Chance-constrained Programming Model Research Articles

A robust possibilistic programming model for simultaneous decision of inventory lot-size, supplier selection and transportation mode selection

As a result of the uncertain changes in global currencies, exchange rates have become a critical part of suppliers' selection problem. In the present paper the problem of supplier, transportation mode selection and inventory replenishment decision (STSI) in the presence of uncertain changes of currency exchange rates and price discounts have been studied. The present article addresses the STSI problem for a buyer sourcing a product from heterogeneous suppliers. Global suppliers' selection procedure is based on the prices that suppliers offer in their local currencies. Late deliveries and rejections also exert influence on sourcing and inventory replenishment decisions. The proposed model is a bi-objective mathematical programming, objective functions of which are minimising the total cost and the late delivery of items. In order to cope with uncertain parameters effectively, a robust possibilistic programming approach is utilised. Finally, to assess the robustness of the solutions obtained by the novel robust optimisation model, our model's results are compared to those generated by the chance-constrained programming model under the data of STAM SANAT Company. Further, the advantages of using this integrated approach versus a sequential approach under currency fluctuations are shown.

A chance-constrained two-stage stochastic programming model for reliable microgrid operations under power demand uncertainty

Abstract In this study, we present a scalable quantitative modeling framework to evaluate the impacts of variability in renewable resources (e.g., solar energy) in the microgrid operation. The problem is formulated as a chance-constrained two-stage stochastic programming model. We use three different policies to ensure that the utilization of renewable energy is high in microgrid operations. More specifically, the first policy enforces a certain percentage of renewable energy utilization from the overall time period while the second and third policies enforce renewable energy utilization for specific hours and every operating hours, respectively. The proposed optimization model is solved using a combined Sample Average Approximation algorithm. Computational results indicate that policy three is more restrictive than the other two policies; thus, we observe high utilization of renewable energy resources in the microgrid operation under this policy. Further, we observe potential benefits of having solar panels under different fuel based generation (FBG) or distributed energy sources (DES) units on fulfilling the required energy demand during a power outage. Finally, the proposed model provides an optimal dispatch of available resources (e.g., renewable energy, DES, FBG units) in such a way that the overall microgrid operational cost is minimized.

A chance-constrained two-stage stochastic programming model for humanitarian relief network design

We consider a stochastic pre-disaster relief network design problem, which mainly determines the capacities and locations of the response facilities and their inventory levels of the relief supplies in the presence of uncertainty in post-disaster demands and transportation network conditions. In contrast to the traditional humanitarian logistics literature, we develop a chance-constrained two-stage mean-risk stochastic programming model. This risk-averse model features a mean-risk objective, where the conditional value-at-risk (CVaR) is specified as the risk measure, and enforces a joint probabilistic constraint on the feasibility of the second-stage problem concerned with distributing the relief supplies to the affected areas in case of a disaster. To solve this computationally challenging stochastic optimization model, we employ an exact Benders decomposition-based branch-and-cut algorithm. We develop three variants of the proposed algorithm by using alternative representations of CVaR. We illustrate the application of our model and solution methods on a case study concerning the threat of hurricanes in the Southeastern part of the United States. An extensive computational study provides practical insights about the proposed modeling approach and demonstrates the computational effectiveness of the solution framework.

Uncertain chance-constrained programming model for project scheduling problem

In this paper, we consider an uncertain project scheduling problem, in which activity durations, with no historical data generally, are estimated by belief degrees and assumed to be uncertain variables. To achieve different management goals, we build three uncertain chance-constrained programming models for project scheduling problem, in which the chance constraint must reach a predetermined confidence level. Moreover, these models can all be transformed to their crisp forms, and an intelligent algorithm is designed to search the optimal schedule. Finally, a numerical example is presented to illustrate the usefulness of the proposed model.

Interval Double-Sided Fuzzy Chance-Constrained Programming Model for Water Resources Allocation

Abstract In this study, an interval double-sided fuzzy chance-constrained programming (IDFCP) approach is developed for identifying water resources allocation strategies under uncertainty. Through incorporating interval parameter programming, double-sided fuzzy programming, and chance-constrained programming into a general framework, IDFCP can effectively deal with uncertainties expressed as intervals, probability distributions, and fuzzy sets. IDFCP can also examine the risk of violating system constraints. IDFCP is then applied to water resources allocation in the middle and upper reaches of Fen River Basin that is associated with multiuser, multiregion, and multisource features. Interval solutions of the compromise decision alternatives are generated under different scenarios in association with different risk levels of violating constraints (i.e., p levels), fuzzy membership degrees (i.e., α-cut levels), credible degrees (i.e., minimum and maximum), and reclaimed water utilization ratios. Results obta...

Inexact nonlinear improved fuzzy chance-constrained programming model for irrigation water management under uncertainty

An inexact nonlinear mλ-measure fuzzy chance-constrained programming (INMFCCP) model is developed for irrigation water allocation under uncertainty. Techniques of inexact quadratic programming (IQP), mλ-measure, and fuzzy chance-constrained programming (FCCP) are integrated into a general optimization framework. The INMFCCP model can deal with not only nonlinearities in the objective function, but also uncertainties presented as discrete intervals in the objective function, variables and left-hand side constraints and fuzziness in the right-hand side constraints. Moreover, this model improves upon the conventional fuzzy chance-constrained programming by introducing a linear combination of possibility measure and necessity measure with varying preference parameters. To demonstrate its applicability, the model is then applied to a case study in the middle reaches of Heihe River Basin, northwest China. An interval regression analysis method is used to obtain interval crop water production functions in the whole growth period under uncertainty. Therefore, more flexible solutions can be generated for optimal irrigation water allocation. The variation of results can be examined by giving different confidence levels and preference parameters. Besides, it can reflect interrelationships among system benefits, preference parameters, confidence levels and the corresponding risk levels. Comparison between interval crop water production functions and deterministic ones based on the developed INMFCCP model indicates that the former is capable of reflecting more complexities and uncertainties in practical application. These results can provide more reliable scientific basis for supporting irrigation water management in arid areas.

Monte Carlo simulation based interval chance-constrained programming for regional ecosystem management – A case study of Zhuhai, China

A Monte Carlo simulation based interval chance-constrained programming (MC-ICCP) model is developed for planning regional ecosystem under uncertainty. MC-ICCP cannot only deal with uncertainties described as discrete intervals and probability distributions, but also examine the risk of violating system constraints. MC-ICCP is then applied to planning regional ecosystem sustainable development in the City of Zhuhai, where different ecosystem service value (ESV) assessment measures, risk levels and eco-environmental indicators are investigated. Solutions of land-use pattern, system benefit, and ecosystem service value under 192 scenarios have been obtained. Results reveal that (i) different ESV assessment measures result in different grass and forest areas; (ii) ESV assessment measure can affect land-use pattern and system benefit; (iii) risk level (p) is the most impact factor. Decisions at a lower risk level (a lower p level) would lead to an increased reliability in fulfilling ecosystem requirements but with a lower system benefit. Results also disclose that forest and grass coverage has more significant effect on land-use pattern than reclamation intension and soil erosion. More attention should be paid on ecosystem land-use that could facilitate regional ecological sustainability. The findings could help decision makers balance system benefit and constraint-violation risk, adjust land-use pattern with consideration of ecosystem sustainability and protect regional ecological environment.

A chance-constrained programming framework to handle uncertainties in radiation therapy treatment planning

A stochastic programming framework is proposed for radiation therapy treatment planning. This framework takes into account uncertainty in setting up or positioning patients identically day-to-day during the treatment period. Because uncertainties are unavoidable, constraint violations are tolerated to some degree in practice. Under this assumption, a chance-constrained programming (CCP) framework is developed to handle setup uncertainties in treatment planning. The proposed framework can be employed under different distributional assumptions. The goal of the proposed approach is to maximize both the statistical confidence level of a treatment plan and the homogeneity of the dose distributions. This novel perspective provides a user-centric and personalized optimization model that allows a trade-off between sufficient tumor coverage and sparing healthy tissues under uncertainty. We describe testing of the performance of the proposed CCP models in terms of plan quality, robustness, and homogeneity and confidence level of the constraints using clinical data for a prostate cancer patient. Optimized CCP plans are also compared to plans developed using a deterministic approach that does not take uncertainties into account. Numerical experiments confirmed that the CCP is able to control setup uncertainties in target coverage and sparing of organs-at-risk.

Development of a fuel management model for a multi-source district heating system under multi-uncertainty and multi-dimensional constraints

In this paper, an interval single-sided fuzzy chance-constrained mixed-integer programming model is developed for the fossil fuel management of a multi-source district heating system under multiple uncertainties, where the heat-supply-capacity expansion planning can also be reflected. The non-dimensional comprehensive equations method is simultaneously improved to quantify the ambiguous heat provisions, utilized as a type of boundary condition inputs to the proposed model. A real-world case study of a heating system located in northeastern China is undertaken to show the feasibility and applicability of the proposed methods. To obtain the reasonable fossil-fuel management schemes, multi-dimensional constraints are incorporated into the model based on a comprehensive consideration in terms of fuel supply and demand, quality and quantity, economic cost and environment protection, as well as their interactions. Results obtained from the case study indicate that the solutions for both continuous and binary variables have been generated, which are useful for identifying suitable fuel-supply patterns and heat-source operational modes for a heating system under different system reliabilities and heating-load distribution states. In addition, the results also reveal that the fossil-fuel management and heating-capacity-expansion pattern, as well as the economic cost and pollutant emission performances are sensitive to the thermal coefficient and system reliability level, which may provide in-depth analyses of tradeoffs for further supporting robust fossil-fuel management under uncertainty.

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 interval chance-constrained fuzzy modeling approach for supporting land-use planning and eco-environment planning at a watershed level

An interval chance-constrained fuzzy land-use allocation (ICCF-LUA) model is proposed in this study to support solving land resource management problem associated with various environmental and ecological constraints at a watershed level. The ICCF-LUA model is based on the ICCF (interval chance-constrained fuzzy) model which is coupled with interval mathematical model, chance-constrained programming model and fuzzy linear programming model and can be used to deal with uncertainties expressed as intervals, probabilities and fuzzy sets. Therefore, the ICCF-LUA model can reflect the tradeoff between decision makers and land stakeholders, the tradeoff between the economical benefits and eco-environmental demands. The ICCF-LUA model has been applied to the land-use allocation of Wujiang watershed, Guizhou Province, China. The results indicate that under highly land suitable conditions, optimized area of cultivated land, forest land, grass land, construction land, water land, unused land and landfill in Wujiang watershed will be [5015, 5648] hm2, [7841, 7965] hm2, [1980, 2056] hm2, [914, 1423] hm2, [70, 90] hm2, [50, 70] hm2 and [3.2, 4.3] hm2, the corresponding system economic benefit will be between 6831 and 7219 billion yuan. Consequently, the ICCF-LUA model can effectively support optimized land-use allocation problem in various complicated conditions which include uncertainties, risks, economic objective and eco-environmental constraints.

Chance-Constrained Dynamic Programming for Multiple Water Resources Allocation Management Associated with Risk-Aversion Analysis: A Case Study of Beijing, China

Water shortage and water pollution have become major problems hindering socio-economic development. Due to the scarcity of water resources, the conflict between water supply and demand is becoming more and more prominent, especially in urban areas. In order to ensure the safety of urban water supply, many cities have begun to build reservoirs. However, few previous studies have focused on the optimal allocation of water resources considering storage reservoirs. In this study, a multi-water resources and multiple users chance-constrained dynamic programming (MMCDP) model has been developed for water resources allocation in Beijing, China, which introduces reservoir and chance-constrained programming into the dynamic programming decision-making framework. The proposed model can distribute water to different departments according to their respective demands in different periods. Specifically, under the objective of maximal benefits, the water allocation planning and the amount of water stored in a reservoir for each season under different feasibility degrees (violating constraints or available water resources situations) can be obtained. At the same time, the model can be helpful for decision-makers to identify the uncertainty of water-allocation schemes and make a desired compromise between the satisfaction degree of the economic benefits and the feasibility degree of constraints.

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.

Two-Stage Stochastic Chance-Constrained Fractional Programming Model for Optimal Agricultural Cultivation Scale in an Arid Area

AbstractIn the agricultural irrigation system, agricultural water availability and demand are significantly affected by seasonal variations. Additionally, agricultural land use in arid regions larg...

Uncertain programming models for fixed charge multi-item solid transportation problem

This paper investigates the fixed charge multi-item solid transportation problem, in which the fixed charges, direct costs, transportation capacities, supply and demand are uncertain variables. Based on the uncertainty theory, expected value programming model and chance-constrained programming model for fixed charge multi-item solid transportation problem are constructed, respectively. We can obtain the optimal solution of two models via solving the relevant deterministic models. Finally, a numerical experiment is implemented to illustrate the application of the models.

Planning and scheduling of the make-and-pack dairy production under lifetime uncertainty

• We consider a planning and scheduling problem in the make-and-pack production. • We propose a chance constrained linear programming model. • The model minimizes the makespan and accounts for the uncertainty in quality decay. • The numerical study reflects real settings from a yoghurt production case. • The proposed production schedule is evaluated by a simulation model. In the dairy processing, the rapid quality decay of milk-based intermediate mixture to make and pack restricts productivity and, forces organizations to carefully plan and schedule their production. Hereby, in this study, we consider a planning and scheduling problem encountered in the dairy industry and propose a chance-constrained programming model accounting for uncertainty in quality decay of intermediate mixture. The aim of the model is to find the optimal lot sizes and production schedule with minimum makespan (total time needed to finish the daily production). The proposed schedule allows the storage duration of intermediate mixture to be within a stochastic lifetime. A case study is presented to illustrate the typical structure of the two-stage semi-continuous make-and-pack production process. The numerical study reflects real settings from a set (Balkan type) yoghurt production process. Accordingly, a simulation of the production process is introduced to evaluate the proposed production plan and schedule in terms of product waste. The model is examined with 32 scenarios consisting of different distribution parameters, confidence levels and demand patterns. Overall in the scenarios, the proposed plan and schedule result in decreasing 2693 l of product waste with 3.24 h increase of makespan in total.

Two uncertain chance-constrained programming models to setting target levels of design attributes in quality function deployment

Quality function deployment (QFD) is widely acknowledged as a customer-oriented product design tool, which is generated by translating consumer demands into design attributes of a product. In order to depict the internal ambiguous factors in the development process more appropriately, uncertain variables with a specialized kind of regular uncertainty distributions based on uncertainty theory are applied. Subsequently, two uncertain chance-constrained programming (CCP) models used for formulating the QFD procedure are set forth, whose objectives are maximizing the consumer satisfaction and minimizing the design cost, respectively. To demonstrate the feasibility of the proposed modelling approach, an example of the motorcycle design problem is illustrated, in which the new target levels of design attributes are selected and analyzed according to the decision-makers’ subjectivity and preference at different confidence levels. Additionally, a comparative study between the uncertain CCP approach and another uncertain expected value modelling approach is conducted. The results indicate that uncertain CCP models are more suitable for optimization in the QFD procedure.

Optimal Latin hypercube sampling-based surrogate model in NAPLs contaminated groundwater remediation optimization process

Abstract A surrogate model based groundwater optimization model was developed to solve the non-aqueous phase liquids (NAPLs) contaminated groundwater remediation optimization problem. To illustrate the impact of sampling method improvement to the surrogate model performance improvement, aiming at a nitrobenzene contaminated groundwater remediation problem, optimal Latin hypercube sampling (OLHS) method was introduced to sample data in the input variables feasible region, and a radial basis function artificial neural network was used to construct a surrogate model. Considering the surrogate model's uncertainty, a chance-constrained programming (CCP) model was constructed, and it was solved by genetic algorithm. The results showed the following, for the problem considered in this study. (1) Compared with the Latin hypercube sampling (LHS) method, the OLHS method improves the space-filling degree of sample points considerably. (2) The effects of the two sampling methods on surrogate model performance were analyzed through comparison of goodness of fit, residual and uncertainty. The results indicated that the OLHS-based surrogate model performed better than the LHS-based surrogate model. (3) The optimal remediation strategies at 99%, 95%, 90%, 85%, 80% and 50% confidence levels were obtained, which showed that the remediation cost increased with the confidence level. This work would be helpful for increasing surrogate model performance and lowering the risk of a groundwater remediation strategy.

Optimizing best management practices for nutrient pollution control in a lake watershed under uncertainty

In this research work, soil and water assessment tool (SWAT) and fuzzy credibility chance-constrained programming (FCCP) model were integrated into an optimization framework for supporting nutrient pollution control in a lake watershed. The framework, so-called SWAT-based FCCP (SFCCP) model, was advantageous in simulating non-point sources (NPS) pollution, optimizing best management practices (BMPs), and addressing system uncertainties. SFCCP was solved by genetic algorithm (GA) for searching optimal placement schemes of BMPs at a lower system cost, where the related uncertainties were addressed as fuzzy parameters. The developed SFCCP model was applied to seek optimal types, sizes and locations of BMPs for nutrient pollution control in a lake watershed system, i.e. Lake Dianchi watershed, China. The study results indicated that when the credibility level increased from 0.55 to 0.95, the total system cost would increase from $8.89 to $12.27×106; meanwhile, the total nitrogen (TN) load discharged into the lake would decrease from 1.22 to 1.19×106kg/yr and the total phosphorus (TP) load would reduce from 51.37 to 50.11×103kg/yr, respectively. It appeared that a higher credibility level would lead to a stricter control requirement, namely a higher reduction of nutrient load by BMPs and a higher system cost. The proposed model could be used to help generate a series of BMPs placement schemes under various credibility levels; this ensures that the nutrient load brought into the lake and tributaries could drop to an acceptable level, with a proper tradeoff between system cost and risk being considered.