Fuzzy EOQ Model for Time Varying Deterioration and Exponential Time Dependent Demand Rate under Inflation

In this study, we have discussed a fuzzy eoq model for deteriorating products with time varying deterioration under inflation and exponential time dependent demand rate. Shortages are not allowed in this fuzzy eoq model and the impact of inflation is investigated. An inventory model is used to determine whether the order quantity is more than or equal to a predetermined quantity for declining items.The optimal solution for the existing model is derived by taking truncated taylor’s series approximation for finding closed form optimal solution. The cost of deterioration, cost of ordering , cost of holding and the time taken to settle the delay in account are considered using triangular fuzzy numbers. In this study, the fuzzy triangular numbers are used to estimate the optimal order quantity and cycle duration. Furthermore, we have used graded mean integration method and signed distance approach to defuzzify these values. To validate our model, numerical examples are discussed for all cases with the help of sensitivity analysis for different parameters. Finally, a higher decay rate results in a shorter ideal cycle time as well as higher overall relevant cost is established. The presented model can be used to predict demand as a quadratic function of time,stock level time dependent demand, selling price, and other variables.


Introduction
An inventory control systems principal goal is to determine when and how much to order.Inventory management is complicated by deterioration. Everything deteriorates over time. Deterioration might be gradual or rapid, thus it's crucial to factor it into your EOQ model. Many scholars are interested in building trade credit-based mathematical models. Goyal, who looked at inventory models with a pre-authorized payment delay. According to numerous research, the cost does not alter over the planning horizon. This assertion may not be accurate, despite the fact that many countries have significant inflation rates. Inflation has an impact on the demand for particular goods. As the value of money decreases, the inflation rate rises. As a result, when selecting the best inventory policy, the impact of inflation and the time worth of money cannot be overlooked.
Dutta and Kumar [1] used a fuzzy trapezoidal number and the Signed distance method to create a fuzzy inventory model without shortages. Jaggi et al [8] created a fuzzy inventory model with deterioration that took demand into account as time-varying. Kumar and Rajput [12] proposed a fuzzy inventory model for deteriorating products with time-dependent demand and partial backlog for deteriorating items with timedependent demand and partial backlog. Economic order quantity in Fuzzy sense for Inventory without backorder Fuzzy sets was developed by Huey -Ming Lee and Jing-shing [6]. Harish Nagar and Priyanka Surana [5] developed a fuzzy inventory model for deteriorating objects that used pentagonal fuzzy numbers as parameters. In a fuzzy world, Dutta and Pavan Kumar [2] built an inventory model without shortages by taking into account keeping costs, ordering costs, and demand. Jershan Chiang-Shing Yao and Huey-Ming Lee [9] deal with backordered inventory. S.K.Indirajit Sinha, P.N.Samantha, and U.K.Mishra [7] use the signed distance approach to solve a fuzzy inventory model of shortages under completely backlogged conditions. Jershan Chiang-Shing Yao and Huey-Ming Lee [9] deal with backordered inventory. S.K.Indirajit Sinha, P.N.Samantha, and U.K.Mishra [7] use the signed distance approach to solve a fuzzy inventory model of shortages under completely backlogged conditions. G.Michael Rosario and R.M.Rajalakshmi [13] used different fuzzy numbers and defuzzified using the signed distance approach to examine an inventory model of allowable shortage. As supplier credit is related to order quantity, Tripathi, R.P., and Mishra, T. [19] developed an eoq model with exponential time-dependent demand rate under inflation.
In this paper an fuzzy eoq model for deteriorating products with time varying deterioration under inflation and exponential time dependent demand rate is adopted for consideration.The main objective is to estimate the optimal order quantity and optimal cycle time using triangular fuzzy numbers. Further more for defuzzification these quantities we use graded mean integration and signed distance method.Finally the model is illustrated by numerical example.

Notations and Assumptions
This paper makes use of the following notations: h : fuzzy keeping cost rate per unit time r : inflation rate is stable over time., where 0 ≤ r < 1 pe rt : selling price per unit at time, where p is the unit selling price at time zero. Ce rt : fuzzy purchase cost per unit at time t, where c is the unit purchase cost at time zero and p > c Ae rt : fuzzy ordering cost per order at time t H : length of planning horizoñ m : the maximum amount of time that can be allowed before settling an account I c : Interest paid per $ in stock per year I d : Interest gained per unit Q : order quantity Q d : minimum order quantity for which payment delays are allowed T : interval between refills T d : the time period due to time-dependent demand in which Q d units are depleted to zero. 5.If Q < Q d , payments for obtained goods must be produced as soon as possible.. 6.If, Q = Q d , a payment delay up to m is allowed.
The generated sales revenue is deposited in an interestbearing account because the account is not settled within the allowed time frame.. The customer pays off all units bought at the end of the credit period and starts paying the interest on the products in stock.

Mathematical formulation
Assume that the horizon length is H = nT , with n indicating the amount of replenishments to be produced during the period H and T indicating the time interval between replenishments. To meet demand, the inventory level I(t) is gradually reduced. As a consequence, the variance in inventory over time can be calculated as With boundary conditions I(0) = Q and I(T ) = 0. Solution of (1) is given by And order quantity is Since the time interval have equal lengths, we have By using the order quantity, we determine the time period during which Q d units are depleted to zero due to demand. Then we have To calculate the overall applicable cost in [0, H], we add the ordering, buying, and keeping costs together.: The interest gained in [0, H] is given by As a result, in (0,H), the total applicable cost is Fuzzy Model: The total cost per unit time in a fuzzy context is calculated using the above triangular fuzzy numbers. rH 2 )) The necessary condition for minimizing the total cost is ∂ Z 1(sd) (T ) ∂T = 0 provided that they satisfy the sufficient condi- Now defuzzifing the total cost Z 1 (T ) by using Graded mean integration method, we have The necessary condition for minimizing the total cost is ∂ Z 1(gm) (T ) ∂T = 0 provided that they satisfy the sufficient con- Now defuzzifing the total cost Z 2 (T ) by using Signed distance method, we have The necessary condition for minimizing the total cost is ∂ Z 4(gm) (T ) ∂T = 0 provided that they satisfy the sufficient con- Now defuzzifing the total cost Z 2 (T ) by using Graded mean integration method, we have The necessary condition for minimizing the total cost is ∂ Z 2(gm) (T ) ∂T = 0 provided that they satisfy the sufficient con- Now defuzzifing the total cost Z 3 (T ) by using Signed distance method, we have