An EOQ Model Under the Condition of Permissible Delay in Payments with Allowed Stock-Out Cost and Lead Time

: From this present study, derive the two consecutive demands between the time intervals: economic order quantity and total annual variable cost. The solution for this inventory model is optimizing the total annual variable cost. Here, given an arithmetical example and sensitivity analysis for the provision of the inventory model, assume the planning horizon and replenishment rate are infinite. To the best of our knowledge, this is the first study to find out the total annual variable cost using various costs under the condition of a permissible delay in payments with an allowed stock-out cost and lead time.


Introduction
By streamlining inventory processes, traders can verify product availability while cutting inventory costs and minimizing superfluous stock.Because of this, the inventory optimization verifies the proper inventory supply to reach the target level while using the least amount of capital.The majority of retailers use deterministic continuous review in accordance with inventory theory.The main goal is to retain inventory items on hand and place further orders as needed.The best model for forecasting client orders and shipping times is this one.
A long time ago, inventories were thought to be regarded as a country's or a person's wealth of measurements and influence.Due to significant advancements in production and fluctuations in product life, inventories are now seen as a far greater risk than a nation's riches of measures and power.The application of essential procedures in inventory management is now required by the idea of inventories, which is referred to as inventory control.If not, inventory control ensures that the right amount and quality of the product is made accessible when needed, taking into account the need to be economical with regard to carrying costs, ordering costs, setup costs, shortfall costs, manufacturing costs, purchase costs, and working capital.The time gap between placing the orders and receiving the orders is called delivery lag or lead time.The time horizon, otherwise called the planning period for which the inventory is to be meticulous, The planning period is divided into two types: one is finite and the other is infinite.
Aggarwal and Jaggi [1] carried out the ordering policies of deteriorating items under permissible delay in payments.Sepehri et al. [2] established the joint pricing and inventory model for deteriorating items with maximum lifetime and controllable carbon emissions under a permissible delay in payments.Taleizadeh et al. [3] recognized an EOQ model with partial delayed payment and partial backordering.Sarkar [4] introduced an EOQ model with delays in payments and stock-dependent demand in the presence of imperfect production.Sarkar [5] obtained an EOQ model with delays in payments and a time-varying deterioration rate.Jaggi et al. [6] framed an EOQ model with a permissible delay in payments in a fuzzy environment.Huang [7] coined an integrated inventory model under conditions of order processing cost reduction and permissible delay in payments.Huang et al. [8] formulated an optimal integrated vendor-buyer inventory policy under conditions of order processing time reduction and permissible delay in payments.Yao and Lee [9] invented fuzzy inventory with a backorder for fuzzy order quantity.Kaushik [10] created an inventory model with a permissible delay in payment and different interest rate charges.Cárdenas-Barrón et al. [11] approached an EOQ inventory model with nonlinear stock-dependent holding costs, nonlinear stock-dependent demand, and trade credit.Meena et al. [12] concluded inventory management in an inflationary environment under delay in payments with shortages.Poswal et al. [13] innovated an investigation and analysis of a fuzzy EOQ model for price-sensitive and stock-dependent demand under shortages.Chen et al. [14] implemented a retailer's economic order quantity when the supplier offers a conditionally permissible delay in payments linked to order quantity.Tripathi [15] computed an innovative approach to the EOQ structure for decaying items with time-sensitive demand, cash discounts, shortages, and a permissible delay in payments.Ghiami et al. [16] organized a two-echelon inventory model for a deteriorating item with stock-dependent demand, partial backlogging, and capacity constraints.Zadeh [17] first introduced the fuzzy set.
The key objective is to formulate the economic order quantity model under the condition of a permissible delay in payments with an allowed stock-out cost and a lead time for finding the optimized total cost.

Assumptions
The following assumptions are made in this developed inventory model: • Infinite planning horizon.
• Stock-out cost is allowed.
• Lead time or delivery lag is allowed.
• During that period, A account is not settled, and the merchant can earn interest.And they have to settle the account at A; for that, they want to pirate money at some specific rate of interest in order to get the balance stocks financed for the period A to T.
For the calculated interest payable and interest earned, there are two possible cases, viz., T ≥ A and T � A.

Case 1: If T ≥ A:
The cost of interest charges for the products retained in stock.The products that are being sold before the replenishment account is established are used to earn interest.The replenishment account is established, the situation is upturned, and efficiently, the products still in stock have to be financed at the interest rate Ic.The stock level at the time of settling the replenishment account is d(T -A) and interest owed during the time is (T -A).The total annual variable cost per unit time is The necessary and sufficient conditions for C(T) to be minimum are ) EOQ: ) Case 2: If T � A: In this condition, the merchant can earn interest on income created from the sales up to T. And take interest paid will be zero.The extreme accumulated total is dTs, if T � A and interest earned during the payment period A. Then, ( ) The total annual variable cost per unit time is The necessary and sufficient conditions for C(T) to be minimum are EOQ:

Numerical example
The

Concluding remarks
It is imperative to note that this is the first study demonstrating that the EOQ inventory model is a study about the circumstances of a permissible delay in payments with an allowed stock-out cost and lead time.From this prescribed inventory model, obtain the values using arithmetical analysis and sensitivity analysis to find the solutions for time interval, EOQ, and the total annual variable cost.In the sensitivity analysis, we change the demand values; we get different values in time, economic order quantity, and total cost.From the sensitivity analysis, we observed that as demand increased, the time interval and total cost decreased.For future studies, this inventory model can extend to stock-dependent demand and deteriorating items in a fuzzy sense.
amount is d.As if T ≥ A and earned interest during the payment period A.

Table 1 .
Sensitivity analysis for varying demand value (d) in Case 1

Table 2 .
Sensitivity analysis for varying demand value (d) in Case 2