Chance-constrained Programming Problem Research Articles

Analyze the Stochastic Solid Fuzzy Transportation Problem with Mixed Constraints through Weibull Distribution

This paper proposed a general formulation of the stochastic solid transportation problem (SSTP) with mixed constraints such as supply, demand and conveyance capacity taken as uncertain under stochastic environment, following the Weibull distribution (WD). The aim of this study is to minimize the transportation cost includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP). SSTP with probabilistic constraints is represented as a chance constrained programming problem. Obtain alpha cut representation from cost coefficient of the fuzzy objective function. We have developed four models for stochastic solid transportation problem. The suggested models are demonstrated by taken as numerical example. A sensitivity analysis is performed to understand parameter’s sensitivity in the proposed model. Introduction: In system of transportation, goods are moved from various sources to destinations using different vehicles and organizational systems, involving both technology and human efforts. Efficient resource allocation in transportation system is crucial for industries and imprecision from factors like fluctuating demand, unreliable supply chains and unpredictable traffic. To address these complexities, advanced mathematical models are needed to manage stochasticity, fuzziness and mixed constraints. The study explores the stochastic solid fuzzy transportation problem with mixed constraint by utilizing the Weibull distribution to model uncertainties inherent in transportation systems. This research addresses the complexity introduced by stochastic variables and fuzzy parameters, particularly in situations where demand, supply and cost of transportation are not deterministic. Objectives: The aim of this study is to minimize the cost of transportation includes probabilistic constraints have inequalities of stochastic solid transportation problem (SSTP). Methods: Obtain alpha cut representation form the cost coefficient of the fuzzy objective function and four models are developed for stochastic solid transportation problem. These models are demonstrating by using a numerical example and a sensitivity analysis is conducted to understand the sensitivity of the parameters in the propose model. Results: Obtained optimal solutions for developed four models of SFSTPMC and sensitivity analysis shows that cost of transportation and flow of unit are sensitive to change in probabilities of demand. Improve transportation system by understanding sensitivity patterns that help decision maker choose appropriate supply availability probabilities. Conclusions: This study presented an approach for solving the SFSTPMC using the Weibull distribution for probabilistic constraints and fuzzy objective functions for transportation cost. Developed and optimized four models, focusing on stochastic parameters. Sensitivity analysis demonstrated the impact of these parameters on transportation cost and unit flow. The results validate the model’s effectiveness in practical resource allocation and decision making under uncertainty.

The Stochastic Transportation Problem with Imprecise Data using Lomax Distribution

The stochastic transportation problem with imprecise data is a probabilistic chance-constrained programming (CCP) problem in which the objective function is fuzzy and the supply and demand are random. Three models for the STP with ID with mixed-type restrictions that follow the Lomax distribution (LD) are created in this research. Optimizing the transportation cost in FTP under probabilistic mixed constraints is the goal of the research project. To do this, the probabilistic mixed constraints are transformed into deterministic form using the LD, and the cost coefficient of the fuzzy objective function is changed with alpha cut representation. Numerical examples are presented to demonstrate the suggested models.

Distributionally robust chance-constrained optimization with Gaussian mixture ambiguity set

Conventional chance-constrained programming methods suffer from the inexactness of the estimated probability distribution of the underlying uncertainty from data. To this end, a distributionally robust approach to the problem allows for a level of ambiguity considered around a reference distribution. In this work, we propose a novel formulation for the distributionally robust chance-constrained programming problem using an ambiguity set constructed from a variant of optimal transport distance that was developed for Gaussian Mixture Models. We show that for multimodal process uncertainty, our proposed method provides an effective way to incorporate statistical moment information into the ambiguity set construction step, thus leading to improved optimal solutions. We illustrate the performance of our method on a numerical example as well as a chemical process case study. We show that our proposed methodology leverages the multimodal characteristics from the uncertainty data to give superior performance over the traditional Wasserstein distance-based method.

Enhancing neutrosophic fuzzy compromise approach for solving stochastic bi- level linear programming problems with right- hand sides of constraints follow normal distribution

In this paper, a bi-level chance constrained programming problem is considered when the coefficients of the objective function is presented as neutrosophic numbers and the right- hand side of the constraints is normal variables and the constraints have a joint probability distribution. While the probability problem and applying the score and accurate functions the problem is converted into an equivalent deterministic non- linear programming problem, a fuzzy programming approach is applied by defining membership function. A linear membership function is used for obtaining optimal compromise solution. A numerical example is given to illustrate the proposed methodology.

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

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

Mixed-Integer Conic Formulation of Unit Commitment with Stochastic Wind Power

Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, the chance-constrained programming (CCP) method is used to study the UC problem with uncertainty wind power generation, and chance constraints such as power balance and spinning reserve are satisfied with a predetermined probability. In order to effectively solve the CCP problem, first, we used the sample average approximation (SAA) method to transform the chance constraints into deterministic constraints and to obtain a mixed-integer quadratic programming (MIQP) model. Then, the quadratic terms were incorporated into the constraints by introducing some auxiliary variables, and some second-order cone constraints were formed by combining them with the output characteristics of thermal unit; therefore, a tighter mixed-integer second-order cone programming (MISOCP) formulation was obtained. Finally, we applied this method to some systems including 10 to 100 thermal units and 1 to 2 wind units, and we invoked MOSEK in MATLAB to solve the MISOCP formulation. The numerical results obtained within 24 h confirm that not only is the MISOCP formulation a successful reformulation that can achieve better suboptimal solutions, but it is also a suitable method for solving the large-scale uncertain UC problem. In addition, for systems of up to 40 units within 24 h that do not consider wind power and pollution emissions, the numerical results were compared with those of previously published methods, showing that the MISOCP formulation is very promising, given its excellent performance.

Solving geometric programming problems with triangular and trapezoidal uncertainty distributions

The geometric programming problem is an important optimization technique that is often used to solve different nonlinear optimization problems and engineering problems. The geometric programming models that are commonly used are generally based on deterministic and accurate parameters. However, it is observed that in real-world geometric programming problems, the parameters are frequently inaccurate and ambiguous. In this paper, we consider chance-constrained geometric programming problems with uncertain coefficients and with geometric programming techniques in the uncertain-based framework. We show that the associated chance-constrained uncertain geometric programming problem can be converted into a crisp geometric programming problem by using triangular and trapezoidal uncertainty distributions for the uncertain variables. The main aim of this paper is to provide the solution procedures for geometric programming problems under triangular and trapezoidal uncertainty distributions. To show how well the procedures and algorithms work, two numerical examples and an application in the inventory model are given.

Chance-constrained energy-reserve co-optimization scheduling of wind-photovoltaic-hydrogen integrated energy systems

An optimal energy-reserve scheduling model of wind-photovoltaic-hydrogen integrated energy systems (WPH-IES) with multi-type energy storage devices including electric, thermal and hydrogen is presented in this paper. The chance-constrained programming (CCP) theory is utilized to model the impact of renewable and load uncertainties on reserve constraints. Considering the non-convex of proposed CCP model, an improved discretized step transformation (DST) method is proposed to transform the CCP problem into a solvable mixed integer linear programming (MILP) formulation. In addition, a critical threshold value selection approach is developed to reduce the number of constraints and improve solution efficiency. Case studies demonstrate that the proposed energy-reserve model can reduce the total operating cost on the premise of ensuring the safety of system operation. The combined improved DST and critical threshold value selection method can reduce computational burden and improve the accuracy of scheduling results.

Solving Stochastic Fuzzy Transportation Problem with Mixed Constraints Using the Weibull Distribution

The development in the industries has necessitated the growth of transportation methods. Due to the variation in the transportation systems, many problems seem to arise in the present times. One such problem is the stochastic fuzzy transportation problem (SFTP). The SFTP is a chance‐constrained programming (CCP) problem with probabilistic constraints where supply and demand are randomness, and the objective function is in fuzzy nature. In this paper, we have developed three models for the SFTP, where the constraints are mixed type following Weibull distribution (WD). The aim of the research work is to optimize the transportation cost in FTP under probabilistic mixed constraints. In order to achieve this, the cost coefficient of the fuzzy objective function is converted to alpha cut representation, and the probabilistic mixed constraints are converted to deterministic form by using the WD. The proposed models are illustrated by providing a numerical example, and the problem is solved using Lingo software. It is worth pointing out that these models are constructed from different points of view. The decision‐maker’s (DM’s) preference has the final say in the usage of the models. A sensitivity analysis (SA) is performed to explore the sensitivity of the parameters in the proposed model.

A Multi-Objective Stochastic Solid Transportation Problem with the Supply, Demand, and Conveyance Capacity Following the Weibull Distribution

This study addresses a multi-objective stochastic solid transportation problem (MOSSTP) with uncertainties in supply, demand, and conveyance capacity, following the Weibull distribution. This study aims to minimize multiple transportation costs in a solid transportation problem (STP) under probabilistic inequality constraints. The MOSSTP is expressed as a chance-constrained programming problem, and the probabilistic constraints are incorporated to ensure that the supply, demand, and conveyance capacity are satisfied with specified probabilities. The global criterion method and fuzzy goal programming approach have been used to solve multi-objective optimization problems. Computational results demonstrate the effectiveness of the proposed models and methodology for the MOSSTP under uncertainty. A sensitivity analysis is conducted to understand the sensitivity of parameters in the proposed model.

Solving multi-objective chance constrained programming problem involving three parameters log normal distribution

The main point of this paper is to find the compromise solution of multi-objective chance constrained mathematical programming problem by considering the parameters of the right hand side restrictions as uncertain variables having three parameters log normal distribution. In the first step, we convert the multi-objective chance constrained mathematical programming problem to a deterministic equivalent multi-objective mathematical programming problem. After converting the multi-objective chance constrained programming problem into deterministic equivalent multi-objective programming problem, fuzzy programming method is used to find the solution which is closest to ideal solution. Numerical example is presented to illustrate the technique. Finally, case study on crop land and water management for irrigation is presented.

Set Squeezing Procedure for Quadratically Perturbed Chance-Constrained Programming

The set squeezing procedure, a new optimization methodology for solving chance-constrained programming problems under continuous uncertainty distribution, is proposed in this paper. The generally intractable chance constraints and unknown convexity are tackled by a novel analyses of local structure of the feasible set. Based on the newly discovered structure, it is proved that the set squeezing procedure converges and local optimality is guaranteed under mild conditions. Furthermore, efficient algorithms are derived for the set squeezing procedure under the widely used quadratically perturbed constraints. The developed method is applied to the mean squared error (MSE) based probabilistic transceiver design as an application example. Simulation results show that the MSE outage probability can be controlled tightly, which leads to lower transmit power, compared to the existing dominant safe approximation method and the bounded robust optimization method.

A chance-constrained energy management in multi-microgrid systems considering degradation cost of energy storage elements

Energy storage devices are very popular in systems with the presence of renewable-based power generation. However, inappropriate usage of them may lead to aging and also the early failure of such systems. The factors affecting the aging of the batteries as the main energy storage devices in electrical systems are frequent changes in charging and discharging status, deep discharge, and a large number of power transactions in a short time. Furthermore, the development of microgrids (MGs) in distribution networks (DNs), leads to more complicated structures known as multi-microgrid (MMG) systems. In this paper, the day-ahead scheduling for the MMG system considering the degradation cost of the energy storage systems has been proposed. To model the uncertainties, chance-constrained programming (CCP) approach has been employed. The CCP helps the system operator to make better decisions without jeopardizing the system security. The degradation cost has been designed in a way that in the energy management process, misuse of the batteries is avoided. A hierarchical three-stage energy management process is designed. First, deterministic equivalents for the CCP problem are made and using these equivalents, in the first stage, MGs perform local scheduling and in the second stage, again considering deterministic equivalents of the CCP, DN performs global management and finally, MGs perform a rescheduling. Simulations with four case studies on a modified version of the IEEE 33-bus test system have shown the effectiveness of the proposed scheduling framework. The results show that decreasing confidence level in CCP decreases the system operational costs, however, threatens the system stability. Also, considering the battery degradation cost prevents the system operator from misapplication of the battery energy storage systems.

Multi-item fuzzy inventory model for deteriorating items in multi-outlet under single management

Multi-item inventory model with stock-dependent demand is developed in fuzzy environment. Items are deteriorated in constant rate and are sold from different outlets in the city under single management. Due to the impreciseness of different parameters, objectives as well as constraints are imprecise in nature. As optimization of fuzzy objectives as well as fuzzy constraints are not well defined, the model is formulated as a multi-objective chance constrained programming problem where optimistic/pessimistic return of the objectives with some degree of possibility/necessity are optimized and constraints are satisfied with some degree of necessity. The model is solved via Multi-Objective Genetic Algorithm (MOGA) when crisp equivalent of the problem is available. In other cases, fuzzy simulation process is proposed to check the constraints as well as to determine the optimistic/pessimistic return of the objectives. The model is illustrated with some numerical examples.

Solution to Chance Constrained Programming Problem in Swap Trailer Transport Organisation based on Improved Simulated Annealing Algorithm

Abstract Swap trailer transport organisation problem originates from the traditional vehicle routing problem (VRP). Most of the studies on the problems assume that the travelling times of vehicles are fixed values. In this paper, the uncertainties of driving times are considered and a chance constrained programming problem is proposed. An improved simulated annealing algorithm is used to solve the problem proposed. The model and algorithm described in this paper are studied through a case study, and the influence of uncertainty on the results is analysed. The conclusion of this study provides theoretical support for the practice of trailer pickup transport.

Stochastic optimal operation of flexible distribution networks for security improvement considering active management

This paper presents a stochastic optimal operation problem of gas turbine integrated distribution networks in the presence of active management schemes, which is formulated as a multi-objective chance-constrained mixed integer nonlinear programming problem. The control variables are the on-load tap-changer tap position, the power provided by the distributed generation (DG), the DG power factor angle, the load participating in demand side management and the switch status. The objectives defined in this paper are to simultaneously minimize the expectation cost and variation coefficient of security distance. Uncertainties related to DG output and load fluctuation and fault power restoration under contingencies are also considered in the optimization problem. The collaboration of normal boundary intersection and dynamic niche differential evolution algorithm is proposed to handle the optimal operation mode. Simulation results are presented and demonstrate the effectiveness of the proposed model. Compared with the operation result without the consideration of security, the security-constrained operation can reduce the expectation cost. So the proposed optimization is reasonable and valuable.

Solving Multi-Objective Chance Constrained Programming Problem involving Three Parameters Log Normal Distribution

The main point of this paper is to find the compromise solution of multi-objective chance constrained mathematical programming problem by considering the parameters of the right hand side restrictions as uncertain variables having three parameters log normal distribution. In the first step, we convert the multi-objective chance constrained mathematical programming problem to a deterministic equivalent multi-objective mathematical programming problem. After converting the multi-objective chance constrained programming problem into deterministic equivalent multi-objective programming problem, fuzzy programming method is used to find the solution which is closest to ideal solution. Numerical example is presented to illustrate the technique. Finally, case study on crop land and water management for irrigation is presented.

Minimizing the total completion time of an urban delivery problem with uncertain assembly time

This paper studies a stochastic programming problem to minimize the total completion time consists of the travel time and the assembly time, for a new class of urban delivery problems. The problem is formulated into a chance-constrained programming problem and the probability distribution characteristics of the uncertainty of product assembly time are obtained using a statistical learning method. Then the original chance-constrained programming problem can be converted into an equivalent deterministic programming problem. Subsequently, an algorithm consisting of two sub-heuristics is proposed to solve the deterministic problem. Finally, comprehensive numerical experiments with real dataset show the superiority of the model and the algorithm.

Resilient Configuration Approach of Integrated Community Energy System Considering Integrated Demand Response Under Uncertainty

Integrated community energy system (ICES) enables multi-energy synergy and impulses interaction of integrated demand response (IDR) from the demand side. Nevertheless, the randomness lies in IDR resources and renewable generation in the ICES configuration issue still lacks thorough analysis. Furthermore, the widely focused concern of resiliency urges more proactive consideration of potential emergencies at the planning stage of ICES. To this end, this paper proposed a resilience-oriented stochastic ICES configuration framework considering IDR influence. First, generalized IDR models are set up in detail with elaborate fuzzy feature analysis of price-responsive multi-energy loads. Then, the vulnerability indicators of tie lines and converting devices of the ICES are first introduced to demonstrate the occasional outage in the normal operation and blackout in the emergent case. In addition, the worst-case conditional value-at-risk (WCVaR) theory is innovatively integrated to the traditional risk-neutral model, which is formulated as a two-stage stochastic chance-constrained programming problem, aiming to combine portfolio with minimizing the worst-case cost caused by a disaster. The models are then transformed into mixed-integer linear programming problems via several linearization techniques. Finally, the results of the case studies showed the established model’s superiority in improving the overall economy (reduce 6.94% of the worst-case cost) and system resiliency (decrease the value-at-risk by 3.20%) against an unforeseen hurricane. Meanwhile, the ICES reliability was also improved by declining 25.65% of unintended load curtailment faced with an emergency. The proposed approach provided an efficient preventive portfolio scheme of the IDR-integrated ICES for decision-makers’ different risk preferences of natural hazards.

Robust Chance Constrained Power Allocation Scheme for Multiple Target Localization in Colocated MIMO Radar System

Taking into account the probabilistic uncertainty on the target radar cross section (RCS) parameter, a robust chance constrained power allocation (RCC-PA) scheme is presented for multiple target localization in colocated multiple-input multiple-output radar system. Such a system adopts a multibeam working mode, in which multiple simultaneous transmit beams are synthesized to illuminate multiple targets independently. We formulate the RCC-PA problem into a chance constrained programming model, where the total power consumption of the multiple beams is minimized to meet a specified multitarget localization accuracy requirement with high probability. Various target RCS fluctuation models have been discussed with different forms of probability distributions. We analytically show that the chance constrained programming problem for each RCS fluctuation model can equivalently be formulated as a deterministic convex optimization problem. Then, by formulating the Karush-Kuhn-Tuckers conditions, we transform the convex optimization problem into a nonlinear equation solving problem, and then solve it by using the bisection method. Simulation results show that our RCC-PA scheme can enhance the power utilization efficiency, and is more robust than the existing deterministic PA schemes.