Generalized Transportation Problem Research Articles

A fuzzy model for generalized transportation problems in China

SummaryThis study underscores the growing significance of multimodal transportation within the cargo sector and its consequential environmental impacts. We present a novel mathematical model for operation scheduling, incorporating variables such as resource availability, customer service benchmarks, and environmental considerations. Our objective is to mitigate transportation expenses and reduce delivery delays. The proposed approach advocates LU decomposition with a pivot strategy for rapid model resolution, adherence to convergence criteria, optimization of cost strategies, and efficient resource utilization. Leveraging the adaptive neural fuzzy inference system (ANFIS) and genetic algorithm (GA), our methodology facilitates learning from past decisions to enhance solutions, aligning supply, and demand efficiently. We evaluate financial and environmental implications across four scenarios, offering insights into the economic and environmental advantages of various transportation modes—trains, ships, airplanes—compared to truck transportation, with a specific focus on CO2 emission impacts. Implementing the ANFIS+GA model in multimodal scenarios yield impressive results: minimal MAPE transportation cost of 0.17%, R2 transportation cost of 0.996, MAPE CO2 emissions of 0.13%, and R2 CO2 emissions of 0.996. By identifying cost‐efficient routes and optimizing resource allocations, our approach enables informed decisions regarding vehicle distribution, supplier selection, and contract negotiations. Additionally, we use LU decomposition to establish the supplier risk threshold, crucial for comparing emission trade variances. Multimodal scenarios typically yield lower emissions, favoring buying emission allowances low and selling them high. Notably, the risk threshold affects low‐emission provider utilization, impacting transportation emissions. With a risk threshold of 0.12 and an emission price of 1.2, our ANFIS+GA‐based multimodal approach achieves a significant −20% deviation in CO2 emissions.

Inventory allocation with full downward substitution and monotone cost differences

We study a single-period multi-product inventory allocation problem with full downward substitution and monotone cost differences. The cost structure with monotone differences is more general than the additive cost structure usually assumed in literature. Using the notion of Monge sequences, we identify conditions under which the problem can be solved efficiently using greedy allocation. For problems that do not meet these conditions, we develop an efficient algorithm that solves the problem to optimality. For this specific problem, our algorithm has substantially lower computational complexity than existing efficient algorithms for the more general transportation problem; we numerically confirm this superior computational efficiency and illustrate the importance of using efficient algorithms at the allocation stage of the inventory management problem.

A MILP model and two heuristics for the Bin Packing Problem with Conflicts and Item Fragmentation

Bin Packing Problem with Conflicts and Item Fragmentation (BPPC-IF) is a variant of the classical bin packing problem in which fragments of the same item can be packed in different bins (item fragmentation), and some pairs of items cannot be packed in the same bin (item conflicts). To solve the problem, we propose a new mixed-integer linear programming (MILP) model and two heuristics. The MILP model represents BPPC-IF as a generalized transportation problem in which sources correspond to items, and sinks are defined for all maximal independent sets in the graph representing conflicts between items. The proposed heuristics, called Sequential Packing Heuristics 1 and 2 (SPH1 and SPH2, respectively) are modifications of the previously proposed Sequential Maximum Degree Packing Heuristic (SMDPH) for BPPC-IF, which fills bins by solving a series of MILP subproblems. Both new heuristics use improved criteria for selecting items to pack. Additionally, in SPH1, bins are filled heuristically without solving any MILP subproblems. Computational experiments carried out on a large set of benchmark problem instances from the literature show that (i) SPH1 obtains good quality solutions and is orders of magnitude faster than the MILP-based heuristics, (ii) compound heuristic SPH1+SPH2 finds on average better solutions than any other heuristic and is on average faster than any other MILP-based heuristic, and (iii) the proposed MILP model greatly outperforms the best previously proposed MILP model.

Development of a heuristic to solve the general transportation problem

The transportation problem is well known and has very important applications. For this well-researched model, there are very efficient approaches for solving it that are available. These approaches include formulating the transportation problem as a linear program and then using the efficient methods such as the simplex method or interior point algorithms. The Hungarian method is another efficient method for solving both the assignment model and the general transportation model. An assignment problem is a special case of the transportation model in which all supply and demand points are 1. Every transportation problem can be converted into an assignment problem since rows and columns can be split so that each supply and each demand point is 1. The transportation simplex method is another method that is also used to solve the general transportation problem. This method is also called the modified distribution method (MODI). To use this approach, a starting solution is required and the closer the starting solution to the optimal solution, the fewer the iterations that are required to reach optimality. The fourth method for transportation models is the network simplex method, which is the fastest so far. Unfortunately, all these approaches for transportation models are serial in nature and are very difficult to parallelize, which makes it difficult to efficiently use the available massively parallel technology. There is a need for an efficient approach for the transportation problem, which is easily parallelizable. This paper presents a See-Saw approach for solving the general transportation problem. This is an extension of the See-Saw approach for solving the assignment problem. The See-Saw moves can be done independently, which makes the approach proposed in this paper more promising than the available methods for transportation models

Using Genetic Algorithms to Solve Discounted Generalized Transportation Problem

Using Genetic Algorithms to Solve Discounted Generalized Transportation Problem

A new ranking technique on heptagonal fuzzy numbers to solve fuzzy transportation problem

In this work, a general fuzzy transportation problem model is discussed and proposed a new ranking procedure to the heptagonal fuzzy numbers using which convert any fuzzy valued transportation problem to a crisp valued transportation problem which then could be solved using the MODI method. The proposed method serves as an efficient method in ranking heptagonal fuzzy numbers which is illustrated through a numerical example. The purpose of this paper is to give a new ranking index to rank fuzzy numbers effectively. Fuzzy numbers cannot be easily ordered as ordinary real numbers. Several proposals have addressed this problem, each with some drawbacks and limitations. This paper renounces to the idea of finding a universal ordering method for fuzzy numbers.

General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary

A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form.

Variance Reduction and Optimization Strategies in a Biased Woodcock Particle Tracking Framework

A novel particle tracking framework is introduced in this paper that utilizes null-collisions to sample distance to collision in Monte Carlo particle transport problems. The sampling process is described in the most general form as it covers all of the main developments concerning the Woodcock method (delta tracking). We show that none of the previously suggested modifications are optimal in terms of either variance or efficiency. Variance analysis is provided for a general transport problem along with the estimation of computational cost. Simplified models with analytic solutions are further investigated and propositions for optimal settings are discussed based on the derived equations. A well-known variance reduction technique, exponential transform, is found to be a limiting case of the biased Woodcock tracking method and comparison shows the proposed framework may outperform the exponential transform in real-case scenarios.

Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces

We present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from $\binom{N+\ell-1}{\ell-1}$ to $\ell\cdot(N+1)$, where $\ell$ is the number of marginal states and $N$ the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to $\ell\cdot(N-1)$ unknowns, and cures the insufficiency of the Monge ansatz; i.e., we show that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg--Kohn density functional theory. In this context $N$ corresponds to the number of particles, motivating the interest in large $N$.

Error estimate for the upwind scheme for the linear transport equation with boundary data

We study the upwind finite volume scheme on a general mesh for the initial and boundary-value problem associated with a linear transport equation. For any BV initial and boundary data we prove the (optimal) convergence rate 1/2 in the L 1-norm. Compared to previous works, our main contribution is to take into account the boundary data and to relax some regularity assumptions on the velocity field. As an intermediate result, we also provide a complete proof of the BV regularity of weak solutions of such a general transport problem.

Preconditioning of Optimal Transport

A preconditioning procedure is developed for the $L_2$ and more general optimal transport problems. The procedure is based on a family of affine map pairs which transforms the original measures into two new measures that are closer to each other while preserving the optimality of solutions. It is proved that the preconditioning procedure minimizes the remaining transportation cost among all admissible affine maps. The procedure can be used on both continuous measures and finite sample sets from distributions. In numerical examples, the procedure is applied to multivariate normal distributions, to a two-dimensional shape transform problem, and to color-transfer problems.

Bi-criteria stochastic generalized transportation roblem: expected cost and risk minimization

In the paper we consider a bi-criteria version of the Stochastic Generalized Transportation Problem, where one goal is minimization of the expected total cost, and the second one is minimization of the risk. We present the model and a solution method for this problem. Keywords Stochastic Generalized Transportation Problem, Bi-criteria Stochastic Gener- alized Transportation Problem, expected cost, variance of the cost, Equaliza- tion Method, branch and bound.

대규모 불균형 수송문제의 간단한 해법

As the Transportation Simplex Method of the general transportation problem are inapplicable to the large-scale unbalanced transportation problem, a commercialized linear programming package remains as the only viable means. There is, however, no method made available to verify the optimality of solutions attained by the package. This paper therefore proposes a simple heuristic algorithm to the large-scale unbalanced transportation problem. From a given problem of supply and demand areas, the proposed algorithm determines the number of demands areas for each supply area and executes on the latter in the ascending order of each of their corresponding demand areas. Next, given a single corresponding demand area, it supplies the full demand volume and else, it supplies first to an area of minimum associated costs and subsequently to the rest so as to meet the demand to the fullest extent. This initial optimal value is then optimized through an adjustment process whereby costs are minimized as much as possible. When tested on the cost matrix, the proposed algorithm has obtained an optimal result improved from the commercial linear programming package by 8.9%.

The Nonlinear Generalized Transportation Problem with convex costs

This paper will investigate the Nonlinear Generalized Transportation Problem (NGTP), where the transportation costs and the costs that depend on the amount of good delivered to the destination points are strictly convex functions. The amounts of goods change during the transportation process. This model presents a congested network and associated time costs. First, we present the NGTIP model, then provide a method of solving the specific problems and thus prove convergence of the model. The numerical experiments prove the effectiveness of the presented algorithm.

Freight car dispatching with generalized flows

In the freight car dispatching problem, empty freight cars have to be assigned to known demands respecting a given time horizon and certain constraints. The goal is to minimize the resulting transportation costs. One of the constraints is that customers can specify the type of cars they want. It is possible, however, that cars of certain types can be substituted by other cars, either in a 1‐to‐1 fashion or at different exchange rates. We show that these substitutions make the dispatching problem hard to solve and hard to approximate. We model the dispatching problem as an integral generalized transportation problem on a bipartite graph. Using rounding techniques, the LP‐relaxation can be transformed to a transportation schedule violating some of the constraints slightly. Under an additional assumption on the cost function, we fix this violation and derive a 4‐approximation of the problem. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(1), 33–39 2015

Modelling of transient heat conduction with diffuse interface methods

We present a survey on different numerical interpolation schemes used for two-phase transient heat conduction problems in the context of interface capturing phase-field methods. Examples are general transport problems in the context of diffuse interface methods with a non-equal heat conductivity in normal and tangential directions to the interface. We extend the tonsorial approach recently published by Nicoli M et al (2011 Phys. Rev. E 84 1–6) to the general three-dimensional (3D) transient evolution equations. Validations for one-dimensional, two-dimensional and 3D transient test cases are provided, and the results are in good agreement with analytical and numerical reference solutions.

An amoeboid algorithm for solving linear transportation problem

Transportation Problem (TP) is one of the basic operational research problems, which plays an important role in many practical applications. In this paper, a bio-inspired mathematical model is proposed to handle the Linear Transportation Problem (LTP) in directed networks by modifying the original amoeba model Physarum Solver. Several examples are used to prove that the provided model can effectively solve Balanced Transportation Problem (BTP), Unbalanced Transportation Problem (UTP), especially the Generalized Transportation Problem (GTP), in a nondiscrete way.

복합상품 운송 문제의 간단한 해법

This paper proposes a heuristic optimal solution of multicommodity transportation problem. The proposed algorithm has 3 steps. First the proposed algorithm transforms multicommodity transshipment problem to a general transportation problem, but if the problem is a multicommodity transportation problem, it is not transformed. And the multicommodity is disassembled to a single commodity. Second if it is a multicommodity transportation problem, the algorithm selects the minimum cost according to commodity, on the other hand if it is a multicommodity transshipment problem, the algorithm directly selects the minimum cost based on demand area. And the algorithm assigns carloadings to be satisfied the supply and demand quantity. The algorithm repeats these processes until a given demand quantity is satisfied. Last if it has a condition that is able to reduce the transportation expense, the proposed algorithm controls the assignment quantity of the initial value that got from the step 2. The proposed algorithm was applied to two multicommodity transportation problem and three multicommodity transshipment problem and it got more good result than an existing linear programming method.

Transport-constrained extensions of collision and track length estimators for solutions of radiative transport problems

In this paper we develop novel extensions of collision and track length estimators for the complete space-angle solutions of radiative transport problems. We derive the relevant equations, prove that our new estimators are unbiased, and compare their performance with that of more conventional estimators. Such comparisons based on numerical solutions of simple one dimensional slab problems indicate the the potential superiority of the new estimators for a wide variety of more general transport problems.

Space Optimization for Warehousing Problem: A Methodology for Decision Support System

This article presents a way of tackling a special class of space optimization problems that arise in a number of practical applications in industry and elsewhere. It presents an elegant solution to a problem that was considered by (Das, 2005) in optimizing storage space in warehouse of a footwear manufacturing company. In (Das, 2005), the problem was formulated as a nonlinear programming problem. In this article, it is shown that the problem can be formulated as a generalized transportation problem which is a special case of generalized network flow problems. Further, an elegant scheme is devised to handle the dynamic situation of warehousing problem which can be easily translated into a decision support system for the warehouse management system. Also, the article points out certain obscurities and gaps in (Das, 2005).