Research Article Two-Level Credit Financing for Noninstantaneous Deterioration Items in a Supply Chain with Downstream Credit-Linked Demand

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1 Discrete Dynamics in Nature and Society Volume 13, Article ID , pages Research Article wo-level Credit Financing for Noninstantaneous Deterioration Items in a Supply Chain with Downstream Credit-Linked Demand Yong He and Hongfu Huang Institute of Systems Engineering, School of Economics and Management, Southeast University, Nanjing 196, China Correspondence should be addressed to Yong He; heyong@16.com Received May 13; Revised 3 July 13; Accepted 31 July 13 Academic Editor: Zhigang Jiang Copyright 13 Y. He and H. Huang. 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. rade credit financing is a useful tool in business today, which can be characterized as the agreement between supply chain members such as permissible delay in payments. In this study, we assume that the items have the property of noninstantaneous deterioration and the demand is a function of downstream credit. hen, an EOQ model for noninstantaneous deterioration is built based on the two-level financing policy. he purpose of this paper is to maximize the total average profit by determine the optimal downstream credit period, the optimal replenishment cycle length, and the optimal ordering quantity per cycle. Useful theorems are proposed to characterize the method of obtaining the optimal solutions. Based on the theorems, an algorithm is designed, and numerical tests and sensitive analysis are provided. Lastly, according to the sensitive analysis, managerial insights are proposed. 1. Introduction Deteriorating products are prevalent in our daily life. According to Shah et al.[1], deterioration is defined as decay, change, or spoilage through which the items are not the same as its initial conditions. here are two categories of deteriorating items. he first one is about the items that become decayed, damaged, or expired through time, such as meat, vegetables, fruits, and medicine. he second category is about the items that lose part or total value through time, such as computer chips, mobile phones, and seasonal products. Both of the two kinds of items have short life cycle. After a period of existence in the market, the items lose the original economic value due to consumer preference, product quality or other reasons. Early research on deteriorating items can be dated back to An EOQ model with exponentially decaying inventory was initially proposed by Ghare and Schrader []. In their study, they show that the inventory level is not only related to the market demand but also to the deteriorating rate, which is a negative exponential function of time. hey proposed the deteriorating items inventory model in which di(t)/dt + I(t) = f(t). Inthefunction,f(t) isthedemandrateat time t, I(t) stands for the inventory level and, refers to the deteriorating rate. Based on the assumption of Ghare and Schrader [], many researchers extended the model by making different assumptions for the deteriorating rate, the demand rate, or shortages allowed, such as Covert and Philip [3], Philip [4],Y.HeandJ.He[5], He et al. [6], Yang et al. [7], and He and Wang [8]. hey all proposed instructive conclusions for real practice. Recently, Goyal and Giri [9] and Li et al. [1] made excellent and detailed review of deteriorating inventory works. In the above mentioned works, most researchers assume that the deterioration of the products in inventory starts fromthearrivalinstock.but,inpractice,somekindsof products may maintain the original condition for a short time, which means during that time, there is no deterioration. For example, the stock of the firsthand vegetables or fruits has a high quality at the beginning time, in which there is no spoilage. Afterward, the stock starts to perish and induce the deterioration cost. Under this circumstance, it is obvious that the assumption that the deterioration starts from the beginning can make the retailer overestimate the inventory cost, which leads to uneconomical inventory policies. his phenomenon is first proposed by Wu et al. [11] as noninstantaneous deterioration. hey proposed the optimal

2 Discrete Dynamics in Nature and Society replenishment policy of an EOQ model when the demand is inventory level dependent and can be partially backlogged. Based on this assumption, there were several interesting and relevant works such as Ouyang et al. [1, 13], Sugapriya and Jeyaraman [14], Chung [15], Yang et al. [16], Wu et al. [17], Geetha and Uthayakumar [18], Singh et al. [19], Chang and Lin [], Changet al. [1], Maihami and Nakhai Kamalabadi [], and Shah et al. [1]. Furthermore, different from traditional EOQ models, in which payment should be made to the supplier after the retailer receiving the stock, many researches are focusing on the application of trade credit financing tools to improve profits or reduce cost of the supply chain. Actually, it is more practical that the supplier/retailer allows for a fixed period to settle the payment without penalty for its retailer/customer toinduceitsdemandrateorreduceonhandinventory. his permissible delay in payment can reduce the capital investment of stock amount, thus reducing the holding cost of inventory. Besides, during the credit period, the retailer can gain interest profit of his sales revenue by the investing or banking business. Over the years, research on this part is prevalent in many works. Goyal [3] wasthefirstto study the EOQ model with permissible delay in payment. hen Aggarwal and Jaggi [4], Jamal et al. [5], Chang and Dye [6], and eng [7] extended Goyal s [3] modelfor deteriorating items, allow for shortages, and so forth. A lot of useful and interesting managerial insights were proposed in their papers. More research on this part can be found in Chung et al. [8], eng et al. [9], Jaber and Osman [3], and Chung and Liao [31]. he above mentioned works are all assumed one-level credit financing, but sometimes this assumption is unrealistic in real business. For car companies, like AA (India) and OYOA (Japan), they not only delay the payment of the purchasing cost until the end of the credit period to their suppliers, but also provide a credit period to their customers. his kind of business style is called two-echelon (two-level or two-part)creditfinancing.huang[3]firstproposedaneoq model with two-level credit financing, and the retailer s credit period is longer than the customer s. ill now, researches on two-level financing can be seen in Ho et al. [33], Liao [34], hangam and Uthayakumar [35], Chen and Kang [36], Min et al. [37], Ho [38], Urban [39], and Chung and Cardenas- Barron [4]. Although the credit financing problems for EOQ or EPQmodelshavebeenstudiedbymanyresearchers,in most works, it is assumed that the credit period offered by the supplier/retailer to the retailer/customer is a constant parameter. Actually, in real business, supplier/retailer can decide the credit period by himself to minimize inventory cost or maximize total profit. Su et al. [41] studiedtheeoq problem of a two-echelon supply chain under two-echelon trade credit financing, where the demand rate is assumed to be dependent on credit period offered by retailer to the customers. he demand rate is an exponential function of thecreditperiod.hesimilarassumptioncanbeseeninho [38]. Besides, Jaggi et al. [4] made a detailed explanation for the property of the credit-lined demand. hey show that demand function for any credit period can be represented as the differential equation: D(N + 1) D(N) = r(s D(N)) where S is the maximum demand, and N is the credit period. hangam and Uthayakumar [35] thenproposeda more general continuous differential equation of the demand functionbasedonjaggietal.[4] and extended their model to deteriorating items. hese effects of the situations imposes us to establish an EOQ model for noninstantaneous deteriorating items with credit-linked demand under two-echelon financing policy, which can be treated as a general framework for several papers such as Ouyang et al. [1] andjaggietal.[4]. here are several useful theorems proposed to illustrate the optimal solution for the model in different conditions. Here we take into account the following factors: (1) noninstantaneous deterioration items; () two-level credit financing is considered; (3) credit-linked demand rate; (4) the credit period offered by the supplier is not necessarily shorter than that offered by theretailertothecustomer. he remainder of this paper is organized as follows. In Section, assumptions and notations are described in detail. In Section 3, theeoqmodelfornoninstantaneous deterioration items under two-level credit financing is made. In Section 4,solutionsforthemodelareproposedanduseful theorems are presented. In Section 5, two special cases are discussed. In Section 6, numerical examples and sensitive analysis are made, and managerial insights are proposed. Conclusions and future research are given in Section 7.. Notations and Assumptions he following notations and assumptions are adopted throughout this paper. (1)heannualdemandratefortheitem,D(N), which is sensitive to the credit offered by the retailer to customers and is a marginally increasing function w.r.t. N. N, is an integer (N = 1,,3,...)anda decision variable throughout this paper. D(N) and D can be used interchangeably in this paper. () Replenishment rate is infinite. (3) Shortage is not allowed. (4) he product life (time to deterioration) has a probability density function f(t) = e (t γ) for t > γ, where γ isthelengthoftimeinwhichtheproducthas no deterioration and is a parameter. he cumulative distribution function of t is given by F(t) = t f(t) = γ 1 e (t γ) for t>γ, so that the deterioration rate is r(t) = (f(t))/(1 F(t)) for t>γ. (5) he length of time in which the product has no deterioration, γ,canbeestimatedbyutilizingtherandom sample data of the product during the past time and statistical maximum method. For simplifying, it is assumedtobeaconstant. (6) M is the permissible delay period in payment for the retailer offered by the supplier (upstream credit). During the period, the retailer can use sales revenue to earn the interest with an annual rate I p up to

3 Discrete Dynamics in Nature and Society 3 the end of M. Attimet = M,thecreditissettled and the retailer has to pay the interest at rate I c for the items in stock. N is the permissible delay period inpaymentforthecustomerofferedbytheretailer (downstream credit). During the period, the retailer hastobeartheopportunitycostoftherevenuewhich is not settled in time N at the rate of I p. (7) ime horizon is infinite. (8) is the length of replenishment cycle. Q is the replenishment quantity per cycle. and Q are decision variables. (9) A, h, c,andp denote the ordering cost per order, the holding cost per time per item excluding interest charges, the purchasing cost per item, and the selling price per item, respectively. All these parameters are constant and positive. (1) For γ<, there is no deterioration in the time to, where the inventory level is I 1 (t). (11) For > γ, there is no deterioration in the time toγ, wheretheinventorylevelisi 1 (t); thereis deterioration in the time γ to,wheretheinventory level is I (t). (1) Z i () is the total average profit which consists of (a) sales profit (SP), (b) the cost of ordering (OC), (c) cost of holding inventory (HC) (excluding interest charges), (d) cost of deterioration (DC), (e) capital opportunity cost (IC), (f) interest earned from the sales revenue (IE), i = 1,, 3, wherei=1indicates M N, i=indicates N M N+γ,andi=3 indicates M N+γ. (13) is the optimal replenishment cycle length. Q is the optimal replenishment quantity. Z i is the minimum of the total annual cost; that is, Z i =Z i ( ). 3. Model Formulation First, we model the demand rate D(N) w.r.t. N. According to Jaggi et al. [4] and hangam and Uthayakumar [35], the marginal effect of credit period on sales is proportional to the unrealized market demand without any delay. Under the assumption, demand can be defined in the following two ways. (1) he demand function of demand can be represented as a differential difference equation D (N+1) D(N) =r(β D(N)). (1) () he demand rate can be depicted by the partial differential equation D (N) N =r(β D(N)). () In both (1) and(), r<1, β is the maximum value of demand rate over the planning horizon. Boundary conditions Inventory level γ ime Figure 1: Inventory system for γ. are D(N ) = α, D(N ) = β.hesolutionsfor(1) and ()are ype 1: D (N) =β (β α)e rn, (3) ype : D (N) =β (β α) (1 r) N. (4) he two types of demand functions are adopted in the following analysis. he inventory system evolves as follows: Q units of the items arrived at the warehouse at the beginning of each cycle. When γ, there is no deterioration in a single cycle. he inventory system is depicted in Figure 1. he inventory level decreases owing to the constant demand rate during the whole cycle. It is given that I 1 (t) =Q Dt, γ. (5) When γ, there is no deterioration in the time interval [, γ]. After that, in the time interval [γ, ], items deteriorates at a constant deterioration rate,which is shown in Figure. he inventory level decreases owing to the demand rate in time [, γ],whichisgivenby I 1 (t) =Q Dt, t γ. (6) In time [γ, ], the inventory level declines owing to both the demand rate and the deterioration. hus, the inventory level is represented by the partial differential equation I (t) = D I t (t), γ t, (7) with the boundary condition I () =. he solution of (7)is I (t) = D [e( t) 1], γ t. (8)

4 4 Discrete Dynamics in Nature and Society (d) Cost of deterioration items (DC): For γ,thereis no deterioration. For γ, the cost of deteriorated items is c(q D)/. So, the deterioration cost is given by Inventory level DC = { c (Q D) γ { { γ. (15) (e) Opportunity cost (IC) and interest earned from sales revenue (IE): In order to establish the total relevant inventory cost function, we consider three cases: Case 1. M N;Case. N M N + γ;andcase3. M N+γ. γ ime Case 1 (M N). In this case, there are two circumstances: γand γ.and,whenm N, there is no interest earned by the retailer. Figure : Inventory system for γ. Considering the continuity of I 1 (t) and I (t) at time t=γ, it follows from (5)and(7)that I 1 (γ) = I 1 (γ) = Q Dγ = D [e( γ) 1], (9) which implies that the ordering quantity per cycle is Q=Dγ+ D [e( γ) 1]. (1) hensubstitute(1)into(6), we have I 1 (t) =D(γ t) + D [e( γ) 1], t γ. (11) hetotalannualrelevantcostconsistsofthefollowingfive parts. (a) Sales profit (SP): SP =(p c)d. (1) (b) Cost of ordering cost per year (OC): OC = A. (13) (c) Cost of holding inventory (HC): here are two possible situations based on the value of and γ. When γ, the inventory system is the first type shown in Figure 1. When γ, the inventory system is the second type depicted in Figure. Consequently, the inventory holding cost is given by h { I 1 (t) dt HC = γ { h { ( I 1 (t) dt + I (t) dt) γ γ γ. (14) (1) γ. he inventory system is depicted in Figure 3. he retailer has the opportunity cost and has no interest earned. he opportunity cost is calculated as IC 11 = ci c ((N M) Q+D ). (16) he total average profit function is Z 11 (, N) = SP (OC + HC + DC + IC 1 ) =(p c)d A (h + ci c)d ci c (N M) D. (17) () γ. he inventory system is depicted in Figure 4. he retailer has the opportunity cost and has no interest earned. he opportunity cost is calculated as IC 1 = ci γ c ((N M) Q+ I 1 (t) dt + I (t) dt). γ (18) he total average profit function is Z 1 (, N) = SP (OC + HC + DC + IC 1 ) =(p c)d A + (h + ci c)d ( γ + γ (e( γ) 1)+ 1 (e( γ) ( γ) 1)) (c + ci c (N M))D (γ + 1 (e( γ) 1)) + cd. (19) he problem of Case 1 is to maximize the function Z 1 (, N) ={ Z 11 (, N) γ () Z 1 (, N) γ.

5 Discrete Dynamics in Nature and Society 5 Paid Inventory level Interest cost M N γ +N ime M N +N ime (a) (b) Figure 3: Inventory system for Case 1 when γ. Paid Inventory level Interest cost M N γ +N ime M N +γ +N ime (a) (b) Figure 4: Inventory system for Case 1 when γ. Case (N M N+γ). In this case, there are three circumstances: M N, M N γ, and γ. (1) M N. he inventory system is depicted in Figure 5. he retailer has no opportunity cost and only has the interest earned. he interest earned per cycle is calculated as IE 1 = pi p (D +D(M N)). (1) he total average profit function is Z 1 (, N) = SP (OC + HC + DC IE 1 ) =(p c)d A hd ( D + pi p +D(M N)). ()

6 6 Discrete Dynamics in Nature and Society Earned Inventory level Interest cost N +N γ ime M N+γ N +N γ ime M N+γ (a) (b) Figure 5: Inventory system for Case when M N. Paid Inventory level Interest cost Earned N M γ N+γ ime N M N+ ime (a) (b) Figure 6: Inventory system for Case when M N γ. () M N γ. he inventory system is depicted in Figure 6. he opportunity cost per cycle is calculated as IC = ci c D(+N M). (3) he interest earned per cycle is calculated as IE = pi p D(M N). (4) he total average profit function is Z (, N) = SP (OC + HC + DC + IC IE ) =(p c)d A hd D(+N M) ci c + pi p D(M N). (5)

7 Discrete Dynamics in Nature and Society 7 Paid Inventory level Interest cost Earned N γ M N+γ ime N M N+γ ime N+ (a) (b) Figure 7: Inventory system for Case when γ. (3) γ. he inventory system is depicted in Figure 7. he opportunitycostpercycleiscalculatedas IC 3 = ci c ( γ M N I 1 (t) dt + I (t) dt). (6) γ he interest earned per cycle is calculated as IE 3 = pi p D(M N). (7) he total average profit function is Z 3 (, N) = SP (OC + HC + DC + IC 3 IE 3 ) = (p c) D (A + (h + ci c)dγ ci c (M N) Dγ + ci c(m N) D +cdγ pi p(m N) D ) 1 ( hγd + ci c (γ M+N)D+cD ) e( γ) 1 (h + ci c)de ( γ) ( γ) 1 +cd. (8) he problem of Case is to maximize the function Z { 1 (, N) Z (, N) = Z { (, N) { Z 3 (, N) M N M N γ γ. (9) Case 3 (M N + γ). In this case, there are three circumstances: γ, γ M N, and M N. (1) γ. he inventory system is depicted in Figure 8.here is no opportunity cost under this circumstance. he interest earned per cycle is calculated as IE 31 = pi p (D +D(M N)). (3) he total average profit function is Z 31 (, N) = SP (OC + HC + DC IE 31 ) =(p c)d A hd ( D + pi p +D(M N)). (31) () γ M N. he inventory system is depicted in Figure 9. here is no opportunity cost under this circumstance. he interest earned per cycle is calculated as IE 3 = pi p (D +D(M N)). (3)

8 8 Discrete Dynamics in Nature and Society Earned Inventory level Interest cost N +N γ N+γ ime M N +N ime M (a) (b) Figure 8: Inventory system for Case 3 when γ. he total average profit function is Z 3 (, N) = SP (OC + HC + DC IE 3 ) =(p c)d A hd ( γ + γ (e( γ) 1)+ 1 (e( γ) ( γ) 1)) cd (γ + 1 (e( γ) 1))+cD+ pi p ( D +D(M N)). (33) (3) M N. he inventory system is depicted in Figure 1. he opportunity cost per cycle is calculated as IC 33 = ci c M N he interest earned per cycle is calculated as I (t) dt. (34) IE 33 = pi p D(M N). (35) he total average profit function is Z 33 (, N) = SP (OC + HC + DC + IC 31 IE 31 ) =(p c)d A hd ( γ + γ (e( γ) 1)+ 1 (e( γ) ( γ) 1)) + cd (γ + 1 (e( γ) 1)) +cd ci c (e( M+N) 1 + pi p(m N) D. +M N ) (36) he problem of Case 3 is to maximize the following function: Z { 31 (, N) Z 3 (, N) = Z { 3 (, N) { Z 33 (, N) 4. Solution Procedure γ γ M N M N. (37) Now, we shall determine the optimal cycle length and downstream credit period for the three cases under maximizing the total average profit function. o find the optimal solution, say (,N ), for Z i (, N) (i = 1,, 3), the following procedures

9 Discrete Dynamics in Nature and Society 9 Earned Inventory level Interest cost N γ N+γ +N ime M N +N ime M (a) (b) Figure 9: Inventory system for Case 3 when γ M N. Paid Inventory level Interest cost Earned N γ N+γ M ime N γ M ime N+ (a) (b) Figure 1: Inventory system for Case 3 when M N. areconsidered.wefirstanalyzethepropertyoftheoptimal replenishment cycle length for any fixed N (N = 1,, 3,...). Case 1 (M N). he problem is to minimize function (). It can be calculated that Z 11 (γ) = Z 1 (γ). Sofunction () is continuous at point =γ. he first-order necessary condition for Z 11 () in (13)tobemaximizedis Z 11 ( N) = A (h + ci c)d =. (38) he second-order sufficient condition is Z 11 ( N) = A 3 <. (39) Consequently, Z 11 ( N)isaconcave function of. hus, there exists a unique value of (say 11 ) which minimize Z 11 ( N)as A 11 = (h + ci c )D. (4)

10 1 Discrete Dynamics in Nature and Society o ensure γ,we substitute(4)into inequality γand we obtain <A (h+ci c )Dγ. (41) Likewise, the first-order necessary condition for Z 1 ( N) in (19)tobemaximizedis Z 1 ( N) = [A+(h+cI c)dγ / + cdγ + ci c (N M) Dγ] [ (h + ci c)dγ ( e( γ) (h + ci c)d + c+ci c (N M) D] e( γ) + 1 ) ( e( γ) e( γ) + 1 γ )=. (4) By using the analogous arguments, we can easily obtain that [A + (h + ci c)dγ [ (h + ci c)dγ (e ( γ) e ( γ) +1) (h + ci c)d +cdγ+ci c (N M) Dγ] + c+ci c (N M) D] (e ( γ) e ( γ) +1 γ)=. (43) It is not easy to find a closed form solution of from (43). But we can show that the value of satisfy (43) not only exists but also is unique. So we have the following lemma. Lemma 1. For a given N,whenM N, (a) if A (h + ci c )Dγ, then the solution of [γ,+ ) (say 1 )in(43) not only exists but also is unique. (b) if <A<(h+cI c )Dγ, then the solution of [γ, + ) in (43) does not exist. Proof. See Appendix A. According to Lemma 1, we have the following result. Lemma. For a given N,whenM N, (a) if A (h+ci c )Dγ, then the total relevant cost Z 1 ( N) has the global maximum value at point = 1, where 1 [γ,+ )and satisfies (43). (b) if <A<(h+cI c )Dγ, then the total relevant cost Z 1 ( N) has maximum value at the boundary point =γ. Proof. See Appendix A. For notational convenience, we mark that Δ 1 = (h + ci c )Dγ. Combining the above mentioned inequality (41), Lemmas 1 and and the assumption M N,wecanobtain the following theorem. heorem 3. For a given N,whenM N, (a) if A < Δ 1,thenZ 1 ( N)=Z 11 ( 11 N)and = 11 ; (b) if A Δ 1,thenZ 1 ( N) = max(z 1 ( 1 N), Z 1 (γ N)). Hence is 1 or γ associated with lower total average profit. Case (N M N + γ). he problem is to maximize function (9). It can be calculated that Z 1 (M M N) = Z (M M N), Z (γ N) = Z 3 (γ N).Sofunction(9) is continuous at point =M Nand =γ. he first-order necessary condition for Z 1 ( N) in () to be minimized is Z 1 ( N) = A (h + pi p)d he second-order sufficient condition is Z 1 ( N) =. (44) = A 3 <. (45) Consequently, Z 1 ( N)isaconcave function of. hus, there exists a unique value of (say 1 ) which minimize Z 1 ( N)as A 1 = (h + pi p )D. (46) o ensure M N,wesubstitute(46)intoinequality M Nand obtain <A (h+pi p )D(M N). (47) he first-order necessary condition for Z ( N)in (5)to be maximized is Z ( N) = A (h + ci c)d =. pi p(m N) D he second-order sufficient condition is Z ( N) + ci c(m N) D (48) = A 3 <. (49) Consequently, Z ( N)is a concave function of. hus there exists a unique value of (say )whichmaximizes Z ( N)as = A + (ci c pi p ) (M N) D. (5) (h + ci c )D

11 Discrete Dynamics in Nature and Society 11 o ensure M N< γ,wesubstitute(5)intoinequality M N< γand obtain (h + pi p )D(M N) <A (h+ci c )γ D (ci c pi p ) (M N) D. (51) Likewise, the first-order necessary condition for Z 3 ( N) in (8)tobemaximizedis Z 3 ( N) =[A+ (h + ci c)dγ ci c (M N) Dγ + ci c(m N) D +cdγ pi p(m N) D ] ( ) 1 [ hγd + ci c (γ M+N)D+cD ] ( e( γ) (h + ci c)d e( γ) + 1 ) ( e( γ) e( γ) + 1 γ )=. (5) By using the analogous arguments, we can easily obtain that [A + (h + ci c)dγ + ci c(m N) D ci c (M N) Dγ +cdγ pi p(m N) D ] [ hγd + ci c (γ M+N)D+cD ] (e ( γ) e ( γ) +1) (h + ci c)d So,wehavethefollowinglemma. (e ( γ) e ( γ) +1 γ)=. Lemma 4. For a given N,whenN M N+γ, (53) (a) if A (h + ci c )γ D (ci c pi p )(M N) D,then the solution of [γ,+ )(say 3 )in(53) not only exists but also is unique. (b) if A < (h + ci c )γ D (ci c pi p )(M N) D, then the solution of [γ,+ )in (53) does not exist. Proof. See Appendix B. According to Lemma 4, we have the following result. Lemma 5. For a given N,whenN M N+γ, (a) if A (h + ci c )γ D (ci c pi p )(M N) D,then thetotalaverageprofitz 3 ( N) has the global maximum value at point = 3,where 3 [γ,+ ) and satisfies (5); (b) if A < (h + ci c )γ D (ci c pi p )(M N) D, then the total average profit Z 3 ( N) has maximum value at the boundary point =γ. Proof. See Appendix B. For notational convenience, we mark that Δ =(h+pi p )D(M N), Δ 3 =(h+ci c )γ D (ci c pi p ) (M N) D. (54) Combining the forementioned mentioned equations (47)and (51), Lemmas 4 and 5, and the assumption N M N+γ, we can obtain the following theorem. heorem 6. For a given N,whenN M N+γ, (a) if <A<Δ,thenZ ( N)=max(Z 1 ( 1 N), Z 1 (M N N)). Hence is 1 or M N associated with higher total average profit; (b) if Δ A<Δ 3,thenZ ( N)=max(Z ( N), Z (γ N)). Hence is 3 or γ associated with higher total average profit. (c) if A Δ 3,thenZ ( N)=Z 3 ( 3 N), = 3. Case 3 (M N+γ). he problem is to maximize function (37).ItcanbecalculatedthatZ 31 (γ N) = Z 3 (γ N) and Z 3 (M N N) = Z 33 (M N N). Sofunction(37) is continuous at point =γand =M N. he first-order necessary condition for Z 31 ( N) in (31) to be minimized is Z 31 ( N) he second-order sufficient condition is Z 31 ( N) = A hd pi p =. (55) = A 3 <. (56) Consequently, Z 31 ( N)is a convex function of. hus, there exists a unique value of (say 31 ) which minimizes Z 31 ( N)as A 31 = (h + pi p )D. (57) o ensure <γ,wesubstitute(57)intoinequality<γand obtain <A<(h+pI p )Dγ. (58)

12 1 Discrete Dynamics in Nature and Society Likewise, the first-order necessary condition for Z 3 ( N) in (33)tobemaximizedis Z 3 ( N) = (A + hdγ / + cdγ) ( e( γ) hd (e( γ) ( e( γ) + 1 ) e( γ) hdγ + cd ) + 1 γ ) pi pd =. (59) By using the analogous arguments, we can easily obtain that (A + hdγ +cdγ) ( hdγ + cd )(e ( γ) e ( γ) +1) hd (e( γ) e ( γ) +1 γ) pi pd =. Let Δ 4 =(hd+pi p D) γ, Δ 5 = (hdγ hdγ + cd +cdγ)+( + hd ) [(M N) e (M N γ) e (M N γ) +1] hdγ +pi p D(M N). hen, we have the following lemma. Lemma 7. For a given N,whenM N+γ, (6) (61) (a) if Δ 4 A Δ 5, then the solution of [γ,m N] (say 3 )in(6) not only exists but also is unique; (b) if A < Δ 4 or A > Δ 5, then the solution of [γ, M N] in (6) does not exist. Proof. See Appendix C. According to Lemma 7, we have the following result. Lemma 8. For a given N,whenM N+γ, (a) if Δ 4 A Δ 5, then the total average profit Z 3 ( N) has the global maximum value at point = 3, where 3 [γ,m N]and satisfies (6); (b) if A < Δ 4, then the total average profit Z 3 ( N) has the maximum value at the boundary point =γ; (c) if A > Δ 5, then the total average profit Z 3 ( N)has themaximumvalueattheboundarypoint=m N. Proof. See Appendix C. Likewise, the first-order necessary condition for Z 33 ( N) in (36)tobemaximizedis Z 33 ( N) = (A + hdγ / + cdγ ci c (M N) pi p (M N) D/) ( + hd hdγ + cd )( e( γ) e( γ) + 1 ) (e( γ) e( γ) + 1 γ ) ci c e( M+N) + 1 )=. (6) By using the analogous arguments, we can easily obtain that (A + hdγ ( hd +cdγ ci c (M N) pi p(m N) D ) hdγ + cd )(e ( γ) e ( γ) +1) (e( γ) e ( γ) +1 γ) ci c (e( M+N) e ( M+N) +1)=. Also, we have the following lemma. Lemma 9. For a given N,whenM N+γ, (63) (a) if A Δ 5, then the solution of [M N,+ )(say 33 )in(63) not only exists but also is unique; (b) if <A<Δ 5, then the solution of [M N,+ ) in (63) does not exist. Proof. See Appendix D. According to Lemma 9,we have the following result. Lemma 1. For a given N,whenM N+γ, (a) if A Δ 5, then the total average profit Z 33 ( N) has the global maximum value at point = 33,where 33 [M N, + ) and satisfies (63); (b) if <A<Δ 5, then the total average profit Z 33 ( N) has the global maximum value at point =M N, where 33 [M N, + ). Proof. See Appendix D.

13 Discrete Dynamics in Nature and Society 13 Combining the forementioned mentioned equation (58), Lemmas 7 1, and the assumption M N+γ, we can obtain the following theorem. heorem 11. For a given N,whenM N+γ, (a) if <A<Δ 4,thenZ 3 ( N)=max(Z 31 ( 31 N), Z 31 (γ N)), Hence is 31 or γ associated with higher total average profit; (b) if Δ 4 A<Δ 5,thenZ 3 ( N)=max(Z 3 ( 3 N), Z 3 (M N N)). Hence is 1 or M N associated with higher total average profit; (c) if A Δ 5,thenZ 3 ( N)=Z 33 ( 33 N),and = 33. For the downstream credit is an integer according to the assumptions,andinteractivealgorithmscanbeusedtofind the optimal solutions for our model. By summarizing the results in heorems 3, 6, and 11, an algorithm to illustrate the optimal solution for the model is proposed as follows. Algorithm 1. Consider the following: (1) Let N=1. () Compare the value of M, N, γ. IfM N,thengo to step 3; If N M N+γ,thengotostep5;If M N+γ,thengotostep7. (3) Calculate Δ 1 (N), (1) If A < Δ 1,then = 11 and Z 1 ( N,N) = Z 1 ( N)=Z 11 ( 11 N). () If A Δ 1,and (i) Z 1 ( 1 N) Z 1 (γ N),then = 1, Z 1 ( N,N)=Z 1( N)=Z 1 ( 1 N); (ii) Z 1 ( 1 N)<Z 1 (γ N), then =γ, Z 1 ( N,N)=Z 1( N)=Z 1 (γ N). (4) If Z 1 ( N 1,N 1) Z 1( N,N), then the optimum solution, say (,N ), is ( N 1,N 1) and Z = Z 1 (,N ).Otherwise,N=N+1,gotostep. (5) Calculate Δ (N) and Δ 3 (N). (1) If A < Δ,then = 1 and Z ( N,N) = Z ( N)=Z 1 ( 1 N). () If Δ A<Δ 3,and (i) Z ( N) Z (M N N),then =, Z ( N,N) = Z ( N)=Z ( N); (ii) Z ( N) < Z (M N N), then =M N, Z ( N,N) = Z ( N)= Z (M N N). (3) If A Δ 3,and (i) Z 3 ( 3 N) Z 3 (γ N),then = 3, Z ( N,N)=Z ( N)=Z 3 ( 3 N); (ii) Z 3 ( 3 N)<Z 3 (γ N),then =M N, Z ( N,N)=Z ( N)=Z 3 (γ N). (6) If Z ( N 1,N 1) Z ( N,N), then the optimum solution, say (,N ), is ( N 1,N 1) and Z = Z (,N ).Otherwise,N=N+1,gotostep. (7) Calculate Δ 4 (N) and Δ 5 (N). (1) If A < Δ 4,then = 31 and Z 3 ( N,N) = Z 3 ( N)=Z 31 ( 31 N). () If Δ 4 A<Δ 5,and (i) Z 3 ( 3 N) Z 3 (γ N),then = 3, Z 3 ( N,N)=Z 3( N)=Z 3 ( 3 N); (ii) Z 3 ( 3 N)<Z 3 (γ N), then =γ, Z 3 ( N,N)=Z 3( N)=Z 3 (γ N). (3) If A Δ 5,and (i) Z 33 ( 33 N) Z 33 (M N N),then = 33, Z 3 ( N,N) = Z 3( N)=Z 33 ( 33 N); (ii) Z ( N) < Z (M N N), then =M N, Z 3 ( N,N) = Z 3( N)= Z 33 (M N N). (8) If Z 3 ( N 1,N 1) Z 3( N,N), then the optimum solution, say (,N ), is ( N 1,N 1) and Z = Z 3 (,N ).Otherwise,N=N+1,gotostep. After obtaining the optimal replenishment cycle,the optimal order quantity can be determined by (1), which is given that Q = Dγ + (D/)[e ( γ) 1]. 5. Special Cases In this section, two special cases are discussed (i.e., [1, 4]) and descriptions are made. Special Case 1 (Ouyang et al. [1]). In this model, they consider an one-level credit financing problem for noninstantaneous deteriorating items with a constant demand, which means, in our model N, r and lim r D(N) = α. If we set N and r,thenforcases and 3 in our paper, the problem is (1) for M γ lim limz 1 () N r =(p c)α [ A + hα lim limz () N r =(p c)α pi p (α +α(m ))] [ A + hα + ci c α( M) pi p αm ]

14 14 Discrete Dynamics in Nature and Society lim limz 3 () N r =(p c)α [(A+ (h + ci c)αγ ci c Mαγ + ci cm α +cαγ pi pm α ) 1 +( hγα + ci c (γ M) α + cα ) e( γ) 1 + (h + ci c)α e( γ) ( γ) 1 cα], lim limz 33 () N r =(p c)α [ A + hα ( γ + γ (e( γ) 1) + 1 (e( γ) ( γ) 1)) + cα (γ + 1 (e( γ) 1)) cα + ci c (e( M) 1 +M ) pi pm α ], Z { 1 () Z () = Z { () { Z 3 () M M γ γ, (64) Z { 31 () Z 3 () = Z { 3 () { Z 33 () γ γ M M. (65) () for M γ lim limz 31 N r =(p c)α [ A + hα lim N lim r Z 3 () =(p c)α [ A + hα pi p (α +α(m ))] ( γ + γ (e( γ) 1) + 1 (e( γ) ( γ) 1)) + cα (γ + 1 (e( γ) 1)) cα pi p (α +α(m ))], In this condition, the relevant cost function is the same as the problem in Case 1 (1) (14) and Case (15) (17) of Ouyang et al. [1]. So Ouyang et al. [1]isaspecialcaseofourmodel. Special Case (Jaggi et al. [4]). In this model, they consider the tow-level financing problem for items without deterioration and with a credit dependent demand rate, which means that and γ in our model.if we set and γ,forcases1 and in our paper, the problem is (1) for M N γ, according to Case 1 in our paper, lim lim Z γ 11 (, N) =(p c)d [ A + (h + ci c)d +ci c (N M) D] Z 1 (, N), () for N M γ, according to Case in our paper, lim lim Z γ 1 (, N) =(p c)d [ A + hd Z (, N) (66) pi p (D +D(M N))]

15 Discrete Dynamics in Nature and Society 15 lim lim Z γ (, N) =(p c)d [ A + hd + ci c D(+N M) pi p D(M N) ] Z 3 (, N), which can be reduced as follows: (67) otal average profit Z { 1 (, N) Z (, N) = Z { (, N) { Z 3 (, N) M N +N N +N M M N +N. (68) In this condition, the total average profit functions (66) (67) are consistent with functions (11), (13), and (15), which are the same as the problem in Jaggi et al. [4]. So Jaggi et al. [4]is also a special case of our model. 6. Numerical Analysis o gain further insights, we conduct the following numerical analysis. Example 13. We consider the first type of demand rate function: D(N) = β (β α)e rn,inwhichα = 36, β= 18 and r=. he values of other parameters are h=4$ per unit year, p=3$perunit,c=$perunit, =.5, γ = /365 year, M = 3/365 year, A=75$, I c =15%per year, and I p =% per year. According to our analysis of the solutionprocedureandthealgorithm,weruntheinteractive numerical results with the value of N=1,,...,9. here are three conditions for M = 3/365 year and γ= /365 year. (a) N 1/365; (b) 11/365 N 3/365; (c) N 31/365. When N 1/365,sayN = 5/365,wehaveΔ 4 = , Δ 5 = 58.94,andA = 15 < Δ 4.Hence, = 31 =.531 year, and Z =Z 31 ( ) = $. When 11/365 N 3/365, sayn = /365, wehave Δ = 63., Δ 3 = , andδ <A <Δ 3.Hence, = 3 =.47 year, and Z =Z 3 ( ) = 888 $. When N 31/365,sayN = 45/365,wegetΔ 1 = and A = 15 < Δ 1.Hence, = 11 =.459 year, and Z =Z 11 ( ) = $. Finally, for every constant N = 1/365, /365,...,9/365, we can get the optimal result which is depicted in Figure 11. From Figure 11 we know that the maximum obtained by the algorithm in this model is indeed the global optimum solution. And the optimal credit offered by the retailer to customers is N = 69/365 year, the optimal length of replenishment cycle is = 16.39/365 year, and the maximum total average profit is Z = 9967 $. Here, we N (year) Figure 11: Optimal total average profit w.r.t. downstream credit period length of the first type demand. show that the optimal replenishment cycle length is less than nondeterioration period, and the retailer has to pay for the opportunity cost and not the interest earned. Example 14. We also consider the first type of demand rate function: D(N) = β (β α)e rn,inwhichα = 36, β = 18, andr =. he values of other parameters are the same except that M = 7/365 year and A = 15 $. Based on the algorithm, the optimal credit offered by the retailer to customers is N = 67/365 year, the optimal length of replenishment cycle is =.74/365 year, and the maximum total average profit is Z = 117 $. In this example, there is interest earned and paid and deterioration cost. Example 15. Here, we consider the second type of demand rate function: D(N) = β (β α)(1 r) N,inwhichα = 36, β = 18,andr =.995. he values of other parameters are thesametoexample 13.heresultisshownasFigure 1. From Figure 1, we know that the global optimum not only exists but also is unique. Based on the algorithm, the optimal credit offered by the retailer to customers is N = 13/365 year, the optimal length of replenishment cycle is = 16.61/365 year, and the maximum total average profit is Z = 9149 $ Sensitive Analysis. Here,weconsiderthefirsttypeof demand rate. Initial parameters are the same to these in Example 13. By varying different values for the parameters, we have the results in able 1. Commentscan be obtained fromable 1 as follows. (1) It can be observed that as A increases, and Q increases and Z decreases. he optimal downstream credit N remains at the threshold. It shows that for a higher ordering cost, the retailer should replenish less frequently and should stock more items at one cycle

16 16 Discrete Dynamics in Nature and Society otal average profit N (year) Figure 1: Optimal total average profit w.r.t. downstream credit period length of the second type demand. to avoid the high ordering cost. As a result, for higher ordering cost, the total average profit decreases. () When h increases,, Q, and Z all decrease. he optimal downstream credit N remains at the threshold. It indicates that for a higher holding cost, the retailer should replenish more often and should reduce the ordering quantity per cycle. Obviously, higher holding cost leads to a lower total average profit. (3) As the purchasing cost per unit c decreases, N,, Q,andZ all increase, which indicates that if the purchasing cost is lower, the retailer should give more credit to customers to induce the market demand, and set a longer replenishment cycle length, a larger orderingquantity.alltheseleadtotheriseoftotal average profit. (4) When selling price p increases, N, Q,andZ increase while decreases. Hence, for a higher selling price, the retailer should give customers more credit to induce market demand. At the same time, he should shorten the replenishment cycle length and order more items to satisfy the demand. As a result, the retailer can earn more due to a higher selling price. (5) As the interest rate for the interest charged I c increases, N,, Q,andZ all decrease. It indicates that to avoid the interest cost, retailer has to shorten the downstream credit, shorten the replenishment cycle length and reduce the ordering quantity. (6) For a higher changing saturation rate of demand r, N,,andQ decrease while Z increases. It means that the retailer can induce the demand by setting alongerdownstreamcredit.atthesametime,to reducetheholdingcost,hemayordermorefrequently and reduce the items ordered per cycle. Anyway, the able 1: Sensitive analysis of Example 13 for parameters A, h, c, p, I c, r, γ, M. Parameter N (Days) (Days) Q (Units) Z ($) A h c p I c r γ M increase of the saturation rate brings more profit to the retailer. (7) As the deterioration starting point γ increases,, Q, and Z increase. It shows that deterioration can cause cost for retailer. If the items are more unwilling to deteriorate, he can earn more by ordering less often and ordering more items per cycle.

17 Discrete Dynamics in Nature and Society 17 able : Sensitive analysis of Example 14 for parameters I p and. Parameter N (Days) (Days) Q (Units) Z ($) I p (8) If M increases, N,,andQ stay at the same threshold while Z increases. Obviously, if the supplier offers the retailer a longer credit, he can earn more from the interest earned. (9) Because there is no deterioration cost and interest earned in Example 13, sothereisnoinfluenceof parameter I p and. o better illustrate the sensitive of parameter I p and, we make another sensitive analysis based on Example 14. he results are shown as able. We also conclude that, (1) As the interest earned rate I p increases, N,,and Q decrease while Z increases. It means that the retailer should shorten the downstream credit length and the replenishment cycle length, and reduce the ordering quantity per cycle. () As the deterioration rate increases,, Q,andZ decrease. It shows that to avoid the deterioration cost, the retailer tries to keep a lower stock level and orders more frequently. 7. Conclusions and Future Research Financing tools play a more and more important role in business today, which provide us with a new method to study the inventory problems. In the inventory problems, credit can have significant influence on the inventory decisions, that is, ordering quantity and ordering cycle length. In this study, we propose an EOQ model of a kind of noninstantaneous deterioration items with two-level credit and creditdependent demand rate. he purpose of this research is to help the retailer determine the optimal replenishment cycle length, optimal ordering quantity, and optimal credit period offered to customers under different situations. It is also a general frame work for many researches, such as Ouyang et al. [1]andJaggietal.[4]. In future research, our model can be extended to more general supply chain structures, for example, decentralized and centralized supply chain. Also, we can regard the price as a decision variable, or we can set assumptions for partial credit and advanced payment discounts. Appendices A. Proof of Lemma 1,Part(a). Motivated by (43), we define a new function F 1 (x) as follows: F 1 (x) = [A+ (h + ci c)dγ +cdγ+ci c (N M) Dγ] [ (h + ci c)dγ (e (x γ) e (x γ) +1) (h + ci c)d + c+ci c (N M) D] (e (x γ) e (x γ) +1 γ), (A.1) for x [γ,+ ). Since the first derivative of F 1 (x) with respect to x [γ, + ) is F (x) = [hγ + ci c (N M) +ci c γ+c+h+ci c ] Dxe (x γ) <, (A.) we obtain that F 1 (x) is a strict decreasing function of x in the interval [γ, + ). Moreover, we have F 1 (x) x =,and F 1 (x) =A (h + ci c)dγ x γ. (A.3) herefore, if A (h + ci c )Dγ,thenF 1 (x) x γ. According to the intermediate value theorem, there exists a unique 1 [γ,+ )such that F 1 ( 1 )=. Proof of Lemma 1, Part (b). If <A<(h+cI c )Dγ,then from (A.3), F 1 (γ) <. SinceF 1 (x) is a strict decreasing function of x in the interval [γ, + ); thus,thereisnovalue of [γ,+ )such that F 1 () =. Proof of Lemma,Part(a). When A (h + ci c )Dγ, 1 istheuniquesolutionof(43) fromlemma 1(a). aking the second derivative of Z 1 () with respect to and finding the valueofthefunctionatthepoint 1,weobtain Z 1 = [hγ+ci c (N M)+cI c γ+c+h+ci c ]De ( 1 γ) 1 <. hus, 1 is the global maximum point of Z 1 (). (A.4)

18 18 Discrete Dynamics in Nature and Society Proof of Lemma, Part (b). From the proof of Lemma 1(b), we know that if < A (h + ci c )Dγ,thenF 1 (x) <, for all x [γ,+ ).huswehave Z 1 = [A+((h+cI c)dγ /) + cdγ + ci c (N M) Dγ] [( (h + ci c)dγ )+( (c + ci c (N M)) )D] (e ( γ) e ( γ) +1) ( ) 1 ((h + ci c)d/ )(e ( γ) e ( γ) +1 γ) = F 1 () <, (A.5) for all [γ,+ ), which implies that Z 1 () is a strict decreasing function of [γ,+ ).So,Z 1 () has a maximum value at the boundary point =γ. B. Proof of Lemma 4,Part(a). Motivated by (53), we define a new function F (x) as follows: F (x) = [A+ (h + ci c)dγ ci c (M N) Dγ + ci c(m N) D +cdγ pi p(m N) D ] [ hγd + ci c (γ M+N)D+cD ] (e (x γ) e (x γ) +1) (h + ci c)d (e (x γ) e (x γ) +1 γ). for x [γ,+ ). (B.1) Since the first derivative of F (x) with respect to x [γ, + ) is F (x) = [hγ + ci c (γ+n M)+c+h+cI c ] (B.) Dxe (x γ) <, we obtain that F (x) is a strict decreasing function of x in the interval [γ, + ). Moreover, we have F (x) x =,and F (x) =A (h + ci c)γ D (ci c pi p ) (M N) D x γ. (B.3) herefore, if A (h + ci c )γ D (ci c pi p )(M N) D,then F (x) x γ. According to the intermediate value theorem, there exists a unique 3 [γ,+ )such that F ( 3 )=. Proof of Lemma 4, part (b). If A < (h + ci c )γ D (ci c pi p )(M N) D,thenfrom(A.3), F (γ) <. SinceF (x) is a strict decreasing function of x in the interval [γ, + );thus, thereisnovalueof [γ,+ )such that F () =. Proof of Lemma 5,Part(a). When A (h + ci c )γ D (ci c pi p )(M N) D, 3 istheuniquesolutionof(53) from Lemma 4(a). aking the second derivative of Z 3 () with respect to and finding the value of the function at the point 3,weobtain Z 3 = [hγ + ci c (N M) +ci c γ + c +h +ci c ]De ( 3 γ) 3 <. hus, 3 is the global minimum point of Z 3 (). (B.4) Proof of Lemma 5, Part (b). From the proof of Lemma 4(b), we know that if <A<(h+cI c )γ D (ci c pi p )(M N) D, then F (x) >,forallx [γ,+ ).huswehave Z 3 =[A+ (h + ci c)dγ ci c (M N) Dγ + ci c(m N) D [ hγd + ci c (γ M+N)D+cD ] (e (x γ) e (x γ) +1) ( ) 1 +cdγ pi p(m N) D ] ( ) 1 ((h + ci c)d/ )(e (x γ) e (x γ) +1 γ) = F () <, [γ,+ ), (B.5) which implies that Z 3 () is a strict decreasing function of [γ,+ ).So,Z 3 () has a maximum value at the boundary point =γ.

19 Discrete Dynamics in Nature and Society 19 C. Proof of Lemma 7,Part(a). Motivated by (6), we define a new function F 3 (x) as follows: F 3 (x) = (A+ hdγ +cdγ) ( (e ( γ) e ( γ) +1) hdγ + cd ) hd (e( γ) e ( γ) +1 γ) pi pd for x [γ,m N]. (C.1) Since the first derivative of F 3 (x) with respect to x [γ,m N] is F + cd 3 (x) = (hdγ + hd )e( γ) pi p D <, (C.) we obtain that F 3 (x) is a strict decreasing function of x in the interval [γ, M N].Moreover, F 3 (x) x M N =(A+ hdγ F 3 (x) x γ =A hdγ + pi pd, (C.3) hdγ + cd +cdγ) ( + hd ) [(M N) e (M N γ) e (M N γ) +1] + hdγ pi pd(m N). (C.4) According to the intermediate value theorem, when Δ 4 A Δ 5,thenF 3 (x) x M N and F 3 (x) x γ,sothere exists a unique 3 [γ,m N]such that F 3 ( 3 )=. Proof of Lemma 7, Part (b). If A < Δ 4 or A > Δ 5,then F 3 (x) x γ <or F 3 (x) x M N >.SinceF 3 (x) is a strict decreasing function of x in theinterval [γ, M N].hus,there is no value of [γ,m N]such that F 3 () =. Proof of Lemma 8,Part(a). When Δ 4 A Δ 5, 3 is the unique solution of (6)fromLemma 7(a). aking the second derivative of Z 3 () with respect to and finding the value of the function at the point 3,weobtain Z 3 () = 3 = ((hdγ + cd) / + hd/ )e ( 3 γ) 3 <. pi pd 3 hus, 3 is the global maximum point of Z 3 (). (C.5) Proof of Lemma 8, Part (b). From the proof of Lemma 7(b), we know that if A < Δ 4,thenF 3 (x) < for all x [γ,m N].hus,wehave Z 3 (A + hdγ / + cdγ) = ((hdγ + cd) /) (e( γ) e ( γ) +1) (hd/ )(e ( γ) e ( γ) +1 γ) = F 3 () <, [γ,m N], pi pd (C.6) which implies that Z 3 () is a strict decreasing function of [γ,m N].So,maxZ 3 () = Z 3 (γ). Proof of Lemma 8,Part(c). From the proof of Lemma 7(b), we know that if A > Δ 5,thenF 3 (x) > for all x [γ, M N].hus,wehave Z 3 (A + hdγ / + cdγ) = ((hdγ + cd) /) (e( γ) e ( γ) +1) (hd/ )(e ( γ) e ( γ) +1 γ) = F 3 () <, [γ,m N], pi pd (C.7) which implies that Z 3 () is a strict decreasing function of [γ,m N].So,maxZ 3 () = Z 3 (M N). D. Proof of Lemma 9,Part(a). Motivated by (53), we define a new function F (x) as follows: F 4 (x) =(A+ hdγ ( hd +cdγ ci c (M N) pi p(m N) D ) hdγ + cd )(e ( γ) e ( γ) +1) (e( γ) e ( γ) +1 γ) ci c (e( M+N) e ( M+N) +1) for x [M N, + ). (D.1)

20 Discrete Dynamics in Nature and Society Since the first derivative of F 4 (x) with respect to x [M N, + ) is F + cd 4 (x) = (hdγ + hd )e( γ) (D.) ci c e( M+N) <, we obtain that F 4 (x) is a strict decreasing function of x in the interval [M N, + ). Moreover, we have F 4 (x) x =, and F(x) x M N = (A+ hdγ hdγ + cd +cdγ) ( + hd ) [(M N) e (M N γ) e (M N γ) +1] + hdγ + pi p(m N) D. (D.3) herefore, if A Δ 5,thenF 4 (x) x M N. According to the intermediate value theorem, there exists a unique 33 [M N, + ) such that F 4 ( 33 )=. Proof of Lemma 9, Part (b). If <A<Δ 5,thenfrom(D.3) F 4 (M N) <. SinceF 4 (x) is a strict decreasing function of x in the interval [M N, + ). hus,thereisnovalueof [M N,+ )such that F 4 () =. Proof of Lemma 1,Part(a). When A Δ 5, 33 is the unique solution of (63)fromLemma 9(a). aking the second derivative of Z 33 () with respect to and finding the value of the function at the point 33,weobtain Z 33 = ((hdγ + cd) / + hd/ )e ( γ) 33 <. hus, 33 is the global maximum point of Z 33 (). ci ce ( M+N) 33 (D.4) Proof of Lemma 1, Part (b). From the proof of Lemma 9(b), we know that if <A<Δ 5,thenF 4 (x) <, forallx [M N, + ).huswehave Z 33 = (A + hdγ /+cdγ ci c (M N) pi p (M N) D/) ((hdγ + cd) /) (e( γ) e ( γ) +1) (hd/ )(e ( γ) e ( γ) +1 γ) (ci c/ )(e ( M+N) e ( M+N) +1) = F 4 () <, [M N, + ), (D.5) which implies that Z 33 () is a strict decreasing function of [M N,+ ). Z 3 () has a maximum value at the boundary point =M Nfor [M N,+ ). Acknowledgments he authors thank the valuable comments of the reviewers for an earlier version of this paper. heir comments have significantlyimprovedthepaper.hisworkissupportedbythe National Natural Science Foundation of China (nos and ). Also, this research is partly supported by the Program for New Century Excellent alents in the University (no. NCE-1-37) and the Ministry of Education of China: Grant-in-aid for Humanity and Social Science Research (no. 11YJCZH139). References [1] N. H. Shah, H. N. Soni, and K. A. Patel, Optimizing inventory and marketing policy for non-instantaneous deteriorating items with generalized type deterioration and holding cost rates, Omega,vol.41,no.,pp.41 43,13. []P.M.GhareandG.F.Schrader, Amodelforexponentially decaying inventory, he Journal of Industrial Engineering, vol. 5, no. 14, pp , [3] R.P.CovertandG.C.Philip, AnEOQmodelforitemswith Weibull distribution deterioration, AIIE ransactions, vol. 5, no. 4, pp , [4] G. C. Philip, A generalized EOQ model for items with Weibull distribution deterioration, AIIE ransactions, vol.6,no.,pp , [5] Y. He and J. He, A production model for deteriorating inventory items with production disruptions, Discrete Dynamics in Nature and Society, vol. 1, Article ID 18917, 14 pages, 1. [6] Y.He,S.Wang,andK.K.Lai, Anoptimalproduction-inventory model for deteriorating items with multiple-market demand, European Journal of Operational Research, vol.3,no.3,pp , 1. [7] H. L. Yang, J.. eng, and M. S. Chern, An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages, International Journal of Production Economics,vol.13,no.1,pp.8 19, 1. [8] Y. He and S. Y. Wang, Analysis of production-inventory system for deteriorating items with demand disruption, International Journal of Production Research, vol.5,no.16,pp , 1. [9] S. K. Goyal and B. C. Giri, Recent trends in modeling of deteriorating inventory, European Journal of Operational Research,vol.134,no.1,pp.1 16,1. [1] R. Li, H. Lan, and J. Mawhinney, A review on deteriorating inventory study, Journal of Service Science and Management, vol.3,no.1,pp ,1.

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