILP version, the Scenario-Price based Decomposition Algorithm (P-DA) is designed based on the key idea of decomposing the above-coupled factors. Further, to mitigate the performance deterioration brought on by large system scale and/or sample size, we expand our base algorithm to the Greedy Scenario-Reduction and Scenario-Price based Decomposition Algorithm (GR&P-DA) through the fast processing of chance constraints by introducing a greedy method. Through computational experiments, we demonstrate the effectiveness of our proposed model and algorithms, and we analyze the impact of various model parameters, including demand level, surge magnitude, surge probability, and rental price of third-party logistics resources, on 4PLN design. Furthermore, our comparative analysis reveals that renting logistics resources to address demand surges can significantly reduce costs and discovers that deploying resources in advance does not always provide an advantage, a temporary “case-by-case” planning approach for renting will give a more cost-saving scheme when surge probability is at a low level as it avoids idle resources by not being stuck with a conservative strategy.

An integrated water resources management model considering carbon source and sink under uncertainty: A case study of agricultural water-dominated basin

The global drive for carbon neutrality promotion through integrated water resource management (IWRM) in the face of burgeoning water demand is of great importance. This study developed an interval fuzzy-credibility chance-constrained multi-objective programming (IFCMP) model for water resources-carbon system (WCS) planning, which not only accounts for multiple uncertainties within system parameters, such as interval, fuzzy, and stochastic, but also effectively balances the conflicting objectives of water scarcity, benefits, emissions, and carbon sinks. The IFCMP-WCS model was applied to an agricultural water-dominated zone in the Tarim River Basin, China. Multiple water-allocation scenarios were examined, encompassing five prefecture-level cities, five departments, and three planning years. The results indicated that: (1) the water deficit was increasing over time, limited by water supply capacity, reaching 0–3.49 × 109 m3 in 2035; (2) as water allocation rises, system benefits and pollutant emissions increase, with the agricultural sector contributing the most economic benefits and pollutant emission; (3) the Tarim River Basin is an important carbon sink area, with the net carbon sink increasing from 8781.7–10,771.1 × 103 to 8851.1–10,937.9 × 103 tonnes during the planning horizon; (4) compared to single-objective models, the IFCMP-WCS model can offer more applicable water resource management options. These findings provide useful information for decision-making regarding IWRM and enhancing the net carbon sink in the Tarim River Basin.

A data-driven chance-constrained BESS planning in distribution networks by a decoupling solution method

In this study, a novel approach for the battery energy storage system (BESS) planning considering the integration of photovoltaic (PV) is proposed. A data-driven chance-constrained BESS programming model including the dynamic network reconfiguration (DNR) is developed. Then, a decoupling solution approach for sitting selection and sizing optimization is proposed to obtain optimal solution. According to the approach, comprehensive cost sensitivity factor (CCSF) is formulated for the sitting selection, and the particle swarm optimization (PSO) combined with the Minty algorithm and the switch operation time reduction algorithm is developed to obtain more accurate sizing. In addition, based on the scenarios generated by the spectral clustering approach, a continuous scenario increment approach is proposed to handle chance-constrained constraints. The numerical simulations on a practical 31-bus distribution network show the superiority of the proposed algorithm over existing methods.

A many-to-many pick-up and delivery problem under stochastic battery depletion of electric vehicles

ABSTRACT The study extends the traditional pick-up and delivery problems (PDPs) to address the specific challenges of urban logistics and electric vehicle (EV) adoption. These challenges include the limited range of EVs, energy consumption along the route, and uncertainty in traffic conditions. To overcome the limited range of EVs, the study includes battery swapping stations to ensure sufficient energy to complete delivery routes. Vehicle energy consumption is considered to reduce range anxiety and optimize energy use. The study also considers the unpredictability of traffic conditions that affect energy consumption and delivery schedules. To address these concerns, the study proposes an approximate Quadratic Chance-Constrained Mixed-Integer Programming (QC-MIP) model with a linear approximation, a constructive heuristic and a meta-heuristic. These quantitative models incorporate comprehensive EV energy estimation approaches, enabling more accurate energy predictions. The proposed approaches provide valuable insights and strategies for improving energy efficiency and delivery performance in urban logistics environments.

Optimal design of low impact development at a community scale considering urban non-point source pollution management under uncertainty

Urban non-point source (NPS) pollution management is receiving increased attention because of the environmental and ecological consequences of urbanization. We developed a Storm Water Management Model (SWMM)-based interval double-sided chance-constrained programming (SWMM–IDCCP) model for the optimal design of low impact development (LID) during urban NPS pollution management at a community scale. Pollution control efficiency and economic costs related to LID practices were considered in the SWMM–IDCCP model, simultaneously addressing uncertainties in pollutant output and decision-making processes. Specifically, by combining SWMM, Monte Carlo simulation, interval linear programming, and double-sided chance-constrained programming, the SWMM–IDCCP model can evaluate export loads of urban NPS pollutants and pollution control efficiencies of LID practices and analyze the corresponding uncertainties. Concurrently, the proposed model can provide reliable and optimal implementation schemes for LID practices under different scenarios, dealing with uncertainties such as discrete intervals and double-sided random parameters. The proposed model was applied to the Dongguan University of Technology in Dongguan City, South China. The random distribution characteristics of pollution load were identified. The ranges of pollution load for total suspended solids, chemical oxygen demand, total nitrogen, and total phosphorus were 76,430–165,698, 44,366–74,155, 2118–3,830, and 97–177 kg in 2021, respectively. Probability distribution characteristics of pollution control efficiencies of three LID practices (i.e., bio-retention cell, green roof, and permeable pavement) for the four urban NPS pollutants were obtained. Using the SWMM-IDCCP model, LID configuration schemes were obtained under diverse pollution control targets and confidence levels, taking into account variations in the decision preference of different administrators. This research indicates that the SWMM-IDCCP model can be used to manage urban NPS pollution at a community scale under uncertainty.

Type-2 fuzzy chance-constrained linear fractional programming model for a water resource management system: A case study of Taiyuan city, China

Abstract Uncertainties arising from extreme climate events and human activities pose a challenge to the efficient allocation of water resources. In this study, a type-2 fuzzy chance-constrained linear fractional programming (T2F-CCLFP) is developed to support the water resource management system under uncertainty by incorporating type-2 fuzzy sets, chance-constrained programming, and fractional programming into a comprehensive multi-objective optimization framework. The model enables the trade-off between economic, social, and environmental sustainability and provides water supply solutions associated with different levels of fuzzy uncertainty and risk of violating constraints. The T2F-CCLFP model is applied to Taiyuan, Shanxi Province, China, to support its water resource management. Results reveal that: (i) the industrial structure is transitioning toward diverse industries from energy and heavy industry dominance; (ii) external water transfer will be the major water-supply sources for the city in the future, accounting for 55 and 50% of the total water supply in 2025 and 2030, respectively; (iii) the water-supply security of the city is enhanced by provoking the utilization of reclaimed water (the annual growth rate is 13.9%). The results are helpful for managers in adjusting the current industry structure, enhancing water supply security, and contributing to the sustainable development of socio-economic and water systems.

Sparrow search algorithm with adaptive t distribution for multi-objective low-carbon multimodal transportation planning problem with fuzzy demand and fuzzy time

In this paper, we present a multi-objective low-carbon multimodal transportation planning problem with fuzzy demand and fuzzy time (MOLCMTPP-FDFT) that minimizes both cost and time while incorporating mandatory carbon emission, carbon tax, carbon trading, and carbon offset policies. Chance constrained programming and interval programming are introduced to formulate the fuzzy demand chance constrained programming model and the fuzzy time interval programming model, respectively. Then, the mathematical model of uncertain MOLCMTPP-FDFT is transformed into a deterministic model. Based on the sparrow search algorithm, t distribution and the concept of Pareto optimality for multi-objective optimization, we also propose a solution strategy for the proposed model. In this algorithm, the number of iterations is used as the degree of freedom of - to update the sparrow location, which strikes a balance between the capabilities of global search and local search. Finally, the proposed algorithm and MOLCMTPP-FDFT are applied to a real case, resulting in a minimum cost of 260730.48 and a time duration of 13.044, which outperform the minimum cost of 268874.88 and minimum time of 18.32 obtained using single-mode transportation. The carbon emissions resulting from the lowest-cost solution obtained using single-mode transportation are 3,198.48, which are significantly more than the allowed emissions. Therefore, the proposed algorithm and mathematical model of MOLCMTPP-FDFT are valuable tools for optimizing multimodal transportation route. Additionally, the experimental results not only validate the superior efficiency and energy-saving benefits of the proposed multimodal transportation routes in comparison to the actual single-modal transportation, but also demonstrate the applicability of different low-carbon policies.

A chance constrained dynamic network reconfiguration based on Minty algorithm in distribution networks

Abstract With high renewable energy sources (RESs) penetration in distribution networks, handling the uncertainties of RESs outputs and multi-time coupling problems in the dynamic network reconfiguration (DNR) is a big challenge. Besides, the existing mathematical and artificial intelligence algorithms for network reconfiguration face the problem of falling into local optima and poor convergence. To address the above challenge and problem, this paper first establishes a chance-constrained programming model to handle the uncertainties. Then the Minty algorithm is applied for efficiency and accurate static network reconfiguration (SNR) in each time interval. Finally, a branch exchange-based method is proposed to eliminate violations for the operation times of switches. Numerical simulations on the IEEE 33 system and an actual 151-bus distribution network show the superiority of the proposed algorithm over existing methods.

Estimation of Electricity Supply and Demand under Dual Carbon Goals: A Chance-constrained Linear Programming Model

Implementing the green and low-carbon development strategy has become a global consensus with the increased severity of global environmental problems. In 2020, Chinese President Xi Jinping put forward China’s carbon peak and carbon neutrality goals (i.e., dual carbon goals), which is a guideline for the development of Chinese power systems. Thus, a supply and demand estimation system that includes dual carbon indicators and considers the long-term supply capacity is needed. To address these gaps, this study put forward an estimation model and provided the solutions. First, a supply and demand estimation indicator system is established to meet the dual carbon goals. Second, with minimizing cost being the main objective and several constraints, a chance-constrained linear programming model is used to estimate the supply and demand of electricity in 2030, 2040, and 2050. Finally, with model relaxation, we obtained the estimation results with Matlab-Cplex, which verified the efficiency and validity of the proposed model.