PRICE-DIRECTED CONTROL OF REMNANT INVENTORY SYSTEMS DANIEL ADELMAN The Unversty of Chcago, Graduate School of Busness, 1101 East 58th Street, Chcago, Illnos 60637, dan.adelman@gsb.uchcago.edu GEORGE L. NEMHAUSER Logstcs Engneerng Center, School of Industral and Systems Engneerng, Georga Insttute of Technology, Atlanta, Georga 30332, gnemhaus@sye.gatech.edu (Receved January 1997; revsons receved January 1998, June 1998; accepted August 1998) Motvated by make-to-order cable manufacturng, we descrbe a remnant nventory system n whch orders arrve for unts of raw materal that are produced-to-stock. As orders are satsed, the partally consumed unts of materal, or remnants, are ether scrapped or returned to nventory for future allocaton to orders. We present a lnear program that mnmzes the long-run average scrap rate. Its dual prces exhbt many ratonal propertes, ncludng monotoncty and superaddtvty. We use these prces n an nteger-programmng-based control scheme, whch we smulate and compare wth an exstng control scheme prevously used n practce. 1. INTRODUCTION In ths paper we ntroduce a model and present polces for operatng an nventory system that generates usable remnants. We rst gve a generc descrpton of ths system. Then we dscuss an ndustral settng where t arses and outlne the rest of the paper. 1.1. System Descrpton Orders for lengths of materal unts arrve to a manufacturng faclty that stocks unts of varous lengths (depcted n Fgure 1). Let C = {1; 2;:::;n} be the set of order lengths, and suppose that orders for length j C are demanded at some rate j 0. If length j s not ordered, then j = 0. An order for a unt of length j can be satsed by any unt n nventory havng length j. At the start of each perod (one shft), unts are allocated to orders. After processng n a producton faclty, each remnant generated n ths case one of length j must be scrapped f t s too short to be reallocated, and otherwse s returned to nventory for future allocaton to another order. Scrapped unts leave the system and are not recycled. Raw unts are produced to replensh length consumed and arrve at aggregate rate wth a fracton P havng length. Let F be the set of raw and remnant lengths, ncludng the null length 0 to represent no scrap. Wthout loss of generalty, we assume F C {0} and let P = 0 f length s not produced as a raw unt. The problem we consder n such remnant nventory systems s to satsfy all orders wth allocatons so as to mnmze the long-run average scrap rate. To llustrate the dynamcs of ths remnant ow, the network n Fgure 2 depcts all possble remanng lengths of a unt for a factory that produces raw unts only at length 20 (.e., P 20 = 1) and has orders only for lengths 3 and 5, at rates 3 = 90 and 5 = 10. Horzontal arcs represent satsfacton of orders for length 5, whle all other arcs represent satsfacton of orders for length 3. The lengths 19; 18; 16; and 13 do not appear because they are not attanable. The bold arcs track a unt of length 20 from the tme t s produced untl t s scrapped. (The dashed arcs wll be dscussed later.) The unt s rst allocated to an order for length 3 and after some producton delay returns to nventory as a remnant of length 17. Next, ths 17 s allocated to another order for length 3 and returns as a remnant of length 14. Upon allocaton to three more orders for length 3, a remnant of length 5 s returned. Ths unt of length 5 can agan be allocated to an order for length 3, returnng a remnant of length 2 that must be scrapped. Alternatvely, ths unt of length 5 can be allocated to an order for length 5 to generate zero scrap. We can summarze the polcy depcted by the bold arcs as requrng that all orders for length 5 be satsed wth unts of length 5, and gven ths requrement, that orders for length 3 be satsed by any unt avalable. Now suppose that when an order for length 5 s to be sats- ed, only two unts are n nventory, and they have lengths 5 and 6 (under some polcy that generates them). To mnmze scrap, clearly we would prefer to allocate the unt of length 5. However, suppose now that the only two unts avalable are lengths 20 and 15. Whch unt s preferable, f ether? In ths case the answer s not so clear because t depends on the scrap that s lkely to be produced n the future by a remnant of length 15 versus a remnant of length 10. In ths paper we provde a methodology for answerng such questons. 1.2. Motvaton and Outlne Ths work s based on our development of an ntegerprogrammng (IP) based system for controllng a large ber-optc cable manufacturng plant (Adelman et al. Subject classcatons: Producton=schedulng, cuttng stock: dynamc. Networks, generalzed networks: dualty theory. Programmng, nteger, applcatons: ber-optc cable manufacturng. Area of revew: MANUFACTURING OPERATIONS. Operatons Research,? 1999 INFORMS 0030-364X/99/4706-0889 $05.00 Vol. 47, No. 6, November December 1999, pp. 889 898 889 1526-5463 electronc ISSN
890 / ADELMAN AND NEMHAUSER Fgure 1. A remnant nventory system. 1999). The new system mplemented n early 1996 has led to more than a 30% reducton n scrap costs. In ths context, unts are optcal bers that are requred by orders for ber-optc cables of customer-speced lengths. The requred transmsson propertes preclude the splcng of optcal bers wthn the cables, and so allocated bers must be at least as long as the ordered lengths. Consequently, remnant bers of varous lengths are generated as cables are manufactured. These bers are stocked n nventory and are contnually replenshed wth new bers,.e., raw unts. Durng the manufacturng process of optcal bers, random breakages and aws occur and so a range of ber lengths s produced (Murr 1992). Consequently, we cannot solve the scrap problem by smply changng the raw ber lengths produced to match orders, but nstead we must control the remnants. In each perod the IP explctly consders only the current perod s decsons, rather than decsons spannng an extended horzon. However, the long-run consequences of these short-term decsons are accounted for usng a functon that values unts accordng to length. To see how such a functon s used, let V 20 ;V 15 ; and V 10 be the values of unts havng lengths 20; 15; and 10, respectvely. Then V 20 V 15 s the net decrease n the total value of the nventory when satsfyng an order for length 5 wth a unt havng length 20. Fgure 2. Possble remanng lengths of a unt for an example factory. If V 20 V 15 V 15 V 10 ; then allocatng the unt of length 20 s preferable to allocatng the unt of length 15. If there s equalty, then we are nderent. We obtan the functon V through an auxlarly lnear program, called the Remnant Network Flow Model (dscussed n 2) that values unts wth respect to the market for them nsde the factory. Orders purchase the unts they need at prces reectng the factory-wde objectve of mnmzng the long-run average scrap rate. Our central goals n ths paper are to: 1. provde a formal methodology for obtanng the value functon, 2. prove many ratonal propertes the value functon satses, and 3. characterze the eect of ts repeated use over tme n our nteger program. The propertes that arse, gven n 3, not only heghten the ntutve economc appeal of our approach, thereby enhancng ts acceptance by management, but as we show n 3.2 are actually essental to proper decson-makng. However, because of lnear programmng degeneracy, there s typcally an nnte set of alternatve optma on whch one or more propertes are volated. To deal wth ths problem, we develop a class of rght-hand-sde perturbatons n 3.3 that guarantee the dervaton of a value functon satsfyng all of the propertes. In 4 we present our nteger programmng model for makng perodc decsons along wth smulaton results demonstratng that our IP-based, prce-drected approach generates sgncantly less scrap than an exstng remnant control algorthm. We also dscuss mplementaton n a stochastc envronment. 1.3. Lterature Revew Our lnear program s related to ones gven n Courcoubets and Rothblum (1991), Krchagna et al. (1998), and Gans and van Ryzn (1997), whch are pathwse formulatons n the tradton of cuttng-stock problems (Glmore and Gomory 1961, Dyckho 1981, Dyckho 1990, Cheng et al. 1994). In constrast wth cuttng-stock problems where remnants are typcally scrapped, our remnants are usable and consumed over tme. Vewng remnants as partally consumed bns, our work s related to on-lne bn-packng (Galambos and Woegnger 1995), where bns are packed sequentally through tme. However, ths lterature explores performance bounds for smple heurstcs, whch s qute derent from our prce-drected approach. The paper by Schethauer (1991) dscusses how to ncorporate remnant values nto the cuttng-stock problem but does not explan how to compute these values. Prce-drected methods, such as the Dantzg Wolfe decomposton (Dantzg and Wolfe 1960), for solvng mathematcal programs have been known for some tme. The derence here s n the use of these prces not to solve a problem nstance, but rather to construct a control polcy for a dynamc system. Roundy et al. (1991) develop a prce-drected methodology for job shop schedulng,
ADELMAN AND NEMHAUSER / 891 Fgure 3. Conservaton of ow at a node n the network G =(F; A). by constrant (2) n the followng lnear program called the Remnant Network Flow Model (PSCRAP), whch mnmzes the long-run average scrap rate : (PSCRAP) Mn = F S ; (1) P + {k:(k; ) A} Y k; = {k:(; k) A} Y ; k + S F; (2) Y ; k = j j C; (3) (; k) A j Y ; k 0 (; k) A; (4) S 0 F; (5) 0: (6) where machne prces come from Langrangan multplers for dualzed constrants of an nteger program. Ther operatng polcy uses these prces heurstcally n solvng local sngle-machne schedulng subproblems. Other work related to allocatng bers n ber-optc cable manufacturng ncludes Johnston (1993) and Northcraft (1974), who present heurstcs smlar to one gven n 4.3. The papers by Gue et al. (1997), Clements et al. (1997), and Nandakumar and Rummel (1998) present other problems that arse n ber-optc cable manufacturng. Constrants (3) ensure that orders are met, statng that the rate at whch unts are allocated to orders of length j must equal the rate j at whch they are demanded. Note that the LP forces all unts to be allocated or scrapped eventually,.e., n the long run there s no nventory holdng. In any feasble soluton, s the consumpton rate of raw unts ether through scrappng or allocaton, as the followng conservaton law expresses: PROPOSITION 1. The Unt Length Flow Conservaton Law F P = j C j j + F S (7) 2. THE REMNANT NETWORK FLOW MODEL 2.1. The Prmal Model We now present a lnear programmng model whose optmal dual prces yeld the value functon V. Dene a network G =(F; A); where the set of nodes s the set of lengths for unts F and the set of arcs s dened by A = {(; k): ( k) C; ; k F}: So A s the set of all possble allocatons. For each (; k) A let the decson varable Y ; k represent the long-run average rate at whch unts of length are transformed nto remnants of length k (by satsfyng an order for length k). Also let A j represent all possble allocatons to an order for length j; dened by A j = {(; k) A: k = j} j C. Note that some nodes represent unts of length F that must be scrapped because they are too short to satsfy any orders. In general, any unt may be scrapped f, for example, unts of that length buld up faster than they can be used. Thus, we gve each node F an outgong arc S ; representng the rate at whch unts of length are scrapped. Each node F also has an ncomng arc wth ow P, where s a global decson varable specfyng the producton rate of raw unts and P s gven. Of course, unts of length may be suppled as remnants from other nodes k and may also be allocated to orders to produce remnants of length k. Ths conservaton of ow, depcted n Fgure 3, s modeled holds for any feasble soluton (; Y; S) to (PSCRAP). PROOF. Multplyng each Equaton (2) by and summng over all F; we obtan F {k: (; k) A} Y ; k {k: (k;) A} Y k; = F (P S ): Now each varable Y ; k appears n the left-hand sde of ths equaton twce: once wth coecent and once wth coecent k. Thus, usng (3) we may rewrte the left-hand sde as ; k = (; k) A( k)y jy ; k = j j : j C (; k) A j j C It follows that mnmzng the long-run average scrap rate s equvalent to mnmzng the rate at whch unts are consumed,.e., (PSCRAP) s equvalent to (PMU) Mn ; subject to (2) (6): Because all nodes F have a scrap arc, there s a trval necessary and sucent feasblty condton, whch we assume holds.
892 / ADELMAN AND NEMHAUSER PROPOSITION 2. Prmal Feasblty: (PMU) and (PSCRAP) are feasble f and only f F; such that P 0 and max { j C: j 0} j: It now follows that (PMU) and (PSCRAP) have optmal solutons. In Fgure 2, because length 20 s the only raw length produced, the ow on the arc enterng node 20 s. The ows on the dashed arcs comng out of nodes 2,1, and 0 are S 2 ;S 1 ; and S 0, respectvely. All ntermedate arcs represent the ows Y ; k. By usng only the bold arcs n Fgure 2, an optmal soluton can be constructed to (PMU) wth = 16:667. Hence, Y20; 17 = Y 17; 14 = Y 14; 11 = Y 11; 8 = Y 8; 5 = 16:667 by ow conservaton (2). At node 5 the ow splts so that Y5; 0 = 10; and Y 5; 2 =6:667. Consequently, S 2 =6:667; S1 =0; and S 0 = 10. It s easly vered that constrants (3) are satsed. The optmal nput rate s = 16:667. Thus, the optmal scrap rate s =2S2 = 13:333; whch can be vered by applyng unt length ow conservaton (7). As a percentage of total length consumed, =20 = 4% s scrap. We dscuss the ntuton behnd ths prmal optmal soluton n 3.2, n the context of a correspondng dual optmal soluton. As we shall see, we can convert ths prmal optmal soluton to another prmal optmal soluton that has postve ow on (8; 3) (3; 0) by shftng ow from (8; 5) (5; 0). Because there are an nnte number of alternatve prmal optmal solutons that use arcs (8,3), (8,5), (5,0), and (3,0) n varous proportons, there s no ratonal bass for restrctng consderaton only to those polces that acheve the partcular rates Y; k n any one of those solutons. These consderatons motvate us to consder the dual. 2.2. The Dual The dual of (PMU) s (DMU) Max j BB j ; (8) j C P V 61; (9) F BB k 6V V k (; k) A; (10) V 0 F: (11) Each node F n the remnant network s gven a potental V ; the value of a unt havng length ; whch s the dual prce assocated wth the unt ow balance constrant (2) for that node. Smlarly, BB j corresponds to the demand satsfacton constrant (3) for length j; and therefore values orders for length j. When (PMU) s nondegenerate these dual prces are unque, and we nterpret them as follows. If raw unts of length are suppled from a secondary source at some small rate 0, then V s the margnal decrease n. Smlarly, f addtonal orders for length j arrve at rate 0, then BB j s the margnal ncrease n. We are nterested n pars ( ;Y ;S ) and (V ; BB ) of optmal solutons to (PMU) and (DMU), respectvely, that satsfy the complementary slackness condtons ( k F Y; k(v P k V k 1 ) =0; (12) S V =0 F; and (13) V k BB k)=0 (; k) A: (14) Trvally, f j C j 0; then 0 n every feasble soluton. Thus by (12), the value of the average raw unt s 1. In the market for unts modeled by (DMU), constrant (10) means that orders for length k are wllng to purchase unts of length ; for V V ; only f they cost no more than BB k k. Ths follows from complementary slackness (14) because otherwse Y ; k = 0. PROPOSITION 3. For all j C such that j 0; BB j = mn (V (; k) A j V j) j C: (15) PROOF. By (10), and by the fact we are maxmzng an objectve functon (8) wth postve coecents, (15) holds. Ths result holds for all j C such that j 0; but may be volated f j = 0. However, n 3.3 we show that a dual optmal soluton can always be found that satses (15). As there may exst more expensve allocatons, but none less expensve, we call BB k the base budget of an order for length k. The objectve (8) then maxmzes the rate at whch value for the factory accumulates from orders purchasng unts. For each length j, the mnmum n (15) can be attaned at multple lengths. We call any (; k) n A j that attans the mnmum a permssble allocaton. DEFINITION 1. The set A 0 {(; k) A : BB k = V Vk } (16) s called the set of permssble allocatons under the optmal dual prces (V ; BB ). DEFINITION 2. F = { F: V =0} s the set of scrappable lengths. Because the arcs n F and A 0 have zero reduced cost, they may have postve ow n an optmal prmal soluton. In 4 we consder IP-based polces that allow only these arcs. The base budget BBj decomposes nto two terms: one for purchasng length j and one for purchasng the resultng change n system scrap. To see ths, let (V ; BB ) be a feasble soluton to (DSCRAP), the dual of (PSCRAP). Then (V ; BB ) s an optmal soluton to (DSCRAP) f and
ADELMAN AND NEMHAUSER / 893 only f V = V + and k F kp k BBj = BB j + j k F kp k F; (17) j C (18) s an optmal soluton to (DMU). Ths mappng gves us an nterestng nterpretaton of the quantty V Vk, the net decrease n the value of the nventory n makng allocaton (; k), because t mples the dentty V Vk = ( k)+(v V f F fp f k ) : (19) Hence, n makng such an allocaton, V (and BB ) accounts for both the order length cut from the unt, k, and the change n the scrap poston of the nventory V Vk. The denomnator on the rght-hand sde smply scales accordng to the average raw unt length. Also, as a consequence of (17), (18), and (19), we are nderent between usng ether (V ; BB ) or (V ; BB ), because ( f F fp f )(BB k (V Vk )) = BB k (V 3. PROPERTIES OF OPTIMAL SOLUTIONS 3.1. The Propertes V k ). We gve sx propertes that are ntutvely desrable for the value functon V to satsfy. Subsequently we wll prove that there always exsts a dual optmal soluton satsfyng these propertes, and we show how to obtan one. PROPERTY 1. Monotoncty: V k. V k ; k F such that Monotoncty states that a unt s at least as valuable as any shorter unt. Ths s ntutve because the unt can handle any set of orders that a shorter one can. PROPERTY 2. Superaddtvty: V that + k F. +k V +V k ; k F such Superaddtvty means that a unt of length 15, for example, s worth at least much as two unts, one havng length 5 and the other havng length 10. The ratonale s that any set of allocatons possble wth the two unts s also possble wth the sngle unt havng length 15. A 15 may even be able to handle other sets of allocatons that the 5 and 10 together cannot, such as ve allocatons to orders for length 3. PROPERTY 3. Scrap valueless: Any length that s scrapped has V =0. We call such lengths scrappable. Any length scrapped should have zero value because t does not satsfy any orders. Ths follows from (13) for lengths such that S 0. However, because there may be some prmal optmal solutons wth S = 0 and others wth S 0;V = 0 s not guaranteed for all optmal solutons of (DMU), even f s less than the mnmum (postvely) ordered length. Ths stuaton llustrates the dculty that can be caused by degeneracy and alternatve optma. We now present three propertes of permssble allocatons. Frst, the set of permssble allocatons ncludes perfect ts. PROPERTY 4. Zero scrap permssblty: It s permssble to generate zero scrap;.e.; BBj = Vj V0 = V j j C. Here we use Property 3 to assume that V0 = 0. Note that ths property should hold even for order lengths j C wth j = 0. PROPERTY 5. Permutablty: If t s permssble to satsfy an order for length j 1 wth a unt of length ; and then permssble to use the remnant to satsfy an order for length j 2 ; then t s also permssble to satsfy the order for length j 2 rst and then j 1. Formally; f BB j 1 = V and BB j 2 = V j 1 BB j 1 = V j 2 V V j 1 j 2 ; then BBj 2 = V j 1 j 2. V j 2 V j 1 and PROPERTY 6. Usablty: For all F; ether s scrappable;.e:; V =0; or there exsts an order length j wth j 0 and j 0 such that BBj = V V j. Usablty states that each unt, regardless of length, has an ecent use;.e., t ether has zero value and s therefore scrappable, or there exsts at least one permssble allocaton to an order length that s postvely demanded n the long run. The usablty of a unt havng length s mmedate from zero scrap permssblty whenever 0. However, usablty should also hold even for lengths F wth = 0. As a consequence of usablty, t s mpossble for a unt to get stuck n the system because t has no use. 3.2. Alternatve Optma Fgure 4 llustrates a value functon assocated wth our example n 1.1 and 2.1 that s monotonc and superaddtve. Also V0 = V 1 = V 2 = 0 because these lengths must be scrapped. Ths V, together wth BB3 = V 3 and BB5 = V 5, consttute a dual optmal soluton, so zero scrap permssblty s satsed. The value functon depcted n Fgure 4 s lsted as soluton #1 n Table 1, whch also contans two alternatve dual optmal solutons. In all three solutons, BB3 = V 3 and BB5 = V 5. The values n solutons #2 and #3 that der from soluton #1 are hghlghted. The reader can verfy that all three dual solutons are feasble and satsfy the complementary slackness condtons (12) (14) wth respect to the prmal soluton presented n 2.1 and are hence optmal. Nevertheless, solutons #2 and #3 volate monotoncty as
894 / ADELMAN AND NEMHAUSER Fgure 4. An example value functon. well as superaddtvty, e.g., soluton #2 has V1 + V 4 V 5. Also, despte the fact that a unt of length 1 would have to be scrapped, V1 0 n soluton #2. In all dual solutons BB3 = V 8 V5 and BB5 = V5 V 0. By (14) ths means that arc ows Y 8; 5 and Y 5; 0 may be postve;.e., each allocaton n the path (8; 5) (5; 0) s permssble. Permutablty says, and t s easy to verfy, that BB5 = V 8 V 3 and BB3 = V 3 V 0 ;.e., each allocaton n the path (8,3) (3,0) s permssble as well. for the exam- Table 1. Multple dual optmal values of V ple problem. V Unt Length Soluton #1 Soluton #2 Soluton #3 0 0 0 0 1 0 0.1667 0 2 0 0 0 3 0.1667 0.1667 0.1667 4 0.1667 0.3333 0.1667 5 0.1667 0.1667 0.1667 6 0.3333 0.3333 0.3333 7 0.3333 0.5 0.3333 8 0.3333 0.3333 0.3333 9 0.5 0.5 0.5 10 0.5 0.6667 0.6667 11 0.5 0.5 0.5 12 0.6667 0.6667 0.6667 13 0.6667 0.8333 0.8333 14 0.6667 0.6667 0.6667 15 0.8333 0.8333 0.8333 16 0.8333 1 1 17 0.8333 0.8333 0.8333 18 1 1 1 19 1 1.1666 1.1666 20 1 1 1 The bold and dashed arcs n Fgure 2 together represent the set of permssble allocatons for soluton #1. As posed earler, suppose an order for length 5 s to be satsed and there are two unts n nventory, one of length 20 and one of length 15. Whch allocaton s preferable, f ether? Because (20,15) s permssble n soluton #1 but (15,10) s not, we may therefore conclude that length 20 s preferable. However, both allocatons are permssble n solutons #2 and #3. Despte the fact that solutons #2 and #3 are optmal, we argue that n practce (15,10) should not be used. To understand why, we must consder what the allocatons not n A 0 for soluton #1,.e., (15,10), (12,7), (10,5), (9,4), (7,2), and (6,1), have n common. Frst note that they all represent satsfacton of orders for length 5. Secondly, each allocates a second or thrd order for length 5 to the unt. Indeed, we may summarze the set of permssble allocatons for soluton #1 by statng that over ts lfetme we may use each raw unt to satsfy at most one order for length 5. Thus, we may satsfy sx orders for length 3 to produce scrap of length 2. Or, we may satsfy one order for length 5 and ve orders for length 3 to produce zero scrap. We would lke to use ths last pattern as much as possble. However, we do not have enough orders for length 5 to use t as often as we wsh, because nne orders for length 3 arrve for each sngle order for length 5 (accordng to 3 and 5 ). Consequently, satsfyng more than one order for length 5 wth a gven unt wastes these precous orders, despte the fact that satsfyng four orders for length 5 results n zero scrap, for example. So then why s (15,10) permssble n solutons #2 and #3? In soluton #2, each allocaton n the path 20 15 10 7 4 1 s permssble. However, because V1 0 by complementary slackness (13) S 1 = 0, and hence there can be no ow traversng ths path. In practce, f we allowed these permssble allocatons, then unts havng length 1 would buld up nntely f not scrapped, and once scrapped would yeld a suboptmal scrap rate. In soluton #3, there s no permssble allocaton for a unt of length 10, nor s t scrappable. Hence, n practce, f we allowed only permssble allocatons, unts of length 10 would buld up nntely as well. In both of these cases Y15;10 =0 n all correspondng prmal optmal solutons even though (15,10) s permssble. The reason n these cases s because usablty s volated by some subsequent remnant length. One way to ensure usablty s to add nntesmal postve nows of each length of unt F, so then the optmzaton must nd a use for each length. When there are also nntesmal nows for each order length j C, we prove n 3.3 that all propertes are satsed. 3.3. Proofs of the Propertes The set A 0 generates a unon of several alternatve prmal solutons, correspondng to a unon of allocaton polces, all of whch satsfy the permutablty property. Of all propertes, that s the only one that does not depend on the -perturbatons we gve next.
ADELMAN AND NEMHAUSER / 895 satses the per- THEOREM 1 (PROPERTY 5). The set A 0 mutablty property: ( 1 ; 2 ) A 0 and ( 2 ; 3 ) A 0 ( 1 ; 1 ( 2 3 )) A 0 and ( 1 ( 2 3 ); 3 ) A 0: PROOF. By (10) and by the denton of A 0. V 1 and V 1 ( 2 3) BB 2 3 = V 2 V 3 V 1 ( 2 3) V 3 BB 1 2 = V 1 V 2 : These nequaltes mply V 1 ( 2 3)6V 1 V 2 + V 3 and V 1 ( 2 3) V 1 V 2 + V 3 ; therefore, V 1 ( 2 3) = V 1 V 2 + V 3 : But then ths equaton can be wrtten n two ways: V 1 V V 1 ( 2 3) = V 2 V 3 = BB 2 3 ; 1 ( 2 3) V 3 = V 1 The concluson then follows. V 2 = BB 1 2 : As demonstrated n 3.2, an arbtrary set of optmal dual prces does not necessarly satsfy Propertes 1 6. However, we now show that under any nntesmal perturbaton contaned wthn a class called -perturbatons, all these propertes are satsed. DEFINITION 3. Choose a small 0 and vectors R F and R C such that 0 F; =1; F j 0 j C wth j =0; j =0 j C wth j 0; and j F: { j C: j6} Perturb each prmal constrant (2) so that t reads + P + = {k:(; k) A} {k:(k; ) A} Y k; Y ; k + S F; (20) and each prmal constrant (3) so that t reads (; k) A j Y ; k = j + j j C: (21) Such a perturbaton s called an -perturbaton. Such perturbatons represent the ntroducton of an exogenous supply of raw unts at rate, wth a fracton havng length. The j s ensure that order lengths j such that j =0 are prced approprately. As 0, the eect of an -perturbaton on the dual s an optmzaton over the dual optmal face. Ths optmzaton can be executed by addng the cut j BB j = ; (22) j C and solvng Max j C j BB j F V (23) subject to (9) (11) and (22). See Greenberg (1986) and Jansen et al. (1996) for more on perturbatons. THEOREM 2 (PROPERTY 1). Under an -perturbaton;v nondecreasng. PROOF. Frst note that V0 = 0 by complementary slackness, because to be feasble S0 = 0 0. Therefore, V0 6V 1. Now suppose that Vk 1 6V k for all k6, for some F. Because V s optmal and we are mnmzng V s n (23), V cannot be decreased n solaton. Note that V can always be feasbly decreased wthout volatng (11), because V 0, and wthout volatng (9). Therefore, there must exst a j C such that V V j = BBj. By dual feasblty we have V V j = BB j 6V +1 V +1 j: But by the nducton hypothess V then that V 6V+1 : j6v j+1 s, whch mples THEOREM 3 (PROPERTY 3). Under an -perturbaton; all lengths F such that mn { j C : j 0} j have V =0. PROOF. Each such length can be allocated to orders at a maxmum rate of { j C : j6} j. Because by Denton 3, { j C: j6} j, ths leaves some excess now from, whch must therefore be scrapped;.e., S 0. The result follows from complementary slackness (13). The converse of ths theorem s not true. For example, suppose P 8 =1; 3 0, and 5 0, but 3 5. Then unts of length 3 must be scrapped, even though length 3 s ordered. Nevertheless, as a corollary to monotoncty, the set of scrappable lengths, F, s a set of consecutve ntegers startng wth 0.
896 / ADELMAN AND NEMHAUSER THEOREM 4 (PROPERTY 4). Under an -perturbaton zero scrap permssblty holds;.e.; BBj = Vj j C n every optmal dual soluton. PROOF. Because j + j 0, prmal feasblty mples that there exsts an F such that Y; j 0, so by complementary slackness V V j = BBj. By permutablty, ths arc can be permuted down untl arrvng at a k j such that Vk j = 0 and V k V k j = BB j. Thus Vk = BB j. If k = j, then we are done; otherwse, by monotoncty Vj 6Vk = BB j. Now by dual feasblty V j BBj, so equalty must hold. THEOREM 5 (PROPERTY 2). Under an -perturbaton; V superaddtve. PROOF. Because BBj = Vj by zero scrap permssblty, dual feasblty says Vj = BBj 6V V j (; j) A: Now we show that all lengths ether have a possble use among the set of permssble allocatons or can be scrapped. THEOREM 6 (PROPERTY 6). Under an -perturbaton; the usablty property s satsed; that s; for all F; ether F or (nclusve) there exsts a k such that k 0; and (; k) A 0. PROOF. Under an -perturbaton each length has an n- ow of, whch must be ether scrapped or allocated. If some of the ow s scrapped, then S 0 and so V =0 by complementary slackness (13). If all of the ow s allocated, then there must be some arc (; k) on whch t s sent such that k 0. Ths follows from Denton 3 because { j C: j6} j ensures that not all of can be consumed by order lengths j6 such that j = 0. Because Y; k 0; (; k) A 0 by complementary slackness (14). Derent choces of and may gve derent dual optmal solutons. From the results gven above, any such soluton satses all propertes. However, t s stll possble, as n 3.2, to have permssble allocatons (; k) such that Y; k = 0 n all correspondng prmal optmal solutons satsfyng complementary slackness. Such an example s gven n Adelman (1997). To avod ths stuaton t s necessary to produce an optmal prmal-dual par satsfyng strct complementary slackness, whch can be done usng an nteror pont algorthm (Jansen et al. 1996). Ths means Y ; k 0 for all permssble allocatons (; k) A 0, and S 0 for all scrappable lengths F. We have shown that under -perturbaton, Propertes 1 6 are satsed by an optmal soluton to (23). We can also show a related result. THEOREM 7. Suppose (V ; BB ) s an optmal soluton of (DMU) satsfyng Propertes 1 6. Then an -perturbaton s can be constructed under whch (V ; BB ) s optmal to (23). PROOF. See Adelman (1997). 4. SIMULATION 4.1. The One-Perod Decson Problem and System Envronment In each perod we have a set of unts avalable for allocaton, each wth a known length, along wth a set of orders requrng allocaton. Each order s for a known length and number of unts. We must (1) select a subset of orders to satsfy, and (2) allocate unts to each order selected. Because there may not be enough unts to satsfy all orders, each order s gven a user-speced prorty bonus. Ths problem s solved perodcally over tme, as new orders arrve and new remnant and raw unts become avalable. In 4.2 we present an nteger program that uses the value functon n makng these perodc decsons. In 4.3 we use smulaton to compare ths approach wth a decson rule prevously used n the cable factory. Although we have analyzed a determnstc system, n practce remnant nventory systems, such as n ber-optc cable manufacturng, experence random arrvals of orders and unts. To test the eectveness of our methodology n such an envronment, the system we smulate has Posson arrvals of orders and unts, thnned accordng to the dstrbutons j and P. We mpose a constrant on the maxmum number of unts that may crculate n the system, so t s mpossble for the nventory to grow ndentely. Although we do not mpose such a hard constrant on the order backlog, n each perod we select a maxmal number of orders wth the unts avalable. In addton, whenever the number of orders awatng allocaton grows too large, we allow a few orders to take nonpermssble allocatons. Although ths negatvely mpacts the scrap rate, we nd that by settng the producton rate of raw unts slghtly above and allowng a large enough number of unts to crculate, the relatve frequency of nonpermssble allocatons s neglgble. In operatng the system, whenever a unt s generated shorter than any length ordered (.e., of length mn { j C: j 0} j) t s mmedately scrapped. Scrappable lengths wth 0 are held n nventory untl the maxmum number of unts allowed n crculaton s reached, at whch tme one s scrapped. If no unt can be scrapped and ths maxmum number of unts s acheved, then producton of raw unts ceases untl a unt can be scrapped. 4.2. IP-Based Operatng Polces In each perod n {0; 1;:::} let O n be the set of orders awatng allocaton and U n be the set of unts avalable. For each o O n let L o C be the length of unts requred by order o. Also, let L u F be the length of unt u U n. Assume that each order o O n requres a 0 unts. (We can nterpret the
ADELMAN AND NEMHAUSER / 897 j s dened n 1.1 as aggregate rates.) Dene n {(u; o) U n O n : L u L o } (24) to be the set of feasble assgnments of unts to orders. Also dene G u; n {o O n :(u; o) n } u U n to be the set of orders that unt u can satsfy, and H o; n {u U n :(u; o) n } o O n to be the set of unts that can satsfy order o. The total base budget of an order requrng a o 0 unts of length L o s a o BB L o. The total budget for the order s then taken to be a o BB L o + o ; (25) where o 0 s the prorty bonus for order o. For each unt u long enough to satsfy order o, we set the assgnment cost to be VL u VL u L o. Let Z o be a decson varable that s 1 f order o s lled, 0 otherwse, and let X u; o be a decson varable equal to 1 f unt u s assgned to order o, 0 otherwse. At the begnnng of each perod n we solve (IP) Max o O n (a o BB L o + o )Z o (VL u VL u L o )X u; o ; (26) (u;o) n o G u;n X u; o 61 u U n ; (27) u H o;n X u; o = a o Z o o O n ; (28) X u; o {0; 1} (u; o) n ; Z o {0; 1} o O n : Constrants (27) ensure that each unt s allocated to at most one order. Constrants (28) state that order o s selected f and only f a o unts are allocated to t. Observe that when a o =1 o O n, substtutng out the Z o varables and rewrtng the rght-hand sde of (28) to be 61 converts ths nto an assgnment problem. By consderng all unts and orders smultaneously, the model globally optmzes the allocatons n each perod, for example satsfyng as many orders as possble wth the lmted permssble allocatons avalable. We assume that the capacty of the faclty that processes the allocated unts s not bndng, so that we are lmted only by the avalablty of unts. If the o s are set small enough, only permssble allocatons would be taken n an optmal nteger soluton. Alternatvely, we could restrct the set of allocatons n n (24). In ether case the objectve functon (26) because of (15) would eectvely reduce to Max o O n o Z o ; (29) Table 2. An emprcal comparson of scrap. SYSTEM LP IP OLD RULE system1 5.92% 5.96% (0.09) 12.97% (0.38) system2 6.25 6.24 (0.07) 7.70 (0.14) system3 5.88 5.88 (0.06) 7.43 (0.10) system4 8.05 8.06 (0.12) 10.23 (0.18) system5 5.60 5.60 (0.06) 14.28 (0.33) system6 6.06 6.04 (0.07) 13.23 (0.42) system7 5.91 5.91 (0.06) 7.63 (0.11) system8 6.74 6.79 (0.14) 11.79 (0.32) so that orders are prortzed accordng to ther bonuses o. Because f o = 0 we would be nderent between selectng order o usng only permssble allocatons, and not selectng t, we set o 0 for all orders o to gve at least some postve ncentve for selecton. 4.3. Results For comparson, we smulated the performance of an exstng remnant nventory control decson rule: (OLD RULE) Begnnng wth the longest order and gvng successve prorty to longest orders rst, gve hghest prorty to generatng the shortest remnant possble n each of the successve ranges [a 1 ;b 1 ); [a 2 ;b 2 ), up to range [a h ;b h ). In Adelman et al. (1999) the authors present a comparson of ths old approach wth our methodology, usng data from an actual ber-optc cable factory. The results gven here are based on a controlled smulaton. Orders ranged from 10 to 25 n length and requred only one unt. Raw unts ranged from 35 to 60 n length, randomly generated by a derent dstrbuton P for each system. The remnant return delay was 2 perods and up to 100 unts were allowed to crculate n the system. Raw unts were produced at the rate 1:1, where was computed by solvng (PMU) for each system nstance. For eght remnant nventory systems, each dened by ts j s and P s, Table 2 compares scrap rates from (PSCRAP), our IP-based approach, and (OLD RULE). Scrap s reported as a percentage of total length of unts produced, wth the half-lengths of 95% condence ntervals for the smulated quanttes collected over 5,000 perods (approxmately 50,000 orders) obtaned usng the method of batch means (Law and Kelton 1991) wth around 30 batches. The scrap rates these systems generate from repeatedly usng our prce-drected nteger program emprcally seem to converge to the mnmum scrap rates gven by our lnear program. Compared wth the old rule, we decrease scrap by 34% on average. Because of varablty n the order and unt arrval streams, the scrap rate uctuates over tme, along wth the order backlog and unt nventory. However, n our experments these uctuatons eventually smooth out to gve us these results.
898 / ADELMAN AND NEMHAUSER 5. CONCLUSIONS We have presented an nventory system n whch the central focus s on the allocaton of remnants over tme. We provded a methodology for valung remnants, and presented many nsghtful propertes satsed by ths value functon. In addton, we presented an nteger program that can be used n practce wth these values to make allocaton decsons. Our smulaton results demonstrated that the emprcal scrap rates attaned by repeated use of ths nteger program through tme, wth our LP-based value functon, seem to converge to the LP s mnmum long-run average scrap rate. ACKNOWLEDGEMENTS Ths paper s based on Danel Adelman s (1997) thess, whch was supported by a Department of Energy Pre- Doctoral Fellowshp for Integrated Manufacturng. Adelman was also supported by the Unversty of Chcago Graduate School of Busness. Both authors were supported by a grant from Lucent Technologes. George Nemhauser was supported by NSF grant DDM-9115768. REFERENCES Adelman, D. 1997. Remnant nventory systems. Ph.D. Dssertaton, Georga Insttute of Technology, School of Industral and Systems Engneerng, Atlanta, GA., G.L. Nemhauser, M. Padron, R. Stubbs, R. Pandt. 1999. Allocatng bers n cable manufacturng. Manufacturng & Servce Oper. Management 1 21 35. Cheng, C.H., B.R. Ferng, T.C. Cheng. 1994. Cuttng stock problem a survey. Internat. J. Producton Econom. 36 291 305. Clements, D.P., J.M. Crawford, D.E. Josln, G.L. Nemhauser, M.E. Puttltz, W.P. Savelsbergh, M. 1997. Heurstc optmzaton: a hybrd AI=OR approach. Proc. CP97: Constrant-Drected Schedulng. Courcoubets, C., U.G. Rothblum. 1991. On optmal packng of randomly arrvng objects. Math. Oper. Res. 16 176 194. Dantzg, G., P. Wolfe. 1960. Decomposton prncple for lnear programs. Oper. Res. 8 101 111. Dyckho, H. 1981. A new lnear programmng approach to the cuttng stock problem. Oper. Res. 29 1092 1104.. 1990. A topology of cuttng and packng problems. Euro. J. Oper. Res. 44 145 159. Galambos, G., G.J. Woegnger. 1995. On-lne bn packng a restrcted survey. ZOR Math. Methods Oper. Res. 42 25 45. Gans, N.F., G.J. van Ryzn. 1997. Optmal control of a mult-class, exble queueng system. Oper. Res. 45 677 693. Glmore, P.C., R. Gomory. 1961. A lnear programmng approach to the cuttng-stock problem. Oper. Res. 9 849 859. Greenberg, H.J. 1986. An analyss of degeneracy. Naval Res. Logst. Quart. 33 635 655. Gue, K.R., G.L. Nemhauser, M. Padron. 1997. Producton schedulng n almost contnuous tme. IIE Trans. 29 391 398. Jansen, B., C. Roos, T. Terlaky. 1996. Introducton to the theory of nteror pont methods. T. Terlaky, ed., Interor Pont Methods of Mathematcal Programmng. Kluwer Academc Publshers, The Netherlands. Johnston, R.E. 1993. Dmensonal ecency n cable manufacturng: problems and solutons. Math. and Comput. Modellng 16 19 35. Krchagna, E.V., R. Rubo, M.I. Taksar, L.M. Wen. 1998. A dynamc stochastc stock cuttng problem. Oper. Res. 46(5) 690 701. Law, A.M., W.D. Kelton. 1991. Smulaton Modelng and Analyss. McGraw-Hll, New York. Murr, M.R. 1992. Some stochastc problems n ber producton. PhD Dssertaton. Rutgers State Unversty of New Jersey, New Brunswck, NJ. Nandakumar, P., J.L. Rummel. 1998. A subassembly manufacturng yeld problem wth multple producton runs. Oper. Res. 46. Northcraft, L.P. 1974. Computerzed cable nventory. Indust. Engrg. 6 45 49. Roundy, R.O., W.L. Maxwell, Y.T. Herer, S.R. Tayur, A.W. Getzler. 1991. A prce-drected approach to real-tme schedulng of producton operatons. IIE Trans. 23 149 160. Schethauer, G. 1991. A note on handlng resdual lengths. Optmzaton 22 461 466.