An Inventory Model for Weibull Distributed Deteriorating Items Under Ramp Type Demand and Permissible Trade Credit Policy

In a classical inventory economic order quantity (EOQ) model, the stock is depleted due to both market demand and deterioration. Many inventory models are developed for items under variable rate of deterioration. The two parameter Weibull distributed term is a representation of constant, time dependent linear and non-linear, increasing and decreasing rate of deterioration. Again the demand rate is assumed here as time dependent in beginning of cycle and then becomes constant as passage of time. Shortages are allowed and fully backlogged. Moreover the trade credit policy is a win-win payment strategy for sharing profit in the inventory system. This present paper deals with a replenishment policy assuming two parameter Weibull distributed deteriorating items, demand rate a ramp type function of time under permissible trade credit policy. Finally several numerical examples are given to illustrate the model and some particular cases are also discussed along with its’ illustrations along with concluding remarks.


Introduction
In a classical EOQ model and in view of a real life situation, inventory is depleted due to both market demand and deterioration. So many decision makers are always confined for making policies to control and maintain inventories of deteriorating items. Researchers like Covert and Philip [1973], A. K. Jalan et al [1996], T. Chakrabarty et al [1998], K.S. Wu (2002), Anil Kumar Sharma[2012], Biswaranjan Mandal[2010],[2020] and many are developing different inventory models assuming items which deteriorate at constant or vary over time. In this paper, the deterioration rate is followed by two parameter Weibull distributed which is a representation of constant, time dependent linear and non-linear, increasing and decreasing function of time.
The assumption of constant demand is not always appropriate for many inventory models. When a new brand of goods like new branded car, dolls, advanced computer devices etc come in the market, the initial demand is mostly increasing with time and then ultimately stabilizes as constant. This pattern of demand is called a ramp type demand of time. It is mostly seen in the present field of economy in any kind of business sectors. In this field, researchers like W.A. Donaldson[1977], E.A. Silver[1979], R.M. Hill[1979], M. Mallick [2018], Biswaranjan Mandal [2020] etc are mentioned a few. The present model assumes such type of ram type demand function of time.
Traditionally, the supplier is paid for the items as soon as the items are received by the purchaser. But a permissible trade credit policy is developed where the supplier may provide a delay period to the customer if the outstanding amount is paid within a fixed settled period. No interest is charged during this period. Moreover beyond this period of time, the interest is charged by the supplier. This trade credit policy is a win-win payment strategy for sharing profit in the inventory system. In this context, few researchers S.P. Aggarwal et al [1995] This present study investigates a situation in which replenishment policy is assumed with two parameter Weibull distributed deteriorating items, demand rate a ramp type function of time under permissible trade credit policy. Shortages are allowed and which is fully backlogged. Finally several numerical examples are given to illustrate the model and some particular cases are also discussed along with concluding remarks.
(v). Deterioration occur when the item is effectively in stock. (vi). The deterioration rate function for two parameter Weibull distribution is When 0 α = , deterioration of items is switched off, when β = 1, θ (t) becomes a constant, when β <1, the rate of deterioration is decreasing with time t and when β >1, the rate of deterioration is increasing with time t.
(vii). The demand rate D(t) is assumed to be a ramp type function of time figure 1).
(viii). Shortages are allowed and fully backlogged.

Notations:
The following notations are made:

Model development:
Let Q be the total amount of inventory purchased or produced at the beginning of each period and S(>0) be the initial inventory assumed after fulfilling backorders. During [0, 1 t ], the stock will be gradually depleted due to the effect of deterioration and market demand, and ultimately falls to zero at t = 1 t . The shortages occur during time period [ 1 t , T] which are fully backlogged. The instantaneous state of the inventory level q(t) at time t governed by the following equations The boundary conditions are (0) q S = and 1 ( ) 0 q t = In the present model, we assume 1 t µ < and so the above two equations become Since 1 ( ) 0 q t = , we get from the equation (3.8) the following (neglecting second and higher order terms of ( 1) Therefore the total amount of on-hand inventory over the entire cycle (0,T) is Putting the value of S obtained from (3.10), we get the following

Inventory Scenarios :
Regarding interests charged and earned the following three distinct cases are considered due to the total depletion of the on-hand inventory at time 1 ( ) t T < Case I : The total interest payable over the entire cycle (0,T) is The total interest earned over the entire cycle (0,T) is The total average cost of the system per unit time is given by The total inventory cost during the period [0, T] contains the following cost components: The average total inventory cost per unit time is given by the following The necessary condition for the minimization of the average cost By the similar procedure as in case I and case II, the optimality equation

Numerical illustration for the particular cases:
Considering the same parametric values as mentioned above, we find the optimum values for the particular cases as follows

Concluding Remarks:
On the basis of the results shown in Table A, Table B and Table C  TC is highly sensitive towards the changes of these parameters c and c t .
(ii) It is also observed that inventory cost ( * TC ) is attained minimum mostly for the case II.
Therefore when the permissible trade credit period c t lies between µ and 1 t , the total average inventory cost attains minimum most in the proposed inventory model.. .
(iii) ** indicates the infeasible solution attained where the condition for the corresponding inventory scenario is violated.
(iv). The results obtained in Table D conclude that there are no major changes in the optimum values of * TC , * S and * Q for the two particular cases of the inventory model assuming absence of deterioration and constant rate of deterioration.