Right-hand Side Of Constraints Research Articles

Modeling urban eco-environmental sustainability under uncertainty: Interval double-sided chance-constrained programming with spatial analysis

In this study, an interval double-sided chance-constrained programming (abbreviated as IDCP) method is developed for dealing with uncertainties expressed as interval values and double-sided randomness in both left- and right-hand sides of constraints. The method is applied to planning urban eco-environmental management of Shenzhen (China), where various eco-environmental indicators are used for ensuring ecological security. Through solving the IDCP-based urban eco-environmental management (abbreviated as IDCP-UEM) model, solutions for land-use pattern, industrial structure, system benefit, and ecosystem service value under different probability levels (i.e. p level) have been generated. Then, the conversion of land use and its effect at small regional extent (abbreviated as CLUE-S) model is used for analyzing the spatial characteristic of land-use pattern, which can further optimize the results from the IDCP-UEM model through providing the spatial allocation of land resources. Results reveal that (i) uncertainties existed in the urban ecosystem have significant impacts on system benefit and land-use pattern; (ii) at the end of the planning horizon, the area of forest land would be raised and the expansion of construction land would slow down; (iii) industries of electronic information and machinery manufacturing should be developed because they contribute higher benefit and discharge relatively less pollutant than other industries; (iv) system benefit would be raised and pollutant emissions (e.g., chemical oxygen demand, ammonia nitrogen, and solid waste) would be mitigated over the planning horizon. The findings are helpful for decision makers to gain insights into tradeoffs between eco-environmental protection and urban-economic development.

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.

A Stochastic Optimization Model for Agricultural Irrigation Water Allocation Based on the Field Water Cycle

Agricultural water scarcity is a global problem and this reinforces the need for optimal allocation of irrigation water resources. However, decision makers are challenged by the complexity of fluctuating stream condition and irrigation quota as well as the dynamic changes of the field water cycle process, which make optimal allocation more complex. A two-stage chance-constrained programming model with random parameters in the left- and right-hand sides of constraints considering field water cycle process has been developed for agricultural irrigation water allocation. The model is capable of generating reasonable irrigation allocation strategies considering water transformation among crop evapotranspiration, precipitation, irrigation, soil water content, and deep percolation. Moreover, it can deal with randomness in both the right-hand side and the left-hand side of constraints to generate schemes under different flow levels and constraint-violation risk levels, which are informative for decision makers. The Yingke irrigation district in the middle reaches of the Heihe River basin, northwest China, was used to test the developed model. Tradeoffs among different crops in different time periods under different flow levels, and dynamic changes of soil moisture and deep percolation were analyzed. Scenarios with different violating probabilities were conducted to gain insight into the sensitivity of irrigation water allocation strategies on water supply and irrigation quota. The performed analysis indicated that the proposed model can efficiently optimize agricultural irrigation water for an irrigation district with water scarcity in a stochastic environment.

A Monte-Carlo-based interval De Novo programming method for optimal system design under uncertainty

In this study, a Monte-Carlo-based interval De Novo programming (MC-IDP) method is developed for designing optimal electricity-allocation system under uncertainty. MC-IDP incorporates Monte Carlo simulation (MCS), interval-parameter programming (IPP), and De Novo programming (DNP) within a general framework. MC-IDP has advantages in (i) constructing optimal system design through introducing the flexibility in the right-hand sides of constraints, (ii) handling uncertainty presented as interval numbers, and (iii) mitigating the influence of decision makers’ subjectivity in optimum-path ratio. MC-IDP is then applied to a case study of planning electricity-allocation system involving multiple conflicting objectives, where various scenarios associated with different optimum-path ratios are examined. Results reveal that different scenarios would lead to varied electricity-allocation patterns, pollutant/ greenhouse gas (GHG) emissions, as well as system benefits. Compared to the traditional interval multiobjective programming (IMOP), MC-IDP can achieve higher system benefits and reduce electricity loss; moreover, the maximum benefit for each objective under MC-IDP can be realized at the same time. Findings are useful to decision makers for evaluating alternatives of system designs as well as for identifying which of these designs can most efficiently achieve the desired system objectives in a more sustainable development manner.

Duality and Correction of Inconsistent Constraints for Improper Linear Programming Problems

We continue the study of approximation properties of alternative duality schemes for improper problems of linear programming. The schemes are based on the use of the classical Lagrange function regularized simultaneously in primal and dual variables. The earlier results on the connection of its saddle points with the lexicographic correction of the right-hand sides of constraints in improper problems of the first and second kind are transferred to a more general type of improperness. Convergence theorems are presented and an informal interpretation of the obtained generalized solution is given.

A copula-based chance-constrained waste management planning method: An application to the city of Regina, Saskatchewan, Canada

ABSTRACTThis study proposes a copula-based chance-constrained waste management planning (CCWMP) method. The method can effectively reflect the interactions between random parameters of the waste management planning systems, and thus can help analyze the influences of their interactions on the entire systems. In particular, a joint distribution function is established using preestimated marginal distributions of random variables and an optimal copula selected from widely used Gaussian, Student’s t, Clayton, Frank, Gumbel, and Ali-Mikhail-Haq copulas. Then a set of joint probabilistic constraints in the chance-constrained programming problems is converted into individual probabilistic constraints using the joint distribution function. Further, this method is applied to residential solid waste management in the city of Regina in Canada for demonstrating its applicability. Nine scenarios based on different joint and marginal probability levels are considered within a multiperiod and multizone context to effectively reflect dynamic, uncertain, and interactive characteristics of the solid waste management systems in the city. The results provide many decision alternatives under these scenarios, including cost-effective and environmentally friendly decision schemes. Moreover, the results indicate that even though the effect of the joint probability levels on the system costs is more significant than that of the marginal probability levels, the effect of marginal probability levels is notable, and there exists a trade-off between the total system cost and the constraint-violation risk. Therefore, the results obtained from the present study would be useful to support the city’s long-term solid waste management planning and formulate local policies and regulation concerning the city’s waste generation and management.Implications: The CCWMP method not only can solve chance-constrained problems with unknown probability distributions of random variables in the right-hand sides of constraints, but also can effectively reflect the interactions between the random parameters and thus help analyze the influences of their interactions on the entire systems. The results obtained through applying this method to the city of Regina in Canada can provide many decision alternatives under different joint probability levels and marginal probability levels, and would be useful to support the city’s long-term solid waste management planning.

Modeling Water Trading under Uncertainty for Supporting Water Resources Management in an Arid Region

AbstractIn this study, a joint-probabilistic interval multistage programming (JIMP) method is developed for planning water resources management under uncertainty. The JIMP method can tackle uncertainties presented in terms of interval parameters in objective function and constraints, in addition to random variables in the left and right-hand sides of constraints. It can also reflect the dynamics in terms of decisions for water resources allocation through transactions at discrete points of a complete scenario set over a multistage context. Moreover, the JIMP method can be used for analyzing various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The JIMP method is applied to a real case of planning water trading for supporting the regional sustainable development of the Kaidu-Qongque River basin, which is one of the most arid regions of China. Monte Carlo simulation is introduced into the JIMP framework for evalua...

An inexact joint-probabilistic programming method for risk assessment in water resources allocation

In this study, an inexact joint probabilistic programming (IJPP) approach is developed for risk assessment and uncertainty reflection in water resources management systems. IJPP can dominate random parameters in the model’s left- and right-hand sides of constraints and interval parameters in the objective function. It can also help examine the risk of violating joint probabilistic constraints, which allows an increased robustness in controlling system risk in the optimization process. Moreover, it can facilitate analyses of various policy scenarios that are associated with different levels of economic consequences when the promised targets are violated within a multistage context. The IJPP method is then applied to a case study of planning water resources allocation within a multi-reservoir and multi-period context. Solutions of system benefit, economic penalty, water shortage, and water-allocation pattern vary with different risks of violating water-demand targets from multiple competitive users. Results also demonstrate that different users possess different water-guarantee ratios and different water-allocation priorities. The results can be used for helping water resources managers to identify desired system designs against water shortage and for risk control, and to determine which of these designs can most efficiently accomplish optimizing the system objective under uncertainty.

Two-stage credibility-constrained programming with Hurwicz criterion (TCP-CH) for planning water resources management

In water-resources management problems, uncertainties exist in a number of impact factors and water-allocation processes, which can bring about enormous difficulties and challenges in generating desired decision alternatives under such complexities. In this study, a two-stage credibility-constrained programming with Hurwicz criterion (TCP-CH) approach is developed for water resources management and planning under uncertainty. TCP-CH can tackle uncertainties presented as probability distributions and fuzzy sets that exist in left- and right-hand sides of constraints and in objective function; it can also permit in-depth analyses of various policy scenarios based on confidence degrees that are associated with different levels of economic penalties when the promised targets are violated. The TCP-CH method is applied to a real case of planning water resources in Kaidu-qongque River basin, which is one of the aridest regions of China. Support-vector-regression (SVR) technique is introduced into TCP-CH for predicting the water demand in the future, which takes advantage of uncertainty reflection and dynamic analysis. Results of water allocation, water shortage, system benefit, and economic penalty are obtained. The results discover that water deficit has brought negative effects on regional economic development, particularly for agricultural production activities. Moreover, tradeoffs between economic benefit and system-failure risk are also examined under different risk preferences of decision makers (i.e., optimistic and pessimistic criteria), which support generating an increased robustness in risk control for water resources allocation under uncertainties. These findings can facilitate the local decision makers in adjusting the current water-allocation pattern based on risk preference option (Hurwicz criterion) to satisfy the increasing water demand.

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.

Interactive Fuzzy Boundary Interval Programming for Air Quality Management Under Uncertainty

This paper presents an interactive fuzzy boundary interval programming (IFBIP) approach for addressing dual uncertainties that exist in the objective function and the left- and right-hand sides of constraints. IFBIP permits an interactive participation of the decision maker (DM) in all steps of the decision process through expressing his/her preferences in linguistic terms. A regional air quality management problem is studied to illustrate the proposed approach. The results indicate that a number of decision alternatives can be obtained under different feasibility degrees of constraints. They can help quantify the relationship between the total operating cost and the constraint violation risk, which is meaningful not only for supporting an in-depth analysis of the tradeoff between the economic objective and the system safety but also for the DM to identify a desired compromise between two conflicting concerns: the satisfaction degree of the goal and the feasibility degree of constraints.

An interior boundary pivotal solution algorithm for linear programmes with the optimal solution-based sensitivity region

We have developed a full gradient method that consists of three phases. The initialisation phase provides the initial tableau that may not have a full set of basis. The push phase uses a full gradient vector of the objective function to obtain a feasible vertex. This is then followed by a series of pivotal steps using the sub-gradient, which leads to an optimal solution (if exists) in the final iteration phase. At each of these iterations, the sub-gradient provides the desired direction of motion within the feasible region. The algorithm hits and/or moves on the constraint hyper-planes and their intersections to reach an optimal vertex (if exists). The algorithm works in the original decision variables and slack/surplus space, therefore, there is no need to introduce any new extra variables such as artificial variables. The simplex solution algorithm can be considered as a sub-more efficient. Given a linear programme has a known unique non-degenerate primal/dual solution; we develop the largest sensitivity region for linear programming models-based only the optimal solution rather than the final tableau. It allows for simultaneous, dependent/independent changes on the cost coefficients and the right-hand side of constraint. Numerical illustrative examples are given.

A coupled factorial-analysis-based interval programming approach and its application to air quality management

In this study, a coupled factorial-analysis-based interval programming (CFA-IP) approach is developed through incorporating factorial analysis within an interval-parameter linear programming framework. CFA-IP can tackle uncertainties presented as intervals that exist in the objective function and the left- and right-hand sides of constraints, as well as robustly reflect interval information in the solutions for the objective-function value and decision variables. Moreover, CFA-IP has the advantage of investigating the potential interactions among input parameters and their influences on lower- and upper-bound solutions, which is meaningful for supporting an in-depth analysis of uncertainty. A regional air quality management problem is studied to demonstrate applicability of the proposed CFA-IP approach. The results indicate that useful solutions have been generated for planning the air quality management practices. They can help decision makers identify desired pollution mitigation strategies with minimized total cost and maximized environmental efficiency, as well as screen out dominant parameters and explore the valuable information that may be veiled beneath their interrelationships. Implications: The CFA-IP approach can not only tackle uncertainties presented as intervals that exist in the objective function and the left- and right-hand sides of constraints, but also investigate their interactive effects on model outputs, which is meaningful for supporting an in-depth analysis of uncertainty. Thus CFA-IP would be applicable to air quality management problems under uncertainty. The results obtained from CFA-IP can help decision makers identify desired pollution mitigation strategies, as well as investigate the potential interactions among factors and analyze their consequent effects on modeling results.

Identifying Optimal Water Resources Allocation Strategies through an Interactive Multi-Stage Stochastic Fuzzy Programming Approach

In this study, an interactive multi-stage stochastic fuzzy programming (IMSFP) approach has been developed through incorporating an interactive fuzzy resolution (IFR) method within an inexact multi-stage stochastic programming framework. IMSFP can deal with dual uncertainties expressed as fuzzy boundary intervals that exist in the objective function and the left- and right-hand sides of constraints. Moreover, IMSFP is capable of reflecting dynamics of uncertainties and the related decision processes through constructing a set of representative scenarios within a multi-stage context. A management problem in terms of water resources allocation has been studied to illustrate applicability of the proposed approach. The results indicate that a set of solutions under different feasibility degrees (i.e., risk of constraint violation) has been generated for planning the water resources allocation. They can not only help quantify the relationship between the objective-function value and the risk of violating the constraints, but also enable decision makers (DMs) to identify, in an interactive way, a desired compromise between two factors in conflict: satisfaction degree of the goal and feasibility degree of constraints. Besides, a number of decision alternatives have been generated under different policies for water resources management, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic penalties when the promised water-allocation targets are violated, and thus help DMs to identify desired water-allocation schemes under uncertainty.

Interactive two-stage stochastic fuzzy programming for water resources management

In this study, an interactive two-stage stochastic fuzzy programming (ITSFP) approach has been developed through incorporating an interactive fuzzy resolution (IFR) method within an inexact two-stage stochastic programming (ITSP) framework. ITSFP can not only tackle dual uncertainties presented as fuzzy boundary intervals that exist in the objective function and the left- and right-hand sides of constraints, but also permit in-depth analyses of various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated. A management problem in terms of water resources allocation has been studied to illustrate applicability of the proposed approach. The results indicate that a set of solutions under different feasibility degrees has been generated for planning the water resources allocation. They can help the decision makers (DMs) to conduct in-depth analyses of tradeoffs between economic efficiency and constraint-violation risk, as well as enable them to identify, in an interactive way, a desired compromise between satisfaction degree of the goal and feasibility of the constraints (i.e., risk of constraint violation).

Identification of filter management strategy in fluid power systems under uncertainty: an interval-fuzzy parameter integer nonlinear programming method

An interval-fuzzy integer nonlinear programming (IFINP) method is developed for the identification of filter allocation and replacement strategies in a fluid power system (FPS) under uncertainty. It can handle uncertainties expressed as interval-fuzzy values that exist in the left- and right-hand sides of constraints as well as in the objective function. The developed method is applied to a case of planning filter allocation and replacement strategies under uncertainty for a FPS with a single circuit. A piecewise linearisation approach is used to convert the nonlinear problem of FPS into a linear one. The generated fuzzy solutions will be used to analyse and interpret the multiple decision alternatives under various system conditions, and thus help decision-makers to make a compromise among the system contamination level, system cost, satisfaction degrees and system-failure risks under different contaminant ingression/generation rates. The results demonstrate that the suction and return filters can effectively reduce the contamination level associated with a low system cost, but the FPS will take lots of failure risk when the contaminant ingression/generation rate is high; and the combination of suction and pressure filters can bring the lowest system cost with more security instead. Furthermore, comparisons for the optimised solutions are made among IFINP, interval-parameter integer nonlinear programming and deterministic linear programming also. Generally, the IFINP method can effectively reduce the total design and operation cost of the filtration system when contaminants ingression/generation rate is high, and it could be extended to the lubricating system.

An interval-based possibilistic programming method for waste management with cost minimization and environmental-impact abatement under uncertainty

Considerable public concerns have been raised in the past decades since a large amount of pollutant emissions from municipal solid waste (MSW) disposal of processes pose risks on surrounding environment and human health. Moreover, in MSW management, various uncertainties exist in the related costs, impact factors and objectives, which can affect the optimization processes and the decision schemes generated. In this study, an interval-based possibilistic programming (IBPP) method is developed for planning the MSW management with minimized system cost and environmental impact under uncertainty. The developed method can deal with uncertainties expressed as interval values and fuzzy sets in the left- and right-hand sides of constraints and objective function. An interactive algorithm is provided for solving the IBPP problem, which does not lead to more complicated intermediate submodels and has a relatively low computational requirement. The developed model is applied to a case study of planning a MSW management system, where mixed integer linear programming (MILP) technique is introduced into the IBPP framework to facilitate dynamic analysis for decisions of timing, sizing and siting in terms of capacity expansion for waste-management facilities. Three cases based on different waste-management policies are examined. The results obtained indicate that inclusion of environmental impacts in the optimization model can change the traditional waste-allocation pattern merely based on the economic-oriented planning approach. The results obtained can help identify desired alternatives for managing MSW, which has advantages in providing compromised schemes under an integrated consideration of economic efficiency and environmental impact under uncertainty.

STABILITY OF EXACT PENALTY FOR NONCONVEX INEQUALITY-CONSTRAINED MINIMIZATION PROBLEMS

In this paper we use the penalty approach in order to study inequality-constrained minimization problems with locally Lipschitz objective and constraint functions in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints.

IFQP: A hybrid optimization method for filter management in fluid power systems under uncertainty

An interval-fuzzy quadratic programming (IFQP) method is developed for the assessment of filter allocation and replacement strategies in fluid power systems (FPS) under uncertainty. It can directly handle uncertainties expressed as interval values and/or fuzzy sets that exist in the left-hand and right-hand sides of constraints, as well as in the objective function. Multiple control variables are used to tackle independent uncertainties in the model's right-hand sides and thus optimize the overall satisfaction of the system performance. The IFQP method is applied to a case of planning filter allocation and replacement strategies under uncertainty for an FPS with a single circuit. A piecewise linearization approach is firstly employed to convert the nonlinear FPS problem into a linear one. The generated decision alternatives can help decision makers to identify desired policies for contamination control under various total costs, satisfaction degrees, and system-failure risks under different contaminant-ingression/generation rates.

Existence and stability of exact penalty for optimization problems with mixed constraints

In this paper we use the penalty approach in order to study minimization problems with mixed constraints in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property holds and is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints.