# Abstract

Many goods undergo decay or deterioration over time which suffer from depletion by direct spoilage while stored. So decay or deterioration of these goods in stock is a very realistic feature and it is necessary to use this factor in inventory models. In this paper we have developed an order level inventory model for constant rate of deterioration. We have also considered a variable type of demand which behaves differently in the given time horizon. The demand rate is constant for a certain fixed time and then the demand varies linearly with time. This paper also deals with different selling prices in two different time periods. The objective of the model is to find the optimal on-hand inventory by considering the profit function.

## Keywords

Deterministic Inventory Model, Deterioration, Profit functions, Two different selling prices, Variable demand rate

## Introduction

One of the most developed fields of Operations Research is inventory modeling. Inventory has been defined as an idle resource that possess economic value by Monks(1987). Keeping an inventory for future sales or use is very common in business. Retail firms, wholesalers, manufacturing companies and even blood banks generally have a stock of goods on hand. Usually the demand rate is decided by the amount of the stock level. The motivational effect on the people may be caused by the presence of stock at times. Large quantities of goods displayed in markets according to seasons motivate the customers to buy more. If the stock is insufficient the customers may prefer some other brands, as shortages will fetch loss to the producers. On the other hand, deterioration is an important natural phenomenon and the consequent loss due to decay of items may be quite significant. Mainly when, physical goods are stocked for future use, in some items such as medicines, foodstuff, dairy items, volatile liquids, the process of deterioration is observed.

Resh et al., (1976) and Donaldson (1977) are the first researchers who considered an inventory model with a linear trend in demand. The time dependent demand patterns reported above are linear, that is, the demand increases continuously with time or decreases continuously along with the time. Dave and Patel (1981), Dutta and Pal (1992), considered time proportional demand. Goyal (1986) considered linear trend in demand. Hariga and Benkherouf (1994) considered exponential time varying demand for deteriorating items. Hill(1995) proposed a time dependent demand pattern by considering it as the combination of linearly time dependent and exponentially time dependent of demand in two successive time periods over the entire time horizon and termed as “ramp- type” time dependent demand pattern.. The works done by Roy(2008), Sabahno(2009), Mirzazadeh(2010), Gayen and Pal(2009) are some of the models for deteriorating items based on different realistic situations. Deterioration is defined as decay, spoilage, loss of utility of the product as defined by Shah and Shukla(2009) . In the present paper demand rate is considered as constant to a fixed time and then it varies linearly with time. We have also considered different selling prices in two time periods since the demand decrease with time.. It is assumed that lead time is zero and shortages are not allowed. The objective of the model is to find the on-hand inventory by maximizing the profit function.

## FUNDAMENTAL ASSUMPTIONS AND NOTATIONS

Following assumptions are made for the proposed model:

Following notations are made for the given model:

Following notations are made for the given model:

I(t) = On hand inventory at time t

R(t) = Demand

θ=The constant deterioration rate where 0 θ 1

I○=Inventory at time t=0

s_{1}= Selling price per unit in (0,t1)

s_{2}=Selling price per unit in(t1,T)

c=Unit cost of the item per unit time

Q=On-hand inventory

H=Holding cost per unit item per unit time

r=Replenishment cost per replenishment which isa constant

T=Duration of a cycle

R_{0}=Initial demand rate

a=Rate of change of demand with respect to t.

## Description of the model

In this model we consider the rate of demand R(t) to be a constant up to a certain time t= and after which it varies linearly with time. If I(t) be the on hand inventory at time t 0, then at time t+Δt, the on-hand inventory will be

I(t+Δt) =I(t)-R(t) Δt – θ. I(t). Δt

Dividing by Δt on both sides,

taking lim on both sides as Δt

where H(t- ) = 1 for t

Now equation (1) becomes,

Take integral with respect to t in equation (3)

When I = I and t = 0,

Now (*) becomes

Again using

## Conclusion

The above model deals with an inventory model of variable demand rate , that is, demand rate is constant up to a time t after which the demand rate varies linearly with time. The paper also deals two different selling prices for two different time periods . During the period (0.t), the demand rate is maintained at a constant level but after that period the amount of inventory decreases continuously with time but the effect of deterioration is maintained throughout the cycle. Hence the inventory level decreases due to the combined effect of demand as well as deterioration. The model is solved by maximizing the profit function and on hand inventory is also found. The above model can also be studied under shortages, backlogging and backordering.

# References

- Abad, P.L.(1996). Optimal pricing and lot-sizing under conditions of perish ability and partial backordering . Managementscience. 42:1093-1104
- Buzacott, J.A.(1975). Economic order quantities with inflation. Operational Research.Quarterly.26:553-558
- Dave,U.; Patel, U.K.(1981). (t,Si) policy inventory model for deteriorating items with time proportional demand. J. of Opl.Res.Soc.32: 137- 142
- Deb,M.; Chaudhuri, K.(1987). A note on the heuristic for replenishment of trended inventories considering shortages. J.opl.Res.soc. 38: 459-463
- Donaldson, W.A. (1977). Inventory replenishment policy for a linear trend in demand an analytical solution. Opl. Res. Qly.28:663-670
- Dutta,T.K.; Pal,A.K.(1990). Deterministic inventory systems for deteriorating items with inventory level dependent demand rate and shortages. Opsearch. 27(4):213-224
- Dutta, T.K.; Pal, A.K.(1992). Replenishment policy for deteriorating items with Timeproportional Demand. An extension of Silver’s Heuristic. AMSE periodicals. Advances in Modelling and simulation(France).3(3): 41-50
- Gayen, M.; Pal, A.K.(2009). A two warehouse inventory model for deteriorating items with stock dependent demand rate and holding cost. Operational Research.9(2):153-165
- Ghare, P.M.; Schrader, G.P. (1963). A model for decaying inventories. J.of Industrial Engineering. 14: 238-243
- Goyal, S.K. (1986). A heuristic for replenishment of trend inventories considering shortages. J.Opl.Res.Soc. 39:885-887
- Mirzazadeh, A.(2010), Effects of variable inflationary conditions on an inventory model with inflation proportional demand rate. J. of Applied sciences 10(7):551-557.
- Monks, J.G. (1987). Operation Management, 3rd Edn.Theory and problems.McGraw-Hill book co. New York Moon, I.
- Paul,K.; Dutta, T.K.; Choudhari, K.S. and Pal, A.K.(1996), An inventory model with two component demand rate and shortages. J. of the operational research socity.47: 1029-1036
- Resh, M.; Friedman; Barbosa, L.C. (1976). On a general solution of the deterministic lot size problem with time proportional demand. Operation Research. 24(4): 718-7218
- Roy, A. (2008). An inventory model for deteriorating items with price dependent demand and time varying holding cost. Advanced modeling and optimization.10(1):25-37
- Roy, T.; Chaudhuri, K.S. (2009). A Production - inventory model under stock -dependent demand, Weibull distribution deterioration and shortage. Int. Federation of Operetional Res. Soc. 16(3):325-346.
- Sabahno, H. (2009). Optimal policy for a deteriorating inventory model with finite replenishment rate and with price dependent demand rate and cycle length dependent price. Int. J. of Business,Economics, Finance and Management Sciences. 1(3):189-193
- Shah, N.H.; Shukla, K.T. (2009). Deteriorating inventory model for waiting time partial backlogging, Applied Mathematical Sciences.3(9): 421-428

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