Robust Sliding-Mode Supply Policy for Periodic-Review Inventory Systems with Time-Varying Lead-Time Delay

Robust Sliding-Mode Supply Policy for Periodic-Review Inventory Systems with Time-Varying Lead-Time Delay P. Ignaciuk*, A. Bartoszewicz** Institute of Automatic Control, Technical University of Łódź Stefanowskiego 18/22 St., 90-924 Łódź, Poland (e-mail: *[email protected],**[email protected]) Abstract: In this paper, the problem of inventory management in production-inventory systems is addressed from a control-theoretic perspective. We analyze the systems where retailers (or customers) are served from a common distribution point. The stock at the distribution center used to satisfy the unknown, time-varying retailers’ demand is replenished either by a single, or multiple suppliers according to the company sourcing strategy. The stock replenishment orders are issued at regular intervals and are realized with delay which differs among the supply options and may change in time. A new, nonlinear inventory policy based on the principles of sliding-mode control is proposed. The policy guarantees that the demand is always entirely satisfied from the on-hand stock (yielding zero lost-sales cost) and the warehouse capacity is not exceeded (what eliminates the risk of high-cost emergency storage). It is also shown to be robust with respect to the unknown, time-varying demand and fluctuations in the orders procurement delay. Keywords: sliding-mode control, inventory control, variable delay, discrete-time systems. 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). While the traditional approaches to inventory control primarily concentrate on the statistical analysis of long-term variables and cost optimization performed on the averaged values of various cost components, recently, the need has arisen for new solutions emphasizing the dynamical nature of goods flow process (Ortega and Lin, 2004). This is mainly attributed to rapidly changing market conditions and technological advances (e.g. Internet ordering, automatic warehouse review) introduced in production-inventory systems to facilitate business activity in this variable environment. Consequently, nowadays, a good inventory control policy should demonstrate not only appropriate steady-state performance but also good transient response. A natural choice for analysis and improvement of inventory systems dynamics is to apply the tools of control theory. A good, comprehensive review of earlier proposals is given in (Ortega and Lin, 2004). The results presented so far involve the use of stochastic-optimal (Neck, 1984), proportionalintegral-derivative (White, 1999), and recently proposed H∞norm-based (Rodrigues and Boukas, 2006) and predictive (Wang et al., 2007) control concepts to regulate either the production rate or the stock level. In this paper, we develop a new supply policy for periodicreview inventory systems using strict control-theoretic methodology. In the considered systems, the on-hand stock used to satisfy the unknown, time-varying demand placed by retailers (or customers) at a distribution center can be

replenished either from a single, or from multiple supply sources. The sources are characterized by different and (possibly) time-varying delay of procuring orders. The aim of the control action is to always satisfy the entire demand from the readily available stock and the currently arriving shipments, thus ensuring no backorder and lost-sales cost. For this purpose, we propose discrete-time sliding-mode (SM) control, which is well known to be efficient and robust regulation technique (Gao et al., 1995; Bartoszewicz, 1998; Young et al., 1999). Since a proper choice of the switching plane is the key part of the design of SM controllers (Bartoszewicz and Nowacka, 2007; Janardhanan and Kariwala, 2008; Ignaciuk and Bartoszewicz, 2008 & 2009), in this paper, we determine the parameters of the plane by solving a linear-quadratic (LQ) optimization problem. The performance index is parameterized with a tuning coefficient used to adjust the controller dynamics so that even in the case of sudden demand changes and lead-time delay variations the control signal does not fluctuate excessively in subsequent review periods. In contrast to the typical approaches to the LQ problem, e.g. (Chu et al., 2006; Janardhanan and Kariwala, 2008), which are mainly suitable for numerical implementations and systems with predefined dimensions, we solve it analytically and obtain the control law in a closed form. Next, in order to ensure the robustness with respect to variations of lead-time delay, a saturation block is introduced into the basic controller operation. It is shown that under the proposed nonlinear policy the stock level never exceeds the assigned warehouse capacity, which means that the potential necessity of expensive emergency storage outside the company premises is eliminated. We also demonstrate that the available stock is never entirely depleted, what guarantees maximum service level.

2. PROBLEM FORMULATION We analyze an inventory system where a common point serves as a distribution center for the network of retailers and customers. The retailers’ requests placed at the center constitute the unknown, bounded demand, which varies with time according to the current market conditions. These requests are realized from the goods available at the distribution center warehouse – the on-hand stock – which is replenished from a supply source (or from several of suppliers if order partitioning is implemented) with certain delay. 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 on-hand stock) with explicit consideration of the latency between placing an order at the supplier(s) and goods arrival at the center. The latency of fulfilling an order will be further referred to as the lead-time delay. The conceptual view of the orders and goods flow process is illustrated in Fig. 1.

definition of the demand is quite general and it accounts for any standard distribution typically analyzed in the considered problem. If there is 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) 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 ) ≤ d max .

(3)

The dynamics of the on-hand stock y depends on the amount of the arriving shipments and on the satisfied demand h. Assuming that the warehouse is initially empty, i.e. y(kT < 0) = 0, and the first orders are placed at kT = 0, the stock level for any k ≥ 0 may be calculated from the following equation m k −1

k −1

p =1 j = 0

j =0

y ( kT ) = ∑∑ γ p u ⎡⎣ jT − L p ( j ) ⎤⎦ − ∑ h ( jT ).

(4)

Fig. 1. Flow of goods and orders in inventory system.

Let us introduce a function ξ ( kT ) = ξ + ( kT ) − ξ − ( kT ) , where

The schematic diagram of the considered periodic-review inventory management system is illustrated in Fig. 2. 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), desired stock level yd, and the history of previous orders. The order quantity u(kT) can be split among m supply options 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 γp is a real number from the interval [0, 1] satisfying

ξ+ represents the sum of the surplus items which arrive at the distribution center by the time kT earlier than expected since their delay experienced in the neighbourhood of kT is smaller than the nominal one, and ξ– denotes the sum of items which should have arrived at the node by the time kT, but which cannot reach the center due to the delay larger than the nominal one. Assuming that the order quantity is bounded by some positive umax then on the basis of (1) the following constraint can be imposed on the values of ξ,

∑

m p =1

γ p = 1 . 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 with time-varying lead-time delay Lp(k) assumed to be a multiple of the review period, i.e. Lp(k) = np(k)T, where np(k) and its nominal value n p are positive integers satisfying

(1 − δ ) n p ≤ n p ( k ) ≤ (1 + δ ) n p .

(1)

m

we can rewrite (4) in the following way m k −1

(2)

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. Notice that this

(

)

k −1

y ( kT ) = ∑∑ γ p u jT − L p + ξ ( kT ) − ∑ h ( jT ), p =1 j = 0

(5)

j =0

which reflects the nominal system operation (items arriving due to the nominal delay) affected by the parameter perturbation ξ. Denoting the share of supply options whose nominal lead time equals jT (j = 1, 2,…, nm) in the total order quantity

The parameter δ ∈ [0, 1) represents the tolerance of delay variation. Notice that because of the time-varying delay certain orders may appear at the distribution center out of order and concurrently with other shipments, causing fluctuations of the stock level. Without the loss of generality, we may order the supply alternatives according to their nominal lead time L p = n pT in the following way L1 ≤ L 2 ≤ … ≤ L m −1 ≤ L m .

∀k ≥ 0 ξ ( kT ) ≤ ξ max = umax δ ∑ p =1 γ p L p . With this notation

by

a j = ∑ p: L

p

= jT

γ p , we have

∑

nm 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 (5) as follows nm

k − j −1

k −1

j =1

i =0

i =0

y ( kT ) = ∑ a j

∑ u ( iT ) − ∑ h ( iT ) + ξ ( kT ) .

(6)

The considered discrete-time system can also be described in the state space as x ⎣⎡( k + 1) T ⎦⎤ = Ax ( kT ) + bu ( kT ) + oh ( kT ) + pξ ( kT ) , y ( kT ) = qT x ( kT ) ,

(7)

Fig. 2. System model. 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 n × n state matrix, b, v, and q are n × 1 vectors ⎡1 an −1 ⎢0 0 ⎢ A=⎢ ⎢ ⎢0 0 ⎢⎣0 0

an − 2 … a1 ⎤ ⎡0⎤ ⎡ −1⎤ ⎡1 ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢0 ⎥ 1 … 0⎥ ⎢0 ⎥ ⎢0⎥ ⎢ ⎥ ⎥ b=⎢ ⎥ v=⎢ ⎥ p=q=⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 … 1⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎢⎣1 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣0 ⎥⎦ 0 … 0 ⎥⎦

(8)

and the system order n = nm + 1 = ( L m / T) + 1 depends on discretization period T and the longest nominal lead time L m among the supply options. 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 desired stock level. 3. CONTROLLER DESIGN In this section, a novel inventory management policy is developed using SM control methodology. First, the nominal system is considered, and the controller parameters are selected by minimizing a quadratic cost functional. Afterwards, the influence of perturbation on the controller performance is analyzed and an enhanced, nonlinear control law is formulated which is robust to delay variations. Let us introduce a sliding hyperplane described by the following equation s ( kT ) = cT e ( kT ) = 0,

(9)

where e(kT) = xd – x(kT) denotes the closed-loop system error and cT = [c1 c2 … cn] is such a vector that cTb ≠ 0. Substituting (7) into equation cTe[(k + 1)T ] = 0 with the disturbances h(kT) ≡ 0 and ξ(kT) ≡ 0, the following feedback control law can be derived u ( kT ) = ( cT b ) cT ⎡⎣ xd − Ax ( kT ) ⎤⎦ . −1

(10)

{

n ⎫ − ∑ ( c1an − j +1 + c j −1 ) x j ( kT ) ⎬ . j =3 ⎭

3.1 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 available stock) to the desired 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 ) = ∑ u 2 (kT ) + w [ yd − y (kT ) ] , k =0

2

(12)

where w is a positive constant applied to adjust the influence of the controller command and the output variable on the cost functional value. Small w reduces excessive order quantities, but lowers the controller dynamics. High w, in turn, implies fast tracking of the reference stock level 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 the value of the command), and the developed controller will become a dead-beat scheme. According to (Zabczyk, 1974), for time-invariant discretetime system (7) the control uopt minimizing criterion (12) can be presented as uopt (kT ) = −gx(kT ) + r ,

(13)

where g = bT K ( I n + bbT K ) A, −1

−1 r = bT ⎡K ( I n + bbT K ) bbT − I n ⎤ k , ⎣⎢ ⎦⎥

(14)

−1 k = − AT ⎡⎢K ( I n + bbT K ) bbT − I n ⎤⎥ k − wqyd , ⎣ ⎦

Using (8) we can rewrite (10) as u ( kT ) = cn −1 c1 ⎡⎣ yd − x1 ( kT ) − an −1 x2 ( kT ) ⎤⎦

It is clear from (11) that the properties of the inventory control policy will be determined by an appropriate choice of the sliding plane parameters c1, c2,..., cn. In the further part of the paper these will be selected in the optimization procedure.

(11)

and semipositive, symmetric matrix K (KT = K ≥ 0) is determined from the following Riccati equation K = AT K ( I n + bbT K ) A + wqqT . −1

(15)

The classical approaches for solving (15), as suggested in literature (Chu et al., 2006), are mainly suitable for numerical calculations and systems with predefined dimensions. In our case, however, it is desirable to formulate the control law in a closed form, which means that an analytical solution of the Riccati equation needs to be found for the system of order n. The analyzed system with arbitrary order distribution can be regarded as a generalization of the one considered in (Ignaciuk and Bartoszewicz, 2009), and optimization problem (13)–(15) can be solved similarly to the case of equal resource allocation addressed in that paper. Consequently, the elements of vector c are determined as, cT = ⎡⎣α α an −1 … α ( an −1 + … + a2 ) 1⎤⎦ cn ,

(16)

(17)

It follows from the system state-space description that the state variables xj (j = 2, 3,…, n) can be expressed in terms of the control signal generated by the controller in the previous With n–1 periods as xj(kT) = u[(k – n + j – 1)T]. x1(kT) = y(kT) we rewrite the control law as follows nm k −1 ⎡ ⎤ u ( kT ) = α ⎢ yd − y ( kT ) − ∑ a j ∑ u ( iT ) ⎥ . j =1 i=k − j ⎣ ⎦

(18)

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 current and the desired stock level (yd – y(kT)) decreased by the amount of open orders (the quantity already ordered at the suppliers but which has not yet arrived at the warehouse due to lead time). The nominal system under the operation of policy (18) is asymptotically stable, if all the roots of the characteristic polynomial of the closed-loop state matrix Ac = [In – b(cTb)–1cT]A are located within the unit circle. The roots of the polynomial det ( zI n − A c ) = z n + (α − 1) z n −1 = z n −1 ⎡⎣ z − (1 − α ) ⎤⎦

(20)

where ⎡

nm

⎣

j =1

ϕ ( kT ) = α ⎢ yd − y ( kT ) − ∑ a j

k −1

⎤

i=k − j

⎦

∑ u ( iT )⎥

(21)

and umax > dmax is a constant denoting the maximum order quantity. As a result, the control signal will always be limited to the interval [0, umax]. 3.3 Properties of the proposed inventory policy

and SM control law becomes n ⎡ ⎤ ⎛ j −1 ⎞ u ( kT ) = α ⎢ yd − x1 ( kT ) − ∑ ⎜ ∑ an −i ⎟ x j ( kT ) ⎥ . j = 2 ⎝ i =1 ⎠ ⎣ ⎦

if ϕ ( kT ) < 0, ⎧0, ⎪ u ( kT ) = ⎨ϕ ( kT ) , if 0 ≤ ϕ ( kT ) ≤ umax , ⎪ if ϕ ( kT ) > umax , ⎩umax ,

(19)

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. 3.2 Robustness issues The quantity calculation performed according to (18) is based on the nominal delay value, which is the estimate of the true (variable) latency, set according to the established contracting agreement with the supplier. The controller designed for the nominal system is robust with reference to delay smaller than the nominal one, yet may generate negative control signal in the presence of longer time lag. In order to eliminate this deficiency, we introduce the following modification into the basic algorithm

The properties of the designed policy will be formulated as two theorems. The first proposition shows how to adjust the warehouse storage space to always accommodate the entire stock and in this way eliminate the risk of (expensive) emergency storage outside the company premises. The second theorem states that the stock level remains positive irrespective of the delay variations in order procurement and demand changes. In consequence, maximum service level is achieved. Theorem 1: If policy (20) is applied, then the stock level at the distribution center is always upper-bounded, i.e.

∀ y ( kT ) ≤ y k ≥0

max

= yd + umax + ξ max .

(22)

Proof: It follows from the algorithm definition and the system initial conditions that the warehouse at the distribution center is empty for any k < (1 − δ ) n1 . Consequently, it is sufficient to show that the proposition holds for all k ≥ (1 − δ ) n . Let us consider some integer k ≥ (1 − δ ) n1 and the value of function φ at the time instant

lT. Two cases ought to be analyzed: the situation when φ(lT) ≥ 0, and the circumstances when φ(lT) < 0. Case 1: We investigate the situation when φ(lT) ≥ 0. Directly from (21), we get nm

y ( lT ) ≤ yd − ∑ a j j =1

l −1

∑ u ( iT ).

(23)

i =l − j

From the algorithm definition, in turn, it follows that u(lT) is always nonnegative, which implies y(lT) ≤ yd. This ends the first part of the proof. Case 2: In the second part of the proof we analyze the situation when φ(lT) < 0. First, we find the last instant l1T < lT when φ was nonnegative. According to (21) φ(0) = αyd > 0, so the moment l1T indeed exists and the value of y(l1T) satisfies inequality analogical to (23). Then, the stock level at the time instant lT can be expressed as nm

l − j −1

l −1

j =1

i = l1 − j

i = l1

y ( lT ) = y ( l1T ) + ∑ a j

∑ u ( iT ) + ξ ( lT ) − ∑ h ( iT ).

(24)

Since y ( l1T ) ≤ yd − ∑ jm=1 a j ∑ i1= l − j u ( iT ) , we get

The value of φ(l1T) < umax. Consequently, following similar reasoning as presented in (27) and (28), we arrive at y(l1T) > 0 and

l −1

n

1

nm

y ( lT ) ≤ yd − ∑ a j j =1

l1 −1

l − j −1

nm

∑ u ( iT ) + ∑ a ∑ u ( iT ) + ξ ( lT )

i = l1 − j

l −1

− ∑ h ( iT ) ≤ yd + i = l1

j =1

∑

j :l − j > l1

aj

j

i = l1 − j

l − j −1

l −1

i = l1

i = l1

(25)

∑ u ( iT ) + ξ ( lT ) − ∑ h ( iT ).

∑

a j ∑ i = l u ( iT ) ≤ u ( l1T ) . On the basis of (3) l − j −1

j :l − j > l1

umax

α

nm

− ∑ aj j =1

l1 −1

∑

i = l1 − j

1

and condition ξ(lT) ≤ ξmax we obtain the following estimate of the stock level y(lT) ≤ yd + umax + ξmax, which concludes the second part of the reasoning and completes the proof of Theorem 1. 

+ ξ ( lT ) − ∑ h ( iT ) = yd − i = l1

nm

+∑ a j j =1

l −1

nm

nm

l − j −1

j =1

i = l1 − j

u ( iT ) + ∑ a j

l −1

The algorithm set a nonzero quantity for the last time before lT at l1T, and this value can be as large as umax. Consequently the sum

y ( lT ) > yd −

umax

α

∑ u ( iT )

+ u ( l1T )

l −1

l −1

i =l − j

i = l1

(29)

∑ u ( iT ) − ∑ a ∑ u ( iT ) + ξ ( lT ) − ∑ h ( iT ).

i = l1 +1

j =1

j

Theorem 2: If policy (20) is applied, and the demand stock level satisfies

Recall that l1T was the last instant before lT when the controller calculated quantity smaller than umax. This quantity could be as low as zero. Afterwards, the algorithm generates the maximum order quantity, and the first sum in (29) reduces to umax(l – 1 – l1). Since for any k, u(kT) ≤ umax, the second sum is upper-bounded by

⎛ m ⎞ yd > umax ⎜ ∑ γ p n p + 1/ α + 1⎟ + ξ max , ⎝ p =1 ⎠

assumption (26) and the fact that h(kT) ≤ dmax, we may estimate stock level (29) in the following way

then

for

kT ≥ (1 + δ ) nmT + Tmax ,

any

(26) where

Tmax = Tymax / (umax – dmax), the stock level is strictly positive. Proof: The theorem assumption implies that we deal with the time instants kT ≥ (1 + δ ) nmT + Tmax . Considering some l ≥ (1 + δ ) nm + Tmax / T and the value of signal φ(lT), we may

distinguish two cases: the situation when φ(lT) < umax, and the circumstances when φ(lT) ≥ umax. Case 1: First, we consider the situation when φ(lT) < umax. Directly from the definition of function φ, we obtain nm

y ( lT ) > yd − umax / α − ∑ a j j =1

l −1

∑ u ( iT ).

(27)

i =l − j

The order quantity is always bounded by umax, which implies ⎛ nm ⎛ m 1⎞ 1⎞ y ( lT ) > yd − umax ⎜ ∑ a j j + ⎟ = yd − umax ⎜ ∑ γ p n p + ⎟ . α⎠ α⎠ ⎝ j =1 ⎝ p =1

(28)

Using assumption (26), we get y(lT) > 0, which concludes the first part of the proof. Case 2: We investigate the situation when φ(lT) ≥ umax. First, we find the last moment l1T < lT when signal φ was smaller than umax. It comes from Theorem 1 that the stock level never exceeds the value of ymax. Furthermore, the demand is limited by dmax. Thus, the maximum interval Tmax during which the controller may continuously generate the maximum order quantity umax is determined as Tmax = Tymax / (umax – dmax), and the instant l1T does exist. Moreover, from the theorem assumption we get l1T ≥ (1 + δ ) nm , which means that by the time l1T the first items from all the suppliers have already reached the distribution center, no matter the delay variation.

umax ∑ jm=1 a j j = umax ∑ p =1 γ p n p . Finally, using the theorem n

m

y ( lT ) > ξ max + umax ( l − l1 ) + ξ ( lT ) − d max ( l − l1 ) .

(30)

Finally, since ξ(lT) ≥ – ξmax, and by assumption umax > dmax, we obtain y ( lT ) > ( umax − d max )( l − l1 ) > 0.

This completes the proof of Theorem 2.

(31) 

4. NUMERICAL EXAMPLE We investigate the properties of the inventory policy proposed in this paper in a simulation test performed for the system described in Section 2. It is assumed that four supply options (m = 4) are used for stock replenishment. The options are characterized by the following nominal lead times, each expressed in the multiples of the review period T = 1 day: L1 = 6T , L 2 = 8T , L3 = 9T , and L 4 = 12T . It is assumed that the true delay can differ from the nominal one by as much as δ = 1/3. The actual delay experienced by orders from supplier p fluctuates according to Lp ( k ) = ⎢ ⎡1 + δ sin ( 2π kT / n p ) ⎤ n pT ⎥ , where ⎢⎣ f ⎥⎦ denotes the ⎦ ⎣⎣ ⎦ integer part of f. Assuming the order partitioning γ1 = γ4 = 1/6, γ2 = 1/4, and γ3 = 5/12, we have a6 = a12 = 1/6, a8 = 1/4 and a9 = 5/12 (a1 = a2 = a3 = a4 = a5 = a7 = a10 = a11 = 0). The maximum daily demand dmax is set as 100 items and the maximum order quantity as 110 items. This results in ξmax equal to 321 items. The actual demand follows the pattern depicted in Fig. 3 (curve a).

Choosing w = 1, we get controller gain α = 0.618 and the desired stock level (selected according to Theorem 2) yd = 1575 > 1571 items. The orders generated by the proposed policy are shown in Fig. 3 (curve b), and the resultant on-hand stock in Fig. 4. It is clear from the graphs that the policy quickly responds to the sudden changes in the

demand (occurring in day 30 and 60), the stock does not increase beyond 2006 items (as dictated by Theorem 1) and it never drops to 0 (for k ≥ n4 + 1 = 12). This means that the warehouse capacity is not exceeded and the entire demand is satisfied. In consequence, no business opportunity is wasted and maximum service level is guaranteed.

Fig. 3. Demand (a), order quantities (b).

Fig. 4. On-hand stock level. 5. CONCLUSIONS In this paper, a novel supply policy for periodic-review inventory systems with multiple supply alternatives was designed using control-theoretic methodology. The obtained policy guarantees that all of the demand is satisfied from the on-hand stock, thus eliminating the risk of missed service opportunities and the necessity for backorders. It demonstrates robustness to both the unpredictable demand changes and the delay variation in order procurement. As the quantity computation involves only simple operations, it can be very efficiently implemented in real systems. ACKNOWLEDGMENT This work has been financed by the Polish State budget in the years 2008–2010 as the research project N N514 300035 “Design of switching surfaces for the sliding mode control of dynamic plants”. P. Ignaciuk is a scholarship holder of the project entitled “Innovative Education without Limits – Integrated Progress of the Technical University of Łódź” supported by the European Social Fund. P. Ignaciuk also gratefully acknowledges financial support provided by the Foundation for Polish Science (FNP).

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