A solid transportation problem with mixed constraint in different environment

In this paper, we have introduced a Solid Transportation Problem where the constrains are mixed type. The model is developed under different environment like, crisp, fuzzy and intuitionistic fuzzy etc. Using the interval approximation method we defuzzify the fuzzy amount and for intuitionistic fuzzy set we use the ($\alpha,\beta$)-cut sets to get the corresponding crisp amount. To find the optimal transportation units a time and space based with order of convergence $O (MN^2)$ meta-heuristic Genetic Algorithm have been proposed. Also the equivalent crisp model so obtained are solved by using LINGO 13.0. The results obtained using GA treats as the best solution by comparing with LINGO results for this present study. The proposed models and techniques are finally illustrated by providing numerical examples. Degree of efficiency have been find out for both the algorithm.     

Solution of a possibilistic multiobjective linear programming problem

The estimate of the parameters which define a conventional multiobjective decision making model is a difficult task. Normally they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from the past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1, 9, 10, 11. The fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy, see 2, 3, and which is defined by its possibility distribution. Once the possibility distribution of the solution has been obtained, if the decision maker wants more precise information with respect to the decision vector, then we can pose and solve a new problem. In this case we try to find a decision vector, which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained. In order to solve this problem we shall develop two different models from the initial solution and based on Goal Programming: an Interval Goal Programming Problem if we define the relation “as accurate as possible” based on the expected intervals of fuzzy numbers, as we showed in [4], and an ordinary Goal Programming based on the expected values of the fuzzy numbers that defined the goals. Finally, we construct algorithms that implement the above mentioned solution method. Our approach will be illustrated by means of a numerical example.     

Separable programming/duality approach to solving the multi-product Newsboy/Gardener Problem with linear constraints

The Gardener Problem is an extension of the multi-product Newsboy Problem with constraints. It deals with situations where not only the demand is random but also the yield (the supply). Separable programming and duality approaches are utilized to solve the constrained Newsboy/Gardener Problem. The solution methodologies are developed for the common probability distribution functions for the demand, and uniform distribution for the supply, rendering exact and approximate solutions to the problem. Numerical examples are given and when applicable, the performance of the developed approach is compared to those of existing works in this arena. The results reveal that the developed solution methods efficiently converge to the optimal or near optimum solutions. Also, a salient feature of the proposed methodologies is that they can utilize readily available commercial software to solve the considered problems. This feature facilitates the portability of the developed models to the classroom environment.     

Fuzzy goal programming for multiobjective transportation problems

Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach.     

A new approach for solving fully intuitionistic fuzzy transportation problems

In this paper, a well-known network-structured problem called the transportation problem (TP) is considered in an uncertain environment. The transportation costs, supply and demand are represented by trapezoidal intuitionistic fuzzy numbers (TrIFNs) which are the more generalized form of trapezoidal fuzzy numbers involving a degree of acceptance and a degree of rejection. We formulate the intuitionistic fuzzy TP (IFTP) and propose a solution approach to solve the problem. The IFTP is converted into a deterministic linear programming (LP) problem, which is solved using standard LP algorithms. The main contributions of this paper are fivefold: (1) we convert the formulated IFTP into a deterministic classical LP problem based on ordering of TrIFNs using accuracy function; (2) in contrast to most existing approaches, which provide a crisp solution, we propose a new approach that provides an intuitionistic fuzzy optimal solution; (3) in contrast to existing methods that include negative parts in the obtained intuitionistic fuzzy optimal solution and intuitionistic fuzzy optimal cost, we propose a new method that provides non-negative intuitionistic fuzzy optimal solution and optimal cost; (4) we discuss about the advantages of the proposed method over the existing methods for solving IFTPs; (5) we demonstrate the feasibility and richness of the obtained solutions in the context of two application examples.     

α-Coverage to extend network lifetime on wireless sensor networks

An important problem in the context of wireless sensor networks is the Maximum Network Lifetime Problem (MLP): find a collection of subset of sensors (cover) each covering the whole set of targets and assign them an activation time so that network lifetime is maximized. In this paper we consider a variant of MLP, where we allow each cover to neglect a certain fraction (1 ? α) of the targets. We analyze the problem and show that the total network lifetime can be hugely improved by neglecting a very small portion of the targets. An exact approach, based on a Column Generation scheme, is presented and a heuristic solution algorithm is also provided to initialize the approach. The proposed approaches are tested on a wide set of instances. The experimentation shows the effectiveness of both the proposed problems and solution algorithms in extending network lifetime and improving target coverage time when some regularity conditions are taken into account.     

A novel GRASP solution approach for the Orienteering Problem

The Orienteering Problem (OP) is a well-known variant of the Traveling Salesman Problem. In this paper, a novel Greedy Randomized Adaptive Search Procedure (GRASP) solution is proposed to solve the OP. The proposed method is shown to outperform state-of-the-art heuristics for the OP in producing high quality solutions. In comparison with the best known solutions of standard benchmark instances, the method can find the optimal or the best known solution of about 70 % of the instances in a reasonable time, which is about 17 % better than the best known approach in the literature. Moreover, a significant improvement is achieved on the solution of two standard benchmark instances.     

Reliable design of a logistics network under uncertainty: A fuzzy possibilistic-queuing model

This paper proposes a bi-objective model for designing a reliable network of bi-directional facilities in logistics network under uncertainties. For this purpose, the model utilizes an effective reliability approach to find a robust logistics network design. The objectives of the model are to minimize the total costs and the expected transportation costs after failures of bi-directional facilities of the logistics network. To solve the model, a new solution approach is proposed by combining queuing theory, fuzzy possibilistic programming and fuzzy multi-objective programming. Finally, the computational experiments are provided to illustrate the effectiveness of the proposed model and solution approach.     

Environmental supply chain network design using multi-objective fuzzy mathematical programming

The concern about environmental impact of business activities has spurred an interest in designing environmentally conscious supply chains. This paper proposes a multi-objective fuzzy mathematical programming model for designing an environmental supply chain under inherent uncertainty of input data in such problem. The proposed model is able to consider the minimization of multiple environmental impacts beside the traditional cost minimization objective to make a fair balance between them. A life cycle assessment-based (LCA-based) method is applied to assess and quantify the environmental impact of different options for supply chain network configuration. Also, to solve the proposed multi-objective fuzzy optimization model, an interactive fuzzy solution approach is developed. A real industrial case is used to demonstrate the significance and applicability of the developed fuzzy optimization model as well as the usefulness of the proposed solution approach.     

Uncertain solid transportation problems

The solid transportation problem arises when bounds are given on three item properties. Usually, these properties are source, destination and type of product or mode of transport, and often are given in a uncertain way. This paper deals with two of the ways in which uncertainty can appear in the problem: Interval solid transportation problem and fuzzy solid transportation problem. The first arises when data problem are expressed as intervals instead of point values, and the second when the nature of the information is vague. Both models are treated in the case in which the uncertainty affects only the constraint set. For interval case, an auxiliary problem is obtained in order to find a solution. This auxiliary problem is a standard solid transportation problem which can be solved with the efficient methods existing. For fuzzy case, a parametric approach which makes it possible to find a fuzzy solution to the former problem is used.     

Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

Multiobjective expected value model for portfolio selection in fuzzy environment

In this paper, we propose a credibilistic framework for portfolio selection problem using an expected value multiobjective model with fuzzy parameters. We consider short term return, long term return, risk and liquidity as key financial criteria. A solution procedure comprising fuzzy goal programming and fuzzy simulation based real-coded genetic algorithm is developed to solve the model. The proposed solution approach is considered advantageous particularly for the cases where the fuzzy parameters of the problem may assume any general functional form. An empirical study is included to illustrate the usefulness of the proposed model and solution approach in real-world applications of portfolio selection.     

Optimal compromise solution of multi-objective minimal cost flow problems in fuzzy environment

Ghatee and Hashemi [M. Ghatee, S.M. Hashemi, Ranking function-based solutions of fully fuzzified minimal cost flow problem, Inform. Sci. 177 (2007) 4271–4294] transformed the fuzzy linear programming formulation of fully fuzzy minimal cost flow (FFMCF) problems into crisp linear programming formulation and used it to find the fuzzy optimal solution of balanced FFMCF problems. In this paper, it is pointed out that the method for transforming the fuzzy linear programming formulation into crisp linear programming formulation, used by Ghatee and Hashemi, is not appropriate and a new method is proposed to find the fuzzy optimal solution of multi-objective FFMCF problems. The proposed method can also be used to find the fuzzy optimal solution of single-objective FFMCF problems. To show the application of proposed method in real life problems an existing real life FFMCF problem is solved.     

Two storage inventory model in a mixed environment

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters—holding costs, ordering costs, purchase costs, etc.—are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it’s Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.     

A Metaheuristic to Solve a Location-Routing Problem with Non-Linear Costs

The paper deals with a location-routing problem with non-linear cost functions. To the best of our knowledge, a mixed integer linear programming formulation for the addressed problem is proposed here for the first time. Since the problem is NP-hard exact algorithms are able to solve only particular cases, thus to solve more general versions heuristics are needed. The algorithm proposed in this paper is a combination of a p-median approach to find an initial feasible solution and a metaheuristic to improve the solution. It is a hybrid metaheuristic merging Variable Neighborhood Search (VNS) and Tabu Search (TS) principles and exploiting the synergies between the two. Computational results and conclusions close the paper.     

Ant colony optimization and local search for bin packing and cutting stock problems

The Bin Packing Problem and the Cutting Stock Problem are two related classes of NP-hard combinatorial optimization problems. Exact solution methods can only be used for very small instances, so for real-world problems, we have to rely on heuristic methods. In recent years, researchers have started to apply evolutionary approaches to these problems, including Genetic Algorithms and Evolutionary Programming. In the work presented here, we used an ant colony optimization (ACO) approach to solve both Bin Packing and Cutting Stock Problems. We present a pure ACO approach, as well as an ACO approach augmented with a simple but very effective local search algorithm. It is shown that the pure ACO approach can compete with existing evolutionary methods, whereas the hybrid approach can outperform the best-known hybrid evolutionary solution methods for certain problem classes. The hybrid ACO approach is also shown to require different parameter values from the pure ACO approach and to give a more robust performance across different problems with a single set of parameter values. The local search algorithm is also run with random restarts and shown to perform significantly worse than when combined with ACO.     

Two-stage fuzzy production planning expected value model and its approximation method

This work develops a novel two-stage fuzzy optimization method for solving the multi-product multi-period (MPMP) production planning problem, in which the market demands and some of the inventory costs are assumed to be uncertainty and characterized by fuzzy variables with known possibility distributions. Some basic properties about the MPMP production planning problem are discussed. Since the fuzzy market demands and inventory costs usually have infinite supports, the proposed two-stage fuzzy MPMP production planning problem is an infinite-dimensional optimization problem that cannot be solved directly by conventional numerical solution methods. To overcome this difficulty, this paper adopts an approximation method (AM) to turn the original two-stage fuzzy MPMP production planning problem into a finite-dimensional optimization problem. The convergence about the AM is discussed to ensure the solution quality. After that, we design a heuristic algorithm, which combines the AM and simulated annealing (SA) algorithm, to solve the proposed two-stage fuzzy MPMP production planning problem. Finally, one real case study about a furniture manufacturing company is presented to illustrate the effectiveness and feasibility of the proposed modeling idea and designed algorithm.     

Scalarization of the fuzzy optimization problems

Scalarization of the fuzzy optimization problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Two solution concepts are proposed by considering two convex cones. The set of all fuzzy numbers can be embedded into a normed space. This motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to solve the fuzzy optimization problems. By applying scalarization to the optimization problem with fuzzy coefficients, we obtain its corresponding scalar optimization problem. Finally, we show that the optimal solution of its corresponding scalar optimization problem is the optimal solution of the original fuzzy optimization problem.     

Multi-armed bandit-based hyper-heuristics for combinatorial optimization problems

There are significant research opportunities in the integration of Machine Learning (ML) methods and Combinatorial Optimization Problems (COPs). In this work, we focus on metaheuristics to solve COPs that have an important learning component. These algorithms must explore a solution space and learn from the information they obtain in order to find high-quality solutions. Among the metaheuristics, we study Hyper-Heuristics (HHs), algorithms that, given a number of low-level heuristics, iteratively select and apply heuristics to a solution. The HH we consider has a Markov model to produce sequences of low-level heuristics, which we combine with a Multi-Armed Bandit Problem (MAB)-based method to learn its parameters. This work proposes several improvements to the HH metaheuristic that yields a better learning for solving problem instances. Specifically, this is the first work in HHs to present Exponential Weights for Exploration and Exploitation (EXP3) as a learning method, an algorithm that is able to deal with adversarial settings. We also present a case study for the Vehicle Routing Problem with Time Windows (VRPTW), for which we include a list of low-level heuristics that have been proposed in the literature. We show that our algorithms can handle a large and diverse list of heuristics, illustrating that they can be easily configured to solve COPs of different nature. The computational results indicate that our algorithms are competitive methods for the VRPTW (2.16% gap on average with respect to the best known solutions), demonstrating the potential of these algorithms to solve COPs. Finally, we show how algorithms can even detect low-level heuristics that do not contribute to finding better solutions to the problem.     

A decomposition heuristics for the container ship stowage problem

In this paper we face the problem of stowing a containership, referred to as the Master Bay Plan Problem (MBPP); this problem is difficult to solve due to its combinatorial nature and the constraints related to both the ship and the containers. We present a decomposition approach that allows us to assign a priori the bays of a containership to the set of containers to be loaded according to their final destination, such that different portions of the ship are independently considered for the stowage. Then, we find the optimal solution of each subset of bays by using a 0/1 Linear Programming model. Finally, we check the global ship stability of the overall stowage plan and look for its feasibility by using an exchange algorithm which is based on local search techniques. The validation of the proposed approach is performed with some real life test cases. This work has been developed within the research area: “The harbour as a logistic node” of the Italian Centre of Excellence on Integrated Logistics (CIELI) of the University of Genoa, Italy