A solid transportation problem with type-2 fuzzy variables

This paper focuses on generating the optimal solutions of the solid transportation problem under fuzzy environment, in which the supply capacities, demands and transportation capacities are supposed to be type-2 fuzzy variables due to the instinctive imprecision. In order to model the problem within the framework of the credibility optimization, three types of new defuzzification criteria, i.e., optimistic value criterion, pessimistic value criterion and expected value criterion, are proposed for type-2 fuzzy variables. Then, the multi-fold fuzzy solid transportation problem is reformulated as the chance-constrained programming model with the least expected transportation cost. To solve the model, fuzzy simulation based tabu search algorithm is designed to seek approximate optimal solutions. Numerical experiments are implemented to illustrate the application and effectiveness of the proposed approaches.     

A solid transportation problem in uncertain environment involving type-2 fuzzy variable

Neural Computing and Applications - The main focus of this paper is to develop a new safety-based restricted fixed charge solid transportation problem with type-2 fuzzy parameter that minimizes...     

Neural networks for solving multicriteria solid transportation problem

In this paper, we present neural networks for solving multicriteria solid transportation problems. The original problem is transformed into an equivalent continuous problem from the continuous-time dynamic system and its optimal solution can be got. The procedure and efficiency of this approach are shown with numerical simulations.     

Iterative functional modification method for solving a transportation problem

We propose a new method for solving transportation problems based on decomposing the original problem into a number of two-dimensional optimization problems. Since the solution procedure is integer-valued and monotonic in the objective function, the required computation is finite. As a result, we get not only a single optimal solution of the original transportation problem but a system of constraints that can yield all optimal solutions. We give numerical examples that illustrate the constructions of our algorithm.     

The collapsing method of defuzzification for discretised interval type-2 fuzzy sets

This paper proposes a new approach for defuzzification of interval type-2 fuzzy sets. The collapsing method converts an interval type-2 fuzzy set into a type-1 representative embedded set (RES), whose defuzzified values closely approximates that of the type-2 set. As a type-1 set, the RES can then be defuzzified straightforwardly. The novel representative embedded set approximation (RESA), to which the method is inextricably linked, is expounded, stated and proved within this paper. It is presented in two forms: Simple RESA: this approximation deals with the most simple interval FOU, in which a vertical slice is discretised into 2 points. Interval RESA: this approximation concerns the case in which a vertical slice is discretised into 2 or more points. The collapsing method (simple RESA version) was tested for accuracy and speed, with excellent results on both criteria. The collapsing method proved more accurate than the Karnik-Mendel iterative procedure (KMIP) for an asymmetric test set. For both a symmetric and an asymmetric test set, the collapsing method outperformed the KMIP in relation to speed.     

A breakable multi-item multi stage solid transportation problem under budget with Gaussian type-2 fuzzy parameters

This paper is based on two mathematical models for multi-item multi-stage solid transportation problem with budget on total transportation cost in Gaussian type-2 fuzzy environment considering the fixed opening charge and operating cost in distribution center. The first model is about transportation of breakable/damageable items, and the second one considers non breakable/damageable items. The main aspect here is to develop the mathematical formulation of multi stage related solid transportation problem where several items are available for transportation. In order to deal with the Gaussian type-2 fuzziness, two chance-constrained programming models are developed based on generalized credibility measures for the objective function as well as the constraints sets with the help of the CV-based reductions method. Finally the reduced model is turned into its equivalent parametric programming problem. The problem is of high complexity and is difficult to find the optimal solution by any classical method and hence a time and space based meta-heuristic Genetic Algorithm has been proposed. Also the equivalent crisp models are solved using GA and LINGO 13.0 and after comparison, GA results are better. The proposed models and techniques are finally illustrated by providing numerical examples. Some sensitivity analysis and particular cases are presented and discussed. Degrees of efficiency is also evaluated for both the techniques.     

A multi-item transportation problem with mode of transportation preference by MCDM method in interval type-2 fuzzy environment

Neural Computing and Applications - In this paper, we employ all the parameters as trapezoidal interval type-2 fuzzy numbers to cope with ambiguity and vagueness problem. There are two issues being...     

Improved genetic algorithm for solving multiobjective solid transportation problem with fuzzy numbers

In this paper, we present improved genetic algorithm for solving the fuzzy multiobjective solid transportation problem in which the coefficients of objective function are represented as fuzzy numbers. The ranking fuzzy numbers with integral value are used in the evaluation and selection. The proposed algorithm is incorporated with problem-specific knowledge and conductive to find out the set of nondominated points in the criteria space based on decision maker degree of optimism.     

New method for solving reviewer assignment problem using type-2 fuzzy sets and fuzzy functions

Reviewer Assignment Problem (RAP) is one of the cardinal problems in Government Funding agencies where the expertise level of the referee reviewing a proposal needs to be optimised to guarantee the selection of good R&D projects. Although many solutions have been proposed for RAP in the past, none of them deals with the inherent imprecision associated with the problem. For instance, it is not possible to determine the “exact expertise level” of a particular reviewer in a particular domain. In this paper, we propose a novel approach for assigning reviewers to proposals. To calculate the expertise of a reviewer in a particular domain, we create a type-2 fuzzy set by assigning relevant weights to the various factors that affect the expertise of the reviewer in that domain. We also create a fuzzy set of the proposal by selecting three keywords that best represent the proposal. We then use a fuzzy functions based equality operator to compute the equality of the type-2 fuzzy set of experts and the fuzzy set of proposal keywords, which is then subjected to a set of relevant constraints to optimize the solution. We consider the four important aspects: workload balancing of reviewers, avoiding Conflicts of Interest, considering individual preferences by incorporating bidding and mapping multiple keywords of a proposal. As an extension to this approach, we further consider the relative importance of each keyword with respect to the submitted proposal by using representative percentage weights to create the FUZZY sets which represent the keywords. Hence, we propose an integrated solution based on the strong mathematical foundation of fuzzy logic, comprised of all the different aspects of expertise modeling and reviewer assignment. An Expert System has also been developed for the same.     

A method for solving a problem of possibilistic programming

The models of possibilistic programming considered in [1–3] are generalized to the case when the binary relations that relate the left and right sides in the model of constraints are fuzzy. The corresponding mathematical formalism for describing models of this class is developed. A method for solving the problem is proposed. The capabilities of the approach are demonstrated in numerical examples.     

Algorithms for solving the single-sink fixed-charge transportation problem

The single-sink fixed-charge transportation problem is an important subproblem of the fixed-charge transportation problem. Just a few methods have been proposed in the literature to solve this problem. In this paper, solution approaches based on dynamic programming and implicit enumeration are revisited. It is shown how the problem size as well as the search space of a recently published dynamic programming method can be reduced by exploiting reduced cost information. Additionally, a further implicit enumeration approach relying on solution concepts for the binary knapsack problem is introduced. The performance of the various solution methods is compared in a series of computational experiments.     

A Quantum-inspired Ant Colony Optimization for solving a sustainable four-dimensional traveling salesman problem under type-2 fuzzy variable

In this paper, a Quantum-inspired Ant Colony Optimization (Qi-ACO) is proposed to solve a sustainable four-dimensional traveling salesman problem (4DTSP). In 4DTSP, various paths with a different number of conveyances are available to travel between any two cities. In this model, we have considered a sustainable 4DTSP in terms of emission as a constraint. Since travel costs and emissions are uncertain/imprecise in nature, so here we consider type-2 variables. Sustainable development in the traveling salesman problem (TSP) sector can be divided into two major sections: economy and environmental. Sustainable TSP development requires balancing to achieve the maximum benefits for these two sectors. For increasing development in sustainable transportation, we need to use some strategies for increasing sustainability. These strategies include improving route and vehicle selection, routing plan, vehicle speed, etc. The novelties of the proposed Qi-ACO algorithm are (i) Qubit generated based on the amount of emission of the vehicle as well as travel cost between two cities, (ii) pheromone initialized and updated depends on the qubit, (iii) quantum-inspired technique makes fast computation. The proposed sustainable 4DTSP is illustrated with some numerical data. The defuzzification of type-2 fuzzy variable based on the Critical value (CV) method is used in this model. The supremacy of the proposed method is established through some statistical tests. The proposed algorithm and its modified form can be easily adapted in ship routing, supply chain problems, and other fields.     

A method for computing heuristics in problem solving

The purpose of this paper is to propose a formal framework for structuring and embedding the heuristic information, in order to allow an algorithmic computation, in quite general cases, of the evaluation function $\text{f}\text{?}\text{(n)}$ of the classical Hart-Nilsson-Raphael algorithm. The notion of semantic graph is first introduced, in which the atomic notion of node is expanded by associating to it an internal structure where the heuristic information is inserted. It is proved that $\text{h}\text{?}\text{(n)}$ can be computed by solving an auxiliary problem, obtained from the original one by adding new arcs, and of smaller complexity than that onw. A new algorithm is then defined for the computation of $\text{h}\text{?}\text{(n)}$ and for the determination of minimal solutions. The validity of the model proposed is discussed in detail.     

A genetic algorithm for solving the fixed-charge transportation model: Two-stage problem

Transportation of goods in a supply chain from plants to customers through distribution centers (DCs) is modeled as a two-stage distribution problem in the literature. In this paper we propose genetic algorithms to solve a two-stage transportation problem with two different scenarios. The first scenario considers the per-unit transportation cost and the fixed cost associated with a route, coupled with unlimited capacity at every DC. The second scenario considers the opening cost of a distribution center, per-unit transportation cost from a given plant to a given DC and the per-unit transportation cost from the DC to a customer. Subsequently, an attempt is made to represent the two-stage fixed-charge transportation problem (Scenario-1) as a single-stage fixed-charge transportation problem and solve the resulting problem using our genetic algorithm. Many benchmark problem instances are solved using the proposed genetic algorithms and performances of these algorithms are compared with the respective best existing algorithms for the two scenarios. The results from computational experiments show that the proposed algorithms yield better solutions than the respective best existing algorithms for the two scenarios under consideration.     

Neural network approach for multicriteria solid transportation problem

In this paper, we propose neural network approach for multicriteria solid transportation problems(STP). First we suggest a neural network architecture to solve single-objective STP according to augmented Lagrange multiplier method. Due to the massive computing unit-neurons and parallel mechanism of neural network approach can solve the large scale problem efficiently and optimal solution can be got. Then we transform the original multicriteria problem into a single objective problem by global criteria method and adopt the neural network approach to solve it. By this way we can get the satisfactory solution of the original problem. The procedure and efficiency of this approach are shown with numerical simulations.     

A fuzzy MCDM method and an application to solid transportation problem with mode preference

In this paper we propose a computationally efficient fuzzy multi-criteria decision making (FMCDM) method. For this purpose we define a ranking function based on credibility measure to rank a fuzzy number over another fuzzy number. A comparative result of our proposed ranking method with the other well known methods is provided. The proposed FMCDM method is successfully applied to find most preferred transportation mode among available modes with respect to some evaluation criteria for a solid transportation problem (STP). Here the evaluation ratings of the alternatives and criteria weights are presented in terms of linguistic variables. The importance weights of the available transportation modes as obtained by this method are then assigned to the STP. Numerical example is provided to illustrate the proposed method and problem.     

用于求解露天矿运输问题的改进差分进化算法

针对露天矿运输问题,以露天矿开采能力和运输能力为约束条件,以运输费用最小为目标函数,建立了露天矿运输问题的数学模型。针对智能优化算法用于求解露天矿运输问题时容易陷入局部最优解的问题,提出了一种改进差分进化算法。该算法通过在差分进化算法中引入归一化操作,使得运输问题中的等式约束能自动成立,有利于跳出局部最优解。应用结果表明,该算法具有较好的可重复性,利用该算法对露天矿运输问题进行优化后,运输成本明显降低。     

A near-exact method for solving the log-truck scheduling problem

We propose a solution method for the log‐truck scheduling problem, which is a generalisation of the pick‐up and delivery problem with time windows. Our approach is based on column generation and pseudo branch and price. Each column in the proposed mathematical model represents one feasible route for one truck. We start by designing a priori an initial set of routes. Then, the subproblem, which is a constrained shortest path problem, is solved by applying a k‐shortest path algorithm. Numerical results from a case study are presented.     

线性–非二次最优控制问题的一种解法

讨论了求解状态终端无约束线性–非二次最优控制问题的拟Riccati方程方法, 并据此提出了计算无约束线性–非二次问题之数值解的方法; 然后将这个方法与一种能近似地化有约束问题为无约束问题的惩罚方法结合起来, 给出了一种算法, 可以计算状态终端有约束的线性–非二次最优控制问题之近似解.     

A method for solving the n-dimensional convex hull problem

A method is presented for finding all vertices and all hyperplanes containing the faces of a convex polyhedron spanned by a given finite set X in Euclidean space Eⁿ. The present paper indicates how this method can be applied to the investigation of linear separability of two given finite sets X₁ and X₂ in Eⁿ. In the case of linear separability of these sets the proposed method makes it possible to find the separating hyperplane.