Two-Storage Fuzzy Inventory Model with Time Dependent Demand and Holding Cost under Acceptable Delay in Payment

If we observe a real business market, the demand for items in each cycle is not in the same pattern, that is, for specific business cycle it may increase, stable or decrease (for instance, cool drinks from end stage of the summer to winter; the demand goes on decreasing, and from the end of winter to peak time of summer; the demand goes on increasing). Also, if the supplier permits for delay in payment, retailer wishes to buy more goods, and for which the retailer may need extra storage (in terms of a rented warehouse). Moreover, the retailer has always wished to sell the items before they expire and accordingly order is placed. Mostly the parameters in a real world inventory model are imprecise. Thus, in the proposed study an inventory model having decreasing time dependent demand pattern with variable holding cost for TwoStorage facility under acceptable delay in payment has been developed. Mathematical model of the problem and its solution procedure is discussed for both crisp and fuzzy environment in order to obtain the optimal replenishment time and cost. Also, numerical examples are discussed to validate the study. Finally, sensitivity analysis is also studied to describe the fluctuating scenario of associated parameters.


Introduction
The main aim of any business firm is to get more profit by expanding their business, creating goodwill in the market, and increasing their brand value. For which they consider many aspects, one of them is the inventory management system which plays an important role in any business affairs. An inventory management depends on different parameters such as demand, deterioration, holding cost, shortages, backlogs, inflation and trade credit, etc.
In the present global scenario, most of the suppliers offer price discount or trade credit financing. This may motivate retailers to purchase items more than that of fixed capacity of existing warehouse. To store these excess items retailer may hire a new warehouse on rental basis. Also, some retailers purchase a huge amount of items that cannot be accommodated in existing storage when a new product having very high demand launched into the market or when seasonal products arrives into the market. In fact, retailers of many products (for instance, apparels, footwear, jewellery, cosmetics, two wheelers, interior decorative items and marbles, etc.) use their primary warehouse at a suitable place in a busy market and that has been decorated with basic facilities to attract the customers for enhancement of their sales. At the same time, they used to rent or own a secondary warehouse to store the items in some other place to avoid heavy rent or maintenance cost.
Trade credit financing also plays an important role in inventory management. Allowing the trade credit finance by the supplier to retailers, supplier gets more quantity orders from existing customers and also attracts the new customers. As a result, sales of the supplier increases and in-hand stock level decreases quickly. Thus, the trade credit policy is helpful for the supplier to get more profit by decreasing the inventory cost. On the other hand, by availing trade credit facility retailer can order more quantity with minimal ordering cost and less investment capital. Also, retailer earns the interest on revenue accumulated by selling the items during the credit period.
While developing inventory models most of the authors assume the holding costs as constant. But it is observed that in some real inventory management, the holding cost of items in a rented warehouse (RW) increases over time or it is proportional to the demand of the rented warehouse (RW). For deteriorating items, the holding cost depends on the availing facility in the rented warehouse.
In the real business world, different costs associated with inventory models are imprecision in nature. That is to say, the holding cost, ordering cost, the interest rate and other associated costs are fluctuating over time. Thus, to deal with such inventory problems, fuzzy set theory has been an excellent tool.
Practically, various deteriorating inventory models have different deterioration patterns. For instance, in cloth items, electronic gadgets, construction materials, fruits, dry fruits, cold drinks, health drinks, medicines, different type of agricultural products, plastic products, metals have different deterioration patterns. Moreover, it has been observed that most of the daily needs have stipulated life span. For example, milk products, vegetables, cool drinks, packed foods, medicines etc. has short life span. Thus, retailer dealing with these type of items has always planned to sell them before they get expire.
Considering the above mentioned facts, here we develop an inventory model having decreasing time dependant demand pattern with variable holding cost for Two-Storage facility under acceptable delay in payment. Moreover, the model is discussed in fuzzy environment by taking the parameters as trapezoidal fuzzy numbers. The objective of the work is to minimize the total cost of inventory by obtaining the optimal inventory time for both the warehouses in crisp and fuzzy environment, and thereafter to study the effect on the optimal solutions subject to the small changes in the associated parameters.

Literature survey
In the year 1976, Hartley [8] initially framed a Two-warehouse inventory problem. Whereas, the concept of trade credit financing in inventory was initiated by Haley and Higgins [7] in the year 1973. The two-warehouse inventory problem together with trade credit financing was developed by many researchers. Recently, Liang and Zhou [18], Liao et al. ( [19,20]), Bhunia et al. [4], Tiwari et al. [28], and Jaggi et al. [10] considered demand and holding cost as constant in their two warehouse inventory problems under permissible delay in payment. Further, the demand function is taken as stock and selling price dependent by Guchhait et al. [6], selling price dependent by Jaggi et al. [12,13] and Sing and Kumar [26], exponentially increasing by Kaliraman et al. [14], and Rajan and Uthayakumar [22] and ramp-type by Chakraborty et al. [5] with constant holding cost in their credit financing problem having two storage facility. But, Sett et al. [24], Yang [29], and Khurana [17] proposed two warehouse inventory models without trade credit financing by considering constant holding cost with different demands. Also, Khanna et al. [15,16] and Jaggi et al. [11] presented inventory models for single warehouse problems under allowable delay in payment with different demand patterns.
The different types of time dependent holding costs have been considered by Barik et al. [2,3], Alfares [1], Mishra and Mishra [21], and Routray et al. [23] for their single warehouse problem. But, Yu [30] considered decreasing holding cost of rented warehouse in his two warehouse inventory model having constant demand under trade credit offer.
In present scenario, most of the researchers using Fuzzy concept in their inventory model to deal the imprecise parameters. Recently, Singh et al. [27] developed a two warehouse inventory model for non deteriorating items with fuzzy demand and fuzzy holding cost without trade credit, Shabani et al. [25] considered fuzzy demand and constant holding cost for their two warehouse inventory problem under permissible delay in payment, Indrajitsingha et al. [9] developed two warehouse problem for items having selling price dependent demand and constant holding cost in fuzzy environment.
From the above mentioned literature, we found that most of the authors considered different types of demand for their single or two warehouse problems with or without trade credit financing. Among all the authors, Jaggi et al. [11] considered exponentially decreasing demand, but no author considered time varying decreasing demand. Again, most of the authors considered constant holding cost. Whereas, Barik et al. [2,3], Alfares [1], Mishra and Mishra [21], Routray et al. [23] assumed variable holding cost for their single warehouse inventory problems and Yu [30] assumed decreasing holding cost for two-warehouse inventory problem. But, no one considered an increasing holding cost. Furthermore, some authors developed their inventory problems in fuzzy environment. Motivated essentially by the above mentioned results, here we investigate an optimal result for inventory items which follow decreasing demand and increasing rental cost with two warehouse facility under trade credit financing. Further, the model is treated in fuzzy environment to overcome the imprecision of associated costs. In Table 1, the precise comparison of the present model and the different models discussed in the literature are provided.

Assumptions
The following assumptions are carried out to develop the present model: (i) The homogeneous (or single) item is considered. (ii) The demand rate is decreasing over time. (iii) The associated items have a greater life span than that of cycle time T , and the number of items deteriorated during cycle is very negligible as compared to the stock in inventory. (iv) The instant Replenishment facility is available, that is, lead time is zero. (v) During the Inventory cycle, stock is available without any backlog, that is, shortages are not allowed. (vi) Twostorage facility is considered. Owned storage (OS) is the retailer's outlet and rented storage (RS) is away from the owned storage(OS). Moreover, the OS has limited storage capacity, and RS has unlimited capacity. (vii) The items in RS depletes first than that of the OS in order to reduce the cost of inventory.
(viii)The holding cost of the OS is constant, but it varies with time for RS. Also, holding cost of RS is more than that of the OS (the holding cost of the RS includes transportation cost from RS to OS, loading and unloading costs).
(ix) Supplier accepts the delay in payment by the retailer.

Notations
The following notations are used in the proposed model:

Mathematical model
In the present model, initially W amount of items ordered to the supplier. As the lead time is zero, the W items (W1 items are kept in OS and remaining (W-W1) in RS) are delivered instantly to the retailer. To reduce the cost of inventory the retailer sells the items in RS first, then OS. So during the interval [0, τ ], the inventory in RS is gradually reduces due to demand of the items, and is vanishing at time t = τ . After RS is empty, the inventory in OS gradually decreases during the interval [τ, T ], and is vanishing at time t = T . That means, both the storages are empty at time T . Figure 1 shows the inventory level at any time t. During the time t ∈ [0, τ ], the inventory in RS is gradually decreases, and reaches at 0 at t = τ . The inventory level in RS is governed by the differential equation under the boundary condition Q r [τ ] = 0. Solving this equation, we have Next, during the time t ∈ [0, τ ], the inventory in OS becomes constant, and for t ∈ [τ, T ] it gradually decreases and vanishes at t = T . The inventory level in OS is governed by the differential equation At time t = τ the amount of inventory in OS is W1. Thus, we have Solving this equation for T , we have .
Based on the assumptions in the model, the total relevant cost Z(τ ) of inventory includes the ordering cost, holding cost, interest earned on sales amount during the trade credit period and interest payable on stock in the inventory after allowable delay time. Now the costs are calculated as follows.
(iv) Interest earned by Retailer; there arise two cases: Case 2 (M > T ): Thus the total relevant cost of the inventory per year is given by, Z(τ ) = Ordering Cost + Total Stock Holding Cost + Interest payable − Interest Earned. Moreover, according to the acceptable delay period given by the supplier, the total cost function Z(τ ) is given by, We have,

Solution procedure
The aim of the present model is to find the optimal value of τ such that Z(τ ) is minimum. The working rule is as follows: (i) Use the expression for T from equation (5.1) in Z1(τ ), Z2(τ ) and Z3(τ ).

Fuzzy model
In the real world business, the different costs associated with any inventory model vary time to time. In this model, the holding cost of items H r (for RS) and H o (for OS), the interest payable θ and interest earned ϑ rates are not constant. Thus, we consider them as Fuzzy numbersH r ,H o ,θ andθ respectively. Taking all these parameters in to consideration, the total cost function under fuzzy environment is given bỹ Here, we havẽ In particular, letθ = (θ1, θ2, θ3, θ4),H r = (HR1, HR2, HR3, HR4),H o = (HO1, HO2, HO3, HO4),θ = (ϑ1, ϑ2, ϑ3, ϑ4) be the trapezoidal fuzzy numbers. Thus, the total cost functions (fuzzy) after defuzzification by Graded Mean Integration Representation (GMIR) Method, we have In the similar lines of the solution procedure for Crisp model, we can obtain the optimal solutions τ * , T * & GM Z * (τ ) for the Fuzzy model.

Sensitivity analysis
Here we consider the Example 1 to study the sensitivity of different parameters. On the basis of results shown in Tables 2 and Figures 6-9, it is concluded that: • The holding cost of OS (H o ) is very sensitive as the increase inH o results the increase in total inventory cost; but decrease in vanishing time (τ ) of RS, and so also the total cycle time T (see Table 2 and Figure 6).
• The holding cost of RS (H r ) is very sensitive as the increase inH r results the increase in total inventory cost; but decrease in vanishing time (τ ) of RS, and so also the total cycle time T (see Table 2 and Figure 7).
• The interest payable (θ) is very sensitive as the increase inθ results the increase in total inventory cost; but decrease in vanishing time (τ ) of RS, and so also the total cycle time T (see Table 2 and Figure 8).
• The interest earned (θ) is very sensitive as the increase inθ results the decrease in total inventory cost, vanishing time (τ ) of RS and so also the total cycle time T (see Table 2 and Figure 9).

Conclusions
Inventory control of products having two-storage facility under acceptable delay in payment is quite relevant in many business organizations. In this paper, a two storage inventory model under acceptable delay in payment for items having life time more than that of the cycle time with decreasing time dependent demand, variable holding cost (in RS) is developed successfully in both crisp and fuzzy environment. It has been observed from the mentioned examples that, the optimal results in fuzzy environment differ from the crisp environment. That is, the imprecision of costs lead the total inventory cost to excess or low. As the fact, the managers of inventory always look for accurate results to minimize the inventory cost. So, we suggest that the fuzzy inventory model is best suitable for the real world inventory problems. Also, the sensitivity of the parameters shows that the optimal replenishment time and cost depends on the cost of those parameters. The corresponding figures (Figure 6-Figure 9) and table ( Table 2) of sensitivity of parameters draw the visible attention justifying our arguments. Thus, the sensitivity analysis section helps the inventory managers in decision making.

Future Research Directions
The present model can be extended in several directions. One is shortage with different backlogging can be incorporated. Second one is different demands and deterioration can be considered. Third one is the model can be considered for imperfect production inventory of deteriorating items. Fourth one is the trade credit period can be linked to ordered quantity. Furthermore, the model can be extended in different direction by considering various inventory constraints.