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A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain
Transportation Research Part E 45 (2009) 409–418
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A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain M.E. Seliaman a,*, Ab Rahman Ahmad b a b
The Office of Planning and Quality, King Fahd University of Petroleum and Minerals, 1317, Dhahran 31261, Saudi Arabia Department of Modeling and Industrial Computing, Faculty of Computer Science and Information System, Universiti Teknologi Malaysia, Malaysia
a r t i c l e
i n f o
Article history: Received 12 April 2008 Received in revised form 4 August 2008 Accepted 12 September 2008
Keywords: Supply chain integration Production-inventory models Without derivatives
a b s t r a c t In this paper, we deal with more generalized inventory coordination mechanism in an nstage, multi-customer, non-serial supply chain, where we extend and generalize pervious works that use algebraic methods to optimize this coordinated supply chain. We establish the recursive expressions for this multi-variable optimization problem. These expressions are used for the derivation of the optimal replenishment policy and the development of the solution algorithm. Further, we describe a simple procedure that can help in sharing the coordination cost benefits to induce all stages to adopt the inventory coordination mechanism. We provide a numerical example for illustrative purposes. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Supply chain management can be defined as a set of approaches utilized to efficiently integrate suppliers, manufactures, warehouses, and stores, so that merchandise is produced and distributed at the right quantities, to the right locations, and at the right time, in order to minimize system-wide cost while satisfying service level requirements (Simchi-Levi et al., 2003). In the past, supply chain production-inventory decisions were not coordinated among the different parties in the supply chain. This lack of coordination leads to weakly connected activities and decisions across the supply chain. Recently firms realized that the global system performance and cost efficiency can be improved through closer collaboration among the chain partners and through high level of coordination of various decision processes. Supply chain inventory-distribution coordination can be achieved by coordinating the cycle time across the chain stages. The simplest way of cycle coordination is the equal cycle under which the same cycle is followed throughout the supply chain. But many supply models achieve coordination by following the integer multipliers mechanism in which the cycle time at each stage is an integer multiple of the cycle time of the adjacent downstream stage, or by integer powers of two multipliers at each stage mechanism (Khouja, 2003). In recent years numerous articles in supply chain modeling have addressed the issue of inventory coordination. Banerjee (1986) introduced the concept of joint economic lot sizing problem (JELS). He considered the case of a single vendor and a single purchaser under the assumption of deterministic demand and lot for lot policy. Goyal and Szendrovits (1986) presented a constant lot size model where the lot is produced through a fixed sequence of manufacturing stages, with a single setup and without interruption at each stage. This model mainly, relaxes the constraint that batches must be of equal size at any particular stage. Goyal (1988) provided a more general model for the case of single vendor single buyer through relaxing the lot-for-lot policy. He showed that his model provides a lower or equal total joint relevant cost compared to Banerjee * Corresponding author. Tel.: +966 509198211; fax: +996 3 8601881. E-mail addresses: [email protected] (M.E. Seliaman), [email protected] (A.R. Ahmad). 1366-5545/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2008.09.012
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(1986). Goyal and Gupta (1989) extensively reviewed the literature which deals with the interaction between a buyer and vendor. They classified the literature dealing with the integrated models into four main classes. The first class represents models which deal with joint economic lot sizing policies. The second class characterizes models which deal with the coordination of inventory by simultaneously determining the order quantity for the buyer and the vendor. The third class is a group of models which deal with integrated problem but do not determine simultaneously the order quantity of the buyer and the vendor. The last class represents models which deal with buyer vendor coordination subject to marketing considerations. Lu (1995) developed a one-vendor multi-buyer integrated inventory model with the objective of minimizing the vendor’s total annual cost subject to the maximum cost that the buyer may be prepared to incur. Goyal (1995) revisited the single-single-buyer where he relaxed the constraint of equal sized shipments of Goyal (1988) and suggested that the shipment size should grow geometrically. Hoque and Goyal (2000) extended the idea of producing a single product in a multistage serial production system with equal and unequal sized batch shipments between stages. Khouja (2003) considered the case a three-stage non-serial supply chain and developed the model to deal with three inventory coordination mechanisms between the chain members. Bendaya and Al-Nassar (2008) relaxed the assumption of Khouja (2003). Regarding the completion of the whole production lot before making shipments out of it and assumed that equal sized shipments take place as soon as they are produced and there is no need to wait until a whole lot is produced. The use of differential calculus to model the integrated production inventory systems is common in the area of operational research. However, several researchers focused on the easy solution methods for the optimization of these types of systems. Cárdenas-Barrón (2001) used algebraic procedure to the EPQ formula taking shortages into consideration within the case of only one backlog cost per unit and time unit. Cárdenas-Barrón (2007) formulated and solved an n-stage-multi-customer supply chain inventory model where there is a company that can supply products to several customers. The production and demand rates were assumed constant and known. This model was formulated for the simplest inventory coordination mechanism which is referred to as the same cycle time for all companies in the supply chain. It was concluded that it is possible to use an algebraic approach to optimize the supply chain model without the use of differential calculus. Chung and Wee (2007) considered an integrated threestage inventory system with backorders. They formulated the problem to derive the replenishment policies with fourdecision-variables algebraically. Wee and Chung (2007) also used a simple algebraic method to solve the economic lot size of the integrated – buyer inventory problem. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure with ease. Chi (2008) presented a simple algebraic method to demonstrate that the lot size solution and the optimal production-inventory cost of an imperfect EMQ model can be derived without derivatives. Kit-Nam Francis Leung (2008) considered the EOQ problem, where the quantity backordered and the quantity received are both uncertain. He used the complete squares method to derive a global optimal expression from a non-convex objective function in an algebraic manner. Cárdenas-Barrón (in press) considered the problem of optimal manufacturing batch size with rework process at single-stag production system. He determined the optimal solution for two different inventory policies. He also established the range of real values of proportion of defectives products for which there is an optimal solution, the closed-form for the total inventory cost for both policies, the mathematical expressions for determining the cost penalty and the additional total cost for working with a non-optimal solution. In this study, we develop an optimal replenishment policy using a simple algebraic method to solve the n-stage, multicustomer, non-serial supply chain inventory problem. Our work is an extension of the three stage supply chain model in Khouja (2003). We consider the integer multiplier coordination mechanism. The remainder of this paper is organized as follows. The next section presents the notation and assumptions made in the model. Section 3 describes the development of the model. A solution procedure is presented in Section 4. Section 5 describes a scheme for sharing the coordination benefits. Section 6 presents a numerical example. Finally, Section 7 contains some concluding remarks.
2. Notation and assumptions The following notations are used in developing the model: T = Basic cycle time, cycle time at the end retailers. Ti = Cycle time at stage i. Sij = Setup cost for firm j at stage i. Si = Total setup cost for all firms at stage i. Ki = Integer multiplier at stage i. hi = Inventory holding cost at stage i. ni = Number of firms at stage i. Dij = The mean demand rate of firm j at stage i. Pij = Production rate of firm j at stage i. A = The product of all production rates for all the companies in the supply chain. Bij = The product of all production rates for all the companies in the supply chain, except for the company j in stage i.
M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
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Assumptions for the multi-stage supply chain production-inventory model: (a) (b) (c) (d) (e) (f)
A single product is produced and distributed through a multi-stage, multi-customer, non-serial, supply chain. Production rates and demand rates are deterministic and uniform. Ordering/setup costs are the same for firms at the same stage. Holding costs cost are the same for firms at the same stage. A lot produced at stage is sent in equal shipments to the downstream stage. The supply chain is vertically integrated and the entire supply chain optimization is acceptable for all partners in the chain. (g) Cycle time at each stage is an integer multiplier of the cycle time used at the adjacent downstream stage.
3. Development of the n-stage-multi-customer supply chain algebraically We consider a multi-stage, non-serial supply chain, where a firm at each stage can have two or more customers. A typical example of such supply chain structure is depicted in Fig. 1. This type of supply chain structure differs from the serial supply chain structure where each stage has only one firm. This supply chain model is formulated for the integer multipliers coordination mechanism, where firms at the same stage of the supply chain use the same cycle time and the cycle time at each stage is an integer multiplier of the cycle time used at the adjacent downstream stage. In this case, the cycle time of an end retailer is T and therefore the total cost per unit time for retailer j is given by:
TC n;j ¼ hn
TDn;j Sn;j þ 2 T
ð1Þ
The holding cost at each stage, except for the final stage (the end retailers’ stage n), is made of two parts: the first one is the carrying cost for the raw materials as they are being converted into finished products during the production portion of the cycle. The second part is the carrying cost of the finished products during the non-production portion of the cycle. During Q the production time, the average annual inventory of raw material as well as finished products is equal to ns¼i K s TD2i;j =2P i;j . During the non-production time, inventory drops by every Ti + 1 years by Ti + 1Dij starting from (Ti–Ti + 1) Dij as shown in Fig. 2. Therefore, the total annual cost for any firm at any stage, except for the final stage, is represented by: n Q
TC i;j ¼ s¼i
n Q
K s TD2i;j hi1 2Pi;j
þ
K s TDi;j
s¼iþ1
2
ðK i ð1 þ Di;j =P i;j Þ 1Þhi þ
Si;j n Q K sT
ð2Þ
s¼i
We assume that Kn = 1. The total cost for the entire supply chain is
TC ¼
X
TC i;j
ð3Þ
i;j
Supplier
Manufactures
Distributors
Retailers 1
1 2
1 1 2 2
2 1
1 3 2
3 1 4 2 Fig. 1. An example of a complex supply chain structure.
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Fig. 2. Raw materials and finished products at two consecutive stages.
We can write TC as
TC ¼ T
9 8 Pn1 P IC i:j A> > =
> : n Q
> ;
2A n Q
K s TD2i;j hi1
þ
1 Xn Si n i¼1 Q T Ks
ð4Þ
s¼i K s TDi;j
þ s¼iþ1 2 ðK i ð1 þ Di;j =P i;j Þ 1Þhi P P We define a0 ¼ ðh0 þ h1 Þ j D21;j B1;J þ A j D1;j h1 ; w0 ¼ 0 and u1 ¼ S1 :
where IC i;j ¼
s¼i
2Pi;j
For i = 1,2,3,. . .,n1 and for all j, we define ai, wi and /i as follows:
ai ¼ ðhi þ hiþ1 Þ
X
D2iþ1;j Biþ1;J þ A
j
X
Diþ1;j hiþ1 A
j
X
Di;j hi
ð5Þ
j
wi ¼ K i1 wi1 þ ai1 u ui ¼ Si þ i1 K i1
ð6Þ ð7Þ
For convenience and handiness, we re-write Eq. (4) in the following manner(see the Appendix A):
TC ¼ TY þ
W T
ð8Þ
where
Y¼
K n1 wn1 þ an1 2A
and W ¼ Sn þ
un1 K n1
ð9Þ ð10Þ
Now applying the algebraic procedure proposed by Cárdenas-Barrón(2007) and (in press), the annual total cost for the entire supply chain in Eq. (8) can be represented by factorizing the term 1/T and completing the perfect square, one has
TC ¼
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 2 T Y 2T YW þ W þ 2T YW T
ð11Þ
Factorizing the perfect squared trinomial in a squared binomial we obtain
TC ¼
pffiffiffiffiffiffiffiffi 1 pffiffiffiffi pffiffiffiffiffiffi2 T Y W þ 2 YW T
ð12Þ
pffiffiffiffi pffiffiffiffiffiffi It is worthy pointing out that Eq. (12) reaches minimum with respect to T when setting ðT Y W Þ2 ¼ 0 * Hence, the optimal basic cycle time T is
rffiffiffiffiffiffi W T ¼ Y
ð13Þ
M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
413
Substituting Eq. (13) into Eq. (8), the minimum value for the annual total cost for the entire supply chain is
pffiffiffiffiffiffiffiffi TC ¼ 2 YW
ð14Þ *
The optimal basic cycle time T is a function of the integer multipliers (Kn1, Kn2, Kn3,. . .K1). We use the method of perfect square to drive the optimal values of these integer multipliers iteratively, or in a recursive fashion. Substituting from Eq. (9) and Eq. (10) for Y and W, respectively into Eq. (14) we get
rffiffiffi 12 2 u ðK n1 wn1 þ an1 Þ Sn þ n1 A K n1 rffiffiffi 1 h p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffii2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 2 2 1 ¼ K n1 wn1 Sn an1 un1 þ wn1 un1 þ an1 Sn A K n1
TC ¼
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 From (15) setting K n1 wn1 Sn an1 un1 ¼ 0; the optimal value of integer multiplier Kn1 is derived as follows
K n1 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
an1 un1
ð16Þ
wn1 Sn
Since the value of Kn1 is a positive integer, the following condition must be satisfied:
ðK n1 Þ:ðK n1 1Þ 6 ðK n1 Þ2 6 ðK n1 Þ:ðK n1 þ 1Þ To drive the optimal value of integer multiplier Kn2, we can rewrite the term
ð17Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn1 un1 in Eq. (15) as follows:
12 u ðK n1 wn1 þ an1 Þ Sn þ n1 K n1 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 2 1 ¼ K n2 wn2 Sn1 an2 un2 þ wn2 un2 þ an2 Sn1 K n2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn1 un1 ¼
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From Eq. (18), setting ½K n2 wn2 Sn1 an2 un2 2 ¼ 0,the optimal value of the integer multiplier Kn2 is derived as follows:
K n2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
an2 un2
ð19Þ
wn2 Sn1
Since the value of Kn2 is a positive integer, the following condition must be satisfied.
ðK n2 Þ:ðK n2 1Þ 6 ðK n2 Þ2 6 ðK n2 Þ:ðK n2 þ 1Þ
ð20Þ
We continue carrying out this iterative process until the optimal value of the integer multiplier K1 is reached
K 1 ¼
rffiffiffiffiffiffiffiffiffiffiffi
a1 u1
ð21Þ
w1 S2
Substituting w1 ¼ a0 ¼ ðh0 þ h1 Þ (21) we obtain
P
2 j D1;j B1;J
P P P P þ A j D1;j h1 ; u1 ¼ S1 and a1 ¼ ðh1 þ h2 Þ j D22;j B2;j þ A j D2;j h2 A j D1;j h1 into Eq.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P P u uððh1 þ h2 Þ j D22;j B2;j þ A j D2;j h2 A j D1;j h1 ÞS1 K 1 ¼ t P 2 P ððh0 þ h1 Þ j D1;j B1;J þ A j D1;j h1 ÞS2
ð22Þ
Since the value of K1 is a positive integer, the following condition must be satisfied:
ðK 1 Þ:ðK 1 1Þ 6 ðK 1 Þ2 6 ðK 1 Þ:ðK 1 þ 1Þ
ð23Þ
fter deriving the optimal value of K1 from Eq. (22) and Eq. (23), the optimal value of K2 can be derived as follows:
K 2 ¼
rffiffiffiffiffiffiffiffiffiffiffi
a2 u2
ð24Þ
w2 S3
Substituting a2 ¼ ðh2 þ h3 Þ
P
2 j D3;j B3;J
P P þ A j D3;j h3 A j D2;j h2 , and u2 ¼ S2 þ KS11 ; we obtain
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u u ðh2 þ h3 ÞP D2 B3;J þ AP D3;j h3 AP D2;j h2 ðS2 þ S1 =K 1 Þ 3;j j j j t K 2 ¼ w2 S3
ð25Þ
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where
$ w2 ¼ K 1 ðh0 þ h1 Þ
X
D21;j B1;J þ A
j
X
%
$
D1;j h1 þ ðh1 þ h2 Þ
X
j
D22;j Bi;J þ A
j
X
D2;j h2 A
j
X
% D1;j h1
j
Since the value of K2 is a positive integer, the following condition must be satisfied:
ðK 2 Þ:ðK 2 1Þ 6 ðK 2 Þ2 6 ðK 2 Þ:ðK 2 þ 1Þ:
ð26Þ
We follow this recursive procedure each time driving the optimal Ki + 1 from the previously derived optimal Ki, until the optimal Kn1 is derived. Then optimal basic cycle time T , which is a function of the integer multipliers (Kn1, Kn2, Kn3,. . .K1), can be derived from Eq. (13).
4. Solution procedure P P Step 1: Set: w0 ¼ 0; a0 ¼ ðh0 þ h1 Þ j D21;j B1;J þ A j D1;j h1 ; K 0 ¼ 0; u1 ¼ S1 . Step 2: for i = 1,2,3,. . .,n1 and for all j compute:
ai ¼ ðhi þ hiþ1 Þ
X
D2iþ1;j Biþ1;j þ A
j
X
Diþ1;j hiþ1 A
j
X
Di;j hi
j
wi ¼ K i1 wi1 þ ai1 u ui ¼ Si þ i1 K i1 rffiffiffiffiffiffiffiffiffiffiffiffiffi a u i i K i ¼ ; s:t:ðK i Þ:ðK i 1Þ 6 ðK i Þ2 6 ðK i ÞðK i þ 1Þ: wi Siþ1 Step 3: Use Eq. (9) to compute pffiffiffiffiffiffiffiffiffiffiffi Y and use Eq.(10) to compute W. Step 4: Compute T ¼ W=Y . 5. A scheme for sharing the coordination benefits In this section, we consider the case where each stage optimizes its own decisions in a decentralized fashion governed by the decisions taken at the adjacent downstream stage. We assume that the whole supply chain is driven by the demand at the downstream retailers, and the partners at the same stage can agree on a common synchronized replenishment cycle time. Under these assumptions, firms at each stage will determine their optimal replenishment cycle as an integer multiple of the replenishment cycle time favored and predetermined by firms at the adjacent downstream stage. In this case, the total cost per unit time for all firms at stage n is given by
T TC n ðTÞ ¼ hn
Jn P
Dn;j
j¼1
þ
2
Sn T
ð27Þ
Eq. (27) can be rewritten as:
TC n ðTÞ ¼
1 2 ðT hn D þ 2Sn Þ 2T
ð28Þ
Using the algebraic method described in Section 3, the optimal common cycle time for all firms at this stage, Tn* can be obtained as:
Tn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sn =hn D
ð29Þ
And the optimal total cost for this stage is
TC n ðT n Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hn Sn D
ð30Þ
Since it is assumed that the replenishment decisions for firms at any stage i are governed by the decisions taken at the adjacent downstream stage i + 1, these firms must have complete knowledge of the information at that stage, and accordingly they will set their replenishment cycle at
Ti ¼ Ki
n Y
K ni T n
s¼iþ1
And they only need to determine the optimal value of the integer Ki.
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Hence, the total annual cost for firms at any stage, except for the final stage, is represented by
TC i ðK i Þ ¼ K i
Ji X j¼1
0 Q 1 n n Q K ns T n D2i;j hi1 K ns T n Di;j Bs¼iþ1 C s¼iþ1 B þ ð1 þ Di;j =P i
Si Ki
n Q s¼iþ1
n
Ki T
n
Ji n X Y
K ni T n Di;j hi
ð31Þ
j¼1 s¼iþ1
Again, we can use the method of perfect squares to obtain the optimal value of Ki as
K ni
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u 0Q n n Q uJ K ns T n D2i;j hi1 K ns T n Di;j uP i C s¼iþ1 u B þ s¼iþ1 2 ð1 þ Di;j =Pi;j Þhi A u @ 2Pi;j uj¼1 u ¼u Si u n t Q n n
ð32Þ
Ki T
s¼iþ1
And since the value of ðK ni Þ:ðK ni
K ni
1Þ 6
is a positive integer, the following condition must be satisfied:
ðK ni Þ2
6 ðK ni Þ:ðK ni þ 1Þ:
ð33Þ
In order to entice all stages to adopt the centralized coordination, we first need to ascertain the individual consequences of adopting this joined coordination instead of following the decentralized fashion governed from downstream stages, as sug gested by Chen and Chen (2005). For this purpose, we define ECTCi fðT n ; K ni Þ ! ðT ; K i Þg as the economic cost consequence of accepting the centralized replenishment and production decision policy by stage i instead of the decentralized policy. This basically represents the difference between the cost of adopting the joined basic cycle time and its integer multipliers with full coordination (centralized policy), and the cost of using the multiple of the cycle time favored by the adjacent downstream stage (decentralized policy). In order to induce all parties to adopt a common coordination mechanism, a saving–sharing mechanism should be provided to offset the additional costs that may be incurred by any party due to adopting the joined policy, and then divide the sharing benefits among all supply members. In the following, we propose a simple savings–sharing mechanism: Step 1: Compute: T n ; TC n ðT n Þ. Step 2: For i = n1–1. Compute: K ni ; TC i ðT n ; K ni Þ.
TCðT n Þ ¼
n X
TC i ðT n ; K ni Þ
i¼1
. Step 3: Compute: For i = 1n.
ECTCi fðT n ; K ni Þ ! ðT ; K i Þg ¼ TC i ðT n ; K ni Þ TC i ðT ; K i Þ n X ECTCi SAVING ¼ i¼1
Step 4: For i = 1n.
SHAREi ¼ SAVING
TC i ðT n ; K ni Þ TCðT n Þ
Implementing this procedure will provide a scheme that will allow all supply chain partners to share the coordination benefits.
6. Numerical example In this section, we consider an example of a four-stage supply chain consisting of one supplier, two manufacturers, four distributors, and six retailers. The relevant data is shown in Table 1. It is also assumed that holding cost for the supplier’s supplier is ho = 0.1. In addition, it is also assumed that manufacturer 1 supplies distributors 1 and 2; manufacturer 2 supplies distributors 3 and 4; distributor 1 supplies retailers 1 and 2; distributor 2 supplies retailer 3; distributor 3 supplies retailer 4; distributor 4 supplies retailers 5 and 6. By applying the solution procedure developed in Section 4, the results of this example for the case of centralized policy are presented in Table 2. We use the same data of this example to analyze the effects of following the decentralized policy and illustrate how the benefits of the coordination can be shared to entice all partners to accept the centralized policy.
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Table 1 Data for the numerical example. Stage i
Firm index j
Parent (i,j)
Holding cost incoming hi
Production rate Pij
Annual demand Dij
Setup cost Si
Suppliers Manufactures
1 1 2 1 2 3 4 1 2 3 4 5 6
– 1 1 1 1 2 2 1 1 2 3 4 4
0.8 2 2 4 4 4 4 7 7 7 7 7 7
396,000 180,000 200,000 90,000 70,000 80,000 100,000 – – – – – –
130,000 60,000 70,000 35,000 25,000 30,000 40,000 20,000 15,000 25,000 30,000 15,000 25,000
1000 200 200 50 50 50 50 10 10 10 10 10 10
Distributors
Retailers
Table 2 Results for the centralized model. Stage
Integer multiplier
Cycle time
Cost
Suppliers Manufactures Distributors Retailers Entire supply chain
2 2 2 – –
0.1071 0.0535 0.0268 0.0134 –
14132.833 14286.579 15111.163 10573.082 54103.657
Table 3 Results for the decentralized model. Stage
Integer multiplier
Cycle time
Cost
Suppliers Manufactures Distributors Retailers Entire supply chain
3 2 2 – –
0.1378 0.0459 0.0230 0.0115 –
14437.362 14555.129 15111.149 10449.880 54553.521
Table 4 Results after sharing the benefits of the centralized mechanism.
Stage
ECTCi fðT n ; K ni Þ ! ðT ; K i Þg
Saving share
The cost after sharing the benefits
Percentage of cost reduction
Suppliers Manufactures Distributors Retailers Entire supply chain
304.529 268.550 0.014 123.202 449.864
119.055 120.026 124.611 86.173 449.864
14318.307 14435.104 14986.538 10363.708 54103.657
0.825 0.825 0.825 0.825 0.825
Table 3 presents the results for the decentralized model. The cost differences between the decentralized and centralized policies for each stage are summarized in Table 4. As shown in Table 4, the centralized replenishment policy increased the costs to the retailers and distributors, while the cost to the suppliers and manufacturers was reduced. Hence, retailers and distributors need to be motivated to adopt the cooperative centralized policy which is found to be superior to the decentralized replenishment policy if the entire supply chain cost is considered. According to the proposed simple saving-sharing scheme, the increased costs at the retailers and manufactures must be covered first. Then the total cost savings of $449.864 is shared to assure equal percentage saving for all stages as shown in Table 4. 7. Conclusion In this paper we proposed an n-stage, multi-customer, non-serial supply chain model. We formulated the model for the integer multipliers inventory coordination mechanism. We extended and generalized pervious works that use the
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simple algebraic method to optimize the coordinated supply chain model without the use of differential calculus. Then, we developed a solution procedure and proposed a simple saving-sharing scheme to entice all the stages to accept the centralized replenishment policy. We illustrated our proposed method by solving a numerical example of a four-stage; non-serial supply chain model. Our generalized algebraic recursive derivation method can be applied to several extensions of the general supply chain inventory coordination model such as planned stock-outs and quantity discount models. Acknowledgement The authors would like to acknowledge the support for this research provided by King Fahd University of Petroleum and Minerals and Universiti Teknologi Malaysia. Appendix A. Explanation of how we derived the structures in Section 3 n Q
Substituting IC i;j ¼
TC ¼ T
K s TDi;j
s¼iþ1
ðK i ð1 þ Di;j =Pi;j Þ 1Þhi in Eq. (4) we get
2
Qn 9 Qn 8 Pn1 P K s TDi;j K s TD2i;j hi1 > s¼iþ1 s¼i > > þ ðK ð1 þ D =P Þ 1Þh A> i i;j i;j i =
> > :
T 2A þ
> > ;
2A (
Xn1 X
Yn
j
i¼1
K s TD2i;j hi1 =2Pi;j s¼i
þ
n Y
þ
n 1X Si Q T i¼1 ns¼i K s
! )
K s TDi;j =2ðK i ð1 þ Di;j =P i;j Þ 1Þhi A
s¼iþ1
þ
1 Xn Si n i¼1 Q T Ks s¼i
(
X X X ADhn þ K n1 ðhn2 þ hn1 Þ j D2n1:j Bn1:j þ K n1 Ahn1 j Dn1:j Ahn1 j Dn1:j þ þK n1 K n2 ðhn3 þ hn2 Þ
X j n Y
n Y s¼1
X
D2n2:j Bn2:j þ K n1 K n2 Ahn2
K s Ah2
X j
s¼2
þ
þ
2P i;j
¼ T=2A ADhn þ
¼
n Q
K s TD2i;j hi1
s¼i
K s Ah1
X j
D2:j
D1:j
n Y
K s Ah2
X j
s¼3 n Y
K s Ah1
j
X
s¼2
j
Dn2:j K n1 Ahn2
D2:j þ )
D1:j
n Y s¼1
þ
X j
Dn2:j þ þ þ
n Y s¼2
X K s ðh1 þ h2 Þ j D22:j B2:j
X K s ðh0 þ h1 Þ j D21:j B1:j
1 Xn Si n i¼1 Q T Ks
ðA:1Þ
s¼i
Rearranging the terms of Eq. (A.1) we get
9 8 P ADhn Ahn1 Dn1:j þ > > > > > > > > j > > > P 2 P P > 3> 2 > > > > > ðh þ h Þ D B þ Ah D Ah D þ > > n2 n1 n1 n2 n1:j n2:j n1:j n1:j > > > > j j j 7 6 > > > > > 7 6 > 3 2 P P P > > 2 < 7 6 ðhn3 þ hn2 Þ Dn2:j Bn2:j þ Ahn2 Dn2:j Ahn3 Dn3:j þ 7 = 1 Xn T Si 6 þ 7 6 j j j 7 6 n i¼1 Q 7> 6 2A > T 7 6 ::::::: > > K n1 6 > Ks 77> 6 2 3 P P P > > > 6 77> 6 > ðh1 h2 Þ D22:j B2:j þ Ah2 D2:j Ah1 D1:j þ s¼i > > 7> 6 K n2 6 7 > > > > 6 77> 6 6 j j j 7 > > 6 77> 6 K26 7::::: > P 2 P > > 4 55> 4 4 5 > > > > ðh þ h Þ D B þ Ah D 0 1 1 1j > > 1j 1:j K1 ; : j j
ðA:2Þ
Now substituting the recursive structures of Eq. (5), Eq. (6), and Eq. (7) into Eq. (A.2) can explain the developed method in Section 3. References Banerjee, A., 1986. A joint economic-lot-size model for purchaser and vendor. Decision Sciences 17 (3), 292–311. Ben-daya, M., Al-Nassar, A., 2008. Integrated multi-stage multi-customer supply chain. Production Planning and Control the Management of Operations 19 (2), 97–104. Cárdenas-Barrón, L.E., 2001. The economic production quantity (EPQ) with shortage derived algebraically. International Journal of Production Economics 70 (3), 289–292. Cárdenas-Barrón, L.E., 2007. Optimizing inventory decisions in a multi-stage multi-customer supply chain: a note. Transportation Research Part E: Logistics and Transportation Review 43 (5), 647–654.
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Cárdenas-Barrón, L.E., in press. Optimal manufacturing batch size with rework in a single-stage production system – a simple derivation. Computers and Industrial Engineering. doi:10.1016/j.cie.2007.07.017. Chen, J.-M., Chen, T.-H., 2005. The multi-item replenishment problem in a two-echelon supply chain: the effect of centralization versus decentralization. Computers and Operations Research 32 (12), 3191–3207. Chi, S.W., 2008. Production lot size problem with failure in repair and backlogging derived without derivatives. European Journal of Operational Research 188 (2), 610–615. Chung, C.J., Wee, H.M., 2007. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. European Journal of Operational Research 183 (2), 933–943. Goyal, S.K., 1988. A joint economic-lot-size model for a purchaser and: a comment. Decision Sciences 19 (1), 236–241. Goyal, S.K., 1995. A one – multi-buyer integrated inventory model: a comment. European Journal of Operational Research 82 (1), 209–210. Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory models: the buyer–vendor coordination. European Journal of Operational Research 41 (3), 261–269. Goyal, S.K., Szendrovits, A.Z., 1986. A constant lot size model with equal and unequal sized batch shipments between production stages. Engineering Costs and Production Economics 10 (1), 203–210. Hoque, M.A., Goyal, S.K., 2000. An optimal policy for a single–single-buyer integrated production-inventory system with capacity constraint of the transport equipment. International Journal of Production Economics 65 (3), 305–315. Khouja, M., 2003. Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transportation Research Part E: Logistics and Transportation Review 39 (3), 193–208. Kit-Nam Francis Leung, 2008. Using the complete squares method to analyze a lot size model when the quantity backordered and the quantity received are both uncertain. European Journal of Operational Research 187 (1), pp. 19–30. Lu, L., 1995. A one – multi-buyer integrated inventory model. European Journal of Operational Research 81 (2), 312–323. Simchi-Levi, D., Kaminsky, P., Simch-Levi, E., 2003. Designing and Managing the Supply Chain: Concepts, Strategies, and Case Studies, second ed. McGrawHill. Wee, H.M., Chung, C.J., 2007. A note on the economic lot size of the integrated-buyer inventory system derived without derivatives. European Journal of Operational Research 177 (2), 1289–1293.
Contents lists available at ScienceDirect
Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain M.E. Seliaman a,*, Ab Rahman Ahmad b a b
The Office of Planning and Quality, King Fahd University of Petroleum and Minerals, 1317, Dhahran 31261, Saudi Arabia Department of Modeling and Industrial Computing, Faculty of Computer Science and Information System, Universiti Teknologi Malaysia, Malaysia
a r t i c l e
i n f o
Article history: Received 12 April 2008 Received in revised form 4 August 2008 Accepted 12 September 2008
Keywords: Supply chain integration Production-inventory models Without derivatives
a b s t r a c t In this paper, we deal with more generalized inventory coordination mechanism in an nstage, multi-customer, non-serial supply chain, where we extend and generalize pervious works that use algebraic methods to optimize this coordinated supply chain. We establish the recursive expressions for this multi-variable optimization problem. These expressions are used for the derivation of the optimal replenishment policy and the development of the solution algorithm. Further, we describe a simple procedure that can help in sharing the coordination cost benefits to induce all stages to adopt the inventory coordination mechanism. We provide a numerical example for illustrative purposes. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Supply chain management can be defined as a set of approaches utilized to efficiently integrate suppliers, manufactures, warehouses, and stores, so that merchandise is produced and distributed at the right quantities, to the right locations, and at the right time, in order to minimize system-wide cost while satisfying service level requirements (Simchi-Levi et al., 2003). In the past, supply chain production-inventory decisions were not coordinated among the different parties in the supply chain. This lack of coordination leads to weakly connected activities and decisions across the supply chain. Recently firms realized that the global system performance and cost efficiency can be improved through closer collaboration among the chain partners and through high level of coordination of various decision processes. Supply chain inventory-distribution coordination can be achieved by coordinating the cycle time across the chain stages. The simplest way of cycle coordination is the equal cycle under which the same cycle is followed throughout the supply chain. But many supply models achieve coordination by following the integer multipliers mechanism in which the cycle time at each stage is an integer multiple of the cycle time of the adjacent downstream stage, or by integer powers of two multipliers at each stage mechanism (Khouja, 2003). In recent years numerous articles in supply chain modeling have addressed the issue of inventory coordination. Banerjee (1986) introduced the concept of joint economic lot sizing problem (JELS). He considered the case of a single vendor and a single purchaser under the assumption of deterministic demand and lot for lot policy. Goyal and Szendrovits (1986) presented a constant lot size model where the lot is produced through a fixed sequence of manufacturing stages, with a single setup and without interruption at each stage. This model mainly, relaxes the constraint that batches must be of equal size at any particular stage. Goyal (1988) provided a more general model for the case of single vendor single buyer through relaxing the lot-for-lot policy. He showed that his model provides a lower or equal total joint relevant cost compared to Banerjee * Corresponding author. Tel.: +966 509198211; fax: +996 3 8601881. E-mail addresses: [email protected] (M.E. Seliaman), [email protected] (A.R. Ahmad). 1366-5545/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2008.09.012
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M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
(1986). Goyal and Gupta (1989) extensively reviewed the literature which deals with the interaction between a buyer and vendor. They classified the literature dealing with the integrated models into four main classes. The first class represents models which deal with joint economic lot sizing policies. The second class characterizes models which deal with the coordination of inventory by simultaneously determining the order quantity for the buyer and the vendor. The third class is a group of models which deal with integrated problem but do not determine simultaneously the order quantity of the buyer and the vendor. The last class represents models which deal with buyer vendor coordination subject to marketing considerations. Lu (1995) developed a one-vendor multi-buyer integrated inventory model with the objective of minimizing the vendor’s total annual cost subject to the maximum cost that the buyer may be prepared to incur. Goyal (1995) revisited the single-single-buyer where he relaxed the constraint of equal sized shipments of Goyal (1988) and suggested that the shipment size should grow geometrically. Hoque and Goyal (2000) extended the idea of producing a single product in a multistage serial production system with equal and unequal sized batch shipments between stages. Khouja (2003) considered the case a three-stage non-serial supply chain and developed the model to deal with three inventory coordination mechanisms between the chain members. Bendaya and Al-Nassar (2008) relaxed the assumption of Khouja (2003). Regarding the completion of the whole production lot before making shipments out of it and assumed that equal sized shipments take place as soon as they are produced and there is no need to wait until a whole lot is produced. The use of differential calculus to model the integrated production inventory systems is common in the area of operational research. However, several researchers focused on the easy solution methods for the optimization of these types of systems. Cárdenas-Barrón (2001) used algebraic procedure to the EPQ formula taking shortages into consideration within the case of only one backlog cost per unit and time unit. Cárdenas-Barrón (2007) formulated and solved an n-stage-multi-customer supply chain inventory model where there is a company that can supply products to several customers. The production and demand rates were assumed constant and known. This model was formulated for the simplest inventory coordination mechanism which is referred to as the same cycle time for all companies in the supply chain. It was concluded that it is possible to use an algebraic approach to optimize the supply chain model without the use of differential calculus. Chung and Wee (2007) considered an integrated threestage inventory system with backorders. They formulated the problem to derive the replenishment policies with fourdecision-variables algebraically. Wee and Chung (2007) also used a simple algebraic method to solve the economic lot size of the integrated – buyer inventory problem. As a result, students who are unfamiliar with calculus may be able to understand the solution procedure with ease. Chi (2008) presented a simple algebraic method to demonstrate that the lot size solution and the optimal production-inventory cost of an imperfect EMQ model can be derived without derivatives. Kit-Nam Francis Leung (2008) considered the EOQ problem, where the quantity backordered and the quantity received are both uncertain. He used the complete squares method to derive a global optimal expression from a non-convex objective function in an algebraic manner. Cárdenas-Barrón (in press) considered the problem of optimal manufacturing batch size with rework process at single-stag production system. He determined the optimal solution for two different inventory policies. He also established the range of real values of proportion of defectives products for which there is an optimal solution, the closed-form for the total inventory cost for both policies, the mathematical expressions for determining the cost penalty and the additional total cost for working with a non-optimal solution. In this study, we develop an optimal replenishment policy using a simple algebraic method to solve the n-stage, multicustomer, non-serial supply chain inventory problem. Our work is an extension of the three stage supply chain model in Khouja (2003). We consider the integer multiplier coordination mechanism. The remainder of this paper is organized as follows. The next section presents the notation and assumptions made in the model. Section 3 describes the development of the model. A solution procedure is presented in Section 4. Section 5 describes a scheme for sharing the coordination benefits. Section 6 presents a numerical example. Finally, Section 7 contains some concluding remarks.
2. Notation and assumptions The following notations are used in developing the model: T = Basic cycle time, cycle time at the end retailers. Ti = Cycle time at stage i. Sij = Setup cost for firm j at stage i. Si = Total setup cost for all firms at stage i. Ki = Integer multiplier at stage i. hi = Inventory holding cost at stage i. ni = Number of firms at stage i. Dij = The mean demand rate of firm j at stage i. Pij = Production rate of firm j at stage i. A = The product of all production rates for all the companies in the supply chain. Bij = The product of all production rates for all the companies in the supply chain, except for the company j in stage i.
M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
411
Assumptions for the multi-stage supply chain production-inventory model: (a) (b) (c) (d) (e) (f)
A single product is produced and distributed through a multi-stage, multi-customer, non-serial, supply chain. Production rates and demand rates are deterministic and uniform. Ordering/setup costs are the same for firms at the same stage. Holding costs cost are the same for firms at the same stage. A lot produced at stage is sent in equal shipments to the downstream stage. The supply chain is vertically integrated and the entire supply chain optimization is acceptable for all partners in the chain. (g) Cycle time at each stage is an integer multiplier of the cycle time used at the adjacent downstream stage.
3. Development of the n-stage-multi-customer supply chain algebraically We consider a multi-stage, non-serial supply chain, where a firm at each stage can have two or more customers. A typical example of such supply chain structure is depicted in Fig. 1. This type of supply chain structure differs from the serial supply chain structure where each stage has only one firm. This supply chain model is formulated for the integer multipliers coordination mechanism, where firms at the same stage of the supply chain use the same cycle time and the cycle time at each stage is an integer multiplier of the cycle time used at the adjacent downstream stage. In this case, the cycle time of an end retailer is T and therefore the total cost per unit time for retailer j is given by:
TC n;j ¼ hn
TDn;j Sn;j þ 2 T
ð1Þ
The holding cost at each stage, except for the final stage (the end retailers’ stage n), is made of two parts: the first one is the carrying cost for the raw materials as they are being converted into finished products during the production portion of the cycle. The second part is the carrying cost of the finished products during the non-production portion of the cycle. During Q the production time, the average annual inventory of raw material as well as finished products is equal to ns¼i K s TD2i;j =2P i;j . During the non-production time, inventory drops by every Ti + 1 years by Ti + 1Dij starting from (Ti–Ti + 1) Dij as shown in Fig. 2. Therefore, the total annual cost for any firm at any stage, except for the final stage, is represented by: n Q
TC i;j ¼ s¼i
n Q
K s TD2i;j hi1 2Pi;j
þ
K s TDi;j
s¼iþ1
2
ðK i ð1 þ Di;j =P i;j Þ 1Þhi þ
Si;j n Q K sT
ð2Þ
s¼i
We assume that Kn = 1. The total cost for the entire supply chain is
TC ¼
X
TC i;j
ð3Þ
i;j
Supplier
Manufactures
Distributors
Retailers 1
1 2
1 1 2 2
2 1
1 3 2
3 1 4 2 Fig. 1. An example of a complex supply chain structure.
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M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
Fig. 2. Raw materials and finished products at two consecutive stages.
We can write TC as
TC ¼ T
9 8 Pn1 P IC i:j A> > =
> : n Q
> ;
2A n Q
K s TD2i;j hi1
þ
1 Xn Si n i¼1 Q T Ks
ð4Þ
s¼i K s TDi;j
þ s¼iþ1 2 ðK i ð1 þ Di;j =P i;j Þ 1Þhi P P We define a0 ¼ ðh0 þ h1 Þ j D21;j B1;J þ A j D1;j h1 ; w0 ¼ 0 and u1 ¼ S1 :
where IC i;j ¼
s¼i
2Pi;j
For i = 1,2,3,. . .,n1 and for all j, we define ai, wi and /i as follows:
ai ¼ ðhi þ hiþ1 Þ
X
D2iþ1;j Biþ1;J þ A
j
X
Diþ1;j hiþ1 A
j
X
Di;j hi
ð5Þ
j
wi ¼ K i1 wi1 þ ai1 u ui ¼ Si þ i1 K i1
ð6Þ ð7Þ
For convenience and handiness, we re-write Eq. (4) in the following manner(see the Appendix A):
TC ¼ TY þ
W T
ð8Þ
where
Y¼
K n1 wn1 þ an1 2A
and W ¼ Sn þ
un1 K n1
ð9Þ ð10Þ
Now applying the algebraic procedure proposed by Cárdenas-Barrón(2007) and (in press), the annual total cost for the entire supply chain in Eq. (8) can be represented by factorizing the term 1/T and completing the perfect square, one has
TC ¼
pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 1 2 T Y 2T YW þ W þ 2T YW T
ð11Þ
Factorizing the perfect squared trinomial in a squared binomial we obtain
TC ¼
pffiffiffiffiffiffiffiffi 1 pffiffiffiffi pffiffiffiffiffiffi2 T Y W þ 2 YW T
ð12Þ
pffiffiffiffi pffiffiffiffiffiffi It is worthy pointing out that Eq. (12) reaches minimum with respect to T when setting ðT Y W Þ2 ¼ 0 * Hence, the optimal basic cycle time T is
rffiffiffiffiffiffi W T ¼ Y
ð13Þ
M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
413
Substituting Eq. (13) into Eq. (8), the minimum value for the annual total cost for the entire supply chain is
pffiffiffiffiffiffiffiffi TC ¼ 2 YW
ð14Þ *
The optimal basic cycle time T is a function of the integer multipliers (Kn1, Kn2, Kn3,. . .K1). We use the method of perfect square to drive the optimal values of these integer multipliers iteratively, or in a recursive fashion. Substituting from Eq. (9) and Eq. (10) for Y and W, respectively into Eq. (14) we get
rffiffiffi 12 2 u ðK n1 wn1 þ an1 Þ Sn þ n1 A K n1 rffiffiffi 1 h p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffii2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 2 2 1 ¼ K n1 wn1 Sn an1 un1 þ wn1 un1 þ an1 Sn A K n1
TC ¼
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 From (15) setting K n1 wn1 Sn an1 un1 ¼ 0; the optimal value of integer multiplier Kn1 is derived as follows
K n1 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
an1 un1
ð16Þ
wn1 Sn
Since the value of Kn1 is a positive integer, the following condition must be satisfied:
ðK n1 Þ:ðK n1 1Þ 6 ðK n1 Þ2 6 ðK n1 Þ:ðK n1 þ 1Þ To drive the optimal value of integer multiplier Kn2, we can rewrite the term
ð17Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn1 un1 in Eq. (15) as follows:
12 u ðK n1 wn1 þ an1 Þ Sn þ n1 K n1 1 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 2 1 ¼ K n2 wn2 Sn1 an2 un2 þ wn2 un2 þ an2 Sn1 K n2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn1 un1 ¼
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi From Eq. (18), setting ½K n2 wn2 Sn1 an2 un2 2 ¼ 0,the optimal value of the integer multiplier Kn2 is derived as follows:
K n2 ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
an2 un2
ð19Þ
wn2 Sn1
Since the value of Kn2 is a positive integer, the following condition must be satisfied.
ðK n2 Þ:ðK n2 1Þ 6 ðK n2 Þ2 6 ðK n2 Þ:ðK n2 þ 1Þ
ð20Þ
We continue carrying out this iterative process until the optimal value of the integer multiplier K1 is reached
K 1 ¼
rffiffiffiffiffiffiffiffiffiffiffi
a1 u1
ð21Þ
w1 S2
Substituting w1 ¼ a0 ¼ ðh0 þ h1 Þ (21) we obtain
P
2 j D1;j B1;J
P P P P þ A j D1;j h1 ; u1 ¼ S1 and a1 ¼ ðh1 þ h2 Þ j D22;j B2;j þ A j D2;j h2 A j D1;j h1 into Eq.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P P u uððh1 þ h2 Þ j D22;j B2;j þ A j D2;j h2 A j D1;j h1 ÞS1 K 1 ¼ t P 2 P ððh0 þ h1 Þ j D1;j B1;J þ A j D1;j h1 ÞS2
ð22Þ
Since the value of K1 is a positive integer, the following condition must be satisfied:
ðK 1 Þ:ðK 1 1Þ 6 ðK 1 Þ2 6 ðK 1 Þ:ðK 1 þ 1Þ
ð23Þ
fter deriving the optimal value of K1 from Eq. (22) and Eq. (23), the optimal value of K2 can be derived as follows:
K 2 ¼
rffiffiffiffiffiffiffiffiffiffiffi
a2 u2
ð24Þ
w2 S3
Substituting a2 ¼ ðh2 þ h3 Þ
P
2 j D3;j B3;J
P P þ A j D3;j h3 A j D2;j h2 , and u2 ¼ S2 þ KS11 ; we obtain
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u u ðh2 þ h3 ÞP D2 B3;J þ AP D3;j h3 AP D2;j h2 ðS2 þ S1 =K 1 Þ 3;j j j j t K 2 ¼ w2 S3
ð25Þ
414
M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
where
$ w2 ¼ K 1 ðh0 þ h1 Þ
X
D21;j B1;J þ A
j
X
%
$
D1;j h1 þ ðh1 þ h2 Þ
X
j
D22;j Bi;J þ A
j
X
D2;j h2 A
j
X
% D1;j h1
j
Since the value of K2 is a positive integer, the following condition must be satisfied:
ðK 2 Þ:ðK 2 1Þ 6 ðK 2 Þ2 6 ðK 2 Þ:ðK 2 þ 1Þ:
ð26Þ
We follow this recursive procedure each time driving the optimal Ki + 1 from the previously derived optimal Ki, until the optimal Kn1 is derived. Then optimal basic cycle time T , which is a function of the integer multipliers (Kn1, Kn2, Kn3,. . .K1), can be derived from Eq. (13).
4. Solution procedure P P Step 1: Set: w0 ¼ 0; a0 ¼ ðh0 þ h1 Þ j D21;j B1;J þ A j D1;j h1 ; K 0 ¼ 0; u1 ¼ S1 . Step 2: for i = 1,2,3,. . .,n1 and for all j compute:
ai ¼ ðhi þ hiþ1 Þ
X
D2iþ1;j Biþ1;j þ A
j
X
Diþ1;j hiþ1 A
j
X
Di;j hi
j
wi ¼ K i1 wi1 þ ai1 u ui ¼ Si þ i1 K i1 rffiffiffiffiffiffiffiffiffiffiffiffiffi a u i i K i ¼ ; s:t:ðK i Þ:ðK i 1Þ 6 ðK i Þ2 6 ðK i ÞðK i þ 1Þ: wi Siþ1 Step 3: Use Eq. (9) to compute pffiffiffiffiffiffiffiffiffiffiffi Y and use Eq.(10) to compute W. Step 4: Compute T ¼ W=Y . 5. A scheme for sharing the coordination benefits In this section, we consider the case where each stage optimizes its own decisions in a decentralized fashion governed by the decisions taken at the adjacent downstream stage. We assume that the whole supply chain is driven by the demand at the downstream retailers, and the partners at the same stage can agree on a common synchronized replenishment cycle time. Under these assumptions, firms at each stage will determine their optimal replenishment cycle as an integer multiple of the replenishment cycle time favored and predetermined by firms at the adjacent downstream stage. In this case, the total cost per unit time for all firms at stage n is given by
T TC n ðTÞ ¼ hn
Jn P
Dn;j
j¼1
þ
2
Sn T
ð27Þ
Eq. (27) can be rewritten as:
TC n ðTÞ ¼
1 2 ðT hn D þ 2Sn Þ 2T
ð28Þ
Using the algebraic method described in Section 3, the optimal common cycle time for all firms at this stage, Tn* can be obtained as:
Tn ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Sn =hn D
ð29Þ
And the optimal total cost for this stage is
TC n ðT n Þ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2hn Sn D
ð30Þ
Since it is assumed that the replenishment decisions for firms at any stage i are governed by the decisions taken at the adjacent downstream stage i + 1, these firms must have complete knowledge of the information at that stage, and accordingly they will set their replenishment cycle at
Ti ¼ Ki
n Y
K ni T n
s¼iþ1
And they only need to determine the optimal value of the integer Ki.
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Hence, the total annual cost for firms at any stage, except for the final stage, is represented by
TC i ðK i Þ ¼ K i
Ji X j¼1
0 Q 1 n n Q K ns T n D2i;j hi1 K ns T n Di;j Bs¼iþ1 C s¼iþ1 B þ ð1 þ Di;j =P i
Si Ki
n Q s¼iþ1
n
Ki T
n
Ji n X Y
K ni T n Di;j hi
ð31Þ
j¼1 s¼iþ1
Again, we can use the method of perfect squares to obtain the optimal value of Ki as
K ni
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u 0Q n n Q uJ K ns T n D2i;j hi1 K ns T n Di;j uP i C s¼iþ1 u B þ s¼iþ1 2 ð1 þ Di;j =Pi;j Þhi A u @ 2Pi;j uj¼1 u ¼u Si u n t Q n n
ð32Þ
Ki T
s¼iþ1
And since the value of ðK ni Þ:ðK ni
K ni
1Þ 6
is a positive integer, the following condition must be satisfied:
ðK ni Þ2
6 ðK ni Þ:ðK ni þ 1Þ:
ð33Þ
In order to entice all stages to adopt the centralized coordination, we first need to ascertain the individual consequences of adopting this joined coordination instead of following the decentralized fashion governed from downstream stages, as sug gested by Chen and Chen (2005). For this purpose, we define ECTCi fðT n ; K ni Þ ! ðT ; K i Þg as the economic cost consequence of accepting the centralized replenishment and production decision policy by stage i instead of the decentralized policy. This basically represents the difference between the cost of adopting the joined basic cycle time and its integer multipliers with full coordination (centralized policy), and the cost of using the multiple of the cycle time favored by the adjacent downstream stage (decentralized policy). In order to induce all parties to adopt a common coordination mechanism, a saving–sharing mechanism should be provided to offset the additional costs that may be incurred by any party due to adopting the joined policy, and then divide the sharing benefits among all supply members. In the following, we propose a simple savings–sharing mechanism: Step 1: Compute: T n ; TC n ðT n Þ. Step 2: For i = n1–1. Compute: K ni ; TC i ðT n ; K ni Þ.
TCðT n Þ ¼
n X
TC i ðT n ; K ni Þ
i¼1
. Step 3: Compute: For i = 1n.
ECTCi fðT n ; K ni Þ ! ðT ; K i Þg ¼ TC i ðT n ; K ni Þ TC i ðT ; K i Þ n X ECTCi SAVING ¼ i¼1
Step 4: For i = 1n.
SHAREi ¼ SAVING
TC i ðT n ; K ni Þ TCðT n Þ
Implementing this procedure will provide a scheme that will allow all supply chain partners to share the coordination benefits.
6. Numerical example In this section, we consider an example of a four-stage supply chain consisting of one supplier, two manufacturers, four distributors, and six retailers. The relevant data is shown in Table 1. It is also assumed that holding cost for the supplier’s supplier is ho = 0.1. In addition, it is also assumed that manufacturer 1 supplies distributors 1 and 2; manufacturer 2 supplies distributors 3 and 4; distributor 1 supplies retailers 1 and 2; distributor 2 supplies retailer 3; distributor 3 supplies retailer 4; distributor 4 supplies retailers 5 and 6. By applying the solution procedure developed in Section 4, the results of this example for the case of centralized policy are presented in Table 2. We use the same data of this example to analyze the effects of following the decentralized policy and illustrate how the benefits of the coordination can be shared to entice all partners to accept the centralized policy.
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M.E. Seliaman, A.R. Ahmad / Transportation Research Part E 45 (2009) 409–418
Table 1 Data for the numerical example. Stage i
Firm index j
Parent (i,j)
Holding cost incoming hi
Production rate Pij
Annual demand Dij
Setup cost Si
Suppliers Manufactures
1 1 2 1 2 3 4 1 2 3 4 5 6
– 1 1 1 1 2 2 1 1 2 3 4 4
0.8 2 2 4 4 4 4 7 7 7 7 7 7
396,000 180,000 200,000 90,000 70,000 80,000 100,000 – – – – – –
130,000 60,000 70,000 35,000 25,000 30,000 40,000 20,000 15,000 25,000 30,000 15,000 25,000
1000 200 200 50 50 50 50 10 10 10 10 10 10
Distributors
Retailers
Table 2 Results for the centralized model. Stage
Integer multiplier
Cycle time
Cost
Suppliers Manufactures Distributors Retailers Entire supply chain
2 2 2 – –
0.1071 0.0535 0.0268 0.0134 –
14132.833 14286.579 15111.163 10573.082 54103.657
Table 3 Results for the decentralized model. Stage
Integer multiplier
Cycle time
Cost
Suppliers Manufactures Distributors Retailers Entire supply chain
3 2 2 – –
0.1378 0.0459 0.0230 0.0115 –
14437.362 14555.129 15111.149 10449.880 54553.521
Table 4 Results after sharing the benefits of the centralized mechanism.
Stage
ECTCi fðT n ; K ni Þ ! ðT ; K i Þg
Saving share
The cost after sharing the benefits
Percentage of cost reduction
Suppliers Manufactures Distributors Retailers Entire supply chain
304.529 268.550 0.014 123.202 449.864
119.055 120.026 124.611 86.173 449.864
14318.307 14435.104 14986.538 10363.708 54103.657
0.825 0.825 0.825 0.825 0.825
Table 3 presents the results for the decentralized model. The cost differences between the decentralized and centralized policies for each stage are summarized in Table 4. As shown in Table 4, the centralized replenishment policy increased the costs to the retailers and distributors, while the cost to the suppliers and manufacturers was reduced. Hence, retailers and distributors need to be motivated to adopt the cooperative centralized policy which is found to be superior to the decentralized replenishment policy if the entire supply chain cost is considered. According to the proposed simple saving-sharing scheme, the increased costs at the retailers and manufactures must be covered first. Then the total cost savings of $449.864 is shared to assure equal percentage saving for all stages as shown in Table 4. 7. Conclusion In this paper we proposed an n-stage, multi-customer, non-serial supply chain model. We formulated the model for the integer multipliers inventory coordination mechanism. We extended and generalized pervious works that use the
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simple algebraic method to optimize the coordinated supply chain model without the use of differential calculus. Then, we developed a solution procedure and proposed a simple saving-sharing scheme to entice all the stages to accept the centralized replenishment policy. We illustrated our proposed method by solving a numerical example of a four-stage; non-serial supply chain model. Our generalized algebraic recursive derivation method can be applied to several extensions of the general supply chain inventory coordination model such as planned stock-outs and quantity discount models. Acknowledgement The authors would like to acknowledge the support for this research provided by King Fahd University of Petroleum and Minerals and Universiti Teknologi Malaysia. Appendix A. Explanation of how we derived the structures in Section 3 n Q
Substituting IC i;j ¼
TC ¼ T
K s TDi;j
s¼iþ1
ðK i ð1 þ Di;j =Pi;j Þ 1Þhi in Eq. (4) we get
2
Qn 9 Qn 8 Pn1 P K s TDi;j K s TD2i;j hi1 > s¼iþ1 s¼i > > þ ðK ð1 þ D =P Þ 1Þh A> i i;j i;j i =
> > :
T 2A þ
> > ;
2A (
Xn1 X
Yn
j
i¼1
K s TD2i;j hi1 =2Pi;j s¼i
þ
n Y
þ
n 1X Si Q T i¼1 ns¼i K s
! )
K s TDi;j =2ðK i ð1 þ Di;j =P i;j Þ 1Þhi A
s¼iþ1
þ
1 Xn Si n i¼1 Q T Ks s¼i
(
X X X ADhn þ K n1 ðhn2 þ hn1 Þ j D2n1:j Bn1:j þ K n1 Ahn1 j Dn1:j Ahn1 j Dn1:j þ þK n1 K n2 ðhn3 þ hn2 Þ
X j n Y
n Y s¼1
X
D2n2:j Bn2:j þ K n1 K n2 Ahn2
K s Ah2
X j
s¼2
þ
þ
2P i;j
¼ T=2A ADhn þ
¼
n Q
K s TD2i;j hi1
s¼i
K s Ah1
X j
D2:j
D1:j
n Y
K s Ah2
X j
s¼3 n Y
K s Ah1
j
X
s¼2
j
Dn2:j K n1 Ahn2
D2:j þ )
D1:j
n Y s¼1
þ
X j
Dn2:j þ þ þ
n Y s¼2
X K s ðh1 þ h2 Þ j D22:j B2:j
X K s ðh0 þ h1 Þ j D21:j B1:j
1 Xn Si n i¼1 Q T Ks
ðA:1Þ
s¼i
Rearranging the terms of Eq. (A.1) we get
9 8 P ADhn Ahn1 Dn1:j þ > > > > > > > > j > > > P 2 P P > 3> 2 > > > > > ðh þ h Þ D B þ Ah D Ah D þ > > n2 n1 n1 n2 n1:j n2:j n1:j n1:j > > > > j j j 7 6 > > > > > 7 6 > 3 2 P P P > > 2 < 7 6 ðhn3 þ hn2 Þ Dn2:j Bn2:j þ Ahn2 Dn2:j Ahn3 Dn3:j þ 7 = 1 Xn T Si 6 þ 7 6 j j j 7 6 n i¼1 Q 7> 6 2A > T 7 6 ::::::: > > K n1 6 > Ks 77> 6 2 3 P P P > > > 6 77> 6 > ðh1 h2 Þ D22:j B2:j þ Ah2 D2:j Ah1 D1:j þ s¼i > > 7> 6 K n2 6 7 > > > > 6 77> 6 6 j j j 7 > > 6 77> 6 K26 7::::: > P 2 P > > 4 55> 4 4 5 > > > > ðh þ h Þ D B þ Ah D 0 1 1 1j > > 1j 1:j K1 ; : j j
ðA:2Þ
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