A FUZZY LEAKAGE INVENTORY MODEL WITH SHORTAGE USING TRIANGULAR AND TRAPEZOIDAL FUZZY NUMBERS

Inventory refers to any kind of economically valued resources that are in various stages of being made ready for sale. With the development of the Economic Order Quantity (EOQ) model by Ford Harris, many models under different conditions and assumptions are proposed. Leakage is a common phenomenon whose occurrence will reduce the profitability of the firm by increasing the minimum operational cost. In majority of real-life inventory problems we face many uncertainties in the key parameters of the corresponding model. These impreciseness and uncertainties in crisp model are improved by using fuzzy set theory. So, in this context, a fuzzy leakage inventory order level is developed by taking holding cost and shortage cost as Triangular as well as Trapezoidal Fuzzy Numbers. Defuzzification is done using Signed Distance Method. A relevant numerical example is illustrated along with sensitivity analysis to justify the proposed notion that the optimal values are improved in fuzzy environment as compared to that of in crisp environment.


INTRODUCTION
Inventory refers to the itemized catalog or list of tangible goods or property, or the intangible attributes or qualities. This refers to the value of materials and goods held by an organization (i) to support production (raw materials, subassemblies, work in process), (ii) for support activities (repair, maintenance, consumables), or (iii) for sale or customer service(merchandise, finished goods, spare parts).
In an inventory system, goods in stock may suffer leakage. Here, leakage refers to the loss in quantity, for example, leakage in case of liquid stock (oil/fuel) or breakages in case of solid goods (glass/bottles), damages made by insects or animals (food items) and transit loss (business) due to malpractices of sales by personnel involved, etc. The occurrence of such leakages will affect the profitability of the organization by way of increasing minimum operational cost, decreasing the optimal quantity to be maintained. Since leakages will exist till it is detected by the management and measures to control it are taken up, it is temporary in nature. Therefore, the solution of inventory problem is a set of specific values of variables under discussion that minimizes the total cost of the system.
The first quantitative treatment of inventory was the simple EOQ model. This model was developed by Harris [16], which was later investigated and applied in academics and industries.
Later on, many researchers developed many inventory models in crisp environment under different parameters and assumptions. But these assumptions do not suit the real world environment and hence there is a great deal of uncertainty and variability. The modern concept of uncertainty evolved with the publication of a seminar paper by Zadeh [8] where he introduced a theory on sets, known as fuzzy set theory, with boundaries that are not specific and precise.
Zimmerman [5] also discussed the concept of fuzzy set theory and its applications in different fields like operations research.
In literature, there are many papers in fuzzified problems on EOQ model and many had studied various cases of fuzzy inventory models. Dhivya and Pandian [9] in their paper drew the attention of the contributions of various researchers in various classes of fuzzy inventory models. Jayjayanti [10] gave a brief note on study of fuzziness in inventory management by different authors. Urgeletti [4] developed EOQ model in fuzzy nature and used triangular fuzzy number. Park [14] proposed a single product inventory model with fuzzy parameters on the basis of Harris [16] model. Dutta and Pawan [3] developed fuzzy inventory model without shortages using trapezoidal fuzzy number and used signed distance method for defuzzification.
Kweimei and Jing [17] fuzzified the order quantity and shortage quantity into triangular fuzzy numbers in an inventory model with backorder. Jaggi et. al [2] developed a fuzzy inventory model for deteriorating items with time varying demand and shortages. Syed and Aziz [7] in their paper developed an inventory model without shortages, representing both the ordering and holding costs by triangular fuzzy numbers and calculating the optimal order quantity using Signed Distance Method of defuzzification. Nabendu and Sanjukta [13] attempted to study inventory with shortages by considering the associated costs involved as different fuzzy numbers. Chandrasiri [1] studied the economic order quantity inventory model without shortages using triangular fuzzy number. Rexlin [15] estimated the fuzzy optimal order quantity and fuzzy total cost of an inventory system with shortage. Uthayakumar and Karuppasamy [11] developed his fuzzy inventory model to reduce the healthcare cost without sacrificing customer service by taking ordering cost, holding cost and order quantity all as triangular fuzzy numbers.
Based on the deterministic leakage inventory model developed by Tomba and Geeta [6], the present fuzzy leakage inventory model is developed by considering holding cost and shortage cost as fuzzy numbers.  (i)Ã is normal, i.e., there exists x ∈ X such that µÃ(x) = 1.

is called a Triangular Fuzzy Number if its membership function is given by
are two trapezoidal fuzzy numbers. Then Defuzzification is the conversion of a fuzzy quantity to a crisp quantity. In this paper, defuzzification is done by using Signed Distance Method.
IfÃ is a Triangular Fuzzy Number, then the signed distance ofÃ is defined as IfÃ is a Trapezoidal Fuzzy Number, then the signed distance ofÃ is defined as (ii) Leakage rate is d 1 units per unit time.
(iv) Holding cost and shortage cost are taken as fuzzy numbers.
(v) Lead time is zero.
(vi) Production is instantaneous and finite.  When the model has shortage and z is the order to which the inventory is raised in the beginning of a run of time interval t ,then the inventory is reduced to zero in time t 1 . Then,

MATHEMATICAL FORMULATION OF INVENTORY PROBLEMS IN DIFFERENT ENVI-
Shortage increase from 0 to Q-z in the remaining time (t-t 1 ).
The inventory also has leakage at a rate of d 1 units per unit time, so the inventory reduces to zero in time t 1 instead of time t 1 .
Shortages then increase from 0 to Q -z in time (t 1 -t 1 ), where Q is the quantity produced per run having leakage in demand pattern.
With the occurrence of leakage, the penalty cost created Then, holding cost (due to leakage) per production run With the occurrence of leakage, the shortage cost is increased.
Then, cost increased in the original shortage cost = C s × area o f ∆CDG ∴ Shortage cost (due to leakage) for one run = C s × area o f ∆DEF +C s × area o f ∆CDG ∴ Average total cost, TC(z) = The optimum order quantity z * and average minimum total cost TC * are obtained by equating the partial derivatives of TC to zero and solving the resulting equations.
Optimum order quantity, Minimum total cost per unit time, When Equations (11) and (12) represent the general model without leakages.

Leakage Inventory Model with Shortage in Fuzzy sense.
The above model is now considered in fuzzy environment under two cases: Case I: When the holding cost and shortage cost are represented by Triangular Fuzzy Numbers.
LetC h : fuzzy carrying or holding cost per unit quantity per unit time, C s : fuzzy shortage cost per unit per unit time.
The total cost given in eqn. (8) is fuzzified as fuzzy average total cost, SupposeC h = (a 1 , b 1 , c 1 ) andC s = (a 2 , b 2 , c 2 ) are triangular fuzzy numbers. Then, from eqn. (13), Then, Defuzzification of fuzzy numberTC is done by using Signed Distance method. So, Differentiating d(TC, 0) w.r.t. z and equating to zero, we get which is the fuzzy optimal order quantity.

NUMERICAL EXAMPLE WITH SENSITIVITY ANALYSIS
To illustrate the developed model, we consider an example when  Table 3. When parameters are considered as Trapezoidal Fuzzy Numbers  From the above three tables, we observe that economic order quantity obtained in fuzzy environment is very much close to the crisp economic order quantity and the minimum total cost in fuzzy sense is less than crisp minimum total cost.

CONCLUSION
In this paper, an inventory leakage model with shortage is studied in fuzzy sense considering the holding cost and shortage cost as fuzzy numbers(triangular and trapezoidal fuzzy numbers).Signed Distance method is opted for defuzzification. Sensitivity analysis indicates that the optimum values obtained in fuzzy environment are close to that of the result from crisp method and are more accurate to real life situations due to the many uncertainties prevailing in practical life.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.