Stochastic Chance-constrained Programming Research Articles

Logistics planning for direct temporary disaster housing assistance under demand uncertainty

In this paper, we propose and study a framework for disaster housing logistics planning under demand uncertainty. Specifically, we utilize a two-stage chance-constrained stochastic programming model to achieve the balance between logistics operational cost and demand fulfillment especially towards extreme disaster scenarios. To do so, we incorporate two operational modalities, one for the ordinary modality and the other for the emergency modality, and the emergency modality is only allowed to be activated for a certain percentage of scenarios that is specified by the decision maker among all scenarios. The set of scenarios is generated according to a spatial regression model for characterizing the disaster housing demand based on a selected number of independent variables related to both the hazard and socioeconomic factors, which is trained offline from historical data. We conduct a numerical experiment based on Hurricane Ian, and our numerical results show the effectiveness of the proposed approach compared to some standard benchmark approaches. We also highlight the managerial insights for disaster housing logistics planning gained through this numerical experiment.

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.

A high-confidence geometric compensation approach for improving downward surface accuracy

Overhangs are usually inevitable in Additive Manufacturing (AM). Moreover, making their downward surfaces reach higher accuracy after printing has an imperative impact on improving the accuracy of them and their corresponding as-printed parts. Although extensive studies have been devoted to improving the accuracy of as-printed parts, the effective, general, reliable, and less-consumption (in time and material) approach to make an overhang (especially on its downward surfaces) reach its ideal accuracy is still rare. Hence, a new machine-leaning-based geometric compensation approach is proposed in this study. First, based on the Taguchi method, a series of (overhang) benchmarks are designed and printed to collect the (geometric) deviations of the downward surfaces. With these data, a new deviation predictor is established based on the gaussian process regression model. The predictor can effectively predict the deviation(s) (as well as its corresponding quantified uncertainty) of each downward surface (after printing) by using a pointwise manner. Meanwhile, to make the downward surface of an overhang meet its ideal accuracy (after printing) with high confidence, a specific stochastic chance-constrained programming problem is first formulated to evaluate the new and suitable overhang height of each point on the downward surface. Based on the above-mentioned predictor, the programming problem gets solved by developing a new overhang-height optimization method that integrates the Monte Carlo simulation with the particle swarm optimization. After that, a support structure-associated compensation method is presented to make a pointwise compensation (according to the above-evaluated overhang height) on the downward surface and determine the support structures’ optimized number and anchor positions. Finally, experiments on several representative overhangs are also implemented based on a typical material extrusion machine to validate the effectiveness of the proposed approach. The results show that the proposed approach reduces the deviations of the downward surface by up to 63.92% on average, saving up to 40% of material (less support structures). In addition, methodological comparisons with state-of-the-art approaches are also implemented. The comparisons show that the proposed approach has the great potential to reliably improve the accuracy of the overhang’s downward surface(s) after printing in a less-consumption manner.

Chance-Constrained Optimization of Photovoltaic System Allocation considering Power Loss, Voltage Level, and Line Current

Changes are emerging that will significantly alter structure and operation of this century’s distribution networks, and photovoltaic (PV) systems will play greater role in energy sector, with implications for power system reliability. Considering uncertainties in solar irradiance and electrical loads and incorporating them into the optimization problem within an appropriate methodology is becoming increasingly important in reshaping distribution networks. In this paper, uncertainty scenarios are handled with Monte Carlo Simulation (MCS) under genetic algorithm (GA) and differential evolution- (DE-) based optimization, and probability distribution functions (pdf) of bus voltages and line current are obtained to be used in chance-constrained stochastic programming. This present study focuses on investigating impact of uncertainties in PV system operating under different irradiance scenarios on power loss with probabilistic constraints in distribution networks instead of precise deterministic limits to contribute more efficient and reliable use of energy. By combining meta-heuristic optimization and MCS technique under one framework, this paper contributes to knowledge base of how to allocate PV plants within distribution networks under chance-constrained strategy. In order to show the effectiveness of the proposed methodology, obtained optimization results are tested using MCS under set of uncertainty conditions and network constraints are evaluated for limit violation probabilities. The effectiveness of this method is investigated based on comparative results of two different optimization methods through probabilistic analysis and simulation.

A case-oriented computational study for sustainable fleet planning in a battery closed-loop supply chain network under uncertainty

PurposeThe purpose of this study is to develop a holistic optimization model for an integrated sustainable fleet planning and closed-loop supply chain (CLSC) network design problem under uncertainty.Design/methodology/approachA novel mixed-integer programming model that is able to consider interactions between vehicle fleet planning and CLSC network design problems is first developed. Uncertainties of the product demand and return fractions of the end-of-life products are handled by a chance-constrained stochastic program. Several Pareto optimal solutions are generated for the conflicting sustainability objectives via compromise and fuzzy goal programming (FGP) approaches.FindingsThe proposed model is tested on a real-life lead/acid battery recovery system. By using the proposed model, sustainable fleet plans that provide a smaller fleet size, fewer empty vehicle repositions, minimal CO2 emissions, maximal vehicle safety ratings and minimal injury/illness incidence rate of transport accidents are generated. Furthermore, an environmentally and socially conscious CLSC network with maximal job creation in the less developed regions, minimal lost days resulting from the work's damages during manufacturing/recycling operations and maximal collection/recovery of end-of-life products is also designed.Originality/valueUnlike the classical network design models, vehicle fleet planning decisions such as fleet sizing/composition, fleet assignment, vehicle inventory control, empty repositioning, etc. are also considered while designing a sustainable CLSC network. In addition to sustainability indicators in the network design, sustainability factors in fleet management are also handled. To the best of the authors' knowledge, there is no similar paper in the literature that proposes such a holistic optimization model for integrated sustainable fleet planning and CLSC network design.

Data-Driven Approximation of Contextual Chance-Constrained Stochastic Programs

Data-Driven Approximation of Contextual Chance-Constrained Stochastic Programs

The new approaches for solving hierarchical Chinese postman problem with stochastic travel times

The hierarchical Chinese postman problem (HCPP) aims to find the shortest tour or tours by passing through the arcs classified according to precedence relationship. HCPP, which has a wide application area in real-life problems such as shovel snow and routing patrol vehicles where precedence relations are important, belongs to the NP-hard problem class. In real-life problems, travel time between the two locations in city traffic varies due to reasons such as traffic jam, weather conditions, etc. Therefore, travel times are uncertain. In this study, HCPP was handled with the chance-constrained stochastic programming approach, and a new type of problem, the hierarchical Chinese postman problem with stochastic travel times, was introduced. Due to the NP-hard nature of the problem, the developed mathematical model with stochastic parameter values cannot find proper solutions in large-size problems within the appropriate time interval. Therefore, two new solution approaches, a heuristic method based on the Greedy Search algorithm and a meta-heuristic method based on ant colony optimization were proposed in this study. These new algorithms were tested on modified benchmark instances and randomly generated problem instances with 817 edges. The performance of algorithms was compared in terms of solution quality and computational time.

A new multi-timescale optimal scheduling model considering wind power uncertainty and demand response

When wind power is connected to a power grid, intermittency and uncertainty increase the difficulty of power system dispatching and operation. A multi-timescale optimal scheduling model based on wind power uncertainty and demand response are proposed to address the uncertainty of wind power integration. The demand response program is first implemented on the customer side to adjust the electricity consumption pattern and reduce the peak-valley load difference. Then, the robust optimization based on extreme scenarios is used in the day-ahead scheduling stage according to the characteristic that the wind power prediction accuracy gradually improves step by step with the refinement of timescales so that the decision variable meets requirements in all scenarios. In the intraday scheduling stage, the method based on stochastic chance-constrained programming is used, and the out-of-bounds phenomenon is allowed in extreme cases. During real-time scheduling, the deviation remaining after intraday scheduling is corrected. Finally, a 10-unit system is used as an example to demonstrate the feasibility and effectiveness of the proposed scheduling model. The simulation results show that the total economic cost, reserve cost and wind power curtailments of the real-time scheduling model are reduced by 18.21%, 66.06% and 66.57%, respectively, compared with the day-ahead scheduling model. In addition, compared with the intraday scheduling model, the wind power curtailments of the real-time scheduling model are reduced by 20.00%, which proves the necessity of multi-timescale coordinated scheduling.

A chance-constrained optimization framework for wind farms to manage fleet-level availability in condition based maintenance and operations

Operations and maintenance (O&M) is a key contributor to wind farm expenditures. To increase competitiveness, wind farm operators are increasingly looking into leveraging real-time sensor data from condition monitoring (CM) systems. CM provides significant insights on evolving asset failure risks for wind turbines. To date, these insights have not been fully leveraged in wind farm O&M due to ad-hoc connections to decision-making. Specifically, CM applications in wind farms have been limited to detection of turbines with imminent failure risks that require immediate replacement. In reality, wind farm maintenance requires a careful proactive orchestration of O&M dependencies across turbines along with multiple sources of uncertainty associated with asset availability, operational and market conditions. This paper proposes a unified condition-based maintenance and operations scheduling approach for wind farms that models uncertainties related to turbine availability, wind power output and market price. The proposed formulation explicitly considers the turbine-to-turbine dependencies in operations and maintenance, such as opportunistic maintenance, to identify the O&M decisions that are optimal for multiple wind farms. The problem is formulated as a chance-constrained stochastic programming model to maximize operational revenue while ensuring high levels of turbine availability and generation. To make the chance constraints tractable, two approximations are proposed with a focus on sample average approximation (SAA) and prominent tail inequalities such as Markov’s inequality and Chernoff bound. Our results on a comprehensive set of experiments demonstrate that the proposed approach provides significant improvements in asset availability, market revenue and maintenance costs in large scale wind farms.

Humanitarian relief network design: Responsiveness maximization and a case study of Typhoon Rammasun

In this article, we study a humanitarian relief network design problem, where the demand for relief supplies in each affected area is uncertain and can be met by more than one relief facility. Given a certain cost budget, we simultaneously optimize the decisions of relief facility location, inventory pre-positioning, and relief facility to affected area assignment so as to maximize the responsiveness. The problem is formulated as a chance-constrained stochastic programming model in which a joint chance constraint is utilized to measure the responsiveness of the humanitarian relief network. We approximate the proposed model by another model with chance constraints, which can be solved based on two settings of the demand information in each affected area: (i) the demand distribution is given; and (ii) the partial demand information, e.g., the mean, the variance, and the support, is given. We use a case study of the 2014 Typhoon Rammasun to illustrate the application of the model. Computational results demonstrate the effectiveness of the solution approaches and show that the chance-constrained stochastic programming models are superior to the deterministic model for humanitarian relief network design.

Nonconvex and Nonsmooth Approaches for Affine Chance-Constrained Stochastic Programs

Chance-constrained programs (CCPs) constitute a difficult class of stochastic programs due to its possible nondifferentiability and nonconvexity even with simple linear random functionals. Existing approaches for solving the CCPs mainly deal with convex random functionals within the probability function. In the present paper, we consider two generalizations of the class of chance constraints commonly studied in the literature; one generalization involves probabilities of disjunctive nonconvex functional events and the other generalization involves mixed-signed affine combinations of the resulting probabilities; together, we coin the term affine chance constraint (ACC) system for these generalized chance constraints. Our proposed treatment of such an ACC system involves the fusion of several individually known ideas: (a) parameterized upper and lower approximations of the indicator function in the expectation formulation of probability; (b) external (i.e., fixed) versus internal (i.e., sequential) sampling-based approximation of the expectation operator; (c) constraint penalization as relaxations of feasibility; and (d) convexification of nonconvexity and nondifferentiability via surrogation. The integration of these techniques for solving the affine chance-constrained stochastic program (ACC-SP) is the main contribution of this paper. Indeed, combined together, these ideas lead to several algorithmic strategies with various degrees of practicality and computational efforts for the nonconvex ACC-SP. In an external sampling scheme, a given sample batch (presumably large) is applied to a penalty formulation of a fixed-accuracy approximation of the chance constraints of the problem via their expectation formulation. This results in a sample average approximation scheme, whose almost-sure convergence under a directional derivative condition to a Clarke stationary solution of the expectation constrained-SP as the sample sizes tend to infinity is established. In contrast, sequential sampling, along with surrogation leads to a sequential convex programming based algorithm whose asymptotic convergence for fixed- and diminishing-accuracy approximations of the indicator function can be established under prescribed increments of the sample sizes.

A novel approach for solving decision-making problems with stochastic linear-fractional models

Stochastic chance-constrained optimization has a wide range of real-world applications. In some real-world applications, the decision-maker has to formulate the problem as a fractional model where some or all of the coefficients are random variables with joint probability distribution. Therefore, these types of problems can deal with bi-objective problems and reflect system efficiency. In this paper, we present a novel approach to formulate and solve stochastic chance-constrained linear fractional programming models. This approach is an extension of the deterministic fractional model. The proposed approach, for solving these types of stochastic decision-making problems with the fractional objective function, is constructed using the following two-step procedure. In the first stage, we transform the stochastic linear fractional model into two stochastic linear models using the goal programming approach, where the first goal represents the numerator and the second goal represents the denominator for the stochastic fractional model. The resulting stochastic goal programming problem is formulated. The second stage implies solving stochastic goal programming problem, by replacing the stochastic parameters of the model with their expectations. The resulting deterministic goal programming problem is built and solved using Win QSB solver. Then, using the optimal value for the first and second goals, the optimal solution for the fractional model is obtained. An example is presented to illustrate our approach, where we assume the stochastic parameters have a uniform distribution. Hence, the proposed approach for solving the stochastic linear fractional model is efficient and easy to implement. The advantage of the proposed approach is the ability to use it for formulating and solving any decision-making problems with the stochastic linear fractional model based on transforming the stochastic linear model to a deterministic linear model, by replacing the stochastic parameters with their corresponding expectations and transforming the deterministic linear fractional model to a deterministic linear model using the goal programming approach

Multi‐agent decentralized microgrids planning considering long‐term demand response model

Demand-side management resources, such as demand response programs (DRPs) are useful options for the reduction of energy cost according to the viewpoint of department of energy. Utilizing these resources in long-term studies like generation and transmission expansion planning (GTEP) needs long-term modeling. In this article, the GTEP was developed considering long-term model of DRPs using a cooperative game theory approach to minimize the total cost of microgrids (MGs). A two-level decision-making method was used to develop this model. On the top level, investment decisions were cooperatively made by MGs and a stochastic chance-constrained mixed integer linear programming formulation was modeled for investment decisions with operational uncertainties on the bottom level. The self-sufficiency index was also considered that guarantees supply of sensitive loads in islanding mode. Effectiveness of the proposed model was shown by numerical studies in three cases where in Case 1, load of each smart MG (SMG) will be supplied using its own resources and trading with the retail market. In Case 2, SMGs can also trade with each other in non-cooperative (profitable) approach and in Case 3, SMGs can trade with each other in cooperative approach. The total cost of SMGs was decreased in Case 2 compared to the previous case by 2.62%, which was due to the use of other SMGs' resources to provide load. Also, the total cost of SMGs was decreased in Case 3 compared to the two previous cases (2.92% and 0.3% decrease compared to the Cases 1 and 2, respectively), which was due to cooperative approach taken by SMGs in this case. In this approach, SMGs provide each other's load without additional cost and try to minimize the total cost of all the SMGs.

A dual-randomness bi-level interval multi-objective programming model for regional water resources management

In this research, a dual-randomness bi-level interval multi-objective programming (DR-BIMP) model was developed for supporting water resources management among multiple water sectors under complexities and uncertainties. Techniques of bi-level multi-objective programming (BMOP), double-sided stochastic chance-constrained programming (DSCCP), and interval parameter programming (IPP) were incorporated into an integrated modeling framework to achieve comprehensive consideration of the complexities and uncertainties of water resources management systems. The DR-BIMP model can not only effectively deal with the interactive effects between multiple decision-makers in complex water management systems through the bi-level hierarchical strategies, but also can characterize the multiple uncertainties information expressed as interval format and probability density functions. It could thus improve upon the existing bi-level multi-objective programming through addressing discrete interval parameters and dual-randomness problems in optimization processes simultaneously. Then, the developed model was applied to a real-world case to optimally allocate water resources among three different water sectors in five sub-regions in the Dongjiang River basin, south China. The results of the model include determining values, interval values, and stochastic distribution information, which can assist bi-level decision-makers to plan future resources effectively to some extent. After comparing the variations of results, it is found that an increasing probability level can lead to higher system benefits, which is increased from [20,786.00, 26,425.92] × 108 CNY to [22,290.84, 27,492.57] × 108 CNY, while the Gini value is reduced from [0.365, 0.446] to [0.345, 0.405]. A set of increased probability levels gives rise to the lower-level objectives. Furthermore, the advantages of the DR-BIMP model were highlighted by comparing with the other models originated from the developed model. The comparison results indicated that the DR-BIMP model was a valuable tool for generating a range of decision alternatives and thus assists the bi-level decision-makers to identify the desired water resources allocation schemes under multiple scenarios.

Analyzing and Quantifying the Intrinsic Distributional Robustness of CVaR Reformulation for Chance-Constrained Stochastic Programs

Chance-constrained program (CCP) is a popular stochastic optimization method in power system planning, and operation problems. Conditional Value-at-Risk (CVaR) provides a convex approximation for chance constraints which are nonconvex. Although CCP assumes an exact empirical distribution, and the optimum of a stochastic programming model is thought to be sensitive in the designated probability distribution, this letter discloses that CVaR reformulation of a chance constraint is intrinsically robust. A pair of indices are proposed to quantify the maximum tolerable perturbation of the probability distribution, and can be computed from a computationally-cheap dichotomy search. An example on the coordinated capacity optimization of energy storage, and transmission line for a remote wind farm validates the main claims. The above results demonstrate that stochastic optimization methods are not necessarily vulnerable to distributional uncertainty, and justify the positive effect of the conservatism brought by the CVaR reformulation.

The interval copula-measure Me based multi-objective multi-stage stochastic chance-constrained programming for seasonal water resources allocation under uncertainty

A copula-measure Me based interval multi-objective multi-stage stochastic chance-constrained programming (CMIMOMSP) model is proposed for water consumption optimization. It can conduct water allocation amid multiple users and multiple stages, and deal with the uncertainties presented as interval numbers, random fuzzy interval numbers, and stochastic variables. It improves upon multi-stage stochastic chance-constrained programming by introducing the multi-objective programming, and it can tradeoff the relationships amid economic benefit, full usage of water resources, and economic loss. It enhances the accuracy of copula function and conditional distribution function through proposing the interval functions. Besides, it can deal with the impact of the decision attitudes of managers on water allocation by formulating the function equation between water demand and the optimistic-pessimistic factor. The CMIMOMSP model is applied to a case study of the Heihe River Basin to verify its application. The results indicate that: (1) the optimistic-pessimistic factors have different degrees of positive influences on water allocation for industrial, domestic and ecological sectors; (2) the joint violated probability and optimistic-pessimistic factor have various range of impacts on agricultural water allocation; (3) tthe objective function values have different variation tendencies with the rise of joint violated probabilities and optimistic-pessimistic factors. Its robustness is enhanced by comparing it with the three single-objective programming models. The CMIMOMSP model can provide various water allocation schemes for managers with different risk attitudes in semi-arid and arid districts.

An Interval Fuzzy, Double-Sided, Chance-Constrained Stochastic Programming Model for Planning the Ecological Service Value of Interconnected River Systems

The western region of Jilin Province is an ecologically fragile area with scarce water resources. The effective allocation of the limited water resources in order to obtain a higher ecological service value is an urgent requirement. In this paper, an interval fuzzy, double-sided chance-constrained, stochastic programming (IFDCP) model was used based on the interconnected river system network project in the western Jilin Province. With the objective of maximizing the value of regional ecological services, the water diversion and supplement schemes were optimized and adjusted. The model results showed that the restored water surface area of all lakes and ponds in the western region of Jilin Province was higher than the initially planned scheme in the high flow year. The water surface area fulfilled the minimum constraints, but did not fulfill the initial scheme in the normal flow year. In the low flow year, the lower limit of some of the regions had to be decreased in order to meet the allocation of the limited water resources. The proportion of floodwater resource utilization gradually increased with an increase in the flood amount. The ecological service value produced in the normal and high flow years was found to be higher than the initial scheme. The marsh wetland can produce higher ecological service value. Therefore, the core of the model optimization was introducing more water to the marsh wetland after fulfilling the basic consumption of ponds and the reed wetland. In addition, the IFDCP model was more flexible in water diversion and supplement as compared to other models that had been developed previously.

Activity Planning for Assistive Robots Using Chance-Constrained Stochastic Programming

In this article, we present a framework for planning an activity to be executed with the support of a robotic navigation assistant. The two main components are the activity and the motion planner. The activity planner composes a sequence of abstract activities, chosen from a given set, to synthesize a plan. Each activity is associated with a point of interest in the environment and with probabilistic parameters that depend on the plan, which are characterized by simulations in realistic scenarios. The low-level action to pass from an activity to the next is handled by the motion planner, which secures the physical feasibility of the chosen actions and their compatibility with the constraints posed by the user and the environment. Indeed, the final plan must respect the user constraints and optimise his/her satisfaction from the activity. We show a possible model for the problem as a chance constrained optimization along with an efficient technique to find high-quality solutions.

A location model for capacitated alternative-fuel stations with uncertain traffic flows

To maximize the total amount of alternative fuel consumption in a traffic network and satisfy the travel demands to the greatest extent possible, this paper addresses the problem of finding the optimal locations in a transportation network in which to construct capacitated alternative-fuel stations with a limited construction budget. Different from most existing studies that treat traffic flows in each origin-destination pair as a constant, a stochastic chance-constrained programming model (SCCPM) is proposed to characterize the problem of interest in this paper, where random variables are employed to capture the uncertainty of traffic flows. Moreover, for solution convenience, a linear equivalent model of the proposed one is deduced. Finally, two sets of numerical experiments are implemented by the Cplex solver to test the performance of the model. Firstly, a series of experiments are conducted on the small scale network to analyze the sensitivities of the parameters in the proposed model. Then we find the optimal locations for alternative-fuel stations in the large-scale network and draw the conclusion that most of the alternative-fuel vehicles are located in demands cluster districts according to the thermodynamic diagram.

Flexible units planning and operational optimization model under large-scale wind power access

After large-scale wind power access to the power grid, the requirements of flexibility of power system become higher, and it is necessary to transform the flexibility of the largest proportion of resources, the thermal units. Firstly, a mathematical model of flexibility requirements and supply of power system is established. Secondly, a supply and demand of flexibility matching probability model based on stochastic chance-constrained programming is built (CCP). Then, a flexible resource planning and operation optimization model is proposed aim to minimize the operational cost of the system. After the transformation of the model, it is solved by using the commercial solver CPLEX. Finally, an example is used to verify the validity of the model. The results show that after the transformation the thermal units can promote the consumption of wind power and reduce the total cost of planning and operation.