The aim of the paper is to develop an Economic order quantity Model (EOQ model) for a single item inventory having an exponential demand. A brief review of the literature dealing with time varying demands is made in the following paragraphs. In formulating inventory model, two factors of the problem have been growing interest to the researchers, one being the deterioration of the items and other being the variation in the demand rate with time. Silver and Meal(1969) developed an approximate solution procedure for the general case of the deterministic, time varying demand pattern. The classical no shortage inventory problem for a linear trend in demand over a finite time horizon was analytically solved by Donaldson (1977). However, Donaldson’s solution procedure was computationally complicated. Silver (1979) derived a heuristic for a special case of a positive, linear trend in demand and applied it to the problem of Donaldson (1977). Ritchie (1980, 1984 and 1985) obtained an exact solution, having the simplicity of the EOQ formula, for Donaldson’s problem for a linear increasing demand. Mitra et al (1984) presented a simple procedure for adjusting the economic order quantity model for the case of increasing or decreasing linear trend in demand. The possibilities of shortage and deterioration in inventory were left out of consideration in all these models. Dave and Paul (1981) developed an inventory model for deteriorating items with time proportional demand. This model was extended by Sachan(1984) to cover the backlogging option. Bahari- Kaushari (1989) discussed a heuristic model for obtaining order quantities when demand is time proportional and inventory deteriorates at a constant rate over time. Deb and Chaudhuri (1987) studied the inventory replenishment policy for items having a deterministic demand pattern with linear (positive) trend and shortages; they developed a heuristic to determine the decision role for soliciting the items and sizes of replenishment over a finite time- horizon so as to keep the total costs minimum. This work was extended my Murdeshwar(1988), Subsequent contribution in this direction came from researchers. Like Goyal (1986, 1988), Dave (1989), Hariga (1994), Goswami and Chaudhari(1991), XU and Wang (1991), Chung and Ting(1993), Kim(1995), Hariga(1995, 1996), Jalan, Giri and Chaudhuri(1996), Jalan and Chaudhuri(1999), Lin, Ton and Lee (2000) etc. These investigations were followed by several researchers like Shah and Jaiswal (1977), Aggarwal (1979), Roy – Chowhuri and Chaudhuri (1983). The Model Developed by Covert and Philip (1973), Philip (1974) and Mishra (1975) did not allowed shortages in inventory and used a constant demand rate.

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Recently Ghosh and Chaudhuri (2004) have obtained an inventory model for a deteriorating item with Weibull Distribution deterioration, time quadratic demand and shortages. In the paper, we assume that time dependence of the demand rate is exponential. Deterioration rate is assumed to be a constant and shortages in the inventory are allowed. An analytical solution of the model is discussed and it is illustrated with the help of numerical example. Sensitivity of the optimal solution with respect to changes in different parameter values is also examined.

The following notations are used in the model. C1- inventory carrying cost per unit per unit time. C2- Shortage cost per unit time C3- ordering cost per order C4- cost of a unit. I0 – Size of initial inventory K – a constant value (0

I0* - Optimal value of I0 t1* – optimal value of t1

K* - Optimal value of K

The following assumptions are used for this model; (i) The Demand rate R(t) varies exponentially with time . i.e R(t)=λ0eαt, λ0, α are constants (ii) Shortage in the inventory are allowed and completely backlogged. (iii) The supply is instantaneous and the lead time is zero. (iv) A deteriorateditem is not repaired or replaced during a given cycle. (v) The holding cost, ordering cost, shortage cost and unit cost remain constant over time. any time t is governed by differential equations With I(0) = I0 and I(t1) =0 and with I(t1)= 0 The Solutions of Eq. (4.1)and Eq.(4.2 )are respectively given by and also Since the length of the shortage interval is a part of the cycle time, we may assume Where K is a constant to be determined in an optimal manner. Now we can write The Inventory level at the beginning of the cycle must be sufficient enough to meet the total demand given by The total quantity of deteriorated item is given by

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Average Shortage Cost The average shortage cost in [t1, T] is given by

The average ordering cost is given by

Cost of deteriorating item per unit time is given by Average variable cost (AVC) per unit time is Now Treating K and T as decision variables, the necessary conditions for the minimization of the average system cost are and After a little calculation, becomes i.e Now (15) yields After a little calculation, becomes

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i.e Now (16) yields We can easily calculate The Optimal value of T* of T and K* of K are obtained by solving (4.18) and (4.20). The sufficient conditions that these values minimize AVC (T,.K) are and Equations (4.18) and (4.20) can only be solved with the help of a computer oriented numerical technique for a given set of parameter values. Once T* and K*

C1=.001, C2=5.0, C3=10.0, C4=4.0, α=0.53,

=0.126

θ=0.05 in appropriate units. Equations (4.18) and (4.20) are now solved using above data. Based on these input data, the computer outputs are as follows: Computer oriented technique provides and at K=K* and T=T* obviously (4.24) and (4.25) are satisfied. The value AVC* is minimal.

C1=.001, C2=6.0, C3=15.0, C4=4.0, α=0.62,

=0.132

θ=0.07 in appropriate units. Based on these input data, the computer outputs are as follows: I0* = .7395 AVC* = 91.9197 in appropriate units. Corresponding to K= K* and T=T*, we have and

The values are satisfied (4.24) and (4.25).

We now study the effects of changes in the values of the system parameters C1, C2, C3, C4, α, θ on the optimal cycle time, stock period, EOQ and Average variable cost derived by the proposed method. The sensitivity analysis is performed by changing each of the parameters by -50%, -20%, +20% and +50% taking one parameter at a time and keeping the remaining six parameters unchanged.

A careful study of above table1 reveals the following: (i) K*, T*, t1*, I0* and AVC* increase (decrease) with increase (decrease) in the value of parameters C1. However K*, T*, t1*, I0* and AVC* are slightly sensitive to changes in C1. (ii) T*, t1*and I0* increase (decrease) with the increase (decrease) in the value of parameters C2, whereas K* and AVC* increase (decrease) with decrease (increase) in the value of C2. However T*, t1*and I0* are moderately sensitive and K*, AVC* are slightly sensitive to changes in C2. (iii) T*, t1*, I0* and AVC* increase (decrease) with the decrease (increase) in the value of parameter C3. Whereas K* increase (decrease) with the increase (decrease) in the value of C3. However T*, t1*, I0* and AVC* are moderately sensitive and K* is slightly sensitive towards changes in C3. (iv) K*, t1*, I0* and AVC* increase (decrease) with the decrease (increase) in the value of parameter C4 whereas T* decrease (increase) with the increase (decrease) in the value of C4. However AVC* is moderately

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(v) K* and I0* increase (decrease) with the increase (decrease) in the value of parameter α. whereas T*, t1* and AVC* decrease (increase) with the increase (decrease) in the value of α. However I0* is moderately sensitive and K*, T*, t1* and AVC* are slightly sensitive towards the changes in α. (vi) T*, t1* and AVC* increase (decrease) with increase (decrease) in the value of parameter λ0 whereas K* and I0* decrease (increase) with increase (decrease) in the value of λ0. However T*, t1*, I0* and AVC* are moderately sensitive and K* is almost insensitive towards changes in λ0. (vii) K*, T*, t1*, I0* increase (decrease) with increase (decrease) in the value of parameter θ whereas AVC* increase (decrease) with the decrease (increase) in the value θ. However K*, T*, t1*, I0* and AVC* are slightly sensitive towards changes in θ.

The chapter is concerned with development of an inventory model for a deteriorating item with constant deteriorating rate having an instantaneous supply an exponential demand. Shortages are completely backlogged. The analytical approach is considered to obtain the optimal solution to minimise the average variable cost per time unit of the inventory system. It is illustrated with the help of numerical examples. Computer Oriented techniques is applied to solve the numerical problem. The effect of changes in the values of different parameters on the decision variable is studied.

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