Automatica 46 (2010) 1982–1993
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Linear–quadratic optimal control strategy for periodic-review inventory systems✩ Przemysław Ignaciuk ∗ , Andrzej Bartoszewicz Institute of Automatic Control, Technical University of Łódź, 18/22 Stefanowskiego St., 90-924 Łódź, Poland
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Article history: Received 7 December 2008 Received in revised form 28 July 2010 Accepted 2 August 2010 Available online 23 October 2010 Keywords: Optimal control Inventory control Riccati equations Discrete-time systems
abstract The paper addresses the problem of efficient inventory management in production–inventory systems focusing on the dynamical nature of goods flow process. In the considered systems, the stock used to satisfy an unknown, time-varying demand is replenished either from a single or from multiple supply sources. The replenishment orders issued in each review period are realized with a delay, which differs among the suppliers and transport alternatives. For the analyzed setting, modeled as a discrete-time nth-order deterministic system, a new inventory policy is developed using a strict control-theoretic methodology. In contrast to the classical, stochastic approaches, the proposed control law is obtained by minimizing a quadratic cost functional, which guarantees the optimal dynamical performance of production–inventory systems with (possibly) different lead-time delays in the supply path. The designed policy ensures that the demand is always entirely satisfied from the on-hand stock (yielding zero lost-sales cost) and the warehouse capacity is not exceeded (which eliminates the risk of high-cost emergency storage). The closed-form solution of the linear–quadratic (LQ) optimization problem allows for a straightforward implementation of the developed control strategy in real systems. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction A properly designed and implemented inventory policy has long been identified as a decisive factor behind the success of production and goods distribution systems (Zipkin, 2000). It is vital both for establishing sound foundations for a new company, or another stage in the production process, as well as for maintaining high effectiveness of the existing business entities and the overall supply chain performance (Silver, Pyke, & Peterson, 1998). The traditional approaches to inventory control primarily concentrate on the statistical analysis of the long-term variables and (static) optimization performed on the averaged values of various cost components. However, the technological advances used to facilitate the flow of information in production–inventory systems (e.g. Internet ordering (Doukidis, Pramatari, & Lekakos, 2008; Teich, Wallenius, Wallenius, & Koopius, 2004), automatic warehouse review
✩ This work was financed by the Polish State budget in the years 2010–2012 as a research project N N514 108638 ‘‘Application of regulation theory methods to the control of logistic processes’’. P. Ignaciuk gratefully acknowledges financial support provided by the Foundation for Polish Science (FNP). The material in this paper was partially presented at 48th IEEE Conference on Decision and Control, December 16–18, 2009, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor Suresh P. Sethi under the direction of Editor Berç Rüstem. ∗ Corresponding author. Tel.: +48 42 6312556; fax: +48 42 6312551. E-mail addresses:
[email protected],
[email protected] (P. Ignaciuk),
[email protected] (A. Bartoszewicz).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.09.010
(Giordano, Zhang, Naso, & Lewis, 2008), RFID enhancements in supply chain (Tajima, 2007)) call for new solutions, which will not only provide good steady-state performance, but will also guarantee prompt response to the changing market conditions. Therefore, in order to answer the challenges of the highly competitive and rapidly varying environment, and in this way safeguard the company prosperity in the 21st century, the traditional approaches to the design of inventory management schemes must be augmented by a solid analysis of the dynamical features of the system. The recognized need for studying the dynamical aspects of production–inventory systems has resulted in a number of research efforts employing the tools of control theory. A good, comprehensive summary of the initial proposals is given in Axsäter (1985), whereas more recent approaches are discussed in the review paper by Ortega and Lin (2004), in the special issue of Automatica (Sethi & Zhang, 2006), and in the extensive survey provided by Sarimveis, Patrinos, Tarantilis, and Kiranoudis (2008). One of the first attempts to apply the control theory to production–inventory systems were the works of Simon (1952) and Vassian (1955). Simon (1952) noticed an analogy between the servomechanism systems and the techniques involved in production planning and inventory ordering decisions, and used this observation to elaborate a continuous-time production rate controller for a single-commodity manufacturing system. Vassian (1955), in turn, extended the servomechanism concept to discretetime inventory control. The early applications of the deterministic maximum principle in production, finance and marketing, were
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
summarized in the extensive review by Sethi (1978). Afterwards various authors proposed the use of stochastic-optimal (Neck, 1984; Vickson, 1982), classical proportional-integral-derivative (PID) (White, 1999), fuzzy PID (Samanta & Al-Araimi, 2001), H∞ norm-based (Rodrigues & Boukas, 2006), predictive (Wang, Rivera, & Kempf, 2007), and adaptive (Aggelogiannaki & Sarimveis, 2008) control concepts to regulate either the production rate or the stock level. In addition to the controller design, the control theory tools have also been applied to conduct a formal analysis of the traditional inventory policies (Grubbström & Wikner, 1996; Hoberg, Bradley, & Thonemann, 2007; Towill, 1982). Towill (1982) described a production–inventory system in a block-diagram form, identifying the principal delay components affecting the system dynamics. He stressed the importance of the feedback loop to counteract the variability of the production delay, and showed that satisfactory performance cannot be obtained by the feedforward (demand averaging) path alone. Grubbström and Wikner (1996) provided a concise mathematical description of the standard ordering rules by means of difference and differential equations involving Dirac delta and Heaviside step functions. Hoberg et al. (2007) applied the transfer functions to describe the inventory system dynamics and addressed the stability issues of the traditional policies in the presence of lead-time delays. They showed that for maintaining the system stability when orders are realized with delay, the reordering decisions of the typical inventory policies must consider the stock in-bound (the quantities already ordered, but not yet delivered due to latency) in addition to the on-hand stock at the warehouse. In this paper, we develop new supply policies for a periodicreview inventory system using strict control-theoretic methodology. The aim of the control action is to always satisfy the entire demand from the readily available stock and currently arriving shipments. In this way any cost associated with backorders and lost sales is eliminated. Additionally, we expect from the policy to generate smoothly varying order quantities in subsequent review intervals. This will constitute a planning benefit for the supplier, and consequently reduce the risk of the supplier nonconforming to the established contracting agreement due to otherwise abrupt and highly unpredictable order changes. For this purpose we apply quadratic cost functional which is well known for its rate smoothening properties in production–inventory systems and a good mathematical framework for conducting the optimization procedure, as discussed by Holt, Modigliani, Muth, and Simon (1960). In contrast to the earlier contributions employing the quadratic cost structure (Bergstrom & Smith, 1970; Dobos, 2003; Kleindorfer, Kriebel, Thompson, & Kleindorfer, 1975; Thompson & Sethi, 1980), which focused primarily on production planning and horizon research, in this paper, we concentrate on the flow of goods in a supply chain. We design a new ordering policy for stock replenishment at the goods distribution center, which is subject to an a priori unknown, time-varying bounded demand. The application of a quadratic cost functional for developing a supply policy allows smoothening of ordering variations caused by demand changes. In addition to the obvious supplier planning benefit, it reduces the influence of demand fluctuations on the order quantity, and, as a result, helps combat the highly undesirable demand amplification in the supply chain known as the bullwhip effect (Geary, Disney, & Towill, 2006). The essential difference of our work as compared to similar results reported in the past is the explicit consideration of the delay which takes place between issuing of an order and its realization. The presence of the delay, which usually spans multiple review periods (especially when foreign supply sources are involved), poses a stability threat, and may be the source of stock and order quantity oscillations (Hoberg et al., 2007). We consider two system configurations depending on the choice of the company sourcing strategy. In the first one, the stock
1983
used to satisfy the unknown, time-varying demand is replenished with a delay from a single supplier. We model the inventory system as an nth-order system and on the basis of its state space description we formulate the linear–quadratic (LQ) optimization problem. The problem is parameterized with a tuning coefficient used to adjust the controller dynamics so that even in the case of sudden demand changes, the control signal (the order quantity) does not fluctuate excessively in subsequent review periods. Contrary to the typical approaches for solving the LQ problem (Arnold & Laub, 1984; Benner & Byers, 1998; Chu, Lin, & Tan, 2006; Dieci, 1991; Guo & Laub, 2000; Higham & Kim, 2001; Kwakernaak & Sivan, 1972; Long, Hu, & Zhang, 2008), which are mainly suitable for numerical implementations and systems with predefined dimensions, we propose an analytical solution of the nth order matrix Riccati equation, and obtain the control law in a closed form. Since in practical settings a company may choose to split an order among several suppliers (Burke, Carrillo, & Vakharia, 2007; Minner, 2003), or use various delivery paths for order procurement (Dullaert, Vernimmen, Raa, & Witlox, 2005), in the second part of the paper we extend the results obtained for the single-supplier case to the inventory system with multiple supply options. We assume that each option may be characterized by a different lead time. Since in the case of multiple supply alternatives the optimal solution may generate negative order quantities, which is inappropriate, as it would imply additional costs of returning the stock to the suppliers, we propose a sub-optimal controller always generating nonnegative quantities. The inventory system governed by either of the proposed supply policies is demonstrated to be asymptotically stable. It is also shown that the stock level never exceeds the assigned warehouse capacity, which means that the potential necessity for an expensive emergency storage outside the company premises is eliminated. The closed-form solution of the optimization problem allows for a straightforward implementation of the developed control strategies in real systems. Moreover, since after a change in demand, the order quantities generated by our solution exponentially converge to steady-state without overshoots or oscillations (which is not easy to achieve in time-delay systems without the deterioration of their dynamic performance) the proposed policies further alleviate the negative impact of demand fluctuations on the supply chain performance. The paper is organized in the following way. First, in Section 2, the inventory system with a single supply source is analyzed. The system model is described in the state space and the optimization problem is formulated. The problem is solved analytically, which allows expressing the control law in a closed form. The favorable properties of the obtained controller are defined and strictly proved. Next, in Section 3, the inventory control problem is addressed in the setting with multiple supply alternatives used to replenish the stock. The state space description of the augmented system is provided, and the optimization task is solved analytically, yielding again a simple in implementation supply policy in a closed form. The asymptotic stability of the system with the designed controller implemented is demonstrated, and its features are discussed and proved. The simulation results illustrating the performance of the proposed strategies are described in Section 4. Finally, the concluding remarks are given in Section 5. 2. Production–inventory system with a single supplier First, we will analyze the case of an inventory system faced by an unknown, bounded, time-varying demand where the stock is replenished from a single supply source. Such a setting, illustrated in Fig. 2, is frequently encountered in production–inventory systems where a common point (distribution center), linked to a
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P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
Fig. 1. Model of inventory system with a single supplier.
The considered discrete-time system can also be described in the state space as x[(k + 1)T ] = Ax(kT ) + bu(kT ) + vh(kT ) y(kT ) = qT x(kT )
Fig. 2. Inventory system with a single supplier.
factory or external, strategic supplier, is used to provide goods for another production stage or a distribution network. The task is to design a stable control strategy which will minimize the lost service opportunities (occurring when only part of the demand can be satisfied from the stock available at the distribution center) with explicit consideration of the delay between placing an order at the supplier and goods arrival at the center. The latency of fulfilling the order will be further referred to as the lead-time delay. 2.1. System model We consider the periodic-review inventory system, whose schematic diagram is depicted in Fig. 1. The stock replenishment orders u(kT ) are issued at regular intervals kT , where T is the review period and k = 0, 1, 2, . . . , on the basis of the on-hand stock (the current stock level in the warehouse at the distribution center) y(kT ), target stock level yd , and the history of previous orders. Each non-zero order placed at the supplier is realized after a lead-time delay Lp , assumed to be a multiple of the review period, i.e. Lp = np T , where np is a positive integer. The demand (the number of items requested from inventory in period k) is modeled as an a priori unknown, bounded function of time d(kT ) 0 ≤ d(kT ) ≤ dmax .
(1)
Notice that this definition of the demand is quite general and it accounts for any standard distribution typically analyzed in the considered problem. If there is a sufficient number of items in the warehouse to satisfy the current demand, then the actually met demand h(kT ) (the number of items sold to customers or sent to retailers in the distribution network) will be equal to the requested one. Otherwise, the demand is satisfied only from the arriving shipments, and the surplus demand is lost (we assume that the sales are not backordered, and the excessive demand is equivalent to a missed business opportunity). Thus, we may write 0 ≤ h(kT ) ≤ d(kT ) ≤ dmax .
(2)
The dynamics of the on-hand stock y depends on the amount of the arriving shipments and on the satisfied demand h. We assume that the warehouse is initially empty, i.e. y(kT < 0) = 0, and the first orders are placed at kT = 0. Applying the definition Lp = np T , the stock level for any k ≥ 0 may be expressed as y(kT ) =
k−1 −
u(jT − np T ) −
j=0
−
k−1 −
k−np −1
h(jT ) =
j =0
k−1 − j=0
k−np −1
h(jT ) =
− j =0
u(jT ) −
−
u(jT )
(4)
where x(kT ) = [x1 (kT ) x2 (kT ) . . . xn (kT )]T is the state vector with x1 (kT ) = y(kT ) representing the on-hand stock in period k and xi (kT ) = u[(k − n + i − 1)T ] for any i = 2, . . . , n equal to the delayed input signal u; A is an n × n state matrix, b, v, and q are n × 1 vectors 1 0
.
A= .. 0 0
1 0
0 1
0 0
0 0
.. .
−1 0 . v= .. 0 0
.. .
... ... .. .
0 0
0
.. . 1
0 . b= .. 0
... ... 0
1
1 0
(5)
.
q= .. 0 0
and the system order n = np + 1 = (Lp /T ) + 1 depends on discretization period T and the order lead-time delay. For convenience of further analysis, we can present the model in the alternative form
x [(k + 1)T ] = x1 (kT ) + x2 (kT ) − h(kT ) 1 x 2 [(k + 1)T ] = x3 (kT ) .. . xn−1 [(k + 1)T ] = xn (kT ) xn [(k + 1)T ] = u(kT )
(6)
where the desired system state xd = [xd1 xd2 . . . xdn ]T = [xd1 0 0 . . . 0]T , and xd1 = yd denotes the demand value of the first state variable, i.e. the target stock level. 2.2. Proposed inventory management policy In this section, a novel inventory management policy is developed using a strict, control-theoretic methodology. The objective is to obtain high customer service levels in the presence of an unknown, time-varying demand and the stock replenishment orders realized with delay. On the contrary to the typical approaches to the inventory control problem, in this paper, we focus on optimizing the dynamical behavior of the system, and not only the static, or average quantities (e.g. long-term cost). After formulating the optimization task, we present a novel, analytical solution to the LQ problem for the considered system with delay. The system stability is demonstrated and the important features of the proposed control scheme are discussed, and strictly proved.
j=−np k−1 − j =0
h(jT ).
(3)
2.2.1. Optimization problem From the point of view of optimizing the system dynamics, the aim of the control action is to bring the system state (the currently
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
available stock) to the target value without excessive control effort, or, alternatively speaking, to reduce the closed-loop system error e(kT ) = xd − x(kT ) to zero using reasonable order quantities. Therefore, we seek for a control uopt (kT ), which minimizes the quality criterion expressed by the following cost functional J (u) =
∞ − {u2 (kT ) + w[yd − y(kT )]2 }
where w is a positive constant applied to adjust the influence of the controller command and the output variable on the cost functional value. A small w reduces excessive order quantities, but lowers the controller dynamics. A high w , in turn, implies fast tracking of the reference stock level yd at the expense of large input signals. In the extreme case, when w → ∞, the term yd − y(kT ) prevails (in this situation the error at the output is to be reduced to zero as quickly as possible, no matter what the value of the command), and the developed controller will become a dead-beat scheme. In economical terms, the coefficient w relates inventory holding cost to order setup cost, i.e. a small w means large setup cost with respect to holding cost, whereas in the case of w > 1 the holding cost dominates over the setup cost. Detailed guidelines for cost coefficient selection can be found in Holt et al. (1960). From the managerial point of view the application of the quadratic cost structure in the considered supply policy design problem has similar effects as discussed in Holt et al. (1960) in the context of production planning. It allows for a satisfactory tradeoff between fast reaction to changes in market conditions (manifesting themselves as bandwidth variations) and smoothness of ordering decisions. As a result, the controller will track the target inventory level yd with good dynamics, yet, at the same time, it will prevent rapid demand fluctuations from propagating in the supply chain. Applying the standard framework proposed in Zabczyk (1974), to system (4), the control uopt (kT ) minimizing criterion (7) can be presented as uopt (kT ) = −gx(kT ) + r
(8)
where
r = bT [K(In + bbT K)−1 bbT − In ]k
(9)
k = −AT [K(In + bbT K)−1 bbT − In ]k − w qyd and semipositive, symmetric matrix K (KT = K ≥ 0) is determined according to the following Riccati equation K = AT K(In + bbT K)−1 A + w qqT .
(10)
The classical approaches for solving (10), as suggested in literature (Arnold & Laub, 1984; Benner & Byers, 1998; Chu et al., 2006; Dieci, 1991; Guo & Laub, 2000; Higham & Kim, 2001; Kwakernaak & Sivan, 1972; Long et al., 2008), are mainly suitable for numerical calculations and systems with predefined dimensions. In our case, however, an analytical solution of the Riccati equation needs to be found for a system of arbitrary order n. The method proposed in this paper involves iterative, analytical substitution of K into the expression on the right hand side of (10) and comparison with its left hand side so that at each step the number of independent variables kij , where kij denotes the element in the i-th row and j-th column of K, is reduced. We begin solving (10) with the most general form of K = K0 , k11 k12
K0 = ..
k12 k22
.
.. .
k1n
k2n
... ... .. . ...
k11
k11 − w k 13 K1 = . ..
k11 − w k11 − w k23
k13 k23 k33
k2n
k3n
.. .
k1n
... ... ... .. .
.. .
k1n k2n k3n .
(12)
.. .
...
knn
Now we substitute K1 given by (12) into the expression on the right-hand side of (10) and compare it with its left-hand side, which allows representing elements ki3 (i = 1, 2, 3) in terms of k11 . We proceed in this way until a general pattern is determined, i.e. until all the elements of K can be expressed as functions of k11 and the system order n. The final form of K is given as follows k11 − w
k11 − w k11 − w
k11 − (n − 1)w
k11 − (n − 1)w
k11
.. .
... ... .. .
.. .
k11 − (n − 1)w k11 − (n − 1)w
.. .
...
(13)
k11 − (n − 1)w
with k11 determined from k11 = nw + 1 − [k11 − (n − 1)w + 1]−1 .
(14)
Eq. (14) has two roots k± 11 =
√ √ √ w[(2n − 1) w ± w + 4]/2.
(15)
Since det(K) = w [k11 − (n − 1)w], only k11 ≥ (n − 1)w guarantees that K is semipositive definite. Having found K, we evaluate g, +
n−1
g = [ 1
1
1
...
1 ]{1 − [k11 − (n − 1)w + 1]−1 }
1 . . . 1 ]α (16) √ where α = ( w(w + 4) − w)/2. Vector k is determined by sub-
=[ 1
1
stituting matrix K, as given by (13), into (9),
k = k1
k1 + w yd
k1 + 2w yd
...
k1 + (n − 1)w yd
T
(17)
where k1 = −w yd {n + [k11 − (n − 1)w]−1 }.
(18)
Then, applying (17) in the second equation in set (9), we calculate r,
g = bT K(In + bbT K)−1 A
After the first iteration we get
(7)
k=0
1985
k1n k2n
r =−
k1 + (n − 1)w yd k11 − (n − 1)w + 1
knn
w yd (19) k11 − (n − 1)w = k+ 11 reduces to r = α yd . Finally,
which after substituting k11 using (16) and (19), the optimal control can be presented in the following way, uopt (kT ) = −gx(kT ) + r = −α
n −
xj (kT ) + α yd
j =1
= α yd − x1 (kT ) −
n −
xj (kT ) .
(20)
j =2
From (6) the state variables xj (j = 2, 3, . . . , n) may be expressed in terms of the control signal generated in the previous n − 1 periods as xj (kT ) = u[(k − n + j − 1)T ].
(21)
Recall that x1 (kT ) = y(kT ) and np = n − 1. Then, substituting (21) into (20), we obtain
u(kT ) = α
yd − y(kT ) −
.. . .
=
n−1 −
u[(k − j)T ]
j =1
(11)
= α yd − y(kT ) −
k−1 − j=k−np
u(jT )
(22)
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P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
which completes the design of the LQ optimal inventory controller for the system with a single supply source. The obtained policy can be interpreted in the following way. The quantity to be ordered in each period is proportional to the difference between the target and the current stock level (yd − y(kT )) decreased by the amount of open orders (the quantity already ordered at the supplier, but which has not yet arrived at the warehouse due to lead time).
The stock level in the l + 1 period can be expressed as y[(l + 1)T ] = y(lT ) + u[(l − np )T ] − h(lT ). Applying (3) and (29), we get l −n p −1
y[(l + 1)T ] =
−
u(jT ) −
l −1 −
j =0
Remark 1. According to Zabczyk (1974), in the circumstances when the control action changes the system state from the initial value x0 = x(0) to zero, the minimum value of quality criterion J for w = 1 can be determined from the following relation. J (uopt ) = x (0)[Kx(0) + 2k]. T
(23)
However, in our case the purpose of the control action is to stabilize the controlled variable y(kT ) = x1 (kT ) at level yd . Therefore, J (uopt ) = −xT (∞)[Kx(∞) + 2k] = −xTd [Kxd + 2k].
h(jT )
j =0
l −n p −1
l−np −1
+ α yd −
−
−
u(jT ) +
j =0
h(jT ) − h(lT )
j =0
= (1 − α)y(lT ) + α yd − α
l−1 −
h(jT ) − h(lT )
j=l−np l −1 −
= yd − (1 − α)[yd − y(lT )] − α
h(jT ) − h(lT ).
(31)
j=l−np
(24)
Substituting (13) and (17) into (24), we get J (uopt ) = −xTd [Kxd + 2k] = −yd (yd k11 + 2k1 ).
(30)
(25)
Applying k11 = k11 given by (15) and k1 given by (18) with w = 1, we arrive at
Since α ∈ (0, 1) and h is always nonnegative, y[(l + 1)T ] ≤ yd . Using the principle of mathematical induction we conclude that the proposition is valid for any review period k ≥ 0. This ends the proof.
+
J (uopt ) = y2d (2n +
√
5 − 1)/2 ≈ y2d (n + 0.618).
(26)
2.2.2. Stability analysis The system is asymptotically stable if all the roots of the characteristic polynomial of the closed-loop system state matrix Ac = A − bg are located within the unit circle. The roots of the polynomial det(zIn − Ac ) = z n + (α − 1)z n−1 = z n−1 [z − (1 − α)]
(27)
are located inside the unit circle, if 0 < α < 2. Since no matter the choice of w the controller gain is positive and smaller than one, the system is stable, and no oscillations appear at the output. 2.2.3. Properties of the proposed policy Further in this section the properties of the proposed inventory policy will be formulated as three theorems. The first theorem defines the warehouse capacity, which needs to be provided to always accommodate the on-hand stock and incoming shipments. The second proposition imposes a constraint on the target stock level necessary to ensure that the entire demand is satisfied. Finally, the third theorem states that the order quantities determined from the algorithm are always nonnegative and bounded, which is a crucial requirement for the practical implementation of an inventory management scheme. Theorem 1. If policy (22) is applied in system (4), then the stock level is always upper-bounded, i.e.
∀ k≥0
y(kT ) ≤ yd .
(28)
Theorem 2. If policy (22) is applied in system (4), and the target stock level satisfies yd > dmax (np + 1/α)
(32)
then for any k ≥ np + 1 the stock level is strictly positive. Proof. It follows from (3) and (31) that y(k = np + 1) > 0. Let us assume that in some period l > np + 1 the stock level is positive. We shall demonstrate that y[(l + 1)T ] is also greater than zero. Since α ∈ (0, 1), then from (31) we get l −1 −
y[(l + 1)T ] ≥ α yd − α
u(lT ) = α yd −
l −1
− j =0
l −1
u(jT ) +
− j =0
h(jT ) .
(29)
(33)
It follows from (2) that for any k 0 ≤ h(kT ) ≤ dmax . Using assumption (32), we obtain y[(l + 1)T ] ≥ α[yd − dmax (np + 1/α)] > 0 which completes the induction proof.
(34)
Theorem 3. If policy (22) is applied in system (4), then the order quantities generated by the algorithm are always nonnegative and bounded, i.e.
∀ k≥0
0 ≤ u(kT ) ≤ max(α yd , dmax ).
(35)
Proof. For k = 0 we have u(0) = α yd . Let us assume that (35) is true for some integer l > 0. We will prove that the proposition holds also for l + 1. Using (29), we can present u[(l + 1)T ] in the following way
u[(l + 1)T ] = α yd −
Proof. The warehouse at the distribution center is empty for any kT ≤ Lp = np T . Hence, it suffices to show that the proposition is satisfied for any k ≥ np + 1. Let us assume that for some integer l ≥ np + 1, y(lT ) ≤ yd . We will demonstrate that the theorem is also true for l + 1. Substituting (3) into (22), we get
h(jT ) − h(lT ).
j=l−np
= α yd −
l −
u(jT ) +
l −
j =0
j =0
l−1 −
l−1 −
j =0
u(jT ) +
h(jT )
h(jT ) − α u(lT )
j =0
+ α h(lT ) = u(lT ) − α u(lT ) + α h(lT ).
(36)
Since α ∈ (0, 1) and 0 ≤ h(lT ) ≤ dmax , then 0 ≤ u[(l + 1)T ] ≤ max(α yd , dmax ). This ends the induction proof.
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
1987
Fig. 3. Model of inventory system with multiple supply alternatives.
So far in this paper we have considered the situation, when only a single supply option is used to refill the stock at the distribution center. However, a company implementing an inventory management scheme may select a different strategy, and use multiple supply alternatives for stock replenishment. Such a strategy may be employed to safeguard business continuity in the case of the supply source experiencing problems in order procurement, or to incite competition among the suppliers for quality and price benefits. In the next section, we analyze the situation of a company using multiple supply options (characterized, in principle, by different lead times) to acquire the stock.
Denoting the share of the supply options whose lead time equals jT (j = 1, 2, .∑ . . , nm ) in the total order quantity by aj = ∑ nm p:Lp =jT γp , we have j=1 aj = 1. Obviously, if no supply option is characterized by lead time jT, then the corresponding share aj in the total order is equal to zero. Using the notion of aj , we can rewrite (38) as follows y(kT ) =
We begin the design with extending the discrete-time model presented in Section 3.1 for the case of multiple suppliers participating in the stock replenishment at the distribution center. Next, the optimization problem is solved for the modified system, and the features of the obtained control policy are evaluated quantitatively. 3.1. System model We consider the inventory control problem illustrated in Fig. 3. Similarly as in the case considered in the previous section, the stock replenishment orders are issued at regular intervals kT , where T is the review period and k = 0, 1, 2, . . . , on the basis of the on-hand stock y(kT ), target stock level yd , and the history of previous orders. This time however, the order quantity u(kT ) can be split among m supply options (being different suppliers, or various transport alternatives for a single supplier) according to the company sourcing strategy. Consequently, in each review period k, γp of the total order is placed at supplier p (p = 1, 2, . . . , m), where ∑m γp is a real number from the interval [0, 1] satisfying p=1 γp = 1. Typically, the values of γp will be obtained using a supplier diversification function (e.g. Burke et al., 2007), or from a cost optimization algorithm, such as the one recently proposed in Dullaert et al. (2005). In the limit case, when γp = 1, a single supplier is selected to deliver the whole order quantity, while γp = 0 implies no replenishment from option p. Each non-zero order placed at supplier p is realized after a lead-time delay Lp , assumed to be a multiple of the review period, i.e. Lp = np T , where np is a positive integer. Without the loss of generality, we may order the supply alternatives according to their lead time in the following way (37)
Assuming that the warehouse is initially empty, i.e. y(kT < 0) = 0, and the first orders are placed at kT = 0, the dynamics of the on-hand stock for any k ≥ 0 may be expressed by the following equation m − k−1 − p=1 i=0
γp u(iT − Lp ) −
k−1 − i=0
h(iT ).
nm −
(38)
k −1 −
k−1 −
u(iT − jT ) −
i =0
h(iT )
i =0
k−j−1
aj
j =1
3. Production–inventory system with multiple suppliers
y(kT ) =
aj
j=1
=
L1 ≤ L2 ≤ · · · ≤ Lm−1 ≤ Lm .
nm −
−
u(iT ) −
k−1 −
h(iT ).
(39)
i =0
i =0
The system can be described in the state space exactly as in (4), with the state matrix modified as follows 1 0
.
A= .. 0 0
an−1 0
.. .
0 0
an − 2 1
.. .
0 0
... ... .. . ... ...
a1 0
.. . . 1
(40)
0
The order of the system with multiple suppliers n = nm + 1 = (Lm /T ) + 1 depends on the longest lead time Lm among the supply options. Similarly as in (6), for convenience of further analysis we can present the multi-source system in the alternative form
x1 [(k + 1)T ] = x1 (kT ) + an−1 x2 (kT ) +an−2 x3 (kT ) + · · · + a1 xn (kT ) − h(kT ) x2 [(k + 1)T ] = x3 (kT ) .. . xn−1 [(k + 1)T ] = xn (kT ) xn [(k + 1)T ] = u(kT )
(41)
where the desired system state xd = [xd1 xd2 . . . xdn ]T = [xd1 0 0 . . . 0]T , and xd1 = yd denotes the target stock level. 3.2. Proposed inventory management policy We seek for a controller minimizing cost functional (7). The desired controller, uopt (kT ) = −gx(kT ) + r, will be determined as a solution of (9) and (10) with the system matrix given by (40). 3.2.1. Solution to the optimization problem The first step in the optimization procedure is to find K from the Riccati equation (10) with matrix A defined by (40). For this purpose, we adopt the analytical method described in Section 2.2, which was applied for the system with one supplier. However, careful investigation of the elements of matrix K and vector g obtained for the multi-supplier case, confirmed by numerical computations, reveals a serious drawback of the obtained optimal control, namely, the controller generates negative order quantities and induces oscillations in the output variable (the stock level). To overcome these deficiencies and to make the scheme attractive to real inventory systems, we appropriately modify the optimization
1988
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
procedure by eliminating the terms ai aj ≪ 1 responsible for the negative quantities. As a result, instead of K we obtain its ˆ In a further part of the paper, we will approximation K. demonstrate that the proposed modification allows us formulate an enhanced control law, which indeed guarantees that the assigned order quantities are always nonnegative and oscillations are eliminated. Similarly as in the case of a single supplier, we begin solving ˆ Kˆ 0 = [kˆ ij ]n×n , and after the (10) with the most general form of K, first iteration we get kˆ 11 an−1 (kˆ 11 − w) kˆ 13 ˆ1 = K .. . kˆ 1n
an−1 (kˆ 11 − w) a2n−1 (kˆ 11 − w) kˆ 23
kˆ 13 kˆ 23 kˆ 33
... ... ... .. .
kˆ 2n
kˆ 3n
...
.. .
.. .
kˆ 1n kˆ 2n kˆ 3n . .. . kˆ nn
kˆ 11 = 1 + w
n −1 −
jaj + 1
− kˆ 11 − w
j =1
n −1 −
.
(42)
(43)
j =1
Solving for kˆ 11 , we arrive at
kˆ ± 11 =
w 2
n−1 −
jaj + 1 ± w(w + 4) /2
(44)
j =1
where only kˆ + 11 guarantees that all the principal minors and the
ˆ are nonnegative and Kˆ is semipositive definite. determinant of K ˆ Substituting K into the first equation in set (9), we obtain the ˆ (In + bbT Kˆ )−1 A, elements of vector gˆ = bT K
. . . (an−1 + · · · + a1 ) −1 n−1 − . × 1 − kˆ 11 − w jaj + 1
1
an − 1
(45)
j =1
ˆ+
Applying k11 and using the fact that
∑n−1 j =1
(46)
into (8), we arrive at uˆ (kT ) = −αˆ x1 (kT ) +
j −1 n − − j=2
an − i
xj (kT ) + rˆ .
(47)
i =1
Applying a similar iterative procedure, we determine constant rˆ from (9) as rˆ = αˆ yd , and the control law u(kT ) = uˆ (kT )
= αˆ yd − x1 (kT ) −
= α yd − y(kT ) −
− − j =2
j −1
an − i
xj (kT ) .
i =1
nm −
j −
(48)
aj
j =1
i =1
nm −
k −1 −
aj
u[(k − n + j − 1)T ]
u[(k − i)T ]
u(iT )
(49)
i=k−j
3.2.2. Stability analysis The characteristic polynomial of the closed-loop system state matrix for the multi-supplier case is the same as in the case of single supplier, i.e. for αˆ = α we have det(zIn − Ac ) = z n−1 [z −(1 − α)]. Since ∀w α is positive and smaller than one, all the roots of the polynomial are contained within the unit circle. Consequently, the closed-loop system with the applied control law is asymptotically stable, and since α < 1, no oscillations appear in the output signal. 3.2.3. Properties of the proposed inventory policy The properties of the developed policy will be formulated as three theorems and strictly proved. First, it will be demonstrated that the stock level does not exceed the warehouse capacity so that any cost associated with the expensive emergency storage is eliminated. Secondly, it will be shown that with the appropriately chosen controller parameters all of the demand at the distribution center is realized from the currently available stock, yielding the maximum service level. Finally, it will be proved that the generated control signal (the order quantities in subsequent review periods) is nonnegative and limited. Theorem 4. If policy (49) is applied in system (4) with matrix A given by (40), then the stock level is always upper-bounded, i.e. y(kT ) ≤ yd .
(50)
Proof. The warehouse at the distribution center is empty for any kT ≤ L1 = n1 T . Consequently, it is sufficient to show that the proposition holds for any k ≥ n1 + 1. Let us assume that for some integer l ≥ n1 + 1, y(lT ) ≤ yd . We will demonstrate that the theorem is also true for l + 1. Substituting (39) into (49), we get
u(lT ) = α yd −
nm −
aj
j=1
l −1 −
l −1 −
u(iT ) +
i =0
u(iT ) +
i =0
an − i
j =2
Remark 2. The gain constant of the obtained controller is identical to the gain of controller (22). Comparing (22) with (49) we conclude that the sub-optimal control law designed for the general case of multiple sourcing alternatives is equivalent to the optimal one when applied to the system with a single supplier.
= α yd − n
which represents an LQ sub-optimal controller, and completes the design of the supply policy for the considered multi-supplier inventory system.
k≥0
√ where αˆ = α = ( w(w + 4) − w)/2. Finally, substituting (46)
yd − y(kT ) −
∀
aj = 1, we get
gˆ = [1 an−1 (an−1 + an−2 ) . . . (an−1 + · · · + a2 ) 1] αˆ
u(kT ) = α
j−1 n − −
j=1
−1 jaj + 1
= α yd − y(kT ) −
ˆ 1 given by (42) into the expression on the Now we substitute K right-hand side of (10) and compare it with its left-hand side, which allows representing elements kˆ i3 (i = 1, 2, 3) in terms of kˆ 11 . ˆ can be expressed We proceed in this way until all the elements of K ˆ is given as functions of kˆ 11 and the system order. The final form of K as follows (for the sake of clarity we present only the upper part of ˆ (see Box I). the symmetric matrix K) ˆ Substituting K into (10), and comparing the first elements of the resultant matrices, we get
in the previous n − 1 periods as xj (kT ) = u[(k − n + j − 1)T ]. With x1 (kT ) = y(kT ) and αˆ = α we rewrite the control law in the following way
l −1 −
h(iT )
i =0 l −1 −
h(iT ) .
(51)
i =0
The stock level in period l + 1 can be expressed as
i =1
From (41) we can get the state variables xj (j = 2, 3, . . . , n) expressed in terms of the control signal generated by the controller
y[(l + 1)T ] = y(lT ) +
nm − j =1
aj u[(l − j)T ] − h(lT ).
(52)
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
2 −
an−1 (kˆ 11 − w)
kˆ 11
kˆ 12 . . . kˆ 1n
an−j [kˆ 11 − (3 − j)w]
j =1
a2n−1 (kˆ 11 − w)
an − 1
2 −
an−j [kˆ 11 − (n − j)w]
ˆ an − 1 an−j [k11 − (n − j)w] . j=1 .. . n −1 n −1 − − ˆ an − j · an−j [k11 − (n − j)w] j =1 n −1
an−j [kˆ 11 − (3 − j)w]
j =1
.. .
n−1 −
...
1989
kˆ 2n
...
.. .
..
kˆ 3n
...
.
−
j=1
j=1
Box I.
Substituting (51) for u[(l − j)T ], we get y[(l + 1)T ] = y(lT ) +
nm −
l−j−1
aj α yd −
−
j =1
−h(lT ) = y(lT ) + α yd
−
i =0
nm −
aj − α
j=1
−
l−j−1
u(iT ) +
nm −
−
h(iT ) − α
nm
i=0
h(iT )
i =0
∀ k≥0
l−j−1
−
aj
j =1
l−1
Theorem 6. If policy (49) is applied in system (4) with matrix A given by (40), then the order quantities are always nonnegative and bounded, i.e.
u(iT )
i=0
l −1
−
aj
−
h(iT ) − h(lT ).
(53)
i=l−j
j =1
It follows from (39) that the term in ∑ the square brackets in (53) n represents y(lT ). Consequently, since j=m1 aj = 1, y[(l + 1)T ] = yd − (1 − α)[yd − y(lT )]
−α
nm −
aj
l −1 −
h(iT ) − h(lT ).
(54)
i=l−j
j =1
Since α ∈ (0, 1) and h ≥ 0, y[(l + 1)T ] does not exceed yd . Using the principle of mathematical induction we conclude that the theorem is true for any period k ≥ 0. This ends the proof. Theorem 5. If policy (49) is applied in system (4) with matrix A given by (40), and the target stock level satisfies
yd > dmax
m 1−
T p=1
γp Lp +
1
(55)
α
0 ≤ u(kT ) ≤ max(α yd , dmax ).
(58)
Proof. For k = 0 we have u(0) = α yd . Let us assume that (58) is true for some integer l > 0. We will prove that the proposition holds also for l + 1. Taking similar steps as in (36) we can represent the control signal generated in period l + 1 as u[(l + 1)T ] = (1 − α)u(lT ) + α h(lT ). Since α ∈ (0, 1) and 0 ≤ h(lT ) ≤ dmax , then 0 ≤ u[(l + 1)T ] ≤ max(α yd , dmax ). This ends the induction proof. Remark 3. Theorems 2 and 5 define the warehouse storage space, which needs to be provided to ensure the maximum service level — Theorem 2 for the single supplier case and Theorem 5 for the multisupplier scenario. The required warehouse capacity is specified following the worst-case uncertainty analysis (for an instructive insight how this methodology relates to production–distribution systems see e.g. Blanchini, Pesenti, Rinaldi, and Ukovich (2000), Sarimveis et al. (2008)). However, since the values given in (32) and (55) scale linearly with the demand, in the situation when the mean value of the demand differs significantly from the maximum one, it may be convenient to substitute dmax with some positive dL < dmax . In such a case a 100% service level is no longer ensured, yet the average stock level, and as a consequence the holding costs, will be reduced.
then for any k ≥ nm + 1 the stock level is strictly positive.
4. Numerical example
Proof. It follows from (38), (54), and the theorem assumption that y(k = nm + 1) > 0. Let us assume that for some integer l > nm + 1 the stock level is positive. We shall demonstrate that y[(l + 1)T ] is also greater than zero. From (54) we get
We verify the properties of the inventory control policies proposed in this paper in a series of simulation tests. The first test demonstrates the features of the controller operating in the inventory setting with a single supply option. Next, we move on to the simulations of the multi-supplier configuration. The second test provides a comparison of the developed (sub-optimal) control law with the optimal one, which is obtained via numerical computation. It is shown how our improved scheme outperforms the LQ optimal control. Finally, in the third test, we investigate the response of controller (49) to different demand patterns, and quantify smoothening of ordering decisions related to demand variations.
y[(l + 1)T ] ≥ α yd − α
nm −
aj
j=1
l −1 −
h(iT ) − h(lT )
i=l−j
≥ α yd − dmax
nm −
jaj +
j =1
1
α
.
(56)
Recall that aj denotes the share of the∑ supply options lead ∑whose nm m time equals jT. Therefore, the sum ja = γ L j p p /T . j =1 p=1 Consequently, we can rewrite (56) in the following way
y [(l + 1)T ] ≥ α yd − dmax
m 1−
T p=1
γp Lp +
1
α
.
(57)
Using assumption (55), we get y[(l + 1)T ] > 0, which completes the induction proof.
Test 1. In the first scenario, we analyze the behavior of the controller designed for the inventory system with a single supplier and given by (22). The system parameters are chosen in the following way: discretization period T = 1 day, lead-time delay Lp = np T = 8 days, and the maximum daily demand at the distribution center dmax = 50 items. The actual demand in the interval of 90 days is shown in Fig. 4. Three simulations are run, each for a different value of weighting coefficient w . The controller gain and the adjusted target stock level used in the simulations are
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P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
600
Table 1 Controller parameters — the case of a single supplier. Gain α
Target stock level yd
0.2 1
0.358 0.618 1
545 > 540 485 > 481 455 > 450
∞
d(k) [items]
60
w = 0.2 w=1 y(k) [items]
Weighting factor w
400
200
0
40
w=∞
0
15
30 45 60 Review period [k]
75
90
Fig. 5. On-hand stock level.
20 60 30 45 60 Review period [k]
75
90
Fig. 4. Demand at the distribution center.
summarized in Table 1. The third simulation is performed for the limit case of w → ∞, i.e. the dead-beat controller. The stock level in the warehouse at the distribution center is shown in Fig. 5, and the order quantities, which are established by the controller, are illustrated in Figs. 6 and 7. It is apparent from the graphs that the order quantities calculated according to (22) are always nonnegative and bounded. From the stock evolution illustrated in Fig. 5 it can also be concluded that the allocated warehouse capacity is not exceeded, which means that no extra (and possibly expensive) storage space needs to be requested. Since the stock level never drops to zero (for kT ≥ (np + 1)T = 9 days), all of the demand is immediately fulfilled at the distribution center. This implies that any cost associated with a backorder discount or unsatisfied customer purchasing goods at another company is eliminated. As a result the maximum service level is achieved. Figs. 5–7 clearly indicate that the controller gain (and the system dynamics) depends on the choice of the weighting coefficient. As w increases, the controller reacts faster to the changes in the demand, and as w is reduced, its responsiveness drops. Moreover, it comes from Theorems 1 and 2 that the increase of w allows for a smaller warehouse capacity while still eliminating the stock-out and emergency storage costs. However, placing more impact on the output error elimination (a large w ) implies bigger values of the initial order quantities, which may be too high for certain suppliers. Therefore, in a majority of practical settings w = 1 would offer a good trade-off between fast system dynamics and supplier capacity constraints. The relation of the controller gain with respect to the weighting coefficient is shown in Fig. 8, and the dependence of the minimum storage space on the gain in Fig. 9. In the subsequent tests, we turn our attention to the inventory setting where multiple suppliers (or various transport alternatives) are exploited for order procurement. It is assumed that four supply options (m = 4) can be used to replenish the stock. The options are characterized by the following lead times, each expressed as a multiple of the review period T = 1 day: L1 = 5T , L2 = L3 = 8T , L4 = 10T . Assuming the order partitioning γ1 = 1/5, γ2 = 1/8, γ3 = 3/8, and γ4 = 3/10, we have a10 = 3/10, a8 = 1/2 and a5 = 1/5 (a1 = a2 = a3 = a4 = a6 = a7 = a9 = 0). Test 2. In the second scenario, we compare the operation of controller (49), which constitutes a sub-optimal solution to the LQ problem in the multi-source case, with the exact, numerically determined LQ optimal control law. Choosing w = 1, we get the
u(k) [items]
15
w=∞ w=1 w = 0.2
40
20
0
8
30
50 Review period [k]
70
90
Fig. 6. Order quantities.
600 w=∞ 400 u(k) [items]
0
w=1 w = 0.2 200
0
0
2
4 6 Review period [k]
8
10
Fig. 7. Order quantities in the initial phase.
1 0.8 Controller gain
0
0.6 0.4 0.2 0
0
5
10 15 Weighting coefficient w
20
Fig. 8. Controller gain vs. w .
gain of the sub-optimal controller α = 0.618 and the desired stock level (selected according to Theorem 5) yd = 485 > 481 items.
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
60
1000
40
yd [items]
u(k) [items]
1500
500
20
0 0 0.05
0.2
0.4 0.6 Controller gain
0.8
1
5
15
30
45 60 Review period [k]
75
90
Fig. 11. Order quantities.
Fig. 9. Warehouse capacity vs. gain. Table 2 Statistics of variables.
400
Variable (X )
300 u(k) [items]
1991
Xµ Xδ 2 Xδ XCV Xδ 2 /dδ 2 XCV /dCV
200 100
a) sub-optimal policy b) LQ optimal policy
0 –100 0
3
6 9 Review period [k]
12
15
Fig. 10. Inventory policy: (a) sub-optimal, (b) LQ optimal.
The parameters of the optimal controller were computed using the Matlab built-in function for discrete-time optimization dlqr. The order quantities calculated by the controllers are shown in Fig. 10. It can be seen from the plots, that the exact solution to the LQ problem (curve b in Fig. 10) exhibits overshoots and oscillations, even for the analyzed situation of the unperturbed system d(kT ) ≡ 0. It is not suitable for application in the considered setting, since the controller generates negative order quantities. The function u(kT ) obtained from our improved, sub-optimal controller (curve a), in turn, although close to the exact solution, never falls below zero. Moreover, the value of quality criterion for the sub-optimal case Japprox = J (ˆu) = 2110,041 exceeds only by 1.3% the optimal one Jdlqr = J (uopt ) = 2081,157. Furthermore, when the proposed scheme is applied, the curve representing the order quantities in subsequent review periods exponentially converges to stationary values without oscillations or overshoots, which is advantageous for establishing long-term predictions of the analyzed inventory system. Test 3. In the third scenario, we verify the operation of controller (49) in the presence of time-varying demand. First, we investigate the system dynamics in response to the demand function which was illustrated in Fig. 4. The order quantities are shown in Fig. 11 and the on-hand stock at the distribution center in Fig. 12. It is obvious from the graph in Fig. 11 that the quantities calculated by the algorithm are always nonnegative and upper-bounded. The policy quickly responds to the sudden changes in the demand trend (occurring in day 40 and 60) without oscillations and overshoots. Moreover, the stock level does not increase beyond the target value and never drops to zero (for k ≥ n4 + 1 = 11). This means that the warehouse capacity is not exceeded and the entire demand is satisfied. In consequence, no business opportunity is wasted and the maximum service level is guaranteed. Next, we test the controller performance in the situation when a stochastic component is incorporated in the demand pattern. The
Demand
Orders
Shipments
25 225 15 0.60 1 1
23.92 8.76 2.96 0.12 0.04 0.2
23.63 4.23 2.06 0.09 0.02 0.15
function used in the simulation is depicted in Fig. 13. Despite the demand frequently exceeding the estimated maximum (which was set at the level of 50 items per day), the order quantities generated by the proposed policy (shown in Fig. 14) are nonnegative and limited, and follow the low-frequency variations of the demand. Although the stock level (illustrated in Fig. 15) does not grow beyond 485 items, it is no longer guaranteed to remain positive and in several periods becomes zero. However, as can be learned from Fig. 16, only a few purchasing requests placed at the distribution center are left unanswered, and the service level is maintained close to its maximum. Finally, we demonstrate the smoothening properties of our strategy in the presence of a highly variable demand, in the simulation scenario when the mean demand value dµ = 25 items, its variance dδ 2 = 225 items2 , standard deviation dδ = 15 items, and coefficient of variation dCV = dδ /dµ = 0.6. Since the mean demand significantly differs from its maximum value, instead of dmax we apply dL = dµ = 25 items in the formula for the target stock level yd , and we appropriately decrease the required warehouse capacity (55). Although it is no longer guaranteed to satisfy all of the customer demand (the service level decreases to 98%), holding costs are nearly halved. The demand pattern together with the generated orders and the incoming shipments are illustrated in Fig. 17. It can be clearly seen from the plots that the demand fluctuations are blocked at the first stage of supply chain. As can be learned from the statistical analysis presented in Table 2, variance and coefficient of variation are diminished both on the forward and the reverse path. According to the most popular (Geary et al., 2006) measure of the bullwhip effect proposed by Chen, Drezner, Ryan, and Simchi-Levi (2000), which is the relation of variances of orders (forward path), or the incoming shipments (reverse path), to the demand, we obtain 25 and 50 fold attenuation of demand variations on the forward and reverse paths, respectively. Bigger attenuation on the reverse path is attributed to the additional benefit of order splitting among supply options with different lead times. This conclusion convincingly justifies the use of the quadratic performance index for the design of control strategies for periodic-review inventory systems.
1992
P. Ignaciuk, A. Bartoszewicz / Automatica 46 (2010) 1982–1993
60
Lost demand [items]
y(k) [items]
500
300
40
20
100 0
0 0
20
40 60 Review period [k]
80
0
15
30 45 60 Review period [k]
75
90
Fig. 16. Unfulfilled demand.
Fig. 12. On-hand stock level.
50
60 d(k), u(k), ur(k) [items]
d(k) [items]
dmax
40
20
40 30 20 10 d
0
0
15
30
45
60
75
0 15
90
Review period [k]
u(k) [items]
60
40
20
5
15
30
45 60 Review period [k]
75
90
Fig. 14. Order quantities.
500
y(k) [items]
45 60 Review period [k]
ur 75
90
Fig. 17. Demand d(k), orders u(k), and shipments received ur (k).
Fig. 13. Demand with stochastic component.
0
30
u
300
systems. As opposed to the typical solutions to the inventory control problem, we focused on the dynamical optimization of the stock replenishment process. The developed controllers quickly react to the changes in the demand without oscillations or overshoots, and generate smoothly varying order quantities. This facilitates efficient planning at the suppliers, and consequently reduces the risk of contract violation. Moreover, the proposed policies guarantee that all of the demand is satisfied from the on-hand stock at the goods distribution center, thus eliminating the necessity of backorders and the losses related to missed service opportunities. A closed-form solution of the linear–quadratic problem allows for a straightforward and computationally effective implementation of the designed controllers in real production–inventory systems. The multi-supplier strategy can also be easily integrated with other optimization algorithms, such as the supplier selection and quantity partitioning among the supply alternatives, e.g. Burke et al. (2007), or Dullaert et al. (2005). We believe that the presented framework can be effectively applied for a broad class of control problems in discrete-time systems with disparate feedback delays, e.g. traffic regulation in communication networks (Ignaciuk & Bartoszewicz, 2008), road traffic control etc. Acknowledgement
100 0
0
15
30
45 Review [k]
60
75
90
Fig. 15. On-hand stock level.
5. Conclusions In this paper, a rigorous control approach was applied to the design of inventory management strategies for periodic-review
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