Optimal ordering policies for perishable multi-item under stock-dependent demand and two-level trade...

Applied Mathematical Modelling xxx (2013) xxx–xxx

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Optimal ordering policies for perishable multi-itemunder stock-dependent demand and two-level trade credit

0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.10.058

⇑ Corresponding author.E-mail addresses: [email protected], [email protected] (M. Jiangtao).

Please cite this article in press as: M. Jiangtao et al., Optimal ordering policies for perishable multi-item under stock-dependent dand two-level trade credit, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.058

Mo Jiangtao a,⇑, Chen Guimei b, Fan Ting c, Mao Hong d

a College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, Chinab Department of Basic Education, Talent International College, Qinzhou, Guangxi, Chinac Information Center, Foshan Power Supply Bureau, Foshan, Guangdong, Chinad Guangshui No.1 High School, Guangshui, Hubei, China

a r t i c l e i n f o

Article history:Received 22 July 2012Received in revised form 9 July 2013Accepted 11 October 2013Available online xxxx

Keywords:Stock-dependent demand rateMulti-itemTwo-level trade credit

a b s t r a c t

In this paper, a multi-item inventory model for perishable items is developed, where thedemand rates of the items are stock dependent, two-level trade credit is adopted andthe restriction of inventory capacity is also considered. The major objective is to determinethe optimal cycle time and order quantities such that the total profit is maximized. Theexistence and uniqueness of the optimal cycle is discussed by Lagrange approach, and linesearch algorithms are developed to find the optimal solution of the model. Furthermore,numerical examples are given to illustrate the methods. The sensitivity of the solution tochanges in the values of different parameters is also discussed.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

In the traditional inventory model, it was assumed that the retailer must pay his/her supplier as soon as he/she receivesthe ordered items. However, this is different with today’s business transactions. Owing to the fierce market competition,credit sale becomes one of the main competitive means of the enterprises. General, the supplier allows a retailer to postponepaying money in a certain period (called a trade credit period). During the period, the retailer does not charge any interestand can earn interest by depositing the generated sales revenue into an interest bearing account. At the end of the period, theretailer settles the payment of the goods and has an interest charged for unsold goods. This is the so called trade credit ordelay in payment. Taking the advantage of trade credit, the retailer reduces the cost and is motivated to order more quan-tities, which will increase the holding cost and the perishable cost. Therefore, the retailer has to balance between his/herrevenue and expenditure. In recent studies on inventory management, several authors have examined the effect of tradecredit policy on the optimal ordering policies. Goyal [1] first developed the economic order quantity (EOQ) inventory modelunder the condition of trade credit. Aggarwal and Jaggi [2] extended Goyal’s model to deteriorating items. Jamal et al. [3]further generalized Goyal’s model to allow for shortages. Chang et al. [4] developed an EOQ model for deteriorating itemswhen the trade credit is linked to the order quantity. Hwang and Shinn [5] developed a model to determine optimal pricingand lot sizing for deteriorating items under condition of permissible delay in payments. Sarkar [6] discussed an EOQ modelwhere demand and deterioration rate are both time dependent, and trade credit is included. There are several relevant pa-pers related to trade credit such as Jamal et al. [7], Sana and Chaudhuri [8], Chung and Liao [9], Balkhi [10], Liao et al. [11],Shah et al. [12] and their references. All the above inventory models implicitly assumed that the supplier would offer theretailer a delay period but the retailer would not offer any delay period to his/her customer. In most business transactions,

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2 M. Jiangtao et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx

this assumption is unrealistic. Usually the supplier offers a credit period to the retailer and the retailer, in turn, passes on thiscredit period to his/her customers [13]. That is so called two-level trade credit. Huang [14] presented an inventory modelassuming that the retailer also permits a credit period to its customer which is shorter than the credit period offered bythe supplier. Huang [15] extended Huang’s model to investigate the inventory policy under two-level trade credit and lim-ited storage space, in which there two warehouses, one is own and another is rented. Kreng and Tan [16] developed an inven-tory model under two-level credit trade policy when the credit period depends on the order quantity. Ho [17] considered asupply chain system under two-level credit trade when the demand depends on price and credit period offered by supplier.Chen and Kang [18] established a supply chain system with the assumption that demand is a negative exponential functionof price and two-level credit trade is adopted. Liao [19] developed an EOQ model with non-instantaneous receipt and expo-nentially deteriorating items under two-level trade credit.

An interesting phenomenon is observed in the supermarket that display of the consumer goods in large quantities attractsmore customers and generates higher demand. Many researchers have given consider able attention to the situation wherethe demand rate is dependent on the level of the on-hand inventory. Baker and Urban [20] established an economic orderquantity model by assuming that the demand rate was a function of the instantaneous stock level. Chang et al. [21] devel-oped inventory model with a deteriorating item and stock-dependent demand rate. Sajadieh et al. [22] reported a supplychain system with stock-dependent demand rate. Jolai et al. [23] established a model under inflation for deteriorating itemswith stock-dependent rate. Hsieh et al. [24] considered a model with stock-dependent demand rate under the assumptionthat backlog rate is a function of the waiting time of customer. Bhattacharya [25] developed a two-item model for deterio-rating items, the demand of one item depended on the other’s stock level. Kar et al. [26] proposed a model that there arefresh and deteriorating items sold from the primary and the secondary shop respectively. The demand of fresh items de-pends on selling price and stock level. Recently, Min et al. [27] established a single item inventory model with deterioratingitem, stock dependent demand rate and two level trade credits. All above models are developed for a single item. However, inreal life, many companies, enterprises or retailers deal with several items and stock them in their showroom/warehouse forsale. There is a restriction either on maximum capital investment in stock at any time, or the maximum warehouse spaceavailable for storage. Padmanabhan and Vrat [28] developed multi-item multi-objective inventory model of deterioratingitems with stock-dependent demand. Ben-Daya and Raouf [29] discussed a multi-item inventory model with stochastic de-mand subject to the restrictions on available space and budget. Saha et al. [30] developed a multi-item inventory model withthe breakability rate and the demand rate both stock-dependent. Tsao [31] considered multi-echelon multi-item channelssubject to supplier’s credit period and retailer’s promotional effort. Tsao and Sheen [32] developed a multi-item supply chainwith credit periods and weight freight cost discounts. Thangam and Uthayakumar [33] developed an EPQ-based model forperishable items with two-level trade credit and demand rate both selling price and credit period dependent. Thangam [34]considered a supply chain for perishable items under advance payment scheme and two-level trade credit. Su [35] estab-lished an integrated inventory system with defective items and allowable shortage under trade credit. The aforesaid mul-ti-item inventory models were developed with either stock dependent or trade credit.

In this paper, we discussed the optimal order policy for retailer with perishable multi-item and stock-dependent demandrate under two-level trade credit and restriction on available space or budget. This is basically an extension of the single iteminventory model by Min et al. [27] to deal with multiple items and restriction on inventory capacity. In more details, we aretaking into account the following factors: (1) a supplier sells multiple items to a retailer; (2) the selling items are perishablesuch as meats, fruits, green vegetables, foodstuffs, etc.; (3) the demand rate of each item is dependent on its instantaneousstock level; (4) the supplier provides the retailer a trade credit period (M) for payments and the retailer offers the partialtrade credit period (N) to his/her customers, and N 6 M; (5) the retailer has a restriction on available space or budget. Underthese conditions, we model the retailer inventory system as a constraint optimization problem. The aim is to determine theoptimal ordering polices to maximize the average system profit. The existence and uniqueness of optimal strategy are dis-cussed by the Lagrange method and line search algorithms are presented to find the optimal cycle length and lot size. Finally,numerical examples are given to illustrate the theoretical results and the methods.

2. Notations and assumptions

2.1. Notations

Pan

i

lease cite this article in press as: M. Jiangtao et al.,d two-level trade credit, Appl. Math. Modell. (201

the index of products, i ¼ 1;2; :::;n
K ordering cost of one order ci unit purchasing price of product i hi unit stock holding cost per year of product i pi unit selling price of product i wi unit capacity of product i W the total storage capacity of inventory
Optimal ordering policies for perishable multi-item under stock-dependent demand3), http://dx.doi.org/10.1016/j.apm.2013.10.058


M. Jiangtao et al. / Applied Mathematical Modelling xxx (2013) xxx–xxx 3

Plan

hi

ease cite this article in press as: M. Jiangtao et al.,d two-level trade credit, Appl. Math. Modell. (201

the perishable rate of product i
Qi the retailer’s order quantity per cycle of product i T inventory cycle length (Decision variable) IiðtÞ stock level at time t of product i, i ¼ 1;2; . . . ;n M the retailer’s trade credit period offered by the supplier in a year N the customer’s trade credit period offered by the retailer in year Ie interest which can be earned per dollar per year Ic interest charges per dollar in stocks per year by the supplier
2.2. Assumptions

(1) The demand rate of product i at time t, Ri(t), is a function of retailer’s stock level Ii(t), that is, Ri(t) = Di + aiIi(t), where Di

and ai are positive constants, i = 1, 2, . . ., n.(2) Shortages are not allowed.(3) Replenish rate is infinite and the lead time is zero.(4) The items are perishable; the residual value is zero.(5) The credit period of retailer is greater than the period of customer, N 6 M.(6) The retailer processes a certain credit period (said M), i.e., The retailer can accumulate revenue and earn interest dur-

ing the credit period without being charged any interest; the retailer needs to settle the account at time t = M and paysfor the interest charges on items in stock.

(7) The credit period of customers is N, i.e., The customer does not need to pay money before t = N, all the fee is settled att = N. But after t = N, the customer needs to pay as soon as possible when he buys items.

(8) In order to simplify discussion, we assume that aipi + aipiIe(M � N) � ci(ai + hi) � hi < 0, i = 1, 2, . . .,n.

3. Model formulations

According to the assumptions, for a product i, i = 1, 2, . . .,n, the inventory level is affected by demand and perishable in acycle, so it can be described by the following differential equation,

dIiðtÞdt¼ �Di � aiIiðtÞ � hiIiðtÞ; 0 6 t 6 T; IiðTÞ ¼ 0: ð1Þ

The solution to the above equation is

IiðtÞ ¼Di

ai þ hieðaiþhiÞðT�tÞ � 1� �

; 0 6 t 6 T: ð2Þ

So the demand rate of product i is

RiðtÞ ¼ Di þ aiIiðtÞ ¼Di

ai þ hihi þ aieðaiþhiÞðT�tÞ� �

: ð3Þ

The retailer’s order size of product i is Q i ¼ Iið0Þ, i.e.

Q i ¼Di

ai þ hieðaiþhiÞT � 1� �

; i ¼ 1;2; . . . n: ð4Þ

Then, the elements comprising the profit function per cycle are as follows.

(a) The ordering cost is K.

(b) The sales revenue is Rsi ¼ pi

R T0 RiðtÞdt ¼ pi

hiDiaiþhi

T þ aiDi

ðaiþhiÞ2ðeðaiþhiÞT � 1Þ

h i.

(c) The purchasing cost is Cpi ¼ ciQi ¼ ciDiaiþhiðeðaiþhiÞT � 1Þ.

(d) The holding cost is Chi ¼ hiR T

0 IiðtÞdt ¼ hiDi

ðaiþhiÞ2eðaiþhiÞT � ðai þ hiÞT � 1� �

.

(e) The retailer earned interest IE, and charged interest IP, are determined by the relationship of T, M and N, so we considerthe following three cases.

Case 1 (T 6 N 6 M). In this case, retailer accumulates revenue and earns interest in [N, M], so interest earned IE1i is

IE1i ¼ piIe

Z T

0ðM � NÞRiðtÞdt ¼ piIeDiðM � NÞ hiT

ai þ hiþ ai

ðai þ hiÞ2ðeðaiþhiÞT � 1Þ

" #: ð5Þ

Since the credit period is longer than the cycle time, there is no interest payable, that is IP1i = 0.

Optimal ordering policies for perishable multi-item under stock-dependent demand3), http://dx.doi.org/10.1016/j.apm.2013.10.058



Case 2 (N 6 T 6 M). In this case, the retailer’s interest earned IE2i contains two parts: the interest in [N, T] and [T, M].

Pleaseand tw

IE2i ¼piIe

Z T

N

Z t

0RiðuÞdudt þ

Z T

0RiðuÞduðM � TÞ

� �¼ piIeDi

hiðT2 � N2Þ2ðai þ hiÞ

� ai

ðai þ hiÞ3ðeðaiþhiÞðT�NÞ � 1Þ þ aiðT � NÞ

ðai þ hiÞ2eðaiþhiÞT

" #

þ piIeDiðM � TÞ hiTai þ hi

þ ai

ðai þ hiÞ2ðeðaiþhiÞT � 1Þ

" #: ð6Þ

Similar to Case 1, the retailer does not need to pay any interest, that is IP2i = 0.

Case 3 (T P M). In this case, retailer’s interest earned IE3i during [N, M] and interest payable IP3i during [M, T] are

IE3i ¼ piIe

Z M

N

Z t

0RiðuÞdudt ¼ piIeDi

hiðM2 � N2Þ2ðai þ hiÞ

þ aiðM � NÞðai þ hiÞ2

eðaiþhiÞT þ aieðaiþhiÞT

ðai þ hiÞ3ðe�ðaiþhiÞM � e�ðaiþhiÞNÞ

" #; ð7Þ

IP3i ¼ ciIc

Z T

MIiðtÞdt ¼ ciIc

Z T

M

Di

ai þ hieðaiþhiÞðT�tÞ � 1� �

dt ¼ ciIcDi1

ðai þ hiÞ2ðeðaiþhiÞðT�MÞ � 1Þ � T �M

ai þ hi

" #; ð8Þ

where the second equation is deduced from (2).So, the average profit per unit time for the inventory system is

APðTÞ ¼AP1ðTÞ; T 6 N 6 M;

AP2ðTÞ; N 6 T 6 M;

AP3ðTÞ; M 6 T;

8><>: ð9Þ

where

APjðTÞ ¼1T

Xn

i¼1

ðRsi þ IEji � Cpi � Chi � IPjiÞ � K

" #; j ¼ 1;2;3: ð10Þ

In summation and simplification, the following results are achieved:

AP1ðTÞ ¼ �KTþ 1

T

Xn

i¼1

ðEieðaiþhiÞT þ FiT � EiÞ; ð11Þ


T

Xn

i¼1

� aipiIeDi

ðai þ hiÞ3eðaiþhiÞðT�NÞ þ EieðaiþhiÞT � hi

2ðai þ hiÞT2 þ LiT þ Si

" #; ð12Þ


T

Xn

i¼1

� ciIcDi

ðai þ hiÞ2eðaiþhiÞðT�MÞ þ ðEi þ HiÞeðaiþhiÞT þ Di

hipi þ hi þ ciIc

ai þ hiT þ Zi

" #; ð13Þ

and

Ei ¼ Diaipi þ aipiIeðM � NÞ � ciðai þ hiÞ � hi

ðai þ hiÞ2; ð14Þ

Fi ¼ DihipiIeðM � NÞ þ hi þ hipi

ai þ hi; ð15Þ

Li ¼ Diðai þ hiÞðhi þ hipi þ hipiIeMÞ þ aipiIe

ðai þ hiÞ2; ð16Þ

Hi ¼aipiIeDi

ðai þ hiÞ3ðe�ðaiþhiÞM � e�ðaiþhiÞNÞ; ð17Þ

Si ¼aipiIeDi

ðai þ hiÞ3þ Di

hi � aipi � aipiIeM

ðai þ hiÞ2þ Di

2ci � hipiIeN2

2ðai þ hiÞ; ð18Þ

cite this article in press as: M. Jiangtao et al., Optimal ordering policies for perishable multi-item under stock-dependent demando-level trade credit, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.058



Pleaseand tw

Zi ¼ Dihi þ ciIc � aipi

ðai þ hiÞ2þ Di

hipiIeðM2 � N2Þ þ 2ci � 2ciIcM2ðai þ hiÞ

: ð19Þ

Notice that by assumptions (5) and (8), we have Ei < 0 and Hi < 0.When the order quantity of product i is Qi, i ¼ 1;2; . . . ;n; the total inventory capacity occupied by all products is
Xn
i¼1

wiQ i ¼Xn

i¼1

wiDi

ai þ hiðeðaiþhiÞT � 1Þ: ð20Þ

Under the limit of the capacity and two levels of credit trade policy, the objective of this paper is to find the replenishmenttime T to maximize AP(T), the average profit per unit time of the system. Thus, the mathematical model of the problem is

Max APðTÞ

s:t:Xn

i¼1

wiDiaiþhiðeðaiþhiÞT � 1Þ 6W ;

T > 0:

ð21Þ

4. Model analysis

Case 1 (0 6 T 6 N 6 M). In this case, AP(T) = AP1(T).First, we maximize AP1(T) with 0 6 T 6 N 6 M. Using the first-order necessary condition, we obtain
Xn
i¼1

ððai þ hiÞT � 1ÞEieðaiþhiÞT þ Ei� �

þ K ¼ 0: ð22Þ

Denoting the left hand side of the above equation as g1(T), we have

g01ðTÞ ¼Xn

i¼1

ðai þ hiÞ2TEieðaiþhiÞT : ð23Þ

Thus g01ðTÞ < 0 for Ei < 0, then g1(T) is decreasing. Furthermore, g1(0) = K > 0 and

g1ðNÞ ¼Xn

i¼1

ðai þ hiÞN � 1ð ÞEieðaiþhiÞN þ Ei� �

þ K: ð24Þ

If g1ðNÞP 0, g1ðTÞP 0, then AP01ðTÞP 0 for 0 6 T 6 N. Thus AP1(T) is an increasing function, the optimal cycleT� ¼minfN; eTg, where eT satisfies the following equation:
Xn
i¼1

wiDi

ai þ hiðeðaiþhiÞT � 1Þ �W ¼ 0: ð25Þ

If g1(N) < 0, then there exists a unique T1 such that 0 < T1 < N and g1(T1) = 0. If T1 satisfies the capacity constraint in (21),then T1 is the optimal cycle. If T1 does not satisfy the capacity constraint in (21), we have
Xn
i¼1

wiDi

ai þ hiðeðaiþhiÞT1 � 1Þ > W: ð26Þ

We now solve problem (21) with 0 6 T 6 N. The Lagrange function of (21) is

L1ðT; kÞ ¼ AP1ðTÞ � kXn

i¼1

wiDi

ai þ hiðeðaiþhiÞT � 1Þ �W

" #; ð27Þ

where k P 0 is the Lagrangian multiplier. The KT conditions of (21) are
Xi ¼ 1n½ððai þ hiÞT � 1ÞEieðaiþhiÞT þ Ei � kwiDiT
2eðaiþhiÞT � þ K ¼ 0; ð28Þ

Xn

i¼1

wiDi


Let the left hand side of (28) be h1(T), we have

h01ðTÞ ¼Xn

i¼1

ðai þ hiÞEi � kwiDi 2þ T2ðai þ hiÞh in o

eðaiþhiÞT : ð30Þ




Thus h01ðTÞ < 0 for Ei < 0, and then h1(T) is decreasing. Note that

Pleaseand tw

h1ðNÞ ¼ g1ðNÞ �Xn

i¼1

kwiDiT2eðaiþhiÞT < 0: ð31Þ

For g1ðNÞ < 0, and h1ð0Þ ¼ g1ð0Þ ¼ K > 0, by the Intermediate Value Theorem and the monotonicity of h1ðTÞ, there exists aunique T2ðkÞ such that 0 < T2ðkÞ < N and h1ðT2ðkÞÞ ¼ 0.

Substituting T ¼ T2ðkÞ into (28) and denoting the left hand side as l1ðkÞ, we have

l1ðkÞ ¼Xn

i¼1

wiDi

ai þ hiðeðaiþhiÞT2ðkÞ � 1Þ �W: ð32Þ

Then

l01ðkÞ ¼Xn

i¼1

wiDieðaiþhiÞT2ðkÞT 02ðkÞ: ð33Þ

Substituting T ¼ T2ðkÞ into (28), and then taking differential with respect to k on the two sides, we get

T2ðkÞXn

i¼1

f½ðai þ hiÞ2Ei � kwiDið2þ ðai þ hiÞT2ðkÞÞ�T 02ðkÞ �wiDiT2ðkÞgeðaiþhiÞT2ðkÞ ¼ 0: ð34Þ

Since k P 0, T2ðkÞ > 0, Ei < 0, we have T 02ðkÞ < 0, together with (33), we have l01ðkÞ < 0.Since T2ðkÞ is a decreasing function, T2ðkÞP 0 and k P 0, so limk!þ1T2ðkÞ exists (said T0). Dividing k on both two sides of

(28) with T ¼ T2ðkÞ and then letting k! þ1, we get �Pn

i¼1wiT20DieðaiþhiÞT0 ¼ 0. This implies that T0 ¼ 0 or limk!þ1T2ðkÞ ¼ 0.

Therefore limk!þ1l1ðkÞ ¼ �W < 0. On the other hand, since L1ðT;0Þ ¼ AP1ðTÞ and T2ð0Þ ¼ T1, according to (26), we havel1ð0Þ > 0. By the Intermediate Value Theorem and the monotonicity of l1ðkÞ, there exists a unique k�1 > 0, such thatl1ðk�1Þ ¼ 0. Thus T2ðk�1Þ is the solution of (29).

Theorem 1. Let T�1 be the solution of problem (21) with 0 6 T 6 N 6 M.

(a) If g1ðNÞP 0, T�1 ¼minfN; eTg;(b) If g1ðNÞ < 0 and T1 satisfies the capacity constraint in (21), then T�1 ¼ T1, where T1 is the unique solution of g1ðTÞ ¼ 0;(c) If g1ðNÞ < 0 and T1 does not satisfy the capacity constraint in (21), then T�1 ¼ T2ðk�1Þ, where k�1 and T2ðk�1Þ are the unique

solution of l1ðkÞ ¼ 0 and h1ðT2ðkÞÞ ¼ 0 respectively.

Based on Theorem 1, we establish the following algorithm to find the optimal solution T�1 for Case 1.

Algorithm 1Step 0: Let k ¼ 0 and e > 0;

Step 1: Calculate g1ðNÞ. If g1ðNÞP 0, T� ¼minfN; eTg, then go to Step 7; else go to Step 2;Step 2: Find T1 satisfies (28). If T1 satisfies (21), let T�1 ¼ T1, and go to Step 7; else go to Step 3;Step 3: Let k ¼ e. Find T2 satisfies (28) and calculate l1ðkÞ by (32). If l1ðkÞ ¼ 0, T� ¼ T2, go to Step 7; else if l1ðkÞ > 0, go

to Step 4; if l1ðkÞ < 0, let kL ¼ k� e, kU ¼ k, go to Step 5;Step 4: Let k ¼ kþ e, go to Step 3;Step 5: Let k ¼ ðkL þ kUÞ=2. Find T2ðkÞ satisfied (28) and calculate l1ðkÞ by (32). If l1ðkÞ ¼ 0, then T� ¼ T2, go to Step 7, or

else go to Step 6;Step 6: If l1ðkÞ > 0, let kL ¼ k; else let kU ¼ k, go to Step 5;Step 7: Output T�1.

Case 2 (N 6 T 6 M). In this case, APðTÞ ¼ AP2ðTÞ. Since the left hand side of the capacity constraint in (21) is an increasingfunction of T . It is obvious that the solution of (21) exists if and only if
Xn
i¼1

wiDi

ai þ hiðeðaiþhiÞN � 1Þ 6W: ð35Þ

First, we maximize AP2ðTÞ with N 6 T 6 M. Using the first-order necessary condition, we obtain
Xn
i¼1

aipiIeDi

ðai þ hiÞ2eðaiþhiÞðT�NÞ 1

ai þ hi� T

� �þ ððai þ hiÞT � 1ÞEieðaiþhiÞT � hi

2ðai þ hiÞT2 � Si

" #þ K ¼ 0: ð36Þ




Let the left hand side of the above equation be g2ðTÞ, then

Pleaseand tw

g02ðTÞ ¼Xn

i¼1

�aipiIeDiTai þ hi

eðaiþhiÞðT�NÞ þ ðai þ hiÞ2TEieðaiþhiÞT � hi

ai þ hiT

� �: ð37Þ

Since Ei < 0, then g02ðTÞ < 0, so g2ðTÞ is a decreasing function of T , and

g2ðNÞ ¼Xn

i¼1

aipiIeDi

ðai þ hiÞ21

ai þ hi� N

� �þ ððai þ hiÞN � 1ÞEieðaiþhiÞN � hi

2ðai þ hiÞN2 � Si

" #þ K: ð38Þ

If g2ðNÞ 6 0, we have g2ðTÞ 6 0 for N 6 T 6 M, then AP02ðTÞ 6 0, so AP2ðTÞ is a decreasing function, according to (35), theoptimal cycle is T� ¼ N.

If g2ðNÞ > 0 and g2ðMÞP 0, we have g2ðTÞP 0 for N 6 T 6 M, then AP02ðTÞP 0, so AP2ðTÞ is an increasing function, theoptimal cycle is T� ¼ M;

If g2ðNÞ > 0 and g2ðMÞ < 0, then there exist a unique T3, such that N < T3 < M and g2ðT3Þ ¼ 0. If T3 satisfies the capacityconstraint in (21), then T3 is the solution; If T3 does not satisfy the capacity constraint in (21), that is
Xn
i¼1

wiDi


We then solve problem (21) with N < T3 < M. The Lagrange function of problem (21) is


i¼1

wiDi


" #: ð40Þ

where k P 0 is the Lagrangian multiplier. According to KT conditions, we have

Xn

i¼1

aipiIeDi

ðai þ hiÞ2eðaiþhiÞðT�NÞ 1

ai þ hi� T

� �þ ððai þ hiÞT � 1ÞEieðaiþhiÞT

" #�Xn

i¼1

hi

2ðai þ hiÞT2 þ Si þ kwiDiT

2eðaiþhiÞT� �

þ K ¼ 0; ð41Þ

Xn

i¼1

wiDi


Next, we will look for k > 0 and N < T3 < M satisfying the above two equations. Let the left hand side of (41) be h2ðTÞ, that is

h2ðTÞ ¼ g2ðTÞ �Xn

i¼1

kwiDiT2eðaiþhiÞT : ð43Þ

Let D1 ¼ g2ðNÞ=ðPn

i¼1wiDiN2eðaiþhiÞNÞ; we have h2ðNÞP 0 for 0 6 k 6 D1. Since g2ðMÞ < 0, so h2ðMÞ < 0. Then, by the Inter-

mediate Value Theorem and the monotonicity, there exists unique T4ðkÞ, such that h2ðT4ðkÞÞ ¼ 0 and N 6 T4ðkÞ < M.Substituting T ¼ T4ðkÞ into (42) and denoting the left hand side as l2ðkÞ, we have

l2ðkÞ ¼Xn

i¼1

wiDi


Then

l02ðkÞ ¼Xn

i¼1

wiDieðaiþhiÞT4ðkÞT 04ðkÞ: ð45Þ

Substituting T ¼ T4ðkÞ into (41) and then taking differential with respect to k on the two sides, we obtain
Xn
i¼1

T4ðkÞ �aipiIeDi

ai þ hie�ðaiþhiÞN þ ðai þ hiÞ2Ei � kwiDið2þ ðai þ hiÞT4ðkÞÞ

� �T 04ðkÞeðaiþhiÞT

�

�Xn

i¼1

hiT4ðkÞai þ hi

T 04ðkÞ þwiDiT24ðkÞeðaiþhiÞT

� �¼ 0: ð46Þ

Since k P 0, T4ðkÞ > 0, Ei < 0, thus it follows that T 04ðkÞ < 0. Then we get l02ðkÞ < 0 from (45).Since T4ðD1Þ ¼ N, we have l2ðD1Þ 6 0 according to (35). On the other hand, when k ¼ 0, L2ðT;0Þ ¼ AP2ðTÞ, so T4ð0Þ ¼ T3.

According to (39), we have l2ð0Þ > 0. By the Intermediate Value Theorem and the monotonicity of l2ðkÞ, there exists a uniquek�2 such that 0 < k�2 6 D1 and l2ðk�2Þ ¼ 0, that is T4ðk�2Þ satisfies (42).




Theorem 2. Let T�2 be the solution of problem (21) with N 6 T 6 M

(a) If g2ðNÞ 6 0, T�2 ¼ N;(b) If g2ðNÞ > 0 and g2ðMÞP 0, T�2 ¼ M;(c) If g2ðNÞ > 0 and g2ðMÞ < 0, T�2 ¼ T3, where T3 satisfies (21);(d) If g2ðNÞ > 0 and g2ðMÞ < 0, T3 does not satisfy (21), then T�2 ¼ T4ðk�2Þ, k

�2 and T4ðk�2Þ are the unique solution of l2ðkÞ ¼ 0 and

h2ðT4ðkÞÞ ¼ 0, respectively.

Algorithm 2Step 0: Calculate g2ðNÞ. If g2ðNÞ 6 0, T� ¼ N, go to Step 6; else go to Step 1;Step 1: Calculate g2ðMÞ. If g2ðMÞP 0, T� ¼ M; else go to Step 2;Step 2: Let k ¼ D1, T ¼ N and calculate l2ðkÞ by (44). If l2ðkÞ ¼ 0, then T� ¼ N, go to Step 6; else let kL ¼ 0, kU ¼ D1, go to

Step 3;Step 3: Let k ¼ ðkL þ kUÞ=2. Find T3ðkÞ by (41). Calculate l2ðkÞ by (44). If l2ðkÞ ¼ 0, T� ¼ T3, go to Step 6; else if l2ðkÞ > 0,

go to Step 4; if l2ðkÞ < 0, go to Step 5;Step 4: Let kL ¼ k, go to Step 3;Step 5: Let kU ¼ k, go to Step 3;Step 6: Output T�2ðkÞ.

Case 3 (M 6 T). In this case, APðTÞ ¼ AP3ðTÞ. Since the left hand side of the capacity constraint in (21) is an increasingfunction of T , it is obvious that the solution of (21) exists if and only if

Pleaseand tw

Xn

i¼1

wiDi

ai þ hiðeðaiþhiÞM � 1Þ 6W: ð47Þ

First, we maximize AP3ðTÞ with T P M. Using the first-order necessary condition, we obtain
Xn
i¼1

ciIcDi

ai þ hieðaiþhiÞðT�MÞ 1

ai þ hi� T

� �þ ððai þ hiÞT � 1ÞðEi þ HiÞeðaiþhiÞT � Zi

� �þ K ¼ 0: ð48Þ

We denote the left hand side of the above equation as g3ðTÞ, then

g03ðTÞ ¼Xn

i¼1

½�ciIcDiTeðaiþhiÞðT�MÞ þ ðai þ hiÞ2TðEi þ HiÞeðaiþhiÞT �: ð49Þ

Since Hi < 0, Ei < 0, g03ðTÞ < 0, then g3ðTÞ is an increasing function of T , and

g3ðMÞ ¼Xn

i¼1

ciIcDi

ai þ hi

1ai þ hi

�M� �

þ ððai þ hiÞM � 1ÞðEi þ HiÞeðaiþhiÞM � Zi

� �þ K: ð50Þ

If g3ðMÞ 6 0, we get g3ðTÞ 6 0 with M 6 T, so AP03ðTÞ 6 0, so AP3ðTÞ is a decreasing function of T , according to (47), theoptimal cycle is T� ¼ M.

If g3ðMÞ > 0, since limT!þ1g3ðTÞ ¼ �1, by the Intermediate Value Theorem and the monotonicity of g3ðTÞ, there exists aunique T5 > M, such that g3ðT5Þ ¼ 0. If T5 satisfies the constraint in (21), then T5 is the solution of problem (21); if T5 does notsatisfy the constraint in (21), we have
Xn
i¼1

wiDi


We then solve problem (21) with M 6 T . The Lagrange function of problem (21) is


i¼1

wiDi


" #; ð52Þ

where k P 0 is the Lagrangian multiplier. According to KT conditions, we have
Xn
i¼1

ciIcDi

ai þ hieðaiþhiÞðT�MÞ 1

ai þ hi� T

� �þ ððai þ hiÞT � 1ÞðEi þ HiÞeðaiþhiÞT � Zi � kwiDiT

2eðaiþhiÞT� �

þ K ¼ 0; ð53Þ

Xn

i¼1

wiDi





Next, we will look for k > 0 and T P M satisfying the above two equations.Let the left hand side of (53) be h3ðTÞ, that is

Table 1Parame

i

12345

Pleaseand tw

h3ðTÞ ¼ g3ðTÞ �Xn

i¼1

kwiDiT2eðaiþhiÞT : ð55Þ

Let D2 ¼ g3ðMÞ=ðPn

i¼1wiDiM2eðaiþhiÞMÞ, we have h3ðMÞP 0 with 0 6 k 6 D2. Since h3ðTÞ ! �1 as T ! þ1, by the Interme-

diate Value Theorem and the monotonicity of h3ðTÞ, there exists a unique T6ðkÞP M such that h3ðT6ðkÞÞ ¼ 0. SubstitutingT ¼ T6ðkÞ into (54), and let the left hand side be l3ðkÞ, that is

l3ðkÞ ¼Xn

i¼1

wiDi


Then

l03ðkÞ ¼Xn

i¼1

wiDeðaiþhiÞT6ðkÞT 06ðk:Þ ð57Þ

Substituting T ¼ T6ðkÞ into (53) and taking differential with respect to k on the two sides, we obtain
Xn
i¼1

T6ðkÞ½�ciIcDie�ðaiþhiÞM þ ðai þ hiÞ2ðEi þ HiÞ � kwiDið2þ ðai þ hiÞT6ðkÞÞ�T 06ðkÞeðaiþhiÞT �Xn

i¼1

wiDiT2eðaiþhiÞT ¼ 0: ð58Þ

Since Hi < 0, Ei < 0, it follows that T 06ðkÞ < 0. Then we get l03ðkÞ < 0 from (57).Since T6ðD2Þ ¼ M, according to (47), we have l3ðD2Þ 6 0. On the other hand, from L3ðT;0Þ ¼ AP3ðTÞ, it follows that

T6ð0Þ ¼ T5, thus l3ð0Þ > 0 from (51). Then, by the Intermediate Value Theorem and the monotonicity of l3ðkÞ, there exists un-ique k�3; such that 0 < k�3 6 D2 and l3ðk�3Þ ¼ 0. Thus T6ðk�3Þ satisfies (54).

Theorem 3. Let T�3 be the solution of problem (21) with M 6 T)

(a) If g3ðMÞ 6 0, then T�3 ¼ M;(b) If g3ðMÞ > 0 and T5 satisfies (21), then T�3 ¼ T5;(c) If g3ðMÞ > 0 and T5 does not satisfy (21), then T�3 ¼ T6ðk�3Þ, k�3 and T6ðk�3Þ are the unique solution of l3ðkÞ ¼ 0 and

h3ðT6ðkÞÞ ¼ 0.

Algorithm 3Step 0: Calculate g3ðMÞ. If g3ðMÞ 6 0, T�3 ¼ M, go to Step 5; else go to Step 1;Step 1: Let k ¼ D2, T ¼ M and calculate l3ðkÞ by (56). If l3ðkÞ ¼ 0, T�3 ¼ M, go to Step 5; else let kL ¼ 0, kU ¼ D2, go to Step

2;Step 2: Let k ¼ ðkL þ kUÞ=2. Find T4ðkÞ satisfies (53). Calculate l3ðkÞ by (56). If l3ðkÞ ¼ 0, T�3 ¼ T4, go to Step 5; else if

l3ðkÞ > 0, go to Step 3; if l3ðkÞ < 0, go to Step 4;Step 3: Let kL ¼ k, go to Step 2;Step 4: Let kU ¼ k, go to Step 2;Step 5: Output T�3ðkÞ.

Based on Theorems 1–3, we can determine the optimal solution of problem (21) as follows.

Theorem 4. Let T� be the optimal solution of problem (21). Then T� satisfies

APðT�Þ ¼maxfAPðT�1Þ;APðT�2Þ;APðT�3Þg:

5. Numerical examples

We consider an inventory system with 5 items ðn ¼ 5Þ, the associated parameters are K ¼ 200, W ¼ 1000, Ie ¼ 0:11,Ic ¼ 0:13, M ¼ 0:6, N ¼ 0:3, e ¼ 0:05 and others are listed in Table 1.

ters of inventory system.

Di ai hi hi ci pi wi

100 0.4 0.05 4 6 12 890 0.3 0.04 3 5 11 10

110 0.5 0.06 5 7 13 980 0.2 0.03 2 4 10 11

120 0.6 0.07 6 8 14 9



Table 2The results of numerical example.

T� Q�1 Q�2 Q�3 Q�4 Q�5 AP�

0 6 T 6 N 0.2050 21.4756 19.1084 23.8956 16.7930 26.3698 2127.6N 6 T 6 M 0.3000 32.1193 28.4250 35.9340 24.8474 39.8731 2399.7M 6 T 0.6000 68.8810 59.9025 78.4416 51.4698 88.6229 2506.6

Table 3Sensitivity analysis of stock-dependent consumption rate parameter.

a1 T� Q�1 Q�2 Q�3 Q�4 Q�5 AP�

0.2 0.2057 21.1077 19.1754 23.9813 16.8512 26.4654 2117.40.3 0.2054 21.2910 19.1420 23.9386 16.8222 26.4178 2122.50.4 0.2050 21.4756 19.1084 23.8956 16.7930 26.3698 2127.60.5 0.2047 21.6614 19.0745 23.8524 16.7635 26.3215 2132.70.6 0.2043 21.8484 19.0405 23.8088 16.7339 26.2729 2138.0

Table 4Sensitivity analysis of deteriorating rate parameter.

h1 T� Q�1 Q�2 Q�3 Q�4 Q�5 AP�

0.04 0.2050 21.4571 19.1118 23.8999 16.7959 26.3746 2128.40.05 0.2050 21.4756 19.1084 23.8956 16.7930 26.3698 2127.60.06 0.2050 21.4941 19.1050 23.8913 16.7900 26.3650 2126.80.07 0.2049 21.5126 19.1016 23.8870 16.7871 26.3602 2126.00.08 0.2049 21.5312 19.0983 23.8827 16.7842 26.3553 2125.2


By applying Algorithms 1, 2 and 3, we obtain the computational results as shown in Table 2, the optimal policy isT� ¼ 0:6000; Q �1 ¼ 68:8810; Q �2 ¼ 59:9025; Q �3 ¼ 78:4416; Q �4 ¼ 51:4698; Q �5 ¼ 88:6229, and the optimal average profit perunit of the system is AP� ¼ 2506:6.

Furthermore, the sensitivity analysis is performed to study the effects of the optimal policy when the value of the stock-dependent consumption rate parameter values and the deteriorating rate are changed. For simplicity, we choose item 1 as anexample with 0 6 T 6 N. The results are shown in Tables 3 and 4.

From Tables 3 and 4, we can see that the larger value of stock-dependent consumption rate parameter leads to shorterordering cycle, lower ordering quantities for items 2, 3, 4 and 5, and higher ordering quantity for item 1. At the same time,the average profit gets higher and lower respectively with the value of stock-dependent consumption rate and deterioratingrate are large.

6. Conclusions

In this paper, an economic ordering quantity model is developed for a two-level supply chain system consisting of multi-ple items, one supplier and one retailer, in which the supplier provides trade credit to the retailer who gives trade credit tohis customers at the same time, the retailer’s warehouse has restricted capacity and each item’s demand is the stock-depen-dent. The order relations between time parameter and credit periods lead to three different models. For each of them, byLagrangian approach, a line search algorithm is proposed to find the optimal replenishment policy and the optimal orderquantity. Some numerical examples are provided to illustrate the proposed model. The sensitivity of the solution to changesin the values of different parameters has also been discussed.

The contributions of this paper to the literature and managerial decision making are now summarized. First, this is anextension of the model by Min et al. [27] to deal with multiple items and restricted capacity, that is, setting n ¼ 1 and Wlarge enough, this model is the same as the one proposed by Min et al. [27]. Second, we suggest optimal properties and de-velop easy to use line searched algorithms for solving the problems described by Lagrangian approach. This is completelydifferent with Min et al. method. We believe our model and solving method can be applied to the following situation. Ina supermarket, multiple items are usually stored at goods shelf to sell in a supermarket. Some items like foodstuffs, greenvegetables, fresh meats, etc. are perishable and have a common characteristic, that is, a large pile attracts more customers.To stimulate demand, the retailer will display each of his/her items in large quantities. Then there arise the problem of allo-cation of storage capacity for each item since the total capacity is restriction.

For further research, this paper may be extended by using the time varying deterioration rate and the order quantity as afunction of credit period. A very interested extension would be different credit periods imposed to different items.

Please cite this article in press as: M. Jiangtao et al., Optimal ordering policies for perishable multi-item under stock-dependent demandand two-level trade credit, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.058



Acknowledgements

The work supported by the Natural Science Foundation of China (Nos: 71261002, 11161003) and Guangxi Science Foun-dation (No: 0991028).

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Please cite this article in press as: M. Jiangtao et al., Optimal ordering policies for perishable multi-item under stock-dependent demandand two-level trade credit, Appl. Math. Modell. (2013), http://dx.doi.org/10.1016/j.apm.2013.10.058

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