Stochastic Transportation Problem Research Articles

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.

Navigating uncertain distribution problem: a new approach for resolution optimization of transportation with several objectives under uncertainty

Amidst uncertainty, decision-making in manufacturing becomes a central focus due to its complexity. This study explores complex transportation constraints and uses novel ways to guide manufacturers. The Multi-objective Stochastic Linear Fractional Transportation Problem (MOSLFTP) is a crucial tool for managing supply chains, manufacturing operations, energy distribution, emergency routes, healthcare logistics, and other related areas. It adeptly addresses uncertainty, transforming efficiency and effectiveness in several domains. Stochastic programming is the process of converting theoretical probabilities into concrete certainties. The artistic compromise programming technique acts as a proficient mediator, reconciling opposing objectives and enabling equitable decision-making. This novel approach also addresses the Multi-objective Stochastic Linear plus Linear Fractional Transportation Problem (MOSLPLFTP), which involves two interconnected issues. The effectiveness of these principles is clearly shown with the help of the LINGO® 18 optimization solver. This study uses a ranking method to compare the similar methods to solve the current problems. A meticulously designed example acts as a significant achievement, shedding light on our method in a practical setting. It serves as a distinctive instrument, leading manufacturers through the maze of uncertainty and assisting them in determining the most advantageous course of action. This journey involves subtle interactions between complexity and simplicity, uncertainty is overcome by decisiveness, and invention is predominant.

New Computer Efficient Approximations of Random Functions for Solving Stochastic Transport Problems

New Computer Efficient Approximations of Random Functions for Solving Stochastic Transport Problems

Multi-choice fractional stochastic multi-objective transportation problem

Multi-choice fractional stochastic multi-objective transportation problem

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Type-2 Fuzzy Stochastic Transportation Problem with Gamma Distribution

Fuzzy Goal Programming Approach for Solving Stochastic Transportation Problem with Additional Budgetary Constraints

Fuzzy Goal Programming Approach for Solving Stochastic Transportation Problem with Additional Budgetary Constraints

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 Mathematical programming approach for a vessel scheduling-transportation problem with multiple sources and destinations, and normal daily demand distributions

The principal focus of this research effort is to investigate a maritime stochastic transportation problem. In this case, crude oil shipments are to be transported from multiple sources to multiple destinations. The demand at the destinations is normally distributed and the violation of certain specified lower and upper daily storage limits can lead to various types of penalties. We aim to utilize (stochastic) mathematical programming approaches to minimize the overall cost of such fleet operation by optimizing the vessel schedules and maintaining acceptable daily storage levels considering the stochastic demand structures. This research effort signifies that the nature of daily demand distributions in the multiple sources-destinations scheduling-inventory scenario significantly impacts the overall fleet schedules and the total expected cost. Therefore, it is crucial to grasp essential stochastic aspects of the daily demands to avoid potential misrepresentation of the operational costs. The robustness of the adopted approach is illustrated by presenting computational results that are based on a wide range of test problems. Moreover, our computational study also examines the impact of variations in demand and the probability of meeting demands on the cost structures.

An Artificial Intelligence Approach for Solving Stochastic Transportation Problems

Recent years witness a great deal of interest in artificial intelligence (AI) tools in the area of optimization. AI has developed a large number of tools to solve the most difficult search-and-optimization problems in computer science and operations research. Indeed, metaheuristic-based algorithms are a sub-field of AI. This study presents the use of the metaheuristic algorithm, that is, water cycle algorithm (WCA), in the transportation problem. A stochastic transportation problem is considered in which the parameters supply and demand are considered as random variables that follow the Weibull distribution. Since the parameters are stochastic, the corresponding constraints are probabilistic. They are converted into deterministic constraints using the stochastic programming approach. In this study, we propose evolutionary algorithms to handle the difficulties of the complex high-dimensional optimization problems. WCA is influenced by the water cycle process of how streams and rivers flow toward the sea (optimal solution). WCA is applied to the stochastic transportation problem, and obtained results are compared with that of the new metaheuristic optimization algorithm, namely the neural network algorithm which is inspired by the biological nervous system. It is concluded that WCA presents better results when compared with the neural network algorithm.

Application of Water Cycle algorithm to Stochastic Fractional Programming Problem

This paper presents an application of Water Cycle algorithm (WCA) in solving stochastic programming problems. In particular, Linear stochastic fractional programming problems are considered which are solved by WCA and solutions are compared with Particle Swarm Optimization, Differential Evolution, and Whale Optimization Algorithm and the results from literature. The constraints are handled by converting constrained optimization problem into an unconstrained optimization problem using Augmented Lagrangian Method. Further, a fractional stochastic transportation problem is examined as an application of the stochastic fractional programming problem. In terms of efficiency of algorithms and the ability to find optimal solutions, WCA gives highly significant results in comparison with the other metaheuristic algorithms and the quoted results in the literature which demonstrates that WCA algorithm has 100% convergence in all the problems. Moreover, non-parametric hypothesis tests are performed and which indicates that WCA presents better results as compare to the other algorithms.

A rolling horizon approach for a multi-stage stochastic fixed-charge transportation problem with transshipment

We study a fixed-charge transportation problem under stochastic and dynamic demand. We propose a multi-stage mixed integer stochastic programming formulation, where the first-stage decision is the delivery from the supplier to the retailers, while transshipment is used, in addition to classical backordering as recourse decision. The objective is the minimization of the total expected cost. We prove that this problem is NP-hard and, through a worst-case analysis, that transshipment can provide significant cost savings. Extensive computational studies are carried out to evaluate the performance of a rolling horizon approach with respect to the optimal cost. Numerical results show that this heuristic provides effective solutions in short computational time. Managerial insights are finally drawn.

Features of transport in non-Gaussian random porous systems

The goal of this work is to employ a semi-analytical framework to investigate key features associated with the transport behavior of an inert solute in non-Gaussian random fields. We focus our analysis on the transport dynamics of a solute plume through a porous medium characterized by spatially heterogeneous non-Gaussian log-conductivity fields, Y. We rest on a stochastic Lagrangian framework to provide semi-analytical formulations to evaluate the statistical moments and cumulative distribution function (CDF) of solute concentration. The heterogeneous structure of the log-conductivity field is modeled as a Generalized Sub-Gaussian process. This model has been shown to capture non-Gaussian and scale-dependent features displayed by several variables, including key parameters of porous media. Our results suggest that the effects of non-Gaussianity in Y on solute concentration statistics are more pronounced at locations near the solute source zone and at early times. The impact of the analyzed non-Gaussian nature of the field of Y is also significant at the lower tails of the distribution. We also explore conditions under which when the concentration CDF in Generalized Sub-Gaussian Y fields can be approximated by the widely used beta distribution. Furthermore, the methodology used in this work is an alternative to the commonly used numerical Monte Carlo method and can be employed as a benchmark tool in computational stochastic mass transport problems in porous media.

Application of Water Cycle Algorithm to Stochastic Fractional Programming Problems

This paper presents an application of Water Cycle algorithm (WCA) in solving stochastic programming problems. In particular, Linear stochastic fractional programming problems are considered which are solved by WCA and solutions are compared with Particle Swarm Optimization, Differential Evolution, and Whale Optimization Algorithm and the results from literature. The constraints are handled by converting constrained optimization problem into an unconstrained optimization problem using Augmented Lagrangian Method. Further, a fractional stochastic transportation problem is examined as an application of the stochastic fractional programming problem. In terms of efficiency of algorithms and the ability to find optimal solutions, WCA gives highly significant results in comparison with the other metaheuristic algorithms and the quoted results in the literature which demonstrates that WCA algorithm has 100% convergence in all the problems. Moreover, non-parametric hypothesis tests are performed and which indicates that WCA presents better results as compare to the other algorithms.

Computation of Some Stochastic Transportation Problems Using Essen Inequality

Computation of Some Stochastic Transportation Problems Using Essen Inequality

Comparison of anticipatory algorithms for a dial-a-ride problem

Progress in digitalization opens opportunities to capture accurate transportation logistics data and provide advanced decision support, which leads into the question of how to efficiently exploit this progress in order to improve solution quality in transportation services. Here we address this issue in the context of a dynamic and stochastic patient transportation problem, where besides considering new events, we also incorporate stochastic information about future events. We propose different anticipatory algorithms and investigate which algorithm performs best according to the given settings in a real-world application. We therefore address different types of dynamic events, appropriate response times, and the synchronization of real-world data with the plan. In order to test and analyze how the algorithms behave and perform, we apply the concept of a digital twin. The implemented anticipatory algorithms compared here are a sample scenario planning approach and two waiting strategies. The question of the value of more sophisticated algorithms compared to algorithms with less computational effort is investigated. The experimental results show that solution quality benefits from incorporating information about future requests, and that simple waiting strategies prove most suitable for such a highly dynamic environment. We find that in highly stochastic environments, a rescheduling should be done whenever a new event occurs, whereas in less stochastic environments it is better to let the optimization engine run a bit longer and not start reoptimization after every new event.

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.

Stochastic optimal transport with free end time

We consider a stochastic transportation problem between two prescribed probability distributions (a source and a target) over processes with general drift dependence and with free end times. First, and in order to establish a dual principle, we associate two equivalent formulations of the primal problem in order to guarantee its convexity and lower semi-continuity with respect to the source and target distributions. We exhibit an equivalent Eulerian formulation, whose dual variational principle is given by Hamilton–Jacobi–Bellman type variational inequalities. In the case where the drift is bounded, regularity results on the minimizers of the Eulerian problem then enable us to prove attainment in the corresponding dual problem. We also address attainment when the drift component of the cost defining Lagrangian L is superlinear L≈|u|p with 1<p<2, in which case the setting is reminiscent of our approach – in a previous work – on deterministic controlled transport problems with free end time. We finally address criteria under which the optimal drift and stopping time are unique, namely strict convexity in the drift component and monotonicity in time of the Lagrangian.

Distributionally robust chance-constrained programs with right-hand side uncertainty under Wasserstein ambiguity

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0–1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the problem sizes that can be handled in such DR-CCP formulations by reducing the solution times from hours to seconds.

Stochastic optimal transport revisited

We prove the Duality Theorems for the stochastic optimal transportation problems with a convex cost function without a regularity assumption that is often supposed in the proof of the lower semicontinuity of an action integral. In our new approach, we prove that the stochastic optimal transportation problems with a convex cost function are equivalent to a class of variational problems for the Fokker-Planck equation, which lets us revisit them. It is done by the so-called superposition principle and by an idea from the mather theory. The superposition principle is the construction of a semimartingale from the Fokker-Planck equation and can be considered a class of the so-called marginal problems that construct stochastic processes from given marginal distributions. It was first considered in stochastic mechanics by E. Nelson, called Nelson's problem, and was proved by E. Carlen first. The semimartingale is called the Nelson process, provided it is Markovian. We also consider the Markov property of a minimizer of the stochastic optimal transportation problem with a nonconvex cost in a one-dimensional case. In the proof, the superposition principle and the minimizer of an optimal transportation problem with a concave cost function play crucial roles. Lastly, we prove the semiconcavity and the Lipschitz continuity of Schrodinger's problem that is a typical example of the stochastic optimal transportation problem.

An optimization of solid transportation problem with stochastic demand by Lagrangian function and KKT conditions

In this paper, a stochastic solid transportation problem (SSTP) is constructed where the demand of the item at the destinations are randomly distributed. Such SSTP is formulated with profit maximization form containing selling revenue, transportation cost and holding/shortage cost of the item. The proposed SSTP is framed as a nonlinear transportation problem which is optimized through Karush–Kuhn–Tucker (KKT) conditions of the Lagrangian function. The primary model is bifurcated into three different models for continuous and discrete demand patterns. The concavity of the objective functions is also presented here very carefully. Finally, a numerical example is illustrated to stabilize the models.