Solution of Matrix Game with Triangular Intuitionistic Fuzzy Pay-Off Using Score Function

Using score function in a matrix game is very rare. In the proposed paper we have considered a matrix game with pay-off as triangular intuitionistic fuzzy number and a new ranking order has been proposed using value judgement index, available definitions and operations. A new concept of score function has been developed to defuzzify the pay-off matrix and solution of the matrix game has been obtained. A numerical example has been given in support of the proposed method.

Colouring of COVID-19 Affected Region Based on Fuzzy Directed Graphs

Graph colouring is the system of assigning a colour to each vertex of a graph.It is done in such a way that adjacent vertices do not have equal colour.It is fundamental in graph theory.It is often used to solve real-world problems like traffic light signalling,map colouring,scheduling,etc.Nowadays,social networks are prevalent systems in our life.Here,the users are considered as vertices,and their connections/interactions are taken as edges.Some users follow other popular users’profiles in these networks,and some don’t,but those non-followers are connected directly to the popular profiles.That means,along with traditional relationship(information flowing),there is another relation among them.It depends on the domination of the relationship between the nodes.This type of situation can be modelled as a directed fuzzy graph.In the colouring of fuzzy graph theory,edge membership plays a vital role.Edge membership is a representation of flowing information between end nodes of the edge.Apart from the communication relationship,there may be some other factors like domination in relation.This influence of power is captured here.In this article,the colouring of directed fuzzy graphs is defined based on the influence of relationship.Along with this,the chromatic number and strong chromatic number are provided,and related properties are investigated.An application regarding COVID-19 infection is presented using the colouring of directed fuzzy graphs.

<i>L</i>(0, 1)-Labelling of Cactus Graphs

An L(0,1)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between the labels assigned to any two adjacent vertices is at least zero and the difference between the labels assigned to any two vertices which are at distance two is at least one. The span of an L(0,1)-labelling is the maximum label number assigned to any vertex of G. The L(0,1)-labelling number of a graph G, denoted by λ0.1(G) is the least integer k such that G has an L(0,1)-labelling of span k. This labelling has an application to a computer code assignment problem. The task is to assign integer control codes to a network of computer stations with distance restrictions. A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we label the vertices of a cactus graph by L(0,1)-labelling and have shown that, △-1≤λ0.1(G)≤△ for a cactus graph, where △ is the degree of the graph G.

Multi-item EPQ model with learning effect on imperfect production over fuzzy-random planning horizon

Uncertainty is certain in the world of uncertainty.This study revisits an economic production quantity(EPQ)model with shortages for stock-dependent demand of the items with reworking and disposing of the imperfect ones over a random planning horizon under the joint effect of inflation and time value of money,where the expected time length is imprecise in nature.Transmission of learning effect has been incorporated to reduce the defective production.The total expected profit over the random planning horizon is maximized subject to the imprecise space constraint.The possibility,necessity and credibility measures have been introduced to defuzzify the model.The simulation-based genetic algorithm is used to make decision for the above EPQ model in different measures of uncertainty.The model is illustrated through an example.Sensitivity analysis shows the impacts of different parameters on the objective function in the model.

A supply chain model for imperfect production system with stochastic lead time demand

This study deals an integrated manufacturer-buyer supply chain system for imperfect production under stochastic lead time demand.Here,defective rate has been followed as a function of production rate.Also,the produced units have been inspected in order to screen the defective units but screening rate is less than the production rate and greater than the demand rate.Buyer purchases the products from the manufacturer.Also,we assume that shortage during the lead time is permitted and demand during the shortage period is fully backordered.The objective is to derive the optimal production rate,ordering quantity and to maximize joint total profit.Basically,two different models for different probability distribution functions of stochastic lead time demand have been developed.Some numerical examples are provided to show the applicability of the proposed models comparing the optimum average profits.Finally,sensitive analysis,conclusion and future researches are presented.

Transportation Problem with Multi-choice Cost and Demand and Stochastic Supply

This paper analyzes the multi-choice stochastic transportation problem where the cost coefficients of the objective function and the demand parameters of the constraints follow multi-choice parameters.Assume that the supply parameters of the constraints in a transportation problem(TP)follow logistic distribution.The main objective of this paper is to select an appropriate choice from the multi-choices for the cost coefficients of the objective function and the demand of the constraints in the TP by introducing Lagrange’s interpolating polynomial in such a way that the total cost is minimized and satisfies the required demand.Using stochastic programming,the stochastic supply constraints of the TP are transformed into deterministic constraints.Finally,a non-linear deterministic model is formulated.Using Lingo software,the optimal solution of the proposed problem is derived.To illustrate the methodology,a real-life problem on the TP is considered.

Time variant multi-objective linear fractional interval-valued transportation problem

This paper studies a time-variant multi-objective linear fractional transportation problem. In reality, transported goods should reach in destinations within a specific time. Considering the importance of time, a time-variant multi-objective linear fractional transportation problem is formulated here. We take into account the parameters as cost, supply and demand are interval valued that involved in the proposed model, so we treat the model as a multi-objective linear fractional interval transportation problem. To solve the formulated model, we first convert it into a deterministic form using a new transformation technique and then apply fuzzy programming to solve it. The applicability of our proposed method is shown by considering two numerical examples. At last, conclusions and future research directions regarding our study is included.

Analytics of an Imperfect Four-Layer Production Inventory Model Under Two-Level Credit Period Using Branch-and-Bound Technique

This paper explains an integrated production inventory supply chain model,which consists of a supplier,a manufacturer and a retailer under two-level credit period.One is manufacturer’s credit period offered by the supplier,and other is retailer’s credit period offered by the manufacturer.Here,the manufacturer replenishes raw materials from the supplier and produces non-deteriorating products in some cycles of equal length.It is noted that here four inventory structures have been considered to show the flow of the materials as either raw materials or finished products from supplier to retailer via manufacturer.Also in this model,manufacturer’s imperfect production has been considered.Here imperfect items are not repairable.The retailer receives good finished products from the manufacturer and sells these to his/her customers during some cycles of equal length in total time horizon.Our main objective is to find the optimal number of production and business cycles of the manufacturer such that the integrated system can get the maximum profit.The above discussed model is elaborated with the help of a numerical example,and a sensitivity analysis is done with respect to some parameters used in this model.

An inventory model under development cost-dependent imperfect production and reliability-dependent demand

This paper considers a model that deals with an imperfect production process where both perfect and imperfect quality items are produced.Here,demand depends on selling price and reliability of the product.Each manufacturing company expects to produce perfect quality items.But due to the long-run process,several kinds of problem such as labor,machinery,and technology arise.As a result,the manufacturing system becomes out-of-control state and consequently produces both perfect and imperfect quality items.Perfect items are ready to sell but imperfect items are reworked at a cost to become perfect.Reworking cost,reliability of the product and reliability parameter of the manufacturing system can be improved by introducing the development cost and also by improving the quality of the raw material of the production system.Under such circumstances,a profit function has been developed to find the optimum values of reliability parameter of the manufacturing system,reliability of the product and duration of production such that a manufacturer gets a maximum profit.Finally,the model has been illustrated with some numerical examples exploring the sensitivity analysis with respect to some parameters.

Bi-level Programming for Stackelberg Game with Intuitionistic Fuzzy Number: a Ranking Approach

This paper introduces a ranking function procedure on a bi-level programming for Stackelberg game involving intuitionistic fuzzy parameters.Intuitionistic fuzzy num-ber is considered in many real-life situations,so it makes perfect sense to address decision-making problem by using some specified intuitionistic fuzzy numbers.In this paper,intuitionistic fuzziness is characterized by a normal generalized triangular intuitionistic fuzzy number.A defuzzification method is introduced based on the pro-portional probability density function associated with the corresponding membership function,as well as the complement of non-membership function.Using the proposed ranking technique,a methodology is presented for solving bi-level programming for Stackelberg game.An application example is provided to demonstrate the applica-bility of the proposed methodology,and the achieved results are compared with the existing methods.

Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator

In this paper, we propose and analyze a three-species predator prey system in the presence of additional food for top predator. It is assumed that the middle predator is acting as a prey as well as a predator and the top predator consumes both prey as well as middle predator. It is also considered that a constant amount of additional food for top predator exists in the ecosystem. The effects of harvesting of top predator are investigated. The existence and stability conditions of the equilibria have been discussed analytically. The Hopf bifurcation analysis of the system with respect to predation rate of prey to the top predator and the harvesting effort have been analyzed both analytically and numerically. Pontryagin＇s maximum principle is used to determine the optimal harvesting of top predator population to maximize the discounted net revenue. From our analysis, it is seen that the additional food has a significant impact to prevent the extinction risk of top predator population and also to increase revenue collection. Finally, some numerical results have been given in support of our analytical findings.