vendor-managed inventory system with partial...

Scientia Iranica E (2017) 24(3), 1483{1492

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringwww.scientiairanica.com

Vendor-managed inventory system with partialbackordering for evaporating chemical raw material

A.A. Taleizadeh�

School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran.

Received 23 December 2013; received in revised form 21 July 2015; accepted 26 September 2015

KEYWORDSSupply chainmanagement;Inventory;Partial backordering;Evaporating product;deterioration.

Abstract. Consider a supply chain including a re�nery producing evaporating chemicalproduct, an exporter, and one or some engine oil producers outside the exporter's country.The exporter uses vendor-managed inventory system implemented between re�nery andexporter to decrease his/her inventory cost. This paper develops two models with partialbackordering for evaporating chemical product developed in a two-layer chain includingsingle re�nery and single exporter with one product before and after utilizing vendor-managed inventory policy. Demand and partial backordering rates are deterministic andconstant. A numerical example is provided to illustrate the applicability of the proposedmodel and solution method.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction and literature review

Vendor-managed inventory is a practice to improvethe inventory management in supply chains. In thissystem, a vendor decides on the appropriate inventorylevels of each product for all his/her retailers andthe appropriate inventory policies to maintain theselevels [1]. The objective of the members of VMIsystem is to decrease both inventory and replenishmentcosts in each supply chain, which bene�ts all membersof the chain. Recently, this policy has widely beenstudied and developed by many researchers. Forinstance, Yang et al. [2] studied the e�ects of adistribution center in a VMI policy including sin-gle manufacturer and multi buyers. Hemmelmayret al. [3] employed vendor-managed inventory policyfor the delivery of blood products to hospitals inEastern Austria. Yu and Huang [4] studied how aproducer and its buyers interact with each other tooptimize their marketing strategies, platform prod-uct con�guration, and inventory management policies

*. Tel.: +98(21) 8208-4486; Fax: +98 (21) 8801-3102,E-mail address: [email protected]

in vendor-managed inventory system utilized in theproposed supply chain. Guan and Zhao [5] devel-oped vendor-managed inventory system under contin-uous review (r;Q) policy. Yao et al. [6] developedVMI system with stochastic demand and backorderedshortage. Kristianto et al. [7] developed an adap-tive fuzzy control application to support a vendor-managed inventory policy in a supply chain. Darwishand Odah [8] developed and utilized vendor-managedinventory policy in a supply chain including singlevendor and several retailers. Egri and Vancza [9]studied the problem of coordination in supply net-works including single supplier and multiple retailerswhere members had asymmetric information aboutdemand and costs. Yu et al. [10] analyzed retailerselection in a vendor-managed inventory system for amanufacturer. Zanoni et al. [11] studied two issueshaving usual interactions in practice, which were the`VMI with consignment' and the `Learning Curve'policies. Chen et al. [12] studied the impact oftransshipment and demand variability on distributionpolicies of the vendor when vendor-managed inven-tory strategy was used. Yu et al. [13] developedVMI policy for replenishing deteriorating raw ma-

1484 A.A. Taleizadeh/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 1483{1492

terial in which shortage was not permitted. Shuet al. [14] designed a logistic network using vendor-managed inventory for a company that was in chargeof managing inventory for its retailers and down-stream warehouses. Taleizadeh et al. [15] studied ajoint optimization of price, replenishment frequency,replenishment cycle, and production rate in vendor-managed inventory system for deteriorating items.Moreover, recently, Cardnas-Barron et al. [16] re-viewed the developments in EOQ models in honorof Ford Whitman Harris. Yang et al. [17] analyzedthe robustness of di�erent supply chain strategiessuch as VMI, electronic point of sales, e-shopping,and emergency transshipments under various uncer-tain environments. Kastsian and Monnigmann [18]studied robustness and stability of a supply chainin which vendor-managed inventory was used. Leeand Ren [19] examined the bene�ts of VMI in aglobal environment under periodic-review stochasticinventory policy, in which the supplier and the retailerfaced uncertain exchange rate and incurred di�erent�xed ordering costs. Almehdawe and Mantin [20]modeled a Stackelberg-game vendor-managed inven-tory framework with two di�erent scenarios. In the�rst one, the manufacturer was the leader and inthe second one, the retailer was leader. Wong etal. [21] studied how a sales rebate contract helpedto achieve supply chain coordination, allowing chainmembers with decentralized decisions to perform acentralized decision for the whole system. Zavanellaand Zanoni [22] developed an integrated inventoryproduction model in which vendor-managed policy wasused. Xu and Leung [23] studied a retail channeland proposed an analytical model for the membersof that channel to determine the inventory policysuch that the net pro�t of channel was maximized.Southard and Swenseth [24] studied VMI policy in aunique chain such that a �rm could justify spendingthe money necessary to create the infrastructure tosupport it. Gumus et al. [25] analyzed the impactsof VMI policy and consignment inventory in a two-layer supply chain. Hariga and Al-Ahmari [26] de-veloped integrated inventory-shelf allocation model fora stock-dependent demand of single product. Theintegrated model was developed for a supply chainoperating under VMI policy and consignment stockagreement. Yu et al. [27] developed a Stackelberggame-theoretic model to optimize inventory, pricing,and advertising decisions in Vendor-Managed Inven-tory (VMI) system. Michaelraj and Shahabudeen [28]developed two supply chain models operating underVMI policy with two di�erent objectives. In the�rst one, distributors had the objective of minimizingthe balance payment considering the �nancial safetyand in the second one, they had the objective ofmaximizing the sales with an aim to go beyond business

survival. Passandideh et al. [29] developed a two-layer supply chain in which VMI and EOQ policieswere used, but shortage was not permitted. Then,Passandideh et al. developed their model to a multi-product multi-constraint case and solved it with ge-netic algorithm. In continuation, Cardenas Barronet al. [30] developed a more e�cient algorithm tosolve their proposed model. Liao et al. [31] devel-oped a multi-objective integrated inventory-locationdistribution network problem operating under vendor-managed inventory policy. Chen et al. [32] studiedthe problem of coordinating a vertically separateddistribution system under vendor-managed inventoryand consignment arrangements.

There is much additional research related to VMIpolicy which is not introduced here. But, the interest-ing point is that none of the studies has developed andutilized VMI policy under partial backordering for anevaporating or deteriorating item. Thus, the aim of thisresearch is to extend and utilize the VMI policy withpartial backordering for evaporating or deterioratingitems. Moreover, in this paper, a real case in petroleumindustry is presented and the developed model is usedto solve the problem on hand for that �rm.

2. Problem de�nition

In this section, we describe a case study in whicha re�nery and an exporter and some retailers of anevaporating chemical product, which are engine oilproducers, exist. Decisions about when and howmuch product should be ordered are so important.Exporter intends to use Vendor-Managed Inventory(VMI) system and asks the re�nery to use VMI policyin order to decrease his/her cost.

The second characteristic of the store's situationis that if the engine oil producers order the productbut cannot buy it because the exporter cannot satisfydemand of the company at appropriated time, theymay do one of two things based on how much of itthey still have. If they have enough of the productto wait until the exporter satis�es their demand, theirorder will be backordered. But if engine oil producersdo not have enough of the product to wait, they willapply to another exporter in the store and will buyit from that one. In this situation, the sale at theoriginal exporter will be lost. If we assume that afraction of the demands can be postponed and theremaining fraction will go elsewhere, which is the case,the shortage at the original exporter will be partiallybacklogged, which is one of the assumptions of theproposed model.

The third characteristic of the store's situationis that the commodity exchange is evaporating rawmaterial of engine oil factory and needs special at-tentions because of evaporation and its e�ects on

A.A. Taleizadeh/Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 1483{1492 1485

mathematical modeling. Moreover, this paper can beused for deteriorating product instead of evaporatingone, because of their similarities in assumptions andmodeling.

In addition to the characteristics of the problem,it is assumed that all parameters of the problem,including evaporation, partial backordering and de-mand rates, and cost parameters, are constant anddeterministic. In the next section, we will model theproblem on hand.

3. Modeling

To model the problem, the following notations are used.

qN The order quantity of exporterq The order quantity of exporter in

vendor-managed inventory system� The constant rate of product

evaporationAR The setup cost per order of the re�neryAE The ordering cost per order of the

exporterC The deterioration cost per unitd The constant demand rate of the

exporter� The partial backordering rate of the

exporterh The inventory holding cost of exporter

in a period per unit per unit timebN The maximum level of backordering

shortage of exporterb The maximum level of backordering

shortage of exporter in vendor-managedinventory system

�0 The �xed cost of lost sale of exporterper unit

�̂ The cost of shortage of exporter perunit per time

TN The time cycleT The time cycle in vendor-managed

inventory systemFN The percentage of cycle length with

positive inventoryF The percentage of cycle length with

positive inventory in vendor-managedinventory system

TCEN Inventory cost of the exporter invendor-managed inventory system

TCE Inventory cost of the exporter invendor-managed inventory system

TCRN Inventory cost of the re�nery

TCR Inventory cost of the re�nery invendor-managed inventory policy

TCN The total cost before vendor-managedinventory policy

TC The total cost of vendor-managedinventory policy

Because of demand and deterioration, during[0; FT ], the inventory level drops to zero. Thus, thedi�erential equation shown in Eq. (1) shows the changein the inventory level during [0; FT ].

dI1(t)dt

= ��I1(t)� d; 0 � t � FT: (1)

Therefore, we have:

I1(t) =d�

�e�(FT�t) � 1

�: (2)

Furthermore, at time FT , shortage occurs andthe inventory level starts dropping below zero suchthat, �nally, the backordered and lost sale quantitiesare respectively:

b = �d(1� F )T; (3)

L = (1� �)(1� F )dT: (4)

Therefore, we have:

Imax = I1(0) =d��e�FT � 1

�: (5)

Since Q = Imax + b, we have:

q =d��e�FT � 1

�+ �d(1� F )T: (6)

The total cost of system for the cycle time, T ,is made up of the exporter's ordering cost, re�nery'sordering cost, the holding cost of product stored inexporter's warehouse in a period, deterioration cost,and cost of shortage. The exporter's holding cost is:

h�FTZ0

I1(t) = h�FTZ0

d�

�e�(FT�t) � 1

�dt

=� d�2

�e�(FT�t)+�t

��FT0

=hd�2

�e�FT��FT�1

�:(7)

Moreover, the �xed cost of exporter is AE and thedeterioration cost will be:

C (I1(0)� dFT ) = C�d�

�e�(FT ) � 1

�� dFT� ; (8)

and the back-ordered and lost sale costs are respec-tively:

��d(1� F )2T 2

2; (9)

�0d(1� �)(1� F )T: (10)


3.1. Analysis of inventory costsBefore utilizing vendor-managed inventory system, thetotal cost function of the exporter and the total costfunction of re�nery are respectively shown in Eqs. (11)and (12). Then, the total cost of the chain is shown inEq. (13):

TCEN =1T

AE +

d�2

�e�FNTN � �FNTN � 1

�+ C

�d��e�FNTN � 1

�� dFNTN�+�̂�d (1� FN )2 T 2

N2

+ �0d(1� �)(1� FN )TN

!; (11)

TCRN =ART; (12)

TCN =TCEN + TCRN =1T

AR +AE

+d�2

�e�FNTN � �FNTN � 1

�+ C

�d��e�FNTN � 1

�� dFNTN�+�̂�d(1� FN )2T 2

N2

+ �0d(1� �)(1� FN )TN

!: (13)

Using approximation of the Taylor series expansion,e�FNTN = 1 + �FNTN + (�FNTN )2

2 , we have:

TCEN =1TN

AE +

(�C + h)d(FNTN )2

2

+�̂�d(1� FN )2T 2

N2

+ �0d(1� �)(1� FN )TN

!; (14)

TCRN =ARTN

; (15)

TCN =1TN

AR +AE +

(�C + h)d(FNTN )2

2

+�̂�d(1� FN )2T 2

N2

+�0d(1��)(1�FN )TN

!:(16)

3.2. Solution methodThe exporter's inventory cost in Eq. (14) is a functionof T and F . Thus, global optimal values of T and F canbe obtained by taking the partial derivative of Eq. (14)with respect to T and F ; then setting them equal tozero (in Appendix`A, we prove that T � and F � give aglobal optimal solution).

@TCEN@TN

=�AET 2N

+(h+C�)dF 2

N2

+�̂�d(1�FN )2

2:(17)

Then, we have:

T �N (FN ) =

s2AE

(h+ C�)dF 2N + �̂�d(1� FN )2 : (18)

and:

@TCEN@FN

=(h+ C�)dFNTN � �̂�dTN (1� FN )

� �0d(1� �): (19)

Finally:

FN =��̂TN + �0(1� �)(h+ C� + ��̂)TN

: (20)

Substituting F �N into T �N (F �N ) yields (see Appendix B):

TN =

s2AE(h+ C� + ��̂)� d (�0(1� �))2

��̂(h+ C�)d: (21)

Shortage quantity before utilizing vendor-managed inventory policy can be calculated as follows(see Eq. (3)):

b�N = �d(1� F �N )T �N : (22)

Moreover, the ordering quantity before utilizingvendor-managed inventory policy using approximationof the Taylor series expansion in Eq. (6) will change to:

q�N =d�FNTN+

�(FNTN )2

2

�+�d(1� FN )TN : (23)

The optimal values of decision variables T and Fof the exporter's total cost shown in Eq. (14) can beobtained using Eqs. (20) and (21) only if � satis�es theinequality given in Eq. (24) (see Appendix C).

� > ��N = 1�p

2AE(h+ C�)dd�0 : (24)


Then, this solution is optimal if this condition ismet and �� > 0. If the condition is met but �� < 0,the cost of the optimal partial backordering solutionmust be compared with the cost of not stocking. Tosummarize: If � > �� > 0 (where �� is givenin Eq. (24)), then the optimal partial backorderingsolution is given by Eqs. (20) and (21); if, however,�� < 0, the cost of that solution must be comparedwith the cost of not stocking at all. If � < ��, theoptimal solution is either to use the classic EOQ modelor not to stock the item at all, whichever costs less [33-36].

But, when vendor-managed inventory policy isused, since re�nery should pay the exporter costs,Eqs. (14) to (16) will change to:

TCE = 0; (25)

TCR =1T

AR +AE +

hd�2

�e�FT � �FT � 1

�+ C

�d��e�FT � 1

�� dFT�+�̂�d(1�F )2T 2

2+�0d(1��)(1�F )T

!; (26)

TC =TCE + TCR =1T

AR +AE

+hd�2

�e�FT � �FT � 1

�+ C

�d��e�FT � 1

�� dFT�+�̂�d(1� F )2T 2

2+ �0d(1� �)(1� F )T

!:(27)

Using approximation of the Taylor series expansion, wehave:

TCR =1T

AR +AE +

(�C + h)d(FT )2

2

+�̂�d(1�F )2T 2

2+ �0d(1��)(1�F )T

!;(28)

TC =1T

AR +AE +

(�C + h)d(FT )2

2

+�̂�d(1� F )2T 2

2+ �0d(1� �)(1� F )T

!:(29)

Once again, since the total inventory cost inEq. (28) is a function of T and F , global optimalvalues of T and F can be obtained by taking the partialderivative of Eq. (28) with respect to T and F , thensetting them equal to zero (as we mentioned in theprevious section, in Appendix A, we prove that T � andF � give a global optimal solution):

@TC@T

=� (AR+AE)T 2 +

d(C�+h)F 2

2+�̂�d(1�F )2

2:

(30)

Then, we have:

T �(F ) =

s2(AR +AE)

d(h+ C)F 2 + �̂�d(1� F )2 ; (31)

and:

@TC@F

= (h+C�)dFT��̂�dT (1�F )��0d(1��):(32)

Finally:

F � =�0(1� �) + ��̂T(h+ C� + ��̂)T

: (33)

Substituting F � into T �(F �), same as when vendor-managed inventory system is not utilized, yields:

T �=

s2(AR+AE)(h+C�+��̂)�d(�0(1��))2

(��̂)(h+ C�)d; (34)

the shortage and order quantities after utilizing vendor-managed inventory policy are respectively:

b� = �d(1� F �)T �; (35)

q� = d�FT +

�(FT )2

2

�+ �d(1� F )T: (36)

Same as the previous case, Eqs. (33) and (34) givethe optimal values of T and F for the cost functionin Eq. (14) only if � satis�es the inequality given inEq. (29).

� > �� = 1�p

2(AR +AE)(h+ C�)dd�0 : (37)

Utilizing the solution method proposed by Penticoet al. [37], the following solution procedure can bedeveloped to determine the optimal values of T andF when VMI policy is used:

1. Calculate �� = 1�p

2(AR+AE)(h+C�)dd�0 ;


2. If � � ��, go to Step 3. If � < ��, determine theoptimal cost of `no stockouts' and compare this withthe cost of losing all demands, �0d, to determinewhether it is optimal to allow no stockouts or allstockouts. If

p2(AR +AE)d(h+ C 0�) � �0d, then

F � = 1, T � =q

2(AR+AE)d(h+C�) , q� = dT �, and b� = 0.

Ifp

2(AR +AE)d(h+ C 0�) > �0d, then T � = 1,F � = q� = 0 and Stop;

3. Using Eq. (33) and (34), calculate the couple of(T �; F �) and go to step 4. If �� < 0, computeTC(F �; T �) using Eq. (29); if (F �; T �) < �0d, goto step 4. Otherwise, T � = 1, and F � = q� = 0.Stop;

4. Determine the optimal value of shortage and orderquantities using Eqs. (35) and (36).

If we would like to solve an EOQ model with partialbackordering for deteriorating item (ignoring VMIpolicy), we should use AS = 0 in the described solutionmethod and all equations will be derived for when VMIpolicy is ignored.

4. Numerical example

In order to illustrate the above solution procedure,let us consider an inventory system with the followingdata:

AR = 100; AE = 100; h = 3; d = 2; 000;

C = 100; � = 0:005; �0 = 1; �̂ = 2:

4.1. First example: � = 0:5Using Eq. (37), we have � = 0:5:

- Step 1.

�� = 1�p

2(AR +AE)(h+ C�)dd�0

= 1�p

2(100 + 100)(3 + 100� 0:005)20002000(1)

= 0:1633;

- Step 2. Since � = 0:5 > �� = 0:1633, go to Step 3;- Step 3. Using Eqs. (34) and (33), we have:

T �=q

2(100+100)(3+100�0:005+0:5�2)�2000(1(1�0:5))2

(0:5�2)(3+100�0:005)2000

= 0:4309;

F � =1(1� 0:5) + 2(0:5)0:4309

(3+100�0:005+2�0:005)0:4309=0:48:

- Step 4. Using Eqs. (35) and (36), we have:

b�=0:5�2000�(1�0:48)�0:4309=224:0689;

q� =2000�

0:43�0:4309 +0:005(0:43� 0:4309)2

2

�+ 0:5�2000(1� 0:48)0:4309 = 638:0366:

4.2. Second example � = 0:1- Step 1. From the �rst step of the �rst example, we

know �� = 0:1633;- Step 2. Since � = 0:1 < �� = 0:1633, the optimal

cost of `no stockouts' and the cost of losing alldemands should be compared. Since the optimalcost of `no stockouts'

p2(AR +AE)d(h+ C 0�) =p

2(100 + 100)2000(3 + 100� 0:005) = 1673:32 isless than the cost of losing all demands �0d =1� 2000 = 2000, we have:

F � = 1;

T � =

s2(AR +AE)d(h+ C�)

=

s2(100 + 100)

2000(3 + 100� 0:005)

= 0:2390

and:q� = dT � = 2000� 0:2390 = 478:0914:

5. Conclusion

In this paper, a supply chain including a re�neryproducing evaporating chemical product and a dis-tributer who exports the products of re�nery, with andwithout vendor-managed inventory policy, is studied.According to the observations of real case, the partialbackordering is assumed where all parameters of themodel are constant and deterministic. The closed-form solutions for decision variables, including optimalperiod length, order and shortage quantities of evap-orating item under vendor-managed inventory policy,after proo�ng the concavity of objective function, arederived. Two numerical examples are provided to illus-trate the applicability of the proposed model and solu-tion method. The result shows that vendor-managedinventory decreases cost in all conditions and is morebene�cial for the coordination system. For futureresearch, considering stochastic partial backordering ordemand rates and several competing re�neries or ex-porters, considering pricing and revenue managementtopic, using time-dependent partial backordering rateor using contacts between the partners can enhance thedeveloped model.

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Appendix A

Proof of the optimality of the solutions(Eqs. (20), (21), (33), and (34))Using the method proposed by Pentico et al. [37], wecan prove that Eqs. (20), (21), (33), and (34) are globaloptimal, although the cost functions, which are shownin Eqs. (16) and (29), are not convex. The cost functioncan be rewritten as follows:

TC(F; T ) =�0

T+��1F 2 � 2�2F + �2

�T

� �3F + �3; (A.1)

where:

�0 = A > 0;

(No VMI : A = AE ; VMI : A = AR +AE); (A.2)

�1 =(�C + h+ ��̂)d

2> 0; (A.3)

�2 =�̂�d

2> 0; (A.4)

�3 = �0d(1� �) > 0: (A.5)

Note that all the �is are positive. We can rewriteEq. (A.1) as follows:

TC(F; T ) =�0

T+ Tr(F ) + q(F ); (A.6)

where:

r(F ) = �1F 2 � 2�2F + �2: (A.7)

and:

q(F ) = ��3F + �3: (A.8)

Our objective is to establish the condition underwhich Eq. (A.6) has a unique interior minimizer.Di�erentiating Eq. (A.6) with respect to T yields:

@TC@T

= � �0

T 2 + r(F ); (A.9)

which equals zero if and only if T satis�es:

T = T �(F ) =

s�0

r(F ): (A.10)

Note that this is the same result, with appropriatechange of notation, given in Eqs. (21) and (33). Sincethe discriminant of r(F ) is negative, r(F ) has no roots.Thus, r(F ) is either all-positive or all-negative. Sincer(0) = �4 > 0, r(F ) is strictly positive in [0; 1]. Thus,Eq. (A.10) gives, for each F , a unique T � = T �(F )


that minimizes the cost function given by Eq. (A.7).Substituting the expression for T �(F ) in Eq. (A.10)into TC(F; T ) given by Eq. (A.6) gives:

TC(F ) � TC(T �(F ); F )=2p�0r(F )+q(F ); (A.11)

which represents minimal possible cost for each valueof F . Note that TC(F ) is continuous; thus, on thecompact interval [0; 1], it has one or more local minima,the smallest of which will be the global minimum of thecost function [37]. To �nd these minima, take the �rstand second derivatives of TC(F ) with respect to F ,yielding:

TC 0(F ) =p�0

r0(F )r(F ) 1

2+ q0(F ); (A.12)

TC 00(F ) =p�0�2r00(F )r(F )� (r0(F ))2�

2r(F ) 32

: (A.13)

Note that TC 0(F ), which is, with the change innotation, same as @TC(F;T )

@F given in Eqs. (19) and (32),is continuous and satis�es TC 0(0) < 0.

r0(F ) = 2�1F � 2�2; (A.17)

q0(F ) = ��3; (A.18)

r00(F ) = 2�1; (A.19)

q00(F ) = 0; (A.20)

TC 0(0) = ��p

�0�3p�2

+ �3

�< 0: (A.21)

The second derivative, TC 00(F ), given in Eq. (A.13) isas below:

TC 00(F )

=p�0[4�1(�1F 2�2�2F+�2)�(2�1F�2�2)2]

2r(F ) 32

=p�0[�8�1�2F + 4�1�2 � 4�2

2 + 8�1�2F ]2r(F ) 3

2

=p�0[4�1�2 � 4�2

2]2r(F ) 3

2> 0; (A.22)

which is positive for all F s, because �1 = (�C+h+��̂)d2

and �2 = �̂�d2 , meaning 4�1�2 > 4�2

2.

Appendix B

Deriving the optimal value of period lengthshown in Eqs. (31) and (33)To proof Eq. (21), from Eq. (18), we know:

T �2N =2AE

(h+ C�)dF 2N + �̂�d(1� FN )2 : (B.1)

Moreover, from Eq. (20) we have:

FN =��̂TN + �0(1� �)(h+ C� + ��̂)TN

: (B.2)

Replacing FN and 1� FN in Eq. (B.1) by:

FN =��̂TN + �0(1� �)(h+ C� + ��̂)TN

;

and:

1� FN =(h+ C�)TN � �0(1� �)

(h+ C� + ��̂)TN;

respectively yields in Eqs. (B.3) and (B.4) as shown inBox B.1. Finally, we have:

TN =

s2AE(h+ C� + ��̂)� d(�0(1� �))2

(��̂)(h+ C�)d: (B.5)

Appendix C

Deriving lower limit of partial backorderingrateSince T � for the partial backordering model must be atleast as large as T � for the basic EOQ model, �rstly, weshould determine T � for the basic EOQ model. Thus,if we consider F = 1, then Eq. (14) (the total cost whenthere is no shortage) will change to:

TCEN =AETN

+(�C + h)dTN

2; (C.1)

and we will have:

dTCENdTN

= �AET 2N

+hd2

+C�d

2; (C.2)

d2TCENd2TN

=2AET 3N� 0: (C.3)

Since the second derivative is positive for allvalues of T > 0, the objective function shown inEq. (C.1) is convex and setting the �rst derivative equalto zero will yield the optimal period length shown inEq. (C.4).

T �N =

s2AE

d(h+ C�): (C.4)

Now, considering T �N for the partial backorderingmodel greater than T �N for the basic EOQ model, wehave:

2AE(h+ C� + ��̂)� d(�0(1� �))2

(��̂)(h+ C�)d� 2AEd(h+ C�)

;

Box B.1

vendor-managed inventory system with partial...

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