Interchanging fill rate constraints and backorder costs in inventory models

Transcription

1 Int. J. Mathematics in Operational Research, Vol. 4, No. 4, Interchanging fill rate constraints and backorder costs in inventory models Jiang Zhang* Department of Management, Marketing, and Decision Sciences, School of Business, Adelphi University, Garden City 11530, New York, USA *Corresponding author Matthew J. Sobel Department of Operations, Weatherhead School of Management, Case Western Reserve University, Euclid Avenue, Cleveland 44106, Ohio, USA Abstract: This paper shows that fill rate constraints and backorder costs are interchangeable in dynamic newsvendor models. We establish monotone mappings between the set of optimal polices with backorder costs and the set of optimal policies with fill rate constraints. This clarifies the extent to which a parametric analysis of optima in one case can be performed via an algorithm directed at the other case. In a sense that is made precise, it is unnecessary to study both cases of a model; either case alone is sufficient. Keywords: inventory policies; fill rate; backorder costs; dynamic programming; discrete demand; continuous demand. Reference to this paper should be made as follows: Zhang, J. and Sobel, M.J. (2012) Interchanging fill rate constraints and backorder costs in inventory models, Int. J. Mathematics in Operational Research, Vol. 4, No. 4, pp Biographical notes: Jiang Zhang is an Associate Professor in the School of Business at Adelphi University. He received his PhD from Case Western Reserve University and his BE from Xidian University in China. His research interests focus on inventory control, supply chain coordination, and RFID applications. His papers have appeared in Decision Support Systems, Operations Research Letters, European Journal of Operational Research, International Journal of Production Economics, International Journal of Operational Research, International Journal of Operations and Quantitative Management, Decision Science Journal of Innovative Education, among others. Copyright 2012 Inderscience Enterprises Ltd.

2 454 J. Zhang and M.J. Sobel Matthew J. Sobel is the William E. Umstattd Professor at Case Western Reserve University in the Department of Operations of the Weatherhead School of Management and the Department of Electrical Engineering and Computer Science (by courtesy). He was educated at Columbia and Stanford Universities, worked in the private sector and government, and was on the faculties of Yale University, Georgia Institute of Technology, and the State University of New York at Stony Brook. His current research includes the coordination of operations and finance, preference theory, and environmental management. He is a Fellow of INFORMS and has authored numerous papers in operations, economics, and environmental management. 1 Introduction Inventories which encounter uncertain demand lead to risks of both excess supply and unsatisfied demand. The associated research on inventory models with stochastic demand studies how best to balance these risks. Initially, the relative importance of the two risks was often parameterised with holding costs and stockout costs. For the past 20 years, research has paid attention to the service role of inventories and focused on a system s fill rate, namely the fraction of demand which is immediately met from on-hand inventory. During this latter period, the relative importance of the two risks has often been parameterised with holding costs and a lower bound on the fill rate. As a result, there are parallel streams of literature which analyse identical models except that one stream has stockout costs and the other has fill rate constraints. These streams of literature correspond in mathematical programming to optimisation subject to constraints and to optimisation of an unconstrained Lagrangean. As in nonlinear deterministic optimisation, in stochastic optimisation the two approaches do not always yield the same results. This paper investigates whether there is redundancy in the two streams of dynamic inventory models with linear purchase costs, namely dynamic newsvendor models. We show the extent to which optimal policies for either kind of model can be inferred from the other. Here, an inventory replenishment policy is called Stockout-optimal, or S-optimal for short, if it minimises the long-run average sum of holding and stockout costs per unit time. So, S-optimality corresponds to an unconstrained Lagrangean formulation. A policy is called Fill-Rate-Optimal, or F-optimal for short, if it minimises the long-run average holding cost per unit time subject to a fill-rate constraint. If demand is continuous, i.e., if the distribution function of demand has a density function, then S-optimality and F-optimality are shown to be equivalent in the following sense: a b Corresponding to any unit stockout cost b, there is a base-stock level y and a fill-rate f such that a base-stock policy with parameter y is both S-optimal with stockout cost b and F-optimal with constraint parameter f. Corresponding to any fill-rate constraint parameter f, there is a base-stock level y and a stockout cost b such that a base-stock policy with parameter y

3 c Interchanging fill rate constraints and backorder costs 455 is both S-optimal with stockout cost b and F-optimal with constraint parameter f. If demand is an integer-valued random variable, the situation is more complicated. Although for every stockout cost b a deterministic base-stock level policy is S-optimal, for most constraint parameters f a randomised base-stock level policy is F-optimal. Nevertheless, a parametric analysis of either kind of optimality can be accomplished via the other kind in the following sense: Corresponding to any unit stockout cost b, there is a base-stock level y and a fill-rate f such that a base-stock policy with parameter y is both S-optimal with stockout cost b and F-optimal with constraint parameter f. d There are sequences of fill-rate constraint parameters 0 <f 1 <f 2 < < 1, base-stock level parameters y 1 <y 2 <, and stockout cost parameters b 1 <b 2 <, with the following property. For any fill-rate constraint parameter f, say f k f<f k+1. Then the base-stock policy with parameter y k is both S-optimal with stockout cost b k and F-optimal with constraint parameter f k. If demand is continuous, the consequence of (a) and (b) is that the structure and parametric analysis of F-optimal inventory policies are implicit in the structure and parametric analysis of S-optimal policies, and conversely. Therefore, it is redundant to analyse one if the other has been solved. If demand is discrete, the consequence of (c) and (d) is that there are discrete sets of fill-rate constraint parameters and stockout costs at which F-optimality and S-optimality are interchangeable. Our methods apply to the interchangeability of pairs of constraints and costs other than fill rate and stockout cost. For example, in Section 7 we sketch results for balancing inventory turnover ratios vs. fill rates inventory costs vs. stockout frequency. We provide a few portals to the large literature that is relevant to these issues. Porteus (2002) and Zipkin (2000) are treatises on stochastic inventory models. Also, see Porteus (1990) for a review of dynamic newsvendor models. Most of the literature on fill-rate constrained models consists of heuristics and approximations. For example, Silver (1970), Yano (1985) and Platt et al. (1997) propose heuristic solutions to fill-rate constrained models using (R, Q) policies. Axsäter (2003) considers a continuous-review fill-rate constrained serial system with batch ordering. The system faces a discrete compound Poisson demand process in which the leadtime demand has a negative binomial distribution. He shows that an optimal policy consists of a mixed multistage echelon stock (R, nq) policy with one of the reorder points varying over time. Schneider (1978) and Schneider and Ringuest (1990) study service-constrained models with setup costs, focus on (s, S) policies where the order quantities are predetermined, and present several approximations to estimate the reorder point s such that the required service level is achieved. Schneider and Ringuest consider a periodic review system with a fixed leadtime.

4 456 J. Zhang and M.J. Sobel Boyaci and Gallego (2001) and Shang and Song (2003) study a periodic review service-constrained serial inventory system with Poisson distributed leadtime demand and use the limiting probability of having positive on-hand inventory at the last stage as the service level which differs from the fill rate. Boyaci and Gallego focus on base-stock policies, develop heuristic solutions, and discuss the relationship between stockout cost and service-constrained models, whereas Shang and Song develop closed-form heuristics to approximate optimal base-stock policies for serial service-constrained systems. Tempelmeier (2010) studies a dynamic multiitem capacitated inventory problem with a fill rate constraint and proposes a heuristic to the lot-sizing problem. The ability to calculate the fill rate is a prerequisite to finding an F-optimal policy. Johnson et al. (1995) review fill rate approximations in periodic-review single-stage models with base-stock level policies. Glasserman and Tayur (1994), Glasserman and Liu (1997), Glasserman (1997), Sobel (2004), Zhang and Zhang (2010), Silver and Bischak (2010), Teunter et al. (2010), Zhang et al. (2010) and Zhang (2012), have other approximations and exact expressions for fill rates of single-stage and multi-stage systems with base-stock level policies. The paper has the following organisation. Section 2 introduces the model and notation and Section 3 characterises F-optimal policies when demand has a density function. Section 4 has the corresponding characterisation when demand is an integer-valued random variable. Section 5 relates F-optimality to S-optimality with characterisations and algorithms. Section 6 contains numerical examples and Section 7 generalises and summarises the results. 2 Model and problem formulations We consider a periodic-review inventory model in which an inventory manager has a single product that faces independent and identically distributed nonnegative demands D 1, D 2, in successive periods. Let D have the same distribution as D 1, let µ = E(D) with 0 <µ<, and let H(y) and B(y) denote the expected excess inventory and excess demand, respectively, in a period which has y units of inventory available to satisfy demand: H(y) =E[(y D) + ] and B(y) =E[(D y) + ]. Given an inventory replenishment policy and initial inventory level, let F N, KI N, and KN S denote, respectively, the N-period fill rate, the average end-of-period inventory level, and the average end-of-period stockout level. Let y n be the inventory which is available to satisfy D n,so [ N ] [ F N = E min {D N ] n,y n } N D KI N = E H(y n ) /N n [ N ] = E B(y n ) /N. K N S We assume that inventory is replenished at the beginning of the period, ordered goods are delivered immediately, excess demand is backordered, and order quantities are nonnegative. Let x n be the inventory level at the start of period n, sox n+1 = y n D n. An inventory replenishment policy π is a nonanticipative

5 Interchanging fill rate constraints and backorder costs 457 decision rule that yields nonnegative order quantities and does not allow planned backorders, so y n max{x n, 0}. Let c, h, and b be the respective unit costs of replenishment, end-of-period inventory, and end-of-period backorders. Let F denote the infinite-horizon fill rate, namely, F = lim inf N F N. These problems are considered in subsequent sections: inf π {lim inf N E π x1 (hk N I ):F f} (1) inf π {lim inf N E π x1 (hk N I + bk N S )} (2) inf π {lim inf N E π x1 (hk N I λf N )}. (3) Problem (1) minimises the average inventory cost subject to a fill rate constraint. Problem (2) minimises the average sum of inventory and stockout costs. Problem (3) is a Lagrangean with terms for the average inventory cost and the fill rate. In equations (1) (3) there is no loss of generality in the assumption, henceforth made, that h =1. In each problem, the primary issues are the form of an optimal policy and the set of optimal policies as a parameter is varied: f in equation (1), b in equation (2), and λ in equation (3). However, the agenda is reduced by showing that equation (3) is a special case of equation (2). The identity z =(z) + ( z) + implies B(y) =H(y) y + µ. Therefore, for N<, the criterion in equation (2) is { N } KI N + bks N [ ] = E B(yn )(1 + b)+y n /N µ. Rewriting F N as [ N ] / N F N = E [D n B(y n )] D n the criterion in equation (3) is { N } { N λf N = E [B(y n )+y n µ]/n λe [D n B(y n )] / } N D n K N I [ N {[ ( N = E 1+λ/ D n /N )]B(y n )+y n }] µ λ. Therefore, varying b in equation (2) is the same as varying λ in equation (3) because N D n/n µ as N. The remainder of the paper concerns only equations (1) and (2). This argument results in the following property.

6 458 J. Zhang and M.J. Sobel Proposition 1: { N } /µ F =1 lim sup N E [B(y n )]/N (4) Let G( ) be the distribution function of demand with m = sup{a : G(a) = 0} and M = inf{a : G(a) =1}. Let G 1 (θ) = inf{a : G(a) θ}, 0 θ 1. It is well known (cf. Porteus, 1990, 2002; Zipkin, 2000) that y n = max{x n,y } is an S-optimal policy with ( ) b y = G 1. (5) b +1 So the parametric analysis of equation (2) consists of solving equation (5) as b spans positive real numbers. In equation (1), the fill-rate constrained minimisation of inventory cost, Proposition 1 permits replacement of the fill rate constraint with a constraint on the average backorder level. Therefore, an F-optimal policy solves the following problem: { [ N ] inf π lim inf N E π x1 H(y n ) /N : lim sup N E π x1 [ N ] } B(y n )/N µ(1 f). (6) The following properties of H( ) and B( ) are useful in Section 3. The straightforward proof is omitted. Lemma 1: H( ) and B( ) are convex functions on (m, M) that are strictly increasing and decreasing, respectively. If M <, then H(M) =M µ and B(m) =µ m. A function is said to be injective (or an injection) if it maps at most one point in the domain to each possible value in the range. Let B 1 (θ) =sup{y : B(y) θ}, 0 θ µ m, and H 1 (η) =sup{y : H(y) η}, 0 η M µ. Since H( ) and B( ) are strictly monotone and continuous (owing to convexity) on (m, M), they are injections; i.e., B[B 1 (θ)] = θ and H[H 1 (η)] = η. In subsequent sections we analyse problem (6) for continuous demand and discrete demand. 3 Continuous demand First we analyse the following N-period version of equation (6) and then let N : { [ N ] [ N ] } inf π E π x1 H(y n ) : E π x1 B(y n ) Nµ(1 f). (7)

7 Interchanging fill rate constraints and backorder costs 459 The single-period newsvendor model with a fill rate constraint is the special case of N =1in equation (7): inf π { Eπ x1 [ H(y1 ) ] : E π x1 [ B(y1 )] µ(1 f) }. (8) The following result states that a base-stock policy is F-optimal for equation (8). Proposition 2: If demand has a density function, then y = max{y,x} is F-optimal in equation (8) with y = B 1 [µ(1 f)]. (9) Proof: Since H( ) is increasing and B( ) is decreasing, the smallest y that satisfies the constraint achieves the constrained minimum of H( ). If x 1 y, then the base-stock policy in equation (9) is optimal in equation (7) regardless of the value of N. Lemma 2: If demand has a density and x 1 y, then y n = y, n =1, 2,...,N, is F-optimal in equation (7) for all finite horizons N I +. Proof: The claim is valid for N =1 (Proposition 2). For any N > 1, rewrite equation (7) as { [ N ] [ N ] } inf π E π H(y n ) : E π B(y n ) Nµ(1 f). (10) This is a convex nonlinear program for which the following Kuhn-Tucker conditions are necessary and sufficient when demand has a density: E[H (y n )] + γ E[B (y n )]=0 for n =1, 2,...,N (11) { N γ E [ B(y n ) ] } Nµ(1 f) =0 (12) N E [ B(y n ) ] Nµ(1 f) and γ 0. Since H ( ) > 0 and B ( ) < 0 on (m, M), if there exists a Kuhn-Tucker point, then equation (11) implies γ 0, and equation (12) yields N E [ B(y n ) ] = Nµ(1 f). (13) Also, from equation (11), E[H (y 1 )] E[B (y 1 )] = E[H (y 2 )] E[B (y 2 )] = = E[H (y N )] E[B (y N )].

8 460 J. Zhang and M.J. Sobel So, y 1 = y 2 = = y N because H( ) and B( ) are convex and monotone. Therefore, with equation (13), E[B(y 1 )] = E[B(y 2 )] = = E[B(y N )] = µ(1 f). Since B( ) is an injection, it follows that y 1 = y 2 = = y N = y = B 1 [µ(1 f)]. Since x 1 y, x n+1 = y n D n and P {D n 0} =1ensure that y n = y x n for all n. Therefore, adding the constraints y n x n for all n reduces the feasibility set of equation (10), but does not affect the optimality of y n = y for all n (given x 1 y ). The transition from equations (7) to (6) is consistent with the large literature which connects finite horizon and infinite horizon inventory models. We exploit the fact that eventually the inventory level is at least as low as the back-stock level y (regardless of the initial inventory level). The proof is brief, straightforward, and omitted. Lemma 3: With probability one there is a period n <, such that x n y for all n n. Proposition 3: If demand has a density, then the base stock policy y n = max{y,x n } for all n, with y specified in equation (9), is F-optimal for equation (6), the infinite horizon problem with a fill rate constraint. Proof: For all N and for all π such that E π x1 [ N B(y n)] Nµ(1 f), if x 1 y, NH(y ) E π x1 [ N H(y ) E π x1 1 N [ N ] H(y n ) ] H(y n ). Therefore, if π is feasible in equation (6), (Lemma 2) [ H(y 1 N ] ) lim inf N E π x1 H(y n ). N Therefore, π is optimal in equation (6) because it is feasible owing to Lemma 3 and [ N ] lim inf N E π x 1 H(y n ) = lim inf N E π x 1 1 N [ n 1 N ] H(x n )+ H(y ) n=n

9 Interchanging fill rate constraints and backorder costs 461 [ 1 N ] = lim inf N E π x 1 H(y ) N n=n = H(y ) Proposition 3 states that an unrandomised base-stock policy, with target base-stock level B 1 [µ(1 f)], is F-optimal when demand has a density. The optimal policy is randomised when demand is discrete. 4 Discrete demand Demand and order quantities are integer-valued in this section. Let ξ j = P {D = j}, j =0, 1,...,M. For a target fill rate f<1, identify k<m such that B(k +1) <µ(1 f) B(k). IfB(k) =µ(1 f), it is clear from Section 3 that y n = k for all n =1, 2,... is F-optimal. The remainder of this section concerns the typical situation in which B(k +1)<µ(1 f) <B(k). Then y n = k for all n is infeasible because the resulting fill rate is lower than f. Ify n = k +1for all n, then the fill rate is strictly higher than f and there is an opportunity to lower the long-run average holding cost per period and still meet the fill rate constraint. That is, a randomised policy dominates the best unrandomised policy. We note that y n = k +1 for all n is F-optimal among unrandomised policies. Recall that x n denotes the inventory level at the beginning of period n. We prove that the following policy, labelled π, is F-optimal among randomised policies. For all n, ifx n = k +1, then y n = k +1.Ifx n k then y n = k +1 with probability β and y n = k with probability 1 β where β = γ(1 ξ 0) B(k) µ(1 f), γ = 1 γξ 0 B(k) B(k +1). (14) We confirm that π is feasible and then prove that it is optimal. Lemma 4: The fill rate of π with parameters in equation (14) is f. Proof: Without loss of generality let x 1 k. Then {y n } is a Markov chain with states k and k +1, transition probabilities p kk =1 β, p k,k+1 = β, p k+1,k =(1 β)(1 ξ 0 ), and p k+1,k+1 =1 (1 β)(1 ξ 0 ), and stationary probabilities λ k = (1 β)(1 ξ 0) λ β k+1 =. 1 (1 β)ξ 0 1 (1 β)ξ 0 Using equation (14), λ k = µ(1 f) B(k +1) B(k) B(k +1) λ k+1 = B(k) µ(1 f) B(k) B(k +1) Therefore, the fill rate, using equations (4) and (15), is F =1 1 [ λ µ k B(k)+λ k+1b(k +1) ] {[ =1 1 ] [ ] } µ(1 f) B(k +1) B(k)+ B(k) µ(1 f) B(k +1) = f µ B(k) B(k +1) (15)

10 462 J. Zhang and M.J. Sobel Let Π(f) be the set of all randomised stationary policies which result in F f, for π Π(f) let {λ π j } be the stationary distribution of {y n} from initial state 0, and write Hλ π for j H(j)λπ j. Proposition 4: Policy π is F-optimal among randomised stationary policies. Proof: Since π Π(f) from Lemma 4, we show that Hλ π = min{hλ π : π Π(f)}. Since there are no planned backorders and P {0 D M} = 1, with probability one max{0,x n } y n M and M x n M for all n. Under any stationary policy π, the inventory position {y n } is a Markov chain with states {0,...,M}. Since ξ 0 < 1, state 0 is recurrent. Therefore, let {λ a : a =0,...,M} be the stationary distribution. The following linear program minimises the average holding cost subject to a fill rate constraint. Find {λ a : a =0,...,M} in order to minimise subject to M H(a) λ a a=0 M B(a) λ a µ(1 f) (16) a=0 M λ a =1 a=0 λ a 0, a =0,...,M. The proof is completed with the following lemma which is proved in Appendix. Lemma 5: The optimal solution of equation (16) is λ k = λ k, λ k+1 = λ k+1, and λ a =0for a k and a k +1. Let w ja be the probability of ordering a j units when the inventory level is j; j = M,...,M; a =0,...,M. Policy π induces w jk =(1 β)(λ k ξ k j + λ k+1 ξ k+1 j), w j,k+1 = β(λ k ξ k j + λ k+1 ξ k+1 j) if j k w k+1,k+1 = λ k+1 ξ 0, and w ja =0, otherwise. We note that π is not uniquely optimal because other policies can induce the same stationary distribution of the inventory position. 5 Interchangeability For b 0 and f [0, 1], let y S (b) =G 1 [b/(b + 1)] and y F (f) =B 1 [µ(1 f)]. If b is the unit stockout cost, then y S (b) =G 1 [b/(b + 1)] is an S-optimal basestock level. If f constrains the fill rate and demand is continuous, then y F (f) = B 1 [µ(1 f)] is an F-optimal base-stock level. If demand is integer-valued, there is a randomisation among y F (f) and y F (f)+1 that is F-optimal. This section focuses on two questions. First, for a unit stockout cost b, is there an f such

11 Interchanging fill rate constraints and backorder costs 463 that y F (f) is an S-optimal base stock level? Second, if f constrains the fill rate, is there a b such y S (b) is an F-optimal base stock level (randomisation among y S (b) and y S (b)+1 if demand is discrete)? Both questions have positive answers and if demand is continuous, there is a formula for b in terms of f and conversely. Although the relation between b and f is more complicated with discrete demand, if B[y F (f)] = µ(1 f), then an F-optimal policy employs base stock level y F (f) (with probability one). We say that demand has a strictly positive density g( ) if g(a) > 0,m<a<M. Let F (y) be the fill rate induced by the base-stock policy with base-stock level y. Then F (M) =1and F (m) =0. Let b U = inf {b 0:G 1 [b/(b + 1)] = M}. Proposition 5: If demand is continuous: y F (f) =y S (b) if equation (17) (equivalently equation (18)) is valid b = G { B 1[ µ(1 f) ]} 1 G { B 1[ ]} (17) µ(1 f) f =1 B { G 1 [b/(b + 1)] } Also, y F ( ) is an injection on [0, 1], µ. (18) y F { F [ y S (b) ]} = y S (b) =G 1[ b/(b +1) ] (19) y S { G[y F (f)] /{ 1 G[y F (f)] }} y F (f) (20) and, if the density is strictly positive, the inequality in equation (20) is satisfied as an equality and y S ( ) is an injection on [0, b U ]. Proof: When the distribution of demand has a density, y F ( ) is an injection on [0, 1] because both B( ) and B 1 ( ) are continuous and strictly monotone. Therefore, equating y S (b) =G 1 [b/(b + 1)] with y F (f) =B 1 [µ(1 f)] and applying B( ) to both sides yields equations (17) and (18). Similarly, Proposition 1 implies F [y S (b)]=1 B{y S (b)}/µ. With the definitions of y F ( ) and y S ( ), this implies equation (19). For equation (20), let a = y F (f) in y S {G(a)/ [1 G(a)]} = G 1 [G(a)] a with equality if G has a strictly positive density. Finally, if demand has a strictly positive density, then y S ( ) is an injection because both G( ) and G 1 ( ) are continuous and strictly monotone. If demand has a strictly positive density function, Proposition 5 states that S-optimality and F-optimality are equivalent in the following sense: a b Corresponding to any unit stockout cost b, a base-stock policy with base-stock level y S (b) is F-optimal if f =1 B[y S (b)]/µ. Corresponding to any fill-rate constraint parameter f, a base-stock policy with base-stock level y F (f) is S-optimal if b = G [ y F (f) ]/ { 1 G [ y F (f) ]}.

12 464 J. Zhang and M.J. Sobel The properties are weaker if demand is discrete or if it is continuous with a density that is not strictly positive. In both instances, G( ) is not strictly monotone which causes discontinuities in G 1 ( ). If demand is discrete, recall the notation ξ j = P {D = j}, j=0, 1,...,M. For expository convenience, in the next result we assume ξ j > 0, j =0, 1,...,M. Only minor modifications are needed if ξ j =0for some j. Proposition 6: If demand is discrete: 1 If G[y F (f) 1]/ { 1 G[y F (f) 1] } <b G[y F (f)]/ { 1 G[y F (f)] }, then y F (f) is S-optimal with parameter b. 2 If f =1 B[y S (b)]/µ, then y S (b) is F-optimal with parameter f. If 1 B[y S (b)]/µ f<1 B[y S (b)+1]/µ, then a randomised policy that employs base-stock level y S (b) with probability 1 β and y S (b)+1 with probability β is F-optimal; β is specified by equation (14). Proof: For part 1, let b 1 = G[y F (f) 1]/ { 1 G[y F (f) 1] } and b 2 = G[y F (f)]/ { 1 G[y F (f)] }. Since G( ) is nondecreasing on [0, M], b 1 <b 2 and ( ) G 1 b1 = G 1 { G[y F (f) 1] } = y F (f) 1 b 1 +1 ( ) G 1 b2 = G 1 { G[y F (f)] } = y F (f). b 2 +1 Since b 1 <b b 2, G 1[ b/(b +1) ] = y F (f). Part 2 follows from Proposition 4. The relationship between S-optimality and F-optimality is more complicated when demand is discrete than when it is continuous because F-optimal policies are random only in the former case. Nevertheless, a parametric analysis of either kind of optimality can be accomplished via the other kind in the following sense: c d Corresponding to any unit stockout cost b, base-stock level y S (b) is F-optimal with constraint parameter f =1 B[y S (b)]/µ. Also, if 1 B[y S (b)]/µ<f<1 B[y S (b)+1]/µ, then a policy which randomises between base-stock levels y S (b) and y S (b)+1 is F-optimal with parameter f. Corresponding to any fill-rate constraint parameter f, base-stock level y F (f) is S-optimal for any stockout cost b in the interval G[y F (f) 1]/ { 1 G[y F (f) 1] } <b G[y F (f)]/ { 1 G[y F (f)] }. 6 Examples This section illustrates the results with numerical examples having continuous demand with a strictly positive density, continuous demand lacking a strictly positive density, and discrete demand.

13 Interchanging fill rate constraints and backorder costs Positive demand density In this subsection, the density function is g(a) =1/10 and G(a) =a/10, a [0, 10]; so µ =5. Therefore, G 1 (θ) =10θ, B(y) =E[(D y) + ] = (10 y) 2 /20, and B 1 (θ) =10 20θ. Using equation (5), an S-optimal policy employs base-stock level ( ) b y S (b) =G 1 b +1 = 10b b +1. In this example, y S ( ) is concave and increasing. Using equation (9), an F-optimal policy has base-stock level y F (f) =B 1 [µ(1 f)]=10 20µ(1 f) = f. In this example, y F ( ) is convex and increasing. Simplifying y S (b) =y F (f) yields (17) and (18): b = 1 1 f 1 f f = b2 +2b (b +1) 2. (21) Each equation in (21) uniquely defines the one-to-one mapping between b and f. Considering b as a function of f, in this example b( ) is convex and increasing. Considering f as a function of b, in this example f( ) is concave and increasing. 6.2 Nonnegative demand density In this subsection, g(a) =1/10 for a [0, 5] and a [6, 11], and otherwise g(a) =0. Therefore, µ =5.5 and G(a) =a/10 if a [0, 5], G(a) =1/2 if a (5, 6], and G(a) = (a 1)/10 if a (6, 11]. SoG 1 (θ) =10θ if 0 θ 1/2 and G 1 (θ) =10θ +1 if 1/2 <θ 1. That is, G 1 ( ) is strictly increasing but discontinuous at θ =1/2. Using equation (5) an S-optimal policy employs base-stock level ( ) { b 10b/(b +1), if 0 b 1 y S (b) =G 1 = b +1 (11b +1)/(b +1), if b>1. Although y S ( ) is strictly increasing as in Section 6.1, it is neither concave nor continuous owing to the discontinuity of G 1 ( ). As Figure 1 shows, no stockout cost b induces y S (b) (5, 6]. However, Y S (b) =G 1 [b/(b + 1)] is not necessarily uniquely optimal. In this case, if b =1, any base-stock level in [5, 6] is optimal. Continuing, B(y) = y (y 2 20y + 110)/20 if 0 y 5 (a y)g(a)da = (17 2y)/4 if 5 <y 6 (11 y) 2 /20 if 6 <y 11.

14 466 J. Zhang and M.J. Sobel Figure 1 Dependence of S-optimal base-stock level on stockout cost: nonnegative density (see online version for colours) Using equation (9), an F-optimal policy has base-stock level f if 0 f 15/22 y F (f) =B 1 [µ(1 f)] = 11f 5/2 if 15/22 <f 17/ f if 17/22 <f 1 As in Section 6.1, y F ( ) is convex and increasing on [0, 1]. Let P = {x :0 x 11} and Q = {x :5<x 6}. Therefore, P = {y F (f) : 0 f 1} and P Q = {y S (b) :0 b}. For y P Q, the one-to-one correspondence between b and f given in equations (17) and (18) is the following: 10b b +1 = f for y [0, 5] 11b +1 b +1 = f for y (6, 11]. Thus, for each b, y F (f) is uniquely S-optimal with f = 10(b 2 + 2b)/[11(b +1) 2 ] [0, 15/22]. Conversely, y S (b) is uniquely F-optimal using b = [ f]/ f [0, 1]. There are similar relationships for b (0, ] and f (17/22, 1], where f = [11(b 2 +2b)+1]/[11(b +1) 2 ] and b = [ f]/ f. In this example, there is no one-to-one mapping between b and f. If f (15/22, 17/22], y F (f) =11f 5/2 is an F-optimal base-stock level but there is no b for which y S (b) =11f 5/2; cf. Figure Discrete demand In this subsection, ξ k =1/10, k =0, 1,...,9; so µ =4.5, G(a) = a a=0 1/10 (0 a 9), and G 1 (θ) = inf{a : G(a) θ} (0 θ 1). Define G 1 (0)=0 to

15 Interchanging fill rate constraints and backorder costs 467 Figure 2 Locus of {(b, f)} with the same optimal base-stock level: nonnegative density (see online version for colours) avoid a triviality. Discrete demand causes G( ) to be piece-wise constant, so equation (5) specifies a many-to-one correspondence between unit stockout costs and S-optimal base stock levels. That is, y S ( ) is piece-wise constant. Here, using equation (9), B(y) =E[(D y) + ]=(9 y )(10 2y + y )/20 { 9 } y F (f) =B 1 [µ(1 f)] = sup y : (a y) 45(1 f). a= y That is, y F ( ) is piece-wise constant (and discontinuous) although B( ) is continuous. Table 1 specifies G( ) and B( ) and Table 2 tabulates G 1 ( ) and B 1 ( ). Tables 3 and 4 illustrate Proposition 6. Table 3 gives y S (b) as b varies, together with the values of f for which y S (b) =y F (f). Table 4 gives y F (f) as f varies, together with the values of b which cause y S (b) =y F (f). It is apparent from Table 1 Distribution function and expected number of backorders y G(y) B(y) =E[(D y) + ] [0,1) y [1,2) y [2,3) y [3,4) y [4,5) y [5,6) y [6,7) y [7,8) y [8,9) y [9, ) 1 0

16 468 J. Zhang and M.J. Sobel Table 2 Values of G 1 ( ) and B 1 ( ) θ G 1 (θ) θ B 1 (θ) [0, 0.1] 0 (3.6, 4.5] 0 (0.1,0.2] 1 (2.8, 3.6] 1 (0.2, 0.3] 2 (2.1, 2.8] 2 (0.3, 0.4] 3 (1.5, 2.1] 3 (0.4, 0.5] 4 (1.0, 1.5] 4 (0.5, 0.6] 5 (0.6, 1.0] 5 (0.6, 0.7] 6 (0.3, 0.6] 6 (0.7, 0.8] 7 (0.1, 0.3] 7 (0.8, 0.9] 8 (0, 0.1] 8 (0.9,1.0] Table 3 S-optimal base-stock levels and fill rates at which they are F-optimal b y S (b) 1 B[yS (b)] µ [ ) 1 B[yS (b)], 1 B[yS (b)+1] µ µ [0,1/9] 0 0 [0,0.2) (1/9, 1/4] [0.2, 0.377) (1/4, 3/7] [0.377, 0.533) (3/7, 2/3] [0.533, 0.667) (2/3, 1] [0.667, 0.778) (1, 3/2] [0.778, 0.867) (3/2, 7/3] [0.867, 0.933) (7/3, 4] [0.933, 0.978) (4, 9] [0.978, 1) (9, ) Table 4 f F-optimal base-stock levels and unit stockout costs at which they are S-optimal ( ] y F (f) G[y F (f) 1], 1 G[y F (f) 1] [0, 0.2) 0 [0, 1/9] [0.2, 0.378) 1 (1/9, 1/4] [0.378, 0.533) 2 (1/4, 3/7] [0.533, 0.667) 3 (3/7, 2/3] [0.667, 0.778) 4 (2/3, 1] [0.778, 0.867) 5 (1, 3/2] [0.867, 0.933) 6 (3/2, 7/3] [0.933, 0.978) 7 (7/3, 4] [0.978, 1.0) 8 (4, 9] (9, ) G[y F (f)] 1 G[y F (f)]

17 Interchanging fill rate constraints and backorder costs 469 Tables 3 and 4 that the parametric analysis of optimal policies can be accomplished either with b or f. For example, when 7/3 <b 4, Table 3 shows that the S-optimal policy has base-stock level 7 and induces a fill rate of Table 4 confirms that the F-optimal base-stock level is indeed 7 when f = However, if < f < 0.977, y F (f) remains 7 which is infeasible! Although y F (f)+1=8is optimal among deterministic policies, a randomised policy is better. For example, if f =0.95, then (cf. Proposition 6), the following randomised policy strictly dominates all deterministic policies. Order nothing when the inventory level is at least 8, and otherwise order up to 7 and 8 with respective probabilities 1 β and β. Using equation (14) and Table 1, γ =[B(7) 4.5(1 0.95)]/[B(7) B(8)] = and β =[0.375(1 0.1)]/[ (0.1)]= Generalisations and summary Numerous pairs of constraints and costs not only fill rate constraints and stockout costs are interchangeable in the sense of Section 5. The key prerequisite is convexity and monotonicity as in Lemma 1. As a first example, consider balancing inventory turnover ratio vs. fill rate. We model the turnover ratio as the ratio of the demand to the average of the inventory levels at the beginning and ending of the period. Accordingly, if τ denotes the constraining upper bound on turnover ratio, then the argument that leads to Proposition 1 shows that the constraint corresponds to { N } /µ lim sup N E [y n + H(y n )]/N 2µ/τ. We consider (i) maximisation of the fill rate subject to this constraint, and (ii) minimisation of the long-run average value of h[h(y n )+y n ]/2+bB(y n ). Let h =1without loss of generality, and let H(y) =H(y)+y. Obvious analogues of the results in Sections 3 5 are valid for these problems because H( ) is convex and strictly increasing on (m, M). For example, if demand has a density, then the following base-stock policy is optimal for problem (i): y n = max{y H,x n } for all n, with y H = H 1 (2µ/τ). The following base-stock policy is optimal for problem (ii): y n = max{y T,x n } for all n, with y T = G 1 [(b 1)/(b + 1)]. Interchangeability follows from the fact that y H and y T span the same set as τ and b span the nonnegative numbers. A second example of the broader applicability of the methods used in earlier sections is the task of balancing inventory costs and stockout frequency. The task can be formalised in several ways including (iii) minimising the long-run average inventory cost subject to an upper bound on the long-run relative frequency of stockout, and (iv) minimising a weighted linear combination of inventory cost and probability of stockout. If demand is continuous with a concave distribution function and m is the upper bound on stockout probability, then the following base-stock policy is optimal for problem (iii): y n = max{y u,x n } for all n, with y u = G 1 (1 m). In problem (iv), if the weight on inventory cost is normalised to unity and c is the weight on stockout probability, then the following base-stock

18 470 J. Zhang and M.J. Sobel policy is optimal for problem (iv): y n = max{y c,x n } for all n, where y c minimises H(y)+c[1 G(y)]. Interchangeability follows from the fact that y u and y c span the same set as m spans [0, 1] and c spans the nonnegative numbers. Generally, in this paper we make precise the extent to which Stockoutoptimality (S-optimality) and Fill-rate-optimality (F-optimality) are interchangeable in dynamic newsvendor models. We also show that F-optimality is achieved by a base-stock policy that is unrandomised if demand is continuous. If demand is discrete, it is achieved by a stationary policy that randomises between two adjacent base-stock levels. Also, there is a weakly monotone mapping from the set of S-optimal policies to the set of F-optimal policies. For practical purposes, the parametric analysis of either kind of problem can be performed via an algorithm directed at the other case. So it is unnecessary to study both cases separately. Although our results apply for the newvendor models, they generally do not apply for the multi-stage inventory systems. It would be an interesting research direction to investigate the relationship of backorder costs and service constraints in multistage systems. Acknowledgements We thank the editor and anonymous reviewers for their constructive comments and suggestions that have greatly improved the quality of this paper. References Axsäter, S. (2003) Note: Optimal policies for serial inventory systems under fill rate constraints, Management Science, Vol. 49, No. 2, pp Boyaci, T. and Gallego, G. (2001) Serial production/distribution systems under service constraints, Manufacturing & Service Operations Management, Vol. 3, No. 1, pp Glasserman, P. (1997) Bounds and asymptchaotics for planning critical safety stocks, Operations Research Vol. 45, No. 2, pp Glasserman, P. and Liu, T. (1997) Corrected diffusion approximations for a multistage production-inventory system, Mathematics of Operations Research, Vol. 22, No. 1, pp Glasserman, P. and Tayur, S. (1994) The stability of a capacitated, multi-echelon production-inventory system under a base-stock policy, Operations Research, Vol. 42, No. 5, pp Johnson, M.E., Lee, H., Davis, T. and Hall, R. (1995) Expressions for item fill rates in periodic inventory systems, Naval Research Logistics, Vol. 42, No. 1, pp Platt, D.E., Robinson, L.W. and Freund, R.B. (1997) Tractable (Q, R) heuristic models for constrained service levels, Management Science, Vol. 43, No. 7, pp Porteus, E.L. (1990) Stochastic inventory theory, in Heyman, D.P. and Sobel, M.J. (Eds.): Handbooks in OR and MS, Chapter 12, Vol. 2, Elsevier Science Publishers, pp Porteus, E.L. (2002) Foundations of Stochastic Inventory Theory, Stanford University Press, Stanford, CA.

19 Interchanging fill rate constraints and backorder costs 471 Schneider, H. (1978) Methods for determining the re-order point of an (s, S) ordering policy when a service level is specified, J. Oper. Res. Soc., Vol. 29, No. 12, pp Schneider, H. and Ringuest, J.L. (1990) Power approximation for computing (s, S) policies using service level, Management Science, Vol. 36, No. 7, pp Shang, H.K. and Song, J. (2003) Analysis of serial supply chain with a service constraint, Working paper, The Fuqua School of Business, Duke University, Durham, NC Silver, E.A. (1970) A modified formula for calculating customer service under continuous inventory reviews, AIIE Transactions, Vol. 2, No. 3, pp Silver, E.A. and Bischak, D.P. (2010) The exact fill rate in a periodic review base stock system under normally distributed demand, Omega, Vol. 39, No. 4, pp Sobel, M.J. (2004) Fill rate of single-stage and multistage supply systems, Manufacturing & Service Operations Management, Vol. 6, No. 1, pp Tempelmeier, H. (2011) A column generation heuristic for dynamic capacitated lot sizing with random demand under a fill rate constraint, Omega, Vol. 39, No. 6, pp Teunter, R.H., Syntetos, A.A. and Babai, M.Z. (2010) Determining order-up-to levels under periodic review for compound binomial (intermittent) demand, Euro. J. of Operational Research, Vol. 28, No. 1, pp Yano, C.A. (1985) New algorithm for (Q, r) systems with complete backordering using a fill-rate criterion, Naval Research Logistics Quarterly, Vol. 32, No. 4, pp Zipkin, P. (2000) Foundations of Inventory Management, McGraw-Hill, New York, NY. Zhang, J. (2012) Analysis of fill rate in general periodic review two-stage inventory systems, Int. J. Operational Research, Vol. 14, No. 4, pp Zhang, J., Bai, L. and He, Y. (2010) Fill rate of general periodic review two-stage inventory systems, Int. J. Operational Research, Vol. 8, No. 1, pp Zhang, J. and Zhang, J. (2007) Fill rate of single-state general periodic review inventory systems, Operations Research Letters, Vol. 35, No. 4, pp Appendix Proof of Lemma 5: Expression (15) is feasible in equation (16). In order to prove optimality, let δ a be the reduced cost of λ a in the simplex algorithm; if δ a 0 for all a k, k +1, then equation (16) is optimal. The corresponding basis matrix of equation (16) is B = [ ] B(k) B(k +1) 1 1 Since det(b) =B(k) B(k +1)> 0, for each λ a the reduced cost is H(a)[B(k) B(k + 1)] H(k)[B(a) B(k + 1)] + H(k + 1)[B(a) B(k)] δ a =. B(k) B(k +1) (22) Let a be the numerator of equation (22); since B(k) B(k +1)> 0, δ a and a have the same signs.

20 472 J. Zhang and M.J. Sobel If a<k, a = H(a)[B(k) B(k + 1)] H(k)[B(a) B(k + 1)] + H(k + 1)[B(a) B(k)] = H(a)[B(k) B(k + 1)] H(k)[B(a) B(k)+B(k) B(k + 1)] + H(k + 1)[B(a) B(k)] =[H(k +1) H(k)][B(a) B(k)] [H(k) H(a)][B(k) B(k + 1)]. We use an induction starting with k =0to show that a 0. If k n 0, for a = k n 1 and n =0,...,k 2 k n 1 =[H(k +1) H(k)][B(k n 1) B(k)] [H(k) H(k n 1)] [B(k) B(k + 1)] =[H(k +1) H(k)][B(k n 1) B(k n)+b(k n) B(k)] [H(k) H(k n)+h(k n) H(k n 1)][B(k) B(k + 1)] =[H(k +1) H(k)][B(k n 1) B(k n)] [H(k n) H(k n 1)][B(k) B(k +1)]+ k n. (23) Since H( ) and B( ) are convex and monotone, k n 1 0 because H(k +1) H(k) H(k n) H(k n 1) 0 B(k n 1) B(k n) B(k) B(k +1) 0. If a>k+1, a = H(a)[B(k) B(k + 1)] H(k)[B(a) B(k + 1)] + H(k + 1)[B(a) B(k +1)+B(k +1) B(k)] =[H(a) H(k + 1)][B(k) B(k + 1)] [H(k +1) H(k)][B(k +1) B(a)] Another induction concludes that a 0 for all a>k+1.