An EOQ Model with Temporary Stock Dependent Demand with Random Lag

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1 International Journal of Computer Science & Communication Vol. 1, No., July-December 010, pp An EOQ Model with Temporary Stock Dependent Demand with Random Lag P.S.R.K.Prasad 1 & K.V.S.Sarma 1 Narayana Engineering College, Nellore Department of Statistics, SV University, Tirupati, India 1 psrkpr@gmail.com, sarma_kvs@rediffmail.com ABSTRACT This paper presents a mathematical model to determine the Economic Order Quantity for a single item inventory situation in which the demand rate depends on the lot size received by the stockiest. The influence of displayed stock on the customers purchase habits is characterized in the model by introducing a random time lag between the display of stock and the realization of customers response. The demand for some items would be at a constant rate initially and the effect of Stock Dependent Demand (SDD starts after the lag. The optimum lot size that minimizes the sum of relevant costs is determined by assuming different probability distributions for the time lag. The working of the model and the sensitivity with respect to input parameters is numerically illustrated. Keywords: Stock Dependent Demand, Economic Order Quantity(EOQ, Uniform Distribution, Truncated Exponential Distribution 1. THE PROBLEM ENVIRONMENT Inventory models with Stock Dependent Demand (SDD have attracted the attention of researchers in the recent years. Following the basic version of the models of this type by Rakesh Gupta and Prem Vrat (1986 a large number of mathematical formulations were contributed by various researchers including Datta and Pal (1990, Gerchak and Wang (1994, Urban (1995, Teng J.T and Chang C.T (003, Zaid T. Balkhi, Lakdere Benkherouf (004, Jinn - Tsair Teng, Chun-Tao Chang (005, Mandal, Roy and Maiti (006, Datta T.K. and Paul K (001 and A.K. Pal, A.K. Bhunia & R.N. Mukherjee, (006. In all these models it is assumed that the SDD will be effective throughout the period which may not be true. In some situations customers react slowly to the advertisement effect of displayed stock. Let t 0 denote the length of the inventory cycle and Q be the displayed stock. The demand initially occurs at the known constant rate but after a lag of u (< t 0 the effect of SDD takes place and the demand rate increases from its current level to a higher level. The parameter u is similar to the concept of lead-time in general inventory models but represents the time-to-respond by the customers instead of the supplier. In a way the SDD effect is temporary and not for the full cycle. In an earlier work Prasad (1997 and G.S.N. Reddy (1998 have studied problems in which the SDD effect occurs at the beginning of the period until time u, after which the normal demand holds good which they called it temporary SDD. In this paper we develop a model to determine the Economic Order Quantity when the SDD takes place after a time lag. We also compare the results with the two cases where SDD occurs before and after the lag. In real world situations the time lag after which the SDD effect starts may be unknown. Some customers may respond quickly to the arrival of stock, while some would postpone their visits or in some extreme cases, people may rush in a festive mood to buy the goods. The stockiest may guess that the SDD comes into effect after one week, 10 days etc., or after a lag of % of the cycle normal cycle length. Thus the lag behaves as a random variable following some probability distribution. Let us consider that the lag U is a continuous random variable with probability density function f (u where u is the realization of U. Earlier Prasad RVS (1996 has reported some results assuming that the SDD effect takes place for u time units at the beginning of the period. The focus in this paper is on the effect of SDD getting realized after a positive lag.. MATHEMATICAL MODEL It is assumed that that rate of replenishment is infinite, lead time is zero, shortages are not permitted and that the SDD effect starts after a random lag of U with probability density function f(u, 0 < u U max. Further E(U = µ and V (U = σ are assumed to be known. The ordering cost is A per order, holding cost is H per unit per unit time, the selling price per unit is P and the unit cost of an item is C and T denotes the cycle length a random variable that depends on U. A lot of size Q units is received at time t = 0 in response to an order placed earlier. The on-hand stock level is Q and the availability

2 44 International Journal of Computer Science & Communication (IJCSC of this stock is made known to customers by suitable means like advertisements. It can be seen that u takes values between 0 and Q*/a when Q* is the lot size. We consider a linear form to describe the SDD given by the function D(Q = where β is a constant indicating the SDD factor. It should be noted that U takes values between 0 and Q/α where Q is the lot size. When U max = Q/α it means there is no effect SDD and when U max = 0 the entire cycle runs with SDD. If u denotes a realization of U, then the demand function is D(Q = α for t u and = for u < t T. The stock on hand will be {Q-αµ at u and this stock will be consumed by time Q αµ α + β Q. The expected cycle length will therefore be E(T = E(u + E(Q αe(u/ = Q(1 Let A 1 and A denote inventory held during the periods (0, u and (u, T. Then the expected holding cost during (0, u is given by α ( µ + σ E(A 1 = H Qµ The expected holding cost during (u, T will be HE( A = ( Q αu H E ( Q = H ( + α µ + σ αqu ( Adding the above two components the expected holding cost for the cycle becomes ( ( ( E A1 + A = E A1 + E A and this reduces to α βq Q ( 1 Q ( E( A1 A H αβ µ + σ + = ( + Now the expected holding cost per unit will be α βq(1 E( A1 + A Q ( 1 αqβ ( µ + σ = H E( T ( + Q H Q = ( ( 1 + βµ αβ Q( µ + σ (1 ( Q (1 ( (3 (4 The ordering cost is A per order so that the expected ordering cost per unit time will be E(A/T = A Q (1 The normal cycle length without SDD would be Q/α but it is reduced to E(T given in (1. The difference {Q αe(t} represents the accelerated sales realized during the cycle due to SDD and fetches a gain of (P C {Q αe(t}. Substituting the expression for E(T and simplifying gives the gain due to SDD as ( P C β ( Q αqu G = (5 The expected gain due to SDD during the cycle will ( P C β ( Q α Qµ be E(G = per unit time becomes E(G/T = ( P C β( Q α Qµ Q(1 so that the expected gain Hence the total cost of inventory is will be ( ( ( + H Q 1 + βµ αβ µ + σ (1 ( Q C(Q = A Q (1 which can be written as (6 Aα HQβµ HQ C(Q = + + Q(1 1 ( (1 Terms independent of Q 1 Q Q(1 (1 Terms independent of Q = { Aα} + ( Putting Ĥ = { H ( 1 ( P C } can be written as Q. Aα ˆ C(Q = Q (1 + HQ (1 ( P C β( Q α Qµ Q(1 ( P C( βq (1 { H 1 ( β P C } β the cost function Terms independent of This is the familiar cost function for the Wilson s Aα formula, from which the EOQ can be obtained as Hˆ which means Aα Q* = { H{1 + βµ } ( β P C }. (7

3 An EOQ Model with Temporary Stock Dependent Demand with Random Lag 45 Here are some special cases of interest. When β = 0 the above expression for Q will be reduced to the simple EOQ formula Q* = A α H which is the classical EOQ formula. Interestingly the randomness in the lag U is replaced by E(U = µ and the variance of the distribution of U is not a component in Q*. However V(U will be a significant component in the calculation of the optimum cost. In the following section a particular cases of uniformly distributed U is examined. 3. THE CASE WHEN U FOLLOWS UNIFORM DISTRIBUTION Suppose that the delay in SDD is a continuous random variable, which follows uniform distribution U(0, U max where U max is the upper limit of the distribution of U. The seller usually calculates the maximum cycle length as Q/α as if there was no SDD. We consider three possible levels for U max namely Q/4α, Q/α and 3Q/4α and see their effect on the EOQ. When U Max = Q/4α it means that the SDD would take place after 1/4 th of the normal cycle length is over and we call this event as Early SDD. Similarly we adopt the notation of Half-way and Delayed SDD for the cases when U Max = Q/α and 3Q/4α respectively. a U-follows Uniform with U max = Q/4α (Early in the cycle: In this case the time lag between the arrival of stock and the SDD to take place is allowed to be anything between 0 and 0.5Q/α, which means that in the worst case the SDD effect takes place form the quarter of the normal cycle. Then we get µ = Q/8α and σ = Q/19α. The equation for Q in this case becomes Q H P C A 4α { ( ( } This is a cubic equation in Q, which can be solved by a numerical method. b U-follows Uniform with U max = Q/α (Half-way through the Cycle: In this case the time lag between the arrival of stock and the SDD to take place is allowed to be anything between 0 and Q/α, which means that in the worst case the SDD effect takes place from the middle of the normal cycle. Then µ = Q/4α an σ = Q/48α. Q H P C A α { ( ( } = 0 which can be solved by a numerical method. c U-follows Uniform with U max = 3Q/4α (Delayed within Cycle: In this case the time lag between the arrival of stock and the SDD to take place is allowed to be anything between 0 and 3Q/4α, which means that in the worst case the SDD effect, takes place from the start of the fourth quarter of the normal cycle. Then we get µ = 3 Q 8α and σ = 9Q 19 α. 3 Q H P C A 4α { ( ( } = 0 This can be solved by a numerical method. It can be seen that when the range of uniform distribution is reduced, the coefficient of Q 3 gets reduced. The following observations can be made on the effect of lag and its distribution on the EOQ. 1. The numerical value in the coefficient of Q 3 is exactly the same as the upper limit of the selected rectangular distribution.. The coefficient of Q is the net holding cost after deducting the potential gain due to SDD. The quantity deducted is proportional to β(p C. 4. NUMERICAL RESULTS WITH UNIFORMLY DISTRIBUTED U Let us consider a situation with A = 100, α = 50, H = 0., C = 5, P = 7. For different values of U max and fixed β the values of EOQ has been Calculated and shown in the following Table. It can be seen from the this simulated trials, that for a given β a higher EOQ is recommended when the lag is only for the first quarter of the cycle and when the lag is allowed to vary from 0 to full cycle length Q/α, there is uncertainty about the lag. It is also possible that the lag may go up to Q/α, which means no SDD. Hence a lower EOQ is obtained. Values of EOQ for various combinations of β and U max Max Delay β = 0.01 β = 0.0 β = 0.03 β = 0.04 Early(0, Q/4α Half-way(0, Q/α Delayed(0, 3Q/4α It can be seen from the above table that for any fixed level of U max, the EOQ is found to increase when β is increased. This is an expected result since in the square root type formula for EOQ an increase in β is known to increase the order quantity. The sensitivity is shown in Figure-4.

4 46 International Journal of Computer Science & Communication (IJCSC Let us now consider the case when U follows Truncated Exponential with U max = Q/α and γ = 4α/Q, It means that In this case we find that E(u = 0.313{Q/α} and V(u = {Q/α} Fig. 4: Behavior of EOQ with and u It is seen from the above analysis that at any estimated level of delay in SDD the model suggests a higher EOQ if is high. Further at any value of the EOQ is found to decrease when the value of U max goes up. This is also an expected result because as the delay increases, the operational duration of SDD becomes small and hence a lower EOQ is recommended. In the following section we use truncated exponential distribution to describe the behavior of U and obtain the EOQ. 5. THE CASE WHEN U FOLLOWS TRUNCATED EXPONENTIAL DISTRIBUTION Suppose U follows exponential with parameter taking values between 0 and U max where Umax is the finite upper limit of U. Then we have to use a Truncated Exponential distribution with density function g( γ, u f(γ, u = P( U U and γ 0. max where g( γ, u = u γe γ for u 0 Hence the truncated density is given by γu γ e f(u = U 1 e γ ; 0 u U max max = 0; Otherwise Suppose the average delay is 0.5 times of the cycle length Q/α. The parameter γ can be taken as 4α/Q which is reciprocal of the average. With parameter γ the truncated Exponential will have Mean and Variance given by E (u = µ = Q 1 γ γq Q e α + e α α γ γ and variance of u is V(u = {E(u} = σ = µ βhq Q = + Q { H β( P C } Aα = α This is a cubic equation in Q, which can be solved by a numerical method. Another case is discussed below. 6. NUMERICAL RESULTS IN THE TRUNCATED EXPONENTIAL CASE Let us consider a situation with A =100, α = 50, H = 0., C = 5, P = 7 and When the lag U follows Truncated Exponential Distribution with U max = Q/α and γ = 4α/Q i.e. early SDD effect the EOQ and optimal inventory costs for Various values of β are tabulated in the following table. β EOQ As in the case of Uniform distribution here also it is observed that the higher delay in SDD (for fixed β value leads to lower EOQ. Further for a fixed level of delay in SDD, the EOQ is found to increase when β increases. Figure 5: Behavior of EOQ With Constant = 0.01 in UD and TED Comparing the EOQ values in the case of Uniform and Truncated exponential cases, it is observed that the EOQ in case of Uniform distribution is higher than that in the truncated exponential case. This is possible because in the Uniform case any value of U occurs with equal chance while in the in truncated exponential case the shape of the distribution determines the

5 An EOQ Model with Temporary Stock Dependent Demand with Random Lag 47 probability; smaller values of U have higher chance of occurrence than larger values. This is shown in the Figure-5 where a specific case when β = 0.01 is considered. 7. OBSERVATIONS In this paper we have studied the effect of delayed SDD on the EOQ assuming that the delay is uncertain. 1. It is observed that a random amount of waiting time (delay in effecting the SDD will alter the EOQ values.. The uncertainty in the delay could be characterized by suitable probability distributions like Uniform or Truncated Exponential. 3. We have compared the effect of uniform and truncated exponential distributions and, it is observed that the EOQ in case of Uniform distribution is higher than that in the truncated exponential case. This is possible because in the Uniform case any value of U occurs with equal chance while in the in truncated exponential case the shape of the distribution determines the probability; smaller values of U have higher chance of occurrence than larger values. REFERENCES [1] A.K. Pal, A.K. Bhunia and R.N. Mukherjee, (006, Optimal Lot Size Model for Deteriorating Items with Demand Rate Dependent on Displayed Stock Level (DSL and Partial Backordering, European Journal of Operational Research, 175(, [] Chang CT.(00, Inventory Model with Stockdependent Demand and Nonlinear Holding Costs for Deteriorating Items, Asia-Pacific Journal of Operational Research. [3] Datta T.K. and Paul K (001, An Inventory System with Stock-dependent, Price-sensitive Demand Rate, Production Planning and Control, 1 (1, [4] Datta, T.K.and Pal, A.K., (1990b A Note on an Inventory Model with Inventory-level-dependent Demand Rate, J.OPL.RES, 41, [5] Giri BC, Chaudhuri KS (1998, Deterministic Models of Perishable Inventory with Stock-dependent Demand Rate and Nonlinear Holding Cost, European Journal of Operational Research. [6] Giri BC, Pal S, Goswami A, Chaudhuri KS (1996, An Inventory Model for Deteriorating Items with Stockdependent Demand Rate, European Journal of Operational Research. [7] Jinn-Tsair Teng, Chun-Tao Chang (005, Economic Production Quantity Models for Deteriorating Items with Price and Stock-dependent Demand, Computers and Operations Research, 3, Issue, (February 005 Pages: [8] Mandal B.N., and Phaujdar, S. (1989a, A Note on an Inventory Model with Stock-dependent Consumption Rate, Opsearch, 6(1. [9] Mandal B.N., and Phaujdar, S. (l989b, An Inventory Model for Deteriorating Items and Stock-dependent Consumption Rate, J. OPL. RES. SOC. [10] Padmanabhan G, Vrat P (1995, EOQ Models for Perishable Items Under Stock Dependent Selling Rate, European Journal of Operational Research. [11] Pal S, Goswami A, Chaudhuri KS.(1993, A Deterministic Inventory Model for Deteriorating Items with Stock-dependent Rate, International Journal of Production Economics. [1] Rakesh Gupta and Prem Vrat (1986, Inventory Model for Stock Dependent Consumption Rate, OPSEARCH. [13] Ray J, Goswami A, Chaudhuri KS (1998, On an Inventory Model with Two Levels of Storage and Stock- Dependent Demand Rate, International Journal of Systems Science. [14] Reddy G.S.N. and K.V.S. Sarma (1998, A Periodic Review Inventory Problem with Variable Stock Dependent Demand, Opsearch, 38(3, [15] Teng J.T. and Chang C.T. (003, Economic Production Quantity Models for Deteriorating Items with Price and Stock Dependent Demand Models. [16] Urban T.L. (1995, Inventory Models with the Demand Rate-dependent on Stock on Shortage Levels, Int. J. Prod Econ.