Distribution Free Bounds for Service Constrained (Q, r) Inventory Systems

Transcription

1 Distibution Fee Bounds fo Sevice Constained (, ) Inventoy Systems Vipul Agawal, Sidha Seshadi Opeations Management Depatment, Leonad N. Sten School of Business, New Yok Univesity, New Yok, New Yok 11 Received June 1998; evised May ; accepted 17 May Abstact: A classical and impotant poblem in stochastic inventoy theoy is to detemine the ode quantity () and the eode level () to minimize inventoy holding and backode costs subject to a sevice constaint that the fill ate, i.e., the faction of demand satisfied by inventoy in stock, is at least equal to a desied value. This poblem is often had to solve because the fill ate constaint is not convex in (, ) unless additional assumptions ae made about the distibution of demand duing the lead-time. As a consequence, thee ae no known algoithms, othe than exhaustive seach, that ae available fo solving this poblem in its full geneality. Ou pape deives the fist known bounds to the fill-ate constained (, ) inventoy poblem. We deive uppe and lowe bounds fo the optimal values of the ode quantity and the eode level fo this poblem that ae independent of the distibution of demand duing the lead time and its vaiance. We show that the classical economic ode quantity is a lowe bound on the optimal odeing quantity. We pesent an efficient solution pocedue that exploits these bounds and has a guaanteed bound on the eo. When the Lagangian of the fill ate constaint is convex o when the fill ate constaint does not exist, ou bounds can be used to enhance the efficiency of existing algoithms. c John Wiley & Sons, Inc. Naval Reseach Logistics 47: , Keywods: inventoy models; stochastic; bounds; algoithm 1. INTRODUCTION A classical and impotant poblem in stochastic inventoy theoy is to detemine the ode quantity () and the eode level () to minimize inventoy holding and backode costs subject to a sevice constaint that the faction of demand satisfied by inventoy in stock (i.e., the fill ate), is at least equal to a desied value. This optimization poblem is usually difficult to solve without making additional assumptions egading the distibution of demand duing the leadtime (see Zipkin [19]). As a consequence, no algoithm othe than exhaustive seach is available fo solving this poblem in its full geneality. In this pape, we deive uppe and lowe bounds fo the optimal values of and fo this poblem and in paticula show that the classical Economic Ode uantity (EO) is a lowe bound fo the optimal. We also pesent an efficient solution pocedue that exploits these bounds and poduces a solution that has a guaanteed eo bound. Coespondence to: S. Seshadi c John Wiley & Sons, Inc.

2 636 Naval Reseach Logistics, Vol. 47 () Seveal heuistics have been poposed fo solving this sevice constained (, ) inventoy poblem. Platt, Robinson, and Fuend in [1] povide a compehensive eview of these pocedues. Heuistics adopt one o both of two appoaches fo solving this poblem. They simplify eithe the expession fo the fill ate, o the expession fo the aveage inventoy. Such simplifications ae intoduced by assuming that thee is at most one ode outstanding at any time (Section 4- of Hadley and Whitin [7], Yano [16], Silve and Wilson [14], and Rosling [1]), o by assuming that the duation of backodes is negligible (Section 4- of [7], [16], and [14]), o by neglecting a pat of the integal in the expession fo the fill ate. The advantage of heuistic pocedues lies in the ease with which a use can detemine the values of and. Fo example, Yano [16] povides an iteative heuistic fo nomally distibuted lead-time demand that is guaanteed to convege. Platt, Robinson, and Feund [1] popose and compae two elegantly motivated closed fom heuistics fo solving this poblem. Ou bounds can be used to enhance the efficiency of these algoithms. The poblem has been found to be intactable when the above simplifications ae not made. This is because the optimization domain shifts fom convex optimization to geneal nonlinea optimization. The difficulty is ceated by the fact that the fill ate constaint is not in geneal a convex function of the poblem paametes, and. Hadley and Whitin [7] ealized the difficulty of this poblem and stated the pogam (algoithm) detemines a elative minimum but does not povide any guaantee that the minimum so obtained is the absolute minimum (p. 188). Obtaining optimal paametes might appea to be inconsequential, Zipkin [19] showed that the penalty of making simplifications in ode to convexify the fill ate expession can be quite substantial in cetain instances. Zipkin [19] poved that the aveage level of backodes was convex in and. He also deived sufficient conditions that can be imposed on the lead-time demand distibution to guaantee that the expession fo the fill ate is convex in (, ). The condition (which is also povided by Zhang in [17]) states that fill ate expession is convex in (, ) if the pobability density function of lead-time demand f(x) is deceasing fo x. This condition esticts to lage positive values. In pactical poblems can be negative o small when eithe the setup cost is high, backode cost is low, demand vaiance is small, o the lead-time is shot. Rosling in a seies of ecent papes ([11] and [1]) has analyzed seveal inventoy cost ate functions. His appoach unifies the analysis of diffeent inventoy models (continuous eview, peiod eview, etc.) with the notion of quasi-convexity (see below) of the cost ate function. Ou model without the fill ate constaint is the closest to model 4 in [11]. In this pape, Rosling shows that the cost (ate) function fo model 4 is quasiconvex if the distibution function of lead-time demand is logconcave. (An accessible summay of logconcavity popety and applications can be found in [4].) He uses this esult in [1], and shows that if the cost ate function is quasiconvex fo all nonnegative cost coefficients, then the Lagangian that incopoates the fill ate constaint is quasiconvex (p. 14). When thee is a sevice constaint, Rosling also comments ([1], p. ): It is also taken fo ganted that an optimal finite policy exists, but the exact conditions fo this ae not well known fo the poblem with a sevice constaint. He too does not establish the existence of a finite optimal solution. In [1], Rosling poposes an algoithm that assumes the quasiconvexity of the cost ate function fo the sevice level constained case and the existence of a finite optimal solution. He also obseves that algoithms poposed in the liteatue ae not always guaanteed to convege, simila to obsevations made by Platt, Robinson, and Feund [1]. Ou bounds ae useful even when the quasiconvexity popety of the Lagangian of the fill ate is established by the lead time demand distibution being logconcave. Notice that a function g(x) fom R n R is said to be stictly quasiconvex if fo all x, y, inr n and θ (, 1),g((1 θ)x + θy) < max{g(x),g(y)}. This popety implies that a local optimum is a global optimum

3 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 637 (see, e.g., []). Theefoe, in ode to use an algoithm that exploits this popety, it is essential to know a good uppe bound fo the decision vaiables (in ou case the ode quantity and the eode level). This explains the need fo the assumption by Rosling that a finite optimal policy exists. As Rosling comments, An algoithm is initiated at w <, it follows...that inceases fo some time until it possibly oveshoots, afte which it might oscillate so that convegence is not monotonic ([1], p. )). Theefoe, establishing the existence of an uppe bound that is not too lage when compaed to the lowe bound inceases the efficiency of algoithms fo solving the fill-ate constained poblem even when the Lagangian is quasiconvex. Ou bounds ae new fo the sevice constained vesion of the poblem. Bounds ae available fo the unconstained vesion of the poblem (see Zheng [18], Axsate [1], and Gallego [6]). Zheng showed that the deteministic EO (modified fo the case when backodes ae allowed) is a lowe bound fo the optimal value of. He also established that the use of the EO as the ode quantity does not lead to substantial deviation fom the optimal cost. Axsate shapened the bound on cost povided by Zheng. Fedeguen and Zheng povided an algoithm in [5] fo detemining the optimal values of and fo the unconstained poblem that is also applicable to the constained poblem in special cases. Discussion of this aspect is given in Section. Gallego povided an uppe bound fo fo the unconstained poblem. The uppe bound given by Gallego gows with the standad deviation of the lead-time demand, wheeas ou bounds ae distibutionfee (and vaiance fee) as explained below. As Nahmias and Smith [9] stess, in many etail settings the vaiance to mean atio can be vey high (anging fom 3 to 5 in thei etail study). Thus ou bounds can be useful in these scenaios even fo the unconstained poblem. Ou analysis is based on the exact fomulation of the objective function, as well as the exact expession fo the fill ate. Inteest in (, ) inventoy systems using the exact fomulation of the objective function, as given by Hadley and Whitin (in Section 4:7 of [7]), was ekindled by the wok of Zipkin and Zheng. The bounds deived by us ae independent of the distibution of the lead-time demand in the following sense. We show that in the wost case the seach fo the optimal values of and can be esticted to fou intevals, the fist of which is given in tems of the eode level and the othes in tems of the ode quantity. In each inteval, egadless of the distibution of the lead-time demand, the atio of the lagest to the smallest value within the inteval is less than 4. The solution algoithm uses this featue by seaching along the values of in the fist inteval and along the values of in the othes. We povide an epsilon-optimal solution algoithm which is guaanteed to convege in polynomial time. Moeove, these bounds can be easily incopoated into existing algoithms in ode to impove the efficiency of the seach fo, even when the fill ate constaint is well behaved (i.e., convex o quasiconvex) o not pesent. We intoduce a novel way of dealing with the fill ate constaint. We expess the cost function as a sum of a convex function of (, ) and a nonconvex function. It has the following special popety that, fo a given, if the eode level () minimizes the convex potion of the cost, then the set of all such (, ()) pais is exactly equal to the set of solutions that satisfy the fill ate constaint. This popety plays an impotant ole in guaanteeing an epsilon-optimal solution within finite time. Anothe novelty in ou analysis is that it does not use the two-point distibution appoach taken by ealie eseaches (such as [1] and [6]) to establish bounds fo the unconstained poblem. The main featue of ou appoach is to bypass the tail integal with egad to the demand distibution. This popety allows us to poduce distibution fee (and vaiance independent) bounds. Ou methods ae attactive when viewed as altenate methods of establishing bounds fo the stochastic inventoy poblem. In Section, we specify the poblem and deive stuctual popeties that ae used to tansfom the poblem. We then deive bounds in Section 3 and povide an algoithm that can be used to detemine (, ) in Section 4.

4 638 Naval Reseach Logistics, Vol. 47 (). PROBLEM FORMULATION We conside the classical fomulation of the continuous eview stochastic inventoy contol poblem. The decision vaiables ae the paametes and. The objective is to minimize the sum of setup, holding and backode costs subject to the constaint that the fill ate should be at least equal to F (as given by Hadley and Whitin in [7]). The setup cost pe ode is given by K. The holding cost pe unit pe unit time is denote as h and the back ode penalty pe unit pe unit time as p. In ode fo the optimal value of to be nonzeo and the optimal values of and to be finite, we assume that F, h, and K ae stictly geate than zeo and that the penalty p is geate than o equal to zeo. The lead-time is assumed to be independent of the inventoy position. The distibution function of the lead-time demand is denoted as F, the complement of F as F c ( ), and the density function of the lead-time demand is denoted as f( ). We assume that the density function f( ) is stictly positive on [, ). The aveage aival ate of demand is denoted as λ and the aveage demand duing the lead-time is given by µ. We assume that the inventoy position is unifomly distibuted in the inteval [, +] and independent of the lead time demand (see Bowne and Zipkin [3], Sefozo and Stidham [13], and Zipkin []) fo discussion of the conditions unde which this assumption holds). In this model all unfilled demand is backodeed. The case when thee ae eithe lost sales o patially lost sales poses geate difficulty in the analysis as descibed in [9] and [8]. We plan to analyze the lost sales model in futue wok. Denote the aveage inventoy and the aveage backode level by Ī(, ) and S(, ). It follows fom the assumption that inventoy position is unifomly distibuted, and Ī(, ) = S(, ) = + + [ y [ y (y x)f(x) dx] dy (x y)f(x) dx] dy. Define [ y ] [ ] G(y) =h (y x)f(x) dx + p (x y)f(x) dx. y The sum of the aveage holding cost and the aveage cost of backode can be expessed in tems of G(y) as The optimization poblem is given by Poblem P: subject to the fill ate constaint, hī(, )+p S(, ) = min C(, ) = min,, 1 ( λk G(y) dy. G(y) dy ) (1) + F. ()

5 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 639 It is known that the objective function of poblem P is jointly convex in and (see Zipkin [19] o Zheng [18]), that G(y) is convex, G( ) =, and that lim y G (y) =h, whee G (y) is the deivative of G(y). In addition, if f( ) > on [, ), then C(, ) is stictly and jointly convex in (, ). REMARK: One appoach to solving P would be to fom the Lagangian and seach fo the optimal multiplie employing the algoithm poposed by Fedeguen and Zheng in [5] (denoted as the FZ algoithm). It can be shown that the Lagangian will be of the same fom as the objective function except that G(y) will be eplaced by G(y) πf(y), whee π is the multiplie. Unfotunately, G(y) πf(y) is not always unimodal a condition that is necessay to ensue the optimality of (, ) found using the FZ algoithm. The condition holds when the lead-time demand is Poisson. It is also woth mentioning that the complexity of an algoithm that uses the Lagangian appoach to incopoate the fill ate constaint is as yet unknown. Moeove, by using the stuctue of the fill ate constaint, it might be possible to use weak duality theoy to develop bounds on the solution poduced by the Lagangian appoach. As mentioned in the Intoduction, Rosling [1] shows that if the distibution of lead-time demand is logconcave, then the Lagangian (that incopoates the fill ate constaint) is quasiconvex, but a finite optimal policy is still not guaanteed. Theefoe, bounds ae equied on the values of and to facilitate the seach fo these paametes. Let (, ) be an optimal solution to Poblem P. Let ( u, u) minimize C(, ) (i.e., ( u, u) achieve the unconstained minimum). Define the fill ate achieved with a given set of (, ) to be Φ(, ). Thus, Φ(, ) =1 + Let u () be the value of that minimizes C(, ) fo a given value of. LEMMA 1: Φ( u, u) F iff p/(p + h) F. PROOF: C(, ) can be expessed as (see Zheng [18]): C(, ) = λk + + [(h + p) y. (3) F(x) dx + p(λl y)] dy. Fom the convexity of C(, ), u () is obtained by solving fo in Theefoe, (h + p) u()+ C(, ) F(x) dx + p(λl u () ) =. (h + p) u() F(x) dx p(λl u ())=.

6 64 Naval Reseach Logistics, Vol. 47 () This implies that h + p p [ u()+ F(x) dx u() F(x) dx ] =. Consolidating the tems in the integals, we obtain h + p p [ ] u()+ (1 F c (x)) dx =. u() This implies that Theefoe, 1 h + p p u()+ u() [ u()+ u() ] =. =Φ(, u ()) = p p + h. (4) This in tun yields p Φ( u, u ( u))=φ( u, u)= p p + h. (5) If p+h F, then fom Eq. (5) we obtain that Φ( u,u) F. On the othe hand, if Φ( u,u) p F, then fom Eq. (5) it follows that p+h F. REMARK: A simila esult to Lemma 1 is deived in [6] in the context of the unconstained poblem, but we need the if and only if agument, which is new in this pape, to chaacteize the solution set fo the constained poblem. Fo a given ode quantity, let () be the eode level that satisfies the fill ate constaint as an equality, i.e., 1 ()+ () =Φ(, ()) = F. (6) Given, let () to be the eode level that minimizes C(, ) and satisfies the fill ate constaint. The function () is chaacteized in the lemma given below. LEMMA : If p/(p + h) <F, then () =(), i.e., () satisfies the fill ate constaint as an equality. If p/(p + h) F, then () = u (). PROOF: Fom equation (3) we know that Φ(, ) is non-deceasing in. Theefoe, the set of eode points, S F, that satisfy equation () fo a given value of is given by S F = { : ()}.

7 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 641 Thus, () can be epesented as: () = ag min S F C(, ). The minimum value of C(, ) fo a given is achieved at u (). It follows fom equation (4) that fo a given value of,ifp/(p + h) <Fthen Φ(, u ()) <F. Fom the fact that Φ(, ) is non-deceasing in, it follows that () > u () when p/(p + h) <F. Theefoe, fom the convexity of C(, ) in, it follows that C(, ) C(, ()) fo S F. This finally implies that () =(). The second pat of the lemma follows fom Lemma 1. REMARK: It should be noted that the set {(, ) :Φ(, ) F, and } need not be convex (see Zipkin [19]). Lemmas 1 and naow down the set in which an optimal solution of P can be found as follows. 1. If p/(p + h) F then the unconstained solution, ( u, u), is optimal fo P.. If p/(p + h) <Fthen the seach fo the optimal solution to poblem P can be esticted to the set {(, ()) : }. We will now use these popeties to modify the objective function in poblem P. As stated in Section 1 we expess the cost as a sum of two functions. The fist function has the popety that fo a given, the eode level () that satisfies the fill ate constaint in equality (as defined ealie), also minimizes this potion of the cost. This set of (, ()) is identical to the set of solutions that satisfy the fill ate constaint (as shown in Lemma 4 below). Moeove we will pove in Lemma 5 that this potion is a convex function of when is eplaced by (). Let K 1 = K p+h and F u = p p+h. LEMMA 3: Poblems P1 and P ae equivalent, whee Poblem P1: ( ) min C λk1 1(, ) = min,, + F S(, )+(1 F )Ī(, )+(F F u)( + / µ) (7) subject to PROOF: [ (p + h) 1 + F. (8) The objective function of poblem P (see equation (1)) can be witten as λk (p + h) + p p + h S(, )+ h ] ) p + hī(, [ ] λk1 = (p + h) + F S(, u )+(1 F u )Ī(, ) [ ] λk1 = (p + h) + F S(, )+(1 F )Ī(, )+(F Fu)(Ī(, ) S(, )).

8 64 Naval Reseach Logistics, Vol. 47 () Futhe, (Ī(, ) S(, )) = LEMMA 4: + + (1) The mapping fom to () is one-to-one. () 1 (). (3) ()+ is inceasing in. (4) lim (()+) =. [ y (y x)f(x) dx (x y)f(x) dx] dy y (+ µ) (y µ) dy ( µ) = = = + / µ. (9) +, PROOF: By assumption f( ) >. Theefoe the fill ate, which is given by 1 is stictly inceasing in. Theefoe given, the fill ate can be satisfied only fo one value of. This poves pat (1) of the lemma. Fo poving pats (), (3) and (4), we know fom Lemma and the fist pat of this lemma that () is the solution to ( ) λk1 + F S(, )+(1 F )Ī(, ) =. (1) Theefoe, the esults obtained in Lemma 3.3 and 3.4 in Zheng [18] coespond to claims (), (3), and (4). 3. BOUNDS ON We will now estict ou attention to the case p/(p+h) <F, that is when the fill ate constaint is binding at the optimal solution to poblem P (o P1). As a esult of Lemmas and 3, poblem P1 can be solved by finding the value of that minimizes: C 1 () = + ()+ G () 1 (y) dy +(F F u )(()+/ µ) (11) whee, [ y ] [ ] G 1 (y) =(1 F ) (y x)f(x) dx + F (x y)f(x) dx. (1) y Notice that, ()+ G () 1 (y) dy = F S(, ())+(1 F )Ī(, ()).

9 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 643 Also, note that by specifying to be equal to () in the cost function (11) we ensue that the fill ate constaint is satisfied. Define, H(, ) = + G 1 (y) dy. As a matte of notation, we denote deivatives using a pime wheeve thee is no scope fo confusion. We shall efe to H(, ()) by H(). The popeties of H() deived in the next lemma will be used to obtain bounds on. LEMMA 5: (1) H (). () H (). (3) H () (1 F )/, fo all. PROOF: Please see Appendix (also see Zheng [18]). THEOREM 1: A lowe bound fo is given by the Economic Ode uantity (EO), i.e., λk h. PROOF: Note that as K>, >, and because h>, <. Theefoe, an optimal solution to P1 has to satisfy the fist-ode condition fo optimality, i.e., C 1() =. Theefoe, we get ( ) = H ( )+(F F u )( ( )+1/). Fom Lemma 4 we know that () ; theefoe, ( ) H ( )+ F F u. (13) Fom pat 3 of Lemma 5, H ( ) (1 F )/; theefoe, ( ) 1 F + F F u = 1 F u. Substituting fo the values of K 1 and F u,weget λk (p + h)( ) h (p + h) o λk h = EO. REMARK: This is an impotant and useful lowe bound on as it is not a function of the backode cost ate p, because in many pactical situation p is difficult to find and many papes in the liteatue (such as [1], [16]) have analyzed the fill ate constained (, ) poblem without a backode cost. It can be shown that when the vaiance of demand appoaches zeo, p, and F 1, then λk/h. Theefoe, the lowe bound is also tight. (At fist sight, this example looks contadictoy because p goes to zeo while the fill ate goes to 1. We wish to

10 644 Naval Reseach Logistics, Vol. 47 () emphasize that this is not unusual if the decision make wishes to only specify a fill ate and does not wish to specify a value fo p.) In contast the lowe bound on fo the unconstained poblem is λk(p + h)/hp (see [6] and [17]) and is not applicable to the constained poblem. We now pesent Lemmas 6 8 that will be used to deive an uppe bound fo the value of. Define whee J() = + H() B, B = H(). It can be veified by the use of L Hospital s ule that H() = G 1 (()). LEMMA 6: (1) J() = J( ) =. () J() is stictly convex. PROOF: (1) As K>, it follows that J() =, and as H( ) = (see Zheng [18]) it follows that J( ) =. () The poof follows fom Lemma 5 given in Zheng [18]. Let m be a value of that satisfies the fist ode condition ( J() )=. LEMMA 7: (1) m is unique. () J()/ is deceasing and convex fo m and inceasing fo > m. Thus, m minimizes J()/. PROOF: Please see the Appendix. LEMMA 8: [H() B]/ is inceasing in. PROOF: Please see the Appendix. As B is constant, it follows fom Eq. (11) that poblem P1 can be solved by finding the value of that minimizes the edefined objective function C n () = + F S(, ())+(1 F )Ī(, ()) B +(F F u )(()+/ µ). (14) Define to be the value of such that ( )=. Denote the lowe bound on deived in Eq. (13) by l. Also, define to be such that ( ) =. The bounds will be deived in two theoems given below. The fist theoem detemines the bounds fo values of the fill ate geate than 6.5%. The second theoem detemines bounds fo fill ates less than 6.5%. THEOREM : If the equied fill ate F exceeds 6.5%, then: (I) If l <, then the optimal solution will be found in one of the thee intevals (1) min(4, ), () 4, o(3)( ) ( ).

11 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 645 (II) If l >, then the optimal solution will fall in one of the two intevals, (1) l min(4 l, ),o() 4. PROOF: CASE I, l : This case will be poved in thee pats. Case I.1 povides the uppe bound on in tems of, while Case I. povides the uppe bound on in tems of ( ). Case I.3 analyzes the egion when is negative and povides the bound fo as a multiple of. CASE I.1: Conside any. Define u to be the smallest value of such that C n ( ) C n ( ) fo u. Let be such that u. Then by assumption ( + H( ) B +(F F u ) ( )+ ) µ Theefoe, + H( ) B +(F F u ) ( ( )+ ) µ. (15) 1 = + H( ) B +(F F u )(( )+ ) + H( ) B +(F F u )(( )+ ) [ λk1 [ λk1 + (H() B) +(F F u )(1+1/)] +(F F u )( () +1/)]. (16) + (H() B) Define A = + (H() B) +(F F u )(1+1/) +(F F u )( () +1/). (17) + (H() B) As A 1 by assumption, an uppe bound on A can be used to bound u in tems of. Assume that ( ). The case whee the eode level is negative is analyzed in I.. To get the uppe bound on A, we substitute ( )/ =in Eq. (17). Theefoe, A + (H() B) + 3 (F F u) + (H() B) + 1 (F F u) = J( ) + 3 (F F u) J() + 1 (F F u). (18) Cases I.1.1, I.1., and I.1.3 will now povide uppe bounds on A depending on the elative values of l,, and m. CASE I.1.1, l m : Fom Lemma 7, m implies J( m )/ m J( )/. Theefoe, we can substitute J( m )/ m fo J( )/ in (18) without affecting the diection of the inequality. Fom the fist-ode condition J( m )/ m = J ( m ), we obtain m + H ( m )= m + (H( m) B) m.

12 646 Naval Reseach Logistics, Vol. 47 () This implies Theefoe, m = 1 J( m ) = m + (H( m) B) = 1 m m ( H ( m ) (H( ) m) B). m ( H ( m )+ (H( ) m) B). (19) m By the definition of l, / / l. Thus, we can substitute / l fo / in (18) without affecting the diection of the inequality. Substituting the value of J( m )/ m fom Eq. (19) in the denominato of (18) and the value of / l fom Eq. (13) in the numeato of (18), we get A H ( l )+ F Fu + (H() B) + 3 (F F u) 1 (H ( m )+ (H(m) B) m )+ 1 (F F. u) As m, it follows fom Lemma 8 that (H( ) B)/ (H( m ) B)/ m.as l m, it follows fom Lemma 5 that H ( l ) H ( m ). Theefoe, 4 A (F F u) 1 (F F u) =4. CASE I.1., l m and CASE I.1.3, m l :As m,it follows fom Lemma 7 that J( )/ J( )/. Theefoe, fom (18) A J( ) + 3 (F F u) + 1 (F F u) J() 3 (F F u) 1 (F F u) =3. CASE I.: We shall now conside the values of fo which the eode point is negative, i.e.,. Simila to inequality (16), we can define such that 1 = + H( ) B +(F F u )(( )+ ) + H( ) B +(F F u )(( )+ ) + H( ) B +(F F u )( ) () + H( ) B +(F F u )(( )+ ), as ( ) =. Fom Lemma 1 given in the Appendix it follows that ( ()+ ) = 1 + F F( + ) fo. (1) Theefoe, fo F.65, (()+/) is inceasing in. Thus, when 1 + H( ) B + H( ) B ()

13 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 647 is tue, then inequality () is also tue. This implies 1 [ λk1 [ λk1 Theefoe, simila to Eq. (17), define A3 as (H( ) B) + ] + (H() B) ]. (3) A3 = (H( ) B) + + (H() B). (4) The fist-ode condition fo the optimality must hold at,so ( ) = H ( )+ ) (F F u) (( )+ ( H ( )+(F F u ) F 1 ). (5) whee inequality (5) follows fom (1). As l and, it follows fom equations (13), (4), and (5) that A3 l + (H( ) B) ( ) + (H() B) H ( l )+(F F u )/+ (H( ) B) H ( )+(F F u )(F 1/) + (H() B). (6) By assumption and as, it follows fom Lemmas 5 and 8 that A3 (F F u )/ (F F u )(F 1/) (7) 1/ =4. (.65 1/) (8) CASE I.3: Conside any () ( ). Fom Lemma 4 we know that if () ( ), then ()+ ( )+. This implies that () ( ). CASE II, l > : The optimal lies in the ange [ l, ). This pat of the theoem will be poved in Case II.1 which povide bounds fo in tems of l fo the case when the optimal is positive and Case II. which povide bounds fo in tems of fo the case when the optimal is negative. CASE II. 1: Define A1 to be such that Cn( l) C = l n() A1. The uppe bound on A1 gives the uppe bound on in tems of l. We obtain, simila to Eq. (18), A1 l + (H( l) B) l + 3 (F F u) + (H() B) + 1 (F F u) = J( l ) l + 3 (F F u) J() + 1 (F F u). (9)

14 648 Naval Reseach Logistics, Vol. 47 () This case will now be poved in thee pats II.1.1, II.1., and II.1.3. CASE II.1.1, m l and Case II.1., m l :As l m it follows fom Lemma 7 that J( l) l J(). Theefoe, fom (9) we obtain A1 J( l ) l + 3 (F F u) + 1 (F F u) J() 3 (F F u) 1 (F F u) =3. CASE II.1.3, l m : The poof of this pat is simila to Case I.1.1. Substituting the value of J( m )/ m fom Eq. (19) in the denominato and the value of / l fom Eq. (13) in the numeato of (9), we get A1 H ( l )+ F Fu + (H( l) B) l + 3 (F F u) 1 (H ( m )+ (H(m) B) m )+ 1 (F F. u) As l m, it follows fom Lemma 8 that (H( l ) B)/ l (H( m ) B)/ m and fom Lemma 5 that H ( l ) H ( m ). Theefoe, A1 4 (F F u) 1 (F F u) =4. CASE II.: If l, the poof of this case is identical to Case I.. If < l, then the poof is simila to Case I., and we bound the value of such that C n ( l )/C n ( ) 1. The eade will note that the bounds on fo positive values of () given in Theoem ae valid fo all values of F>. Theefoe, only bounds on when () is negative ae pesented in Theoem 3. THEOREM 3: When the optimal is negative and the equied fill ate is less than 6.5%, then the optimal falls in one of the following intevals: (1) If µ/4, then 4 () If >4µ, then 4 (3) If µ/4 < 4µ, then 16µ. PROOF: (1) We conside the values of fo which the eode point is negative, i.e.,. Simila to inequality (16) we define such that 1 = + H( ) B +(F F u )(( )+ µ) + H( ) B +(F F u )(( )+ µ) (H( ) B) (F F u) (F F u )( µ ) + 1 (F F u) (F F u )( () + (H() B). (3) + µ )

15 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 649 Fom the poof fo Case I.1.1 of Theoem we know that (H( ) B) (F F u) + (H() B) + 1 (F F u). (31) Using inequality (31), it can be veified 1 that inequality (3) cannot be tue fo if Fom Eq. (6) given in the Appendix, we get () =(1 F ) Using inequality (33) it follows that inequality (3) holds tue if Reaanging tems in inequality (34), we obtain µ. (3) ( () + µ ) ()+ 1 F. (33) µ. (34) ( (1 F )+µ) 1 (35) (1 F ). µ As µ 1/4 it follows that 1 1 µ (1 F ) >. The maximum value of 1 ( /µ)(1 F ) is 4 when /µ =1/4 and F =. Theefoe, if / 4, inequality (3) cannot hold tue fo any value of. Thus, when /µ 1/4, inequality (3) cannot hold fo values of 4. 1 () 4µ: Simila to inequality () in the poof of Theoem, we get 1 + H( ) B +(F F u )(( )+ ) (36) + H( ) B +(F F u )(( )+ ). Fom Eq. (6) in the Appendix, we get () =(1 F ) ()+ (37) 1 Fo positive A, B, X, Y, A X and B Y,ifA/B and X/Y, then (A X)/(B Y ).

16 65 Naval Reseach Logistics, Vol. 47 () and =(1 F ) Subtacting Eq. (37) fom (38), we obtain ( )+. (38) ( ) = ( )+ + ( )+ µ ()+ (39) (4) Since 4µ in this case it follows fom inequality (4) that ( )/ 1 4. Using this in the denominato of inequality (36), the poof now follows in a manne simila to Case I.1.1 in Theoem. (3) µ/4 4µ: The poof follows fom pat 1 and. 4. ALGORITHM We utilize the bounds deived in the pevious section in the algoithm given below fo detemining the optimal values of and. The algoithm is fo the case when the equied fill ate is geate than 6.5%. A simila algoithm can be constucted when the fill ate equied is less than 65.5%; and the details ae omitted due to consideations of space. Define C con () = λk ( 1 + H()+(F F u) ()+ ). This function diffes fom the oiginal cost function in only a constant that is C con () =C 1 ()+(F F u )µ. Let ɛ be the maximum desied eo in C con (). Let N = log() log(1 + ɛ) (41) and =ɛ /. (4) Algoithm fo Detemining (, ) to Satisfy a Given Fill Rate F 1. If p/(p + h) F, utilize the FZ algoithm to compute the optimal values of (, ). Othewise, poceed to step.

17 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 651. Detemine the EO = λk/h and the value of. A simple line seach can be used to detemine the value of, i.e., to find the value of such that () =. Detemine the exact value of the lowe bound fo by solving the convex pogam { λk1 l = ag min + H()+(F F u) }. 3. If EO<, then pefom a seach (see step 3.1 and 3.) in the two intevals, I 1 =[( ), ( )] and I =[, 4 ]. Othewise, let I 3 =[EO,4EO] and execute step Seach in I 1 : Let i = ( ) (i/n),i =1,,...,N. Detemine i s coesponding to each of the i s such that the fill ate constaint is met [see Eq. ()]. Thee ae thee cases to conside. If l > N, go to step. If l < 1, set i =1. Else, if 1 l N, detemine i such that i l < i+1. Evaluate C con () at each of the ( i, i ) pais, fo i = i,i +1,...,N. 3.. Seach in I : Let i = (i/n),i =1,,...,N. Detemine i s coesponding to each of the i s such that the fill ate constaint is met [see Eq. ()]. Detemine the value of i as done in step 3.1. If ( N ), then let M = N. Else, let N = max i : ( i ), and M = N + 6, ɛ i = +[i 1 N ], i N +1. Evaluate C con () at each of the ( i, i ) pais, fo i = i,i +1,...,M. Poceed to step Seach in I 3 : Let i = EO (i/n),i =1,,...,N. Detemine i s coesponding to each of the i s such that the fill ate constaint is met [see Eq. ()]. Detemine the value of i as done in step 3.1 and the value of M and i as in step 3.. Evaluate C con () each of the ( i, i ) pais, fo i = i,i +1,...,M. 4. Choose the (, ) pai that gives the lowest value of C con () in steps 3.1 and 3. o step 3.3. end Let the algoithm etun the ode quantity k. Define the eo fom using k as δ, i.e., δ = C con( k ) C con ( ).

18 65 Naval Reseach Logistics, Vol. 47 () LEMMA 9: The (, ) pai poduced by the above algoithm gives an eo with espect to C con ( ) that is at most ɛ pecent fo equied fill ates geate than 6.5%. PROOF: Theoem guaantees that the optimal values of (, ) will lie in the anges I 1 and I o in I 3 as the case might be. Suppose the optimal ode quantity lies between two adjacent values of s as detemined by the algoithm. Let these values of be j and j+1, with j < < j+1. Notice that by convexity and the choice of l the function ( / + H()+(F F u )(()+/)) is inceasing in fo l. Thus, δ λk/ j + H( j )+(F F u )( j + j /) λk/ + H( )+(F F u )( + /) 1 (F F u )( j ) λk/ + H( )+(F F u )( + /). (43) If the cost minimizing solution lies in I 1 then the eo bound follows fom the choice of the values of i s, i.e., fom (41) and (43) δ j j+1 j+1 ɛ. If the cost minimizing solution lies in I o I 3, we note that fom Lemma 4 that j + j + j j j+1 j. (44) If ( N ), then fom (41), (43), and (44), δ ( j+1 j ) j ( 1+ɛ 1) ɛ. If ( N ), then, fom Lemma 1 in the Appendix and fom the fact that F 6.5%, it follows that (( i )+ i /) is inceasing in i fo i N +1, whee N is as defined in steps 3. and 3.3 of the algoithm. Also, by definition (step 3. of the algoithm) N+1 = and ( ) =. Thus, ( i )+ i Theefoe, fom (4), (43), (44), and (45), ( )+ = fo i N +1. (45) δ (F F u)( j+1 j ) (F F u )( /) = ɛ / / ɛ.

19 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems CONCLUSION In the peceding sections, we consideed a geneal fom of the poblem of detemining the optimal values of (, ) subject to a fill ate constaint. The bounds ae also applicable to two special cases. Fist, the majoity of the liteatue deals with the special case in which thee is no backode penalty, i.e., p =. This implies that F u =. All ou bounds ae applicable fo this case as well since the bounds ae independent of p. Second, if the fill ate constaint wee not imposed, then F could be allowed to appoach F u fom above in Theoem. As a consequence and fom Lemma 1, the bounds will continue to hold. APPENDIX PROOF OF LEMMA 5: (1) By diffeentiating H() with espect to we get ( ()+ ) H G 1 (()+) G () 1 (y) dy () = + (G 1(()+) G 1 (())) (). (46) As () is the solution to ( + + G 1 (y) dy ) =, it follows that G 1 (()+) G 1 (())=. (47) Theefoe, H () = ( G 1 (()+) ()+ G () 1 (y) dy ). (48) Fom Eq. (47) and because G 1 ( ) is convex, it follows that G 1 (()+) G 1 (y) fo y [(),()+]. This poves that H () is nonnegative. () By definition, H(, ) =F S(, )+(1 F )Ī(, ). Fom Eq. (9), we get Ī(, ) = S(, )+/+ µ. Theefoe, H(, ) = S(, )+(1 F )( + µ). Zipkin [19] shows that if f( ) >, then S(, ) is stictly convex in (, ). Theefoe, H(, ) is stictly convex fo in (, ), i.e., fo a given α (, 1), ( 1,( 1 )), and (,( )), αh( 1,( 1 )) + (1 α)h(,( )) >H(α 1 +(1 α),α( 1 )+(1 α)( )). (49) Fom Eq. (1) we know that () minimizes H(, ), foagiven; hence H(α 1 +(1 α),α( 1 )+(1 α)( )) H(α 1 +(1 α),(α 1 +(1 α) )). (5) It follows fom (49) and (5) that αh( 1,( 1 )) + (1 α)h(,( )) >H(α 1 +(1 α),(α 1 +(1 α) )). This poves that H() is a stictly convex function of. (3) It follows fom the convexity of H() that H () lim H (), fo all >. Fom Eq. (48) ()+ lim H () = lim G 1 (()+) lim () G 1 (y) dy. (51)

20 654 Naval Reseach Logistics, Vol. 47 () We know fom Lemma 4 that lim G 1 (() +) = G( ) =. Theefoe, we use L Hospital s ule fo computing one sided limits (see Theoem 4, page 64 in [15]) to compute the limit fo the fist quantity on the ight-hand side of Eq. (51). Diffeentiating the numeato and denominato sepaately, we obtain Similaly, lim ()+ () Fom Eqs. (5) and (53), G 1 (()+) lim G G 1 (y) dy = lim = lim G 1 (()+)(1 + ()). (5) 1 G 1 (()+) = lim G 1 (()+)(1 + ()). (53) lim H () = lim G 1 (()+)(1 + ()). (54) It follows fom Zheng [18] that lim G 1 () = 1 F. Fom Lemma 4 we get 1+ () 1. Theefoe, lim H () (1 F )/. PROOF OF LEMMA 7: (1) By the definition of m, ( ) J() = 1 [ J ( m) J(m) ] =. = m m m This implies that J ( J(m) m)=. (55) m Let us suppose that m is not unique and that thee ae two values 1 and that satisfy Eq. (55). Assume without loss of geneality that > 1. It follows fom Eq. (55) that J ( )=J( ) and 1 J ( 1 )=J( 1 ). These equations imply that J( ) J( 1 )= J ( ) 1 J ( 1 ). (56) Fom Lemma 6, J() is stictly convex. Theefoe, J( ) J( 1 ) <J ( )( 1 ). (57) Subtacting Eq. (57) fom Eq. (56), we get > (J ( ) J ( 1 )) 1. This is a contadiction as J ( ) J ( 1 ) due to the convexity of J(). This shows that m is unique. () To pove the second pat of this lemma ( ) J() = J () + [ ] J() J (). (58) Fom Lemma 6, J ()/. Fom the fist pat of this lemma m is unique; theefoe ( / )(J()/) can change sign only once. Theefoe, ( / )(J()/) fo m, which implies that J()/ J () fo m. Hence using (58) we get that ( / )(J()/) >, fo m. Finally, fom the fist pat of this lemma, ( / )(J()/), fo m. This poves the second pat of the lemma. PROOF OF LEMMA 8: ( ) H() B = 1 ( H () H() B ). As H() is an inceasing convex function (fom Lemma 5); theefoe, fo any >, weget H () (H() H()). (59) As B = H(), it follows fom (59) that ( ) H() B.

21 Agawal and Seshadi: Sevice-Constained (, ) Inventoy Systems 655 LEMMA 1: When, F () = 1+ F( + ) 1+F. PROOF: Fom the definition of () in Eq. (6) we obtain ()+ (1 F ) = () = ()+ + () ()+ = ()+. (6) Taking the deivative of both sides of Eq. (6) with espect to, we obtain (1 F ) = ()+F c (()+)(1 + ()) = ()F(()+)+F c (()+). (61) Reaanging the tems in (61), we get () = 1+ F 1+F. (6) F( + ) REFERENCES [1] S. Axsate, Using the deteministic EO fomula in stochastic inventoy contol, Manage Sci 4(6) (1996), [] M.S. Bazaaa, H.D. Sheali, and C.M. Shetty, Nonlinea pogamming theoy and algoithms, Wiley, New Yok, [3] S. Bowne and P. Zipkin, Inventoy models with continuous, stochastic demand, Ann Appl Pobab 1 (1991), [4] S. Dhamadhikai and K. Joag-Dev, Unimodality, convexity, and applications, Academic, New Yok, [5] A. Fedeguen and Y.S. Zheng, An efficient algoithm fo computing an optimal (, ) policy in cont. eview stochastic inventoy systems, Ope Res 4 (199), [6] G. Gallego, New bounds and heuistics fo (, ) policies, Manage Sci 44() (1998). [7] G. Hadley and T.M. Whitin, Analysis of inventoy systems, Pentice-Hall, Englewood Cliffs, NJ, [8] M. Moses and S. Seshadi, Policy mechanisms fo supply chain coodination, IIE Tans Ope Eng 3(3) (), [9] S. Nahmias and S.A. Smith, Optimizing inventoy levels in a two-echelon etaile system with patial lost sales, Manage Sci 4(5) (1994), [1] D.E. Platt, L.W. Robinson, and R.B. Feund, Tactable (, R) heuistic models fo constained sevice level, Manage Sci 43(7) (1997). [11] K. Rosling, Applicable cost ate functions fo single item inventoy contol, woking pape, Depatment of Industial Engineeing, Lund Univesity, P.O. Box 118, S-1 Lund, Sweden, [1] K. Rosling, The squae-oot algoithm fo single-item inventoy optimization, woking pape, Depatment of Industial Engineeing, Lund Univesity, P.O. Box 118, SE-1 Lund, Sweden, [13] R. Sefozo and S. Stidham, Semi-stationay cleaing pocesses, Stochastic Pocess Appl 6 (1978), [14] E.A. Silve and T.G. Wilson, Cost penalties of simplified pocedues fo selecting eode points and ode quantities, Int Tech Conf Poc, Ameican Poduction and Inventoy Contol Society, Washington, DC, 197, pp [15] D.W. Widde, Advanced calculus, nd ed., Dove, New Yok, [16] C.A. Yano, New algoithms fo (, ) systems with complete backodeing using a fill-ate citeion, Nav Res Logistics uat 3 (1985),

22 656 Naval Reseach Logistics, Vol. 47 () [17] H. Zhang, A note on convexity of sevice-level measues of the (, q) system, Manage Sci 44(3) (1998), [18] Y.S. Zheng, On popeties of stochastic inventoy systems, Manage Sci 38 (199), [19] P. Zipkin, Inventoy sevice-level measues: Convexity and appoximations, Manage Sci 3 (1986), [] P. Zipkin, Stochastic leadtimes in continuous-time inventoy models, Nav Res Logistics uat 33 (1986),