Chance Constraint Programming Research Articles

Chance-Constrained optimal scheduling of power and hydrogen systems in power distribution network with significant photovoltaic and carbon offsetting strategies

The urgent need to address climate issues increases the significance of sustainable strategies in power networks. This study focuses on the optimal operation of a hydrogen-based microgrid incorporated with photovoltaics (PVs), time-of-use demand response (TOU-DR) program, distributed generation (DG) units, hydrogen fuel stations (HFSs), and battery storage units. Employing a multi-objective approach, the study aims to minimize emissions and system operation costs, being validated by implementing it on the IEEE 33-bus test network. To address the carbon emissions issue, the Power-to-gas (P-to-G) facility is utilized to capture and recycle CO2 pollutions. The implementation of a carbon trading market mechanism is also adopted to further enhance the generation capacity of the PV units and facilitate the achievement of sustainable development goals. Besides, the presence of uncertainties in renewable generation, demand, operational characteristics of hydrogen vehicles, and market prices pose challenges for the system operator, which is addressed by employing the chance constraint programming (CCP) method. Results show that using the P-to-G system has led to a 10 % emissions reduction. However, this is accompanied by a corresponding 25 % increase in operational costs. Additionally, when operating under the CCP constraints, the system's storage level has been increased by 10 % to ensure optimal performance.

Water–Food Nexus System Management under Uncertainty through an Inexact Fuzzy Chance Constraint Programming Method

This study discusses the planning of a regional-scale water–food nexus (WFN) system using an inexact fuzzy chance constraint programming (IFCCP) method. The IFCCP approach can handle uncertainties expressed as interval and fuzzy parameters, as well as the preferences of decision makers. An inexact fuzzy chance constraint programming-based water–food nexus (IFCCP-WFN) model has been developed for the City of Jinan with the consideration of various restrictions related to water and land availability, as well as food and vegetable demands. Solutions for the planting areas for different crops in different periods have been generated under the different preferences of decision makers. The water resource availability would be the priority factor affecting the WFN system under demanding conditions, in which wheat cultivation would be dominated by this factor under fuzzy confidence levels of 0.2 and 0.5, and the planting area of corn would be determined by this factor under high fuzzy confidence levels (e.g., 0.8). In addition, the reliability of irrigation would decrease with increasing fuzzy confidence levels under demanding conditions, limiting the planting areas for crops and leading to a decreasing trend of the system benefit. Adequate water resources would be available for irrigation under optimistic conditions, implying no significant contributions to the planting schemes. Nevertheless, increasing food loss rates would result in more planting areas to satisfy food requirements and thus a greater system benefit under advantageous conditions. Compared with the developed IFCCP-WFN model, the interval-linear-programming-based water–food nexus (ILP-WFN) model can merely reflect the lower and upper bounds of uncertain parameters and neglects the inherent distributional information within the fuzzy parameters. Thus, the ILP-WFN model is unable to reveal the inherent impacts of the fuzzy parameters on the resulting planting strategies.

Optimal deployment of dynamic wireless charging facilities for electric bus route considering stochastic travel times

Installing dynamic wireless charging (DWC) facilities on dedicated bus lanes endows electric buses (EBs) with the ability to charge while in motion, effectively overcoming the limitations of short driving ranges. However, if the facilities are deployed along the entire EB route, it would incur significant economic costs and result in resource wastage. In this study, we divide the EB route into segments including station segments, intersection segments, and elementary segments of road section. The fluctuations in travel times of different trips on these segments are described. To optimize the DWC deployment plan, a chance constraint programming model is established with the objective of minimizing the costs of purchasing power transmitters and inverters. A combined algorithm using stochastic simulation and particle swarm optimization is designed to solve the model. A real EB route is used to conduct the case study, and results indicate that it is more suitable to deploy DWC facilities in adjacent elementary segments of road section compared to individually distributed station segments or intersection segments. Additionally, as wireless charging power or battery capacity increases, the number of required inverters and the length of power transmitters decrease, resulting in a nonlinear decrease in the deployment costs of charging facilities.

Event-triggered reactive power tracking optimization for second-level power fluctuations of renewables and stochastic loads

The power fluctuations of renewable energy sources and stochastic loads (RESL) makes it difficult for system operators to achieve optimal control. This paper proposes an event-triggered reactive power optimization strategy based on the event-triggered algorithm to rapidly track the optimal operating states for accommodating the second-level power fluctuations of RESL, with less computation and communication. Firstly, on a longer temporal scale, event-triggered multi-stage global optimization (MGO) is performed on a rolling basis to obtain the time-interval optimal operating states according to chance constraint programming, and the system operation security is improved by narrowing the voltage-feasible regions. Next, at each moment within the time interval, event-triggered local tracking optimization (LTO) is performed at each controller of RESL by rapidly regulating reactive power to adapt to its second-level active power fluctuations. The LTO is completed at the local level by using the sensitivity information from MGO, without running the optimal power flow program, to relieve the computation and communication burdens. Then, event-triggered LTOs performed at all controllers synchronously form the system-level real-time optimization. The unnecessary MGO and LTO are skipped by using the event-triggered algorithm when small power fluctuations are occurring. Finally, a case study on the IEEE-39 bus power system validates the feasibility of the proposed strategy.

An assets maintenance-workforce planning problem under uncertainty: A chance constraints assisted simulation-optimization approach

Preventive and corrective maintenance-workforce planning for assets is a growing area of research, given the presence of uncertainty. We model a Simulation-Optimization framework to solve this challenging large-scale maintenance-workforce planning problem while considering its stochastic nature inside both the simulation and optimization models. A holistic simulation model is proposed through multiple modules to imitate different aspects of the maintenance process, i.e., spare facilities/storage and workforce. The workforce module represents technicians’ types, skills and levels, and employment life-cycle, including recruitment, promotion, and separation/retirement. The present uncertainty in the simulation model demands its corresponding optimization model to cope with the nature of the problem and guarantee a degree of confidence level of the optimal decisions. Therefore, a Chance Constraints Programming (CCP) method is integrated with an evolutionary optimization algorithm to limit the risk level of violating the random constraints while minimizing the total maintenance cost. Discrete Event Simulation (DES) and Self-Adaptive Differential Evolution assisted with Chance Constraints (SA-DE-CC) algorithm are coupled to solve the current problem. The optimization model is proposed to optimize the preventive maintenance frequency and workforce planning problem while implicitly considering corrective maintenance in the simulation model. The Taguchi method performs scenario and risk analysis for multiple CC probability distributions (i.e., Weibull and Normal), confidence levels, and critical parameter combinations. The model’s sensitivity analysis and managerial insights are also discussed.

The role of precise management in mitigating non-point source nitrogen pollution of a reservoir region: Environmental profits and economic benefits

Non-point source pollution is a worldwide water environment pollution problem, which could be effectively solved with the concept and method of precision management. In this study, an integrated optimization decision system (IODS) was developed for supporting precise management of non-point source nitrogen pollution under multiple uncertainties. Several kinds of methods were innovatively coupled into the IODS, including material flow analysis (MFA), Gumbel copula model (GCM), path analysis (PA), and interval multi-objective chance constraint programming (IMOCCP). The IODS could (i) identify the main sources of non-point source pollution through the MFA; (ii) estimate the pollution status of different zones within the entire watershed using the GCM and PA; (iii) incorporate the pollution sources and pollution status to the IMOCCP as decision variables and development scenarios (i.e., S1 to S3), respectively. The settlement of the IMOCCP model could provide targeted optimal solutions to zones of different pollution status, with minimized non-point source nitrogen pollution and maximized economic benefit. Our research demonstrates great advantage of precise management in obtaining more environmental profits and economic benefits. Specifically, the precise management makes the system benefit increased by about [89.86%, 96.10%], [87.35%, 95.00%], and [28.23%, 60.40%] from S1 to S3, respectively. While the nitrogen loading decreased by about [23.40%, 81.84%], [66.51%, 145.72%], and [385.05%, 623.16%], correspondingly. This work may offer inspiration and guidance for the practice of precise management in mitigating the non-point source pollution.

Crash Frequency Minimization with Severity Mitigation in Road Geometric Design Using Chance Constraint Programming Optimization

Crash frequency and crash severity are two major aspects of transportation safety. In this paper, we propose a decision-making scheme combining statistical analysis and optimization modeling to be used in transportation safety study. We conduct a safety analysis, a travel speed analysis, and an optimization analysis to develop a two-stage decision scheme to minimize crash frequency while mitigating crash severity, using data collected in urban environments in Lincoln, Nebraska. In the safety analysis and the travel speed analysis, we study the impact of lane width and other related road geometric design parameters on annual crash frequency and vehicle travel speed using count models and linear regression models, respectively. In the optimization analysis, the proposed two-stage stochastic programming model determines the lane width and other road geometric design parameters in the first stage, and then the posted road speed limit in the second stage for each scenario. To mitigate crash severity, we use a chance constraint to restrict a certain percentile of the vehicle travel speed to comply with the posted road speed limit. This two-stage decision scheme is shown to be effective for the data collected in Lincoln, Nebraska, when restricting the vehicle travel speed of up to two positive standard deviations from the mean travel speed to be under the posted speed limit. The application of a stochastic programming model that utilizes regression analysis results serves as an innovative decision scheme that effectively connects statistical analysis and optimization studies in road geometric design for transportation safety. Its objective is to minimize crash frequency while simultaneously mitigating crash severity. This methodology has extensive potential for application in various environments to assist in the reduction of both crash frequency and crash severity.

Project Portfolio Selection Considering the Fuzzy Chance Constraint of Water Environmental Restoration

The water environment restoration project portfolio (WERP) selection is discussed in this paper. By complying with the analysis of the project’s multidimensional property and operation mode, this paper develops the chance constraint and the management constraint of the WERP from the perspectives of public service and enterprise operation. In addition, the multi-objective mixed integer linear programming model is constructed by combining the expectation method and the fuzzy chance constraint programming method. The results demonstrate that: (1) Our proposed method successfully circumvents the occurrence of local objective optimization within a specific confidence interval, thereby achieving a balance between economic and water environment restoration objectives; (2) including fuzzy chance constraints in our proposed method significantly diminishes the risk of exceeding the WERP capacity, thereby ensuring the effectiveness of water environment restoration by adopting a market-based approach. However, further examination of the impact of various sub-projects in WERP is necessary, along with the integration of novel evolutionary algorithms to enhance the efficiency of our model.

Risk-assessment of carbon-dioxide recycling in a gas-fired power plant using CVaR-based convex optimization

With the rise of renewable resources’ penetration, the use of clean excessive renewable energy as the primary source of a Power to Gas (PtG) system for the reduction of CO2 pollution is quite advantageous. This paper presents a mixed integer linear formulation for modeling the optimal operation of a gas-fired power plant (GFPP)-PtG system that utilizes an excessive renewable resource, wind power, as the primary mover of the proposed PtG system for capturing and recycling the produced CO2 pollution. CO2 and CH4 gas storages are considered in the proposed PtG system to boost the operational efficiency of the system. Additionally, the scenario generation method models both wind power generation and electrical demand. The effective conditional value at risk (CVaR) method, which is a convex risk modeling approach, models the risks imposed by the generated scenarios for wind power and electrical demand. Due to the convexity of the proposed formulation for the PtG system and the risk modeling method, the ultimate model can be simply solved by using any available LP solver. To evaluate the outcomes of an optimization problem based on CVaR, a comparative analysis is conducted with chance constraints programming, which demonstrates the optimality and computational efficiency of the CVaR results due to convexity. The effectiveness of the proposed model is approved by modeling a test system based on real data, and the obtained numerical results are discussed comprehensively.

Optimal Modeling of Sustainable Land Use Planning under Uncertain at a Watershed Level: Interval Stochastic Fuzzy Linear Programming with Chance Constraints

In this paper, an uncertain interval stochastic fuzzy chance constraint land use optimal allocation method is proposed and applied to solve the problem of land use planning in river basins. The UISFCL-LUP method is an aggregation of interval parametric programming, fuzzy linear programming and chance constraint programming which can cope with uncertain problems such as interval value, fuzzy set and probability. In this paper, the uncertain mathematical method is explored and studied in the optimal allocation of land use in the next two planning periods of Nansihu Lake Basin in China. Moreover, it was proved that ISFCL-LUP can deal with the uncertainty of interval, membership function and probability representation and can also be used to solve the land use planning and land use strategy analysis under uncertain conditions. On the basis of model calculations, we obtained the optimal allocation results for six types of land use in four regions over two planning periods based on different environmental constraints. The results show that the optimized λ value (that is, the degree of satisfaction with all the model conditions) is in the range of [0.54, 0.79] and the corresponding system benefits are between [18.4, 20.4] × 1012 RMB and [96.7, 109.3] × 1012 RMB. The results indicate that land managers can make judgments based on the different socio-economic development needs of different regions and determine strategic land use allocation plans under uncertain conditions. At the same time, the model obtained interval solutions under different system satisfaction and constraint violation probabilities, which helps land managers to analyze the importance of land system optimization and sustainable development more deeply.

Simulated Annealing-Based Heuristic for Multiple Agile Satellites Scheduling Under Cloud Coverage Uncertainty

Agile satellites are the new generation of Earth observation satellites (EOSs) with stronger attitude maneuvering capability. Since optical remote sensing instruments cannot see through the cloud, the cloud coverage brings a significant influence on the satellite observation missions. The scheduling of multiple agile EOSs (AEOSs) is already complicated due to strong satellite maneuverability and onboard satellite energy constraints. Moreover, introducing cloud coverage uncertainty further increases scheduling complexity. Motivated by these challenges, we address the multiple AEOSs scheduling problem under cloud coverage uncertainty where the objective aims to maximize the entire observation profit. A chance constraint programming model is adopted to describe the uncertainty initially, and the observation profit under cloud coverage uncertainty is then calculated via a sample approximation method. To overcome the solving difficulty, an improved simulated annealing (ISA)-based heuristic including a fast insertion strategy is proposed for large-scale observation missions. Finally, extensive experiments are conducted to verify the effectiveness and efficiency of the proposed method. Experimental results show that the ISA-based heuristic outperforms other algorithms for the multiple AEOSs scheduling problem under cloud coverage uncertainty, which validates the proposed algorithm.

Modeling a hydrogen-based sustainable multi-carrier energy system using a multi-objective optimization considering embedded joint chance constraints

Hydrogen-based power generation could increase the sustainability and efficiency of future energy systems due to the higher storing capabilities resulting from the higher gravimetric density of hydrogen energy. This work proposes a hydrogen-based multi-carrier energy system (HMES) comprising renewable energy resource (RES), electricity, and hydrogen markets as input energy carriers and power, cooling, and heating as demands. Thermal water storage and hydrogen tank storage are considered to improve energy efficiency. This system uses the full capacity of intermittent wind and solar energy to increase RES utilization and decrease carbon emissions simultaneously. The environmental and economic goals of the proposed HMES are satisfied through the weighted-sum multi-objective method. The trade-off solution between operation and emission costs is ultimately obtained through the max-min fuzzy method. Chance constraint programming (CCP), which offers decision-makers a variety of risk-taker strategies, is utilized as a tool to manage the risk associated with stochastic optimization. The confidence level of the CCP determines the risk tolerated by the decision-maker, improving the optimization process by creating a more realistic approach to risk management in real-world models. The results show the benefits of hydrogen tank by reducing operation costs by 4.5% and importance of CCP leading to operation costs reduction.

A joint chance constrained optimization algorithm with robust reconstruction for multi-echelon and multi-period closed-loop manufacturing adjustable system design under multi-source uncertainty

The complex electronic assembly scheduling problem under uncertainty poses a new challenge. The joint chance constraint programming method based on robust reconstruction is used to hedge the endogenous and external uncertainty in the closed-loop manufacturing scheduling model of multi-echelon and multi-product electronic assembly process under multi-source uncertainty in this study. The numerical cases of electronic factory in assembly and closed-loop manufacturing mode are studied, the sensitivity of cost to different types of uncertain parameters is analyzed. The results show that different types of uncertainty have different effects on economic benefits, and endogenous uncertainty has a greater impact on manufacturing costs. Decision makers can adjust the adjustable parameters according to external policies and economic situation to control the conservatism and economic benefits of the scheme, and at the same time reduce the demand for uncertain data and improve the efficiency of solution. The model provides a new sustainable production method for electronic assembly manufacturing practitioners, and makes the best production decision when weighing economic benefits and risks.

Coupling Fuzzy Bi-Level Chance Constraint Programming and Spatial Analysis for Urban Ecological Management

In this study, a fuzzy bi-level chance constraint programming (FBCP) model is developed for urban ecological management in Xiamen, China. FBCP has advantages in balancing trade-offs between multiple decision makers and can address fuzzy and stochastic uncertainty in ecosystem management. It also can reflect the impact of different violation risk levels and emission reduction measures on system benefit, ecosystem service value, and land resource allocation. Then, the conversion of land use and its effects at small regional extent (CLUE-S) model is employed to provide the spatial allocation of future land resources under different scenarios. Results reveal that (i) carbon fixation and climate regulation are the major contributors to the ecosystem service value, with a proportion of [15.4, 15.6]% and [18.5, 18.8]%, respectively; (ii) the main environmental problem in Xiamen is the water pollution caused by the excessive discharge of commercial and residential land, with COD and NH3-N account for [68.81, 69.33]% and [67.65, 68.20]% of the total discharge of the city, respectively; (iii) the violation risk p level is the most impact factor, and the schemes with high system benefit would face greater default risk and lower ecological quality; (iv) FBCP model considers the trade-off between economic benefit and ecological quality, while the fuzzy chance constraint programming (FCP) model achieves a high system benefit at the expense of the environment. These findings help decision makers to understand the impact of parameter uncertainty and pollutant discharge policies on system benefit, and adjust land-use patterns to weigh ecological environment protection with urban economic development.

Joint Power Allocation and Route Planning Scheme for Multitarget Tracking in Airborne Radar Network Under Multiuncertainty

In this article, a joint power allocation and route planning (JPARP) scheme is proposed for multitarget tracking in airborne radar network (ARN) in the presence of multiuncertainty. The main mechanism of the proposed scheme is to utilize the optimization technique to collaboratively plan route and allocate power for each airborne radar, with the aim of minimizing the power consumption of the ARN while satisfying the predetermined multiple target tracking (MTT) performance, inter-radar communication performance, and inter-radar safe distance. The Bayesian Cramer–Rao lower bound (BCRLB) and the communication data rate (CDR) are derived and adopted as metrics to evaluate the MTT performance and inter-radar communication performance, respectively. To ensure the robustness of the algorithm under multiuncertainty, we formulate the JPARP scheme as a chance-constraint programming (CCP) problem. To efficiently solve this no-convex and nondeterministic CCP problem, we relax it into an unconstrainted optimization problem. Then, a hybrid intelligent algorithm based on stochastic simulation and improved genetic algorithm (GA) is proposed to obtain a near-optimal solution. Simulation results are provided to demonstrate that the JPARP scheme is more effective in saving ARN power consumption and more robust under uncertainty than the conventional algorithm.

An Extended Robust Chance-Constrained Power Allocation Scheme for Multiple Target Localization of Digital Array Radar in Strong Clutter Environments

The traditional power allocation method for multi-target localization adopts the Robust Chance Constrained Power Allocation Scheme (RCC-PA), which does not consider the strong clutter characteristics in the current radar detection environment. However, how to reasonably allocate the power required for the radar to locate the target in the strong clutter is a fundamental challenge for improving the combat capability of the radar. Since Simultaneous Multi-beam Digital Array Radar (SM-DAR) can provide high-resolution information on targets in strong clutter, this paper extends the RCC-PA scheme to strong clutter and introduce an extended target model suitable for high-resolution SM-DAR. At the same time, the Gamma distribution is used to reflect the statistical characteristics of the Radar Cross Section (RCS) so that the extended RCC-PA scheme (ERCC-PA) can be suitable for all the scatterers whose RCS belong to Gamma distribution families. In the ERCC-PA scheme, the Strong Clutter Information Reduction Factor (SCIRF) of the extended target is first derived. Then, a Gamma Chance-constraint Programming Model (Γ-CCP model) is constructed to optimize the power allocation for locating multiple extended targets in strong clutter. The extended dichotomy method for power allocation is also given. Theoretical analysis shows that the power of multi-target localization in strong clutter can still be reasonably allocated under the target with multi-measurement characteristics. In addition, the power distribution of multi-target localization in the strong clutter is negatively correlated with the shape parameter of the Gamma distribution. Specifically, the larger the shape parameter characterizes the larger individual strong scatterer, resulting in the SM-DAR requiring less positioning power. The experimental results verify the theoretical analysis and show that the ERCC-PA scheme can improve the utilization of power compared with the benchmark and has the advantage of robustness to the fluctuation of the target RCS.

A new supply chain design to solve supplier selection based on internet of things and delivery reliability

In this study, an attempt is made to examine the planning for supply, production, and distribution with an integrated approach to reduce waste and returned products in such a way that the supplier and manufacturer exchange information in supply chain. In this paper, manufacturer share information with suppliers through IoT and blockchain. For best implementation of supply chain based on these technologies, a manufacturer must chose the optimal combination of suppliers that can better meet the criteria of technologies implementation. In this paper, IoT and blockchain are examined to supply chain management by considering the impactful criteria of implementation in suppliers' selection and optimization. For this purpose, this paper applies the multi criteria decision making (MCDM) methods for prioritizing the suppliers. Then, the obtained weights are plugged into the model as the inputs of the designed model. In this study chance constraint programming (CCP) is leveraged to develop optimization models that take the uncertainties and randomness into account. Since the developed model is a non-differentiable and non-convex, genetic algorithm (GA) and particle swarm optimization (PSO) are leveraged to solve the formulated model. Finally, the efficiency of the presented method is verified through a case study from the food supply chain.

Credibility based chance constrained programming for parallel machine scheduling under linear deterioration and learning effects with considering setup times dependent on past sequences

The industry has expressed significant concern regarding the issue of parallel machines and the influence of learning and deterioration. This research investigates non-identical parallel machine scheduling, taking into account the simultaneous consideration of learning effects, deterioration, and past-sequence-dependent setup times. Due to the existence of uncertain parameters in real-world scenarios, the processing times and due dates are assumed to be triangular fuzzy numbers. A fuzzy nonlinear mathematical model with two objective functions is presented and solved using the fuzzy Chance Constraint Programming approach. The two objectives are the summation of earliness and tardiness, as well as makespan. To achieve an efficient near-optimal Pareto front for the problem, a hybrid NSGA-II and VNS multi-objective meta-heuristic is proposed and the results are discussed. Finally, the augmented ε-constraint method is utilized to address issues with small dimensions. The computational analysis demonstrates the effectiveness of this proposed algorithm in tackling problems, especially those with substantial dimensions.

A memetic algorithm for a relocation-routing problem in green production of gas considering uncertainties

• A fuzzy chance constraint programming model for relocation-routing problem is formulated. • The uncertainty of amount of generated sewage and travel time are considered in the model. • Memetic algorithm and Monte Carlo simulation are integrated to solve the models. • Several series of experiments are conducted to evaluate the proposed model and algorithm. This study introduces a relocation-routing problem with a fuzzy amount of sewage and stochastic travel time in natural gas production. In this study, a set of sewage treatment plants (STPs) and a certain number of gas-gathering stations (GGSs) are distributed on the field. With the increasing amount of production, however, the current STPs cannot satisfy the production level. Policymakers propose several location candidates to build new STPs and aim to minimize the total cost of running the newly built STPs and the original STPs. The practical attributes of the sewage return logistics, capacity of vehicles and STPs, uncertain amount of sewage, stochastic travel times, and other constraints are taken into account. The new problem proposed in this study is defined as Relocation-Routing Problem with Fuzzy Sewage and Stochastic Travel Time (RLRPFSSTT), which has never been investigated before. To minimize the total cost, including the construction cost of newly opened STPs and transportation cost between STPs and GGSs, this paper designs a memetic algorithm to optimize location and routing problems simultaneously. Benchmark-based experimental data is designed, and the computational results demonstrate the effectiveness of the proposed memetic algorithm. Sensitivity analysis and comparisons are also carried out to validate the advantage of considering uncertainties. The proposed model and algorithm are meant to further supplement and extend the location and routing models, as well as have great significance for the decision-makers of industrial logistics in oil fields and coal mines.

Distributionally robust location–allocation models of distribution centers for fresh products with uncertain demands

This paper studies the location–allocation problem of distribution centers (DCs) for fresh products, in which it requires to determine the number and locations of DCs and design the allocation scheme for fresh products. The main challenge in optimizing this problem is to handle the uncertain demands with only partially known probability distribution information. To address this challenge, this paper first characterizes the ambiguous probability distributions of fresh products demands by an ambiguity set. Then, from different perspectives, this paper proposes two distributionally robust (DR) location–allocation models to find the optimal decisions that perform best in view of the worst-case distribution within the ambiguity set. The first model is with ambiguous expectation, which is transformed into a mixed-integer linear programming (MILP) model via employing the linear decision rule technique. The second model is with ambiguous chance constraints. This model is a semi-infinite chance constraint programming problem, which is transformed into a mixed-integer second-order cone programming (MISOCP) model by deriving safe approximations of the ambiguous chance constraints. To verify the feasibility and effectiveness of the proposed models, a case study about Hema fresh chain stores in Beijing is addressed. The experimental results demonstrate that the optimal decision provided by the proposed optimization methods can immunize against ambiguous probability distributions of demands with a small cost price.