Optimal Assignment

In Linear Programming, the transportation problem is a special class of model. It deals with the situation in which a commodity from several sources is shipped to different destinations with the main objective to minimize the total shipping cost. In this paper, we present an alternative method to North West Corner method by using Statistical tool called Coefficient of Range. Numerical examples are explained to justify the results.

This paper presents a platform to create and manage virtual computing laboratories using Cloud resources. Using this platform a professor can create a customized laboratory according to the class needs. The laboratory is composed of a set of virtual machines that students may use to get access to the necessary computing resources to attend the class. The platform aims at the creation of a solution to avoid proprietary lock in's, and it was designed to be agnostic to the cloud infrastructure. The machines of the lab can be accessed using some remote desktop protocol and managed by non-experts users.

This paper proposes an algorithm for solving multi-objective assignment problem (MOAP) through interactive fuzzy goal programming approach. A mathematical model has been established to discuss about multi-objective assignment problem which is characterized by non-linear(exponential) membership function. The approach emphasizes on optimal solution of each objective function by minimizing the worst upper bound, which is close to the best lower bound. To illustrate the algorithm a numerical example is presented.

The solid transportation problem (STP) is a particular type of linear programming problem. This paper presented an approach for solving STP in a highly efficient and a few iterations until finding an optimal solution. The proposed method provides an initial solution near to optimum solution to the solid transportation problem close to the optimal solution. Using Lingo software, a numerical example was used to compare the solution from the proposed method to the optimal solution.

T he fractional programming is a generalization of linear programming where the objective function is a ratio of two linear functions. Similarly, in fractional transportation problem the objective is to optimize the ratio of two cost functions or damage functions or demand functions. As the ratio of two functions is considered, the fractional programming models become more suitable for real life problems. Keeping in view the complexities associated with real life transportation problem like vagueness and uncertainty involved with the parameters. The implementation of fuzzy techniques can be very useful. Therefore, in this article a Fully Fuzzy Multi-objective Fractional Transportation Problem (FFMOFTP) is considered. All the coefficients of the parameters, demands and supplies are considered as fuzzy numbers. The purpose of using fuzzy numbers is to deal with the uncertainties and vagueness associated with the parameters. Two cases are considered, one with triangular and other with ...

In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem whereas all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP. On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore,the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN.The main advantage of this...

In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

In today‟s daily life situations TP, we frequently face the situation of unreliability in addition to unwillingness due to various unmanageable components. To handle with unreliability and unwillingness multiple researchers have recommended the intuitionistic fuzzy (IF) delineation for material. So, here, we contemplate a fuzzy TP of type – 1 IFN‟s i.e., availability and demand are TIFN‟s and costs are real numbers. We initiate transportation method to find optimum solution of a IFTP of type-1 make use of ranking function. A relevant numerical example is also included.

In Linear Programming, the transportation problem is a special class of model. It deals with the situation in which a commodity from several sources is shipped to different destinations with the main objective to minimize the total shipping cost. In this paper, we present an alternative method to North West Corner method by using Statistical tool called Coefficient of Range. Numerical examples are explained to justify the results.

This article introduces a transportation model with two objective functions. The first objective function is the function which minimizes the total cost and the second objective function is the function which minimizes the total time. Parameters of source, destination and maximum capacity of transportation means are assumed to follow the Pareto distribution. The multiobjective transportation model is the development of a single objective transportation model. Parameters of the objective function are expressed by triangular fuzzy numbers. Fuzzy multi objective problems are transformed into a deterministic single objective using the convex combination method. The formulated model is applied to the problem of shipping metal crates. There are 3 types of conveyances namely HDL, Engkel and Wingbox. Obtained the optimal total cost of Rp. 3,770,294 and metal crates delivery time to be for 13 hours.

Problems with qualitative, quantitative and uncertain information can be modelled better using trapezoidal intutionistic fuzzy numbers (TrIFNs) than fuzzy numbers. Due to the partial ordering of TrIFNs, many ranking methods are available in the literature for comparing fuzzy and intuitionistic fuzzy numbers (Li in

In Linear Programming, the transportation problem is a special class of model. It deals with the situation in which a commodity from several sources is shipped to different destinations with the main objective to minimize the total shipping cost. In this paper, we present an alternative method to North West Corner method by using Statistical tool called Coefficient of Range. Numerical examples are explained to justify the results.

Nowadays, resource allocation for virtual networks (VNs) is brought as an imperative problem. For the characteristics of virtual networks, multiple virtual networks with different topo can co-exist on a shared infrastructure. A difficult point of problem is how to use the resource effectively and to satisfy the requirement of virtual network request. This problem is NP-hard. In this paper, we introduce a heuristic algorithm to solve this problem. To simulate a virtual network mapping problem in real world, we using two input data: an infrastructure network which is modeled by a connected graph and a set of virtual network request graphs with each graph contains their constraints, time and duration. The main purposes are to maximize the revenue and to minimize the cost when allocate the virtual networks to a substrate network. The experimental results are reported to show the efficiency of propose algorithm comparing to the Enhanced Greedy Node Mapping (EGNM) algorithm.

Gercek problemlerde, miktarlanrun kesin olarak bilinmedigi tasima problemleri ile sik sik karsilasihr. Mevcut  stok ve talep miktarlan bazt kontrol edilemeyen etmenlerden dolayi belirsiz olabilir.  Bu cahsrnada, birim tasima maliyetleri ile stok ve talep miktarlan bularuk sayilar oldugunda, bularuk sayilann  iiyelik fonksiyonlanm kullanarak bularuk tasirna problemini cozen bir algoritma sunduk. Onerilen bu cozurn  algoritmasinda, tasima probleminin optimal uygun cozumleri elde edildi. Onerilen cozumtln etkinligini  gostermek icin, sayisal ornek verildi. Verilen ornek WINQSB[16] optimizasyon prograrru ile cozuldu.

A lot of studies in different fields are researching, creating and developing principles, methods and tools to find the solutions which are possible for all tasks and problems that occur in management, technical, production and other systems. In practice, we might be facing with different situations where several ways to manage them are in disposal. The aim is to find the optimal solution, which depending on the situation can be cost effective, will realise the highest possible profit, and will use the shortest possible time to reach the main goals through optimal usage of all human, technical and financial capacities. These solutions will be the main preconditions prior to undertake the managerial decision, its implementation, continuity or in planning activities. In this paper we will try to find the optimal way to design the production systems for small enterprises, in order to find the best appropriate solution that fulfils the economic and technological criteria for material, energy and information flow, including production capacities based on the market needs and the flexibility of labour force. Our aim is to apply variety of methods for finding the initial and optimal solutions for the problem of transport, by comparing different methods, including well known degeneration methods to eliminate the obstacles that arise during the implementation of different activities for finding the optimal solution.

In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

In this paper, we investigate an optimal more-for-less solution of an intuitionistic fuzzy transportation problems with mixed constraints in a single stage. An algorithm called intuitionistic fuzzy zero point method is used to find an optimalmore-for-less solution in terms of triangular intuitionistic fuzzy numbers.This method is very simple, easy to understand and apply. An optimal more-for-less solution procedure illustrated with the new relevant numerical example.

Blending operation of different grades of ores is performed to maintain the rich part of ore as much as possible. The purpose of this paper is to find the optimum solution of blending operation of phosphate ore applying linear programming using Excel Solver Spreadsheet Software Packages. Optimization process was applied to three phosphate mines based on selling price as a control factor. The same mines were optimized in another way taking profit in the objective function of the blending process. The obtained results gave the same blending percentages with the two options. There is a big difference in the gained profit up to 18 million dollars when the selling price option was applied as a control factor. On the other hand, seven stockpiles of phosphate ore; each has quantity and average grade extracted and accumulated from different mines. The final profit calculated from the option of selling price as a control factor was 360 thousand dollars higher than the one estimated when profit control factor was used.

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In this paper, a multi-level model for the solid transportation problem having uncertain variables is presented. Multi-level programming deals with the situation where more than one decision maker is available to model decentralized planning problem in any hierarchical system. In the model presented in this paper, all the parameters involved viz. transportation costs, supply availabilities, demands and conveyance capacities are considered as uncertain parameters. As the problem is based on uncertainty theory, the idea of minimizing (or maximizing) the expectation of objective functions under the set of chance constraints is followed to formulate the problem. A crisp model equivalent to the multi-level uncertain solid transportation problem is also given. A fuzzy based solution approach for solving multi-level programming problems is discussed to solve the presented model. Further, a numerical example of a three-level uncertain solid transportation problem with two conveyance options is given in order to understand the model more clearly.

This note tries to answer issues raised in Bhardwaj and Kumar (J Optim Theory Appl 163(2): 685-696, 2014). The research summarizes that the results obtained in Khan et al. (J Optim Theory Appl 159: 536-546, 2013) are sound and correct and it fulfills all the necessary requirements of its scope and objectives.

This article presents an adaptive integral sliding mode control (SMC) design method for parameter identification and hybrid synchronization of chaotic systems connected in ring topology. To employ the adaptive integral sliding mode control, the error system is transformed into a special structure containing nominal part and some unknown terms. The unknown terms are computed adaptively. Then the error system is stabilized using integral sliding mode control. The controller of the error system is created that contains both the nominal control and the compensator control. The adapted laws and compensator controller are derived using Lyapunov stability theory. The effectiveness of the proposed technique is validated through numerical examples.

The increase in the size of the problems facing humans, their overlap, the division of labor, the multiplicity of departments, as well as the diversity of products and commodities, led to the complexity of business and the emergence of many administrative and production problems. It was necessary to search for appropriate methods to confront these problems. The science of operations research, with its diverse methods, provided the optimal solutions. It addresses many problems and helps in making scientific and thoughtful decisions to carry out the work in the best way within the available capabilities. Operations research is one of the modern applied sciences that uses the scientific method as a basis and method in research and study, and its basic essence is to build a model that helps management in making decisions related to difficult administrative problems. For example, the military field, financial aspects, industry, in construction for building bridges and huge projects to evaluate the time taken for each project and reduce this time, financial markets and stocks and forecasting economic conditions, in hospital management and controlling the process of nutrition and medicines within the available capabilities, in agriculture Agricultural marketing and many other problems that have been addressed using classical operations research methods.

Abstract: In the present paper, we introduced the concept of single valued trapezoidal neutrosophic number, which is generalization of single valued neutrosophic number. A generalization of crisp, fuzzy and intuitionistic fuzzy sets represents as neutrosophic sets, which have uncertainty, inconsistent, and incompleteness information in real world problem. De-neutrosophication is a process to convert neutrosophic number into a crisp number for practical applications. For unbalanced neutrosophic transportation problem, we also use here minimum row column method and set a comparison among crisp and neutrosophic optimal solutions. Here we use two models of transportation problems to understand the applications in neutrosophic environment.

In the current times of the predominance of COVID-19, almost all the countries are conducting inoculation drives. Given the market's inability to compute how much to manufacture, how to transport and the frequently changing demand, the cost of safely and timely transporting the vaccines from factory to syringe is currently indeterminate. In this paper, we formulate this situation using a bilevel transportation problem with neutrosophic numbers (BLTP-NN). The problem comes from a vaccine manufacturing company where the vaccine is produced and then transported to different distribution centres from where it is further transported to various health centres for the conduction of their vaccination drive. The authors have tried to perceive this situation from two perspectives by formulating two different problems. The first problem is a bilevel linear fractional transportation problem which aims at minimizing the transportation cost in proportion to per unit maximization of quantity transported. The second problem is a bilevel indefinite quadratic transportation problem which aims at minimizing the transportation cost and depreciation cost. In both problems, cost coefficients are neutrosophic numbers along with availabilities and demands in the constraint set. These formulated bilevel transportation problems in neutrosophic environment are solved using goal programming strategy to arrive at a satisfactory solution. The relevance of this work is to help the decision makers in budgeting their finances related to the transportation by strategic disbursement leading to a smooth administration of vaccination program.

Cuneiform scripts constitute an immense source of information about ancient history, dating back almost four thousand years. Documents were written by imprinting wedgeshaped impressions into wet clay tablets, and current scholarly practice typically transcribes the resulting markings by hand with ink on paper. This work develops algorithmic methods for cuneiform script, combining feature extraction for cuneiform wedges with prior work on segmentation-free word spotting using part-structured models. We adapt the inkball model used for word spotting to treat wedge features as individual parts arranged in a tree structure. The geometric relationship between query and target is measured by the energy necessary to deform the tree structure. We also introduce an optimizing method for wedge feature extraction based on optimally assigning tablet structuring elements to hypothesized wedge models. Finally, we evaluate the method on a real-world dataset, and show that it outperforms the state of the art in cuneiform character spotting.

Network virtualization is a promising technique for building the Internet of the future since it enables the introduction of new features into network elements at low cost. An open issue in virtualization is how to search for an efficient mapping of virtual network elements onto those of the existing physical network. Mapping is an NP-hard problem and existing solutions take long time to find a solution. This paper presents four new approximated algorithms based on two integer linear programming formulations that runs fast and, also, consider various real network characteristics, which is neglected by other proposals in the literature.

Network virtualization is a promising technology for the Internet of the Future. An open issue in virtualization is the management of network resources in a way that energy savings are achieved without compromising the Quality of Service (QoS) requirements of the virtual networks. The dynamic allocation and deallocation of virtual networks can lead the state of the substrate to a less than optimum energy consumption. This paper introduces two algorithms for the migration of virtual routers and/or links which aims to allocate resources so that energy consumption is minimized. The efficacy of the migration of virtual routers and/or links and its impact on energy consumption are analyzed based on results derived via simulations.

Network virtualization is a promising technique for building the Internet of the future since it enables the low cost introduction of new features into network elements. An open issue in virtualization is how to search for an efficient mapping of virtual network elements onto those of the existing physical network, also called the substrate network. Mapping is an NP-hard problem and existing solutions ignore various real network characteristics in order to solve problem in a reasonable time frame. This paper introduces two new algorithms for the solution of the mapping problem, both based on 0-1 integer programming, for the solution of the mapping problem which consider a whole new set of network parameters not taken into account by previous proposals. Simulation experiments confirm the efficiency of the proposed algorithms.

The Future Internet demands energy efficient communication to cope with the ever increasing power consumption. Virtualization techniques have proved to be effective in reducing power consumption of network devices. An open issue in virtualization for green networking is the search for an energy-efficient mapping of virtual networks onto physical networks. This paper introduces a new model for the mapping of virtual networks which aims at reducing the energy consumption. This model is based on an integer linear programming formulation and several parameters, corresponding to characteristic of real networks, are considered. Simulation results attest the efficacy of the proposal.

In this paper, we investigate an assignment problem in which cost coefficients are triangular intuitionistic fuzzy numbers. In conventional assignment problem, cost is always certain. This paper develops an approach to solve an intuitionistic fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the costs (profits) matrix of the assignment problem are triangular intuitionistic fuzzy numbers. Then its triangular shaped membership and non-membership functions are defined. A new ranking procedure which can be found in [4] and is used to compare the intuitionistic fuzzy numbers so that an Intuitionistic Fuzzy Hungarian method may be applied to solve the intuitionistic fuzzy assignment problem. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

Network virtualization is a promising technique for building the Internet of the future since it enables the low cost introduction of new features into network elements. An open issue in virtualization is how to search for an efficient mapping of virtual network elements onto those of the existing physical network, also called the substrate network. Mapping is an NP-hard problem and existing solutions ignore various real network characteristics in order to solve problem in a reasonable time frame. This paper introduces two new algorithms for the solution of the mapping problem, both based on 0-1 integer programming, for the solution of the mapping problem which consider a whole new set of network parameters not taken into account by previous proposals. Simulation experiments confirm the efficiency of the proposed algorithms.

In this article, the crisp, fuzzy and intuitionistic fuzzy optimization problem is formulated. The basic definitions and notations related to optimization problems are given in the preliminaries section. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set is presented in this article. Then, with the help of the proposed algorithm the optimal solution of the crisp, fuzzy and intuitionistic fuzzy optimization problems are determined. A new theorem related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems is proposed and proved. Some new and concrete results related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems are presented. To illustrate the proposed method, some real-life numerical examples are presented. The proposed article provides seven fully worked examples with screenshots of output summaries from the software used in the computations for better understanding. The advantages of the proposed approach as compared to other existing work are also specified. Detail analyses of the comparative study as well the discussion are given. To show the advantages of the proposed approach, superiority analysis is discussed. Comparison analysis and the advantages of the proposed operators are also discussed. Some managerial applications and the advantages of the proposed approach are given. Finally, conclusion and future research directions are also given. Keywords Fuzzy set Á Fuzzy number Á TFN Á TrFN Á Intuitionistic fuzzy set Á Intuitionistic fuzzy number Á TIFN Á TrIFN Á Type-2 FTP Á Type-2 IFTP Á Type-2 IFSTP Á FAP Á IFAP Á IFSAP Á Linear programming method Á Integer programming problem Á Optimal solution Á Optimal assignment Á PSK theorem Electronic supplementary material The online version of this article (

This article demonstrates a fuzzy goal programming (FGP) approach with the use of genetic algorithm (GA) for proper deployment of patrol manpower to various road-segment areas in urban environment in different shifts of a time period to deterring violation of traffic rules and thereby reducing the accident rates in a traffic control planning horizon. To expound the potential use of the approach, a case example of the city Kolkata, West Bengal, INDIA, is solved.

The aim of this paper is to study multi-objective assignment problem with imprecise costs, time and ineffectiveness instead of its precise information. Here, elements of cost matrix, consumed time matrix and ineffectiveness level matrix have been represented by generalized trapezoidal fuzzy numbers as it is suitable way to represent the impreciseness of values provided by the decision makers due to time pressure or limited information and poor information processing capabilities. A priority based fuzzy goal programming method has been developed for generalized trapezoidal fuzzy numbers and it is applied for multi-objective assignment problem. Euclidean distance function is used to identify the most appropriate priority structure of fuzzy goals among the different priorities of the fuzzy goals. An illustrative numerical example is provided to demonstrate the effectiveness of the proposed approach.

Network Virtualization is a key component of the Future Internet, providing the dynamic support of different networks with different paradigms and mechanisms in the same physical infrastructure. A major challenge in the dynamic provision of virtual networks is the embedding approach taking energy efficiency into account, while not affecting the overall Virtual Network (VN) acceptance ratio. Previous research focused on either designing heuristic-based algorithms to address the efficient embedding problem or to address the energy impact. This paper proposes an integer linear programming formulation, Energy Aware-Virtual Network Embedding-Node-Link Formulation (EA-VNE-NLF), that solves the online virtual network embedding as an optimization problem, striving for the minimum energy consumption and optimal resource allocation per VN mapping. Two different objective functions are proposed: i) addressing primarily the resource consumption problem-Bandwidth Consumption Minimization (BCM); ii) addressing primarily the energy consumption problem-Energy Consumption Minimization (ECM). The performance of each objective function is evaluated by means of simulation and compared with an existing objective function, Weighted Shortest Distance Path (WSDP), that is considered state of the art of the resource allocation problem. The simulation results show that the objective function BCM reduces the energy consumption of the physical network by 14.4%, and improves the embedding factor by 4.3%, consuming almost the same amount of resources as requested, and slightly worsening the VN acceptance ratio by 2.3%. ECM reduces the energy consumption of the physical network by 31.4% and improves the embedding factor by 4.1%, without affecting the VN acceptance ratio when compared to WSDP.

The Future Internet demands energy efficient communication to cope with the ever increasing power consumption. Virtualization techniques have proved to be effective in reducing power consumption of network devices. An open issue in virtualization for green networking is the search for an energy-efficient mapping of virtual networks onto physical networks. This paper introduces a new model for the mapping of virtual networks which aims at reducing the energy consumption. This model is based on an integer linear programming formulation and several parameters, corresponding to characteristic of real networks, are considered. Simulation results attest the efficacy of the proposal.

Network virtualization is a promising technology for the Internet of the Future. An open issue in virtualization is the management of network resources in a way that energy savings are achieved without compromising the Quality of Service (QoS) requirements of the virtual networks. The dynamic allocation and deallocation of virtual networks can lead the state of the substrate to a less than optimum energy consumption. This paper introduces two algorithms for the migration of virtual routers and/or links which aims to allocate resources so that energy consumption is minimized. The efficacy of the migration of virtual routers and/or links and its impact on energy consumption are analyzed based on results derived via simulations.

In this paper, an algorithm for solving interval time-cost tradeoff transportation problemsis presented. In this problem, all the demands are defined as intervalto determine more realistic duration and cost. Mathematical methods can be used to convert the time-cost tradeoff problems to linear programming, integer programming, dynamic programming, goal programming or multi-objective linear programming problems for determining the optimum duration and cost. Using this approach, the algorithm is developed converting interval time-cost tradeoff transportation problem to the linear programming problem by taking into consideration of decision maker (DM).

In this paper, we investigate an assignment problem in which cost coefficients are triangular intuitionistic fuzzy numbers. In conventional assignment problem, cost is always certain. This paper develops an approach to solve an intuitionistic fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the costs (profits) matrix of the assignment problem are triangular intuitionistic fuzzy numbers. Then its triangular shaped membership and non-membership functions are defined. A new ranking procedure which can be found in [4] and is used to compare the intuitionistic fuzzy numbers so that an Intuitionistic Fuzzy Hungarian method may be applied to solve the intuitionistic fuzzy assignment problem. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

The Birkhoff-von Neumann Theorem shows that any bistochastic matrix can be written as a convex combination of permutation matrices. In particular, in a setting where n objects must be assigned to n agents, one object per agent, any random assignment matrix can be resolved into a deterministic assignment in accordance with the specified probability matrix. We generalize the theorem to accommodate a complex set of constraints encountered in many real-life market design problems. Specifically, the theorem can be extended to any environment in which the set of constraints can be partitioned into two hierarchies. Further, we show that this bihierarchy structure constitutes a maximal domain for the theorem, and we provide a constructive algorithm for implementing a random assignment under bihierarchical constraints. We provide several applications, including (i) single-unit random assignment, such as school choice; (ii) multi-unit random assignment, such as course allocation and fair division; and (iii) twosided matching problems, such as the scheduling of inter-league sports matchups. The same method also finds applications beyond economics, generalizing previous results on the minimize makespan problem in the computer science literature.

Transportation problems are a certain class of mathematical programming problem that arises frequently in various practical applications. But in real life, the transportation parameters may not be precise always due to lack of information, environmental factors, imprecision in judgment etc. Therefore, it is very interesting to deal with transportation problems under uncertainty. The best way to denote the imprecise data is fuzzy number. In the literature, there are several methods to find the optimal intuitionistic fuzzy cost of intuitionistic fuzzy transportation problem of type-2 i.e., availability and demands taken as real numbers and the transportation cost of unit product taken as intuitionistic fuzzy numbers. In this paper, we propose an efficient method to find intuitionistic fuzzy optimal solution of intuitionistic fuzzy transportation problem of type-2. The proposed method is quite easy and achieve its goal speedily as compare to the existing methods in the literature. There is no need to find the initial basic feasible solution, proposed method directly gives the optimal solution. Numerical example illustrate the methodology.

Network virtualization allows multiple heterogeneous virtual networks (VNs) to coexist on a shared infrastructure. Efficient mapping of virtual nodes and virtual links of a VN request onto substrate network resources, also known as the VN embedding problem, is the first step toward enabling such multiplicity. Since this problem is known to be-hard, previous research focused on designing heuristic-based algorithms that had clear separation between the node mapping and the link mapping phases. In this paper, we present ViNEYard-a collection of VN embedding algorithms that leverage better coordination between the two phases. We formulate the VN embedding problem as a mixed integer program through substrate network augmentation. We then relax the integer constraints to obtain a linear program and devise two online VN embedding algorithms D-ViNE and R-ViNE using deterministic and randomized rounding techniques, respectively. We also present a generalized window-based VN embedding algorithm (WiNE) to evaluate the effect of lookahead on VN embedding. Our simulation experiments on a large mix of VN requests show that the proposed algorithms increase the acceptance ratio and the revenue while decreasing the cost incurred by the substrate network in the long run.

In published works on fuzzy linear programming there are only few papers dealing with stability or sensitivity analysis in fuzzy mathematical programming. To the best of our knowledge, till now there is no method in the literature to deal with the sensitivity analysis of such fuzzy linear programming problems in which all the parameters are represented by LR flat fuzzy numbers. In this paper, a new method, named as Mehar's method, is proposed for the same. To show the advantages of proposed method over existing methods, some fuzzy sensitivity analysis problems which may or may not be solved by the existing methods are solved by using the proposed method.

T he fractional programming is a generalization of linear programming where the objective function is a ratio of two linear functions. Similarly, in fractional transportation problem the objective is to optimize the ratio of two cost functions or damage functions or demand functions. As the ratio of two functions is considered, the fractional programming models become more suitable for real life problems. Keeping in view the complexities associated with real life transportation problem like vagueness and uncertainty involved with the parameters. The implementation of fuzzy techniques can be very useful. Therefore, in this article a Fully Fuzzy Multi-objective Fractional Transportation Problem (FFMOFTP) is considered. All the coefficients of the parameters, demands and supplies are considered as fuzzy numbers. The purpose of using fuzzy numbers is to deal with the uncertainties and vagueness associated with the parameters. Two cases are considered, one with triangular and other with ...

In this paper, we investigate an assignment problem in which cost coefficients are triangular intuitionistic fuzzy numbers. In conventional assignment problem, cost is always certain. This paper develops an approach to solve an intuitionistic fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the costs (profits) matrix of the assignment problem are triangular intuitionistic fuzzy numbers. Then its triangular shaped membership and non-membership functions are defined. A new ranking procedure which can be found in [4] and is used to compare the intuitionistic fuzzy numbers so that an Intuitionistic Fuzzy Hungarian method may be applied to solve the intuitionistic fuzzy assignment problem. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

In this paper, we investigate an assignment problem in which cost coefficients are triangular intuitionistic fuzzy numbers. In conventional assignment problem, cost is always certain. This paper develops an approach to solve an intuitionistic fuzzy assignment problem where cost is not deterministic numbers but imprecise ones. Here, the elements of the costs (profits) matrix of the assignment problem are triangular intuitionistic fuzzy numbers. Then its triangular shaped membership and non-membership functions are defined. A new ranking procedure which can be found in [4] and is used to compare the intuitionistic fuzzy numbers so that an Intuitionistic Fuzzy Hungarian method may be applied to solve the intuitionistic fuzzy assignment problem. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.

proposed a method for solving a special type of interval-valued intuitionistic fuzzy transportation problems (IVIF-TPs) (transportation problems (TPs) in which the quantity of the product to be supplied is represented as a real number, whereas, all the other parameters are represented as interval-valued triangular intuitionistic fuzzy numbers (IVTIFNs)). In this note, an interval-valued intuitionistic fuzzy transportation problem (IVIF-TP) is solved by Bharti and Singh's method and shown that more than one IVTIFNs, representing the optimal interval-valued intuitionistic fuzzy (IVIF) transportation cost is obtained, which is mathematically incorrect as the obtained distinct IVTIFNs has different physical meanings. Also, it is pointed out that to resolve this flaw of Bharti and Singh's method may be considered as a challenging open research problem.

Network Virtualization is claimed to be a key component of the Future Internet by enabling the coexistence of heterogeneous (virtual) networks on the same physical infrastructure, providing the dynamic creation and support of different networks with different paradigms and mechanisms in the same physical network. A major challenge in the dynamic provision of virtual networks resides on the efficient embedding of virtual resources into physical ones. Since this problem is known to be N P-hard, previous research focused on designing heuristic-based algorithms; most of them do not consider a simultaneous optimization of the node and the link mapping, leading to non-optimal solutions. This paper proposes an integer linear programming formulation to solve the virtual network embedding problem, as a simultaneous optimization of virtual nodes and links placement, providing the optimal boundary for each virtual network mapping. A link − node formulation is used and the multi-commodity flow constrain is applied. In addition, a heuristic algorithm for virtual network embedding is also proposed and compared against the optimal formulation. The performance of the integer linear programming formulation and of the heuristic is evaluated by means of simulation. Simulation experiments show significant improvements of the virtual network acceptance ratio, in average additional 10% of the virtual network requests are accepted when using the integer linear programming formulation, which corresponds, in average, to more 7 virtual networks accommodated on the physical network.

A transportation problem can be involving multiple objectives, multiple products, and multiple conveyances. These kinds of transportation problems are named multi-objective multi-attributes solid transportation problems (MMSTP). In this study, a solution based on goal programming has been proposed for MMSTP, in which the supply and demand are uncertain. Moreover, to handle uncertainty, different uncertainty parameters, which are between 0.6 and 0.9, have been used. Then, the results with obtained these parameters are compared by using cost function values. The results indicate that when the uncertainty parameter decreases, the cost increases. Finally, it is shown that an optimal solution can be found using this model through an example.