An EOQ model for perishable products with discounted selling price and stock dependent demand

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1 CEJOR DOI /s z ORIGINAL PAPER An EOQ model for perishale products with discounted selling price and stock dependent demand S. Panda S. Saha M. Basu Springer-Verlag 2008 Astract A single item economic order quantity model is considered in which the demand is stock dependent. After a certain time the product starts to deteriorate and due to visualization effect and other aspects of deterioration the demand ecomes constant. In that situation a discount on selling price provides significant increment in demand rate. In this paper we investigate how much discount on selling price may e given during deterioration to maximize the profit per unit time and whether a pre-deterioration discount affects the unit profit or not. A mathematical model is developed incorporating oth pre- and post deterioration discounts on unit selling price, where analytical results reveal some important characteristics of discount structure. A numerical example is presented and sensitivity analysis of the model is carried out. Keywords Inventory Stock dependent demand Discounted selling price Deterioration List of symols C 0 S r 1 r 2 h set up cost constant selling price of the product per unit discount offer per unit efore deterioration discount offer per unit after deterioration holding cost per unit per unit time S. Panda B Department of Mathematics, Bengal Institute of Technology, 1.no. Govt. Colony, Kolkata , West Bengal, India shiaji_panda@yahoo.com S. Saha M. Basu Department of Mathematics, University of Kalyani, Kalyani 745, West Bengal, India

2 S. Panda et al. d c Q 1 Q 2 Q 3 Q 4 Q 41 Q 5 Q 51 T 1 T 2 T 3 T 4 T 41 T 5 T 51 t 1 disposal cost per unit per unit purchase cost of the product order level for pre- and post deterioration discount on selling price order level for only post deterioration discount on selling price order level for no discount on selling price order level for post deterioration discount on selling price with instant deterioration order level for no discount on selling price with instant deterioration order level for pre-deterioration discount on selling price for fixed life time product order level for no discount on selling price for fixed life time product cycle length for pre- and post deterioration discount on selling price cycle length for only post deterioration discount on selling price cycle length for no discount on selling price cycle length for post deterioration discount on selling price with instant deterioration cycle length for no discount on selling price with instant deterioration cycle length for post deterioration discount on selling price for fixed life time product cycle length for post deterioration discount on selling price for fixed life time product a decision variale representing the time from which pre-deterioration discount starts 1 Introduction Classical economic order quantity EOQ model was developed y considering static demand rate. But demand of physical goods in reality may e stock dependent, time dependent, price dependent or their cominations. According to Levin et al at times, the presence of inventory has a motivational effect on people around it. It is a common elief that large piles of goods displayed in a departmental store leads the customers to uy more. It is also investigated y Silver and Peterson 1985 that sales at retail level tend to e proportional to inventory displayed. To quantify this Baker and Uran 1988 proposed an EOQ model with power-form inventory level dependent demand. In the last two decades the variaility of inventory level dependent demand rate on the analysis of inventory system was descried y researchers like Pal et al. 1993, Phelps 1980, Mondal and Phaujdar 1989, Goswami and Choudhuri 1995, Ritchie and Tsado 1985, Silver 1979, Silver and Meal 1979 and others. They had descried the demand rate as the power function of on hand inventory. Datta and Pal 1990 investigated an inventory system with two components demand rate where the consumption of an item is dependent on the instantaneous inventory level until a given inventory level is reached after which the demand is constant and it reaches zero level at the end of the cycle. Later Uran 1992 modified the model y relaxing the zero ending inventory. There is a vast literature on stock dependent inventory and it s

3 An EOQ model for perishale products outline can e found in the review article y Uran 2005 where he unified two types of inventory level dependent demand y considering a periodic review model. Product perishaility is an important aspect of inventory control. Deterioration, in general, may e considered as the result of various effects on stock, some of which are damage, spoilage, osoletes, decay, decreasing usefulness and many more. While kept in store fruits, vegetales, foodstuffs, etc. suffer form depletion y decent spoilage. Through a gradual loss of potential or utility with the passage of time, electronic goods, grain, radio active sustances deteriorate. Gasoline, alcohol etc undergo physical depletion over time through the process of evaporation. Ghare and Schrader 1963 were the first proponent of deterioration in inventory literature. They developed EOQ model for items with exponential decay and deterministic demand. Liu and Shi 1999 classified perishaility and deteriorating inventory models into two major categories, namely decay models and finite lifetime models. Finite lifetime models assume a limited lifetime for each item. Blood cells, cans of fruit, foodstuffs, cosmetics, drugs, etc are examples of the items having fixed lifetimes. Decaying products are of two types. Products which deteriorate from the very eginning and the products which start to deteriorate after a certain time. Lot of articles are availale in inventory literature considering deterioration. Interested readers may consult the survey papers of Nahmias 1982, Raafat 1991 and Goyal and Giri Every organization dealing with inventory faces a numer of fundamental prolems. Pricing decision is one of them. It has to decide how much to ask for each units and when to drop the price as the season rolls on. It always wants the price to e marginal, not so high that it put off potential uyers and not so low that it losses out on potential profits. Therefore, price can e considered as an important tool to influence demand. This led many researchers to investigate pricing strategy on inventory models in details Arcelus and Srinivasan 1998; Shah and Shah 1993; Wee and Law Khouja 2000 studied an inventory prolem under the condition that multiple discounts can e used to sell excess inventory. Dave et al investigated a deterministic production lot-size model in which demand is a convex function of price and time. The nature of independent effects of temporary price discounts on inventory policies is well documented in literature Weatherford and Bodily 1992; Petruzzi and Dada 1999 considering various assumptions. Neff 2000 mentioned that discount is considered as a way of oosting sales. The resulting increase in demand is expected to generate higher revenues and to accelerate inventory depletion rates with corresponding decrease in holding cost. Examples of this type of ehavior is common among supermarkets, fashion apparel industry, electronic industry, high-tech products, periodicals, etc. More over question of accelerating the inventory depletion rate is relevant for any price-sensitive item, ut specially so when perishaility is a concern ecause of the relatively short span of life for gradual deterioration. However, most of the studies except few, do not attempt to unify the two research streams: temporary price reductions and deterioration. As a result, the effect of discount for deteriorating items on revenue has een ignored and the models deal with cost minimization rather than with profit maximization ojective function. Not only that to the est of author s knowledge, no one has tried to introduce a temporary discount on selling price efore the start of deterioration as well as a discount on selling price as the deterioration starts to enhance the demand in order to oost the inventory depletion rate. This paper represents the

4 S. Panda et al. issue in details. The rest of the paper is organized as follows. In Sect. 2 assumptions and notations are provided for the development of the model. The mathematical model is developed in Sect. 3. In Sect. 4 a numerical example is presented to illustrate the development of the model. Finally Sect. 5 deals with summary and some concluding remarks. 2 Assumptions and notations A replenishment cycle of an infinite time horizon EOQ type model is developed under the following notations and assumptions. 1. Replenishment rate is infinite. 2. The deterioration rate is assumed as = Ht τ, where t is the time measured from the instant arrivals of a fresh replenishment indicating that the deterioration of the items egins after a time τ from the instant of the arrival in stock. is a constant 0 <<<1 and Ht τ is the well known Heaviside s function defined as Ht τ = { 1, if t τ 0, otherwise 3. Demand depends on the on-hand inventory up to time τ from time of fresh replenishment, eyond which it is constant and defined as follows RI t = { a + It, if t <τ a, if τ t where a > 0 is the initial demand rate independent of stock level and condition of inventory. > 0 is the stock sensitive demand parameter. It is the instantaneous inventory level at time t. 4. r 2 0 r 2 1 is the percentage discount offer on unit selling price during deterioration. α 2 = 1 r 2 n 2 n 2 ɛ R, the set of real numers, is the effect of discounted selling price on demand during deterioration. α 2 is determined from priori knowledge of the seller such that the demand rate is influenced with the reduction rate of selling price. It is ovious that when r 2 0, α 2 1, i.e., the demand of decreased quality items remains same. r 1 0 r 1 1 is the percentage pre-deterioration discount offer on unit selling price. α 1 = 1 r 1 n 1, n 1 ɛ R is the effect of pre-deterioration discount on demand. If r 1 0, i.e., for no pre-deterioration discount the demand is assumed a + It. It is a common phenomenon that demand rate of fresh goods is oosted for discount on selling price and demand of decreased quality items increases significantly for a discount offer on selling price. Since the demand rate is enhanced for the discounts offer, therefore, it is partially stock dependent and partially stock and selling price dependent if discounts offer are given. Otherwise, it is two-component stock dependent.

5 An EOQ model for perishale products 3 Model formulation 3.1 Formulation of the asic model with pre- and post deterioration discounts on selling price At the eginning of the replenishment cycle the inventory level raises to Q 1.As time progresses it decreases due to instantaneous stock dependent demand up to the time τ.afterτ deterioration starts and the inventory level decreases for deterioration and constant demand. Ultimately inventory reaches zero level at T 1. We assume that efore the start of deterioration from time t 1, r 1 % discount on unit selling price of the product is given in order to oost the demand of fresh items. Clearly, this discount is continued for the period of time τ t 1. As deterioration starts from τ, r 2 % discount on unit selling price is provided to enhance the demand of decreased quality items. This discount is continued for the rest of the replenishment cycle. Then, the ehavior of inventory level is governed y the following system of linear differential equations dit dt = a It, 0 t t 1 1 = α 1 a + It, t 1 t τ 2 = α 2 a I t, τ t T 1 3 with the initial and oundary conditions I 0 = Q 1, I T 1 = 0. Solving the equations we get I t = a [ e t 1 + Q 1 e t, 0 t t 1 4 = a [ e α 1t 1 t t Q 1 e α 1t 1 t t 1, t 1 t τ 5 = aα [ 2 e T1 t 1, τ t T 1 6 Now, at the point t = τ we have from Eqs. 4 and 5 Q 1 = [ aα2 et1 τ 1 + a e α 1τ t 1 +t 1 a 7 Holding cost and disposal cost of inventories in the cycle is HC + DC = h t 1 0 τ I tdt + h Purchase cost in the cycle is given y t 1 I tdt + h + d T 1 τ I tdt PC = cq 1

6 S. Panda et al. Total sales revenue in the order cycle can e found as SR = S t 1 0 a + Itdt + α 1 1 r 1 Thus total profit per unit time of the system is τ t 1 a + Itdt + α 2 1 r 2 π 1 r 1, r 2, t 1, T 1 = TP 1 T 1 = 1 T 1 [SR PC HC DC C 0 T 1 τ adt On integration and simplification of the relevant costs, the total profit per unit time ecomes, π 1 = 1 [Sat 1 + S1 r 2 α 2 at 1 τ T 1 + S h at 1 Q a 1 e t 1 at 1 τ +[S1 r 1 α 1 h + Q 1 + a e t 1 1 eα 1t 1 τ α 1 + S1 r 1 α 1 aτ t 1 h + d aα 2 e T1 τ 1 T 1 τ cq 1 C 0 Note that two discounts r 1 and r 2 are given on constant unit selling price S of the product. There may raise another case: the discount on unit selling price of the product from the start of deterioration may e given on the pre- deterioration discounted selling price 1 r 1 S. The pre-deterioration discount on selling price is to e given in such a way that the discounted selling price is not less that the unit cost of the product, i.e., S1 r 1 c > 0. Similarly, S1 r 2 c > 0. Applying these constraints on the unit total profit function we have the following maximization prolem maximize π 1 r 1, r 2, t 1, T 1 suject to, {r 1, r 2 } < 1 S c r 1, r 2, t 1, T Our ojective here is to determine the optimal values of r 1, r 2, t 1 and T 1 to maximize the unit profit function. It is very difficult to derive the results analytically. Thus some numerical methods must e applied to derive the optimal values of r 1, r 2, t 1 and T 1, hence the unit profit function. There are several methods to cope with constraint optimization prolem numerically. But here we use penalty function method De 2000 to derive the optimal values of the decision variales. 8

7 An EOQ model for perishale products 3.2 Some special cases Model with only post deterioration discount on unit selling price We now consider the asic model y relaxing the assumption of discounted selling price from t 1 efore deterioration. Only discount on selling price will e given as soon as the deterioration starts. In that case t 1 = τ, r 1 = 0. From Eq. 7 the initial inventory level is found as [ aα2 Q 2 = et2 τ 1 + a e τ a 10 Sustituting t 1 = τ and r 1 = 0, we have the unit profit function of the system from Eq. 8 as [ π 2 r 2, T 2 = TP 2 = 1 Saτ + S1 r 2 α 2 at 2 τ T 2 T 2 + S h aτ Q a 1 e τ h + d aα 2 e T2 τ 1 T 2 τ cq 2 C 0 There is only post deterioration discount on selling price. Therefore, we have the maximization prolem Model for no discount on unit selling price 11 maximize π 2 r 2, T 2 suject to, r 2 < 1 S c 12 r 2, T 2 0 In this case we consider that there is no pre-deterioration as well as no post deterioration discount on unit selling price. Sustituting t 1 = τ and r 1 = r 2 = 0wehavefrom Eq. 7 the initial inventory level as Q 3 = [ a et 3 τ 1 + a e τ a 13 And from Eq. 8 total profit per unit time ecomes, π 3 T 3 = 1 [Saτ + SaT 3 τ+ S h + d aτ Q T a 1 e τ h + d a e T3 τ 1 T 3 τ cq 3 C 0 14

8 S. Panda et al. Since, no discount is provided on the unit selling price of the product, no constraint will e imposed on 14. The only constraint is the nonnegativity restriction for T Model for instant deterioration If the product starts to deteriorate as soon as it is received in the stock, then there is no option to provide pre-deterioration discount. Only we may give post deterioration discount. In that case, τ = t 1 = 0 and r 1 = 0. Then form Eqs. 7 and 8 the order quantity and unit profit function for constant demand and post deterioration discount can e found respectively as Q 4 = aα 2 et π 4 r 2, T 4 = TP 4 = 1 [ S1 r 2 at 4 α 2 h + d aα 2 T 4 T 4 e T 4 1 T 4 cq 4 C 0 16 Thus, we have to determine r 2 and T 4 from the maximization prolem maximize π 4 r 2, T 4 suject to, r 2 < 1 S c 17 r 2, T 4 0 Order level and unit profit function for model with instant deterioration and constant demand with no discount are otained from 15 and 16 y sustituting r 2 = 0as Q 41 = a et From Eq. 8 total profit per unit time ecomes, π 41 T 41 = 1 [ SaT 41 h + d a T Model for fixed life time products e T 41 1 T 41 cq 41 C 0 19 If the product has a fixed self life then question of post deterioration discount will not arise. If T 5 e the cycle length then replacing T 1 y T 5 and for τ T 5, r 2 = 0, Eqs. 7 and 8 provide the order quantity and unit profit function respectively for fixed life time product with pre deterioration discount as follows Q 5 = a [ e α 1T 5 t 1 +t

9 An EOQ model for perishale products and π 5 r 1, t 1, T 5 = 1 T 5 [Sat 1 + S h Q 5 + a 1 e t 1 at 1 + S1 r 1 α 1 at 5 t 1 + [S1 r 1 α 1 h at 1 T 5 + Q 5 + a e t 1 1 eα 1t 1 T 5 cq 5 C 0 α 1 21 There is only pre-deterioration discount on selling price and cycle length must e less than life time of the product, therefore, we have the maximization prolem maximize π 5 r 1, t 1, T 5 suject to, r 1 < 1 c S T 5 τ r 1, T Note that this case may e viewed in another perspective instead of considering it for fixed life time products. The product starts to deteriorate after τ ut the replenishment cycle ends efore the start of deterioration. And only pre-deterioration discount is provided after t 1. Thus it also represents the scenario for time to deterioration products with the relationship 0 t 1 T 5 τ. Profit function and order level for finite life time product with no pre-deterioration discount can e found from Eqs. 21 and 20, y letting t 1 T 5 and r 1 0 and replacing T 5 and Q 5 y T 51 and Q 51, respectively as profit per unit time ecomes, Q 51 = a [ e T π 51 t 1, T 51 = 1 T 51 [ SaT 51 + S h at 51 + Q 51 + a 1 e T 51 cq 51 C 0 24 with the restriction T 51 τ. 3.3 Model analysis In last section we have derived all possile unit profit functions arise from the associativity of deterioration and discount on unit selling price. In this section, we verify the applicaility of the proposed discount structure. Let us consider the following theorem.

10 S. Panda et al. Theorem 1 For n 1 = n 2 = n, π 1 >π 2 if { r 1 < min 1 c S, S c } h eτ S Proof The values of π 1 and π 2 for fixed r 1 and r 2 are always less than those for optimal r 1 and r 2. Thus it is sufficient to show that π 1 >π 2 for fixed r 1 and r 2. Note that T 1 and T 2 are cycle lengths for the models with pre- and post deterioration discounts and only post deterioration discount on unit selling price. Since the pre-deterioration discount on selling price is additional, demand of fresh items must e enhanced and hence the inventory depletion rate must increase. Thus, it is ovious that T 2 is always greater than T 1.FromEqs.8 and 11, we have, π 1 π 2 = TP 1 T 1 TP 2 T 2 TP 1 TP 2 T 2 Therefore, it is sufficient to show that TP 1 TP 2 /T 2 > 0. Now, TP 1 TP 2 = 1 T 2 T 2 + [ aα 2 T 2 τ+ a S c h e τ + s1 r 2 S h aα 2 T 1 τ+ a S c h e α 1τ t 1 +t 1 +S1 r 2 + [S1 r 1 α 1 h eα 1τ t 1 1 α 1 e α 1τ t 1 S h + h + d aα 2 [T 2 τ 2 T 1 τ Thus TP 1 TP 2 /T 2 > 0if, 1 S c h eτ + S1 r 2 S h + S1 r 1 α 1 h eα 1τ t 1 1 α 1 [ S c h e α 1τ t 1 +t 1 + S1 r 2 e α 1τ t 1 S h > aα 2T 2 τ+ a aα 2 T 1 τ+ a 26 But T 2 > T 1, therefore, [aα 2 T 2 τ+ a /[aα 2T 1 τ+ a > 1. Thus from 26 we have 1 S c h eτ + S1 r 2 S h + S1 r 1 α 1 h eα 1τ t 1 1 α 1 [ S c h e α 1τ t 1 +t 1 + S1 r 2 e α 1τ t 1 S h > 1

11 An EOQ model for perishale products After simplification we have, S c h e τ > eα1τ t1 1 S h S1 r 1 α 1 h e α 1 1τ t 1 1 α 1 27 But e α 1τ t 1 > e α 1 1τ t 1, i.e., [e α 1τ t 1 1/e α 1 1τ t 1 1 > 1. Therefore, 27 yields, S c h e τ S h S1 r 1 α 1 h α 1 S c h r 1 < e τ S > 1 S c h e τ + h Sr 1 > h α 1 However, y assumption, r 1 < 1 c S. Therefore, π 1 >π 2 if { r 1 < min 1 c } S c h S, e τ S 28 Theorem 2 For n 1 = n 2 = n, π 2 >π 3 if r 2 < min 1 c Sn 1 + n S, S h Sn 1 e τ 1 ce τ Proof The value of π 2 for fixed r 2 is always less than optimal value of r 2. Thus it is sufficient to show that π 2 >π 3 for fixed r 2.HereT 2 is the cycle length when post deterioration discount is applied on unit selling price to enhance the demand of decreased quality items. For the enhancement of demand the inventory depletion rate will e higher and consequently the cycle time will reduce. T 3 is the cycle length when no discount is applied on selling price. Oviously T 3 is greater than T 2. Without loss of generality let oth the profit functions π 2 and π 3 are positive. Then π 2 π 3 = TP 2 T 2 TP 3 T 3 TP 2 TP 3 T 3 Therefore, it is sufficient to show that [TP 2 TP 3 /T 3 > 0. If it can e shown that [TP 2 TP 3 /T 3 > 0 is an increasing function of r 2 then our purpose will e served. Now, differentiating it with respect to r 2 we have

12 S. Panda et al. [ π 2 π 3 = 1 Sn 11 r 2 + n h + d at2 τ r 2 T 3 1 r 2 n+1 + S h eτ 1 ce τ h + d ane T 2 τ 1 1 r 2 n+1 Therefore, [TP 2 TP 3 /T 3 > 0if π 2 π 3 / r 2 > 0, i.e., if Sn 11 r 2 + n h + d + S h eτ 1 ce τ h + d net 2 τ 1 T 2 τ Now, et 2 τ 1 T 2 τ > 0 29 > 1. Thus from 29 wehave, Sn 11 r 2 + n S h eτ 1 ce τ > 0 i.e., r 2 < Sn 1 + n S h eτ 1 ce τ Sn 1 However, we have the restriction r 2 < 1 c S Hence π 2 >π 3 if r 2 < min 1 c Sn 1 + n S, S h eτ 1 Sn 1 ce τ 30 Theorem 1 indicates that for same n 1 and n 2 pre- and post deterioration discounts on unit selling price produce higher profit than that with only post deterioration discount on unit selling price, if the percentage post deterioration discount on unit selling price is less than min { 1 S c, S c h eτ S }. Whereas, Theorem 2 demonstrates that only post-deterioration discount on unit selling price is more profitale than profit corresponding to no discount on selling price { if the percentage post deterioration } discount on unit selling price is less than min 1 S c, Sn 1+n. S h eτ 1 ce τ Sn 1 A simple managerial indication is as follows: in pure inventory scenario if the product deteriorates after a certain time then it is always more profitale to apply oth preand post deterioration discount on unit selling price. And the amount of percentage

13 An EOQ model for perishale products discount must e less than the limit provided in 28 for oth pre and post deterioration discount. The upper limit for the amount of only post deterioration discount on unit selling price is given in 30. { } Theorem 3 For n 1 = n 2 = n = 1, π 4 >π 41 if r 2 < min 1 S c, 1 Sn 1 cn Proof Proof of this theorem is very simple. Letting τ 0in30 the upper limit for r 2 can e otained immediately. Lemma 1 π 5 is an increasing function of T 5. Proof Differentiating π 5 partially with respect to T 5 we get π 5 = T TP 5 5 T 5 TP 5 T 5 T 2 5 Thus, in order to show that π 5 is an increasing function of T 5 it is sufficient to show TP that T 5 5 T 5 TP 5 > 0 After a little simplification we have [ TP 5 a T 5 TP 5 = [α 1 T 5 1e α 1T 5 t S1 r 1 h T 5 α 1 c +S h c eα 1T 5 t 1 e t1 1 T 5 α 1 1 α 1 T 5 1e α 1T 5 t Right hand side of 32 is greater than zero if T 5 a 1 1 > 0. Since, 0 r 1 1, T 5 a 1 1 > 0 for all r TP 1 > 0. Hence, T 5 5 T 5 TP 5 > 0 and π 5 is an increasing function of T 5. Theorem 4 For fixed r 1 > 0 and for any r 2, π 1 >π 5 if T 1 <τ+ 2 S1 r 2 c c + h + d Proof From Lemma 1 it is found that π 5 is an increasing function of T 5. Thus it is not possile to otain optimal T 5 within the range 0 t 1 T 5 τ to determine the optimal value of π 5. Clearly, π 5 attains it s maximum value at T 5. Now using the same logic as in Theorem 1 we have π 1 π 5 T5 =τ = TP 1 TP 5 T5 =τ T 1 τ TP 1 TP 5 T5 =τ τ

14 S. Panda et al. Since τ>0, it is sufficient to show that TP 1 TP 5 T5 =τ > 0. TP 1 TP 5 T5 =τ = S c h + d aα2 et1 τ 1e α 1τ T 1 et1 1 caα 2 e T1 τ 1 + S1 r 1 α 1 c h + d aα2 α 1 et1 τ 1e α 1τ T S1 r 2 α 2 at 1 τ h + daα 2 e T1 τ 1 T 1 τ 32 Now 32 is greater than zero if ca α2 S1 r 2 α 2 at 1 τ e T1 τ 1 > 0 i.e., if S1 r 2 + h + d T 1 τ > et1 τ 1 c + h + d 33 Since is very small, taking first order approximation of the exponential function and simplifying we have from 33 S1 r 2 c c + h+d > T 1 τ 2 Which implies, T 1 <τ+ 2 S1 r 2 c c + h + d = T 1, say Hence the theorem. From lemma 1 it is found that π 5 is an increasing function of T 5. Therefore, π 5 attains it s maximum value at T 5. Thus, it may e concluded that for time to deteriorating products to gain higher amount of unit profit replenishment cycle would e continued up to the start of deterioration and in that case optimal value of T 5 cannot e determined. However, Theorem 4 indicates that if the replenishment cycle length for the case of oth forms of discount, is less than T 1 then it provides higher unit profit for any positive value of r 2. Since, S1 r 2 c/c + h + d > 0, T 1 is always greater than τ. Hence it is not desirale to terminate the replenishment cycle efore the start of deterioration, in particular, for time to deteriorating products. From Theorem 4 it is also found that there is a functional relationship etween T 1 and r 2 and the upper ound of r 2 can e otained as r 2 < 1 S c T 1 τh+d 2S = r 2 say. Conversely, it may e concluded that for any T 1 >τ, model with oth forms of discount provides higher unit profit if r 2 < r 2.

15 An EOQ model for perishale products Theorem 5 For n 1 = n 2 = n, π 5 >π 51 if t 1 > 1 ln α 1 Sr 1 hα 1 1 [S c hα 1 1 Proof Using the same logic as in Lemma 1 and Lemma 2, we find that π 5 π 51 = TP 5 T 5 TP 51 T 51 TP 5 TP 51 T 51 Now differentiating TP 5 TP 51 /T 51 with respect to t 1 we get, TP 5 TP 51 /T 51 t 1 [ S h ce t 1 1 α 1 T 51 + α 1 S h S1 r 1 α 1 h = aeα1t51 t1 TP 5 TP 51 /T 51 t 1 i.e., > 0ifS h ce t 11 α 1 + α 1 S h S1 r 1 α 1 h >0. Thus π 5 π 51 t 1 > TP 5 TP 51 /T 51 t 1 > 0, if e t 1 > [α 1 Sr 1 hα 1 1/[S c hα 1 1 i.e., t 1 > 1 ln α 1 Sr 1 hα 1 1 [S c hα Theorem 5 indicates that for fixed life time product pre-deterioration discount on unit selling price produces higher profit if the time to start discount is ounded aove y the limit provided y 34. However, from 34 it is found that there is a functional relationship etween t 1 and r 1. Hence ound on t 1 always implies a ound on r Model for positive terminal inventory level In the previous section we developed the model with oth forms of discounts and several sucases under the assumption that I T 1 = 0. That is the terminal inventory level of the replenishment cycle is zero. Because for deteriorating product it is a common practice in reality that product is not carried forward to the next replenishment cycle. But it does not imply that the ending inventory level is zero. Thus, in this section relaxing the restriction of zero ending inventory level we formulate the model and verify whether it provides higher profit or not, though the remanning inventory is not passed to the next replenishment cycle. The governing differential equations remain same as in Sect. 3.1 with the initial condition except the oundary condition. The

16 S. Panda et al. oundary condition is I T 1 = Q 0. Solving the differential equations we have I t = a [ e t 1 + Q 1 e t, 0 t t 1 35 = a [ e α 1t 1 t t Q 1 e α 1t 1 t t 1, t 1 t τ 36 = aα [ 2 e T1 t 1 + Q 0 e T1 t, τ t T 1 37 Sustituting t = τ in the Eqs. 36 and 37 we get Q 1 = [ aα2 et 1 τ 1 + a + Q 0e T 1 τ e α 1τ t 1 +t 1 a 38 Proceeding in the same way as in the previous case we have the profit per unit time π 6 r 1, r 2, t 1, Q 0, T 1 = 1 [Sat 1 +S1 r 2 α 2 at 1 τ+ S h at 1 Q T a 1 e t 1 at 1 τ +[S1 r 1 α 1 h + Q 1 + a e t 1 1 eα 1t 1 τ α 1 + S1 r 1 α 1 aτ t 1 h + d aα 2 e T1 τ 1 T 1 τ e T1 τ 1 + Q 0 cq 1 C 0 39 And we have the constraint maximization prolem maximize π 61 r 1, r 2, t 1, T 1, Q 0 suject to, {r 1, r 2 } < 1 S c 40 r 1, r 2, t 1, T 1, Q 0 0 Note that if Q 0 = 0 then from 39 we get the unit profit function 8. Proceeding in the same way as in the previous su section all the su cases may e derived easily for non-zero terminal inventory level. Since the demand of the product is stock dependent, higher volume of inventory in shelf always results in higher depletion rate. But how much inventory would e replenished initially for maximum utilization of stock dependent demand structure, that depends partially on proper amount of end inventory level. Because the product deteriorates and it is not possile to pass the excess amount of inventory to the next replenishment cycle. To quantify this, differentiating π 6 partially with respect to Q 0 and simplifying we have

17 An EOQ model for perishale products [ π 6 = 1 S h et 1 1 ce t 1 + s1 r 1 α 1 h Q 0 T 1 1 e α 1τ t 1 α 1 h + d 1 e T τ e α 1τ t 1 41 The unit profit function increases for the increment of terminal inventory level if π 6 / Q 0 > 0, i.e., if [ S h et 1 1 ce t 1 + s1 r 1 h 1 e α 1τ t 1 h α 1 + d > 0, i.e., if t 1 > 1 ln S h + h + d S h c But τ t 1. Thus 1 S h τ> ln + h + d S h c Therefore, π 6 increases for the increment of Q 0 if [ 1 S > C + e τ c d + h eτ 1 42 If condition 42 is not satisfied then unit profit does not increase for the increment of terminal inventory level. That is unit profit increases as Q 0 decreases and it attains maximum value when terminal inventory level is zero. Hence we have the following theorem Theorem 6 For deteriorating item under two-component stock dependent demand structure non-zero terminal inventory level is profitale if 42 is satisfied. Note that if condition 42 is satisfied, then theoretically π 6 attains maximum value for infinite units of terminal inventory. Consequently the amount of order quantity is infinite. From practical point of view this is quite impossile ecause shelf-space in a departmental store is always limited. Hence, finite order quantity always results in finite terminal inventory level. Thus to otain a finite optimal profit there must e an upper ound of Q 0 such that restriction on shelf-space is not violated. Conversely, depending on the availaility of shelf-space the upper limit of Q 0 is to e determined. If Q s is the availale shelf-space then Q 1 Q s.using38 and simplifying we have Q 0 [ Q s + a = Q 0, say e [α 1τ t 1 +t 1 aα 2 et 1 τ 1 a Therefore, prolem 40 is defined with the additional constraint 43. e T 1 τ 43

18 S. Panda et al. Theorem 6 indicates that non-zero ending inventory is profitale if the selling price is at least [τ d + h/[e τ 1+h/ higher than the purchase cost. Except S, all the parameters involved in the system are uncontrollale in the decision making context. Thus lower values of, h and higher values of τ,, sometimes may encourage the decision maker to adopt non zero terminal inventory policy. This is quite justified ecause in such situation inventory depletion rate is higher and it is continued for larger time period efore deterioration and volume of holding cost over the replenishment cycle is low. However, in majority of the literature, related to stock dependent demand structure for deteriorating items, the terminal inventory level is assumed as zero. Perhaps, i for it s simplicity for the asence of an additional variale, ii in the immensely competitive and highly informative market it is not possile to fix unit selling price in almost all the cases to avail the enefit of positive ending inventory level. Because it is highly sensitive to the uncontrollale system parameters, iii even if, it is possile to fix a competitive unit selling price, the availaility of shelf-space may not lead to apply it for the restriction 43, iv as we mentioned aove that people dealing with usiness generally does not go eyond the common practise of zero ending inventory, though sometimes it s possile to e enefitted. Although assumption of variale ending inventory level does not require it s positiveness, still in the decision making context once the competitive unit selling price is determined, for a given set of parameter values, the decision maker should verify whether 42 is satisfied or not. Because for deteriorating items, positive inventory level seldom happens in reality, as inventory is not carried forward to the next cycle. Thus for the simplicity regarding the applicaility of the model, if 42 is satisfied then only keeping in mind the availaility of shelf-space the decision maker would go for positive terminal inventory level. Under stock dependent demand to avail the effect of stock sensitivity more, throughout the replenishment cycle ending inventory level may e assumed as positive. But, if the demand is constant after a certain time period as in this model, then positive effect of stock dependency of demand is further diminished and hence chance of otaining higher unit profit than zero ending inventory level is further reduced. In the next section we consider two numerical examples and perform some quantitative measurements of this issue specially with Example 2. 4 Numerical example Example 1 The parameter values are a = 80, = 0.3, h = 0.6, d = 2.0, S = 10.0, C 0 = 100.0, c = 4.0, = 0.03, τ = 1.2, n 1 = n 2 = 2.0. The optimal values are listed in Tale 1. For the set of parameter values and S, the condition 42 is not satisfied. Thus for all the models terminal inventory levels are zero. For model with oth pre- and post deterioration discounts, the pre-deterioration discount on unit selling price is 38.9% and discount starts at time t 1 = and continued to τ=1.2. Then the product start to deteriorates. During this time in order to enhance inventory depletion rate, 56.4% discount is provided for remaining time of replenishment cycle. Profit per unit time is The optimal order quantity and cycle length are and respectively. The unit profit and order quantity for only post deterioration discount are and less y 23.5 and 136.2%, respectively.

19 An EOQ model for perishale products Tale 1 Optimal values of the decision variales, order quantity and profit per unit time of the models Model Nature of r 1 r 2 t 1 Cycle Order Unit profit discounts length quantity Time to deterioration With all discounts Post deterioration discount No discount Instant deterioration Post deterioration discount No discount Fixed life Pre-deterioration discount No discount The cycle length is , 6.3% higher than that for model with oth types of discounts. The post deterioration discount in this case is 45.1% and is less than 11.3% from the model with oth types of discounts. Whereas the model with no discount provides least profit per unit time among it s class. The results are quite justified and agree with the model analysis of last section. Though higher amount of percentage discounts on unit selling price in the form of pre- and post deterioration discounts for larger periods of time results in lower per unit sales revenue, still it is more profitale. Because the inventory depletion rate is much higher for discount enhanced demand resulting lower amount of inventory holding cost and deteriorated items. Therefore, it is always profitale to apply oth forms of discounts on unit selling price to earn more profit. The results of two sucases, for instant deteriorating items and products having fixed life time are also given in Tale 1. For instant deteriorating products using the same set of data, 7.1% discount is provided on unit selling price to earn 0.77% more profit than that with no discount. For fixed life time products also, discounted selling price provides 24.14% more profit if 38.9% discount is given from time Note that the upper limits for r 1 and r 2 as well as for t 1 and cycle lengths found analytically are satisfied y the numerical values found in this section. In Tale 2 some sensitivity analysis of the model with oth types of discounts is performed y changing the parameter values 40, 20, 20 and 40%, taking one parameter at a time and keeping the remaining parameters values unchanged. It is found from Tale 2 that the model is highly sensitive for the error in the estimation of the parameter values S and c. It is moderately sensitive for the change in the parameter values a, and h. Low sensitivity if found for the change in parameter values C 0, d and. Sensitivity analysis reveals some common characteristics, e.g., profit increases as selling price increases. For the decrement of holding cost or unit purchase cost unit profit increases, and so on. From Tale 2 it is found that, if the selling price decreases, percentage discount of oth types as well as the pre-deterioration discount starting time decreases. Whereas decrement of unit cost leads to the increment of r 1, r 2 and t 1. This is ovious ecause lower unit selling price always provides less revenue and in that case maximum profit is achievale if the discounts and t 1 are low. Conversely, if the unit cost is low then y providing more discounts and more time pre-deterioration

20 S. Panda et al. Tale 2 Sensitivity analysis of the model with oth types of discounts Parameter % change % change % change % change % change % change π % change in r 1 in r 2 in t 1 in T in Q in π S c h d a C

21 An EOQ model for perishale products Tale 3 Optimal values of the decision variales, order quantity and terminal inventory level for example 2 Model r 1 r 2 t 1 Terminal inventory Cycle Order Unit level length quantity profit Zero ending inventory Positive ending inventory Tale 4 Least unit selling prices to e applied for profitale positive terminal inventory level for different parameter values for example 2 h S τ S S discount the inventory depletion rate can e increased, resulting lower amount of holding cost and deterioration cost, hence higher unit profit. Example 2 We consider the parameter values same as Example 1 except, h and τ.itis assumed that = 0.4, h = 0.02 and τ = 1.6. The availale shelf-space is Q s = The optimal values are given in Tale 3. It is found from Tale 3 that profit for non-zero terminal inventory level is with order quantity and terminal inventory level The profit is 4.65% higher than unit profit for zero-terminal inventory level. Note that the unit selling price is 10 and it satisfies the condition 42. But higher unit profit for non-zero ending inventory level is otained for sufficiently small value of h and high values of and τ. Since, in the decision making context adoptaility of non-zero terminal inventory level is dependent on the unit selling price of the product, in Tale 4 some sensitivity analysis of the unit selling price is performed for different values of the parameters. Because, if the unit selling price satisfies 42 then it is always preferale to apply non-zero terminal inventory policy. Tale 4 indicates that when h and τ are moderately high and is low, extraordinarily high unit selling price is to e applied to otain more profit with positive ending inventory policy. Which is not at all competitive in the decision making context. Thus for very low value of h and high values of τ and one may apply a competitive unit selling price and adopt positive terminal inventory policy. Otherwise, applying a non-realistic unit selling price we have to adopt non-zero terminal inventory policy. Which is not desirale in competitive decision making environment.

22 S. Panda et al. 5 Summary and concluding remarks In this paper an inventory model is developed for products, which start to deteriorate after a certain time, with two components stock dependent demand. Pre- and post deterioration discounts are provided on unit selling price of the product. When discounts are provided the demand is partially stock dependent and partially stock and selling price dependent. The mathematical model is developed allowing pre- and post deterioration discounts on unit selling price. Then the amount of allowale oth form of discounts and when the pre-deterioration discount should e started to earn more profit are determined analytically. It is found that, if the amount of discounts are restricted elow the limits provided in the section model analysis, then the unit profit is always higher. It is also derived analytically the upper limit for only post deterioration discount on unit selling price to earn more revenue than revenue earned for no discount. However, Theorem 1 indicates the upper limit for post deterioration discount on unit selling price and hence the upper limit for pre-deterioration discount on unit selling price to earn maximum profit y applying oth form of discounts among its class. Then two sucases are considered for instant deteriorating items and for fixed life time products. The upper limit of r 2 for instant deteriorating products and upper limit of r 1 for fixed life time products are also derived analytically. The model for non-zero terminal inventory level is also derived. It is found analytically that, if the unit selling price is higher than the purchase cost y a certain amount then positive terminal inventory level is profitale. Numerical examples are presented to justify the claim of model analysis. It is also noted that the model for fixed life time products is equivalent to the model if the decision maker wishes to terminate the replenishment cycle efore the start of deterioration for time to deterioration products. From numerical example it is found that for exceptionally low value of h and high value of τ and non-zero ending inventory level is profitale suject to the shelf-space constraint. This is rarely found to happen in reality. For normal and high values of the parameters zero-terminal inventory level provides higher unit profit. Thus for simplicity regarding the applicaility of the model it is always preferale to adopt zero-terminal inventory level policy. It is shown analytically that for time to deteriorating products the replenishment cycle should e continued after τ to achieve higher unit profit. Temporary price discounts for perishale products to enhance inventory depletion rate for profit maximization is an area of interesting research, that needs further investigation. The model proposed here may e extended in several ways. For example, one may allow shortage and partial acklogging, demand fluctuations, etc. Acknowledgments Authors are thankful to the anonymous referees for their comments and suggestions on the earlier versions of the manuscript for the improvement of the paper. References Arcelus FJ, Srinivasan G 1998 Ordering policies under one time discount and price sensitive demand. IIE Trans 30: Baker RC, Uran TL 1988 A deterministic inventory system with an inventory level dependent demand rate. J Oper Res Soc 39:

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