Equivalent Deterministic Problem Research Articles

Optimal Power Flow Models With Probabilistic Guarantees: A Boolean Approach

The legacy Optimal Power Flow (OPF) dispatch in electric power grids with high proliferation of renewables can be at risk due to the lack of awareness on major uncertainties, and sudden changes in renewable outputs. This may, in turn, result in conditions where transmission line power flows are significantly exceeded, and subsequent automatic protective actions take place. This letter presents a new generalized joint chance-constrained model for the OPF problem that effectively captures the stochasticity in renewable power generation in the system. In dealing with the complexity, and non-convexity of the proposed optimization model with probabilistic guarantees, we propose a novel tractable Boolean method to transform the model into an equivalent deterministic mixed-integer linear problem, which can be solved quickly, and efficiently by off-the-shelf solvers. Numerical results verify the effectiveness of the proposed model, and the suggested Boolean methodology.

Direct plastic structural design under random strength and random load by chance constrained programming

A new formulation to calculate shakedown limit load of structures under stochastic conditions of strength and loading is developed. Direct structural reliability design is based on the required failure probabilities by chance constrained programming, which is an effective approach of stochastic programming if it can be formulated as an equivalent deterministic optimization problem.

An efficient inverse algorithm for load identification of stochastic structures

Force identification of stochastic structures is very important in science and engineering, which also leads to the challenges in the field of computational mechanics. Monte-Carlo simulation (MCS) method is a robust and effective random simulation technique for the dynamic load identification problem of the stochastic structure. However, the MCS method needs large computational cost and is also inefficient for practical engineering applications because of the requirement of a large quantity of samples. In this paper, in order to improve computational efficiency of MCS, a novel algorithm is proposed based on the modified conjugate gradient method and matrix perturbation method. First, the new developed algorithm exploits matrix perturbation method to transform dynamic load identification problems for stochastic structures into equivalent deterministic dynamic load identification problems. Then the dynamic load identification can be realized using modified conjugate gradient method. Finally, the statistical characteristics of identified force are analyzed. The accuracy and efficiency of the newly developed computational method are demonstrated by several numerical examples. It has been found that the newly proposed algorithm can significantly improve the computational efficiency of MCS and it is believed to be a powerful tool for solving the dynamic load identification for stochastic structures.

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.

Chance-constrained LQG production planning problem under partially observed forward-backward inventory systems.

Abstract The target system of our research is a reverse logistics system with imperfect information of inventory variables. This system is affected by two independents and uncorrelated random variables that represent demand and return fluctuations. A Discrete-time, chance-constrained, Linear Quadratic Gaussian Problem under imperfect information of inventory systems (DCLQG) is formulated in order to develop an aggregate manufacturing and remanufacturing plan. Technically, an optimal closed-loop solution for this stochastic problem is possible, but it is not easy to get it, particularly for large size problems. Thus, an open-loop updating approach that provides a quasi-optimal solution is investigated here. This approach considers an equivalent deterministic problem to the DCLQG problem. It is based on the conditional mean value and on variances of inventory variables, which are estimated from a Kalman filter procedure. Such an approach allows managers to build an aggregated production plan, periodically revised, that helps them to make decisions. An open-loop updating approach is compared to a no-updating approach, which depends only on the initial condition of states of the system. An example shows the importance of information gathering to provide sub-optimal solutions for stochastic problems with imperfect information of states. It is also shown that sub-optimal production policies can improve the company’s profitability.

Sustainable Waste Management Drilling Process in Fuzzy Environment

Sustainable management issues of waste during drilling oil wells in marine conditions, the process of disposal of drill cuttings in the conditions of deficiency, and fuzzy initial information using fuzzy inference system are investigated. Based on the conducted system analysis, the main criteria for controlling the process of re-injection of suspended drill cuttings were analyzed and selected. We described the technology of preparation and injection of drill cuttings slurry into the underground horizon. The method of modeling and management of the process of disposal of drilling cuttings in the marine environment in a fuzzy environment with the use of fuzzy inference system, which helps to overcome the problems of scarcity and fuzziness of the original information due to the knowledge and experience of experts are proposed. The scheme and structure of the elements of the fuzzy inference system based on the Mamdani algorithm are given. The implementation of the fuzzy output system procedure was carried out in MatLab using Fuzzy Logic Toolbox. For the purpose of sustainable waste management in the process of oil production of marine fields, waste management tasks are formulated as a fuzzy mathematical programming problem, which takes into account economic and environmental criteria and many production constraints that may be fuzzy. Since the vector of such criteria is characterized by inconsistency, the developed methods for solving the set tasks of sustainable management are based on various tradeoff schemes modified to work in a fuzzy environment. The novelty and originality of the developed methods lies in the fact that, unlike the well-known methods of similar methods for solving fuzzy problems, they are set and solved without conversion to a system of equivalent deterministic problems, with-out losing the main part of the collected fuzzy information. This allows, through the full use of the original fuzzy information, to obtain a more adequate solution to the fuzzy problem of the real problem under production conditions.

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.

Multi‐objective linear fractional inventory model with possibility and necessity constraints under generalised intuitionistic fuzzy set environment

This study presented a multi-objective linear fractional inventory (LFI) problem with generalised intuitionistic fuzzy numbers. In modelling, the authors have assumed the ambiances where generalised trapezoidal intuitionistic fuzzy numbers (GTIFNs) used to handle the uncertain information in the data. Then, the given multi-objective generalised intuitionistic fuzzy LFI model was transformed into its equivalent deterministic linear fractional programming problem by employing the possibility and necessity measures. Finally, the applicability of the model is demonstrated with a numerical example and the sensitivity analysis under several parameters is investigated to explore the study.

Multiple-criteria cash-management policies with particular liquidity terms

Abstract Eliciting policies for cash management systems with multiple assets is by no means straightforward. Both the particular relationship between alternative assets and time delays from control decisions to availability of cash introduce additional difficulties. Here we propose a cash management model to derive short-term finance policies when considering multiple assets with different expected returns and particular liquidity terms for each alternative asset. In order to deal with the inherent uncertainty about the near future introduced by cash flows, we use forecasts as a key input to the model. We express uncertainty as lack of predictive accuracy and we derive a deterministic equivalent problem that depends on forecasting errors and preferences of cash managers. Since the assessment of the quality of forecasts is recommended, we describe a method to evaluate the impact of predictive accuracy in cash management policies. We illustrate this method through several numerical examples.

Heuristic Solutions for a Class of Stochastic Uncapacitated p-Hub Median Problems

In this work, we propose a heuristic procedure for a stochastic version of the uncapacitated r-allocation p-hub median problem with nonstop services. In particular, we assume that the number of hubs to which a terminal can be allocated is bounded from above by r. Additionally, we consider the possibility of shipping traffic directly between terminals (nonstop services). Uncertainty is associated with the traffic to be shipped between nodes and with the transportation costs. If we assume that such uncertainty can be captured by a finite set of scenarios, each of which with a probability known in advance, it is possible to develop a compact formulation for the deterministic equivalent problem. However, even for small instances of the problem, the model becomes too large to be tackled by a general solver. This fact motivates the development of a heuristic procedure, whose starting point is the determination of a feasible solution to every (deterministic) single-scenario problem. These solutions are then embedded into a process based on the path relinking methodology: gradually an initial solution to the overall problem is transformed by the incorporation of attributes from some guiding solutions. In our case, the guiding solutions are those found for the single-scenario problems. We report and discuss the results of the numerical experiments performed using instances randomly generated for the new problem from the well-known CAB and AP data sets.

Benders decomposition approach with heuristic improvements for the robust foodgrain supply network design problem

We investigate the problem of designing a supply chain for foodgrains distribution under uncertainty, which is motivated from the Indian public distribution system (PDS). Initially, a deterministic mixed-integer programming (MIP) model is formulated to determine the strategic decisions of storage and transportation capacity hiring and the tactical decisions of procurement, inventory, and transportation planning to minimise the total cost over a finite planning horizon. The storage capacity hiring decision is characterised by the options of hiring under long-term contract for the entire planning horizon and from spot market for individual period. Next, we extend the model to consider the uncertainties associated with supply, demand and procurement parameters, and propose a scenario-based robust optimisation model to minimum the total relative regret. An MIP-based fix-and-optimise heuristic is proposed to efficiently solve the deterministic equivalent problem of each scenario. To solve the robust model, we implement Benders decomposition algorithm (BDA) with several heuristic techniques including warm-start strategy to build an initial feasible solution, trust region for master problem, and logical inequalities to accelerate the performance of the BDA. Finally, the effectiveness of the solution approach is demonstrated through extensive computational experiments using the test instances simulating the Indian PDS.

Multi-objective stochastic transportation problem involving three-parameter extreme value distribution

In this paper, we considered a multi-objective stochastic transportation problem where the supply and demand parameters follow extreme value distribution having three-parameters. The proposed mathematical model for stochastic transportation problem cannot be solved directly by mathematical approaches. Therefore, we converted it to an equivalent deterministic multi-objective mathematical programming problem. For solving the deterministic multi-objective mathematical programming problem, we used an ?-constraint method. A case study is provided to illustrate the methodology.

Stochastic project management: multiple projects with multi-skilled human resources

This paper presents two stochastic optimization approaches for simultaneous project scheduling and personnel planning, extending a deterministic model previously developed by Heimerl and Kolisch. For the problem of assigning work packages to multi-skilled human resources with heterogeneous skills, the uncertainty on work package processing times is addressed. In the case where the required capacity exceeds the available capacity of internal resources, external human resources are used. The objective is to minimize the expected external costs. The first solution approach is a “matheuristic” based on a decomposition of the problem into a project scheduling subproblem and a staffing subproblem. An iterated local search procedure determines the project schedules, while the staffing subproblem is solved by means of the Frank–Wolfe algorithm for convex optimization. The second solution approach is sample average approximation where, based on sampled scenarios, the deterministic equivalent problem is solved through mixed-integer programming. Experimental results for synthetically generated test instances inspired by a real-world situation are provided, and some managerial insights are derived.

Deterministic Conversion of Uncertain Manpower Planning Optimization Problem

Manpower planning is a very important component of human resource management. However, there are many indeterminate factors that should be taken into consideration in manpower planning. For example, the decision of employees to quit the job is determined by their preference, which is beyond the control of human resource department. It can be realistically modeled as a random variable when the historical data of quitting rate are large enough. Otherwise, it can only be regarded as an uncertain variable when the historical data are inadequate. In this paper, we discuss a manpower planning optimization problem for a manufacturing company with hierarchical system, where the quitting rate of employees is modeled as an uncertain variable. First, we formulate a mathematical model for this uncertain manpower planning optimization problem, where the influence on the production outputs by employees is taken into consideration. Second, we present a deterministic conversion method to transform this uncertain manpower planning optimization problem into an equivalent deterministic discrete-time optimization problem. It is further converted into an equivalent linear programming model with an equality constraint and an inequality constraint. Finally, we use the real data from Singapore, Denmark, and China to carry out a numerical simulation and make a comparison with the results obtained based on stochastic model to show the advantages of our method.

A new regularization method for the dynamic load identification of stochastic structures

For the dynamic load identification for stochastic structures, ill-posedness and randomness are main causes that lead to instability and low accuracy. Monte-Carlo simulation (MCS) method is a robust and effective random simulation technique for the dynamic load identification problems of stochastic structures. However, it needs large computational cost and is also inefficient for practical engineering applications because of the requirement of a large quantity of samples. In order to improve its computational efficiency, this paper proposes a novel computational algorithm for the dynamic load identification of stochastic structures. First, the newly developed algorithm transforms dynamic load identification problems for stochastic structures into equivalent deterministic dynamic load identification problems. Second, a new regularization method is proposed to realize the deterministic dynamic load identification. Third, the assessments of the statistics of identified loads are obtained based on statistical theory. Finally, the stability and robustness of the proposed algorithm are well validated by two engineering examples. It is demonstrated that the newly developed regularization method outperforms the traditional Tikhonov regularization method in computational accuracy. Moreover, the newly proposed algorithm can significantly improve the computational efficiency of MCS and is very stable and effective in solving the dynamic load identification for stochastic structures.

A new approach based on possibilistic programming technique and fractile optimization for bilevel programming in a hybrid uncertain circumstance

Fuzzy random bilevel programming is very important for modeling hierarchical decision processes consisting of two decision makers with a hybrid uncertainty of fuzziness and randomness. This type of problem is usually much more difficult to handle than either purely fuzzy bilevel programming or purely stochastic bilevel programming due to complexity of hybrid uncertainty and NP-hardness of bilevel programming. In this paper, a novel approach based on Me-based possibilistic programming technique and fractile optimization is proposed to transform and cope with the fuzzy random bilevel programming problem. On the basis of the Me-based possibilistic programming method, the original problem is first converted into a Me-based bilevel chance constrained programming model. Then fractile criterion optimization and probabilistic chance constrained conditions are employed to transform the Me-based bilevel chance constrained model into an equivalent deterministic bilevel nonlinear programming problem. Furthermore, different satisfactory solutions associated with varying optimistic-pessimistic attitudes of the decision makers can be searched by the fuzzy interactive solution approach, which are helpful for the decision makers to gain more desirable information under uncertainty. Finally, several numerical examples are provided to illustrate the feasibility of the proposed model and solution methodology.

Application of stochastic programming for iron ore quality control

Stockpiling and blending play a major role in maintaining the quantity and quality of the raw materials fed into processing plants, especially the cement, iron ore and steel making, and coal-fired power generation industries that usually require a much uniformed feed. Due to the variable nature of such materials, they even come from the same source and the produced ores or concentrates are seldom homogeneous enough to be directly fed to the processing plant ore furnaces. Processing plants in iron ore mines need uniform feed properties in terms of each variable (in this work, iron phosphorous ratio and Fe content in magnetite phase) grade of ore, and therefore, homogenization of iron ore from different benches of an open pit or ore dumps has become an essential part of modern mine scheduling. When ore dumps are considered as an ore source, the final grade of the material leaving the dump to the blending bed cannot be easily determined. This difficulty contributes to mixing the materials of different grades in a dump. In this work, the ore dump elements were treated as normally distributed random variables. Then a stochastic programming model was formulated in an iron ore mine in order to determine the optimum amount of ore dispatched from different bench levels in open pit and also four ore dumps to a windrow-type blending bed in order to provide a mixed material of homogenous composition. The chance-constrained programming technique was used to obtain the equivalent deterministic non-linear programming problem of the primary model. The resulting non-linear model was then solved using LINGO. The results obtained showed a better feed grade for the processing plant with a higher probability of grade blending constraint satisfaction.

Stochastic programming solution for placement of satellite ground stations

Disaster recovery efforts are enhanced through the collection and dissemination of satellite data, which is downloaded from satellites to ground stations. The optimal ground station locations vary depending on the location of the disaster, but ground station construction occurs before the realization of a disaster. Thus, a stochastic optimization problem arises: decide the location of ground stations before disasters with uncertain locations. We use a stochastic programming approach to select the location of ground stations given a set of potential disaster scenarios. The objective is to maximize the expected amount of data downloaded. The problem formulation consists of a two-stage stochastic program where the first-stage determines the locations of the ground stations and the second-stage schedules the uploading and downloading of data. We solve the problem using the L-shaped method; we find that it significantly outperforms solving the deterministic equivalent problem directly. We also find that an alternate second-stage formulation significantly improves solution time. The optimized set of ground stations found by our algorithm is compared to the set of ground stations operated by the National Oceanic and Atmospheric Administration’s; results confirm that the current placement is effective and demonstrate the benefit in adding additional ground stations.

Effects of Probability Function on the Performance of Stochastic Programming

Stochastic programming is a valuable optimization tool where used when some or all of the design parameters of an optimization problem are defined by stochastic variables rather than by deterministic quantities. Depending on the nature of equations involved in the problem, a stochastic optimization problem is called a stochastic linear or nonlinear programming problem. In this paper,a stochastic optimization problem is transformed intoan equivalent deterministic problem,which can be solved byany known classical methods (interior penalty method is applied here).The paper mainly focuseson investigatingthe effect of applying various probability functions distributions(normal, gamma, and exponential) for design variables. The following basic required equations to solve nonlinear stochastic problems with various probability functionsfor random variables are derived and sensitivity analyses to studythe effects of distribution function typesand input parameterson the optimum solution are presented as graphs and in tables by studyingtwoconsidered test problems. It is concluded that thedifference between probabilistic and deterministic solutions toa problem, when the normal distribution ofrandom variables isused, is very different fromthe results when gamma and exponential distribution functions are used. Finally, it is shownthat the rate of solution convergence tothe normal distribution is faster than the other distributions.

Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure

We consider a scalar stochastic linear optimization problem subject to linear constraints. We introduce the notion of deterministic equivalent formulation when the underlying probability space is equipped with a probability multimeasure. The initial problem is then transformed into a set-valued optimization problem with linear constraints. We also provide a method for estimating the expected value with respect to a probability multimeasure and prove extensions of the classical strong law of large numbers, the Glivenko–Cantelli theorem, and the central limit theorem to this setting. The notion of sampling with respect to a probability multimeasure and the definition of cumulative distribution multifunction are also discussed. Finally, we show some properties of the deterministic equivalent problem.