Research Article An Inventory Model for Perishable Products with Stock-Dependent Demand and Trade Credit under Inflation

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1 Mathematical Problems in Engineering Volume 213, Article ID 72939, 8 pages Research Article An Inventory Model for Perishle Products with Stock-Dependent Demand and rade Credit under Inflation Shuai Yang, 1 Chulung Lee, 2 and Anming Zhang 3 1 Graduate School of Information Management and Security, Korea University, Anam-dong 5-ga, Seongbuk-gu, Seoul , Republic of Korea 2 School of Industrial Management Engineering and Graduate School of Management of echnology, Korea University, Anam-dong 5-ga, Seongbuk-gu, Seoul , Republic of Korea 3 Sauder School of Business, University of British Columbia, 253 Main Mall, Vancouver, BC, nada V6 1Z2 Correspondence should be addressed to Chulung Lee; leecu@korea.ac.kr Received 5 July 213; Accepted 1 October 213 Academic Editor: Dongdong Ge Copyright 213 Shuai Yang et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider an inventory model for perishle products with stock-dependent demand under inflation. It is assumed that the supplier offers a credit period to the retailer, and the length of credit period is dependent on the order quantity. he retailer does not need to pay the purchasing cost until the end of credit period. If the revenue earned by the end of credit period is enough to pay the purchasing cost or there is budget, the balance is settled and the supplier does not charge any interest. Otherwise, the supplier charges interest for unpaid balance after credit period, and the interest and the remaining payments are made at the end of the replenishment cycle. he objective is to minimize the retailer s (net) present value of cost. We show that there is an optimal cycle length to minimize the present value of cost; furthermore, a solution procedure is given to find the optimal solution. Numerical experiments are provided to illustrate the proposed model. 1. Introduction After the global economic crisis, developing countries (and even some developed countries) have suffered from large scale inflation. Meanwhile, the inflation in food market is especially severe. For example, according to the BBC Online, the global grain price increased 1% in July 212 because of thehotanddryweatherintheusaandeasterneurope.since the demand for food is quite rigid, the inflation increases thepovertylevelindevelopingcountries.purchasewith large amount may reduce the cost and bring out more sales,butitmayincreasethemarketriskandspoilagedue to the characteristic of perishle products. In traditional EOQ model, the payment time does not affect the profit and replenishment policy. If the inflation is considered, the order quantity and payment time can influence both the supplier and retailer s decisions. he supplier may offer a credit period to promote the market, and the retailer may order more since the funding pressure is less. herefore, we should consider all these factors in order to make a reasonle replenishment policy under inflation. his research tries to determine the optimal order quantity for the inventory management of perishle products under inflation when the supplier offers a credit period. raditional inventory models assume that the demand rate is independent of the inventory level. However, the observationsinsupermarketsshowthatalargepileofgoodsinduce consumers to buy more. his occurs because larger stockpiles receive more visibility; especially for some perishle food (e.g., vegetles, fruits, and bread), high inventory may also suggest that they are fresh and popular. Silver and Peterson [1] observed that the consumption rate could be proportional to the stock level displayed. Later, Baker and Urban [2] considered a more practical assumption that the demand rate is a polynomial function of instantaneous stock level. Sana and Chaudhuri [3] considered a deterministic EOQ model with delays in payments and price discount offers, incorporating stock-dependent demand and other demand patterns. Since the perishle products may deteriorate over time, Mandal and Phaujdar [4] presented an inventory model for perishle

2 2 Mathematical Problems in Engineering products with a constant deterioration rate, assuming that the demand rate is a linear function of instantaneous stock level. Some of the relevant works for inventory management of deteriorating and stock-dependent demand items are by Sana [5], Padmanhan and Vrat [6], Mandal and Maiti [7], Dye and Ouyang [8], Chang et al. [9], and others. Moreover,inflationandtimevalueofmoneyareignored because they may not influence the inventory policy significantly. However, after global financial crisis, many countries have suffered from large scale inflation. herefore, inflation and time value of money should be considered when the inventory policy is made. he pioneer researcher in this area was Buzacott [1], who developed an EOQ model under inflation subject to different types of pricing policies. Vrat and Padmanhan [11] considered an inventory model with initial-stock-dependent demand under a constant inflation rate. Bose et al. [12] presented an inventory model for deteriorating items with time-dependent demand and shortages under inflation and time discounting. Recently, Sarkar et al. [13] developed an economic manufacturing quantity model for an imperfect production system with inflation and time value of money to determine the optimal production reliility and production rate that maximize the profit. Sana [14] proposed a control policy for a production system under inflation, assuming that the demand is dependent on the stock and the sales team s promotional effort. Some relatedworkscanbefoundinchungandlin[15], Hou [16], Jaggi et al.[17], Roy et al. [18], and others. When the inflation is considered, the credit period may be employed in replacement of price discounts or financial service and also affect the replenishment policy. herefore, some suppliers are willing to offer a credit period to promote the market competition. Chang [19] developed an inventory model for deteriorating items under inflation and time discounting, assuming that the supplier offers a trade credit if the retailer s order size is larger than a certain level. Although there are many exiting research works on the inventory management of perishle products, few papers have discussed the inventory management for perishle products with stock-dependent demand under inflation and time discounting. his paper deals with this problem, and it provides an optimal solution to minimize the retailer s present value of cost. 2. Model Formulation his section introduces the assumptions, notations, and mathematical formulation used in our perishle product inventory model Assumptions. he following assumptions are used throughout the paper. (1)hesuppliersellsonesingleitemtotheretailerin quantity. (2)heitemsarereplenishedwhenthestocklevelbecomes zero. (3) he supplier provides a credit period, which is dependent on the order quantity. (4) he deteriorating rate is constant. he deteriorating items cannot be repaired and the salvage value is zero. (5) he demand rate D(t) at the retailer s end is dependent on the instantaneous inventory level q(t),which means D(t) = a bq(t), a, 1 b. (6)heleadtimeiszeroandshortagesarenotallowed. (7) he planning horizon of the inventory system is finite. he number of cycles must be integer in the planning horizon Mathematical Model. he replenishment cycle starts with the initial inventory level Q and ends with zero stock. Since the inventory is depleted by the effect of both stockdependent consumption and deterioration, we can describe the retailer s inventory level q(t) by the following differential equation: dq (t) = [q(t) θq(t)], t. (1) dt By integrating both sides of (1)withrespecttot, 1 q(t) θq(t) dq (t) = dt. (2) With the boundary condition q() =, the solution of the integral equation is q (t) = a bθ [e(bθ)( t) 1]. (3) With q() = Q,weget Q= a bθ [e(bθ) 1]. (4) On the ove assumptions, there are two scenarios to arise: Scenario A, M ;ScenarioB,M< Scenario A: When M. Since the credit period M is longer than the replenishment cycle length, theretailer can sell all the items before the end of credit period, as shown in Figure 1. herefore, there is no interest charged by the supplier. he elements of the retailer s cost are as follows: ordering cost, purchasing cost, and holding cost. Since the replenishment is made at the beginning of each cycle,thepresentvalueoftheorderingcostduringthefirst cycle is A. he purchasing cost is paid at the end of credit period M; thepresentvalueofthepurchasingcostduringthefirstcycle is CQe rm = bθ [e(bθ) 1] e rm. (5) Because the holding cost occurs all over the replenishment cycle, the present value of the holding cost during the first cycle is h q (t) e rt dt = (bθ) r (e r 1) (bθ)(rbθ) [e(bθ) e r ]. (6)

3 Mathematical Problems in Engineering 3 Inventory level Q Proposition 1. When there exists a unique at which dpvh 1 ()/d = =for (, M], thenpvh 1 () is minimized at = if PV 1 (M)(erM 1) rpv 1 (M). Otherwise, =Misthe optimal solution. Proof. Let PV 1 () and PV 1 () represent the first and the second derivatives of PV 1 () with respect to, respectively. aking the first derivative of PVH 1 () with respect to, M ime Figure 1: he retailer s inventory level of the first cycle when M. Inventory level Q ime 2 L = m Figure 2: he retailer s inventory cycles in the planning horizon. herefore, the net present value of the cost during the first cycle is PV 1 () =ACQe rm h q (t) e rt dt =A bθ [e(bθ) 1]e rm [e r 1 bθ r e(bθ) e r ]. rbθ As shown in Figure 2, therearem cycles in the planning horizon. herefore, the present value of the total cost over the planning horizon L is PVH 1 () = m 1 n= PV 1 () e rn 1 e r PV 1 () {A 1 e r bθ [e(bθ) 1]e rm (bθ) r (e r 1) (7) (bθ)(rbθ) [e(bθ) e r ]}. (8) dpvh 1 () = e r (1 e rl ) d (1 e r ) 2 [PV 1 () (er 1) rpv 1 ()]. (9) Let f 1 () = PV 1 ()(er 1) rpv 1 ();thendpvh 1 ()/d = (e r (1 e rl )/(1 e r ) 2 )f 1 (). e r (1 e rl )/(1 e r ) 2 = (1 e rl )(1/(e r e r 2))>,andtheyaredecreasingon. aking the first derivative of f 1 () with respect to, we get f 1 () =PV 1 () (er 1)rPV 1 () (er 1). (1) In f 1 (), PV 1 () =e(bθ) rm (bθ)(rbθ) (bθ) e r (bθ) e (bθ) re r )] PV 1 () =(bθ) e(bθ) rm r (bθ) e r (bθ)(rbθ) (bθ) 2 e (bθ) r 2 e r ]. (11) Obviously, PV 1 () and PV 1 () arebothmorethanzero. herefore, f 1 () > ; f 1() is increasing on.fromf 1 () = ra and lim f 1 () =, the Intermediate Value heorem (homas and Finney [2]) implies that there exists auniquesolution,whichmakesf 1 ( )=.Hence, dpvh 1 () d = e r (1 e rl ) < if << { (1 e r ) 2 f 1 () = if = { { > if >. (12) herefore, if f 1 (M) = PV 1 (M)(erM 1) rpv 1 (M), is the optimal solution. Otherwise, PVH 1 () is decreasing on (, M] and =Mis the optimal solution Scenario B: When M< se 1. Letting Pe rm M D(t)e rt dt CQ, whichmeansat time M,therevenueearnedismorethanthepurchasingcost,

4 4 Mathematical Problems in Engineering then the revenue is enough to pay the purchasing cost. here is no interest charged by the supplier, although the credit period M is shorter than the replenishment cycle length, as shown in Figure 3. he objective function is the same with that under Scenario A: PVH 1 () = m 1 n= PV 1 () e rn Inventory level Q 1 e r PV 1 () {A 1 e r bθ [e(bθ) 1]e rm (bθ) r (e r 1) (bθ)(rbθ) [e(bθ) e r ]}. (13) se 2. Let Pe rm M D(t)e rt dt < CQ, butthereisbudget. hat means, although the revenue earned by time M is less than the purchasing cost, there is budget to pay the short purchasing cost at time M. herefore, there is still no interest charged by the supplier. he objective function is the same with that under Scenario A: PVH 1 () = m 1 n= PV 1 () e rn 1 e r PV 1 () {A 1 e r bθ [e(bθ) 1]e rm (bθ) r (e r 1) (bθ)(rbθ) [e(bθ) e r ]}. (14) se 3. Let Pe rm M D(t)e rt dt<cq,andthereisnobudget. Asaresult,alltherevenueearnedbytimeMis used to pay the purchasing cost and the supplier charges interest rate I c from M to for the unpaid balance. he interest and the remaining payments should be made at the end of the replenishment cycle. herefore, there are four elements in the retailer s cost: ordering cost, holding cost, the purchasing cost paid at time M, and the interest and the remaining payments made at the end of replenishment cycle.wooftheelementsare different from Scenario A: the purchasing cost paid at time M and the interest and the remaining payments made at the end of the replenishment cycle. M ime Figure 3: he retailer s inventory level of the first cycle when M<. he present value of the purchasing cost paid at time M during the first cycle is equal to the present value of the revenue earned by time M: M P D (t) e rt dt =P[ (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]. (15) he present value of the remaining payments and interest paid at the end of the replenishment cycle during the first cycle is [CQ Pe rm M D (t) e rt dt] [1 I c ( M)]e r ={ bθ [e(bθ) 1] Pe rm (bθ) r (1 e rm ) (bθ)(bθr) e (bθ) (1 e (bθr)m )]} 1I c ( M)]e r. he net present value of the cost during the first cycle is PV 2 () =Ah q (t) e rt dt [CQ Pe rm M D (t) e rt dt] 1I c ( M)]e r M P D (t) e rt dt (16)

5 Mathematical Problems in Engineering 5 =A (bθ) r (e r 1) (bθ)(bθr) [e(bθ) e r ] { bθ [e(bθ) 1] Pe rm (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]} 1I c ( M)]e r P[ (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]. (17) he present value of the total cost over the planning horizon L is PVH 2 () = m 1 n= PV 2 () e rn 1 e r PV 2 () 1 e r {A h q (t) e rt dt [CQ Pe rm M D (t) e rt dt] I c ( M) 1] e r M P D (t) e rt dt} {A 1 e r (bθ) r (e r 1) (bθ)(rbθ) [e(bθ) e r ] { bθ [e(bθ) 1] Pe rm (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]} 1I c ( M)]e r P[ (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]}. (18) Proposition 2. When there exists a unique at which dpvh 2 ()/d = =for (M, ),thenpvh 2 () is minimized at = if PV 2 (M)(erM 1) rpv 2 (M). Otherwise, =Misthe optimal solution. Proof. Let PV 2 () and PV 2 () represent the first and the second derivatives of PV 2 () with respect to, respectively. aking the first derivative of PVH 2 () with respect to, dpvh 2 () d = e r (1 e rl ) (1 e r ) 2 [PV 2 () (er 1) rpv 2 ()]. (19) Letting f 2 () = PV 2 ()(er 1) rpv 2 (), then dpvh 2 ()/d = (e r (1 e rl )/(1 e r ) 2 )f 2 (). aking the first derivative of f 2 () with respect to, we get f 2 () = PV 2 ()(er 1)rPV 2 ()(er 1).Inf 2 (), PV 2 () = (bθ) e r (bθ)(rbθ) (bθ) e bθ re r ] [e (bθ) bθr (1 e (bθr)m )] I c ( M) 1]e r { bθ [e(bθ) 1] Pe rm (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]} I c r I c r ( M)]e r P bθr e(bθ) (1 e (bθr)m )> PV r 2 () = (bθ) e r (bθ)(rbθ) bθ) 2 e (bθ) r 2 e r ] [(bθ) e (bθ) (bθ) bθr (1 e (bθr)m )e (bθ) ] 1I c ( M)]e r 2[e (bθ) bθr (1 e (bθr)m )e (bθ) ]

6 6 Mathematical Problems in Engineering I c r I c r ( M)]e r { bθ [e(bθ) 1] Pe rm (bθ) r (1 e rm ) (bθ)(bθr) e(bθ) (1 e (bθr)m )]} r 2 2I c ri c r 2 ( M)]e r P (bθ) bθr e(bθ) (1 e (bθr)m )>. (2) herefore, we can get f 2 ()=PV 2 ()(er 1)rPV 2 ()(er 1) >. Fromf 2 (M) = PV 2 (M)(erM 1) rpv 2 (M) and lim f 2 () =,wecangetthatthereisaunique solution,atwhichf 2 ( )=,andpvh 2 () is minimized from the Intermediate Value heorem ([2]). herefore, if f 2 (M) = PV 2 (M)(erM 1) rpv 2 (M), is the optimal solution. Otherwise, PVH 2 () is increasing on (M, ),and =Mis the optimal solution. 3. Solution Procedure In this section, we develop two algorithms to find the optimal solution under the condition of whether there is budget to pay thepurchasingcostattheendofthecreditperiod. se 1. If there is budget, the interest will never be charged by the supplier. Algorithm A. We have the following steps. Step 1. Input all the initial data, and set the optimal cycle length, the optimal present value of the total cost, and the number of cycles to be opt =, PVH() =,and m =. Step 2. Set m=m1.let=h/mand get Q from Q= (a/(b θ))[e (bθ) 1]. Find the corresponding M i from Q. hen, we can get PVH 1 (). Step 3. If PVH 1 () PVH(),update opt, PVH(),and m,andgotostep2. Otherwise, the current opt, PVH(), and m are the optimal solutions. se 2. here is no budget. herefore, when the revenue earned by the end of credit period is not enough to pay the purchasing cost, the supplier charges interest for the unpaid balance. Algorithm B. We have the following steps. Step 1. Input all the initial data, and set the optimal cycle length, the optimal present value of the total cost, the optimal PVH 2 (), and the number of cycles to be opt =, PVH() =, PVH 2 () =,andm =. Step 2. Set m = m 1.Let = H /m and get Q from Q = (a/(b θ))[e (bθ) 1]. Find the corresponding M i from Q. If M i and Pe rm M D(t)e rt dt < CQ, PVH 2 () is the optimal solution. Otherwise, PVH 1 () is the optimal solution. If we get PVH 1 (), gotostep3.otherwise,goto Step 4. Step 3. If PVH 1 () PVH(),update opt, PVH(), and m, and then go to Step 2. Otherwise, the current opt, PVH(),andm are the optimal solutions. Step 4. If PVH 2 () PVH(),update opt, PVH(), PVH 2 (),andm, and then go to Step 2. If PVH() < PVH 2 () PVH 2 (),updatepvh 2 () and go to Step 2. Otherwise, the current opt, PVH(), and m are the optimal solutions. 4. Numerical Example his section presents two cases where the results are illustrated. he following parameters are used in the first case. a = 2, b =.3, θ =.1, A = 1, h=1, L=2,and I c =.2. We assume the supplier offers a credit period of M [.5,.15,.25] when the retailer orders more than 3 items per time. here is no budget. From le 1, one can see that when the credit period is short, the retailer prefers to order less to decrease the interest charged by the supplier. When the credit period is so long enough that the retailer could earn enough revenue to pay the purchasing cost, the order quantity increases significantly. herefore, the credit period is a good promotion means to attract more orders. hen we consider a special case where the planning horizon is infinite. he following parameters are used in this case: a = 2, b =.3, θ =.1, A = 1, h=1, L=, and I c =.2. We assume the supplier offers a credit period of M [.5,.15,.25] when the retailer orders more than 2 items per time. here is budget to pay the purchasing cost at the end of the credit period. le 2 showsthat,asthediscountrateincreases,the retailer chooses to shorten the replenishment cycle and accelerate the fund flow. hat explains why people prefer short term and low risk investment when inflation is significant. he credit period could also increase the order quantity when theretailerhasbudget,buttheinfluenceissmall.heretailer just regards the trade credit as a discount. herefore, the credit period policy is much more attractive for small retailers or the ones who are in financial distress. 5. Conclusions In this paper, an inventory model for perishle products with stock-dependent demand and credit period under inflation and time discounting has been proposed. he credit period is dependent on the purchasing quantity. If the purchasing cost is totally paid at the end of the credit

7 Mathematical Problems in Engineering 7 le 1: Effect of r and M on decisions without budget. r M Q Present value of total cost.5 2/ PVH 2 () = / PVH 1 () = / PVH 1 () = / PVH 2 () = / PVH 1 () = / PVH 1 () = / PVH 2 () = / PVH 1 () = / PVH 1 () = le 2: Effect of r and M on decisions with budget and infinite L. r M Q Present value of total cost PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = PVH 1 () = period, the supplier does not charge any interest. Otherwise, the supplier charges interest for unpaid balance after credit period. All remaining payments should be made at the end of each cycle. From the results we can see that, as inflation rate goes up, the cycle length and order quantity decrease. he longer credit period offered by the supplier encourages the retailer to buy more, especially for these small retailers. he inflation could restrain the consumption for the perishle products with stock-dependent demand, and offering a trade credit is a good promotion for the supplier to enlarge the market under inflation. he results show that additional cost savings may be obtained by adjusting the order quantity with consideration of the inflation and time value of money. herefore, this research proposes a better replenishment policy than the basic EOQ model in terms of the total cost when inflation and time value of money variation are considered. he proposed model may be extended in several directions. First, we may further incorporate the pricing strategy into the analysis. Second, shortage is allowed and the unsatisfied demand could be lost, totally backordered, or partially backordered. hird, the deterministic demand may bechangedtoastochasticdemand. Notations P: Selling price per unit q(t): Inventorylevelattimet Q: Initialinventorylevel(q() = Q) C: Purchasing cost per unit, with P>C : Replenishment cycle length (decision varile) H: Holdingcostperunit A: Ordering cost L: R: Planning horizon Discount rate (i.e., opportunity cost) per unit time, which is related to the time valueofmoneyandinflationrate θ: Deteriorating rate M: Credit period, M={M,q Q< q 1 ;M 1,q 1 Q<q 2 ;...;M k,q k Q} I c : he interest charged per $ per unit time by the supplier when >M,withI c >e r. Acknowledgments his work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF D137). his work was also supported by NSFC (no ); Coordination and Disruption Coping for Customer Oriented Coopetition Supply Chain Networks. References [1] E. A. Silver and R. Peterson, Decision Systems for Inventory Management and Production Planning,JohnWiley&Sons,New York, NY, USA, 2nd edition, [2] R. C. Baker and. L. Urban, A deterministic inventory system with an inventory-level-dependent demand rate, the Operational Research Society, vol. 39, no. 9, pp , 1988.

8 8 Mathematical Problems in Engineering [3] S. S. Sana and K. S. Chaudhuri, A deterministic EOQ model with delays in payments and price-discount offers, European Operational Research, vol.184,no.2,pp , 28. [4] B. N. Mandal and S. Phaujdar, An inventory model for deteriorating items and stock dependent consumption rate, Operational Research Society, vol.4,no.5,pp , [5] S. S. Sana, Optimal selling price and lotsize with time varying deterioration and partial backlogging, Applied Mathematics and Computation,vol.217,no.1,pp ,21. [6] G. Padmanhan and P. Vrat, EOQ models for perishle items under stock dependent selling rate, European Operational Research,vol.86,no.2,pp ,1995. [7] M. Mandal and M. Maiti, Inventory of damagle items with varile replenishment rate, stock-dependent demand and some units in hand, Applied Mathematical Modelling, vol. 23, no. 1, pp , [8] C.-Y. Dye and L.-Y. Ouyang, An EOQ model for perishle items under stock-dependent selling rate and time-dependent partial backlogging, European Operational Research, vol. 163, no. 3, pp , 25. [9] C.-. Chang, J.-. eng, and S. K. Goyal, Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand, International Production Economics,vol.123,no.1,pp.62 68,21. [1] J. A. Buzacott, Economic order quantities with inflation, Operational Research Quarterly,vol.26,no.3,pp ,1975. [11] P. Vrat and G. Padmanhan, An inventory model under inflation for stock dependent consumption rate items, Engineering Costs and Production Economics, vol.19,no.1 3,pp , 199. [12] S. Bose, A. Goswami, and K. S. Choudhuri, An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting, Journal of the Operational Research Society, vol.46,no.6,pp , [13] B. Sarkar, S. S. Sana, and K. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money an EMQ model, Expert Systems with Applications, vol. 38, no. 11, pp , 211. [14] S. S. Sana, Sales team s initiatives and stock sensitive demand aproductioncontrolpolicy, Economic Modelling, vol.31,pp , 213. [15] K.-J. Chung and C.-N. Lin, Optimal inventory replenishment models for deteriorating items taking account of time discounting, Computers and Operations Research,vol.28,no.1,pp.67 83, 21. [16] K.-L. Hou, An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting, European Operational Research,vol.168,no.2,pp ,26. [17] C. K. Jaggi, K. K. Aggarwal, and S. K. Goel, Optimal order policy for deteriorating items with inflation induced demand, International Production Economics, vol.13,no.2, pp ,26. [18] A. Roy, M. K. Maiti, S. Kar, and M. Maiti, An inventory model for a deteriorating item with displayed stock dependent demand under fuzzy inflation and time discounting over a random planning horizon, Applied Mathematical Modelling, vol. 33, no. 2, pp , 29. [19] C.-. Chang, An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity, International Production Economics,vol.88,no.3,pp , 24. [2] G. B. homas and R. L. Finney, lculus with Analytic Geometry, Addison-Wesley, Reading, Mass, USA, 9th edition, 1996.

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