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Predictive Control of a Decentralized Supply Chain Unit Pin-Ho Lin Department of Chemical Engineering, Nanya Institute of Technology, Tao-Yuan, Taiwan
Shi-Shang Jang* and David Shan-Hill Wong Department of Chemical Engineering, National Tsing-Hua University, Hsin-Chu 30043, Taiwan
In a supply chain system, it is very important to forecast the changes in the market in order to maintain an inventory level that is just enough to satisfy customer demand. However, demand forecasting is responsible for the so-called “bullwhip effect”, the exaggeration of demand fluctuation toward upstream nodes. In this study, we present a minimum variance control (MVC) approach to solve this problem. First, customer demand trends are described by a general ARIMA model. The customer demand forecast is used to determine two inventory targets: inventory at hand and inventory on the road. The changes in inventory level are predicted using a model of material balance and information flow. The order policy is obtained by minimizing the errors between predicted inventory levels and set points and using a function that penalizes large changes in orders. Two controller parameters, the penalty cost factor and the relative weight between the changes in inventory and absolute inventory, are used to optimize the excess inventory and backorder subject to the constraint of no “bullwhip”. Simulation results show that this approach can track customer demand and maintain a proper inventory level without causing a bullwhip effect, whether the customer demand trend is stationary or not. Furthermore, the performance of the MVC is found to be superior to those of other approaches such as orderup-to-level, proportional and integral (PI) control, and smoothing order policy found in the literature. 1. Introduction During the past decade, there has been increasing interest in modeling the time-series behavior of a supply chain. The purpose of these studies is to design ordering policies that can minimize inventory without creating back order that sacrifices customer satisfaction. A supply chain system is essentially a dynamic balance of material and information flow with an ordering policy serving as the control system. Towill1 presented a classical control system formulation using a Laplace transform to analyze the performance of the inventory management. Perea-Lo´pez et al.2,3 analyzed the impact of several heuristic control laws, again using continuous time domain simulation. However, the real logistical operation of a supply chain is inherently a discrete dynamic system, and it is more convenient to use a time series model or a z-transform.4-7 A special phenomenon in supply chain dynamics is the “bullwhip effect”, which is the increase in the fluctuations of demand toward the upstream of the supply chain. The causes, methods, and profits of the reduction of “bullwhip” have been actively investigated.8-15 One of the main causes of the bullwhip effect is forecasting. The random noise of customer demand introduced into inventory set-point adjustments through demand forecasting will be amplified by aggressive control action. It is customary to reduce the fluctuations of demand forecasting by using simple filters.8,9 Lin et al.7 suggested a cascade control scheme to reduce bullwhip. Disney and Towill5,6 proposed that inventory be divided * To whom correspondence should be addressed. Phone: +886-3-571-3697. Fax: +886-3-571-5708. E-mail: ssjang@ mx.nthu.edu.tw.
into inventory on hand and inventory on the road. In these control policies, the filtered demand forecast acts as a set-point adjustment only; the controller parameters are not linked to the demand fluctuations explicitly. Note that, in a supply chain unit, in addition to the above bullwhip effect, some other constraints, such as the shortages of the suppliers, should be included. Some model based approaches16,17 are derived to handle these problems. It is very clear that the customer demand variations can be viewed as a stochastic process. A mature control scheme that could be used for stochastic process control is the minimum variance control (MVC) scheme.18-20 The objective of this work is to show that, by using MVC to control inventory at hand and inventory on the road, changes in demand can be successfully tracked, excess inventory and backorder are minimized, and the ordering bullwhip can be effectively suppressed as well. 2. Theory The relation of information and material exchange between an independent distributing unit and its supplier and customer in a decentralized supply chain system is depicted in Figure 1. 2.1. Balances of Material and Information Flow. Let IP(t) be the inventory position of a distributor at time instant t. It includes the actual inventory on hand IH(t) plus the material already delivered on the road from the supplier IR(t). Let YS(t) be the product delivered from its supplier and YC(t) be the product delivered to its customer. Given current time t, a time delay of L is assumed for delivery from the supplier so that goods dispatched at time t will arrive at time t + L. However, due to the need for examination and administrative
10.1021/ie0489610 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/18/2005
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predicted values of the controlled variables. The parameter F is the penalty factor to suppress the aggravated control actions. The variable dH is the time delay of the delivery action from its supplier. We assume that the customer demand takes the form of an ARIMA time series as
UC(t) )
Figure 1. Material and information flow of a simple three-node supply chain.
processing, this new delivery is only available to the customer at t + L + 1. It is also assumed that ordering information is communicated instantaneously. However, an order placed at time t to the supplier will only be processed at time t + 1, but a customer order is fulfilled instantly. Let US(t) be the amount of orders placed by the distributor to its supplier and UC(t) be the amount of orders placed by the customer. The following inventory balance can be derived:7
1 (YS(t) - YC(t)) 1 - z-1
(1)
1 (z-LYS(t) - YC(t)) 1 - z-1
(2)
1 - z-L YS(t) 1 - z-1
(3)
IP(t) ) IH(t) )
IR(t) )
IH(t + 1) )
B(z-1) C(z-1)
(4)
1 (z(-L-1)US(t) - UC(t)) -1 1-z
(5)
z (1 - z ) US(t) 1 - z-1
(10)
Θ(z-1)
(11)
Φ(z-1)∆r+1
B ˆ (z-1) D ˆ (z-1) Ei(z-1) US(t + i - L P(z ) IˆH(t + i) ) A ˆ (z-1) C ˆ (z-1) Fi(z-1) 1) + IH(t) (12) C ˆ (z-1) -1
ˆ (z-1) P(z-1) C D ˆ (z-1)
-1
) Ei(z ) +
F (z-1) -i i z -1 D ˆ (z )
(13)
For the other controlled variable IR(t), by comparing to eq 6, we have no stochastic term in the model,
i)mH nR
∑ (P(z i)m
1 ∆
(6)
(P(z-1) IˆH(t + i) - Pg SPH(t + i))2 + -1
)
)
The polynomials Ei(z-1) and Fj(z-1) can be obtained from the following Diophantine equation
-L
nH
∑
(9)
and dH ) L. The i-step-ahead predictor of IH is given by
2.2. Minimum Variance Control. To apply MVC for such a system, we define an objective function as
J)
z-dHB(z-1) C(z-1) U ξ(t) (t) + S A(z-1) D(z-1)
D(z-1)
The relationships between different inventories and order and demand are given by the following equations:
IR(t) )
where ξ(t) is a white noise having a mean of zero and unity variance. If we assume that the relation of controlled variable IH to manipulated variable US(t) and random part ξ(t) of the disturbance input, i.e., the customer demand UC(t), can be described by a general linear stochastic equation, then
A(z-1)
YS(t) ) z-1US(t) YC(t) ) UC(t)
-1
(8)
ξ(t) Φ(z-1)∆r
By comparing eq 9 to eqs 5 and 8, we have
If we also assume that there is always enough stock at hand for the supplier and distributor, then
IH(t) )
Θ(z-1)
IR(t + 1) )
2
) IˆR(t + i) - Pg SPR(t + i)) +
z-dRQ(z-1) US(t) R(z-1)
(14)
R
nH-dH
∑ i)1
F(∆US(t))2 (7)
The control objective function J consists of two controlled variables IH and IR. SPH and SPR denote their set points, respectively. The parameters mH (or mR) and nH (or nR) are the minimum cost horizon and prediction horizon, respectively. For simplicity, we assume that nH ) mH ) L + 1 and nR ) mR ) 1. The head “∧” symbol represents the predictive estimation of the variables. P(z-1) is a polynomial in z-1, and Pg is the gain of the polynomial P(z-1). It performs a moving average of the
with
Q(z-1) R(z-1)
)
1 - z-L ∆
(15)
and dR ) 0. Its j-step-ahead predictor equation is
P(z-1) IˆR(t + j) ) z-1Wj(z-1)US(t + j - 1) + Vj(z-1) IR(t) (16) Q ˆ (z-1)
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Similarly, the polynomials Wj(z-1) and Vj(z-1) can be found from the following Diophantine equation:
P(z-1) Q ˆ (z-1)
) Wj(z-1) + z-j
-1
R ˆ (z )
Vj(z-1)
(17)
-1
R ˆ (z )
The optimization for J can be solved to give the control law US(t) )
(
Pg SPH(t + L + 1) - zL+1 P(z-1) + 1 + F∆ -
)
EL+1(z-1) Φ(z-1)∆r+1 Θ(z-1)
EL+1(z-1) Φ(z-1)∆r Θ(z-1)
(
IH(t) +
)
∆ Pg SPR(t + 1) - z P(z ) IR(t) 1 - z-L 1 + F∆ -
-1
EL+1(z-1) Φ(z-1)∆r
UC(t)
)
P(z-1)(2 - z-L) + F∆2
(
)
EL+1(z-1) Φ(z-1)∆r+1 Θ(z-1)
-1
-L
2
P(z )(2 - z ) + F∆
(19)
In general, most supply chain researchers use a forecaster to predict customer demand. However, in this paper, we use a minimum variance predictor instead of a forecaster to handle the customer demand. Hence, we can set SPH(t) ) UC(t) and SPR(t) ) L × UC(t). Substituting into eqs 18 and 19, we get
US(t) UC(t)
(
∆LPg + zL+1 P(z-1) + )
)
EL+1(z-1) Φ(z-1)∆r+1 Θ(z-1)
P(z-1)(2 - z-L) + F∆2
(20)
2.3. Performance Measure Criteria. The performance of a supply chain is reflected by two factors. One is the average excess inventory (AEI), which is the cost of too much stock
EI(t) ) IH(t - 1) - UC(t) IH(t - 1) g UC(t) AEI )
1 τH
∫0τ EI(t) dt H
(21) (22)
where τH is a time horizon. The other factor is the average backorder (ABO), which measures the depletion of the safety stock and may endanger customer satisfaction
BO(t) ) UC(t) - IH(t - 1) IH(t - 1) g UC(t) ABO )
1 τH
|US(z-1)| -1
|UC(z )|
)
var(US(t)) var(UC(t))
)
∫0τ BO(t) dt H
(23) (24)
It should be pointed out that different weights can be assigned to these factors. We will use a performance
var(US(t))/var(ξ(t)) var(UC(t))/var(ξ(t)) 1 π 1 π
(18)
+
(25)
where η is a weighting factor. 2.4. Bullwhip Effect. By definition, the bullwhip effect can be measured by the ratio of the orders to the supplier to the demand from its customer. Hence, the following bullwhip constraint must be considered when trying to maximize performance. In the recent literature,6 in the case of stationary demand, the following analytical equation for the bullwhip effect can be derived:
Θ(z-1)
∆Pg(SPH(t + L + 1) + SPR(t + 1))
zL+1 P(z-1) +
Ψ ) η × AEI + (1 - η)ABO
BW )
Alternatively, the above equation can also be expressed in terms of customer demand.
US(t)
index that incorporates both excess inventory and backorder in a balanced manner:
∫0π|FU (jw)|2 dw S
∫0 |FU (jw)|2 dw π
)
(26)
C
where FUS(jw) ) US(jw)/ξ(jw), FUC(jw) ) UC(jw)/ξ(jw), and ξ is a zero mean white noise input. Note that, in this paper, it is assumed that the demand UC is following the model of ARIMA, eq 8. Let UC′ ) UC∆r and US′ ) US∆r. Hence,
BW )
|US(z-1)| -1
|UC(z )|
)
|US′(z-1)| -1
|UC′(z )|
)
var(US′(t)) var(UC′(t))
1 π ) var(UC′(t))/var(ξ(t)) 1 π var(US′(t))/var(ξ(t))
)
∫0π|FU ′(jw)|2 dw S
∫0 |FU ′(jw)|2 dw π
(27)
C
where FUS′(jw) ) US′(jw)/ξ(jw) and FUC′(jw) ) UC′(jw)/ ξ(jw); that is, the ramp effect of the ARIMA model is neglected in the bullwhip effect. Note that this extension is only valid in cases of the ARIMA nonstationary demand model. 2.5. Performance Optimization Problem. If we assume a one-step look ahead where
P(z-1) ) 1 - p1z-1
(28)
there will be two controller tuning parameters, F and p1. The parameter F represents the importance we assign to violent control action. Note that p1 ) 0 implies that we are concerned with the predicted inventory level only; p1 ) 1 implies we are concerned with the changes in predicted inventory level. Hence, the optimal control problem becomes
minΨ F,p1
s.t. BW e 1
(29)
3. Other Ordering Policies In this work, three other ordering policies are compared with the above MVC approach. 3.1. Order-up-to-Level Policy. The most common and simplest ordering policy used is the following orderup-to-level policy:
US(t) ) SPP(t) - IP(t)
(30)
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Figure 2. Magnitude ratio |US/UC| for a stationary demand process using various values of the weighting factor F with the same value of p1 ) 0.5.
Figure 3. Magnitude ratio |US/UC| for a stationary demand process using various values of p1 with the same penalty factor F ) 0.
Here, SPP(t) is the set point of the inventory position. If we assume that SPp(t) ) (L + 2)UC(t), the above equation can be modified as the following equation:
US(t) ) (L + 2)∆UC(t) + UC(t)
(31)
3.2. Proportional and Integral (PI) Control Policies. A traditional controller that guarantees no offset is the PI controller.
(
US(t) ) Kc 1 +
1 (SP(t) - IP(t)) τ∆
)
(32)
where Kc is the proportional gain and τ is the integral constant. It is obvious that the order-up-to-level policy is a proportional (P) controller with Kc ) 1. 3.3. Smoothing Order Policy (SOP). Towill and coworkers5,6 proposed the following general replenishment rule:
˜ C(t) + US(t) ) U
Figure 4. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of F and p1 of MVC for a stationary demand process.
1 1 (U ˜ (t) - IH(t)) + (LU ˜ C(t) Tn C Tw IR(t)) (33)
U ˜ C is the demand forecast,
U ˜ C(t) ) FF(z-1) UC(t)
(34)
where FF(z-1) is a filter forecaster. Tn and Tw are the parameters of the ordering decision rule. It is easily found that the net stock (IH) plus products on order (IR) equals the inventory position (IP). In cases where Tn ) Tw ) T, eq 33 will be reduced as
1 ˜ C(t) + ((L + 1)U US(t) ) U ˜ C(t) - IP(t)) T
(35)
In other words, the smoothing order rule becomes a feedforward controller and a proportional controller. If Tn ) Tw ) 1, the smoothing law will be reduced to the order-up-to-level policy. 4. Results and Discussions Two cases are considered in this work to demonstrate the capabilities of this approach to track the change in demand and to avoid the bullwhip effect. One is the case of stationary process demand; the other is the case of a nonstationary process demand.
Figure 5. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of Kc and Kc/τ of PI control for a stationary demand process.
4.1. Stationary Demand. A demand is stationary when the parameter r in eq 8 is zero. A first-order autoregressive (AR1) model for customer demand is used as the example.
UC(t) )
1 ξ(t) 1 - 0.6z-1
(36)
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Figure 6. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of Tn and Tw of SOP control for a stationary demand process.
Figure 2 plotted the frequency domain response of the above equation for the case with p1 ) 0.5 and various values of F using eq 20. The bullwhip can be reduced if
we increase F. Similar effects can be observed if we fix F and increase p1 (Figure 3). Table 1 compares the optimized performance indices obtained using the proposed MVC approach to those obtained using the PI and SOP schemes. Order-up-tolevel results are also presented as a standard. Notably, the bullwhip effect cannot be suppressed using the order-up-to-level policy, since there is no parameter that can be tuned in the whole ordering strategy. On the other hand, the MVC approach provides performance indices superior to those of the PI and SOP methods. Figure 4 gives the contour plots of both Ψ and BW as functions of p1 and F. Note that all the numerical values of the objective function Ψ are found through numerical simulations, since there are some stochastic terms in eq 36. The bullwhip effect is reduced as the smoothing factor p1 is increased. However, the total cost function Ψ is decreased as p1 is increased. On the other hand, the cost function Ψ is also deceased as the penalty factor F is decreased. The optimum value for the balanced consideration of excess inventory and the backorder cost function is found at F ) 0.42 and p1 ) 0.65. Figure 5 gives the contour plots of both Ψ and BW as functions of Kc and Kc/τ for PI control. It is found that BW increases with the increase of both the proportional
Figure 7. Dynamic simulation results of a supply chain unit for a stationary demand process with (a) MVC using F ) 0.42 and p1 ) 0.65, (b) PI control using Kc ) 0.38 and Kc/τ ) 0, and (c) SOP control using Tn ) 2.40 and Tw ) 1.21.
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Figure 8. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of F and p1 of MVC for a nonstationary demand process.
and integral actions of the PI controller. However, as shown in Figure 5, the cost function decreases as we increase the proportional action of the PI controller and decrease the integral action. The optimum value is found when a small I controller is used. Figure 6 gives the contour plots of both Ψ and BW as functions of Tn and Tw for the SOP method. It is found that the contour of BW ) 1 exhibits a turning point. Hence, the tuning of the parameters in the smoothing order policy is bound by this constraint and becomes independent of the performance indices used. Figure 7 displays the dynamic simulation results for an optimal MVC with F ) 0.42 and p1 ) 0.65 (Figure 7a), a PI controller with Kc ) 0.38 and Kc/τI ) 0 (Figure 7b), and a SOP controller with Tn ) 2.40 and Tw ) 1.21 (Figure 7c). It can be seen that the variations of demand are tracked well without any bullwhip by the MVC controller. The PI and SOP controllers can also follow the set point changes without bullwhip, but the performance in terms of Ψ is inferior to that of the MVC, as listed in Table 1. 4.2. Nonstationary Demand. Consider a nonstationary model.
UC(t) )
1
ξ(t)
(1 - 0.6z-1)(1 - z-1)
(37)
Table 1 gives the optimal simulation results of the objective function by implementing the above-mentioned controllers. Again, the performance indices obtained by the optimized MVC are better than those obtained using optimized PI and SOP control. Figure 8 gives the contour plots of both Ψ and BW as functions of F and p1 for the MVC approach. The trends
Figure 9. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of Kc and Kc/τ of PI control for a nonstationary demand process.
Figure 10. Contour diagram of BW (solid lines) and Ψ (dotted lines) as functions of Tn and Tw of SOP control for a nonstationary demand process.
of both Ψ and BW for the nonstationary MVC case are about the same for the stationary case, as shown in Figure 4. The optimum value can be found at a value of F ) 2.60 and p1 ) 0.86, as shown in Figure 8. Figure 9 gives the contour plots of both Ψ and BW as functions of Kc and Kc/τ of PI control. It is found that both Kc and Kc/τ must be kept very small to avoid the bullwhip effect. Similarly, Figure 10 shows that optimization of Tw and Tn in the SOP approach is also constrained by the contour of BW ) 1, as the data exhibit a turning point.
Table 1: Comparisons of the Performance of Different Ordering Policies for Stationary and Nonstationary Demand while η ) 0.5 demand type stationary
nonstationary
control law order-up-to-level SOP PI MVC order-up-to-level SOP PI MVC
parameters Tn ) 2.40, Tw ) 1.21 Kc ) 0.38, τ ) ∞ F ) 0.42, p1 ) 0.65 Tn ) 2.85, Tw ) 1.98 Kc ) 0.37, τ ) 370 F ) 2.60, p1 ) 0.86
BW
Ψ
AEI
ABO
2.25 1.006 0.996 0.997 2.25 0.999 0.987 1.000
1.649 1.831 1.858 1.545 22.36 25.12 34.12 24.41
1.631 1.804 1.856 1.585 35.15 12.89 17.69 12.02
1.668 1.859 1.859 1.504 9.58 37.35 50.53 36.80
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Figure 11. Dynamic simulation results of a supply chain unit for a nonstationary demand process with (a) MVC using F ) 2.60 and p1 ) 0.86, (b) PI control using Kc ) 0.37 and Kc/τ ) 0.001, and (c) SOP control using Tn ) 2.85 and Tw ) 1.98. Table 2: Comparison of AEI and ABO Based on Different Weightings and Two Factors of the Objective Function demand type stationary nonstationary
weighting factor
parameters
BW
Ψ
AEI
ABO
η ) 0.9 η ) 0.5 η ) 0.1 η ) 0.9 η ) 0.5 η ) 0.1
F ) 0.75, p1 ) 0.61 F ) 0.42, p1 ) 0.65 F ) 0.21, p1 ) 0.67 F ) 3.58, p1 ) 0.83 F ) 2.60, p1 ) 0.86 F ) 1.80, p1 ) 0.89
0.990 0.997 1.007 1.000 1.000 0.995
1.553 1.545 1.493 14.90 24.41 31.70
1.556 1.585 1.686 10.83 12.02 13.34
1.528 1.504 1.472 51.51 36.80 32.39
Figure 11 displays the dynamic simulation results of a supply chain unit with nonstationary customer demand using MVC, PI, and SOP approaches. The optimal settings and performance are given in Table 1. It can be seen that the variations in demand are tracked well without any bullwhip by the MVC and other controllers, but the excess inventory and backorder are much smaller for MVC, as shown in Figure 11 and Table 1. 4.3. Different Objective Function Weightings and Robustness Tests of MVC. Note that all analyses displayed above are based on the assumption that the importance of excess inventory and backorder are equal in the supply chain system. However, in many cases, the importance of AEI and ABO may not be equal under the special considerations of each supply chain. Table 2 displays the AEI and ABO under different weighting
Table 3: Robustness Tests of the MVC Strategy demand type
model
perfect (φ ) 0.60) with 20% modeling error (φ ) 0.72) nonstationary perfect (φ ) 0.60) with 20% modeling error (φ ) 0.72)
stationary
BW
Ψ
AEI
ABO
0.997 1.027
1.55 1.91
1.59 1.97
1.50 1.86
1.000 24.41 12.02 36.80 1.147 34.471 16.81 52.13
factors of the objective function (eq 25). The control result shows that the importance of backorder and excess inventory can be tuned using the MVC law. The other important consideration of MVC is the case of modeling error. Table 3 gives the simulation results of the performance of MVC based on 20% modeling errors of the above stationary and nonstationary ex-
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assume that the demands of these two retailers are of the following model:
1 ξ(t) 1 - 0.6z-1 1 ξ(t) UC,RB(t) ) (1 - 0.6z-1)∆ UC,RA(t) )
(38) (39)
The distribution center models its demand based on the historical orders from these two retailers, such that
UC,DC(t) ) Figure 12. Multinode supply chain structure.
amples. The results show that both BW and the objective function are only slightly degraded. 4.4. Multinode Simulation Example. In real supply chain cases, there are many supply nodes such as retailers, distribution centers, and warehouses. Figure 12 shows a three-node supply chain system with a single distribution center and two retailers. We assume that the distribution center takes the orders from two different retailers: one exihibits stationary demand while the other exihibits nonstationary demand. We will
1 ξ(t) (1 - 0.62z-1)∆
(40)
The MVC controllers are, hence, designed for the above three nodes separately. Figure 13 displays the tracking of the demands of these three controllers. Table 4 gives the comparisons of the performance of MVC and all of the other control strategies mentioned above. Once again, the MVC policy is superior to all other wellknown ordering policies. 5. Conclusion A method for determining the ordering policy for a given demand pattern is derived using minimum vari-
Figure 13. Dynamic order actions for a multinode supply chain using different order policies. Table 4: Comparisons of MVC to Other Ordering Policies for the Example of Three Decentralized Nodes control law
node
order-up-to-level
retailer A retailer B distribution center retailer A retailer B distribution center retailer A retailer B distribution center retailer A retailer B distribution center
SOP PI MVC
parameters
BW
Ψ
AEI
ABO
Tn ) 2.40, Tw ) 1.21 Tn ) 2.85, Tw ) 1.98 Tn ) 4.38, Tw ) 3.00 Kc ) 0.38, τ ) ∞ Kc ) 0.37, τ ) 370 Kc ) 0.24, τ ) ∞ F ) 0.42, p1 ) 0.65 F ) 2.60, p1 ) 0.86 F ) 2.90, p1 ) 0.85
2.25 2.25 7.89 1.006 0.999 1.008 0.996 0.987 1.004 0.997 1.000 1.004
1.649 22.36 23.97 1.831 25.12 29.31 1.858 34.12 43.26 1.545 24.41 27.08
1.631 35.15 11.36 1.804 12.89 17.87 1.856 17.69 28.20 1.585 12.02 16.95
1.668 9.58 36.59 1.859 37.35 40.75 1.859 50.53 58.33 1.504 36.80 37.21
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ance control theory. An expression of the bullwhip effect can be obtained. Fine-tuning of this controller is obtained by minimizing excess inventory and backorder subject to the constraint that there should be no bullwhip. The optimized performance of MVC is found to be superior to the optimized performances of the control schemes reported in the literature. Acknowledgment The authors thank the National Science Council, Taiwan, and Top Information System Co. for financial support for this work through Grant NSC91-2622-E007006. Literature Cited (1) Towill, D. R. Dynamic analysis of an inventory and order based production control system. Int. J. Prod. Res. 1982, 20, 671687. (2) Perea, E.; Grossmann, I.; Ydstie, E.; Tahmassebi, T. Dynamic modeling and classical control theory for supply chain management. Comput. Chem. Eng. 2000, 24, 1143-1149. (3) Perea-Lo´pez, E.; Grossmann, I. E.; Ydstie, B. E.; Tahmassebi, T. Dynamic modeling and decentralized control of supply chains. Ind. Eng. Chem. Res. 2001, 40, 3369-3383. (4) Dejonckheere, J.; Disney, S. M.; Lambrecht, M. R.; Towill, D. R. Transfer function analysis of forecasting induced bullwhip in supply chains. Int. J. Prod. Econ. 2002, 78, 133-144. (5) Dejonckheere, J.; Disney, S. M.; Lambrecht, M. R.; Towill, D. R. Production, Manufacturing and Logistics - Measuring and avoiding the bullwhip effect: A control theoretic approach. Eur. J. Oper. Res. 2003, 147, 567-590. (6) Disney, S. M.; Towill, D. R. On the bullwhip and inventory variance produced by an ordering policy. Omega 2003, 31, 157167. (7) Lin, P. H.; Wong, D. S. H.; Jang, S. S.; Shieh, S. S.; Chu, J. Z. Controller design and reduction of bullwhip for a model supply chain using z-transform analysis. J. Process Control 2004, 14, 487-499.
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Received for review October 27, 2004 Revised manuscript received July 7, 2005 Accepted September 16, 2005 IE0489610
