Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order. Such a cos elemen adds a concave componen ino he cos funcion. The convexi argumen for he newsvendor model is no longer applicable; neiher does he opimali of he base-sock polic. Wha is he effec of he fixed ordering cos? Compared o he case of no fixed ordering cos, here is no poin o order if is larger han he expeced shorage cos for no ordering. Thus, we end o order less ofen, and order a larger invenor posiion han he case of no fixed ordering cos. If we order, we end o order larger quani. The (s, S) polic is a simple invenor managemen scheme ha maches he requiremen. Under he (s, S) polic, we order o make an invenor level S if he invenor on hand is less han or equal o s, and do nohing oherwise. We are going o give sufficien condiions under which he (s, S) polic is indeed opimal. 8.1. Inuiion on he Opimali of he (s, S) Polic To ge he inuiion of he problem, consider a one-period problem wih all coss discouned o he beginning of he period. Adop he same noaion as in he newsvendor problem. Le v T (x) be he erminal cos if here are x iem lef; le = c + he(-d) + + E(D-) + + v T (-D)). (1) is he expeced oal discouned cos less he fixed ordering cos if we sar a period wih no on-hand iem and we bu iems for he period. Suppose ha we have x iems on hand a he beginning of he period. If we decide o bu -x iems for he period, our expeced discouned cos for he period is hen + cx. If we decide no o bu an iem, our expeced discouned cos for he period is hen he(x-d) + + E(D-x) + + v T (x-d)) = G(x) cx. Compared o doing nohing, i is beer o bu -x iems if + < G(x); oherwise, i is beer no o order. If ordering is beer, he opimal invenor posiion is given b ha minimizes in (1). The argumen shows ha i suffices o work wih G and o make he ordering decision, and o decide he ordering quani laer from G, if we order a all. The approach is wha we ake below wihou firs explicil worring abou he acual ordering quani x. Suppose ha is convex as shown in Fgiure 1. If we sar he period wih e iems on hand, we will no place an order. Such an acion leads o a higher cos ( > G(e) for > e, no o menion he addiional fixed ordering cos ). The same argumen applies o an > e. If we sar he period wih b iems on hand, i is beneficial o place a new order if here is 1
no fixed ordering cos, bu is no so wih he given. The same argumen applies o an s S and i is no opimal o re-order for in his range. If we sar he period wih a iems on hand, ordering up o S saves G(a) G(S) > 0. This applies o an < s. G 0 (x) a s b S e s S x Figure 1. Convex Figure 2. The shape of he acual cos funcion G 0 when (s, S) polic is Le G 0 be he cos funcion afer he fixed and variable ordering li d coss have been included. As shown in Fgiure 2, once he (s, S) is adoped, he expeced discouned cos is no longer convex. If we have a muli-period problem, he convexi of he expeced cos funcion is los afer one period, and he argumen for he newsvendor problem is no longer applicable. In he following, le us ge a feel of he shape of G for (s, S) o be opimal. Figure 3 shows ha unlike he case of zero fixed ordering cos, (s, S) polic can be opimal even if he global minimum is no righ mos. On he oher hand, as shown in Figure 4, (s, S) ma no be opimal even if he global minimum is righ mos. s S Figure 3. Opimal (s, S) polic when he global minimum of G is no righ mos a b d e l Figure 4. Opimal inv. level = b or l, depending on he invenor level, wheher i is less han a or beween d and e a b d e l On closer examinaion, i is he relaive posiions and magniudes of he minima ha maer. Check ha for he following, he (s, S) polic is opimal. s S a Figure 5. The relaive posiions and magniudes of he minima 2
The opimali of he (s, S) polic is characerized b he following sufficien condiions (2) and (3) ha mus hold a he same ime: Take S o be he global minimum of. Se s = min{u: G(u) = +G(S)}. Then For an local minimum a of G such ha S < a, +G(S) for s S. (2) +G(a) for S a. (3) For < s, he shapes and values (which mus be no less han +G(S)) are irrelevan o he opimali of he (s, S) polic. The above sufficien condiions ensure he opimali of he (s, S) polic for a period. As discussed for Figure 2, when he opimal (s, S) polic is applicable, he fixed ordering cos and he variable purchasing cos will disor funcion G o urn i o he acual cos funcion G 0. Even if G saisfies (2) and (3), G 0 ma no be so. If his G 0 serves as par of he cos funcion for an earlier period in a muli-period problem, he disorion of G 0 ma make (s, S) polic non-opimal for he earlier period. From he above discussion, (s, S) poic is opimal for a muli-period problem if for an period, condiions (2) and (3) can be perserved. If so, he srucural proper of G can be perserved in he dnamic programming recursion and in ever period, he (s, S) polic is opimal (hough he values of s and S ma change across periods). In he following, we discuss -convex funcions ha ensure he perservaion of condiions (2) and (3) across periods. In fac, -convex funcions saisf condiions sronger han (2) and (3). 8.2. -Convexi and -Convex Funcions The -convexi is an exension of convexi such ha a -convex funcion saisfies (2) and (3) and hese condiions are preserved b he dnamic programming recursion. The original definiion of a -Convex funcion b Scarf is: Definiion 8.2.1. A funcion f is -convex if for an 0 < < 1, x, f( x + (1- )) f(x) + (1- )(f() + ). The inerpreaion is ha for an x, such ha x, funcion f lies below f(x) and f()+ for all poins on (x, ). An equivalen definiion of -convexi is Definiion 8.2.2. A funcion f is -convex if (i) for an 0 < a and 0 < b, f(x) + ( f ( x) f ( x b)) f(x+a) +, or a b 3
f ( b) f ( a) b a f ( c) f ( b) c b (ii) for an a b c,. A differeniable funcion f is -convex if for an x, f(x) + f '(x)(-x) f() +. (4) Exercise 8.2.1. Show ha Definiions 8.2.1 and 8.2.2 are equivalen, and ha (4) holds for differeniable -convex funcions. Exercise 8.2.2. Show ha a -convex funcion saisfies (2) and (3). A -convex funcion needs no be coninuous; i simpl canno have a posiive jump, nor oo big a negaive jump (Figure 6). (a) (b) (c) Figure 6 (a): A -convex funcion; (b) and (c) non--convex funcions Exercise 8.2.3. Properies of -Convex Funcions (a). A convex funcion is 0-convex. (b). If 1 2, a 1 -convex funcion is 2 -convex. (c). If f is -convex and c > 0, hen cf is k-convex for all k c. (d). If f is 1 -convex and g is 2 -convex, hen f+g is ( 1 + 2 )-convex. (e). If f is -convex and c is a consan, hen f+c is -convex (f). If f is -convex and c is a consan, hen h where h(x) = f(x+c) is -convex. (g). If f is -convex and D is random, hen h where h(x) = E[f(x-D)] is -convex. (h). If f is -convex, x <, and f(x) = f() +, hen for an z [x, ], f(z) f()+. The las proper is a srong one. I sas ha for a -convex funcion f, for an, f crosses f() + onl once (from above) in (-, ). 4
The -convexi is sufficien bu no necessar for he opimali of (s, S) polic, i.e., an (s, S) polic is opimal even if is no -convex. Figure 7. Opimal (S, s) polic for non--convex funcions 8.3. An Opimal (s, S) Polic The dnamic programming recursion can be pu as f (x) = cx + min{g (x), min[ G ( )] }. (5) We will show ha if v T is -convex, hen he (s, S) polic is opimal for all he periods when here is a fixed ordering cos. (Periods ma have differen (s, S)-values.) As in he newsvendor problem, we assume ha h 0. Le us firs exend (1) o he general sage where x G () = c + he(-d) + + E(D-) + + E[f +1 (-D)], (6) and f +1 is -convex. We are going o show ha G () is -convex. Lemma 8.3.1. If f +1 is -convex, hen G () is -convex in (5). Proof. From Exercise 8.2.3 (g) and (c), E[f +1(-D)] is a -convex funcion of. As discussed for he newsvendor problem, c + he( D) + + E(D ) + is a convex funcion, which is 0-convex b Exercise 8.2.3 (a). Furher b Exercise 8.3.1 (d), he sum of he wo erms, G (), is -convex. Lemma 8.3.2. If G is -convex, hen an (s, S) polic is opimal. Proof. Remember ha in single period-problem, i suffices o work wih G () o deermine he opimal ordering polic; he expeced oal cos discouned o he beginning of he period is hen G () cx for an iniial invenor x of he period. The onl difference is ha we need o consider he fixed cos if we order. To find he opimal polic for min{g (x), min[ G ( )] }, le he global minimum of G () as S and se s = min{u: G (u) = G (S)+}. There are hree cases, depending on (s, S), < s, or > S. For all (s, S), G (S) G () G (S)+; S being he minimum gives he x 5
firs inequali, and G being -convex gives he second inequali. I is no opimal o reorder for (s, S). For < s, from Exercise 8.2.3(h), G crosses G (S)+ onl once, from above a S. Thus, G () > G (S)+, and i is opimal o order up o an invenor posiion S. For > S, we claim ha i is no opimal o order. Suppose oherwise, i.e., a 1 > S, i is opimal o order up o 2. Then G ( 1 ) > G ( 2 )+. However, in his case, G ( 1 ) lies above he segmen joining G (S) and G ( 2 )+, which conradics he fac ha G () is -convex. Le opimal, and x G ( ) min{g (x), min[ G ( )] }. From Lemma 8.3.2, he (s, S) polic is x ( S), for x s, ( x) (7) ( x), o. w. x Lemma 8.3.3. If G () is convex, G ( ) is -convex. Proof. Consider wo values 1 and 2 such ha 1 < 2. We are going o show ha for all ( 1, 2 ), G ( ) lies below he segmen joining ( 1, ( 1)) and ( 2, ( 2)). From (7), here are hree cases: (i) no order a boh 1 and 2 ; (ii) an order a 1 bu no a 2, and (iii) an order a boh 1 and 2. Check ha i is impossible o order a 2 bu no 1. Case (i): When here is no order a 1, G ( ) = G () for all 1. Because G () is - convex, Naurall, for all ( 1, 2 ), G () lies below he segmen joining ( 1, G ( 1 )) and ( 2, +G ( 2 )). Case (ii): Le 1 < s < 2, i.e., i is opimal o order in [ 1, s] bu no in s, ]. Wihin ( 2 (, s 2], G ( ) = G (). The -convexi of G dicaes ha G lies below he segmen joining (x, G (x)) and ( 2, +G ( 2 )) for an x (s, 2 ]. This shows ha G lies below he segmen joining (x, ( x )) and ( 2, ( 2)) for an x (s, 2 ]. To esablish he resul for [ 1, s], recall ha s = min{u: G (u) = G (S)+}; ha S is he global minimum of G ; ha G lies below (s, G (s)) and (S, G (S)+). B consrucion G (S) G ( 2 ), i.e., G (s) = G (S)+ G ( 2 )+, i.e., G lies below (s, G (s)) and ( 2, G ( 2 )+). Noe ha (s, G (s)) = (s, ()) s and ( 2, G ( 2 )+) = ( 2, ( 2) +), i.e., G lies below (s, ()) s and ( 2, ( 2) +). For x [ 1, s), ( x ) = G (S)+ = (). s Cerainl G lies below (x, ( x )) and ( 2, G ( ) +). Case (iii): When boh 1 and 2 order, G ( ) = G (S) for all [ 1, 2 ]. Thus, ( ) ( 1 ( 2 2 G lies below he segmen connecing ( 1, G )) and ( 2, + G )). The inuiion of Lemma 8.3.3 is illusraed graphicall as in Figure 8. 6
G () 1 s 2 S 2 G ( ) 1 s S 2 Figure 8. The inuiion of Lemma 8.3.3 2 Theorem 8.3.4. Suppose ha v T is coninuous and -convex. Then for each period, he opimal reurn funcion is -convex and here exiss an opimal (s, S) polic. Proof. When v T is coninuous and -convex, Lemma 8.3.1 shows ha G N is -convex; Lemma 8.3.2 shows ha an (s, S) polic is opimal; Lemma 8.3.3 shows ha f N is coninuous and -convex. All hese properies preserve in ever period. 7