THE BULLWHIP EFFECT IN SUPPLY CHAINS WITH STOCHASTIC LEAD TIMES. Zbigniew Michna, Izabela Ewa Nielsen, Peter Nielsen

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1 M A T H E M A T I C A E C O N O M I C S No. 9(16) 013 THE BUWHIP EFFECT IN SUPPY CHAINS WITH STOCHASTIC EA TIMES Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse Absrac. I his aricle we cosider a simple wo sage supply chai. We quaify he variace amplificaio of orders he bullwhip effec i a model wih sochasic lead imes. Employig he movig average forecasig mehod for lead ime demads we obai a exac value of he bullwhip effec measure. We aalyze he formula usig umerical examples. Keywords: supply chai, bullwhip effec, iveory policy, sochasic lead ime, lead ime demad forecasig. JE Classificaio: F15, F3,. OI: /me Iroducio Supply chais are eworks of firms (supply chai members) which ac i order o deliver a produc o he ed cosumer. Supply chai members are cocered wih opimizig heir ow objecives ad his resuls i he poor performace of he supply chai. I oher words, he local opimum policies of members do o resul i he global opimum of he chai ad hey yield he edecy of repleishme orders o icrease i variabiliy as oe moves up i a supply chai. Such a pheomeo is called he bullwhip effec. This effec was recogized by Forreser (Forreser 1958) i he middle of he weieh ceury. The erm, bullwhip effec, was coied by Procer & Gamble maageme. The bullwhip effec is cosidered harmful because of Zbigiew Micha eparme of Mahemaics, Wrocław Uiversiy of Ecoomics, Komadorska Sree 118/10, Wrocław, Polad. zbigiew.micha@ue.wroc.pl Izabela Ewa Nielse, Peer Nielse eparme of Mechaical ad Maufacurig Egieerig, Aalborg Uiversiy, Fibigersræde 16 Room: 4103, 90 Aalborg, emark izabela@m-ech.aau.dk; peer@m-ech.aau.dk

2 7 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse is cosequeces which are (see e.g. Buchmeiser e al. 008): excessive iveory ivesme, poor cusomer service level, los reveue, reduced produciviy, more difficul decisio-makig, sub-opimal rasporaio, sub-opimal producio ec. This makes i criical o fid he roo causes of he bullwhip effec ad o quaify he icrease i demad variabiliy a each sage of he supply chai. I he curre sae of research ypically five mai causes of he bullwhip effec are cosidered (see e.g. ee e al. 1997a ad 1997b): demad forecasig, o-zero lead ime, supply shorage, order bachig ad price flucuaio. To decrease he variace amplificaio i a supply chai (i.e. o reduce he bullwhip effec) we eed o ideify all he facors causig he bullwhip effec ad o quaify heir impac o he effec. May researchers assumig a deermiisic lead ime have sudied he ifluece of differe mehods of demad forecasig o he bullwhip effec such as simple movig average, expoeial smoohig, ad miimum mea-squared-error forecass whe demads are idepede, ideically disribued or cosiue a iegraed movig-average, auoregressive process or auoregressive-movig average (see Graves 1999, ee e al. 000, Che e al. 000a ad 000b, Alwa e al. 003, Zhag 004 ad uc e al. 008). Moreover, hey quaify he impac of a deermiisic lead ime o he effec which, as follows from heir works, is oe of he major facors causig he effec. Sochasic lead imes were iesively ivesigaed i iveory sysems see Bagchi e al. 1986, Harihara ad Zipki 1995, Mohebbi ad Poser 1998, Sarker ad Zagwill 1991, Sog 1994a ad 1994b, Sog ad Zipki 1993 ad 1996 ad Zipki Mos of hese works cosider he so-called exogeous lead imes, ha is hey do o deped o he sysem e.g. he lead imes are idepede of he orders ad he capaciy uilizaio of a supplier. Moreover, hese aricles sudied how he variable lead imes affec he corol parameer, he iveory level or he coss. Recely he impac of sochasic lead imes o he bullwhip effec has bee iesively ivesigaed. Oe ca ivesigae he so-called edogeous lead imes, ha is depedig o he sysem. This is aalyzed i So ad Zheg (003) showig he impac of edogeous lead imes o he amplificaio of he order variace. I he paper of Kim e al. (006) he bullwhip effec performace is give uder he codiio ha lead ime demads are prediced usig he movig average forecasig mehod. Aoher work ivesigaig sochasic lead imes i he coex of he bullwhip effec is he paper of uc e al. (008). They assume ha lead imes are idepede ad ideically disribued ad demads cosiue a firs-order auoregressive AR(1) process or a mixed firs-order auoregressive-movig average

3 The bullwhip effec i supply chais wih sochasic lead imes 73 ARMA(1,1) process. The disadvaage of his approach is ha he lead imes are o prediced o make a order. I pracice his is o feasible. Micha ad Nielse (013) prese a model where lead imes ad demads are forecased separaely usig he movig average mehod. They idicae ad quaify he impac of lead ime forecasig o he bullwhip effec. The mai aim of his aricle is o quaify he bullwhip effec i a model wih sochasic idepede ideically disribued lead imes where orders are made by he lead ime demads which are forecased usig he movig average mehod. We will modify he model of Kim e al. (006) because heir approach ad assumpios seem o be ifeasible i real supply chais.. Supply chais ad he bullwhip effec I rece sudies a supply chai is a sysem of orgaizaios, people, aciviies, iformaio, ad resources ivolved i movig a produc or service from suppliers o cusomers. More precisely a supply chai cosiss of cusomers, reailers, warehouses, disribuio ceers, maufacures, plas, raw maerial suppliers ec. They are members or sages (echelos) of a give supply chai. A supply chai has a liear order which meas ha a he boom here are cusomers, above cusomers here is a reailer, above he reailer here is for example a maufacurer ad so o. The liear order is deermied by he flow of producs which sream dow from he supplier hrough he maufacurer, warehouse, reailer o he cusomers. Fiacial ad iformaio flows ca accompay he flow of producs. The simples supply chai ca cosis of cusomers (cusomers are o regarded as a sage i may aricles), a reailer ad a maufacurer (a supplier). A every sage (excep cusomers) a member of a supply chai possesses a sorehouse ad uses a cerai sock policy (a repleishme policy) i is iveory corol o fulfill is cusomer s (a member of he supply chai which is immediaely below) orders i a imely maer. Commoly used repleishme policies are: he periodic review, he repleishme ierval, he order-up-o-level iveory policy (ou policy), (s, S) policy, he coiuous review, he reorder poi (see e.g. Zipki 000). A member of a supply chai observes he demads from he sage immediaely below ad lead imes from he sage immediaely above. Based o he previous demads ad lead imes ad usig a cerai sock policy, he member of a chai makes a order o is supplier which is he sage immediaely above. Thus a every sage oe ca observe he demads from he sage below ad repleishme orders se o he sage above. The pheomeo of he variace amplificaio i reple-

4 74 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse ishme orders if oe moves up i a supply chai is called he bullwhip effec (see isey ad Towill 003 ad Geary e al. 006 for he defiiio ad hisorical review). Muso e al. (003) asser: Whe each member of a group ries o maximize his or her beefi wihou regard o he impac o oher members of he group, he overall effeciveess may suffer. The bullwhip effec is he key example of a supply chai iefficiecy. May researchers use he raio of variaces as he bullwhip effec measure ha is if q is a radom variable describig he demads (orders) of a member of he supply chai o he member above ad is a radom variable resposible for he demads of he member below (e.g. q describes he demads of a reailer o a maufacurer (supplier) ad shows he cusomer s demads o he reailer) he he measure of performace of he bullwhip effec is he followig Var(orders) / E(orders) Varq/ Eq BM. Var(demads) / E(demads) Var / E Usually i mos models E = Eq. The value of BM is greaer ha oe i he presece of he bullwhip effec i a supply chai. If BM is equal o oe he here is o variace amplificaio, whereas BM smaller ha oe idicaes dampeig which meas ha he orders are smoohed compared o he demads. Aoher very impora parameer of he supply chai members is he measure of he e sock amplificaio of a give supply chai member. More precisely le N s be he level of he e sock of a supply chai member (e.g. a reailer or a supplier) ad be he demads observed from is dowsream member (cusomers or a reailer), he he followig measure Var(e sock) Var( N s ) NSM Var(demads) Var is a very impora parameer of he supply chai performace. I may models i is assumed ha he coss are proporioal o Var (orders) ad o Var ( N s ). 3. Models wih sochasic lead imes The lead ime is regarded as he secod cause of he bullwhip effec, afer demad forecasig. ead imes are made of wo compoes, physical delays ad iformaio delays. I he models oe does o disiguish bewee hose compoes. The lead ime is he ime bewee whe a order

5 The bullwhip effec i supply chais wih sochasic lead imes 75 is placed by a member of a supply chai ad he period whe he produc is delivered o he member. The assumpio ha he lead ime is cosa is raher urealisic. Udoubedly i may supply chais physical ad iformaio delays are radom which meas ha a member of a supply chai does o kow he values of he fuure lead imes ad i he pas he/she observed ha heir values were varied i a sochasic maer. For isace i he paper of So ad Zheg (003), he model of a supply chai wih sochasic lead imes is moivaed by he semicoducor idusry where he dramaic boom-ad-bus cycles cause delivery lead imes o be highly variable, ragig from several weeks durig he low demad seaso o over several mohs durig he high demad seaso. Moreover, i he models ivesigaig he bullwhip effec oe ca decide how ime is represeed. There are wo choices, discree or coiuous ime. We will aalyze he sochasic echiques which use discree ime. We assume ha he observaios are made a ieger momes of ime which meas ha ime is represeed i uis of he review periods ad ohig is kow abou he sysem i he ime bewee observaios. The mai differece i models wih sochasic lead imes lies i he defiiio of he lead ime demad forecas which is ecessary o make a order. e us recall ha he lead ime demad a he begiig of a period (a a cerai sage of he supply chai) is defied as follows i i0..., (1) where, +1, deoe demads (from a sage below) durig, + 1,... periods ad is he lead ime of he order placed a he begiig of he period (order made o a sage above). This value ses dow he demad durig a lead ime. The demads come from he sage immediaely below ad he lead imes come from he supplier immediaely above, ha is hey are he delivery lead imes of he supplier which is immediaely above he member. This quaiy is ecessary o make a order by he member of he supply chai o a supplier which is immediaely above. I pracice he member of he supply chai does o kow is value a he mome bu he/she eeds o predic is value o make a order. Thus if we wa o aalyze he bullwhip effec we eed o look closer a he defiiios of he lead ime demad forecasig ˆ. Some of hem are less or more realisic or eve ifeasible i real supply chais wih sochasic lead imes. May aricles o he bullwhip effec ivesigae differe mehods of demad fore-

6 76 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse casig uder he assumpio ha he lead imes are cosa. The problem of he lead ime demad predicio is much more complicaed if he lead imes are sochasic. The mere demad forecasig is o sufficie o make a order. 4. A model wih lead ime demad forecasig e us recall he work of Kim e al. (006). I heir approach, lead ime demad forecasig is defied as follows ˆ 1 j, j1 where is he delay parameer of he predicio ad is he previous kow lead ime demad of he order placed a he begiig of he ime j. This mehod is pracically feasible. The problem of he approach of Kim e al. (006) lies i he impracical defiiio of he pas lead ime demads. Namely hey coiue j j j ji ji j1 j1 i0 i1 j1 ˆ, where is a lead ime. Firsly, if we assume ha lead imes are sochasic he wih every lead ime demad we associae a differe lead ime j. Moreover heir defiiio does o work eve i he case of a deermiisic lead ime because a he begiig of he mome we do o kow he values of demads j i if j i (hey explai i is a mirror image ad equivale i erms of a priori saisical aalysis ). e us aalyze he bullwhip effec uder he above seig bu wih esseial modificaios. More precisely le us cosider he simples supply chai, ha is oe cosisig of cusomers, a reailer ad a supplier. We assume ha he cusomers demads cosiue a iid sequece. Moreover, lead imes are deermiisic ad equal o where is a posiive ieger, ha is = 1,,... I is assumed ha he reailer s repleishme order policy is he order-up-o-level policy ad his/her lead ime demad forecasig is based o he movig average mehod. Thus he forecas of he j

7 The bullwhip effec i supply chais wih sochasic lead imes 77 lead ime demad a he begiig of he period based o he movig average mehod is as follows 1 1 i0 ˆ. () i I his approach we have o ge back wih lead ime demads a leas o he period because we kow he demads ill he period 1. Moreover, le us recall ha he demad forecas aloe usig he movig average is as follows ˆ 1 j. j1 Thus subsiuig io eq. () he kow values of he previous lead ime demads we ge i j i j j1 j i0 j0 j0 i0 j0 j0 (3) ˆ ˆ ˆ. Applyig he order-up-o-level policy we fid ha he iveory level of he reailer a ime is ˆ S zˆ, (4) where ˆ is he error of he lead ime demad forecas which is usually defied as follows ˆ ˆ Var (5) ad z is a cosa called he ormal z-score. I some aricles, ˆ is defied more pracically, ha is isead of variace he empirical variace of ˆ is ake. This complicaes calculaios very much bu we mus meio ha he esimaio of ˆ icreases he size of he bullwhip effec. These wo approaches coicide if z=0. I is easy o oice ha uder he above assumpios ˆ is idepede of. Thus he order quaiy q placed by he reailer a he begiig of a period is q S S ˆ ˆ ˆ ˆ ˆ ˆ, j 1 j 1 1 j0 j0 where i he secod las equaliy we use equaio (3).

8 78 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse To calculae he value of q we eed o cosider wo cases, ha is ad <. Thus i he case he order q placed by he reailer is as follows ad q 1 1 j j j j Varq 1 1. Var Var I he case of < we ge ad q 1 1 j j j j1 1 1 Varq 1 1. Var Var Proposiio 1. If he lead imes are deermiisic ad posiive ieger valued ha is = 1,,... ad lead ime demads are forecased usig he movig average mehod he he bullwhip effec measure is Varq 1 if BM. Var 4 1 if I Figure 1 we ploed he bullwhip effec measure for deermiisic lead imes whe lead ime demads are prediced by he movig average mehod (see Proposiio 1). e us oice ha he bullwhip effec fucio BM() as a fucio does o have ay jump a ha is i smoohly ges across he poi = (compare Proposiio 1 wih he similar resul of Kim e al. 006).

9 The bullwhip effec i supply chais wih sochasic lead imes 79 Fig. 1. The plo of he bullwhip effec measure as a fucio of where = 7 Source: ow elaboraio. Now we follow he work of Kim e al. (006) wih cerai modificaios o fid he bullwhip effec measure i he presece of sochasic lead imes. We assume ha he cusomers demads cosiue a iid sequece ad he lead imes are also idepede ad ideically disribued ad he sequeces are muually idepede. e us pu E, Var, E ad Var. Addiioally we eed o assume ha lead imes are bouded radom variables ha is M where M is a posiive ieger. This assumpio is o adoped i Kim e al. (006) which makes heir resuls slighly impracical because his is ecessary o make he predicio of lead ime demads. More precisely we ge back a leas M periods o forecas he lead ime demad a ime, ha is we kow he values of lead ime demads a imes M, M 1,... which meas ha we may o kow he values of lead imes M + 1, M +,... a ime (ad furher demads) because he orders placed a imes M + 1, - M +,... ca be urealized a mome. This assumpio is he mai differece bewee our approach ad he model of Kim e al. (006), ad his resuls i a differe bullwhip effec performace measure as we ca see

10 80 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse laer. Moreover i our approach we eed o kow he disribuio of o calculae he bullwhip effec measure, ha is we assume ha P k p, where k = 1,,..., M ad k is he umber of periods (i pracice we esimae hese probabiliies). aer we will see ha oly he value of p M is ecessary i he bullwhip effec measure. Thus he predicio of he lead ime demad a ime usig he mehod of movig average wih he legh is as follows ˆ 1 1 M j j0 k. (6) e us repea ha he lead ime demads M 1, M,... we may o kow a ime, which is why i he lead ime forecasig we egage M, M 1,... ha is he lead ime demads up o ime M. Thus by equaio (1) we ge 1 M j 1 1 M ji j0 i0 ˆ. (7) We assume ha he reailer uses he order-up-o-level policy, hus he level of he iveory a ime is give i equaio (4). By he saioariy ad idepedece of he sequeces of demads ad lead imes oe ca show ha give i (5) does o deped o. Hece we obai ˆ 1 1 ˆ ˆ M j 1 M j 1 j0 j0 q M 1 M M M 1 M i M i 1 i0 i0 Theorem 1. Uder he above assumpios ad for M he bullwhip effec measure is he followig Varq pm BM 1. Var Proof. Usig he law of oal variace we have Varq E Var( q, ) VarE q,. M M M M. (8)

11 The bullwhip effec i supply chais wih sochasic lead imes 81 By equaio (8) we ge Thus M M E q M, M 1. Var E q M, M. (9) Var q,. I he firs We eed o cosider wo cases o fid case M < M we ge Var q M M, 1. M M M M If M = M we have M M Var q M, M 1. Fially we obai Var, 1 I. M M q M M M M where I is he idicaor fucio. Thus we ge EVar p M M, M 1 q which ogeher wih equaio (9) give he asserio. The formula for he bullwhip effec measure i he case < M is more complicaed ad is derivaio is raher cumbersome. I pracice he case M is more ieresig, because we require o pu large i he forecas o ge a more precise predicio. e us oice ha if = M = (which gives p M = 1 ad = ) is deermiisic, he formula of Theorem 1 is cosise wih Proposiio 1. Compare Theorem 1 wih he resuls of Kim e al. (006). We have o meio ha i he formula of Theorem 1 he erm pm gives he larges coribuio i he bullwhip effec for large because i is O(1/). This meas ha i reducig he bullwhip effec he probabiliy of

12 8 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse he larges lead ime is very impora. I is asoishig ha if p M = 0 ad we sill ge back M periods i he predicio of lead ime demads, he bullwhip effec measure is reduced by he erm O(1/) ad is of he form Varq BM 1. Var e us compare he values of he bullwhip effec measure uder p M > 0 ad p M = 0. More precisely, le have he discree uiform disribuio o {1,, 3} ha is p k = 1/3 for k = 1,, 3 he M = 3, = ad / 3. I he case p M = 0 we assume ha has he discree uiform disribuio o {1, } ha is p k = 1/ for k = 1, he = 1.5 ad 1/ 4 (ad we sill ge back a leas M = 3 periods o predic he lead ime demad). The resuls are i Table 1. Source: ow elaboraio. Table 1. The measure of he bullwhip effec as a fucio of for M = 3 ad / 0.5 p M > 0 p M = Similarly, we ca calculae he bullwhip effec measure for loger lead imes. More precisely, le have he discree uiform disribuio o {1,,, 7} ha is p k = 1/7 for k = 1,,, 7 he M = 7, = 4 ad 4. I he case p M = 0, we assume ha has he discree uiform disribuio o {1,,..., 6} ha is p k = 1/6 for k = 1,,..., 6 he = 3.5 ad

13 The bullwhip effec i supply chais wih sochasic lead imes (ad we sill ge back a leas M = 7 periods o predic he lead ime demad). The resuls are show i Table. Source: ow elaboraio. Table. The measure of he bullwhip effec as a fucio of for M = 7 ad / 0.5 p M > 0 p M = Fig.. The plo of he bullwhip effec measure as a fucio of ad where p M = 0.1, / 0.5 ad = 3 Source: ow elaboraio.

14 84 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse Fig. 3. The plo of he bullwhip effec measure as a fucio of ad where p M = 0.1, / 0.5 ad = 3 Source: ow elaboraio. Fig. 4. The plo of he bullwhip effec measure as a fucio of ad where p M = 0.1, / 0.5 ad = 10 Source: ow elaboraio.

15 The bullwhip effec i supply chais wih sochasic lead imes 85 Moreover, we ploed he bullwhip effec measure as a fucio of wo variables. I Figure we visualized he bullwhip effec measure as a fucio of he delay parameer ad he mea value of lead imes for p M = 0.1, / 0.5 ad = 3. Similarly, i Figure 3 we preseed BM as a fucio of he delayed parameer ad he sadard deviaio of lead imes for p M = 0.1, / 0.5 ad = 3. I Figure 4 he plo of bullwhip as a fucio of he expeced value ad he sadard deviaio of lead imes is show for p M = 0.1, / 0.5 ad = Coclusios ad fuure research opporuiies Oe of he mai objecives of he supply chai maageme is o quaify he bullwhip effec which is a sadard example of a iefficiecy i supply chais. The quaiaive descripio of he bullwhip effec eables o fid all causes of he effec ad o reduce is egaive resuls which are a subsaial icrease i coss ad disorders i he coiuiy of deliveries. Thus he bullwhip effec is he mos severe pheomeo observed i supply chais ad is aalysis is he mai challege of supply chai maageme. I his aricle we have quaified he bullwhip effec i he presece of sochasic lead imes where he order is made by he predicio of lead ime demads usig he simple movig average mehod. We have oiced ha he bullwhip effec measure depeds esseially o he delay parameer of he forecasig, he probabiliy of he larges lead ime, ad he expecaio ad variace of lead imes. These facors affec he size of he bullwhip effec, ad a proper uig of hem ca dampe he effec. Theorem 1 provides a lo of iformaio o he facors of he bullwhip effec i a quaiaive maer which is ecessary i supply chai maageme. Alhough he bullwhip effec has bee ivesigaed sice he middle of he weieh ceury, here are sill a lo of pahs which have o be examied ad aalyzed. A member of a supply chai placig a order has may ools ad policies o choose from. So assumig ha lead imes are radom, which is usually observed i supply chais, he member of a supply chai mus predic he ex lead imes o make a order, which is more aural ha he lead ime demad forecasig iself. More precisely, whe makig a order oe ca forecas he ex lead ime demad by predicig lead imes ad demads separaely. Thus i furher approaches o he lead ime

16 86 Zbigiew Micha, Izabela Ewa Nielse, Peer Nielse forecasig problem oe eeds o ivesigae srucures oher ha iid of lead imes ad demads. Eve if oe cosiders a more complicaed srucure of demads, for example auoregressive-movig average leavig iid srucure of lead imes, he his will complicae he derivaios of he bullwhip effec measure o a sigifica degree. Oher opporuiies lie i differe forecasig mehods for lead imes ad demads. Oe ca employ simple movig average, expoeial smoohig, ad miimum-mea-squared-error forecass o predic lead imes ad demads. Here oe ca apply differe mehods for lead ime forecasig ad demad forecasig or he same mehods bu oher ha he movig average mehod. I oher direcios of research oe ca ivesigae muli-echelo supply chais i he presece of sochasic lead imes beig prediced a every sage where he iformaio o demads ad lead imes is shared or is o shared. The value of he bullwhip effec measure i hese siuaios will be surely valuable for heoriss ad praciioers i he field of supply chai maageme. Ackowledgeme The auhors wish o hak he Polish Naioal Cere of Sciece for heir suppor uder he gra UMO-01/07/B/HS4/0070. Refereces Agrawal S., Segupa R.N., Shaker K. (009). Impac of iformaio sharig ad lead ime o bullwhip effec ad o-had iveory. Europea Joural of Operaioal Research 19. Pp Alwa.C., iu J., J., Yao.Q. (003). Sochasic characerizaio of upsream demad processes i a supply chai. IIE Trasacios 35. Pp Bagchi U., Hayya J., Chu C. (1986). The effec of leadime variabiliy: The case of idepede demad. Joural of Operaios Maageme 6. Pp Buchmeiser B., Pavlijek J., Palcic I., Polajar A. (008). The Bullwhip effec problem i supply chais. Advaces i Producio Egieerig & Maageme 3(1). Pp Chafield.C., Kim J.G., Harriso T.P., Hayya J.C. (004). The bullwhip effec -Impac of sochasic lead ime, iformaio qualiy, ad iformaio sharig: a simulaio sudy. Producio ad Operaios Maageme 13(4). Pp Che F., rezer Z., Rya J.K., Simchi-evi. (000a). Quaifyig he bullwhip effec i a simple supply chai. Maageme Sciece 46(3). Pp Che F., rezer Z., Rya J.K., Simchi-evi. (000b). The impac of expoeial smoohig forecass o he bullwhip effec. Naval Research ogisics 47(4). Pp isey S.M., Towill.R. (003). O he bullwhip ad iveory variace produced by a orderig policy. Omega 31. Pp

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