Chance-constrained Linear Programming Research Articles

Inexact Chance-Constrained Linear Programming Model for Optimal Water Pollution Management at the Watershed Scale

An inexact chance-constrained linear programming ICCLP model for optimal water pollution management at the watershed scale was developed. We selected the net expenditures of the alternative strategies, including initial capital investment and operating costs, as the objectives of water pollution management. The total environmental capacity of the water bodies at different probability levels qi was considered a key constraint; other constraints included in the model were government minimum requirements on farmland area, land cover, treatment rate of domestic wastewater and rural wastes, and certain technical constraints. The ICCLP model was applied to Lake Qionghai watershed in China for water quality improvement with the goal of achieving a minimum total cost. Alternative strategies were incorporated following discussions with shareholders and experts. A three-period optimization was conducted based on the alternative strategies; the model parameters were based on field investigations. Five probability levels were considered in the model: qi=0.01, 0.25, 0.50. 0.90, and 0.99. The model results showed that the total optimized costs were between US$55,710.86,80,691.8110 4 and US$72,151.39,101,338.610 4 under different probability levels. The model results indicate that soil erosion treatment, nonpoint source control measures, and rural waste treatment have much higher costs than other strategies, and our findings indicate that the ICCLP model can effectively deal with optimal water pollution management under uncertainty at the watershed scale.

IFTCP: An Integrated Method for Petroleum Waste Management under Uncertainty

Petroleum waste management has been of much concern in recent years since pollution from petroleum industries may lead to various impacts and risks to environmental systems. In this study, an interval fuzzy two-stage chance-constrained linear programming (IFTCP) method is developed for planning petroleum waste management systems. The IFTCP improves upon the existing optimization methods by allowing uncertainties presented in terms of intervals, fuzzy sets, and probability distributions to be effectively incorporated within the optimization framework. Moreover, it can support the analysis of policy scenarios that are associated with economic penalties when the promised targets are violated. The developed method is then applied to a case of long-term petroleum waste management planning. Interval solutions, which are associated with different levels of constraint-violation risk and system satisfaction degree, have been obtained by solving two submodels based on an interactive algorithm. They can be used to generate decision alternatives and support an in-depth analysis of the tradeoff between system cost and system-failure risk.

ICCLP: An Inexact Chance-Constrained Linear Programming Model for Land-Use Management of Lake Areas in Urban Fringes

Lake areas in urban fringes are under increasing urbanization pressure. Consequently, the conflict between rapid urban development and the maintenance of water bodies in such areas urgently needs to be addressed. An inexact chance-constrained linear programming (ICCLP) model for optimal land-use management of lake areas in urban fringes was developed. The ICCLP model was based on land-use suitability assessment and land evaluation. The maximum net economic benefit (NEB) was selected as the objective of land-use allocation. The total environmental capacity (TEC) of water systems and the public financial investment (PFI) at different probability levels were considered key constraints. Other constraints included in the model were land-use suitability, governmental requirements on the ratios of various land-use types, and technical constraints. A case study implementing the system was performed for the lake area of Hanyang at the urban fringe of Wuhan, central China, based on our previous study on land-use suitability assessment. The Hanyang lake area is under significant urbanization pressure. A 15-year optimal model for land-use allocation is proposed during 2006 to 2020 to better protect the water system and to gain the maximum benefits of development. Sixteen constraints were set for the optimal model. The model results indicated that NEB was between $1.48 x 10(9) and $8.76 x 10(9) or between $3.98 x 10(9) and $16.7 x 10(9), depending on the different urban-expansion patterns and land demands. The changes in total developed area and the land-use structure were analyzed under different probabilities (q ( i )) of TEC. Changes in q ( i ) resulted in different urban expansion patterns and demands on land, which were the direct result of the constraints imposed by TEC and PFI. The ICCLP model might help local authorities better understand and address complex land-use systems and develop optimal land-use management strategies that better balance urban expansion and grassland conservation.

On Distributionally Robust Chance-Constrained Linear Programs

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class.

ITCLP: An inexact two-stage chance-constrained program for planning waste management systems

An inexact two-stage chance-constrained linear programming (ITCLP) method is developed for planning waste management systems. The model is derived by incorporating the techniques of two-stage and chance-constrained programming within a general interval-optimization framework. It can tackle uncertainties presented as both probability distributions and discrete intervals. Moreover, it can be used for analyzing various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated. It can also help examine the reliability of satisfying (or risk of violating) system constraints under uncertainty. The developed method is applied to a case of long-term waste management planning. Interval solutions associated different risk levels of constraint violation are obtained. They can be used for generating decision alternatives and thus help waste managers to identify desired policies under various environmental, economic, and system-reliability constraints.

Optimal crop planning and water resources allocation in a coastal groundwater basin, Orissa, India

Intensive rice cultivation in a coastal groundwater basin in Balasore district of Orissa province (eastern India) during monsoon and winter seasons has resulted in extensive pumping of groundwater by a network of shallow, mini-deep and deep tubewells. Particularly, shallow tubewell owners using centrifugal pumps are unable to lift groundwater during winter seasons due to rapid drawdown of groundwater table below suction lift caused by mini-deep and deep tubewell owners. The seawater intrusion front is also progressing inland in an alarming rate. To develop a long-term sustainable land and water management strategies for the aforementioned issues in humid regions, the district administration realized the need for crop planning and water resources management policies in deterministic and stochastic regimes. As non-structural measure, the Deterministic linear programming (DLP) and chance-constrained linear programming (CCLP) models were developed to allocate available land and water resources optimally on seasonal basis so as to maximize the net annual return from the study area, considering net irrigation water requirement of crops as stochastic variable. These models were solved using the quantitative systems for business (QSB) software package. Sensitivity analysis of the models has been carried out by varying three ranges of cropping scenarios (20, 40 and 50% deviation from the existing cropping pattern) and combinations of surface water and groundwater at various risk levels (10, 20, 30 and 40%). The total groundwater available and withdrawal in the region are 655.87 × 10 3 and 255.03 × 10 3 km 3, respectively. The study reveals that 40% deviation of the existing cropping pattern is the optimal that satisfies the minimum food requirement and maintain geo-hydrological balance of the basin. The sensitivity analysis of conjunctive use of surface water and groundwater shows that 20% surface water and 30% groundwater availability as the optimum water allocation level. The proposed cropping and water resources allocation policies of the developed models were found to be socio-economically acceptable that maintained the balance of the entire system, considering all the constraints and restrictions imposed.

Marketing of Cotton Fiber in the Presence of Yield and Price Risk

AbstractAn expected-utility model and a chance-constrained linear programming model were used to analyze four marketing strategies and seven crop insurance alternatives for cotton marketing in Georgia. The results suggest that existing marketing tools and insurance alternatives can be used to reduce cotton producers' revenue risk. The optimal level of yield and price insurance coverage depends on an individual producer's risk aversion.

The DLP decision support system and its extension to stochastic programming*

A brief overview is given of a new decision support system for solving deterministic resource decision problems. It is based on a synthesis of techniques of dynamic programming and linear programming — hence the acronym D L P The main purpose of the article is to develop an extension that incorporates uncertainty of information in production and objective coefficients. The resulting model/algorithm — a new method for chance-constrained stochastic linear programming — unites techniques of stochastic dynamic programming with techniques of linear programming, and it can be implemented by building very directly on the D L Pdecision support system. A further extension is introduced more briefly. Its combination of chance constraints and recourse (simple or complete) constitutes a fundamental new approach to stochastic programming, one that is likely to be appropriate for many multiperiod, resource-planning problems under uncertainty The discussion is illustrated throughout using rangeland resource decision problems. A self-contained appendix provides a supplementary illustration for a highway pavement maintenance and rehabilitation problem

Reservoir operation for hydropower optimization: A chance-constrained approach

This paper presents a chance-constrained linear programming formulation for reservoir operation of a multipurpose reservoir. The release policy is defined by a chance constraint that the probability of irrigation release in any period equalling or exceeding the irrigation demand is at least equal to a specified valueP (called reliability level). The model determines the maximum annual hydropower produced while meeting the irrigation demand at a specified reliability level. The model considers variation in reservoir water level elevation and also the operating range within which the turbine operates. A linear approximation for nonlinear power production function is assumed and the solution obtained within a specified tolerance limit. The inflow into the reservoir is considered random. The chance constraint is converted into its deterministic equivalent using a linear decision rule and inflow probability distribution. The model application is demonstrated through a case study.

Chance-constrained linear programming with location scale distributions

M. Kress, in 1984, studied the chance-constrained critical path problem. The author proved that if the project time random variables follow a class of location-scale probability distributions, then there exists a specific threshold confidence level, such that the critical path for the system remains the same for all higher confidence levels. The purpose of this article is to study the similar properties for a general chance-constrained linear programming (C2LP) problems with location-scale probability distributions. We present results for chance-constrained linear programming which parallel those in Kress's article. © 1992 John Wiley & Sons, Inc.

Sequential optimization and revision of production plans over time

Production plans generated by linear programming technique over the planning horizon are not responsive to the actual fluctuations of stochastic demand at each time period. The deterioation in the total-cost performance is due to absence of predicting-error feedback for sequential revision of the production targets. In this paper, a simple sequential optimization procedure using minimum mean-square-error recursive forecast is introduced. The total cost performance of this proposed technique is compared with the chance-constrained linear programming method for sequential production plan generation.

Planning for alkali land reclamation under rainfall uncertainty

A chance-constrained linear programming model with rainfall as stochastic input has been developed to plan optimal land and water use in alkali soils under reclamation. The model is based on the water balance of a typical alkali catchment in the command area of the Western Jamuna Canal in Haryana (India). The zero-order decision rule has been used to obtain the deterministic equivalent of chance constraint and optimal solutions have been obtained for four levels of rainfall probability. Available water supplies at low rainfall probabilities were found to be inadequate to sustain the high cost reclamation technology. Exploitation of the full potential of agricultural land calls for the augmentation of available water supplies.

Reservoir management: A reliability programming approach

A reliability programming model is developed to determine the optimum monthly releases from a multipurpose reservoir. The model is based on chance‐constrained linear programming (CCLP) and dynamic programming (DP). The flood and drought reliabilities are represented in the CCLP in the form of chance constraints. These reliabilities are varied parametrically from their minimum required levels to their maximum possible values. The trade‐offs between flood and drought reliabilities are made using a forward two‐dimensional DP routine. The model does not require equal flood and/or drought reliabilities during the year. Folsom reservoir of the California Central Valley Project is modeled to demonstrate the use of this reliability programming approach. The resulting operation policy shows high reliabilities of flood and drought during winter and summer months, respectively. The variable risk levels avoid unnecessary releases during summer (for flood protection) and unnecessary storage of water during winter (for drought protection).

A Chance Constrained Approach to Regional Wastewater Management Systems

A chance-constrained linear programming model has been developed for regional wastewater management systems which involve stochastic quantities. The objective of the model is to minimize the annual operating costs by finding the monthly optimal biochemical oxygen demand (BOD) removal to maintain a desired water quality (dissolved oxygen, DO) in the stream with a certain degree of satisfaction (reliability). A zero-order decision rule is then used to find deterministic equivalents for chance-constraints. The possibility of a tradeoff between a desired water quality and the degree of satisfaction for a given annual budget is also shown.

Chance-Constrained Linear Programming with Chi-Square Type Deviates

The implications of replacing the assumption of normal distribution of the parameters (A, b, c) by chi-square and other related nonnegative distributions are discussed here in the framework of chance-constrained linear programming.

Chance-Constrained Linear Programming under Truncation and Varying Sample Sizes

Chance-Constrained Linear Programming under Truncation and Varying Sample Sizes

A Generalization of Some Distribution Aspects of Chance-Constrained Linear Programming

AN ORDINARY LINEAR PROGRAMMING MODEL is said to be chance-constrained if its linear constraints are associated with a set of probability measures indicating the extent of violation of the constraints. When partial violation of the constraints is allowed for, the chance-constrained approach may be viewed as a method for providing appropriate safety margins. This approach has been generalized [9] in recent years in several directions. of which two are specially worth mentioning. First, although for reasons of simplicity solutions restricted to linear decision rules only have often been employed in chance-constrained programming, it is now possible to have solutions of a more general functional form, and this considerably enhances the scope of application of the chance-constrained approach in dynamic models with nonlinear objective functions. Second, it is not necessary in the chance-constrained approach to make the assumption that the decision maker's utility function is quadratic (or of a specific form), as it is required, for example, in the portfolio selection studies by Markowitz and others who base the analysis on the mean and variance of the probability distribution of net returns. Although the extent of violation of the constraints that would be tolerated is preassigned subjectively in this approach by the decision maker before the: actual solutions are computed, the tolerance measure may be parametrically varied, as in the revealed preference theory, and the resulting optimal solutions may help the decision maker move to the most preferred solution. This. formulation (see [8], [19]) also considerably helps the scope of applicability of the chance-constrained approach by developing suitable criteria for decisions under risk. Our objective here is to analyze the implications of a non-normal distribution of the random elements (A, b, c) of a linear program in the framework of probabilistic linear programming, where only the two approaches of chance-constrained programming (CCP) and stochastic linear programming (SLP) are considered. It is interesting to note that although the CCP and the SLP approaches are developed with different objectives in mind, they have frequently been applied under the assumption of normality of relevant

Safety-First Rules under Chance-Constrained Linear Programming

The approach of chance-constrained linear programming is analyzed here in the context of safety-first principles based on Tchebycheff-type inequalities. The analysis attempts to define relatively distribution-free tolerance levels and the incidence of nonnormality in chance-constrained linear programming.