In this chapter, a model is proposed that considers the optimum level of collaboration associated with a given number of suppliers, the expected inventory cost, and the supply chain management costs under a long-term partnership. Because the level of collaboration affects these costs, a contractor can reach the lowest cost position with a certain level of collaboration. A high level of collaboration enables the contractor to respond more efficiently to demand uncertainty, but involves more complexity due to an additional exercise of collaboration and control than would result with a low level of collaboration.
6.3.1.1 Inventory Cost
The deviation of the actual quantity ordered from the target causes economic losses (i.e., holding and shortage costs). Unit holding costs ofCh for each unsold unit in inventory and shortage costs of CS for each unit of unsatisfied demand were set. The salvage value at the end of each period is assumed to equal zero. Let I{D(t)>F(r,t)} = 1 if D(t) > F(r,t) andI{D(t)>F(r,t)} = 0 if D(t) < F(r,t); then the terms CS{D(t)−F(r,t)}I{D(t)>F(r,t)} impose penalties for a stock out (a loss of goodwill) and the termCh{F(r,t)−D(t)}I{D(t)<F(r,t)}represents overstock (in- ventory holding cost). An asymmetric loss function would be appropriate if the loss differs for the values of F(r,t) that are equidistant from the targetD(t). The
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104 Seong-Hyun Nam et al.
total inventory loss,l(t), using an asymmetric loss function could be expressed as follows:
l(t)=
CS{D(t)−F(r,t)} if D(t)> F(r,t) Ch{F(r,t)−D(t)} if D(t)<F(r,t)
The expected loss (penalty cost) under the underling process can be denoted by L S(t)= E[l(t)]. To derive L S(t), the probability of the forecasted demand being less than the actual demand can be derived as follows. For notational simplicity, let S(r,t)= P[F(r,t)< D(t)]; then
Proposition 6.2
S(r,t)= P[F(r,t)< D(t)]= 1
√2Q(r,t)
Z(r,t)
−∞ e −x
2 2Q(r,t)d x
(6.6) where
Q(r,t)= t
S
exp{2R(x)}d x, R(t)= t
s
r(x)d x
and
Z(r,t)=(1/)
D(t) exp{R(t)} −FS− t
s
r(x)D(x) exp{R(x)}d x
Proof
S(r,t) = P[F(r,t)< D(t)]
= P
exp{−R(t)}
FS+
t S
r(x)D(x) exp{R(x)}d x
+ t
S
exp{R(x)}d W(x)
≤D(t)
= P t
s
exp{R(x)}d W(x)
≤(1/)
D(t) exp{R(t)} −FS− t
s
ãr(x)ãD(x)ãexp{R(x)}d x
= 1
√2Q(r,t)
Z(r,t)
−∞ e −x
2 2Q(r,t)d x
using Theorem 9.2.3 in Arnold (1974) (Q.E.D.).
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A Determination of the Optimal Level of Collaboration 105
For a simple expression of the following equation, let Qr =∂Q(r,t)/∂r =2
t
s
(x−s) exp{2R(x)}d x and
Zr = (/)
D(t)(t−s) exp{R(t)}
− t
s
D(x) exp{R(x)}{1+r(x)(x−s)}d x
Proposition 6.3 Givenr(t)=r ∈[0, 1]
Sr(r,t) = ∂S(r,t)/∂r = −(QT/2Q)
Z(r,t)
−∞ (1/
2Q) exp{−y2/2Q}d y
+(2Q)−0.5
(Qr/2Q2) Z(r,t)
−∞ y2exp{−y2/2Q}d y +(Zr)
×exp{−Z2/2Q}
Proof Leibniz’s rule for differentiating an integral can facilitate the proof (Q.E.D). The expected loss function at timet can be derived as follows:
Proposition 6.4
L S(t) = E[l(t)]=CSãS(r,t)ã {D(t)−B(t)}
+Chã {1−S(r,t)} ã {A(t)−D(t)} (6.7) where:
A(t) = E[F(r,t)|F(r,t)> D(t)]= K(r,t)+V(r,t)[(u)/{1−(u)}]
B(t) = E[F(r,t)|F(r,t)< D(t)]= K(r,t)−V(r,t)[(u)/(u)]
ỉ(u) = (1/√
2) exp{−u2(t)/2},(u)={u(t)} = u(t)
−∞ ỉ(x)d x u(t) = {D(t)−K(r,t)}/
V(r,t)
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106 Seong-Hyun Nam et al.
Proof
L S(t) = E[l(t)]=CS ãE[(D−F)|D> F]ã P[D> F] +ChãE[(F −D)|D<F]ãP[D< F]
= CS ãE[D|D>F]ãS−CSãE[F|D> F]ãS
+ChãE[F|D< F]ã(1−S)−ChãE[D|D< F]ã(1−S) D(t) is assumed to be a bounded continuous real value function,E[D|D> F]=D (Chung 1974). Therefore,
L S(t) = CS ãS(r,t)ã[D(t)−E{F(r,t)|D(t)>F(r,t)}]
+Chã {1−S(r,t)} ã[E{F(r,t)|D(t)<F(r,t)} −D(t)]
Assuming that the initialF(t =s)=FSare both constant or normally distributed, F(r,t) is a Gaussian stochastic process with expectationK(r,t) and varianceV(r,t).
See Theorem 8.2.10 in Arnold (1974) and Proposition 6.1. Accordingly, the ex- pected value can be obtained using the truncated normal distribution (Ryan 2000) as follows:
Hence,
E{F(r,t)|F(r,t)> D(t)} = K(r,t)+V(r,t)
(u(t)) 1−(u(t))
= A(t)
and
E{F(r,t)|F(r,t)< D(t)} = K(r,t)−V(r,t)
(u(t)) (u(t))
= B(t)
Hence,
L S(t) = E[l(t)]=CSãS(r,t)ã {D(t)−B(t)}
+Chã {1−S(r,t)} ã {A(t)−D(t)}
6.3.1.2 The Contractor’s Acquisition Price and Suppliers’ Production Cost
Assume that the unit wholesale price charged by a supplier falls between an up- per bound WU and a lower boundWL. Because the process and production cost
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A Determination of the Optimal Level of Collaboration 107
reductions are the benefits of supply chain collaboration (Mentzer et al. 2000, McLaren et al. 2002), in this section, Homburg and Kuester’s (2001) purchas- ing price was used as the contractor’s acquisition price. The relationship between the number of suppliers and the level of collaboration can be portrayed as follows:
W(n,r,t)={r(t)} ã {WL+(WU−WL)/(bn−1)}, {r(t)}>0, b>1, andn1.
The unit production cost,C(t), of the supplier can be defined as a markup based on the given wholesale price. Then the retail price P(t) andC(t) can be calculated as follows:
P(n,r,t) = (t)W(n,r,t), 1≤(t)<∞,C(n,r,t)
= (t)W(n,r,t), 0<(t)1.
6.3.1.3 Collaborative SCM Cost
Collaboration costs are composed of all the costs associated with managing a rela- tionship with specific suppliers. These costs are associated with the exchange of information, decision making, system implementation and integration, process col- laboration and integration, and data transaction and integration costs (McLaren et al. 2002). Recent studies (Bakos and Brynjolfsson 1994, Homburg and Kuester 2001, Choi and Krause 2006) indicate that collaboration costs are increasing as the numbers of suppliers increase and collaboration levels become more complex. Choi and Krause’s (2006) Proposition 1.1 and 1.2 suggests that there is a positive linear relationship between the number of suppliers and collaboration costs. Hence this paper defines the collaboration costs (CL) at time t as a continuous function of the collaboration level {r(t)∈(0, 1]} and the number of suppliers {n∗(t)=n}.
It follows that by letting f C L(,) be a continuous function representing cost, then
C L(n,r,t)= fC L{r(t),n},C Lr =∂fC L/∂r >0 and
C Ln=∂fC L/∂n>0
6.3.1.4 Transaction Risk
Other transaction costs are related to transaction risks (Clemons et al. 1993, Choi and Krause 2006) and the number of suppliers. If a contractor increases the number of suppliers, then the probability of unreliable delivery is higher (Handfield and
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108 Seong-Hyun Nam et al.
Nichols 1999, Nishiguchi T. 1994, Choi and Krause 2006). Additional problems between different suppliers might be that downtime results and managerial resources must be increased to maintain the contractor’s goals (Homburg and Kuester 2001).
In contrast, if the contractor uses fewer suppliers, then there are several possible risks involved, such as the suppliers becoming potential competitors after acquiring core technology from the contractor. The cost of changing partners is huge when the contractor needs to change suppliers, and poor supplier performance causes even more serious harm. Both Proposition 2.1 of Choi and Krause’s study (2006) and operations costs cited by Homburg and Kuester (2001) suggest that there is a positive quadratic relationship between the number of suppliers and a given contractor.
Supply risk could be reduced through collaboration. Supply risk cost (SR) can be formulated as follows:
S R(n,r,t)= fS R(n,r),S Rnn =∂2fS R/∂n2 ≥0 for alln and
S Rr =∂fS R/∂r ≤0
6.3.1.5 Loss Related to Supply Responsiveness
Response time is an important issue in supply chain management. Some benefits of fast response time are reduced cycle time, higher quality, fewer internal problems, and better service. Supply responsiveness is viewed as the suppliers’ accurate and quick responses to the contractors’ requests for new requirements. Previous research indicates that single sourcing has more effective responsiveness than multiple sourc- ing because single sourcing has a better chance of promoting a close relationship and open communication between the contractor and the supplier (Liker and Choi 2004, Treleven and Schweikhart 1988, Handfield and Bechtel 2002, Larson and Kulchitsky 1998). Choi and Krause (2006) suggested that there should be a negative associa- tion between the number of suppliers and suppliers’ responsiveness. Collaboratively sharing information with supply chain members dramatically improved customer responsiveness (Horvath 2001). Assume that supplier responsiveness decreases as the number of suppliers increases, but decreases as ther(t) increases. Hence,supply re- sponsiveness cost(SR) related to the number of suppliers and the level of collaboration can be represented as follows:
RC(n,r,t)= fRC{r(t),n} with RCn=∂fRC/∂n>0 and
RCr =∂fRC/∂r <0
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