Two-level lot-sizing with inventory bounds

Transcription

1 Two-level lo-sizing wih invenory bounds Siao-Leu Phourasamay, Safia Kedad-Sidhoum, Fanny Pascual Sorbonne Universiés, UPMC Univ Paris 06, CNRS, LIP6 UMR 7606, 4 place Jussieu Paris. {siao-leu.phourasamay,safia.kedad-sidhoum,fanny.pascual}@lip6.fr May 24, 2018 Absrac We sudy a wo-level uncapaciaed lo-sizing problem wih invenory bounds ha occurs in a supply chain composed of a supplier and a reailer. The firs level wih he demands is he reailer level and he second one is he supplier level. The aim is o minimize he cos of he supply chain so as o saisfy he demands when he quaniy of iem ha can be held in invenory a each period is limied. The invenory bounds can be imposed a he reailer level, a he supplier level or a boh levels. We propose a polynomial dynamic programming algorihm o solve his problem when he invenory bounds are se on he reailer level. When he invenory bounds are se on he supplier level, we show ha he problem is NP-hard. We give a pseudo-polynomial algorihm which solves his problem when here are invenory bounds on boh levels. In he case where demand lo-spliing is no allowed, i.e. each demand has o be saisfied by a single order, we prove ha he uncapaciaed lo-sizing problem wih invenory bounds is srongly NP-hard. This implies ha he wo-level lo-sizing problems wih invenory bounds are also srongly NP-hard when demand lo-spliing is considered. Keywords : Dynamic lo-sizing, invenory bounds, NP-hardness, dynamic programming 1 Inroducion We consider a wo-level supply chain wih a supplier and a reailer. The reailer has o saisfy a demand for a single iem over a finie planning horizon of discree periods. In order o saisfy he demand, he reailer has o deermine a replenishmen plan over he horizon, i.e. when and how many unis o order. In order o saisfy he reailer s replenishmen plan, he supplier has o deermine a producion plan. Ordering unis induce a fixed ordering cos and a uni ordering cos for boh acors. Carrying unis in he invenory induce a uni holding cos for boh acors as well. Moreover, he quaniy ha can be held in invenory a each period can be limied, since invenory bounds can be imposed a he reailer level, a he supplier level or a boh levels. The cos of he supply chain is given by he sum of he supplier and he reailer oal coss. The wo-level Uncapaciaed Lo-Sizing (2ULS) problem wih invenory bounds consiss in deermining he order and he invenory quaniies a each period for boh replenishmen and producion plans in order o saisfy he exernal demand while minimizing he oal cos of he supply chain. Lieraure review For many pracical applicaions, i is unreasonable o suppose ha he invenory capaciy is unlimied. In paricular, he producs ha need emperaure conrol or special sorage faciliies may have a limied sorage capaciy. This is for example he case in he pharmaceuical indusry [2]. These consrains have led o he sudy of lo-sizing problems wih invenory bounds. The single level Uncapaciaed Lo-Sizing problem wih Invenory Bounds (ULS-IB) was firs inroduced by Love [11]. He proves ha he problem wih piecewise concave ordering and holding coss and backlogging can be solved using an O(T 3 ) dynamic programming algorihm. Aamürk and Küçükyavuz [2] sudy he ULS-IB

2 2 problem under he cos srucure assumed in Love s paper [11], considering in addiion a fixed holding cos. They propose an O(T 2 ) algorihm o solve he problem. They also make a polyhedral sudy of he ULS-IB problem [1] by considering wo cos srucures: linear holding coss, linear and fixed holding coss. They provide an exac separaion algorihm for each problem. Toczylowski [14] addresses his problem by solving a shores pah problem in O(T 2 ) ime. More recenly, Hwang and van den Heuvel [8] propose an O(T 2 ) dynamic programming algorihm o solve he ULS-IB problem wih backlogging and a concave cos srucure by exploiing he so-called Monge propery. Guiérrez e al. [6] improved he ime complexiy by developing an algorihm ha runs in O(TlogT) using he geomeric echnique of Wagelmans and van Hoesel [17]. However, van den Heuvel e al. [15] show ha heir algorihm does no provide an opimal soluion for he ULS-IB problem. Liu [10] proposes an O(T 2 ) algorihm based on he geomeric approach in [17] bu Önal e al. [13] prove ha his algorihm does no compue an opimal soluion for he ULS-IB problem. Recenly, Aamürk e al. [3] propose a polyhedral sudy of he capaciaed fixed-charge nework flow problem. The capaciaed lo-sizing problem wih invenory capaciies can be represened by his nework. They generalize he flow cover and flow pack inequaliies of he fixed-charge nework flow problem by proposing new valid inequaliies based on he pah srucure of he nework and provide face condiions. They also show ha hese inequaliies are effecive by using hem in a branch-and-cu algorihm. Zangwill [19] proposes an O(LT 4 ) dynamic programming algorihm for he muli-level uncapaciaed lo-sizing problem (where L is he number of levels). In paricular, van Hoesel e al. observe ha Zangwill s algorihm runs in O(T 3 ) when L = 2 [16]. More recenly, Melo and Wolsey [12] improve his complexiy by proposing an O(T 2 logt) dynamic programming algorihm. Zhang e al. [20] propose a polyhedral sudy of he muli-level lo-sizing problem where each level has is own exernal demand. They give an O(T 4 ) dynamic programming algorihm o solve he wo-level problem. A few papers deal wih he 2ULS problem wih invenory bounds. Jaruphongsa e al. [9] sudy his problem wih demand ime window consrains and saionary invenory bounds a he supplier level. They impose some assumpions on he cos parameers (among hem, he uni producion cos is non-increasing). These assumpions make he problem solvable in O(T 3 ) using a dynamic programming algorihm. They also prove ha when each demand is saisfied by a single dispach, he problem is NP-hard. Hwang and Jung [7] propose a dynamic programming algorihm ha solves he 2ULS-IB problem wih invenory bounds a he reailer level and concave coss in O(T 4 ) which has he same complexiy as he one provided in his paper. However, conrary o heir resul, we presen a dynamic programming algorihm based on some srucural properies specific o he invenory bounds for which we give correcness proofs. Conribuions In his paper, we sudy he complexiy of single-iem 2ULS problems wih invenory bounds. We consider ha eiher he supplier, he reailer, or boh of hem, have a limied invenory capaciy. A polynomial dynamic programming algorihm is provided o solve he problem wih invenory bounds a he reailer level. The problem is shown o be weakly NP-hard when he invenory bounds are imposed a he supplier level. A complexiy analysis for his class of problem is also proposed under he no lo-spliing assumpion where each demand has o be saisfied by a unique order. In he sequel, we will denoe 2ULS-IB R (resp. 2ULS-IB S ), he problem where a each period, he invenory quaniy a he reailer (resp. supplier) level canno exceed he invenory bound. Finally, he 2ULS-IB SR problem is he problem where boh he supplier and he reailer have a limied invenory capaciy. The following ables summarize he complexiy resuls for he varians of he 2ULS-IB problems considered: Table 1: Complexiy resuls wih lo-spliing Problem Complexiy ULS-IB polynomial [2], [11] 2ULS-IB R polynomial (Secion 3) 2ULS-IB S polynomial wih paricular cos srucure [9] NP-hard (Secion 4) 2ULS-IB SR NP-hard (Secion 5)

3 3 Table 2: Complexiy resuls wihou lo-spliing (NLS) Problem Complexiy ULS-NLS srongly NP-hard (Secion 6) 2ULS-IB R -NLS srongly NP-hard (Secion 6) 2ULS-IB S -NLS weakly NP-hard wih demand ime windows [9] srongly NP-hard (Secion 6) 2ULS-IB SR -NLS srongly NP-hard (Secion 6) This paper is organized as follows. A mahemaical formulaion for he single-iem 2ULS problem wih invenory bounds is provided in Secion 2. Secions 3, 4 and 6 follow he resuls described in Tables 1 and 2. In Secion 5, we show ha he 2ULS-IB SR problem is solvable using a pseudo-polynomial ime algorihm. 2 Mahemaical formulaions In his secion, we describe he mahemaical formulaion of he 2ULS problem as well as he invenory bound consrains for he addressed problems. Le T be he number of periods over he planning horizon. We denoe by d he demand a each period for {1,, T}. The reailer s (resp. supplier s) coss are defined by a fixed ordering cos f R (resp. f S ), a uni ordering cos p R (resp. p S ) and a uni holding cos hr (resp. h S ) for {1,, T}. The reailer s (resp. supplier s) invenory bound a each period is denoed by u R (resp. u S ) for {1,, T}. We denoe by x R (resp. x S) he quaniy ordered by he reailer (resp. supplier) a period, sr (resp. s S ) he reailer s (resp. supplier s) invenory level a he end of period and y R (resp. y S ) he reailer s (resp. supplier s) seup variable, which is equal o 1 if an order occurs a period a he reailer (resp. supplier) level and 0 oherwise. The 2ULS problem can be formulaed as follows: min T =1 ( f S ys + p S xs + h S ss + f R y R + p R x R + h R s R ) (1) s.. s R 1 + xr = d + s R {1,..., T}, (2) s S 1 + xs = x R + s S {1,..., T}, (3) x R M R y R {1,..., T}, (4) x S M S ys {1,..., T}, (5) x S, x R, s S, s R 0 (6) y S, y R {0, 1} T (7) where M R = M S = i= T d i. The supply chain oal cos is given by (1). Consrains (2) (resp. (3)) are he invenory balance consrains a he reailer (resp. supplier) level. The supplier demand is he amoun ordered a he reailer level a each period. Consrains (4) and (5) force he seup variables o be equal o 1 if here is an order, i.e. if x R > 0 or x S > 0 respecively. The 2ULS problem can be viewed as a fixed charge nework flow problem (see Figure 1) where he nodes represen he periods a each level. A source node is also considered in order o represen he oal supplied quaniy i=1 T d i. For each node, he verical inflows are he ordering quaniies and he horizonal ouflows represen he invenory quaniies. In addiion, arcs represening he exernal demand a each period a he reailer level are considered. In he sequel, we will no represen he dummy node, and he arcs will be represened only if hey are acive (i.e. a verical arc will be represened if he corresponding ordering quaniy is posiive, and a horizonal arc is represened if he corresponding invenory quaniy is no null).

4 4 T d =1 Supplier T x S s S x R Reailer T d 1 d 2 d d +1 d T Figure 1: The 2ULS problem as a fixed charge nework flow. s R In addiion o his classical problem, we inroduce invenory bounds consrains. The invenory bounds consrains for he 2ULS-IB R problem are given by: s R u R {1,..., T}. (8) The mahemaical formulaion can be srenghened by seing M R o min(d + u R, T i= d i) in he consrain (4). Similarly, he invenory bounds consrains for he 2ULS-IB S problem are given by: s S u S {1,..., T}. (9) Similarly, parameer M S can be replaced by min(d + u S + ur, T i= d i) in he consrain (5). The mahemaical formulaion of he 2ULS-IB SR problem is obained by adding he consrains (8) and (9) o he mahemaical formulaion of he 2ULS problem. 3 The 2ULS-IB R problem The 2ULS-IB R problem has been firs sudied by Hwang and Jung [7]. We presen srucural properies of an opimal soluion for he problem and propose an O(T 4 ) algorihm o solve i. Since he invenory bounds are only se a he reailer level, he superscrip R will be omied in he invenory bound parameer u R ha will be denoed by u. Zangwill [19] shows ha here exiss an opimal soluion for he 2ULS problem ha verifies he Zero Invenory Ordering (ZIO) propery a each level, i.e. s i 1 xi = 0 for all {1,..., T} and i {S, R}. As shown by [1, 14], he following assumpion can be saed wihou loss of generaliy: Assumpion 1. u 1 u + d for all {1,..., T}. 3.1 Dominance properies In his secion, we propose some dominance properies in order o deermine an efficien solving approach such ha here exiss an opimal soluion for he 2ULS-IB R problem ha saisfies hese properies. We know ha he ZIO propery does no hold for he ULS-IB problem [11, 10]. Le us firs show ha for he 2ULS-IB R problem, he cos of he bes soluion in which he ZIO propery is fulfilled a he reailer level may be arbirarily large compared o he cos of an opimal soluion in which he ZIO propery is no required. Propery 1. For he 2ULS-IB R problem, he cos of he bes ZIO policy a he reailer level may be arbirarily large compared o he cos of an opimal policy. Proof. Consider he following insance I: T = 2, h S = p S = f R = h R = [0, 0], f S = [0, 1], p R = [0, 1], d = [0, B + 1] and u = [B, B], where B is a large consan. The bes soluion saisfying he ZIO propery a he reailer level is given by x S = [d 2, 0], x R = [0, d 2 ]. The corresponding cos is B + 1 whereas he opimal non-zio soluion is given by x S = [d 2, 0], x R = [B, 1] inducing a cos equals o 1 (see Figure 2).

5 5 B+1 B+1 B+1 1 Supplier B+1 B 1 B Reailer B+1 0 B+1 Bes ZIO soluion of cos B+1 Opimal (non-zio) soluion of cos 1 f S = [0, 1] h S = [0, 0] p S = [0, 0] f R = [0, 0] h R = [0, 0] p R = [0, 1] u = [B, B] Figure 2: Soluions for he insance I of he 2ULS-IB R problem. Le us now give he definiion of a block (Definiion 3), previously inroduced in [2, 1] for he single level case. Definiion 1 (Subplan). Le i and j be wo periods such ha 1 i j T. A subplan [i, j] is a parial soluion a he reailer level of he 2ULS-IB R problem beween he periods i and j defined by x R i,..., xr j. Definiion 2 (Regular subplan). Le i and j be wo periods such ha 1 i j T. A regular subplan [i, j] is a subplan [i, j] such ha s R i 1 {0, u i 1} and s R j {0, u j }. Definiion 3 (Block). Le i and j be wo periods such ha 1 i j T. Le α {0, u i 1 } and {0, u j }. A block [i, j] α is a regular subplan [i, j] where sr i 1 = α, sr j = and 0 < s R < u for all {i,..., j 1}. In oher words, a block [i, j] α is a regular subplan [i, j] where he invenory quaniies for each period beween i and j are sricly posiive bu no equal o he invenory bound. A regular subplan is made of one or several blocks. Definiion 4 (Order quaniy). Le d k = k i= d i be he cumulaive demand beween periods and k. The order quaniy a he reailer level in a subplan [i, j] is given by X ij = d ij s R i 1 + sr j. Observe ha for a block [i, j] α, X ij = d ij α +. Thereafer, we give some properies observed by an opimal soluion for he 2ULS-IB R problem. Theorem 1. Le P be he se of poins ha saisfy Consrains (2)-(8) of he 2ULS-IB R problem. A poin (y R, x R, s R, y S, x S, s S ) P is an exreme poin if and only if: 1. here is a mos one ordering period in every block [i, j] α, for all 1 i j T, α {0, u i 1}, {0, u j }, 2. he ZIO propery holds a he supplier level. This heorem follows from he properies relaed o he opimal flows in a fixed-charged nework wih concave coss [18]. As he 2ULS-IB R problem is a single source fixed-charged nework wih linear coss, Theorem 1 is a direc applicaion of he characerizaion of exreme poins in hese neworks. Propery 2. An exreme poin of P saisfies he following properies a he reailer level for all 1 i j T, α {0, u i 1 }, and {0, u j }: (i) If [i, j] 0 is a block and if here is an ordering period in his block, hen his ordering period is i. (ii) If [i, j] α u j is a block and if here is an ordering period in his block, hen his ordering period is j. Proof. (i) If d i > 0, hen period i is necessarily an ordering period since si 1 R = 0. If d i = 0 and here is an ordering period in he block [i, j] 0, hen we have sr i > 0 and period i is necessarily an ordering period since s R i 1 = 0.

6 6 (ii) Assume ha he (unique) ordering period is k in he block [i, j] α u j wih i k j 1. From Assumpion 1, we have u k u k+1 + d k+1 u k+2 + d k+2 + d k+1 u j + d j + j 1 i=k+1 d i. Thus u k u j + d k+1,j. There are wo possible cases: Case 1: u k = u j + d k+1,j. In his case, since s R j = u j, hen sk R = u k which is no possible since [i, j] α u j is a block. Case 2: u j 1 < u j + d j. In his case, i is no possible o have s R j = u j wihou having an addiional ordering period in he block, which conradics Theorem 1. So, he ordering period has o be a period j in a block [i, j] α u j. Using Theorem 1 and Propery 2, we propose a polynomial ime algorihm o solve he 2ULS-IB R problem. 3.2 Recursion formula In his secion, we derive a polynomial backward dynamic programming algorihm o solve he 2ULS-IB R problem. The raionale of his algorihm is o compue a block decomposiion of he reailer s replenishmen plan such ha he oal cos of he supply chain is minimized using he dominance properies of he opimal soluions of he problem. Le i, j be wo periods such ha 1 i j T. Le us consider a regular subplan [i, j] of a soluion of he 2ULS-IB R problem. Noice ha by definiion [i, j] is no necessarily a block unless propery 0 < sk R < u k for all k {i,..., j 1} holds. Assume ha a period, an order quaniy X ij = d ij si 1 R + sr j (see Definiion 4) is available a he supplier level, i.e. i is eiher ordered a period or sored a period 1 assuming he ZIO policy. The aim is o decompose he regular subplan [i, j] ino blocks saisfying Propery 2. An example is given in Figure 3. The graph represens subplans of a soluion for an insance of he 2ULS-IB R problem where T = 4. A period = 1, a quaniy X 11 = d 1 + u 1 is available and a period = 2, a quaniy X 24 = d 24 u 1 is available a he supplier level (i is also available a period 3). In his example, [2, 4] is a regular subplan composed of he blocks [2, 3] u 1 0 and [4, 4]0 0. X 11 X 24 X 24 X 44 Supplier X 11 X 23 X 44 Reailer 1 u 1 2 u 1 d d 1 d 2 d 3 d 4 Figure 3: Subplans decomposiion for an insance of he 2ULS-IB R problem where T = 4. The recursion formula will be defined in he following order given ha [i, j] is an inerval of periods: he cos of a block (φ α ), he cos funcions required o compue he cos of a regular subplan including he supplier s holding cos (w αγ, vα and G α ) and finally he cos of he supply chain (Cα i ) Compuaion of he cos of a block Le φ α be he cos of saisfying he demands of a block [i, j]α wih a single ordering a period k if i exiss (Theorem 1), 1 i k j T. We will denoe by φ α ij he cos of he block [i, j]α wihou an ordering period. Using Propery 2(i), he cos φ 0 is defined as follows: φ 0 = f R i + pi RX ij + j hn R (d n+1,j + ), n=i if k = i and 0 < d ij + u i 0, if i = j and d ij + = 0 +, oherwise.

7 7 If [i, j] u i 1 u j is a block such ha u i 1 = d ij + u j, hen he demands of his block can be saisfied wihou seing any order beween period i and j since si 1 R = u i 1. Oherwise, using Propery 2(ii), a quaniy X ij has o be ordered a period j. Moreover, we have o ensure ha he invenory bounds consrains are no violaed, and ha he demands d i,j 1 can be covered by he invenory quaniy a he end of period i 1, i.e. u i 1 > d i,j 1. Thus, he cos φ u i 1u j of he block is given by: φ u i 1u j = j n=i hr n (u i 1 d in + u j ), fk R + pr k X ij + j 1 n=i hr n (u i 1 d in ) + h R j u j, if k = j and +, if u i 1 = d ij + u j d ij + u j > u i 1 > d i,j 1 oherwise. In a block [i, j] u i 1 0, if u i 1 < d ij, hen he quaniy X ij can be ordered a any period k beween i and j. In his case, we have o ensure ha he invenory bounds consrains are no violaed, ha he invenory quaniy u i 1 covers he demands before period k (u i 1 > d i,k 1 ), and ha he demands afer period k can be saisfied (u k d k+1,j ). The cos φ u i 10 is hen given by: φ u i 10 = Compuaion of he cos of a regular subplan f R k + pr k X ij + k 1 n=i hr n (u i 1 d in ) + j 1 n=k hr n d n+1,j, if d ij > u i 1 > d i,k 1 and w αγ = j 1 n=i hr n (u i 1 d n ), +, u k d k+1,j if u i 1 = d ij oherwise. Le G α be he opimal cos o cover he demands d ij of he regular subplan [i, j] where a quaniy X ij is available a period a he supplier level wih 1 i j T and 1 j, si 1 R = α and sr j =. Compuing G α requires he compuaion of he coss of he blocks ha compose he subplan [i, j]. Therefore, in order o compue G α efficienly, we firs need o find a suiable decomposiion of he subplan [i, j]. Le w αγ be he opimal cos of he regular subplan [i, j] where a quaniy X ij is available a period a he supplier level, k is a period of he firs block of [i, j], and if here is an order in his firs block, hen his order occurs a period k. The aim is o find he las period l of he firs block of he regular subplan [i, j] wih a possible order a period k in an opimal soluion. For sake of clariy, since he las period l of he firs block is no known, we use he noaion γ {0, 1} o represen he ougoing sock a he end of he firs block. Parameer γ = 0 (resp. γ = 1) means ha he ougoing sock a he end of he firs block is null (resp. equal o he invenory bound). The index, i, j, k, α, γ, are such ha 1 i k < j T, 1 k, si 1 R = α, sr j = and γ {0, 1}. The cos w αγ is given by: In he case where γ = 1, he firs erm φ αu l ilk min k l<j {φα0 ilk + G0 k,l+1,j } + k 1 p= hs px ij, if γ = 0, min k l<j {φαu l ilk + Gu l k,l+1,j } + k 1 p= hs px ij, if γ = 1. (10) in Equaion (10) represens he cos of saisfying he demands of he block [i, l] α u l wih a possible order a period k. The second erm G u l k,l+1,j in Equaion (10) represens he opimal cos for saisfying he demands of he regular subplan [l + 1, j] where sl R = u l and s R j = and assuming ha he quaniy X l+1,j is available a period k a he supplier level. Finally, he las erm k 1 p= hs px ij represens he cos of carrying X ij unis from period o period k a he supplier level. In he sequel, we define by l αγ achieving he minimum of he cos w αγ, i.e. lαγ = argmin wαγ. he period A represenaion of he cos w α1 is depiced in Figure 4. There are X ij unis available a period a he supplier level. A he reailer level, [i, l] α u l is he firs block of he regular subplan [i, j], where s R i 1 = α and sr j =. A quaniy X il is ordered a period k in his block. A he supplier level, a quaniy X ij is sored from period o

8 8 period k and an amoun X l+1,j is available a period k o saisfy he demands of he regular subplan [l + 1, j]. The differen erms of w α1 are shown in Figure 4. X ij unis are available Supplier k 1 p= hs p X ij X ij X... ij X l+1,j k k+1 X il Reailer α u i... k... l l l+1... j d i d k d l d l+1 d j φ αu l ilk G u l k,l+1,j Figure 4: Illusraion of he cos w α1 where [i, j] is a regular subplan and an amoun X ij is available a he supplier level a period. Le v α, wih 1 i < j T and 1 j, be he minimum cos of a regular subplan [i, j] composed of a leas wo blocks such ha si 1 R = α and sr j = and assuming ha a quaniy X ij is available a period a he supplier level, 1 j. The cos v α is given by: v α = min i k<j;γ {0,1} {wαγ }. (11) From he definiion of he cos v α, we can hen compue he cos Gα where he fixed ordering cos f S a he supplier level is no included. The cos G α is given by: G α = min{ min k j {φα + k 1 l= hs l X ij}, v α }, if X ij > 0, φ α ij, if X ij = 0. In Equaion (12), he erm min k j {φ α + k 1 l= hs l X ij} represens he opimal cos of he regular subplan [i, j] when i is made of a single block, and v α is he opimal cos of [i, j] when i is composed of a leas wo blocks. If [i, j] α is no a block or if k < i hen he cos φα will be equal o +. Moreover, Gα will be equal o + if i > j or > j. (12) Compuaion of he cos of he supply chain Le Ci α be he opimal cos of he supply chain for saisfying he demands d it of he regular subplan [i, T] where st R = 0 and he firs ordering period a he supplier level is larger han or equal o, wih 1 i T, 1 T, α {0, u i 1 }. The oal ordering quaniy of he subplan is equal o X it. The aim is o deermine he ordering periods saisfying he ZIO propery a he supplier level in order o saisfy he demands of he regular subplan [i, T]. If X it = 0, hen no order is required a he supplier level. The cos is hen equal o he cos GiT α0 of he subplan [i, T] wih si 1 R = α: Ci α = Gα0 it. (13) If X it > 0, hen he quaniy X it is compleely or parially ordered a period or a a subsequen period if no order occurs a period a he supplier level. The cos Ci α is given by he following equaion where 1(x) is a

9 9 funcion which is equal o 0 if x = 0 and + oherwise (see Figure 5). C α i = min{cα +1,i, f S + p S X it + G α0 it, (14) min {min( f S + p S X il, 1(X il )) + G αγ il + C γ i l<t;γ {0,u l } +1,l+1 }}, where is he las ordering period a he reailer level in he regular subplan [i, l] ( is deermined and sored when he cos G αγ il is compued). The period + 1 is he earlies ordering period from which he supplier can order for saisfying he demands of he regular subplan [l + 1, T]. If here is no ordering period in he regular subplan [i, l], hen we se =. The firs erm in Equaion (14) corresponds o he case where here is no order a period a he supplier level. The second erm in Equaion (14) corresponds o he case where a quaniy X it is ordered a period a he supplier level. Finally, he las erm in Equaion (14) represens he case where he quaniy X it is parially ordered a period a he supplier level: a quaniy X il > 0 is ordered a period o saisfy he demands of he regular subplan [i, l] wih i l < T. Because of he ZIO propery a he supplier level, he supplier orders he quaniy X l+1,t afer period. A represenaion of he las erm of he cos Ci α is provided in Figure 5. In his figure, a quaniy X il of unis is ordered a period a he supplier level for saisfying he demands of he regular subplan [i, l] where si 1 R = α and sl R = γ {0, u l }. The period corresponds o he las ordering period in he regular subplan [i, l]. Since he ZIO propery holds a he supplier level, we know ha s S = 0. Then, he nex likely candidae for an ordering period a he supplier is he period + 1 if i exiss. The componens in he definiion of he cos Ci α are depiced in he figure. X il min(f S + ps X il, 1(X il )) Supplier x R > 0 Reailer α i γ l l+1... T d i d d l d l+1 d T G αγ il C γ +1,l+1 Figure 5: Illusraion of he cos Ci α where [i, T] is a regular subplan and is an ordering period a he supplier level. Opimal cos. The opimal cos of saisfying he demands of he regular subplan [1, T] is given by C11 0 since sr 0 earlies order period a he supplier level is = 1. = 0 and he Complexiy analysis A pre-processing phase will consis of he compuaion of d 1j for all j {1,..., T} in O(T). Therefore, each d ij for all i, j {1,..., T} can be compued in consan ime. Moreover, he holding coss required in he compuaion of each cos componen is pre-compued and sored in O(T 2 ). Therefore, he cos φ α can be compued and sored in O(T3 ) for all i, j, k {1,..., T}. Besides, i akes O(T 4 ) ime o compue and o sore he coss G α and v α. Finally, he cos wαγ is compued in O(T5 ), and hen he ime complexiy of he dynamic programming algorihm based on he recursion formula (14) o compue C11 0 is O(T5 ). In wha follows, we show how he ime complexiy of compuing he cos w αγ can be improved from O(T5 ) o O(T 4 ) by generalizing he resul of Aamürk and Küçükyavuz [2] for he 2ULS case. To his end, we firs need

10 10 o recall he observaion of Aamürk and Küçükyavuz [2] for he reailer level. We provide a deailed explanaion of he observaion in A. Observaion 1. For all 1 i k j T, α {0, u i 1 } and {0, u j }, we have: (i) if φ α = +, hen φα i 1,j,k = + where α = 0 if α = 0 and α = u i 2 if α = u i 1. (ii) if φ 0 = +, hen φ0 i 1,j,k = +. (iii) if φ u i 1 φ u i 2 i 1,j,k = = +, hen: φ u i 1 + 1, if u i 2 > d i 1,k 1, u i 2 < u i 1 + d i 1 and d ij + > u i 1 φ u i 1 + 2, if u i 2 > d i 1,k 1, u i 2 < u i 1 + d i 1 and d ij + = u i 1 +, oherwise u i 2 + d i 1 ) and 2 = fk R + 1. where 1 = h R i 2 u i 2 + (p R k k 1 l=i 1 hr l )(u i 1 The observaion below is deduced from Observaion 1 and Assumpion 1. I will be used o compue efficienly he cos w u i 1γ. Observaion 2. For fixed periods, j, k such as 1 k j T and 1 k, l u i 2γ,i 1,j,k = lu i 1γ for all 1 i k. Proof. Le, i, j, k be periods such as 1 i k j T and 1 k. Le l be a period beween k and j. Two cases have o be considered: Case 1: d il + > u i 1 where {0, u l }. Le us show ha for each period n {l + 1,..., k}, we have d in + > u i 1 where {0, u n }. Le n > l and = u n (he proof for = 0 is similar). We have d in + u n = d i,n 1 + d n + u n d i,n 1 + u n 1... d il + u l > u i 1 from Assumpion 1. Thus, for fixed periods, j, k and γ = 0 (he proof is similar for γ = 1), from Observaion 1 and he previous remark, if u i 2 > d i 1,k 1, u i 2 < u i 1 + d i 1 and here exiss a period l beween k and j such ha d il > u i 1, we have: w u k 1 i 20,i 1,j,k = min k l<j {φu i 20 i 1,l,k + G0 k,l+1,j } + h S px i 1,j p= = min k l<j {φu i 10 ilk + G 0 k 1 k,l+1,j } + h S px i 1,j + 1 (15) = w u k 1 i 10 + p= p= h S p(u i 1 u i 1 + d i 1 ) + 1 (16) Equaion (15) comes from Observaion 1 and he fac ha 1 is independen of l. The ordering quaniy X i 1,j of a regular subplan [i 1, j] consiss of he ordering quaniy X ij of a regular subplan [i, j] plus he quaniy u i 1 u i 2 + d i 1 which leads o Equaion (16). Case 2: d il + = u i 1 where {0, u l }. In ha case, for all periods n < l, we canno have d in + > u i 1 oherwise we fall ino he previous case. Moreover, for all periods n > l, since d il + = u i 1, he regular subplan [i, n] where s R i 1 = u i 1 and s R n {0, u n } canno be a block. Then, similarly o he previous case, for fixed periods, j, k and γ = 0, if u i 2 > d i 1,k 1, u i 2 < u i 1 + d i 1 and here exiss a period l beween k and j such ha d il = u i 1, we have: w u k 1 i 20,i 1,j,k = wu i 10 + p= h S p(u i 1 u i 1 + d i 1 ) + 2

11 11 If u i 2 > d i 1,k 1 or u i 2 < u i 1 + d i 1 or he assumpion of Case 1 or he one of Case 2 are no saisfied, hen we se w u i 20,i 1,j,k = +. Therefore, for fixed periods, j, k and 1 i k, we have lu i 2γ,i 1,j,k = lu i 1γ. The cos w αγ,i 1,j,k can be compued from wαγ independenly of period l by using Observaion 1 and 2. For all 1 i T and given k,, j, wih i k < j T and 1 k, for α {0, u i 1 }, γ {0, 1} and {0, u j }, he cos w αγ can be done in O(T) ime using he following equaions: (i) w 0γ,i 1,j,k = + for any value of w0γ (ii) w u i 2γ,i 1,j,k = w u i 1γ k 1 h S l (u i 1 u i 2 + d i 1 ), l= if u i 2 > d i 1,k 1 and d i,l u i 1 γ w u k 1 i 1γ l= if u i 2 > d i 1,k 1 and d u i,l i 1 γ +, oherwise. + γ u l u i 1 γ h S l (u i 1 u i 2 + d i 1 ), Consequenly, for fixed periods, k, j, he cos w αγ + γ u l u i 1 γ > u i 1 = u i 1 wih 1 i k can be compued in O(T) ime. So, for all periods i, k,, j such ha 1 i k j T and 1 k, he cos w αγ is compued in O(T4 ) ime. This implies ha he algorihm which solves he 2ULS-IB R problem runs in O(T 4 ) ime. 4 The 2ULS-IB S problem Jaruphongsa e al. [9] propose a polynomial ime algorihm o solve he 2ULS-IB S problem wih demand ime window consrains and saionary invenory bounds. They consider ha h S h R and ha he fixed ordering cos and he uni ordering cos are decreasing. These specific coss make he problem solvable in polynomial ime. In his secion, we consider he 2ULS-IB S problem under a general cos srucure and we prove ha his problem is NP-hard. Theorem 2. The 2ULS-IB S problem is NP-hard. Proof. We prove ha he 2ULS-IB S problem is NP-hard hrough a reducion from he subse sum problem, which is an NP-complee problem [5]. An insance of he subse sum problem is given by an ineger S and a se S of n inegers (a 1,..., a n ). The quesion is: does here exis a subse A S such ha ai A a i = S? We ransform an insance of he subse sum problem ino an insance of he 2ULS-IB S problem in he following way: - T = 2n + 1. Le us denoe by T 1 (resp. T 2 ) he se of odd (resp. even) periods in he se {1,..., 2n}. - d = 0 for all T 1 T 2, d T = S - f S = 1 for all T 1, f S = 2S for all T 2 {T} f R = 2S for all T 1 {T}, f R = 0 for all T 2 - h S = h R = 0 for all {1,..., T} - p R = 0 for all {1,..., T} p S = 1 1/a 2 for all T 1 T 2, p S T = 0 - u S = a 2 for all T 1 T 2, u S T = 0

12 12 A represenaion of his insance is given in Figure 6. The fixed ordering coss and he uni ordering coss of he supplier (resp. reailer) are indicaed a he op (resp. boom). A he supplier level, he quaniies on he horizonal edges represen he invenory bounds. f S 1 2S 1 2S 1 2S 2S p S a 1 a 1 a 2 a 2 a 3 a 3 Supplier a 1 a 1 a 2 a 2 a 3 a Reailer S f R 2S 0 2S 0 2S 0 2S p R Figure 6: Insance A of he 2ULS-IB S problem in he proof of Theorem 2 wih n = 3, S = {a 1, a 2, a 3 }. Observaion 3. Noe ha if we order x S = a 2 a period T 1 hen he oal ordering cos is equal o f S + p S xs = 1 + (1 1/a 2 )a 2 which is exacly equal o xs (in his case, he average cos of ordering one uni is equal o 1). If x S < a 2 a period T 1, hen we have ha he oal ordering cos f S + p S xs = 1 + (1 1/a 2 )xs = x S + 1 xs /a 2 > xs (in his case, he average cos of ordering one uni is larger han 1). From his observaion, le us prove ha here exiss a soluion for he 2ULS-IB S problem of cos a mos S if and only if here exiss a soluion for he subse sum problem. Assume ha here exiss a soluion A of he subse sum problem. The following soluion for he 2ULS-IB S problem is of cos a mos S: for each elemen a i in he se A, he supplier orders a quaniy a i a period = 2i 1 and sore i unil period + 1 (see Figure 7). The invenory bound is no exceeded since i is exacly equal o a i. From Observaion 3 above, he cos of ordering a i unis for each a i I a he supplier level is equal o a i. Since ai I a i = S, he oal cos a he supplier level is S. A period = 2i, he reailer orders all he unis and sore hem unil period T. Since f R = 0 for all T 2 and h R = p R = 0 for all {1,..., T}, he oal cos a he reailer level is equal o 0. So, here exiss a soluion for he 2ULS-IB S problem of cos S. a 1 a 3 a 1 a 3 Supplier a 1 a 3 a 1 a 1 a 1 a 1 S Reailer S Figure 7: Soluion for he 2ULS-IB S problem in he proof of Theorem 2 wih n = 3, S = {a 1, a 2, a 3 } and a 1 + a 3 = S. Assume ha here exiss a soluion for he 2ULS-IB S problem wih a cos of a mos S. Since f S = 2S for all T 2, he supplier has o order a period T 1, oherwise he cos will exceed S. Likewise, since f R = 2S for all T 1, he reailer has o order a period T 2. In order o no exceed he invenory bounds, he supplier can sore a mos u S = a 2 unis from period o period + 1. Thus, he quaniy ordered by he supplier a period T 1 is a mos a 2. A period T 2, he reailer orders he unis in he supplier s invenory and sores hem

13 13 unil period T wih a cos equal o 0. From Observaion 3, if he supplier orders a period, hen x S = a 2 (his is he only way o order one uni wih a cos of a mos 1 so ha he oal cos is a mos S). Thus, S = T a 2 where T is he se of periods where he supplier orders. This implies ha here exiss a soluion o he subse sum problem. The relaed lo-sizing problem wih producion capaciy consrains insead of invenory bounds has been proved o be NP-hard [4]. The insance parameer p S used in he proof of Theorem 2 is based on [4]. Moreover, i is worh noicing he invenory bound a period acs as a producion capaciy since he supplier canno supply a he ordered unis. 5 The 2ULS-IB SR problem We have proved ha he 2ULS-IB S problem is NP-hard. By seing u R = =1 T d, we can ransform an insance of he 2ULS-IB S problem ino an insance of he 2ULS-IB SR problem. Thus, he 2ULS-IB SR problem is a leas as hard as he 2ULS-IB S problem. In his secion, we describe a pseudo-polynomial dynamic programming algorihm o solve he 2ULS-IB SR problem. This proves ha his problem is no srongly NP-hard. Le s R {0, 1,..., u R} (resp. ss {0, 1,..., u S }) be he invenory quaniy available a he end of period a he reailer (resp. supplier) level. The principle of he algorihm is o consider all he possible values of he invenory quaniy s R (resp. s S ) a he reailer (resp. supplier) level. Noice ha he ZIO propery does no hold neiher a he supplier nor a he reailer levels for he 2ULS-IB SR problem. Le C i (X) be he cos of ordering X unis a level i {R, S} a period, where he level R (resp. S) corresponds o he reailer (resp. supplier) level. The cos C i (X) is given by: { f i C(X) i + p i X, if X > 0 = 0, oherwise. We define V (s R 1, s, ss 1, s) as he cos of saisfying he demand d when: - s 1 R (resp. ss 1 ) unis are sored a period 1 and s (resp. s) unis are sored a period a he reailer (resp. supplier) level, - X R = s + d s 1 R (resp. XS = s + X R s S 1 ) unis are ordered a period a he reailer (resp. supplier) level. The cos V (s 1 R, s, ss 1, s) is defined by: V (s R 1, s, ss 1, s) = { C R (X R ) + C S (X S ), if s u R, s u S and s + s d T +, oherwise. Le H (s 1 R, ss 1 ) be he minimum cos of saisfying he demands d T where s 1 R (resp. ss 1 ) unis are sored a period 1 a he reailer (resp. supplier) level. From he definiion of he cos V (s 1 R, s, ss 1, s), we can compue he cos H (s 1 R, ss 1 ) as follows: H (s R 1, ss 1 ) = min {V (s s S R 1 R, s, ss 1, s) + H +1( s, s)},,s SS where S R = {max(0, s 1 R d ),..., M R}, wih MR = min(u R, d T), and S S = {max(0, s S 1 XR ),..., M S }, wih M S = min(u S, d T). Opimal cos The opimal cos of saisfying he demands d 1T assuming ha s0 R = ss 0 = 0 is given by H 1(0, 0). We iniialize he recursion by seing H T+1 (s R, ss ) = 0 for all he values sr 1 and ss i 1 ensuring feasibiliy.

14 14 Complexiy analysis Compuing he cos V (s 1 R, s, ss 1, s) can be done in O(uS 1 ur 1 MR MS ) for each period. Therefore, i akes O( i=1 T (us i 1 ur i 1 MR i Mi S)) o compue he opimal cos H 1(0, 0). This bound consiues he complexiy of he dynamic programming algorihm. This is pseudo-polynomial, implying ha he 2ULS-IB SR problem is no srongly NP-hard. In he nex secion, we consider he 2ULS problems wih invenory bounds assuming ha he demand a he reailer level has o be covered by a single order. 6 Analysis of lo-sizing problems wihou lo-spliing Jaruphongsa e al. [9] inroduce he problem where each demand mus be saisfied by exacly one dispach, i.e. he demand lo-spliing is no allowed a he reailer level. We called his consrain he No Lo-Spliing (NLS) consrain. In pracice, his sudy is moivaed by raceabiliy requiremens for he produc where he managemen of he invenory and he ranspor can be improved if he demand is supplied from he supplier o he reailer by a single delivery. We noe x R k 0 he quaniy of demand d which is ordered a period k o saisfy a demand d a he reailer level. We have i=1 xr i = d. Definiion 5 (NLS consrain). An ordering plan x R fulfills he NLS consrain if here does no exis wo periods l and k wih l < k such ha xl R > 0 and xr k > 0 for all periods. The 2ULS-IB R and he 2ULS-IB S problems wih he NLS consrain are denoed by 2ULS-IB R -NLS and 2ULS-IB S -NLS respecively. Before sudying he complexiy of he laer problems, i is ineresing o analyze he complexiy of he single level problem wih NLS consrain, ha we denoe by ULS-IB-NLS. The proofs of he following resuls are given in Appendix. We consider T periods {1,..., T}. In he ULS-IB-NLS problem, ordering unis a period induces a fixed ordering cos f and a uni ordering cos p. Carrying unis from period o period + 1 induces a holding cos h. The oal cos is given by he sum of he ordering and holding coss. The aim is o deermine an ordering plan which saisfies he demands and which minimizes he oal cos. We denoe by x he ordering quaniy a period, s he invenory quaniy a he end of period and y he binary (seup) variable which is equal o 1 if here is an order a period and 0 oherwise. We say ha he invenory bound is saionary if u is consan hroughou he planning horizon. Theorem 3. The ULS-IB-NLS problem is srongly NP-hard, even if he invenory bound is saionary. Noe ha ULS-IB problem can be solved in polynomial ime [2, 11]. Theorem 3 shows ha adding he NLS consrain o his problem makes i srongly NP-hard. Corollary 1. The 2ULS-IB R -NLS problem is srongly NP-hard, even if he invenory bound is saionary. Jaruphongsa e al. [9] prove ha he 2ULS-IB S -NLS problem wih demand ime window consrains is weakly NP-hard. We show ha his problem is also weakly NP-hard wihou demand ime window consrains, and ha i is even srongly NP-hard. Corollary 2. The 2ULS-IB S -NLS is srongly NP-hard, even if he invenory bound is saionary. Consider he case where he supplier and he reailer have invenory bounds. We prove ha he 2ULS-IB SR -NLS problem is srongly NP-hard. Corollary 3. The 2ULS-IB SR -NLS problem is srongly NP-hard, even if he invenory bound is saionary.

15 15 7 Conclusion and fuure work This paper considers wo-level uncapaciaed lo-sizing problems wih invenory bounds, and provides a complexiy analysis of hese problems. We presen an O(T 4 ) dynamic programming algorihm which solves he problem where he invenory bounds are se a he reailer level. When he invenory bounds are se a he supplier level, we prove ha he problem is weakly NP-hard. We also presen a pseudo-polynomial dynamic programming algorihm which ensures ha his problem is no srongly NP-hard. Considering ha lo-spliing is no allowed, we prove ha he ULS problem wih invenory bounds and he 2ULS problems where he invenory bounds are se eiher a he reailer level, or a he supplier level or a boh of hem are srongly NP-hard. I would be ineresing for a fuure work o improve he running ime of he algorihm solving he 2ULS-IB R problem. Moreover, he complexiy of he 2ULS-IB S problem where he invenory bounds of he supplier are saionary is an open problem. Anoher ineresing perspecive is o consider ha he supplier and he reailer share he same invenory faciliy. In his case, a each period, he invenory quaniy of he supplier plus he one of he reailer canno exceed a given invenory bound. The lo-sizing problems ha have been sudied is his paper consider a single iem. I would also be ineresing o sudy he case where here are several iems. Finally, invesigaing efficien algorihms o solve he NP-hard 2ULS problems wih invenory bounds is also a promising issue for pracical applicaions. In paricular, i would be ineresing o consider he valid inequaliies proposed by [3] for solving he wo-level case. Acknowledgemens This work was suppored by FUI projec RCSM Risk, Credi Chain & Supply Chain Managemen, financed by Région Ile-de-France. Appendix A Proof of Observaion 1. (i) If φ α = +, hen he regular subplan [i, j] wih a single order a period k, sr i 1 = α and s R j = is no a block. The violaion(s) observed in he regular subplan [i, j] will also hold for he regular subplan [i 1, j]. (ii) If φ 0 = +, hen [i, j]0 is a block, and by Propery 2(i) here is an ordering period a k = i. We consider he regular subplan [i 1, j] wih an order a period k, si 2 R = 0 and sr j =. If d i 1 = 0, hen he regular subplan [i 1, j] is no a block since si 2 R = 0. If d i 1 > 0, hen d i 1 could no be covered and hus he regular subplan [i 1, j] is no a block. (iii) If φ u i 1 = +, hen [i, j] u i 1 is a block wih an ordering period k if i exiss. We consider he regular subplan [i 1, j] wih a single order a period k, si 2 R = u i 2 and s R j =. We wan o deermine if his regular subplan is a block. We know ha u i 2 u i 1 + d i 1 (Assumpion 1). If u i 2 = u i 1 + d i 1, hen his regular subplan is no a block because in ha case si 1 R = u i 1. If u i 2 < u i 1 + d i 1, he we have si 1 R < u i 1 d ij +, and here mus be an ordering period a k in he subplan [i 1, j]. Moreover, if u i 2 > d i 1,k 1, hen we have a block [i 1, j] u i 2. The reailer has o order a quaniy u i 1 u i 2 + d i 1 > 0 in addiion o X ij = d ij u i 1 + a period k. The invenory quaniies beween periods k and j remain unchanged in he block [i 1, j] u i 2. Since he demand d i 1 has o be covered by u i 2, here are u i 1 u i 2 + d i 1 less unis in he invenory beween periods i 1 and k 1. The cos φ u i 2 i 1,j,k of he block [i 1, j]u i 2 can be derived from φ u i 1 by considering hese wo cases: Case 1: Assume ha a quaniy X ij > 0 is ordered a period k in he block [i, j] u i 1. Then, he cos of he block [i 1, j] u i 2 is given by: φ u i 1 + hi 2 R u i 2 + (pk R k 1 l=i 1 hr l )(u i 1 u i 2 + d i 1 ) = φ u i Case 2: Assume ha no ordering period occurs in he block [i, j] u i 1. Then, an addiional fixed ordering cos fk R mus be considered o compue he cos of he block [i 1, j] u i 2, which will be given by: φ u i 1 + fk R + 1 = φ u i

16 16 Appendix B Proof of Theorem 3. We show ha he 3-Pariion problem, which is srongly NP-hard [5], can be reduced o he ULS-IB-NLS problem in polynomial ime. Recall ha an insance of he 3-Pariion problem is given by an ineger b and 3m inegers (a 1,..., a 3m ) such ha 3m i=1 a i = mb and b/4 < a i < b/2 for all i {1,..., 3m}. The quesion is: does here exis a pariion A 1... A m of {1,..., 3m} such ha i Aj a i = b for all j {1,..., m}. We ransform an insance of he 3-Pariion problem ino an insance of he ULS-IB-NLS problem in he following way: - T = 5m periods. Le us denoe T 1 (resp. T 2 ) he se of odd (resp. even) periods in he se {1,..., 2m}. - d = 0 for all T 1 d = (m /2)b for all T 2 d = a 2m for all {2m + 1,..., T} - f = 0 for all T 1 f = b + 1 for all T 2 {2m + 1,..., T} - h = 0 for all {1,..., T} - p = 0 for all T \{2m 1} and p 2m 1 = 1 - u = mb for all {1,..., T} The insance is illusraed in Figure 8. The fixed ordering coss are indicaed a he op of each period. The invenory bounds are represened on he horizonal edges. f 0 b b b + 1 b + 1 b + 1 p mb mb mb mb mb mb 2m-1 2m 2m+1... T 0 (m-1)b 0 (m-2)b 0 0 a 1 a 3m Figure 8: Insance of he ULS-IB-NLS problem in he proof of Theorem 3. Le us show ha here exiss a soluion o he ULS-IB-NLS problem of cos a mos b if and only if here exiss a soluion o he 3-Pariion problem. Assume ha here exiss a soluion (A 1,..., A 3m ) of he 3-Pariion problem. The cos of he following ( soluion ) of he ULS-IB-NLS problem is b: a each period T 1, we order x = i A(+1)/2 a i + d +1 = b + b m +1 2 unis. Since p = 0 for all T 1 \{2m 1} and p 2m 1 = 1, i coss p 2m 1 x 2m 1 = p 2m 1 b = b o order hese unis. A each period T 2, he demand d is saisfied and b unis are sored which implies ha here is exacly s = 2 b unis ( in sock ) a he end of period. A each period T 1, we sore exacly a quaniy s 1 + x = 1 2 b + m 1 2 b = mb and he invenory bound u is no exceeded. Each demand d for all < 2m is saisfied and here is mb unis in sock a period 2m for saisfying he demands a period {2m + 1,..., T}. Since here is no holding cos, he cos of his soluion is b. Noe ha his soluion fulfills he NLS consrain since each demand is saisfied by a single order. Assume now ha here exiss a soluion o he ULS-IB-NLS problem of cos a mos b (see Figure 9).

17 17 mb (m-1)b b mb b mb 2b mb mb 2m-1 2m 2m+1... T 0 (m-1)b 0 (m-2)b 0 0 a 1 a 3m Figure 9: Soluion for he ULS-IB-NLS problem in he proof of Theorem 3. Since he fixed ordering cos is equal o b + 1 for all T 2 {2m + 1,..., T}, we canno order a hese periods. Thus, all orders are se a period T 1. Since for each period T 2, d = (m /2)b, and since he invenory bound is mb, a mos 2 b unis can be sored from period T 2 o a period in T 1. Since mb unis have o be available a period 2m (oherwise he cos will be greaer han b), hen 2 b unis have o be sored from period T 2 o period + 1. So, we have o order b unis a each period T 1 for saisfying he demands d 2m+1,T (we canno order all he unis a period 2m 1 since p 2m 1 = 1 and he cos will be greaer han b). Assuming he NLS consrain, each demand d for all {2m + 1,..., T} is saisfied by a single ordering period a T 1. So, here is a pariion of he periods {2m + 1,..., T} ino m ses (A 1,..., A m ) such ha i Aj d i = b for all j {1,..., m}. Since each demand d for all {2m + 1,..., T} corresponds o an ineger of (a 1,..., a 3m ), his means ha here exiss a soluion o he 3-Pariion problem. Proof of Corollary 1. We do a reducion from he ULS-IB-NLS problem, ha is srongly NP-hard, as shown by Theorem 3. We ransform an insance of he ULS-IB-NLS problem ino he following insance of he 2ULS-IB R -NLS problem. The coss of he reailer are he ones of he ULS-IB-NLS problem, i.e. u R = u, f R = f, p R = p and h R = h for all {1,..., T}. The supplier coss are given by f S = h S = p S = 0 for all {1,..., T}. The demands are he same as he ones of he ULS-IB-NLS problem. Since all he supplier s coss are 0, he cos of an opimal soluion for he ULS-IB-NLS problem is equal o he opimal cos of is corresponding 2ULS-IB R -NLS insance (see Figure 10). mb (m-1)b b Supplier Reailer m-1 2m 2m+1... T mb (m-1)b b mb b mb 2b mb mb 2m-1 2m 2m+1... T 0 (m-1)b 0 (m-2)b 0 0 a 1 a 3m Figure 10: Soluion for he 2ULS-IB R -NLS problem in he proof of Corollary 1. By Theorem 3, he 2ULS-IB R -NLS problem is also srongly NP-hard. Proof of Corollary 2. As in he proof of Corollary 1, we do a reducion from he ULS-IB-NLS problem, which is srongly NP-hard, as shown in Theorem 3. We ransform an insance of he ULS-IB-NLS problem ino he following insance of problem 2ULS-IB S -NLS. The supplier s coss are he ones of he ULS-IB-NLS problem, i.e. u S = u, f S = f, p S = p and h S = h for all {1,..., T}. The reailer s coss are given by f R = p R = 0 for all {1,..., T} and, for all {1,..., T}, h R = M, where M is a large number (we can fix M = =1 T (h + p )). By his way, in an opimal soluion of he 2ULS-IB S -NLS problem, no quaniy will be sored a he reailer level. The demands are he same as he ones of he ULS-IB-NLS problem. Since f R = p R = 0, he cos of an opimal soluion of he 2ULS-IB S -NLS problem is equal o he opimal cos of is corresponding ULS-IB-NLS problem. Figure 11 illusraes such a soluion.

18 18 mb (m-1)b b mb b mb 2b Supplier mb mb 2m-1 2m 2m+1... T (m-1)b (m-2)b a 1 a 3m Reailer m-1 2m 2m+1... T 0 (m-1)b 0 (m-2)b 0 0 a 1 a 3m Figure 11: Soluion for he 2ULS-IB S -NLS problem in he proof of Corollary 2. Therefore, by Theorem 3, he 2ULS-IB S -NLS problem is also srongly NP-hard. Proof of Corollary 3. The proof of his corollary is he same as he one of Corollary 2 for he 2ULS-IB S -NLS problem by adding any invenory bound a he reailer level (in an opimal soluion no quaniy will be sored a he reailer level). References [1] A. Aamürk and S. Küçükyavuz. Lo sizing wih invenory bounds and fixed coss: Polyhedral sudy and compuaion. Operaions Research, 53(4): , [2] A. Aamürk and S. Küçükyavuz. An o(n 2 ) algorihm for lo sizing wih invenory bounds and fixed coss. Oper. Res. Le., 36(3): , May [3] A. Aamürk, S. Küçükyavuz, and B. Tezel. Pah cover and pah pack inequaliies for he capaciaed fixed-charge nework flow problem. SIAM Journal on Opimizaion, 27(3): , [4] M. Florian, J. K. Lensra, and A. H. G. R. Kan. Deerminisic producion planning: Algorihms and complexiy. Managemen Science, 26(7): , [5] M. R. Garey and D. S. Johnson. Compuers and Inracabiliy: A Guide o he Theory of NP-Compleeness. W. H. Freeman, Jan [6] J. Guiérrez, A. Sedeño-Noda, M. Colebrook, and J. Sicilia. An efficien approach for solving he lo-sizing problem wih ime-varying sorage capaciies. European Journal of Operaional Research, 189(3): , [7] H.-C. Hwang and H.-G. Jung. Warehouse capaciaed lo-sizing for a wo-sage supply chain. In The 2011 KORMS/KIIE Spring Join Conference on 26 May, pages , Incheon, Korea, [8] H.-C. Hwang and W. van den Heuvel. Improved algorihms for a lo-sizing problem wih invenory bounds and backlogging. Naval Research Logisics (NRL), 59(3-4): , [9] W. Jaruphongsa, S. Çeinkaya, and C.-Y. Lee. Warehouse space capaciy and delivery ime window consideraions in dynamic lo-sizing for a simple supply chain. Inernaional Journal of Producion Economics, 92(2): , [10] T. Liu. Economic lo sizing problem wih invenory bounds. European Journal of Operaional Research, 185(1): , [11] S. F. Love. Bounded producion and invenory models wih piecewise concave coss. Managemen Science, 20(3): , [12] R. A. Melo and L. A. Wolsey. Uncapaciaed wo-level lo-sizing. Oper. Res. Le., 38(4): , July 2010.