Deterministic EOQ Models for Non-Linear Time Induced Demand and Different Holding Cost Functions

This paper presents an Economic order quantity (EOQ) model for deteriorating items. The demand rate is non-linear function of time. In this paper two models have been derived for different holding costs (i). The holding cost is linear function of the on hand inventory level and (ii). A non-linear function of time for which the item is kept in the stock. Optimization is done for both the models and numerical examples are presented to check the feasibility of the optimal solutions. Sensitivity analysis is also presented with respect to the various parameters used in the numerical example.


INTRODUCTION
Controlling and managing the inventory is among the biggest concern for any business regardless of its level. This concern leads the researchers to make inventory models for the better management of inventory. But while dealing with the real life problems it is not possible to consider all the factors affecting the depletion of inventory. Yet researchers have been able to consider most of the phenomenon like deterioration, demand rate etc.
As most of the physical goods undergo deterioration due to spoilage and many other factors. Most of the eatables that are available in market use preservatives. So they cannot be use after a definite time. So deterioration is an important factor to consider while developing an inventory model. Balkhi and Benkherouf (2004) developed an inventory model for deteriorating items with stock dependent and time varying demand rates. Lee and Dye (2012) established inventory model for deteriorating items under stock dependent demand rate and controllable deterioration rate. Arinadav and Herbon (2013) presented optimal inventory policy for a perishable item with demand function sensitive to price and time. Chang et al. (2010) presented optimal replenishment for non-instantaneous deteriorating items with stock-dependent demand. Giri and Chaudhari have developed many inventory models for the deteriorating items. Moon and Giri (2005) developed Economic order quantity models for ameliorating or deteriorating items under inflation and time discounting. Giri, Chaudhari and Goswami (1996) presented an inventory model for deteriorating items with stock-dependent demand rate. Giri and Chaudhari (1998) established deterministic model of perishable inventory with stock dependent demand rate and non-linear holding cost.
Most of the inventory models have been developed with constant holding cost. But this is not a realistic case. Weiss (1982) has taken no-linear holding cost in his paper. Goh (1994) also presented EOQ model with general demand and holding cost functions. Muhlemann and Valris (1980) have also taken variable holding cost rate in formulating the EOQ model. Singh, Tripathi and Mishra (2013) developed inventory model with deteriorating items and time-dependent holding cost.  presented an inventory model with stock-dependent demand and different holding cost function. Other studies that have been done in this area can be marked for Alfares (2007), Pando (2013), Tripathi (2015) and Roy (2008).
In real life it is observed that the demand rate is often influenced by the amount of on-hand inventory. Soni and Shah(2008) presented a mathematical model to formulate optimal ordering policies for retailer when demand is partially constant and partially stock-dependent and the supplier offer progressive permissible delay to settle the account. Silver  Gupta and Vrat (1986) established EOQ model for demand rate is a function of initial stock level.
In this paper the main aim is to find optimal cycle time which minimizes the total relevant cost. The rest of the paper is organized as follows. Assumptions and notations are given in section 2 followed by mathematical formulation. Numerical examples are discussed in section 4. In section 5 we provide sensitivity analysis, Conclusions and future research directions have been marked in the last section 6.

ASSUMPTIONS AND NOTATIONS
Following assumptions are made throughout the manuscript 1. The demand is a function of power of time.
2. Shortages are not allowed.

MATHEMATICAL MODEL
At the initial level of cycle time T the inventory level is Q which is depleted during the cycle time T due to constant rate of deterioration and time dependent demand rate and becomes zero at the end of cycle time T.
The differential equation describing the changes in the inventory level   qt over the period ( 0 tT  ) is given by: With the boundary condition   0 qQ  and   0 Solving (1) and neglecting higher powers of  we get     The order quantity for one cycle is

Model A:
In this model, the holding cost is taken to be the linear function of on-hand inventory level   qt .
Therefore, the holding cost is Substituting (2) in (4) gives The deterioration cost is given by 11 0 Using (3) in (6), we have Deterioration cost is 2 11 The total relevant cost per unit time is given by In this paper our main concern is to find the optimal order quantity Q  , which minimizes the total relevant cost TCU of the inventory model.
The necessary condition for the TCU to be minimum is Substituting the value of HC and DC from equation (5) and (7) From the above expression we can calculate the value of T  , that can be used to calculate the value of Q  by substituting in (3), which minimizes the total relevant cost TCU of the inventory system, provided The second derivative of (9) w.r.t T is given by It can be seen from (12)

Model B: Non-linear time dependent holding cost
In this model holding cost is non-linear function of time ( 0 tT  ).
The holding cost per order will be Holding cost is  (16) Differentiating (14) w.r.to cycle time T and equating it to zero, we will get the expression 2 1 11 Differentiating (14) w.r.to cycle time T, twice yields By putting various values of the parameters, we will be able to find the value of

NUMERICAL EXAMPLE
Following data is used in their appropriate units to get the optimal values for the inventory system. To obtain the minimum value further calculation is required.

SENSITIVITY ANALYSIS
The sensitivity analysis has been performed here based on above example 1, changing one parameter at a time and keeping all other parameters constant.        (ii). From table A2 it is clear that the increasing effect of deterioration rate  increases TCU* but decreases Q * and T * .
(iii). Table A3 shows the variation in the value of Q * , T * and TCU * with respect to the per unit holding cost h. There is no change in optimal order quantity corresponding to unit holding cost h. Whereas increment in the value of h decreases the optimal cycle time T* and increase in the total relevant cost TCU * .
(iv). Table A4 indicates that the increment in the scale parameter D1 results the decrement Q* and T * but increment in TCU * .
(v). From Table A5, we see that a significant decrease in the unit ordering cost K leaves no change in Q * but produces a significant decrease in T* and TCU * . Table A6 shows that as the value of per unit item cost C1 increases, TCU * increases and T * decreases, whereas no change is seen in Q * .

CONCLUSION AND FUTURE RESEARCH
In this paper, we have developed inventory models with non-linear time dependent demand. Holding cost rate is taken as quantity dependent for model A and non-linear time-dependent for model B. This type of assumption is valid with time retailers which sells products like green vegetables, breads and seasonal fruits, whose quality decreases with time due to direct spoilage or physical decay. The time-dependent holding cost is realistic assumption because to arrange greater storage facilities to cease spoilage and to keep the freshness of the commodities in the stock cannot let the holding cost constant.
Mathematical models have been developed for two different situations. Sensitivity analysis with respect to variation of different parameters revealed significant changes in T  The model discussed in this paper may be generalized to allow for shortages. We may also extend the present model for exponential demand as well as inflation dependent demand.