Equivalent Deterministic Problem Research Articles

Solving Neutrosophic Multi-Dimensional Fixed Charge Transportation Problem

In the decision making problems, transportation problem (TP) is one of the most significant applications that aim to optimize the objective function of products shipped from different sources to different destinations. When an additional charge such as toll tax, parking fee, permit fee and so on are included in the transportation cost then the problem becomes a fixed charge TP. If the shipped products are damageable, they are affected by the type of vehicle and the distance of route. So it is quite evident that the vehicle type and the distance of route play a significant role in TP. In reality, along with sources and destinations, the decision maker (DM) desires to consider the parameters such as the mode of transport and routes of transportation that are used for shipment of the products from sources to destinations in order to minimize the loss of damageable products and to maximize the profit. As a result, mode of transport and routes of transportation are added to TP, then the TP becomes a four-dimensional TP. Sometimes, the data/parameters of the problems may not be always precise due to incomplete information, inadequate data or shortage of evidence. To deal with these obstacles, the parameters of the problem can be represented as the single-valued trapezoidal neutrosophic numbers (SVTrNNs). The neutrosophic data facilitate a reasonable and practicable way for DMs to tackle decisionmaking problems by managing indeterminacy and providing an effective framework for analysis and synthesis of complex decision scenarios. In view of this, in this paper we have considered the single objective four-dimensional fixed charge transportation problem (4DFCTP) with parameters, supply, demand, conveyance, transportation cost, fixed charge as single-valued trapezoidal neutrosophic numbers and the distance of routes as real numbers. First, the score function is utilized to transform the neutrosophic parameter into its deterministic parameter in order to avoid the negative values for the decision variable. Then the deterministic problem comprises deterministic parameters such as supply, demand, conveyance, transportation cost with fixed charges and distance of routes. Secondly, a novel approach namely, min zeromin cost approach is introduced for finding the optimal solution to the equivalent deterministic problem in polynomial time. The main objective of this paper is to optimize the breakable products and routing plan of vehicles in a way to minimize the total transportation cost with fixed cost of the business organizations using the proposed min zero-min cost approach. To demonstrate the problem’s validity and relevance, two numerical examples are solved using our proposed approach. To highlight the proposed approach, comparison of the solution with the LINGO software is performed. The obtained optimal solution from the proposed approach is the same as the LINGO software. At last, conclusions as well as future work related to the study are presented.

SOLVING STOCHASTIC TRANSPORTATION ELECTRICITY PROBLEM WITH FUZZY INFORMATION ON PROBABILITY DISTRIBUTION USING MATLAB PROGRAM

This study focuses on MATLAB code programs of the entire stages of solving Stochastic Transportation Linear Programming Problems with Fuzzy Uncertainty Information on Probability Distribution Space (STLPPFI) with its algorithm outlines. A MATLAB code program of STLPPFI problem solver with algorithm outlines are proposed to solve STLPPFI model problems, and it utilizes many concepts as Alpha-Cut technique, Truth Degrees technique, Linear Fuzzy Membership Function (LFMF), Trapezoidal Fuzzy Number , Triangular Fuzzy Number , Linear Fuzzy Ranking Function (LFRF), Expectation Weighted Summation technique (EWS) and analyzing cases via second condition test of alpha-cut technique. The STLPPFI problem solver is utlized to convert STLPPFI into its corresponding equivalent Deterministic Transportation Linear Programming Problem (DTLPP) via defuzzifying from fuzziness on probability distribution space and derandomization randomness of problem formulation respectively. In addition, Dual-Simplex algorithm method with Vogel Approximation Algorithm Method (VAM) are used to obtain optimal solution from DTLPP. All MATLAB code programs with their proposed algorithm outlines are new except Dual-Simplex and VAM. The MATLAB code program of STLPPFI problem solver are more efficient along with a numerical example on electricity field illustrating practicability of this proposed MATLAB code program with its algorithm. Finally, the solution procedure illustrates the MATLAB code program of the proposed method is practical and applicable in the fields of energy and industry as it facilitates the method of transforming the energy at the lowest cost, least time running and is commercially applicable. Comparative comments are provided between Dual-Simplex and VAM in solution process.

Chance-constrained co-optimization of peer-to-peer energy trading and distribution network operations

Peer-to-Peer (P2P) energy markets enable participants to trade locally produced electrical energy at a mutually agreed-upon price. Distributed Energy Resources (DERs) and flexible loads are fundamental components of P2P energy markets, facilitating the market participants to supply or consume energy in accordance with trading agreements. Consequently, it is vital to account for the stochastic nature of DERs and load demand while designing the P2P energy trading model in order to attain optimal operation of P2P energy markets and secure operation of the distribution network. The scenario-based and robust optimization techniques commonly used to characterize uncertainty suffer from data requirement, computational burden, and solution conservatism standpoints. This paper, therefore, proposes a chance-constrained model for P2P energy trading considering renewable energy uncertainty and distribution network constraints. The uncertain behaviour of renewable generation resources is modelled using chance constraints and integrated with the decision-making process of the P2P energy market and distribution network operation. The chance-constrained P2P energy trading problem is reformulated as a deterministic equivalent problem that can be solved by readily available commercial solvers. The ex-ante and ex-post-performance of the proposed chance-constrained formulation is evaluated on a 15-bus radial distribution network, while the scalability of the proposed formulation is demonstrated on modified IEEE 33-bus and 69-bus test networks. The results showcase the effectiveness of the proposed chance-constrained model in handling the uncertainty and respecting the network constraints while deriving the P2P energy trading outcomes.

Convex Programming-Based Robust Optimization Procedure for Blast-Excited Structures with Bounded Uncertainty

Application of the robust optimization (RO) technique to ensure the least possible deviation of structural performance is gaining increasing attention in recent years. However, most RO studies have been conducted in the probabilistic domain, where sufficient uncertainty information is available to construct probability distribution functions of response parameters. But, in many cases, such as for structures under underground blast excitation, hazard parameters like charge weight and charge center distance do not have any definite probability density function. Only their ranges of variations are known to an engineer. Hence, it is more logical to model these parameters as uncertain-but-bounded (UBB) type, which requires only bounds of variations to characterize the uncertainty. The available RO studies in the UBB-domain mainly deal with static loads. Studies on time-variant dynamic loads, such as blast load, are scarce in the existing literature. Also, blast load time-histories show record-to-record variations in their signatures, even with the same setup of hazard parameters. This aspect cannot be considered by the conventional convex programming approaches of RO in the UBB domain. To address all these issues, a new RO procedure is proposed in the present study, which is based on the dual response surface method. The proposed RO procedure can transform the complex simulation-based RO procedure under dynamic blast load to an equivalent deterministic problem with explicit constraints. Thereby, the RO can be easily solved by a gradient-based optimizer. The effectiveness of the proposed approach is elucidated by the complex real-world problems of a multistory building and an underground reinforced concrete bunker. The results are compared with conventional RO approaches. The results show that the proposed approach consistently yields more economic as well as robust solutions in a computationally efficient way.

Solving a new bi-objective mathematical model for a hybrid flow shop scheduling problem with robots and fuzzy maintenance time

This paper presents a bi-objective mathematical model of a hybrid flow shop scheduling problem (HFSSP) with robots and fuzzy maintenance time. The proposed model minimizes the mean completion time of jobs and the total cost of the processes based on maintenance costs and incurred transportation costs via robots. A fuzzy chance-constrained programming (FCCP) model is used to present its deterministic-equivalent problem. It is then solved by two multi-objective decision-making approaches, namely LP-metric and goal attainment (GA). A sensitivity analysis of the weights of the objective functions is conducted and the behavior of the model is studied for each instance. Also, a TOPSIS (the technique for order of preference by similarity to ideal solution) method is used to rank the efficiency of the proposed solution approaches. Future research areas are also suggested.

A response band-based method for time-dependent reliability-based robust design optimization

In this paper, a response band-based method for time-dependent reliability-based robust design optimization is proposed. The proposed method provides a novel alternative framework, consist of a two-step transformation stage and a solving stage, to solve the time-dependent reliability-based robust design optimization problem. The original time-dependent reliability-based robust design optimization problem is transformed into an equivalent deterministic robust design optimization problem in the transformation stage, and the equivalent problem is settled in the solving stage. In the transformation stage, the dynamic modal decomposition technique and the kriging technique are combined to overcome the problem that there is no standard for both time division and observation sampling in the commonly used transformation methods. In the solving stage, an approach for constructing the response band of the objective function is presented, which significantly reduces the computational consumption of the variation evaluation of the objective function. Five cases are employed to verify the effectiveness of the proposed method.

Drones for relief logistics under uncertainty after an earthquake

This study presents a post-disaster delivery problem called the relief distribution problem using drones under uncertainty, in which critical relief items are distributed to disaster victims gathered at assembly points after a disaster, particularly an earthquake. Because roads may be obstructed by debris after an earthquake, drones can be used as the primary transportation mode. As the impact of an earthquake cannot be easily predicted, the demand and road network uncertainties are considered. Additionally, the objective is to minimize the total unsatisfied demand subject to a time-bound constraint on the deliveries, as well as the range and capacity limitations of drones. A two-stage stochastic programming and its deterministic equivalent problem formulations are presented. The scenario decomposition algorithm is implemented as an exact solution approach. To apply this study to real-life applications, a case study is conducted based on the western (European) side of Istanbul, Turkey. The computational results are used to evaluate the performance of the scenario decomposition algorithm and analyze the value of stochasticity and the expected value of perfect information under different parametric settings. We additionally conduct sensitivity analyses by varying the key parameters of the problem, such as the time-bound and capacities of the drones.

Uncertain optimal control problem with the first hitting time objective and application to a portfolio selection model

This paper discusses an uncertain time optimal control problem by considering time efficiency, which is to optimize the objective function about the first hitting time subject to uncertain differential equations. According to the definition of the α-path, the uncertain time optimal control problem is transformed into an equivalent deterministic optimal control problem. Two kinds of time optimal control models are presented where optimistic value and reaching index are chosen as the optimality criteria, respectively. Applying the proposed uncertain optimal control model to a portfolio selection problem, we obtain the uncertainty distribution of the first hitting time (the investors’ first profit time). Meanwhile, sufficient conditions of the optimal control strategy of such models are provided. Numerical simulations are provided which reveal the change for our optimal control strategy.

A Transformation-Based Improved Kriging Method for the Black Box Problem in Reliability-Based Design Optimization

In order to overcome the drawbacks of expensive function evaluation in the practical reliability-based design optimization (RBDO) problem, researchers have proposed the black box-based RBDO method. The algorithm flow of the commonly employed RBDO method for the black box problem consists of the outer construction loop of the surrogate model of the constraint function and the inner surrogate model-based solving loop. To improve the solving ability of the black box RBDO problem, this paper proposes a transformation-based improved kriging method to increase the effectiveness of the two loops identified above. For the outer loop, a sample distribution-based learning function is suggested to improve the construction efficiency of the surrogate model of the constraint function. For the inner loop, a paired incremental sample-based limit reliability boundary construction approach is suggested to transform the RBDO problem into an equivalent deterministic design optimization problem that can be efficiently solved by classical optimization algorithms. The test results of five cases demonstrate that the proposed method can accurately construct the surrogate model of the constraint function and efficiently solve the black box RBDO problem.

An open-loop solution for a stochastic problem with imperfect state information and chance-constraint adjusted by an optimal gain

This study aims to develop an optimal aggregate production planning policy for a reverse logistics system with imperfect state information and chance constraints. The system consists of a forward channel for storing and distributing products and a backward channel for collecting, remanufacturing, or discarding products. The paper presents a chance-constrained linear-quadratic Gaussian optimization model that considers imperfect state information and formulates an equivalent deterministic problem. It also introduces a minimum variance problem to address the uncontrolled variance of the serviceable inventory variable, whose result is an optimal gain to balance the conditional variances of inventory and production over time, making the solution more cost-efficient. The open-loop solution with the minimum variance gain shows its efficiency through a simple example, reducing the total cost of production by controlling the growth of inventory and production variances. Furthermore, this approach offers a practical way for managers to create inventory-production scenarios for decision-making in a reverse logistics system with imperfect state information and chance constraints.

A neural network approach to solve geometric programs with joint probabilistic constraints

We study a dynamical neural network approach to solve nonconvex geometric programs with joint probabilistic constraints (GPPC for short) with normally distributed coefficients and independent matrix row vectors. The main feature of our framework is to solve the biconvex deterministic equivalent problem of GPPC without the use of any convex approximation unlike the state-of-the-art solving methods. The second feature of our approach is to solve the dynamical system without the use of any iterative method but only by solving an ordinary differential equation system. We prove the stability together with the convergence of our neural network in the sense of Lyapunov. We also prove the equivalence between the optimal solution of the deterministic equivalent problem of GPPC and the solution of the dynamical system. We provide numerical experiments to show the performances of our approach compared to the state-of-art.

Chance constrained dynamic optimization approach for single machine scheduling involving flexible maintenance, production, and uncertainty

Chance constraints are suitable for industrial process modeling under uncertain conditions, where constraints cannot be strictly satisfied or do not need to be fully satisfied. In this paper, a single machine scheduling problem involving flexible maintenance, production, and uncertainty is modeled as a chance constrained dynamic optimization problem (CCDOP). A novel method is proposed for transforming the CCDOP into an equivalent deterministic dynamic optimization problem (DOP) with fixed state jump times. Furthermore, by using the idea of l1 penalty function and a smooth approximation technique, the resulting deterministic DOP becomes a smoothing penalty problem, which is a non-convex nonlinear parameter optimization problem (NNPOP) with simple bounds on the variables. To solve the NNPOP, a gradient-based stochastic search algorithm (GSSA) is developed based on a gradient-based adaptive search algorithm (GASA) and a novel stochastic search algorithm (NSSA). The convergence analysis result shows that the GSSA is a globally convergent algorithm. Finally, two numerical examples are used to illustrate the effectiveness of the proposed method. Numerical results show that the GSSA has excellent convergence behavior with robust computation feature, providing better results compared with the other typical methods.

Risk-averse hub location: Formulation and solution approach

In this study, we present risk-neutral and risk-averse two-stage stochastic formulations for the uncapacitated multiple allocation p-hub median problem and discuss the impact of risk-aversion on the optimal solution. Although stochastic models are useful to tackle the uncertainty in problem parameters, the solution of these models requires higher computational effort than their deterministic counterparts. Therefore, we present a scenario decomposition algorithm for the stochastic formulations. To evaluate the performance of the proposed solution algorithm, a set of computational experiments is conducted on real data sets. The results show that the proposed algorithm is very effective in finding optimal or near-optimal solutions in significantly shorter computation time than that of deterministic equivalent problems.

Optimal control of nonlinear Markov jump systems by control parametrisation technique

This paper considers an optimal control problem of nonlinear Markov jump systems with continuous state inequality constraints. Due to the presence of continuous-time Markov chain, no existing computation method is available to solve such an optimal control problem. In this paper, a derandomisation technique is introduced to transform the nonlinear Markov jump system into a deterministic system, which simultaneously gives rise to an equivalent deterministic dynamic optimisation problem. The control parametrisation technique is then used to partition the time horizon into a sequence of subintervals such that the control function is approximated by a piecewise constant function consistent with the partition. The heights of the piecewise constant function on the corresponding subintervals are taken as decision variables to be optimised. In this way, the approximate dynamic optimisation problem is an optimal parameter selection problem, which can be viewed as a finite dimensional optimisation problem. To solve it using a gradient-based optimisation method, the gradient formulas of the cost function and the constraint functions are derived. Finally, a real-world practical problem involving a bioreactor tank model is solved using the method proposed.

Robust optimization of control command for aerospace vehicles with aerodynamic uncertainty

To reduce the design burden of Aerospace Vehicles (ASVs) control systems, this paper proposes a multi-constrained robust trajectory optimization method, which provides a good front-end input for the control system. Differ from the conventional aircraft, some control performance of ASVs is not only related to the model parameters, but also affected by the flight status. Therefore, the robust optimization method combines this characteristic of ASVs, sets the control performance as one of the optimization objectives, and considers the influence of parameter uncertainty. In this method, the polynomial chaos expansion algorithm is used to transform the trajectory optimization problem with uncertain parameters into the equivalent deterministic robust trajectory optimization problem. Finally, compared with traditional deterministic trajectory optimization methods to illustrate the effectiveness of proposed control optimization method.

An Efficient Neurodynamic Approach to Fuzzy Chance-constrained Programming

In both practical applications and theoretical analysis, there are many fuzzy chance-constrained optimization problems. Currently, there is short of real-time algorithms for solving such problems. Therefore, in this paper, a continuous-time neurodynamic approach is proposed for solving a class of fuzzy chance-constrained optimization problems. Firstly, an equivalent deterministic problem with inequality constraint is discussed, and then a continuous-time neurodynamic approach is proposed. Secondly, a sufficient and necessary optimality condition of the considered optimization problem is obtained. Thirdly, the boundedness, global existence and Lyapunov stability of the state solution to the proposed approach are proved. Moreover, the convergence to the optimal solution of considered problem is studied. Finally, several experiments are provided to show the performance of proposed approach.

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.

Multiperiod transshipment location–allocation problem with flow synchronization under stochastic handling operations

AbstractThe transshipment location–allocation problem consists of locating transshipment facilities (e.g., intermodal hubs) of a transportation network and allocating freight flows through them, from several origins to several destinations, to satisfy demand and supply constraints. The objective is to maximize the total net transportation utility given by the total shipping utility minus the total cost to locate the facilities. Moreover, flow synchronization at the facilities must also be ensured. Unfortunately, the flow synchronization depends on a broad set of unknown events, which could cause both unexpected reductions of the facility capacity and uncertain utility of handling operations. In this paper, we first want to evaluate how uncertainty on facility capacity and handling operations utility affects the transshipment location–allocation problem in terms of complexity, net gain, and optimal solutions. Moreover, we extend the problem from a single to a multi‐period setting to have a wider view of future scenarios realizations and consequently synchronize the flows by using different facilities on different periods. We propose a two‐stage stochastic programming formulation with recourse and analyze, over a ground set of instances, some well‐known economic indicators to derive managerial insights on the importance of addressing uncertainty for the problem. Finally, given the computational burden of solving the deterministic equivalent problem, we propose several heuristics based on progressive hedging and test their performance.

Optimizing grinding operation with correlated uncertain parameters

ABSTRACT Parameters appearing in deterministic optimization (DO) models are kept constant during optimization, though uncertainty in these parameters is an inherent and unavoidable. Investigations about the effect of these uncertain parameters (UP) on the final outcomes are, therefore, necessary. Chance-Constrained Programming (CCP) can handle such situations by introducing the probability of constraint satisfaction and converting optimization under uncertainty problems into an equivalent deterministic optimization problem (EDOP). However, the most popular variant of this method neglects the aspects of parameter dependency for the ease of the analysis. Under the conditions when the correlations exist among UPs, the use of joint CCP with parameter dependency information is necessary to avoid misleading results arising from the assumption of no correlation. To show the impact of dependency of UPs on the outcomes of optimization, joint CCP has been adopted in this study for an industrial grinding process (IGP), which is non-linear in nature in terms of its system states as well as UPs. The effects of both confidence level and correlation coefficient have been shown on the Pareto optimal (PO) front for different sets of distributions among UPs while handling a multi-objective optimization problem (MOOP) of the Grinding Process.

Multi-objective enhanced interval optimization problem

In this paper, we consider a multiple objective optimization problem whose decision variables and parameters are intervals. Existence of solution of this problem is studied by parameterizing the intervals. A methodology is developed to find the $$t\omega $$ -efficient solution of the problem. The original problem is transformed to an equivalent deterministic problem and the relation between solutions of both is established. Finally, the methodology is verified in numerical examples.