Stochastic Chance-constrained Programming Model Research Articles

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 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.

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.

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.

Chance‐constrained maintenance scheduling for interdependent power and natural gas grids considering wind power uncertainty

Considering the increased interactions between power grids and natural gas grids, this study presents a chance-constrained maintenance scheduling model for integrated gas-electric grids with wind energy integration. Given the uncertainties of wind power, the loss of wind power probability is modelled as a chance constraint, ensuring the high utilisation of wind power. To overcome the adversities caused by the non-linear and non-convex models of natural gas systems, a piecewise linear approximation method is employed to transform the non-linear models into a group of mixed integer linear models. A big-M formulation method is used to construct inequality constraints for lines/pipelines to be under maintenance. In addition, unit commitment is also coordinated to achieve the best maintenance strategies. The proposed chance-constrained stochastic programming model is converted into an equivalent deterministic programming model via a strong extended formulation for the sample average approximation by leveraging the star-inequalities. Several tests on a four-node natural gas system with a six-bus power system and a 20-node natural gas system with a modified IEEE 118-bus power system demonstrate the effectiveness of the proposed model.

Application of a Fuzzy Two-Stage Chance Constrained Stochastic Programming Model for Optimization of the Ecological Services Value of the Interconnected River System Network Project in the Western Jilin Province, China

Incremental ecological service value is an important assessment indicator for ecosystem restoration projects, such as the interconnected river system network project in western Jilin Province, China. In this paper, a fuzzy two-stage chance-constrained stochastic programming (FTSCCP) model is proposed for the optimization of regional incremental ecological service value through adjustments to the original water diversions and ecological water supplement schemes for ecosystem restoration under different management requirements and scenarios. The proposed method can effectively address uncertainties such as fuzzy parameters, probability distributions, and random variables in optimizing processes simultaneously. The results reveal that, under the requirement of meeting the original plan, a decrease in restoration scale would mainly appear under low-flow scenarios, and for the other scenarios, the original plan would be fully realized with an increase in flood utilization and a decrease in normal water supplementation. For the requirement of maximizing the incremental ecological service value, the adjustments would mainly be increases in water supplementation for reed wetlands and marsh wetlands and decreases in water surface area, including fish ponds and crab ponds. The ecological service values would be 11,256.79 million CNY (Chinese Yuan), 16,465.11 million CNY and 17,639.59 million CNY under low-, normal- and high-flow scenarios, respectively, with increases of 3.2%, 6.9% and 14.5% compared to the values based on the requirement of meeting the original plan. The results would be helpful for guiding regional ecosystem restoration and water resource management in the western Jilin Province.

Towards sustainable water resources planning and pollution control: Inexact joint-probabilistic double-sided stochastic chance-constrained programming model

This study presents an inexact joint-probabilistic double-sided stochastic chance-constrained programming (IJDSCCP) model for sustainable water resources planning and pollution control in water quality management systems under uncertainty. Techniques of interval parameter programming (IPP), joint-probabilistic programming (JPP) and double-sided stochastic chance-constrained programming (DSCCP) are incorporated into a modeling framework. The IJDSCCP can not only address uncertainties presented as interval parameters and double-sided randomness (i.e. both left-hand and right-hand sides) that are characterized as normal distributions, but also examine the reliability level of satisfying the entire system constraints. It further improves upon conventional stochastic chance-constrained programming for handing random uncertainties in the left-hand and right-hand sides of constraints. Moreover, a non-equivalent but sufficient linearization form of the IJDSCCP is presented to solve such a problem. Then, the model is applied to a representative case for water resources planning and pollution control. The results including water resources planning solutions, pollution control plans and system benefits under the combinations of different joint and individual probability levels will be obtained. The solutions are expressed as combinations of deterministic, interval and distributional information, which can facilitate analysis of different forms of uncertainties. After investigating and comparing the variations of results, it is found that an increasing joint probability level can lead to higher system benefits, i.e., [13,841.68, 21,801.81] × 106 Yuan (p = 0.01, p1 = 0.0033, p2 = 0.0033 and p3 = 0.0033), [14,150.26, 22,260.06] × 106 Yuan (p = 0.05, p1 = 0.0166, p2 = 0.0166 and p3 = 0.0166) and [14,280.55, 22,415.52] × 106 Yuan (p = 0.10, p1 = 0.033, p2 = 0.033 and p3 = 0.033). A set of decreased individual probability levels gives rise to the maximum system benefits at the same joint probability level. Furthermore, the results of the IJDSCCP are compared with a general interval-based optimization framework as well. Therefore, the results from the IJDSCCP are valuable for assisting managers in generating and identifying decision alternatives under different scenarios.

Note on “A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning”

A recent article by Zaghian et al. (2018) proposed reformulations, proposed reformulations of stochastic chance-constrained programming models for radiation therapy treatment planning. This note questions the validity of the proposed reformulations and shows that they are not equivalent to the original formulations. Two numerical examples illustrate that the approach proposed by Zaghian et al. (2018) provides approximation problems and not reformulations.

A two-stage chance-constrained stochastic programming model for a bio-fuel supply chain network

This study presents a two-stage chance-constrained stochastic programming model that captures the uncertainties due to feedstock seasonality in a bio-fuel supply chain network. The chance-constraint ensures that, with a high probability, Municipal Solid Waste (MSW) will be utilized for bio-fuel production. To solve our proposed optimization model, we use a combined sample average approximation algorithm. We use the state of Mississippi as a testing ground to visualize and validate the modeling results. Our computational experiments reveal some insightful results about the impact of MSW utilization on a bio-fuel supply chain network performance.

An inexact log-normal distribution-based stochastic chance-constrained model for agricultural water quality management

ABSTRACTIn this study, an inexact log-normal-based stochastic chance-constrained programming model was developed for solving the non-point source pollution issues caused by agricultural activities. Compared to the general stochastic chance-constrained programming model, the main advantage of the proposed model is that it allows random variables to be expressed as a log-normal distribution, rather than a general normal distribution. Possible deviations in solutions caused by irrational parameter assumptions were avoided. The agricultural system management in the Erhai Lake watershed was used as a case study, where critical system factors, including rainfall and runoff amounts, show characteristics of a log-normal distribution. Several interval solutions were obtained under different constraint-satisfaction levels, which were useful in evaluating the trade-off between system economy and reliability. The applied results show that the proposed model could help decision makers to design optimal production patterns under complex uncertainties. The successful application of this model is expected to provide a good example for agricultural management in many other watersheds.

A Chance-constrained operating room planning with elective and emergency cases under downstream capacity constraints

Abstract Operating room planning is an important and challenging decision making problem that should be tackled by hospitals. The present work is aimed at investigating this planning problem by accounting for the availability of two scarce and costly resources; the operating rooms and the Intensive Care Unit (ICU) which are shared between elective and emergency patients. The consideration of ICU beds is particularly important for operating room planning including cases where the patient requires an ICU bed after surgery. Indeed, in this case, without the availability of both an operating room time and an ICU bed, the surgery cannot be performed. A novel two-stage chance-constrained stochastic programming model is proposed for operating room planning by considering random surgery duration, random patient Length Of Stay (LOS) in the ICU and random resource capacity reserved for emergency cases. The objective is to minimize patient-related costs and expected operating room utilization costs and penalty costs for exceeding ICU capacity while ensuring a low risk level on surgery cancellation. A featured Sample Average Approximation (SAA) algorithm is developed to solve the model. Numerical experiments are carried out to verify the convergence property of the proposed algorithm and evaluate its performance. The results demonstrate the superiority of the operating room plans obtained by the proposed approach in terms of robustness. However, it is shown that this robustness is achieved at the expense of higher costs and lower operating room utilization.

Multi‐objective stochastic optimal planning method for stand‐alone microgrid system

To achieve economic and environmental benefit for the stand-alone microgrid consisting of diesel generators, wind turbine generators, photovoltaic generation system and lead-acid batteries, a multi-objective stochastic optimal planning method and a stochastic chance-constrained programming model are presented. In the model, the optimal objective is to simultaneously minimise the total net present cost and carbon dioxide emission in life cycle; the type and capacity of distributed generation units have been selected as the optimal variables; the loss of capacity is adopted as probability index constraint; the coordinated operation strategies between diesel generators and battery, the multi-unit operation constraints of diesel generators and the reserve capacity have been considered in the hard-circle operation strategy. Considering the uncertainties of wind speed, clearness index and load demand, Markov process transition probability matrix is adopted to synthesise those time series data. Optimal planning for an island microgrid system has been carried out by the planning system for microgrid (PSMG), a self-developed optimal planning software based on the multi-objective stochastic optimal planning method for stand-alone microgrid system.

Supply Chain Risk Management in Chinese Chemical Industry based on Stochastic Chance-Constrained Programming Model

It is necessary to research on how to manage risks in Chinese chemical industry supply chain effectively. After analyzing supply chain risks and mitigation strategies in Chinese chemical industry supply chain, firstly this paper defines the supply chain risk management problem in Chinese chemical industry. Then this paper develops a stochastic chance-constrained programming model to select optimal mitigation strategies effectively to achieve risk reduction goa ls without exceeding the expenses budget. Finally, this paper gives a numerical example and discusses how the change of model parameter affects the optimal solution.

Gamma distribution approach in chance-constrained stochastic programming model

In this article, a method is developed to transform the chance-constrained programming problem into a deterministic problem. We have considered a chance-constrained programming problem under the assumption that the random variables a ij are independent with Gamma distributions. This new method uses estimation of the distance between distribution of sum of these independent random variables having Gamma distribution and normal distribution, probabilistic constraint obtained via Essen inequality has been made deterministic using the approach suggested by Polya. The model studied on in practice stage has been solved under the assumption of both Gamma and normal distributions and the obtained results have been compared.