International Journal of Supply and Operations Management

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1 International Journal of Supply and Operations Management IJSOM May 014, Volume 1, Issue 1, pp ISSN-Print: ISSN-Online: EOQ Model for Deteriorating Items with exponential time dependent Demand Rate under inflation when Supplier Credit Linked to Order Quantity Rakesh Prakash Tripathi a, Dinesh Singh b and Tushita Mishra b a Department of Mathematics, Graphic Era University, Dehradun (UK) India b Department of Mathematics, S.G.R.R. PG College, Dehradun (UK) India Abstract In paper (004) Chang studied an inventory model under a situation in which the supplier provides the purchaser with a permissible delay of payments if the purchaser orders a large quantity. Tripathi (011) also studied an inventory model with time dependent demand rate under which the supplier provides the purchaser with a permissible delay in payments. This paper is motivated by Chang (004) and Tripathi (011) paper extending their model for exponential time dependent demand rate. This study develops an inventory model under which the vendor provides the purchaser with a credit period; if the purchaser orders large quantity. In this chapter, demand rate is taken as exponential time dependent. Shortages are not allowed and effect of the inflation rate has been discussed. We establish an inventory model for deteriorating items if the order quantity is greater than or equal to a predetermined quantity. We then obtain optimal solution for finding optimal order quantity, optimal cycle time and optimal total relevant cost. Numerical examples are given for all different cases. Sensitivity of the variation of different parameters on the optimal solution is also discussed. Mathematica 7 software is used for finding numerical examples. Keywords: Inventory, inflation, exponential time dependent, credit, finance 1. Introduction The main aim of any inventory control system is that when and how much to order. A large number of research papers and books have been published presenting models for doing this under various assumptions and conditions. Inventories are often replenished periodically at certain production rate, which is seldom infinite. Deterioration plays an important role in inventory management. In real life, almost all items deteriorate over time. Deterioration may be slow or fast. Thus, during the development of EOQ model, deterioration cannot be ignored. Corresponding address: tripathi_rp031@rediffmail.com 0

2 EOQ Model for Deteriorating Items with exponential time dependent Demand Rate Some of deteriorating items are such as volatile liquids, blood bank, medicines, fashion goods, radioactive materials, photographic films, and green vegetables etc. At present, most of the items have significant rate of deterioration. The analysis of deteriorating inventories began with Ghare and Schrader (1963), who established the classical inventory model with constant rate of deterioration. Covart and Philip (1973) extended Ghare and Schrader (1963) model and obtained an economic order quantity model for a variable rate of deterioration by assuming a two parameter Weibull distribution. Researchers like Philip (1974), Wee (1997), Misra (1975), Chakraborty et al. (1998), Mukhopadhyay et al.(004), and Tadikamalla (007) established inventory models that focused on deteriorating products. Goyal and Giri (001) and Raafat (1991) established a complete survey of the published inventory literature for deteriorating inventory models. Balkhi and Benkherouf (1996) developed a method for obtaining an optimal production cycle time for deteriorating items in a model where demand and production rates are functions of time. At present, great interest has been shown in developing mathematical models in the presence of trade credits. Kingsman (1983), Chapman et al. (1985) and Daellenbach (1986) have developed the effect of the trade credits on the optimal inventory policy. Trade credit has been a topic of interest for many authors in inventory policy like Hayley and Higgings (1973), Davis and Gaither (1985), Ouyang et al. (004), Ward and Chapman et al. (1988). Recently, Khanna et al. (011) developed an EOQ model for deteriorating items with time dependent demand under permissible delay in payments. Inventory models with permissible delay in payments were first studied by Goyal (1985). Shinn et al. (1996) extended Goyal's (1985) model by considering quantity discounts for freight cost. Chu et al. (1998) and Chung et al. (001) also extended Goyal's model for the case of deteriorating items. Sana and Chaudhury (008) developed a more general EOQ model with delay in payments, price discounting effect and different types of demand rate. All the above articles are based on the assumption that the cost is constant over the planning horizon. This assumption may not be true in real life, as many countries have high inflation rate. Inflation also influences demand of certain products. As inflation increases, the value of money goes down. As a result, while determining the optimal inventory policy, the effect of inflation and time value of money cannot be ignored. Buzacott (1975) discussed EOQ model with inflation subject to different types of pricing policies. Wee and Law (1999) established the problem with finite replenishment rate of deteriorating items taking account of time value of money. Chang (004) developed an inventory model for deteriorating items under inflation under a situation in which the supplier provides the purchaser with a permissible delay of payments if the purchaser orders a large quantity. Recently, Tripathi (011) developed an inventory model under which the supplier provides the purchaser a permissible delay in payments if the purchaser orders a large quantity. This paper is the extension of Tripathi (011) paper in which demand rate is time dependent and deterioration rate is zero. Liao (007) established the inventory replenishment policy for deteriorating items in which the supplier provides a permissible delay in payments if the purchaser orders a large quantity. Hon and Lin (009) developed an inventory model to determine an optimal ordering policy for deteriorating items with delayed payments permitted by the supplier under inflation and time discounting. Teng et al. (01) proposed an EOQ model in which the constant demand to a linear, non- decreasing demand functions of time, which is suitable not only for the growth stage but also for the maturity stage of the product life cycle. Khanra et al. (011) developed an EOQ model for deteriorating item having time dependent developed an EOQ model for deteriorating item having time- dependent demand when delay in payment is permissible. Int J Supply Oper Manage (IJSOM), Vol.1, No.1 1

3 Tripathi, Singh and Mishra Yang et al. (010) developed an EOQ model for deteriorating items with stock- dependent demand and partial backlogging. Ouyang and Chang (013) extended the effects of the reworking imperfect quality item and trade credit on the EPQ model with imperfect production process and complete backlogging. Soni (013) developed an EOQ model considering (i) the demand rate as multivariate function of price and level of inventory (ii) delay in payment is permissible. Taleizadeh and Nematollahi (014) investigated the effect of time- value of money and inflation on the optimal ordering policy in an inventory control system. Sarkar et al. (014) developed an economic manufacturing quantity (EMQ) model for the selling price and the time dependent demand pattern in an imperfect production process. Teng et al. (013) developed an EOQ model extending the constant demand to a linear, nondecreasing demand function of time and incorporate a permissible delay in payment in payment under two levels of trade credit into the model. Tripathi and Pandey (013) presented an inventory model for deteriorating items with Weibull distribution time dependent demand rate under permissible delay in payments. Sarkar (01) presented an EOQ model for finite replenishment rate where demand and deterioration rate are both time dependent. Tripathi (011) established an inventory model for non- deteriorating item and time dependent demand rate under inflation when the supplier offers a permissible delay to the purchaser, if the order quantity is greater than or equal to a predetermined quantity. Some articles related to the inventory policy under delay in payments can be found in Mirzazadeh and Moghaddam (013), Mirzazadah et al. (009), Teheri et al. (013) and their references. In this paper, an attempt has been made to develop an inventory model for deteriorating items with exponential time dependent demand rate in which inflation and time value of money are taken into account. Optimal solution for the proposed model is derived by taking truncated Taylor's series approximation for finding closed form optimal solutions. Numerical examples and sensitivity analysis have been performed to observe the effect of different parameters on the optimal inventory replenishment policy. The rest of this paper is organized as follows. In section notations and assumptions are mentioned, which have been used throughout the manuscript. In section 3, the mathematical models are derived under four different circumstances in order to minimize the total cost in planning horizon. In section 4, determination of optimal solution is presented. Numerical examples are provided in section 5 to demonstrate the applicability of the proposed model. We characterize the effect of the values of parameters on the optimal replenishment cycle, order quantity and total relevant cost in section 6. In section 7 conclusion and future research is given.. Notations and Assumptions The following notations are used throughout this paper: h : holding cost rate per unit time r : constant rate of inflation per unit time, where 0 1 rt pe : selling price per unit at time t, where p is the unit selling price at time zero rt ce Ae H m I c rt : purchase cost per unit at time t, where c is unit purchase cost at time zero and p > c : ordering cost per order at time t, where A is the ordering cost at time zero : length of planning horizon : permissible delay in settling account : Interest charged per $ in stock per year Int J Supply Oper Manage (IJSOM), Vol.1, No.1

4 I d EOQ Model for Deteriorating Items with exponential time dependent Demand Rate : Interest earned per unit Q : order quantity Qd : minimum order quantity at which the delay in payments is permitted T : replenishment time interval Td : the time interval that Qd units are depleted to zero due to time dependent demand I(t) : level of inventory at time t R(t) : annual demand as a R() t e t, λ > 0, 0 < α 1 Z(T) : total relevant cost over (0,H) The total relevant cost consists of (i) cost of placing order (ii) cost of purchasing (iii) cost of carrying inventory excluding interest charges (iv) cost of interest charges for unsold items at the initial time or after credit period m and interest earned from sales revenue during credit period m..1. Assumptions 1. The inflation rate is constant.. Shortages are not allowed. 3. The demand for item is exponentially increasing function of time. 4. Replenishment is instantaneous. 5. If Q < Qd, then the payments for items received must be made immediately. 6. If Q Qd, then the delay in payments up to m is permitted. During permissible delay period the account is not settled, generated sales revenue is deposited in an interest bearing account. At the end of credit period, the customer pays off all units ordered and begins payment of the interest charged on the items in stock. 3. Mathematical Formulation Let us consider the length of horizon H = nt, where n is an integer for the number of replenishment to be made during period H, and T is an interval of time between replenishment. The level of inventory I(t) gradually decreases mainly to meet demand only. Hence the variation of inventory with respect to time is given by di() t t. I( t) e, 0 t T = H/n ; 0 1, (1) dt With boundary conditions I(0) = Q and I(T) = 0. Solution of (1) with boundary condition I(T) = 0, is given by ( ) T t t I( t) e e, 0 t T () ( ) And order quantity is ( ) T Q e 1 (3) ( ) Since the lengths of time interval are all the same, we have ( ) T t t I( kt t) e e, 0 t T (4) ( ) From the order quantity, we can obtain the time interval that Qd units are depleted to zero due to demand only as Int J Supply Oper Manage (IJSOM), Vol.1, No.1 3

5 Tripathi, Singh and Mishra 1 ( ) Td log 1 Qd ( ) Qd ( ) ( ) Or Td 1 Qd Q d 3 (approximately) (5) To obtain total relevant cost in [0, H], we obtain ordering cost, purchasing cost and holding cost as follows: (a) Ordering cost n1 rh e 1 OC A( kt ) A rt k0 e 1 (ref. appendix) (6) (b) Purchasing cost n1 rh c ( ) T e 1 PC I(0) C( kt ) e 1 rt k0 ( ) e 1 (ref. appendix) (7) (c) Holding cost n1 T ( ) T T T rh hc e e e 1 e 1 HC h C( kt ) I( kt t) dt rt k0 ( ) 0 e 1 (8) Now the following four cases arise based on the values of T, m and Td for finding interest charged and interest earned. Case I. 0 < T < Td Since cycle time interval T is less than Td i.e. T < Td (i.e. Q < Qd), the delay in payments is not permitted in this case. The supplier must be paid for the items as soon as the customer receives them. Since the interest is charged for all unsold items start at the initial time, the interest payable in (0, H) is given by n1 T ( ) T T T rh cic e e e 1 e 1 IC1 I c C( kt ) I( kt t) dt rt k0 ( ) 0 e 1 (9) Thus the total relevant cost in (0, H) is Z ( T) OC DC HC IC 1 1 ( ) T T T rh c ( ) T c( h Ic ) e e e 1 e 1 A e 1 rt ( ) ( ) e 1 Case II: Td m < T Since Td m < T, we know that there is a permissible delay m which is greater than the cycle time interval T. As a result, there is no interest charged i.e. IC = 0, but the interest earned in (0, H) is given by n1 T T t t IE Id P( kt ) e tdt m T e dt 0 k0 0 rh pid T 1 e 1 e 1m T rt e 1 So the total relevant cost in this case in (0, H) is (10) (11) Int J Supply Oper Manage (IJSOM), Vol.1, No.1 4

6 EOQ Model for Deteriorating Items with exponential time dependent Demand Rate Z ( T) OC DC HC IE ( ) T T T c ( ) T hc e e e 1 A e 1 ( ) ( ) rh pid T 1 e 1 e 1m T rt e 1 Case III: Td m T (1) Since T m Td, the delay in payments is permitted and the total relevant cost includes both the interest charged and the interest earned. The interest charged in (0, H) is 3 n1 IC I C( kt ) I( kt t) dt c k0 T m ( ) T m T T m rh cic e e e e e 1 rt ( ) e 1 The interest earned in [0, H] is n1 m m rh t pid m e 1 e 1 IE3 Id P( kt ) e tdt me rt k0 0 e 1 Hence the total relevant cost Z ( T) OC DC HC IC IE (14) ( ) T T T c ( ) T hc e e e 1 A e 1 ( ) ( ) ( ) T m T T m m rh cic e e e e pid m e 1 e 1 me rt e 1 Case IV: m Td T Since T Td m, case IV is similar to case III. Therefore, the total relevant cost in (0, H) is ( ) T T T c ( ) T hc e e e 1 Z4 ( T ) A e 1 ( ) ( ) (16) ( ) T m T T m m rh cic e e e e pid m e 1 e 1 me rt e 1 The graphs of all four cases are given in the appendix for better explanation. Since it is difficult to find the solutions for finding the exact value of T, therefore, we make use of the second order approximation for the exponential terms in equation (10), (1), (15) and (16), which follows as: (13) (15) Int J Supply Oper Manage (IJSOM), Vol.1, No.1 5

7 Tripathi, Singh and Mishra rt rt e 1 rt rh rh e 1 rh (17) rm rm e 1 rm The above approximation is valid for smaller values of deterioration rate θ and inflation rate r such that rt <1, rm < 1 and rh < 1. In reality, the values of deterioration rate θ or inflation rate r is usually very small. Hence, equations (10), (1), (15) and (16) reduces to ( ) A c 1 1 rt 1 rh Z T c h I. c T H T A c ( m 1) T rt rh Z ( T ) c h T pid m 1 H 1. T A ct cic pidm (1 m) rt rh Z3( T) c h T m 1 H 1. T T T A ct cic pidm (1 m) rt rh Z4 ( T) c h T m 1 H 1. T T T Similarities and Differences among case I, II, III and IV If interest earned IE3 becomes zero then case III becomes case I for limit t = 0 to t = T. If interest charges IC3 becomes zero then case III becomes II. (18) (19) (0) (1) 4. Determination of optimal solution To find the optimal solution for the problem, we minimize Zi(T) for Case I, II, III, IV respectively and then compare them to obtain minimum value. Our aim is to find minimum relevant cost for all cases i.e. Case I, II, III, IV respectively with respect to T. The necessary and sufficient condition to minimise Zi(T) ; i =1,,3,4 for given values of T are respectively dz1 ( T) dz ( T) dz3( T) dz4 ( T) d Z1( T) d Z ( T) d Z3( T) d Z4 ( T ) 0, 0, 0, 0, and 0, 0, 0, 0 dt dt dt dt dt dt dt dt Differentiating equations (18), (19), (0) and (1) with respect to T and equating to zero, we get c ( h I )(1 rt) r) T A 0. () c (1 rt) c( h ) pid( m 1) r( c pidm) T A 0. (3) c (h + I )(1 rt) r(1 I m) T m ci pi (1 m) A 0. (4) c c c d And c (h + I )(1 rt) r(1 I m) T m ci pi (1 m) A 0. (5) c c c d And second derivative of equations (18), (19), (0), (1) are given by d Z1 A = rc rh 3 h Ic H 1 > 0. dt T (6) Int J Supply Oper Manage (IJSOM), Vol.1, No.1 6

8 EOQ Model for Deteriorating Items with exponential time dependent Demand Rate pi ( m 1) r = 1 > 0. T d Z A d c r( h ) rh H 3 dt dz3 A m cr rh = 3 3 ci c pid (1 m) h Ic H 1 > 0. dt T T dz A m cr rh = (1 ) 1 > 0. ci 3 3 c pid m h Ic H dt T T 5. Numerical examples 4 And (7) (8) (9) Example 1: Let H = 1 year, λ= 500 units, h = $/unit/year, Ic= 0.10/$/year, Id = 0.05/$/year, A= $00 per order, r = 0.05 per unit, θ = 0./ unit/year, α = 0.5, c = $ 5 per unit. Substituting these in (4), we get Optimal cycle time T = T1* = year, optimal total relevant profit Z1 (T) = Z1*(T1*) = $3114, and optimal order quantity Q = Q1* = units, which shows that if Q* < Qd then T1 < Td, this proves case I. Example : Let H = 1 year, λ= 50 units, h = $/unit/year, Ic= 0.16/$/year, Id = 0.05/$/year, A= $150 per order, r = 0.05 per unit, θ = 0./ unit/year, α = 0.8, c = $ 10 per unit, m = 90 days, p = 40. Substituting these in Equation (4), we get Optimal cycle time T = T* = year, optimal total relevant profit Z (T) = Z*(T*) = $ , and optimal order quantity Q = Q* = units, which shows that T m, if Q*(T) Qd, then T Td, we get Td T m, this proves case II. Example 3: Let H = 1 year, λ= 100 units, h = $5/unit/year, Ic= 0.08/$/year, Id = 0.01/$/year, A= $50 per order, r = 0.05 per unit, θ = 0./ unit/year, α = 0.9, c = $ 30 per unit, m = 90 days, p = 40. Substituting these in (4), we get Optimal cycle time T = T3* = year, optimal total relevant profit Z3 (T) = Z3*(T3*) = $ , and optimal order quantity Q = Q3* = units, which shows that T3 m if Q*(m) Qd, then m Td, we get, T3 m Td, this proves case III. Example 4: Let H = 1 year, λ= 100 units, h = $5/unit/year, Ic= 0.08/$/year, Id = 0.01/$/year, A= $50 per order, r = 0.05 per unit, θ = 0./ unit/year, α = 0.9, c = $ 30 per unit, m = 90 days, p = 40. Substituting these in (4), we get Optimal cycle time T = T4* = year, optimal total relevant profit Z4 (T) = Z4*(T4*) = $ , and optimal order quantity Q = Q4* = units, which shows that T4 m if Q*(m) Qd, then m Td, we get, T4 Td m, this proves case IV. 6. Sensitivity Analysis We have performed sensitivity analysis by changing A, c, h, m and θ and keeping the remaining parameters at their original values. The corresponding variations in the cycle time, the order quantity and total relevant cost are given in Table 1 (Table 1.a, Table 1.b, Table 1.c, Table 1.d), Table (Table.a, Table.b, Table.c, Table.d, Table.e), Table 3 (Table 3.a, Table 3.b, Table 3.c, Table 3.d, Table 3.e) for case I, II and III respectively. Int J Supply Oper Manage (IJSOM), Vol.1, No.1 7

9 Tripathi, Singh and Mishra Case I A Table 1.a. Variation of ordering cost A. T T * 1 1 Z1 T C Table 1.b. Variation of purchasing cost c. T T * 1 1 Z1 T H Table 1.c. Variation of holding cost h. T T * 1 1 Z1 T Table 1.d. Variation of deterioration rate θ. Θ T T * 1 1 Z1 T Int J Supply Oper Manage (IJSOM), Vol.1, No.1 8

10 Case II A EOQ Model for Deteriorating Items with exponential time dependent Demand Rate Table.a. Variation of ordering cost A. T T * Z T C Table.b. Variation of purchasing cost c. T T * Z T H Table.c. Variation of holding cost h. T T * Z T m (in days) Table.d. Variation of permissible delay period m. T T * Z T Int J Supply Oper Manage (IJSOM), Vol.1, No.1 9

11 Tripathi, Singh and Mishra Θ Table.e. Variation of deterioration rate θ. T T * Z T Case III A Table 3.a. Variation of ordering cost A. T T * 3 3 Z3 T C Table 3.b. Variation of purchasing c. T T * 3 3 Z3 T H Table 3.c. Variation of holding cost h. T T * 3 3 Z3 T Int J Supply Oper Manage (IJSOM), Vol.1, No.1 30

12 M EOQ Model for Deteriorating Items with exponential time dependent Demand Rate Table 3.d. Variation of permissible delay period m. T T * 3 3 Z3 T Θ Table 3.e. Variation of deterioration rate θ. T T * 3 3 Z3 T All the above observations can be sum up as follows: From Table 1.a, increase of ordering cost A results increase in optimal cycle time T = T1*, order quantity Q = Q1*, and total relevant cost Z1 (T1*). From Table 1.b, increase of unit purchasing cost c results decrease in optimal cycle time T = T1*, order quantity Q = Q1*, and increase in total relevant cost Z1 (T1*). From Table 1.c, increase of holding cost h results decrease in optimal cycle time T = T1*, order quantity Q = Q1*, and increase in total relevant cost Z1 (T1*). From Table 1.d, increase of deterioration rate θ results decrease in optimal cycle time T = T1*, order quantity Q = Q1*, and increase in total relevant cost Z1 (T1*). From Table.a, increase of ordering cost A results increase in optimal cycle time T = T*, order quantity Q = Q*, and total relevant cost Z (T*). From Table.b, increase unit purchase cost c results decrease in optimal cycle time T = T*, order quantity Q = Q*, and increase in total relevant cost Z (T*). From Table.c, increase of holding cost h results decrease in optimal cycle time T = T*, order quantity Q = Q*, and increase in total relevant cost Z (T*). From Table.d, increase of credit period m results slight increase in optimal cycle time T = T*, order quantity Q = Q* and total relevant cost Z (T*). From Table.e, increase of deterioration rate θ results decrease in optimal cycle time T = T*, slight decrease in order quantity Q = Q*, and increase in total relevant cost Z (T*). From Table 3.a, increase of ordering cost A results increase in optimal cycle time T = T3*, order quantity Q = Q3*, total relevant cost Z3 (T3*). From Table 3.b, increase of unit purchasing cost c results decrease in optimal cycle time T = T3*, order quantity Q = Q3* and increase in total relevant cost Z3 (T3*). Int J Supply Oper Manage (IJSOM), Vol.1, No.1 31

13 Tripathi, Singh and Mishra From Table 3.c, increase of holding cost h results decrease in optimal cycle time T = T3*, order quantity Q = Q3* and increase in total relevant cost Z3 (T3*). From Table 3.d, increase of credit period m results increase in optimal cycle time T = T3*, order quantity Q = Q3*, total relevant cost Z3 (T3*). From Table 3.e, increase of deterioration rate θ results increase in optimal cycle time T = T3*, order quantity Q = Q3* and slight decrease in total relevant cost Z3 (T3*). Note: If α = 0, this paper reduces to Chang [4] and if θ = 0, demand rate is time dependent this paper reduces to Tripathi [41]. 7. Conclusion and Future Research The present model is based on exponential time dependent demand rate. Some researchers adopted an exponential functional form like a.e bt, a > 0, b in not equal to zero implying exponential increase ( b > 0 ) or decrease ( b < 0) in the demand rate. An exponential rate being very high, it is doubtful whether the real market demand of any commodity can rise or fall exponentially, it means accelerated rise or fall in demand. Accelerated increase in the demand rate takes place in the case of the aircraft, computer, machines and their spare parts. Accelerated decrease in the demand rate is found to be in case of obsolete aircraft, computer, and spare parts in machines. In this paper, we developed an inventory model for deteriorating items under inflation to determine the optimal ordering policy when the supplier provides a permissible delay in payment linked to order quantity. In order to obtain closed form solution of the optimal cycle time, truncated Taylor s series approximation is used in exponential terms. Numerical examples are studied to illustrate the theoretical results for all four cases. A sensitivity analysis is given to the variation of different parameters. There are some managerial phenomena (i) a higher value of purchasing cost causes, lower value of cycle time, order quantity and higher values of total relevant cost (ii) a higher value of holding cost causes lower values of cycle time, order quantity and higher values of total relevant cost. (iii) a higher value of deterioration rate causes, lower values of optimal cycle time, order quantity and higher values of total relevant cost. The proposed model can be extended in several ways. For instance, we may extend the constant deterioration rate to a time dependent deterioration rate. In addition, we could consider the demand as a function of stock level, function of time, selling price and others. Finally, it can be generalized for shortages, quantity discount, freight changes and others. Appendix (i) n1 n1 n1 rkt rkt rt rt 3 rt ( n1) rt C( kt ) ce c e c 1e e e e k0 k0 k0 rnt rh e 1 e 1 c c ( since nt = H) rt rt e 1 e 1 Int J Supply Oper Manage (IJSOM), Vol.1, No.1 3

14 EOQ Model for Deteriorating Items with exponential time dependent Demand Rate (ii) (iii) n1 n1 n1 rkt rkt rt rt 3 rt ( n1) rt A( kt ) Ae A e A 1 e e e e k0 k0 k0 rnt rh e 1 e 1 A A ( since nt = H) rt rt e 1 e 1 n1 n1 n1 rkt rkt rt rt 3 rt ( n1) rt P( kt ) pe p e p 1 e e e e k0 k0 k0 rnt rh e 1 e 1 p p (since nt = H) rt rt e 1 e 1 (iv) Taylor s series expansion of exponential terms is e rt rt 1 rt, etc. Inventory Level Q Time 0 T T d T T d(1) -----(n-1) T T d (n-1) nt=h Inventory Level Q Time 0 T d T m T d (1) T m(1) T d (n-) (n-1) T m (n-) nt=h Case II. T d m T Int J Supply Oper Manage (IJSOM), Vol.1, No.1 33

15 Tripathi, Singh and Mishra Inventory Level Q Time 0 T d m T T d (1) m (1) T T d (n-) m (n-) (n-1) T nt=h Case III. T d m T Inventory Level Q Time 0 m T d T m(1) T d (1) T m (n-) T d (n-) (n-1) T nt=h Case IV. m T d T Figure 1 : Graph between inventory level and length of planning horizon Acknowledgements We would like to thank the editor and the referees for the valuable and constructive comments that have led to a significant improvement in the original paper. References Balkhi, Z. T. And Benkherouf, L. (1996). A production lot size inventory model for deteriorating items and arbitrary production and demand rate. European Journal of Operational Research, Vol.9, pp Buzacott, J.A. (1975). Economic order quantity with inflation. Operations Research Quarterly, Vol. 6(3), pp Int J Supply Oper Manage (IJSOM), Vol.1, No.1 34

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