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2 Some Convexity and Sensitivity Properties of the (R, T ) Inventory Control Policy for Stochastic Demand Uday S. Rao Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA March 1994, Revised June Abstract We show that long-run average cost function for the (R, T ) policy with normally distributed demand has a structure similar to that of the corresponding (Q, r) model so that many of the useful properties of the latter model are applicable. In particular, (i) the optimal cost is relatively insensitive to the choice of the reorder interval, T, provided a near-optimal R corresponding to the chosen T is used. (ii) The relative percentage increase in costs due to use of a suboptimal T obtained via a deterministic analysis is at most 6.125%. In addition we show that, (i) Use of an (R, T ) policy instead of the optimal policy increases costs by at most 41.42%. (ii) In computational experiments, the (R, T ) policy exhibits substantially superior insensitivity and near-optimality properties when compared to the above worst case guarantees. We also characterize certain problem instances for which the empirical performance of the (R, T ) policy may be expected to deteriorate. (iii) Simple Power-of-Two, (R, T )-based heuristics for the multi-item joint replenishment problem and the single-item, two-stage serial system have a worst-case performance guarantee of 1.5 and respectively. 1

3 1 Introduction One of the commonly used inventory control doctrines is the \order-up-to" or (R, T ) policy in which the inventory status is reviewed every T periods and, if required, an order is placed to increase the inventory position (on-order + on-hand? backorders) up to R. Such policies have been successfully exploited for situations with deterministic demand where both order quantities and reorder intervals can be maintained constant (Roundy, [R86]). When demand is stochastic, typically either the order quantity remains constant [(Q, r) policy] or the reorder interval remains constant [(R, T ) policy]. We focus on the latter model treating both R and T as decision variables. For some early references on this problem, see Hadley & Whitin [HW63] or Silver & Peterson [SP85]. Most recent research on the single-item, single-stage, periodic review, stationary, stochastic demand inventory problem has focused on the optimal (S, s) policy. However, since this optimal policy has a complex cost function, computation and analysis of the optimal parameters is often impractical. We analyze the simpler but sub-optimal (R, T ) policy as a means for controlling inventory of certain classes of items. Note that this policy allows control of the timing of order placements which facilitates coordination of order epochs in multi-item, multi-echelon systems, such as the Joint Replenishment Problem (JRP). Computational experimentation, reported in Atkins & Iyogun's [AI88], on the use of a \power-oftwo", (R; T )-based heuristic for joint replenishments has been quite encouraging. Thus, our research is partially motivated by the possibility of using (R; T ) policies to extend some of Roundy's algorithms for complex deterministic inventory systems ([R86]) to their stochastic counterparts. We assume that time is continuous, that the order delivery lead time is a known constant and that backorder (resp. holding) costs are charged proportional to the long-run, timeweighted, average backorders (resp. on-hand stock). Demand is modeled using the Poisson process or the continuous normal approximation. Our analysis of the (R, T ) policy is akin to Zheng's work ([Z92]) on the (Q; r) policy. Setting r = r(q), the optimal reorder point corresponding to Q, Zheng has shown that the long-run average cost function, C(Q; r), is relatively insensitive to the choice of Q, and that the optimality conditions for Q and r have an appealing and intuitive graphical interpretation. In this paper, we extend Zheng's results to the (R, T ) policy and study their potential implications for the joint replenishment problem and a two stage serial inventory system. 2

4 In Section 2, we demonstrate that the sensitivity results proven by Roundy and by Zheng, respectively for deterministic systems and for the (Q; r) model, extend naturally to the (R; T ) model. Further, use of an Economic Order Interval from a deterministic analysis can itself generate a good approximation for T, provided the optimal order-up-to level corresponding to the chosen T is used. We also present a worst-case bound on the relative increase in cost of using a simple, periodic review, (R; T ) policy instead of the more responsive, continuous review (Q; r) policy. Some numerical experimentation is conducted in Section 3 to evaluate the empirical insensitivity and near-optimality of the (R, T ) policy when demand is Poisson. We conclude by illustrating possible extensions of this research to certain multi-item / multistage systems. 1.1 The Demand Model et X(t) be a random variable denoting cumulative demand over an interval of length t. We assume that this demand process, X(t), is non-decreasing in t and has stationary and independent increments. Further, for simplicity, demand is assumed to be continuous, so that the parameters R and T can take any non-negative, real values. However, we note that the only monotonic stochastic process with stationary, independent increments and continuous sample paths is the deterministic process, X(t) = t (see Feller [F71], Chapter IX). Consequently, the continuous model must be viewed as a convenient tool for approximating the real system. A commonly used stochastic demand model with continuous sample paths approximates demand by assuming that X(t) is independent normal with mean t and variance 2 t. That is, the cumulative demand process, X(t), is a Brownian Motion with drift. In our analysis we adopt this model and denote the density function of demand by (x; t) = x? t p t! = " p 1 2t exp? (x? t)2 2 t # : (1) Since this demand process is not monotonic in t, we assume that there is a smallest time unit, say t = 1, and that is suciently larger than, e.g., > 3:5, so that t p t for all t 1, and the probability of negative demand is suciently small. 1.2 The Cost Function and Optimality Conditions et denote the xed lead time for order delivery from a perfectly reliable, external supplier. The cost parameters are K, which represents order / setup cost, and h, p, which denote 3

5 respectively, the inventory holding and backordering cost rate. Under the general conditions on demand described in Serfozo & Stidham, [SS78], the long-run average cost function for the (R, T ) policy with continuous demand is given as follows: C(R; T ) = K T + 1 T Z T 0 G(R; + t)dt; (2) where, following Zheng [Z92], we have pooled the inventory holding and backordering costs into the function G. Clearly, if x + = max(0; x), G(R; t) = he[(r? X(t)) + ] + pe[(x(t)? R) + ] = h(r? t) + (h + p) Z 1 x=r (x? R) (x; t)dx: Since, in general, C(R; T ) may not be jointly convex in R and T, we note the following: Proposition 1.1 G(R; + t) is convex in R for xed t and convex in t for xed R. Proof. The required results follow 2 G(R; + t) = (h + p) (R; + t) 2 G(R; + t) = (h + p) (R; + t)[? 2 +t + ( R +t + )2 ] 0; where the last non-negativity follows from R 0 and the assumption that t p t. The above proposition implies that C(R; T ) is convex in R or T if the other variable is xed. Consequently, the optimal R and T can be found by a simple iterative search procedure. et R(T ) = argmin R C(R; T ) denote the optimal R corresponding to reorder interval T. By equation (3) below, R(T ) is unique whenever there is a positive probability of positive demand over any non-zero time period. Further, R(T ) may be obtained by solving a \newsboy" problem with random demand having density 1 T )=@R = 0 R +T so that Z +T Z h xr(t G(R; t)dt = h R RxR (x; t)dxdt? p R +T Z +T Z (x; t)dxdt = p xr(t ) By dierentiating equation (3) we can show that R 0 (T ) = R +T R T R xr(t ) [ (x; + T )? (x; t)] dxdt T R +T (R(T ); t)dt (x; t)dt. That is, at R(T ), R xr (x; t)dxdt; (x; t)dxdt = hpt h + p : (3) > 0; (4) where the inequality follows because R xr(t )[ (x; + T )? (x; t)]dx > 0 for all t < T +. Thus, as expected, R(T ) is non-decreasing in T and R(T )! 1 as T! 1. By equations (3) and (4), we see that R 0 (T ) R +T (R(T ); t)dt = R xr(t ) (x; + T )dx? h p+h, 4

6 so that, for normal demand with () denoting CCDF of a standard normal distribution: ( lim T!1 R0 (T ) lim T!1 R(T )?(+T p ) +T R(T )?(+T p ) +T ( )? h p+h )= = 1? h p+h 1= = p p + h ; (5) where we have used the fact (Appendix 4, [HW63]) that R +T (R; t)dt ( R?(+T ) )=, R(T )?(+T ) and that ( p )! 1 as T! 1. +T Now dene C(T ) = C(R(T ); T ) and U(T ) = R +T G(R(T ); t)dt so that C(T ) = Proposition 1.2 The cost function C(T ) can be written in the form: C(T ) = K T + 1 T where u() is a non-decreasing, convex function dened by u(y) = G(R(y); + y). p +T K+U (T ). T R T 0 u(y)dy, Proof. Observe that U(0) = 0, U(T ) is dierentiable and U 0 (T ) is continuous. Consequently, by the rst fundamental theorem of calculus, we can write U(T ) = R T 0 u(y) = U 0 (y) = G(R(y); + y) + R 0 (y) R +y u(y)dy, G(R(y); t)dt = G(R(y); + y), since the integral is zero at R(y). By Proposition 6.1 (Appendix) u 0 (T ) 0, so u(t ) is non-decreasing. Convexity of u(t ) follows from Proposition 6.2. Proposition 1.3 C(T ) is convex. If T minimizes C(T ), then C(T ) = u(t ) and hence, by Proposition 1.2, K + R T 0 u(y)dy = T u(t ). R T Proof. By Proposition 1.2, C 0 (T ) =? K? T 2 0 u(y)dy + u(t ) T 2 T 2u(T ) + u0 (T ) T 2 T and C 00 (T ) = 2K T R T 0 u(y)dy T 3?. Dene f(t ) so that C00 (T ) = 2K T 3 + f (T ) T 3. Since f(0) = 0 and f 0 (T ) = u 00 (T ) 0, f(t ) 0 for all T 0. Thus C 00 (T ) 0, so C(T ) is convex. Now since C 0 (T ) =? C(T ) T + u(t ) T, at T = argmin C(T ), we have C 0 (T ) = 0 or C(T ) = u(t ). We interpret u(y) as the rate at which expected inventory costs accumulate at time + y, if the order-up-to level is R(y). By convexity of C(T ), the optimal T may be found using a simple one-dimensional golden section search. An interesting graphical interpretation of the optimality condition for T may be developed as follows: A typical graph of u(t ), for continuous demand, is shown in Figure 1, where we have indicated that, as per Proposition 1.3, u(t ) = C(T ) and the area under u() between 0 and T measures the inventory cost per cycle. The total cost per cycle is given by the rectangular area u(t )T and is comprised of a setup cost portion, K, which lies above u(t ), and the inventory cost per cycle. (Note: The function u d (T ) in Figure 1 will be dened shortly.) 2 Sensitivity Properties of C(R; T) All sensitivity properties of interest follow from Proposition 1.2 and the following: 5

7 u(t) C(T*) K u (T) d Inventory Costs / Cycle T* T Figure 1: Graph of u(t ) showing T Proposition 2.1 et H = hp h+p. Then, for normal demand, u0 (T ) lim T!1 u 0 (T ) = H. Proof. By (5), lim T!1 R 0 (T ) p. By equation (12) in the Appendix, h+p u0 (T ) = "! # (? R 0 (T )) (h + p)? h + (h + p) R(T )? ( + T ) p + T Hence, lim T!1 u 0 (T ) (? p hp )[(h + p) 1? h] + 0 = h+p h+p u 0 (T ) lim T!1 u 0 (T ), which yields the required result. 2 p + T R(T )? ( + T ) p + T! = H. By convexity of u(t ), et u d (T ) denote the inventory cost rate function, u(t ), for the deterministic demand model. From Hadley & Whitin [HW63], we know that C d (T ) = K T + Z T 0 u d (y)dy for u d (T ) = hpt h + p = HT; with u0 d(t ) = H: (6) By the convexity of u(t ) as a function of X( + T ) & T and Jenson's inequality ([R70]), u(t ) u d (T ) = HT: (7) We now use the above results to extend the sensitivity properties from Roundy [R86] and Zheng [Z92] to the (R; T ) inventory model. 2.1 The Main Sensitivity Results Given the structure of C(T ), the following two results follow directly from the arguments used by Zheng [Z92] to prove the corresponding results for the (Q; r) model. emma 2.2 For any > 0, T > 0, Z T T u(y)dy 2? 1 2 T u(t ) 6

8 Theorem 2.3 ((R; T ) counterpart of Roundy [R86] & Zheng [Z92]) If T minimizes C(T ) then, for any > 0, C(T ) C(T ) 1 2 ( + 1 ) p We now compare the (R, T ) policy having optimal parameter values with an (R( 2T d ), p 2T d ) policy, where T d is the optimal solution to a deterministic Economic Order Interval problem. From Hadley & Whitin [HW63], s 2K T d = H ; R d = + pt d h + p : (8) R( p 2T d ) solves equation (3). Since u d (T ) u(t ) and 0 u 0 (T ) u 0 d(t ), note that, as per Figure 1, u(t ) requires a larger T to trap an area K above it than does u d (T ). Hence, T T d. R T R T d 0 u(y)dy. Consequently, we have Further, by the monotonicity of u(y), 1 T 0 u(y)dy 1 T d the following result, whose proof follows from Gallego's [G93] proof of the corresponding result for the (Q, r) policy. Theorem 2.4 If T minimizes C(T ), and T d dened by equation (8) then, T T d C(T d )? C(T ) C(T 12:5% ) is the optimal deterministic order interval and, as in Zheng [Z92] and Gallego [G93], resp.: and C( p 2T d )? C(T ) C(T ) 6:125% We now bound the relative dierence in cost of using an (R; T ) policy vs. cost of using the optimal policy (for the continuous review case, the (Q; r) policy is optimal) emma 2.5 (Gallego [G93]) For the (Q; r) model with G(y) def = E[h(y?X()) + +p(x()? y) + ], G(y ) = min y G(y), and long-run average cost C(Q; r) = 1 [K + R r+q Q r G(y)dy], we have C(Q ) = C(Q ; r(q )) q 2KH + G(y ) 2 : emma 2.6 et G(y ) be as dened in the above lemma. Then C(T ) = C(R(T ); T ) p 2KH + G(y ) Proof. Consider the decomposition C(T ) = C 0 (T ) + u(0) where C 0 (T ) = 1 T [K + R T 0 u 0 (y)dy] with u 0 (y) = u(y)? u(0). Then T C 0 (T ) = T u 0 (T ) 2K = T d p 2KH; where the inequality follows from the convexity of u 0 (y). Thus C 0 (T ) T d T p 2KH. Since T d T, C(T ) p 2KH + u(0). Now observe that u(0) = G(R(0); ) = G(y ). In the above lemma, it may be noted that u(0) is a constant which corresponds to the \newsboy" cost resulting exclusively from variability in demand. This cost is incurred even for K = 0, in which case we place orders as often as required. 7

9 Theorem 2.7 For the continuous demand model, in the worst-case, C(R(T ); T )? C(Q ; r(q )) C(Q ; r(q )) 0:4142 That is, use of an (R; T ) policy can cost at most % more than use of an (r; Q) policy. Proof. Observe that C(R(T ); T ) p 2KH + G(y ) p 2:0 q 2KH + G(y ) 2 1:4142 C(Q ; r(q )) Note that a weak worst-case result such as Theorem 2.7 is of little practical signicance. However, our computational experimentation suggests that the actual dierence between costs of the two policies is often of the order of a few percent. Furthermore, it is sometimes possible to guess if this dierence will be signicant by examining the specic problem data and extrapolating from our computational tests. We conclude this section by presenting a worst-case bound on the ratio C(R; T )=C(Q ) for a certain sub-optimal choice of R. et T (R), T d (R) denote the optimal T corresponding to order-up-to-level R, resp. for the stochastic and deterministic demand problem. As before, it is possible to show that T (R) is monotonic increasing in R (i.e., T 0 (R) 0) and that T (R) T d (R). et e C(R) = C(R; T (R)). Proposition 2.8 For e R = r + Q, e C(e R)?C(Q ) C(Q ) 1:0. Proof. The logic used in this proof is similar to that used for results pertaining to C(T ). As in Proposition 1:3 (by denition (2) and the optimality condition for T ( e R)), we have ec( e R) = G( e R; +T ( e R)). Since G( e R; +t) is convex in t, as in emma 2.6, G( e R; +T ( e R)) G( e R; )+ 2K T (e R) = G(r +Q )+ T d T (e R) p 2KH. Since e R R d and T ( e R) T (R d) T d (R d) = T d, the required result follows from the well-known facts ([Z92]) that C(Q ) = G(r + Q ) and C(Q ) p 2KH. 3 Computational Results For consistency and comparability, the set of test problems we experimented with are the same as the 135 problems used in Zheng, [Z92]. In all problem instances, the lead time is normalized to = 1. Demand is Poisson with rate taking on values 5, 25 and 50; K is set to be 1, 5, 25, 100, or 1000; h is 1, 10 or 25; p is 5, 10, 25 or 100. As is common in practice (Zheng, [Z92]), we restrict attention to p h. Note that Zheng's data set contains 45 problem instances with p < h. In our experimentation on these problems, results similar to the case of p h were obtained. Table 1 lists the computed parameter values and costs 8

10 for three typical numerical problem instances which, for each K value, dier only in the h=p ratio. Table 1: Some Numerical Examples: Policy Parameters and Costs %Dev1 = C(R(T d ); T d )? C(R(T ); T ) C(R(T ); T ) 100; %Dev2 = C(R(T ); T )? C(r ; Q ) C(r ; Q ) Poisson Demand, = 50, = 1, h = 10, p = 25. K T d T R(T d ) R(T ) r Q C(R ; T ) %Dev1. %Dev Poisson Demand, = 50, = 1, h = 20, p = 20 K T d T R(T d ) R(T ) r Q C(R ; T ) %Dev1. %Dev ? Poisson Demand, = 50, = 1, h = 15, p = 100 K T d T R(T d ) R(T ) r Q C(R ; T ) %Dev1. %Dev In Table 1, % Dev1 represents the relative increase in cost due to \error in optimization of T ", i.e., due to incorrectly choosing T = T d ; % Dev2 is the relative \error in policy" which results due to use of an (R; T ) policy instead of a (Q; r) policy. In general, the relative \error in optimization of T " is small (at most 3.0%). The relative dierence between using an (R; T ) policy and a (Q; r) policy is more pronounced, but still within six percent for the tabulated problem instances. Further experimentation has revealed that the cost of using an (R; T ) policy is relatively insensitive to the T value (rounding to a power-of-two multiple of a base planning period did not increase costs substantially). However, it is more important to choose the optimal R for the specied T. For the data in Table 1, % Dev1 is highest for K = 5 and % Dev2 is highest for K p. Although it may not be obvious from Table 1, we have also found that both % Dev1 and % 9

11 Dev2 tend to become smaller for very small or very large K. When compared to % Dev1, a larger change (increase or decrease) in magnitude of K is required to make % Dev2 \small." In addition, the dierence between % Dev1 and % Dev2 appears to be smaller for very small K, for instance K = 1 in Table 1. Our preliminary experimentation suggests that: 1. Whether demand is modeled using a continuous distribution or a discrete distribution, it is relatively easy to compute the optimal R and T using a simple search technique such as the golden section method. (Requisite upper bounds on T follow from the fact that HT C(R ; T ) M where M is a demand distribution-free upper bound on C(R ; T ) which may be obtained as in [G93].) 2. As expected, T and R(T ) increase when K or p increases or when h decreases. However, no discernible monotonicity in parameters/costs resulted from increasing the h=p ratio. In general, R(T ) is close to r + Q and T is close to Q =. The relative dierence in parameter values, T d & T and R(T d ) & R(T ), decreases as K increases. (This is consistent with Figure 1 which indicates that T T d for large K. Note that for the Poisson demand model u(t ) still retains the same general shape depicted in Figure 1, however, since R takes on discrete values, u(t ) is non-smooth.) 3. The actual gap in long-run average costs between using the optimal T and the suboptimal T d is much smaller in practice than the worst case bound of 12.5%. In all cases, a near-optimal order-up-to level, R(T ), corresponding to the reorder interval T must be used. (Note that although use of a reorder interval of p 2T d results in an improved worst-case performance of %, in practice p 2T d may be far from optimal, particularly for relatively large values of K for which T T d.) 4. The cost of using the periodic review (R; T ) policy with optimal parameters is often reasonably \close" to the cost of a continuous review (Q; r) policy. From our computational testing (on dierent sets of test problems), it appears that the performance of the (R, T ) policy deteriorates for small values of and with h K p, where denotes \less than by an order of magnitude." For instance, with h = 1, K = 10, p = 100, = 1, = 1, the (R, T ) policy was about 41 % worse than a (Q, r ) policy. Results for the 135 problems tested may be summarized as follows: The error in optimization, indicated by the percent deviation, % Dev1, was within 0.25% for over 50% of the 10

12 problems and between 0.25% and 0.75% for 20% of the problems. All the problems had % Dev1 within 3.0%, with an average % Dev1 value of 0:63%. As expected, the relative error in policy was much higher with % Dev2, having min, mean and max values of 0.36%, 7.31% and 23.32% respectively. Overall, the computational testing supports our belief that, in practice, an (R; T ) policy may oer advantages of co-ordination which make it viable for controlling inventory of certain classes of items, particularly in multi-item, multi-echelon systems. 4 Discussion and Extensions 4.1 Compound Poisson Demand In modeling costs for the (R, T ) policy we have implicitly assumed that demand follows a evy process (i.e., it has stationary and independent increments). Note that any such innitely divisible distribution may be written as the limit of an appropriate sequence of compound Poisson distributions (Feller, [F71]). Consequently, the compound Poisson demand model is of special signicance. By [SS78], the long-run average cost function becomes C(R; T ) = K T P (X(T ) > 0) + 1 T Z +T h h(r? t) + (h + p)e[(x(t)? R)+ ] i dt: When demand is discrete, E[(X(t)? R) + ] = P 1 x=r(x? R) (x; t); and X N (t) (x; t) = P (X(t) = x) = P ( i=0 Y i = x) = X n0 P ( nx i=0 Y i = x) P (N(t) = n); (9) where N(t) is a Poisson point process independent of any Y i and the Y i are independent, identically distributed, non-negative, discrete random variables. In this case, P (X(T ) > 0) = 1? e?t with = [1? P (Y i = 0)], and, we have the following result: Proposition 4.1 C(R; T ) is convex in R for xed T and convex in T for xed R. Proof. For ease of exposition, let p(n; x; t) = P ( P i=n i=0 Y i = x)p (N(t) = n). Convexity in R follows from the fact that C(R; T )?C(R?1; T ) is increasing in R. Now C(R; T 2 = K T 3 [2? 2e?T? 2T e?t? 2 T 2 e?t ] + (h+p)? 2T P x>r T 3 [2 R +T Pn0(x? R)p(n; x; + T ) + T 2 P x>r K f(t ) + (h+p) g(t ): T 3 T 3 Now note that f(0) = 0 and f 0 (T ) = 3 T 2 e?t P x>r P n0(x? R)p(n; x; t)dt Pn>0(x? p(n; x; + T 0, so f(t ) 0 for all T 0. Similarly, using the P (N(t) = n) = P (N(t) = n?1)?p(n(t) = n) and that, for iid Y i 0, 11

13 E[(X( + T ) + Y 1 + Y 2? R) + ]? 2E[(X( + T ) + Y 1? R) + ] + E[(X( + T )? R) + ] 0, the interested reader can verify that g(0) = 0, and g 0 (T ) 0, which completes the proof. As before, let C(T ) = C(R(T ); T ). Clearly, by the above proposition, C(T ) is convex over each T -interval for which R(T ) remains constant. Although proving convexity of C(T ) is complicated by the discontinuity in R(T ), it is possible to argue that C(T ) K T (1? e?t ) + 1 T R +T G(R(T ); t)dt is unimodular, so that the optimal R and T may be easily found using a golden section search. We conjecture that sensitivity results similar to those for the normal demand model should apply for most commonly used Y i (discrete or continuous). However, for general Y i, the loss of dierentiability makes rigorous analysis a challenging area for future research. Another possible research area would be the approximation of demand by distributions such as the logistic distribution, for which R(T ) may be expressed using a simple closed form expression in T. 4.2 Multi-Product / Multi-Echelon Problems We consider a single stage, N-item system and then a single-item, two-echelon serial system. The Joint Replenishment Problem (JRP): In this problem, we incur a major setup cost, K, every time any item is ordered and a minor setup cost, K i, associated with each individual item, i, that is included in the order. This order/setup cost structure aims to co-ordinate order placement epochs for dierent items so as to save on joint setups relating to full-truckload transportation costs or quantity discounts for items ordered from the same supplier. The presence of non-zero minor setup costs considerably complicates the problem since the optimal policy is not easily characterizable. Many heuristics, such as the (S, c, s) \can-order" policy (Federgruen et al [FGT84]) and the (Q; S i ) policy (Pantumsinchai [P92]), have been proposed for this complex stochastic problem, but no worst-case analysis exists for any of these heuristics. Heuristic for the JRP: et denote the \base planning period." Allocate the joint setup cost, K, to the dierent items in such a way that item i's set-up cost equals i K + K i for some i satisfying P i i = 1. (See [AI88] for more on the choice of i.) et T i be the optimal re-order interval for the resulting single-item (R, T ) problem. As in [R86], let k i be the smallest non-negative integer satisfying k i log 2 (T i =)? 0:5. The reorder interval and order-up-to level for item i are specied by T i = 2 k i and R i = R i (T i ), where the latter denotes the optimal item i order-up-to level corresponding to re-order interval T i. 12

14 et C JRP and C H JRP denote respectively, the optimal cost and the cost resulting from the above heuristic. et C i ( i ) denote the optimal cost for the single item problem with setup cost i K + K i. Since, for any set S f1; 2; : : : ; Ng of ordered items, K + P i2s K i X P i2s( i K + K i ): Furthermore, since 1 p 2 T i T i It is easy to allocate the i C JRP by [AI88] p 2, by Theorems 2.3 and 2.7, i C i ( i ) (10) C i (T i ) 1 2 (p p 2 )C i (T i ) = 1:06C i (T i ) 1:06 p 2C i ( i ) (11) in such a manner that only the items with re-order interval = min T i are assigned positive i values. For such an allocation, P i C i (T i ) = C H JRP and: Proposition 4.2 The worst case performance of the above heuristic for the JRP is given by: C H JRP 1:06 p 2C JRP = 1:5C JRP : Power-of-Two based (R, T ) policies may also be useful for a dierent single stage, multiitem production/inventory system, in which resource constraints make co-ordination of order epochs for dierent items undesirable. For instance, there could be limited warehouse storage space, so that, you prefer to stagger order delivery epochs. Heuristics with worst-case performance guarantees for this problem environment with stochastic demand and dierent types of capacity constraints is another possible area for future research. Two-Stage Serial System: Recent research on the single end-item, two stage serial system has focused on developing eective heuristics based on the (Q, r) model. Under certain special circumstances, remarkable worst-case performance guarantees have been obtained by Atkins & De [AD93] and Chen [C94]. We apply the (R, T ) policy to this two stage system. A typical inventory prole for such a system is shown in Figure 2. Clearly, the complicating factor is the interaction between echelons, e.g., the amount that can be delivered to the downstream stage is restricted by the availability of inventory at the upstream stage. et stage 2 denote the upstream stage that interacts with a perfectly reliable supplier; and let downstream stage 1 experience stationary stochastic customer demand at rate. The parameters at stages 1 and 2 are respectively: delivery (transportation) lead times, 1, 2 ; order/setup cost, K 1, K 2 ; echelon holding cost rates, h 1, h 2 ; and backordering cost rate p 1 at stage 1. As is common in practice, we assume that p 1 h 1. et I i (t) denote the echelon inventory level (net inventory downstream from or) at stage i and let B(t) denote the 13

15 T 2 = 4 T 1 Reliable Supplier Stage 2 Echelon Inventory 2 1 Stage 1 Inventory Customer T 1 Time Figure 2: Typical Inventory Prole for Two-Stage Nested, (R, T ) Policy backorder level at time t. Then, I 2 (t) I 1 (t) and, inventory costs at time t are incurred at rate h 2 I 2 (t) + h 1 I 1 (t) + (p 1 + h 1 + h 2 )B(t). Consequently, we have the following lower bounding result pertaining to re-allocation of inventory holding (shortage) costs from stage 2 to stage 1 (from stage 1 to stage 2). Proposition 4.3 et h e 1, h e 2, ep 1, ep 2 be such that 0 h e 2 h 2 with h e 1 + h e 2 = h 1 + h 2 and 0 ep 1 p with ep 1 + ep 2 = p. et P denote any feasible inventory policy for the two stage system. If C = C(h 1 ; h 2 ; p; 0; P) and C e = C( h e 1 ; h e 2 ; ep 1 ; ep 2 ; P) denote the long-run average cost incurred by two systems with inventory cost rates resp. h 1 ; h 2 ; p; 0 & h e 1 ; h e 2 ; ep 1 ; ep 2 and all other cost parameters identical, then C e C. Consequently, a valid \parameter-allocation" lower bound (Chen and Zheng [CZ94]) on the two stage problem is B = C 1(K 1 ; e h 1 ; ep 1 ; 1 ; ) +C 2(K 2 ; e h 2 ; ep 2 ; ; ) + h 2 1, where C i represents the optimal cost for a single stage problem with the specied parameters. Henceforth, we restrict attention to policies for which, on average, stage 1 places orders at least as frequently as stage 2. This assumption is valid for \nested" policies ([R86]) that send a shipment to stage 1 every time stage 2 receives a delivery from the external supplier. Nested policies are known to be optimal only for deterministic systems and for stochastic systems with 2 = 0. For simplicity, we assume that the latter is true. For such systems, re-allocation of order costs from stage 1 to stage 2 cannot increase costs. That is, similar to Proposition 4.3, C( f K 1 ; f K 2 ; P) C(K 1 ; K 2 ; P) for 0 f K 1 K 1 and K 1 + K 2 = f K 1 + f K 2. Since reorder intervals increase with xed order costs, wlog we assume that T 1 T 2. Now consider the following single stage systems with parameters specied by: System2: K 2, h 2, p 1, 1, ; System1: K 1, h 1, p 1, 1, ; and \collapsed" System0: K 1 +K 2, h 1 +h 2, p 1, 1,, with inventory holding cost suitably modied to reect expected cost, h 2 1, of inventory 14

16 in transit from stage 2 to stage 1. Under continuous review, let the corresponding minimum costs for these three systems be C 2 = C 2 (r2; Q 2), C 1 = C 1 (r1; Q 1) and C 0 = C 0 (r0; Q 0) resp. R r(q)+q P (X() y)dy = p = R 1 +T r(q) p+h T et = p 1 =(p 1 + h 2 ). Since 1 P (X(t) R(T ))dt, Q for both (Q; r) and (R; T ) policies, represents the expected fraction of the ordering cycle at stage 2 during which stage 2's net echelon inventory, I 2, is non-negative. Intuitively, the following lower bound results from the observation that stage 2 incurs cost at rate at least C 2; stage 1 incurs cost at rate at least C 1; further, to avoid double counting of shortage penalty, stage 1 incurs non-zero costs only over a fraction of stage 2's order cycle. Proposition 4.4 (logically similar to [AD93]; [CZ94]; [C94]) A valid lower bound on the long-run average cost of any policy for the two stage problem is C 2 + C 1. If r 2 + Q 2 < r 1 + Q 1, the two stages are collapsed into a single stage with optimal cost C 0. Heuristic & worst-case analysis for a two stage serial system: Compute T 1 by rounding to the nearest power-of-two multiple of a base planning period,, the optimal reorder interval corresponding to System1. (Set = T 1, if no base planning period has been specied.) et R 1 (T 1 ) be the optimal order-up-to level corresponding to T 1. Similarly, using parameters for System2, compute T 2 and R 2 (T 2 ). Wlog, T 2 T 1. (A) If R 1 (T 1 ) R 2 (T 2 ) or T 1 = T 2, collapse the two stages into one stage with parameters, K 1 + K 2, h 1 + h 2, p 1, 2 (System0) and compute (R 0, T 0 ), the power-of-two (R, T ) policy for the resulting single stage system. Both stages follow an (R 0, T 0 ) policy with order placement synchronized so that stage 2 carries no inventory. (B) Otherwise, T 2 2T 1 and stage 2 follows an (R 2 (T 2 ), T 2 ) policy. et t be any epoch at which stage 2 receives a delivery from the supplier. By nestedness, at time t, stage 2 sends a shipment to raise stage 1's inventory position up to R(T 1 ). Subsequently, stage 1 plans to place orders at time epochs t + T 1, t + 2T 1, : : :, using a policy of ordering up to min(i 2 (order epoch), R 1 (T 1 )), provided the order quantity is strictly positive. Proposition 4.5 For a two stage serial system with p 1 T )-based policy has a worst-case performance guarantee of Outline of Proof. Case (A): et (Q B i h 1, the above power-of-two, (R, ; r B i ) denote the optimal solution to the lower bounding problems in proposition 4.3. Clearly, since 2 = 0, when R 1 (T 1 ) > R 2 (T 2 ) or T 1 = T 2, the best, feasible (R, T ) policy should set T 1 = T 2 with R 1 (T 1 ) = R 2 (T 2 ). By computing (R 0 ; T 0 ) 15

17 for System0, the heuristic nds the best such solution, with cost C 0. Note that for each stage i, it is possible to achieve optimal single stage (R, T ) values satisfying R 1 (T 1 ) = R 2 (T 2 ) and T 1 = T 2 by adjusting inventory costs (and possibly xed order costs) to some e h 1, e h 2, ep 1, ep 2 values from Proposition 4.3. Since these are optimal R, T values for the corresponding single stage problem, by Theorem 2.7 and Proposition 4.3, the cost of the corresponding feasible (R, T ) policy, P, is C( e h 1 ; e h 2 ; ep 1 ; ep 2 ; P) p 2[C 1 ( e h 1 ; ep 1 ; e Q 1; er 1) + C 2 ( e h 2 ; ep 2 ; e Q 2; er 2) + h 2 1 ] p 2[C 1 ( h e 1 ; ep 1 ; Q B 1 ; r B 1 ) +C 2 ( h e 2 ; ep 2 ; Q B 2 ; r B 2 ) + h 2 1 ] p 2[C 1 (h 1 ; p 1 ; Q B 1 ; r B 1 ) + C p 2 (h 2 ; p 2 ; Q B 2 ; r B 2 ) + h 2 1 ] = 2B, where the second inequality follows from the optimality of ( e Q i ; er i ) for the modied inventory cost rates; the third inequality follows from Proposition 4.3. Since R 1 (T 1 ) = R 2 (T 2 ) with T 1 = T 2, the actual cost of policy P is C(h 1 ; h 2 ; p; 0; P) = C( e h 1 ; e h 2 ; ep 1 ; ep 2 ; P). Now C 0 C(h 1 ; h 2 ; p; 0; P) yields C 0 p 2B. Case (B): In this case, T 2 2T 1 and the heuristic's long-run average cost is no more than C 2 (T 2 ) + C 1 (T 1 ) + (1? p 1 p 1 +h 1 )C 1 (T 1 )T 1 =T 2. Note that the heuristic's cost was obtained by spreading a portion of the cost incurred during the last (partial) order at stage 1 over the entire cycle, T 2, as this ensures that the residual costs eectively incurred at stage 1, over a time interval T 2, accrue at rate C 1 (T 1 ). By Theorems 2.3 and 2.7, C(T 2 )+C(T 1 ) 1:5(C 2 +C 1) and (1? p 1 p 1 +h 1 )C 1 (T 1 )T 1 =T 2 (0:5)1:5C 1 (0:5) = 0:375C 1 0:375(C 2 + C 1) which yields a performance guarantee of An empirical investigation of power-of-two (R, T )-based heuristics for general multi-item, multi-echelon inventory systems is a possible area for future research. 5 Conclusions In this paper we have shown that, for normal demand, the long-run average cost for the (R; T ) inventory control policy is convex in T when R = R(T ), the optimal order-up-to-level corresponding to T. R(T ) is obtained by solving a newsboy problem with an appropriately dened demand distribution; T may be found by an ecient golden section search (with an intuitive graphical interpretation of the optimality condition). For normal demand, the ratio of the cost of using T = T instead of the optimal T is bounded by 1( + 1 ), 2 provided R = R(T ). Preliminary computational testing suggests that, in practice, use of the deterministic economic order interval, T d, or a powers-of-two approximation, instead of the optimal T does not increase costs signicantly. As expected, the (R; T ) policy is more expensive than the more responsive, (Q; r) policy. However, the dierence in cost 16

18 of the two policies may be bounded in the worst-case and is often of the order of a few percent, implying that the (R; T ) policy could be useful in situations where scheduling and coordination issues dominate. Based on the above results, we have developed power-of-two, (R, T )-based heuristics, with worst-case performance guarantees, for the joint replenishment problem and a two stage serial system. Acknowledgements. This research was partially supported by NSF Grant DDM to Cornell University. The author is grateful the area editor and two anonymous referees for many suggestions that improved this paper. 6 Appendix Wlog, we scale time so that T 1. et v v(t ) = R(T )?(+T ) and w w(t ) = p v(t ). +T Note that v 0 = R 0 (T )?, and w 0 = p 1 +T (R0 (T )?? R(T ) ). 2 2(+T ) Proposition 6.1 u(t ) G(R(T ); + T ) is increasing in T. Proof. Since X( + T ) = ( + T ) + p + T Normal(0; 1) and R za z(z) = (a), u(t ) = h(r(t )? ( + T )) + (h + p) Z 1 R(T ) = hv + (h + p)[ p + T (w)? v(w)] and (x? R(T )) (x; + T )dx u 0 (T ) = [h? (h + p)(w)]v 0 + (h + p) 2 p (w) 0; (12) + T where the last inequality follows from the fact that v 0 = R 0 (T )? 0, and (w) = R P (X( + T ) R(T )) 1 +T T P (X( + t) R(T ))dt (3) = h. h+p Proposition 6.2 For normal demand, u(t ) is convex. Proof. u 00 (T ) depends on R 00 (T ). Given that R 0 (T ) R +T (R(T ); t)dt = (w)? h h+p, Z +T R 00 (T ) (R(T ); t)dt + R 0 (T ) d dt Z +T (R(T ); t)dt =?(w)w 0 : (13) Since (R(T ); t) = p 1 R?t ( p ), from Appendix 4 in [HW63], R +T +T t (R; t)dt = 1 [W (R; + T )? W (R; )] where W (R; t) = ( R?t p )? t e2r(t )=2 ( R+t p ). Thus, t d dt Z +T (R(T ); t)dt = (R(T ); + T )? R 0 (T )[ 2 2 e2r(t )=2 ( R + t p t )]t=+t t= : 17

19 The last term in the right hand side of the above equation is zero because for > 3:5 and t 1, ( R+t p ) ( p t t ) (3:5) 0. Hence, R 00 (T ) [(w)? h ] Z h+p (4) +T R 0 = R 00 (T ) (T ) (R(T ); t)dt (13) =?(w)w 0? R 0 (w) (T ) p + T Dierentiating equation (12) and substituting v 00 (T ) = R 00 (T ), we get u 00 (T ) h+p =?R 00 (T )[(w)? h ] + h+p [(w)w0 ]v 0? (14) = R 0 (T )[(w)w 0 + R 0 1 (T ) p (w)] + +T (w)w0 [v 0? It is easy to verify that [v 0? u 00 (T ) h+p = 1 p 2 +T w(w)w0? (w); 4(+T ) 3=2 p w] = p + T w 0. Consequently, 2 +T p (w)[p + T R 0 (T )w 0 + R 0 (T ) 2 + [ p + T w 0 ] 2? +T 1 = p (w)[(p + T w 0 + R0 (T ) ) T 2 4 R0 (T ) 2? 2 ] 0: 4(+T ) where the non-negativity follows by considering two cases (i) 3R 0 (T ) 2 p w]? (w) 2 +T 4(+T ) 3=2 2 ] 4(+T ) (14) 2, in which case 3 4 R0 (T ) 2 2 4(+T ), and (ii) 3R0 (T ) 2 < 2, in which case, ( p + T w 0 + R0 (T ) 2 ) 2 2 4(+T ). References [AD93] [AI88] [C94] [CZ94] [FGT84] [F71] Atkins, D. and S. De, \94%-Eective ot-sizing for a Two Stage Serial Production / Inventory System with Stochastic Demand," Working Paper 92-MSC- 001, Faculty of Commerce, UBC, Canada, Atkins, D. and P. Iyogun, \Periodic Versus `Can-Order' Policies for Coordinated Multi-Item Inventory Systems," Management Science 34, , Chen, F., \94%-Eective Policies in a Two-Stage Serial System with Stochastic Demand," Working Paper, Graduate School of Business, Columbia University, New York, NY, Chen, F., and Y-S. Zheng, \ower Bounds for Multi-echelon Stochastic Inventory Systems," Management Science, 40, 11, pp , Federgruen A., H. Groenvelt, and H. C. Tijms, \Coordinated Replenishments in a Multi-Item Inventory System with Compound Poisson Demands," Management Science, 30, 3, pp , Feller, W., \An Introduction to Probability Theory and its Applications," Volume II, 2nd Edition, John Wiley & Sons,

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