Int. J. Pure Appl. Sci. Technol., 22(2) (2014), pp. 25-35
International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online at www.ijopaasat.in Research Paper
An Intuitionistic Approach to an Inventory Model without Shortages Prabjot Kaur1,* and Mahuya Deb2 1
Department of Mathematics, Birla Institute of Technology, Mesra, Jharkhand, India
2
Usha Martin Academy, Ranchi, Jharkhand, India
* Corresponding author, e-mail: (
[email protected]) (Received: 1-4-14; Accepted: 6-6-14)
Abstract: Inventory management policies are crucial for the successful operations of firms involving inflow, storage and outflow of physical goods. However, the parameters associated with inventory problems often deal with uncertainties, and as such it is justifiable to consider these factors in elastic form as deterministic values may fail to give the correct approximations. This paper deals with a basic Economic Order Quantity(EOQ)inventory model where the objective is to determine the optimal cost and an optimum order quantity of inventory by taking certain non-deterministic parameters as triangular intuitionistic fuzzy numbers. Two case studies of the mathematical model have been given in order to show the applicability and robustness of the proposed model. Sensitivity analysis has been carried out which shows the linear relation between holding cost, EOQ and total cost .The advantage of the proposed intuitionistic approach is that it is a robust model which deals with the varying parameters in a general business inventory consistent with human behaviour by reflecting and modelling the hesitancy present in real life situations.
Keywords: Inventory management, Triangular intuitionistic fuzzy number (TIFN), Economic order quantity (EOQ).
1. Introduction: In any sort of business, a certain extent of inventory of resources is held to provide desirable services to the customers and to achieve the sales turnover target. Inventory control is the art of controlling an optimized amount of stock held in various forms within a business to economically meet the demands
Int. J. Pure Appl. Sci. Technol., 22(2) (2014), 25-35
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placed upon that business. Inventory management policies are crucial to the successful operations of the firms. Any disturbance in the inventory can lead to an overall misbalance in the management policies. The most important task of inventory management is making a trade-off between the minimization of the total cost and maximization of the customer satisfaction. Two fundamental questions that must be answered in controlling the inventory of physical goods are when to replenish the inventory and how much to order for replenishment. The earliest derivation of what is often called the simple lot size formula was obtained by Ford Harris of the Westinghouse Corporation in 1915 [4] in which the objective is to obtain an optimal and economic order quantity in such a way that the total yearly inventory cost is minimized, using mathematical relations (given afterwards). The EOQ model not only takes into consideration the cost price of the inventory, but also the other expenses involved in maintaining the inventory, like holding costs (which may include various aspects such as electricity, storage facility etc.). It is also referred to as Wilson formula since it was derived by R.H Wilson[15] as an integral part of the inventory control scheme thereby arising interest in the EOQ model in academics and industries [8]. Shortly after the World War II, a stochastic version of the simple lot size model was developed by Whitin [19]. Later, Hadley et al [5] analysed many inventory systems. Research work on static lot size model has been reported in the literature by Arrow [7]. The parameters related to classical inventory model are all crisp. But, in real life situations, these parameters may have slight deviations from the exact value which may not follow any probability distribution. In such situations, if they are treated as fuzzy parameters, then such a model becomes more realistic. The theory of fuzzy set introduced by Zadeh [13] in 1965 has achieved successful applications in various fields including inventory control. Recently, the concept of fuzzy parameters has been introduced in the inventory problems by several researchers. Park [10], Kacpryzk and Staniew [9] introduced fuzzy sets in the inventory problem. Consequently researchers began to make use of fuzzy numbers to consider these uncertain factors in elastic form. Hsieh [2] discussed an inventory model where demand and lead time are assumed to be fuzzy trapezoidal numbers. De and Rawat [12], proposed an EOQ model without shortage cost by using triangular fuzzy number. Bai and Li. [16], Dutta. et.al. [3] have also used Triangular and Trapezoidal Fuzzy number in building inventory models for determining the optimal order quantity and the optimal cost. However the fuzzy set theory was extended to the intuitionistic fuzzy sets by Atanassov [11] by adding an additional nonmembership degree. Among various extensions of fuzzy sets, IFSs have captured the attention of many researchers in the last few decades. This is mainly due to the fact that IFSs are consistent with human behaviour, by reflecting and modelling the hesitancy present in real life situations. Therefore in practice, it is realized that human expressions like perception, knowledge, and behaviour are better represented by IFSs rather than fuzzy sets The concept of an IFS can be seen as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of conventional fuzzy sets. The IF-set may express information more abundant and flexible than the fuzzy set when the uncertain information is involved. Banarjee and Roy [17] generalized the application of the intuitionistic fuzzy optimization in the constrained multi objective stochastic inventory model. Susovan Chakraborty et al. [18] gave the solution for the basic EOQ model using intuitionistic fuzzy optimization technique. Mahapatra [6] gave a multiobjective inventory model of deteriorating items with some constraints in an intuitionistic fuzzy environment. This paper is a novel approach in building up the intuitionistic inventory model considering the demand and ordering cost as TIFN and for defuzzification an accuracy function defined by [1] has been considered. The total variable cost is expressed as a function of these two decision variables and then the reorder point is obtained which minimizes the total cost. As the model parameters are generally imprecise or may be unable to exhibit the variability in such situations IFS theory can come as a rescue. The organisation of the paper is as follows: In section 2 the preliminaries on Intuitionistic fuzzy sets is detailed. Section 3 deals in establishment of the Inventory Model in the Intuitionistic fuzzy environment. Section 4 deals in optimising and finding a solution for the Inventory problem using TIFN. Section 5 deals with a numerical and a sensitive analysis to illustrate the result. Lastly the concluding remark is stated in Section 6.
Int. J. Pure Appl. Sci. Technol., 22(2) (2014), 25-35
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2. Preliminaries on Intuitionistic Fuzzy Sets: Intuitionistic fuzzy set was introduced by Atanassov (1986) and it has found applications in various areas of research. In this section some basics and notations on intuitionistic fuzzy sets are reviewed. 2.1 Definition 1: A fuzzy set A/in X = {x} is given by (Zadeh [13]): A/= {< x, µA(x) >|x ∈X}
Where µA: X → [0, 1] is the membership function of the fuzzy set A; µ A ∈ [0, 1] 2.2 Definition 2: An intuitionistic fuzzy set A in X is given by (Atanassov [11]):
A = {< x, µA(x), νA(x) >|x ∈X}
µA: X → [0, 1] νA: X → [0, 1]
Where
With the condition
00)
4. ′ = 5 ̿′ is a TIFN {(ka1,ka2,ka3)(ka1',ka2,ka3' )}
(4)
5. Division of two TIFN is = {67 , 7
A B
′ ′ a3 a1 ,7 89 , , : } is also a TIFN ′ 7 ′ b3 b1
(5)
2.5 Accuracy Function for Defuzzification:
/ / Let ̿ = (a1, a2 , a3 );(a1 , a2 , a3 ) be a TIFN, then accuracy function [1] for defuzzification is defined
{
as
̿′
=
(
}
;
;
); (a1
