Max/Min Fuzzy Function

In this paper, economic order quantity (EOQ) models without shortages for single item and multi-items are presented. Here, the holding cost of the item is a continuous function of the order quantity. The costs involved in this model are imprecise in nature. The main contributions of this research are as follows: The proposed EOQ model is discussed in two cases by describing the model in an uncertain environment. In case-1, EOQ models with fuzzy parameters (like ordering cost, holding cost, and unit product cost) are considered. Here all the fuzzy parameters are represented by trapezoidal fuzzy numbers. The said EOQ model is carried out by using the signed-distance method. In case-2, EOQ models with interval parameters (like ordering cost, holding cost, unit product cost, and the total money investment for the quantities) are considered. This proposed model is solved by using interval linear programming problem (ILPP) technique based on the best and the worst optimum values of the objective function. Numerical examples are given to exemplify the proposed model and also the results of different models are compared.KeywordsEOQTrapezoidal fuzzy numberInterval numberSigned-distance methodInterval linear programming problem (ILPP)

The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.

The track of developing Economic Order Quantity (EOQ) models with uncertainties described as fuzzy numbers has been very lucrative. In this paper, a fuzzy Economic Production Quantity (EPQ) model is developed to address a specific problem in a theoretical setting. Not only is the production time finite, but also backorders are allowed. The uncertainties, in the industrial context, come from the fact that the production availability is uncertain as well as the demand. These uncertainties will be handled with fuzzy numbers and the analytical solution to the optimization problem will be obtained. A theoretical example from the process industry is also given to illustrate the new model.

Circular economy is the one of the most targeted development in which remanufacturing, recycle and reuse the materials that creates job with minimised resource used. Application of circular economy is increasing day by day for economic growth and healthy business. Government encourages the use of circular economy instead of linear economy by providing subsidies through regulations and taxes. In this paper, a fuzzy economic order quantity model in circular economy is presented. Since some parameters are uncertain in nature, they are considered as triangular fuzzy number. Defuzzification is done by using graded mean integration representation method (GMIR). In this paper, ordering quantity and circularity index level are decision variables. The concavity of the average profit function is shown theoretically. The optimality of the parameters is determined analytically. It is further endorsed by numerical example with an intension to validate the proposed model. The CE gives a remarkable gain in profit, which is the focus of any industry that is concerned about environmental conservation. The financial advantage of the company is given due weight age in this model. For this, sensitivity analysis is independently carried out by deploying circular economy in inventory modelling.

The fuzzy optimal path under uncertainty is one of the basic network optimization problems. Considering the uncertain environment, many fuzzy numbers are used to represent the edge weights, such as interval number and triangular fuzzy number. Then, these fuzzy numbers are converted to real numbers directly. This converting makes the optimal path the shortest path selection problem. However, much information of uncertainty get lost when converting fuzzy numbers to real numbers. In order to ensure all the origan data complete, in this paper, a fuzzy optimal path solving model based on the Monte Carlo method and adaptive amoeba algorithm is proposed. In Monte Carlo process, a random number which belongs to the fuzzy number is generated. Then, Physarum polycephalum algorithm is used to solve the shortest path every time and record the result. After many times calculation, many shortest paths have been found and recorded. At last, by analysing the characters of all the results, the optimal path can be selected. Several numerical examples are given to illustrate the effectiveness of the proposed method, the results show that the proposed method can deal with the fuzzy optimal path problems effectively.

Inventory management is critical for many industries. A proper control of inventory can considerably enhance a company's profit margins. Determination of the Economic Order Quantities (EOQ) is important in order to achieve optimal operating condition

Group-oriented service composition is still challenging for big group and personalized QoS requirements. Therefore, this paper presents a Hybrid QoS-aware Web Service Composition Strategies for Group Pareto Optimal Plan (HOWSCS_GGOP) based on multi-attribute group decision making (MAGDM) theory. It includes five main steps: calculating group preference, clustering weighted cluster weights, constructing cluster weighted normalized decision-making matrix, determining the ideal solution, and evaluating alternatives for group synthetically. In addition, a hybrid Web service QoS model is introduced to describe QoS values with real numbers, interval numbers, triangular fuzzy numbers and intuitionistic fuzzy numbers. Experimental results show that the proposed algorithm can well solve the problem of Web service composition for big group and personalized QoS requirements.

The applicability of a fuzzy approach to a problem originating from administrative accounting, namely to determine an economic order quantity (EOQ) in a variable competitive environment with imprecise and vague data, is presented. For this purpose, the model of Ordered Fuzzy Numbers developed by the first author (W.K.) and his two co-workers is used. The present approach generalizes the one developed within the framework of Convex Fuzzy Numbers and stays outside the probabilistic one.

Based on the generalized defuzzifying approach derived in this paper, we establish the fuzzy economic order quantity (EOQ) model with backorder. When the order quantity is a crisp number, we use the direct derivation method to obtain the optimal solution. When the order quantity is a fuzzy number, we use the extension of the Lagrangean method to solve the inequality constraints. The results indicate that the optimal decision-making is determinate. In addition, if all fuzzy parameters are crisp numbers or the backorder cost is extremely large, this model will degrade into the classical inventory model with backorder and the simple inventory model without backorder, respectively

EOQ models with stockouts for disassembly systems

In this paper, an inventory model has been developed with trade credit financing and back orders under human learning. In this model, it is considered that the seller provides a credit period to his buyer to settle the account and the buyer accepts the credit period policy with certain terms and conditions. The impact of learning and credit financing on the size of the lot and the corresponding cost has been presented. For the development of the model, demand and lead times have been taken as the fuzzy triangular numbers are fuzzified, and then learning has been done in the fuzzy numbers. First of all, the consideration of constant fuzziness is relaxed, and then the concept of learning in fuzzy under credit financing is joined with the representation, assuming that the degree of fuzziness reduces over the planning horizon. Finally, the expected total fuzzy cost function is minimized with respect to order quantity and number of shipments under credit financing and learning effect. Lastly, sensitive analysis has been presented as a consequence of some numerical examples.

Optimization in Fuzzy Economic Order Quantity (FEOQ) Model with Promotional Effort Cost and Units Lost Due to Deterioration

This paper extends a fruitful fuzzy EOQ (Economic Order Quantity) track of research with a multi-item model that takes storage limitation into consideration as well as a finite production rate (i.e. the products are produced to stock and not purchased). The order cycle time and the demand are allowed to be triangular fuzzy numbers. In the process industry, this is a situation that often occurs, since the production equipment typically manufactures a large amount of only one product at a time. The model is developed and solved with Lagrange multipliers. Two illustrative examples are also given.

An intuitionistic fuzzy economic order quantity (EOQ) inventory model with backlogging is investigated using the score functions for the member and non-membership functions. The demand rate is varying with selling price and promotional effort (PE). A crisp model is formulated first. Then, intuitionistic fuzzy set and score function (or net membership function) are applied in the proposed model, considering selling price and PE as fuzzy numbers. To obtain the best inventory policy, ranking index method has been adopted, showing that the score function can maintain the ranking rule also. Moreover, optimization is made under the general fuzzy optimal (GFO) and intuitionistic fuzzy optimal (IFO) policy. Finally, a graphical illustration, numerical examples with sensitivity analysis and conclusion is made to justify the model.

The present paper proposes a fuzzy economic order quantity (EOQ) model for deteriorating items with stock dependent demand rate where shortages are not allowed. The cycle time is divided into two parts. The first interval demand is considered as stock dependent and the second part is assumed to be constant. We first develop a crisp model. Since most of the parameters of the inventory model are uncertain, we consider the demand, holding cost, deterioration cost and deterioration rate as triangular fuzzy numbers. Graded mean representation, signed distance method and centroid method are used to defuzzify the total cost function. The results obtained by these methods are compared with the help of a numerical example. Sensitivity analysis is also carried out to explore the effect of changes in the values of the system parameters.