The multi-item capacitated lot-sizing problem with setup times and shortage costs

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1 The multi-item capacitated lt-sizing prblem with setup times and shrtage csts Nabil Absi 1,2, Safia Kedad-Sidhum 1 1 Labratire LIP6, 4 place Jussieu, Paris Cedex 05, France 2 Dynasys S.A., 10 Avenue Pierre Mendes France, Schiltigheim, France Abstract We address a multi-item capacitated lt-sizing prblem with setup times and shrtage csts that arises in real-wrld prductin planning prblems. Demand cannt be backlgged, but can be ttally r partially lst. The prblem is NP-hard. A mixed integer mathematical frmulatin is presented. Our apprach in this paper is t prpse sme classes f valid inequalities based n a generalizatin f Miller et al. [26] and Marchand and Wlsey [24] results. We als describe fast cmbinatrial separatin algrithms fr these new inequalities. We use them in a branch-and-cut framewrk t slve the prblem. Sme experimental results shwing the effectiveness f the apprach are reprted. Keywrds: multi-item, capacitated lt-sizing, setup times, shrtage csts, prductin planning, mixed integer prgramming, branch-and-cut. Intrductin The Multi-item Capacitated Lt-sizing Prblem with with Setup times and Shrtage csts called MCLSSP is a prductin planning prblem in which there is a time-varying demand fr a set f N items dented I = {1, 2,,N} ver T perids. The prductin shuld satisfy a restricted capacity and must take int accunt a set f additinal cnstraints. Indeed, launching the prductin f an item i at a given perid t fr a demand requirement d it invlves a variable capacity v it and a fixed cnsumptin f resurce f it usually called setup time in lt-sizing literature. The ttal available capacity at perid t is c t. The prductin shuld als satisfy lt-sizing cnstraints. Fr each perid t, an inventry cst γ it is attached t each item i as well as a variable unit prductin cst α it andasetupcstβ it. The prblem has the distinctive feature f allwing requirement shrtages because we deal with prblems with tight capacities. Indeed, when we are in lack f capacity t prduce the ttal demand, we try t spread the capacity amng the items by minimizing the ttal amunt f demand shrtages. Thus, we intrduce in the mdel a unit cst parameter ϕ it fr item i at perid t fr the requirement nt met regarding the demand. These csts shuld be viewed as penalty csts and their values are very high in cmparisn with ther cst cmpnents. T try t meet the demand fr an item i at perid t, we culd anticipate the prductin ver sme perids f time. Therefre, it dente the last perid at which an item i prduced at perid t can be cnsumed. The prblem MCLSSP is t find a prductin planning that minimizes the demand shrtage, the setup, the inventry and the prductin csts. Originally, the mtivatin fr designing a branchand-cut algrithm t slve the MCLSSP was t try t deal with real-wrld instances where the capacities were tight and were the mst imprtant bjective was t try t meet the maximum This wrk has been partially financed by DYNASYS S.A., under research cntract n. 588/2002. Crrespnding authr. Safia.Kedad-Sidhum@lip6.fr 1

2 amunt f client s needs. In these industrial applicatins, pstpning the demand is frequently prhibited. Our results are integrated in an APS 1 sftware. Flrian et al. [15] and Bitran and Yanasse [7] have shwn that the single-item capacitated lt-sizing prblem is NP-hard, even fr many special cases. Chen and Thizy [10] have prved that multi-item capacitated lt-sizing prblem (MCLSP) with setup times is strngly NP-hard. Since the seminal papers by Wagner and Within [38] and Manne [23] in the late 1950s, a lt f research has been dne n lt-sizing prblems. The single-item prblem has been given special interest fr its relative simplicity and fr its imprtance as a sub-prblem f sme mre cmplex lt-sizing prblems. Fr a cmplete review, the reader can refer t [9]. Althugh prductin planning mdels invlving multiple items, restrictive capacities and significant setup times ccur frequently in industrial situatins and have ften been studied in the literature, btaining ptimal and smetimes even feasible slutins remains challenging. Trigeir et al. [36] were amng the firsts t try t slve such mdels. They prpsed a lagrangean relaxatin based heuristic t slve the single-machine, multi-item, capacitated lt-sizing prblem with setup times t btain near-ptimal slutins. Since the lagrangean slutins are usually infeasible, they used a smthing heuristic in rder t btain feasible prductin plans. Hwever, we can ntice that fr all the instances with tight capacities, they were nt able t find feasible slutins. Belvaux and Wlsey [6], Leung et al. [19] and Pchet and Wlsey [32] prpsed exact methds t slve multi-item capacitated lt-sizing prblems by strengthening the LP frmulatins with valid inequalities and then using a mixed integer prgramming (MIP) slver. Barany et al. [4] have defined sme inequalities fr the uncapacitated lt-sizing prblem. Miller et al. [26] have studied the plyhedral structure f sme capacitated prductin planning prblems with setup times. We can als mentin the wrk f Marchand and Wlsey [24] fr the 0-1 knapsack prblem which appears as a relaxatin f a number f structured MIP prblems such as the MCLSP prblem. There are few references dealing with lt-sizing prblems with shrtage csts. Recently, Sandbthe and Thmpsn [35] addressed a single-item uncapacitated lt-sizing prblem with shrtage csts. The authrs prpsed an O(T 3 ) frward dynamic prgramming algrithm t slve the prblem. Aksen et al. [3] prpsed a dynamic prgramming methd t slve the same prblem in O(T 2 ). Lparic et al. [21] prpsed valid inequalities fr the single-item uncapacitated lt-sizing prblem with sales instead f fixed demands and lwer bunds n stck variables. T the best f ur knwledge, this is the first paper that deals with setup times cnstraints and shrtage csts fr the multi-item capacitated lt-sizing prblem. Nevertheless, we can cite the fllwing results fr slving prblems where demand cannt be met at every perid. Dixn et al. [13] deal with lack f capacity by cnsidering vertimes. The capacity cnstraint is expanded by making extra capacity available at a certain cst. The multi-item capacitated lt-sizing prblem with setup times and vertime decisins is investigated by Diaby et al. [12], Özdamar and Birbil [28] and Özdamar and Bzyel [29]. Anther class f methds allws backlg. Here demand must be satisfied, but the items can be prduced later at an extra cst. We can cite the wrk f Pchet and Wlsey [31] and Zangwill [41]. In all these cases, the demand must be satisfied and the amunt f lst sales fr each item at each perid is nt given. The nly infrmatin that we have is the amunt f missing capacity at each perid t satisfy the amunt f riginal and backlgged demands. The main cntributins f this paper are twfld. First, we shw that the results btained frm cnsidering relaxatins based n single-perid sub-mdel can be used t derive new valid inequalities fr the MCLSSP prblem. These results are derived frm Miller et al. [26] previus wrk n the plyhedral structure f the single-perid relaxatin f the multi-item capacitated lt-sizing prblem. Secnd, we use these inequalities within a branch-and-cut framewrk t find near ptimal slutins. An utline f the remainder f the paper fllws. Sectins 1 and 2 describe MIP frmulatins f the MCLSSP prblem and its single-perid relaxatin. In Sectins 3 and 5 we state results cncerning the generalizatin f the (l, S), cver and reverse cver valid inequalities. In Sectin 6, we shw that these inequalities can be strengthened using a lifting prcedure. Separatin heuristics are presented in Sectins 4, 7 and 8. Finally, cmputatinal results are given in Sectin 9 t shw the effectiveness f using these inequalities in a branch-and-cut algrithm. 1 Advance Planning and Scheduling. 2

3 1 Frmulatin f the MCLSSP prblem In this sectin we present a MIP frmulatin f the MCLSSP prblem, which is an extensin f the classical frmulatin f the MCLSP prblem previusly studied by Miller [25] and Trigeir et al. [36]. This mdel is usually called aggregated mdel, see [9]. Other frmulatins are studied in the literature. We can mentin the facility lcatin-based frmulatin intrduced by Krarup and Bilde [18] and the shrtest path frmulatin prpsed by Evans [14]. In the sequel f the paper, we cnsider that i =1,...,N and t =1,...,T. We set x it as the quantity f item i prduced at perid t. T deal with the fixed setup times and csts, we need als t define y it as a binary variable equal t 1 if item i is prduced at perid t (i.e. if x it > 0). The variable s it is the inventry value fr item i at the end f perid t. The demand shrtage fr item i at perid t is mdeled by a nn-negative variable r it added t the prductin variables x it with a very high unit penalty cst in the bjective functin, because the main gal is t satisfy the custmer and thus t have the minimum amunt f the requirements nt met. We can ntice that r it = (s i,t 1 + x it)+d it if r it > 0and0therwise. subject t: min N T i=1 t=1 α itx it + β ity it + γ its it + ϕ itr it (1) x it + r it s it + s i,t 1 = d it, i =1,...,N,t=1,...,T. (2) x it min N i=1 v itx it + N i=1 ( c t f it it, v it f ity it c t, t =1,...,T. (3) d it ) y it, i =1,...,N,t=1,...,T. (4) r it d it, i =1,...,N,t=1,...,T (5) x it,s it,r it 0, i =1,...,N,t =1,...,T (6) y it {0, 1}, i =1,...,N,t=1,...,T (7) The bjective functin (1) minimizes the ttal cst induced by the prductin plan (unit prductin csts, inventry csts, shrtage csts and setup csts). Cnstraints (2) are the flw cnservatin f the inventry thrugh the planning hrizn. Cnstraints (3) are the capacity cnstraints, the verall cnsumptin must remain lwer than the available capacity. If we prduce an item then the prductin must nt exceed a maximum prductin level, this cnditin is ensured by cnstraints (4). Indeed, the maximum prductin is the minimum between the maximum quantity f the item that we can prduce and the ttal requirement n sectin [t,..., it] fthe hrizn. We recall that it dente the last perid at which an item i prduced at perid t can be cnsumed. Cnstraints (5) define upper bunds n the requirement nt met fr item i n perid t. Cnstraints (6) and (7) characterize the variable s dmain: x it, s it and r it are nn-negative fr i =1,...,N and t =1,...,T and y it is a binary variable fr i =1,...,N and t =1,...,T. In the sequel f the paper, we refer t valid inequalities fr the set defined by (2) (7) as valid fr MCLSSP. 3

4 2 Single-perid relaxatin f the MCLSSP prblem Based n the frmulatin f the MCLSSP prblem described in sectin 1, we define a simplified sub-mdel btained by cnsidering a single time perid relaxatin. This is particularly useful t derive valid inequalities fr the MCLSSP prblem. The gal f this relaxatin is nt t slve each perid separately by cnsidering nly the demand f the current perid but is t prvide strng valid inequalities fr the single-perid prblem that are als valid fr the initial prblem taking int accunt aggregated demands. This is dne by allwing anticipatins n prductin. This mdel is called the single-perid relaxatin f the MCLSSP with preceding inventry [25]. Our apprach is similar t the ne used by Cnstantin [11] and Miller [25] t derive a set f valid inequalities fr the MCLSP prblem based n a single-perid relaxatin. In this relaxatin, the prductin ver a given perid culd satisfy the requirement f a sectin f cnsecutive perids. Cnsequently, fr each perid t = 1,...,T and each item i = 1,...,N we use the parameter it previusly defined with it =1,...,T. This will enable us t create a mathematical mdel fr each perid t = 1,...,T which captures the interactin between the tight capacity in ne hand and the requirements, the prductins and the setups n the ther hand frm perid t t it, fr each item i =1,...,N. Here ur gal is t derive valid inequalities fr MCLSSP by cnsidering simplified mdels btained frm a single time perid relaxatin with preceding inventry. P Let us dente: δa,b i b = t=a dit. One simple family f valid inequalities is given by Prpsitin 1. The inequalities x it + it are valid fr MCLSSP. 0 r it si,t 1 + it +1 1 δ i t, it y it A δ i t,it, i =1,...,N,t=1,...,T. (8) Prf. Summing the cnstraints (2) ver the sectin f hrizn [t,..., it] gives: it (x it + r it ) s i,it + s i,t 1 = it d it, i =1,...,N,t=1,...,T. (9) The variable x it can be redefined by cnsidering the perid where the prductin is really cnsumed. This refrmulatin is called the facility lcatin-based frmulatin intrduced initially by Krarup and Bilde [18]. Therefre, we dente w itt with t [t, it] the quantity f the item i prduced at perid t (t 0) and cnsumed at perid t. The variables w i0t then represent the pening inventry f item i at the beginning f the hrizn which will be cnsumed at perid t. We will have: and s it = x it = t T T t =0 t =t+1 w itt, i =1,...,N,t=1,...,T. (10) w it t, i =1,...,N,t=1,...,T. (11) By replacing (10) and (11) in (9), we get fr each i =1,...,N and t =1,...,T: Mrever: s i,t 1 + x it + it +1 T t =t w it t + it r it it T t =0 t = it +1 it w it t = d it (12) 4

5 it +1 T t =t w it t = it +1 it t =t w it t + it T +1 t = it +1 w it t (13) and: it T t =0 t = it +1 w it t = t T t =0 t = it +1 w it t + it T +1 t = it +1 w it t (14) By replacing (13) and (14) in (12), we get fr each i =1,...,N and t =1,...,T: it it s i,t 1 + x it + w it t +1 t =t t =0 By definitin f variables w it t, we knw that: 1. w it t d it y it 2. w i0t d it P t P 3. T t =0 t = it +1 w it t =0 t T t = it +1 it w it t + r it = Cnsequently, frm (15), we get fr each i =1,...,N and t =1,...,T: Furthermre, Finally, s i,t 1 + x it + x it + it it +1 it it t =t d it y it + t =t d it = δ i t, it r it + s i,t 1 + it +1 it r it it d it δ i t, it y it δ i t, it it d it (15) In the sequel f the paper, we dente by SPMCLSSP the Single-Perid relaxatin f the prblem MCLSSP where (2) is replaced by (8). As previusly mentined, we refer t valid inequalities fr the set defined by (3) (8) as valid fr SPMCLSSP. P The expressin s i,t 1 + it +1 δi t, it y it can be cnsidered as being the ending P inventry f item i at perid t P 1 and dented es i,t 1. Thus, we have es i,t 1 = s i,t 1 + it +1 δi t, it y it. Similarly, we nte it r it by erit and δi t, it by dit. e The inequalities (8) are equivalent t: x it + er it + es it e dit, i =1,...,N,t=1,...,T. (16) Since we wrk n a single-perid in SPMCLSSP and given that each perid will be cnsidered separately, it may be mre cnvenient t remve the tempral index in the previus expressin t facilitate the reading f the remaining mathematical frmulatins. The inequalities (8) are written: x i + er i + es i e di, i =1,...,N. (17) 5

6 3 Valid (l, S) inequalities fr the prblem SPMCLSSP T intrduce the (l, S) inequalities fr SPMCLSSP, let us define the fllwing prblem dented MCLSP by: subject t: min N T i=1 t=1 α itx it + β ity it + γ its it (18) x it s it + s i,t 1 = d it, i =1,...,N,t=1,...,T (19) N x it + N i=1 i=1 f ity it c t, t =1,...,T. (20) x it,s it 0, i =1,...,N,t=1,...,T (21) y it {0, 1}, i =1,...,N,t=1,...,T (22) The prblem MCLSP is a simplified versin f MCLSSP with v it equalt1andndemand shrtage allwed, s that the variables r it are set t zer. We dente: SPMCLSP the single-perid relaxatin f MCLSP with (19) replaced by: x i + s i d i.we recall that the tempral index is remved. ULSP the uncapacitated versin f the single-item relaxatin f MCLSP. In this prblem, the cacapity cnstraints linking the items are remved. Thus, each item is cnsidered separately and the item index is useless. Barany et al. [4, 5] prved that a cmplete plyhedral descriptin f the cnvex hull f the ULSP is given by sme inequalities frm the basic LP relaxatin f the standard MIP frmulatin tgether with the (l, S) inequalities. The (l, S) inequalities are expressed as : t S x t + t S d tl y t d 1l Where l {0, 1,...,T}, S {0, 1,...,l}, S P = {0, 1,...,l}\S and t dtt = k=t d k. The authrs reprted gd cmputatinal results fr multi-item capacitated lt-sizing prblems using the (l, S) inequalities within a branch-and-cut scheme. Prpsitin 2. (l, S) inequalities fr the SPMCLSP, (Miller et al. [26]) If c f i + d i then the inequalities are facet-inducing fr the SPMCLSP. s i + d iy i d i,i=1,...,n (23) Pchet and Wlsey [33] intrduced the (k, l, S, I) inequalities fr the single-item lt-sizing prblem with cnstant capacity and they shwed that these are nntrivial facets f its cnvex hull. The (k, l, S, I) inequalities can be expressed in the fllwing general frm: s t 1 + x t + B ty t B 0 t U t V where fr any k and l such that 1 k l T,(U, V ) is any partitin f [k,...,l], B 0 R + and B t R + (t V ). B t are defined with respect t the demand and the capacity. Fr mre details, the reader can refer t [33]. 6

7 Withut lss f generality, we can mdify the inequalities (8) in rder t have a structure similar t the (k, l, S, I) inequalities, by replacing δ i t, it y it by x it fr values f t in any subset f [t +1,..., it]. Prpsitin 3. Given a partitin (U, V ) f the interval [t +1,..., it], theinequalities x it + it r it + ψ s i,t 1 + U are valid fr MCLSSP. δ i t, it y it + V x it Prf. The prf is similar t the prf f prpsitin 1.! δ i t, it, i =1,...,N,t=1,...,T (24) The inequalities (24) are called the (l, S) inequalities fr the prblem SPMCLSSP. 4 Separatin heuristic fr (l, S) inequalities In this sectin, we present a fast cmbinatrial separatin heuristic t create (l, S) inequalities fr the MCLSSP prblem. Accrding t the prpsitin 3, the (l, S) inequalities (24) are valid fr MCLSSP. We recall the expressin f these inequalities: x it + it r it + ψ s i,t 1 + U δ i t, it y it + V with (U, V ) a partitin f [t +1,..., it]. x it! δ i t, it, i =1,...,N,t=1,...,T The idea f the separatin heuristic is t create a set U [t,..., it] fr each item i and fr each perid t fr generating an (l, S) inequality fr the MCLSSP prblem. We add t t U if δ i t, it y it <x it r t the set V therwise. We illustrate this principle in the fllwing algrithm: Algrithm 1 Separatin heuristic fr (l, S) inequalities 1: t 1, i 1 2: while (i N) d 3: while (t T ) d 4: t t +1 5: while (t it) d 6: if δ i t, it y it <x it then 7: U U {t } 8: else 9: V V {t } 10: end if 11: t t +1 12: end while 13: if (The inequality (24) based n S, U and T is vilated) then 14: Add the inequality (24) at the current nde. 15: end if 16: t t +1 17: end while 18: i i +1 19: end while We can ntice that the separatin heuristic fr the (l, S) inequalities is in O(NT 2 ). 7

8 5 Cver and reverse cver inequalities fr the SPM- CLSSP In this sectin, we generalize sme results n the cver and reverse cver inequalities defined by Miller et al. [26]. Definitin 1. (Cver) AsubsetfitemsS f I is knwn as cver f the prblem SPMCLSSP if: λ S = f i + v i e di c 0 (25) Fr the cver S, λ s expresses the lack f capacity when all the items f S are prduced. Indeed, if λ s > 0 then the ttal requirements f all the items f S are strictly higher than the available capacity. Prpsitin 4. (Cver inequalities) The inequality is valid fr SPMCLSSP. v i (es i + er i) λ S + max n f i,v idi e λ S (1 y i) (26) Prf. The prf is similar t the ne presented in Miller et al. [26] by adding the demand shrtage variables er i as well as the variable resurce cnsumptin v i. The inequalities (17) can be written: Then: es i + er i e di x i, i =1,...,N. v ies i + v ier i v i e di v ix i If all the items f S are prduced, y i =1 i S, frm (3) we get: Then: By replacing P v ies i + v ier i v ix i c v i e di ψ c v i e di + f i c by λ S we get: f i f i! = v i e di + f i c v i (es i + er i) λ S (27) We define a set S 0 = {i S : y i =0} that represents the items in S that are nt prduced. If fi fi S 0 fi fi =1,wehaveexactlyneitemi S such that y i =0. Frm (27) we can write: We knw that: v i (es i + er i) λ S f i (28) 8

9 v i (es i + er i) v i (es i + er i ) v i e di (29) Thus, frm (28) and (29) we can cnclude that: v i (es i + er i) λ S +max n f i,v i di e λ S (30) Let us cnsider nw the case where fi fi S 0 fi fi > 1. The inequality (30) can easily be generalized by cnsidering the items in S 0 nebyne.hence,weget: v i (es i + er i) λ S + 0 max n f i,v i e di λ S The inequality (31) can be generalized fr the set S by intrducing the term (1 y i)ttake int accunt the prductin f the item. Hence, we have: v i (es i + er i) λ S + max n f i,v i e di λ S (1 y i) (31) The previus inequalities can be strengthened by a lifting prcedure described in what fllws. Prpsitin 5. (A secnd frm f cver inequalities) Given a cver S f SPMCLSSP, and an rder f items i S such that f [1] +v [1] e d[1] f [ S ] + v [ S ] e d[ S ].LetT = I\S, and(t,t ) be any partitin f T. We define µ 1 = f [1] + v [1] e d[1] λ S. If S 2 and f [2] + v [2] e d[2] λ S,theinequality v i (es i + er i) λ S + is valid fr SPMCLSSP. max n f i,v idi e λ S λ S (1 y i)+ f [2] + v [2] d[2] e i T (v ix i (µ 1 f i) y i) (32) Prf. Let (x,y,es, er ) any pint f the cnvex hull f SPMCLSSP. Let S 0 = {i S : yi =0}, S 1 = {i S : yi =1} and T = {i T : yi =1}. We cnsider three cases: If fi fi T fi fi = 0, then we have frm prpsitin (4) that the inequality (32) is valid. If fi T fi = 1, then we assume that T = {i }; t shw that the pint (x,y,es,er ) satisfies the inequality (32), it is sufficient t shw that: v i (es i + er i )+ We knw that: v i (es i + er i )+ max n f i,v idi e λ S yi λ S + max n f i,v idi e λ S + λ S (v f [2] + v [2] d[2] e i x i (µ1 f i )) (33) ( max n f i,v idi e ) λ S yi min v i (es i + er i)+ max n f i,v idi e λ S y i x i =x i,y i =1 Let us cnsider the fllwing prblem: ( ) min v i (es i + er i)+ max n f i,v idi e λ S y i x i =x i,y i =1 (34) 9

10 We will prve that the ptimal slutin f this prblem has a value higher r equal t the right memberfi f the inequality (33). Let r S fi = fi n fi i S : f i + v idi e >λ S fifi P and A b b a = a f [i] + v [i] d[i] e. If we define: ϕ S (u) = 8 >< >: u, if 0 u A S r S +1 A S r S +1 +(rs j) λs, if A S j+1 u A S j λ S,j =1,...,r S (35) A S r S +1 u +(rs j) λs + A S j λ S, if A S j λ S u A S j,j =2,...,r S then the ptimal slutin f the minimizatin prblem (34) is given by: ( ) min v i (es i + er i)+ max n f i,v i e di λ S y i = v idi e + ϕ S (c (f i x i =x + v i x i,y i =1 i )) (36) The prf is an bvius generalizatin f the result presented in [26]. We refer the reader t the paper f Miller et al. [26] fr mre details. A simple representatin f ϕ S is given in Fig. 1. ϕ S (u) A S r S +(r S k +1)λ S A S r S +(r S k)λ S A S r S + λ S A S r S A S A r S r S +1 S A S r S λ S A S r S 1 λ S A S k+1 A S u k A S k λ S A S k 1 λ S Nw, we need t prve that: v i e di ϕ S (c (f i + v i x i )) λs + Figure 1: Functin ϕ S. max n f i,v idi e λ S + λ S (v f [2] + v [2] d[2] e i x i (µ1 f i )) (37) T d that, we use the fllwing prperty, which is als a generalizatin f a result presented in Miller et al. [26]): ( max ϕ S (c (f i + v ix i)) + 0 x i c f i v i Mrever, we have: v idi e = v [1] d[1] e + v idi e = v [1] d[1] e + ) λ S (v f [2] + v [2] d[2] e ix i (µ 1 f i)) = \[1] \[1] Using the expressin (38), we have: v i e di max n f i,v idi e λ S + \[1] \[1] min nf i + v idi,λ e S (38) min nf i + v idi,λ e S 10

11 v idi e = v [1] d[1] e + max 0 x i c f i v i ( \[1] v idi e v [1] d[1] e + \[1] max n f i,v idi e λ S + ϕ S (c (f i + v i x i )) + (v f [2] + v [2] d[2] e i x i (µ 1 f i )) max n f i,v i e di λ S + ϕ S (c (f i + v i x i )) + (v f [2] + v [2] d[2] e i x i (µ1 f i )) v idi e v [1] d[1] e λ S + λ S + \[1] ϕ S (c (f i + v i x i )) + λ S max λ S n f i,v i e di λ S + λ S (v f [2] + v [2] d[2] e i x i (µ1 f i )) (39) Since v [1] d[1] e λ S f [1] (we knw that : f [1] + v [1] d[1] e λ S), v [1] d[1] e λ S can be replaced n f [1],v [1] d[1] e λ S. By rewriting the expressin (39), we get (37). Then, we derive the by max inequality (33). If fi fi T fi fi > 1, then the expressin (33) can be easily generalized by cnsidering the items which belng t T ne by ne. We get then the inequality (32). ) In what fllws, we describe anther class f valid inequalities based n the reverse cver set. Definitin 2. (Reverse Cver) AsubsetSfI is knwn as reverse cver f SPMCLSSP if: µ S = c f i + v i e di 0 (40) Fr a reverse cver S, µ S expresses the available capacity left when the ttal requirement fr each item f S is prduced. Prpsitin 6. Let S be a reverse cver f SPMCLSSP, T = I\S and (T,T ) be any partitin f T.Theinequality v i (es i + er i) ψ is valid fr SPMCLSSP. f i + v i e di! i T y i f i (1 y i) i T ((c f i) y i v ix i) (41) Prf. The prf presented here is similar t the ne described in Miller et al. [26]. In the fllwing, we take int accunt the demand shrtage variables er i as well as the variable resurce cnsumptin v i. Let (x,y,es, er ) be any pint f the cnvex hull f SPMCLSSP. We have t cnsider three cases: If y i =0fralli T, then the inequality is valid, because P vi (es i + er i ) P fi (1 ey i ). Let T = Φ j T : y j =1 Ψ If fi T fi = 1, we assume that T = {i } Frm (3) we have: c f i (v ix i + f iyi )+v i x i Frm (17) we als have: 11

12 Cnsequently, we get: Which gives: The inequality v i (es i + er i ) c f i x i e di es i er i v i (es i + er i ) v i e di + ψ v edi i es i er i + f iyi + v i x i f iy i ((c f i ) v i x i ) f i + v idi e! f i (1 yi ) ((c f i ) v i x i ) (42) is thus valid fr SPMCLSSP. If fi fi T fi fi > 1, the inequality (42) can be easily generalized by cnsidering the items f T ne by ne. The inequality (41) fllws. 6 Lifting cver and reverse cver inequalities In this sectin, we will strengthen the valid inequalities by using superadditive functins fr an iterative imprvement. We refer the reader t [17] and [39] fr a detailed descriptin f lifting prcedures using superadditive functins. Our wrk is based n Marchand and Wlsey [24] wrk n the cntinuus knapsack prblem, as well as the adaptatins carried ut by Miller et al. [26] fr the prblem ( Pr) in rder t lift cver and reverse cver inequalities fr the SPMCLSSP prblem. 6.1 The 0-1 cntinuus knapsack prblem Let us define the fllwing prblem: ( ) Y = (y, s) {0, 1} n R+ 1 : a jy j b + s. j J (43) With: J = {1,...,n},a j Z +,j J and b Z +. Let ({j },C,D)beacverpairfrY such that: C D P = {j },C D = J λ C = j C aj b>0 a j >λ C We can ntice that µ D = a j λ C = P j D aj P j J aj b > 0. C is thus a cver set and D is a reverse cver set. We will nw recall main results n the cver and reverse cver inequalities defined fr this prblem. Prpsitin 7. (Cntinuus cver inequalities, Marchand and Wlsey [24]) Let ({j },C,D) be a cver pair fr Y. We cnsider an rder f the elements f C such that a [1] a P [rc ] where r C is the number f elements f C with a j >λ C. Let us dente A 0 =0 j and A j = p=1 a [p],j =1,...,r C.Weset: φ C (u) = 8 >< >: (j 1)λ C, if A j 1 u A j λ C,j =1,...,r C (j 1)λ C +[u (A j λ C)], if A j 1 λ C u A j,j =1,...,r C (r C 1)λ C +[u (A rc λ C)], if A rc λ C u (44) 12

13 The inequality j C min (λ C,a j) y j + D\j φ C (a j) y j is valid fr Y and defines a facet f cnv(y ). A simple representatin f φ C is given in Fig. 2. j C\j min (λ C,a j)+s (45) φ C (u) (k 1)λ C λ C A 0 A 1 λ C A 1 A 2 λ C A k 1 A k λ C u Figure 2: Functin φ C. Prpsitin 8. (Cntinuus reverse cver inequalities, Marchand and Wlsey [24]) Let ({j },C,D) be a cver pair fr Y. We cnsider an rder f the elements f D such that a [1] a [rd ] where r D is the number f elements f D with a j >µ D,whereµ D = a j λ C. Let A 0 =0and A j = P j p=1 a [p],j =1,...,r D.Weset: ψ D (u) = The inequality j D 8 >< >: u jµ D, if A j u A j+1 µ D,j =0,...,r D 1 A j jµ D, if A j µ D u A j,j =1,...,r D 1 A rd r Dµ D, if A rd µ D u (a j µ D) y j + j C\j ψ D (a j) y j is valid fr Y and defines a facet f cnv(y ). A simple representatin f ψ C isgiveninfig.3. (46) j C\j ψ D (a j)+s (47) 6.2 Lifting cver inequalities fr the SPMCLSSP prblem In what fllws, we use the results f Marchand and Wlsey [24] t btain valid inequalities strnger than (26). Let us recall that: S is a cver fr the SPMCLSSP prblem. T = I\S. T,T is a partitin f T. U T. 13

14 ψ D (u) A k+1 (k +1)µ D A k kµ D A 2 2µ D A 1 µ D A 0 A 1 µ D A 1 A 2 µ D A 2 A k A k+1 µ D A k+1 u Figure 3: Functin ψ D. Frm the cnstraints (3) and (17), we can write: U f iy i + v i e di v ies i v ier i P By adding e U vi d iy i t bth sides f this inequality, we see that: f iy i + v idi e v ies i v ier i c + Thus: U U v i e diy i + If we dente u = P U U f i + v i e di y i c + U c U v ies i + v ier i + v i e diy i v i e di v ies i + v ier i + v i e diy i v i e di, it results that: U v i e diy i (48) f i + v i e di y i c + u (49) The inequality (49) can thus be cnsidered as a cnstraint f a 0-1 cntinuus knapsack prblem. S, we get the fllwing prperties. Prpsitin 9. Given S a cver f SPMCLSSP, and U asubsetfi\s, the inequality (where φ S is defined by (44)) U v i (es i + er i) λ S + is valid fr SPMCLSSP. max n f i,v idi e λ S (1 y i)+ φ S f i + v idi e y i + v idi e (1 y i) (50) i U i U Prf. Let ({j },U {j },S) be a cver pair fr SPMCLSSP such that: f j +v j e dj >λ S. Accrding t the prpsitin 7, the fllwing inequality is valid fr SPMCLSSP min λ S,f i + v idi e y i + φ S f i + v idi e y i i U \{j} + U min λ S,f i + v idi e v ies i + v ier i + v i e diy i v i e di 14

15 Since f j + v jdj e >λ S,wehavemin(f j + v jdj,λ e S)=λ S. By adding min(f j + v jdj,λ e S) λ S t the right side f the previus inequality, we get: min λ S,f i + v idi e y i + φ S f i + v i e di y i min λ S,f i + v idi e λ S i U v ies i + v ier i + v idiy e i v idi e + U which is valid fr SPMCLSSP. We btain the inequatin (50) by simplificatin. Prpsitin 10. Let S be a cver f SPMCLSSP. We cnsider an rder [1],...,[ S ] such that f [1] + v [1] d[1] e f [ S ] + v [ S ] d[ S ] e.letussett = I\S and (T,T ) any partitin f T.We define µ 1 = f [1] + v [1] d[1] e λ S. If S 2 and f [2] + v [2] d[2] e λ S,theinequality v i (es i + er i) λ S + max n f i,v i e di λ S (1 y i)+ φ S f i + v idi e y i U i U + v i e λ S di (1 y i)+ i U f [2] + v [2] d[2] e is valid fr SPMCLSSP. Prf. The prf is similar t the prf f prpsitin 5. i T (v ix i (µ 1 f i) y i) (51) 6.3 Lifting reverse cver inequalities fr the SPMCLSSP prblem In the same way, we can ntice that the inequality (48) is a cnstraint f a 0-1 cntinuus knapsack prblem. The fllwing prpsitins hld. Prpsitin 11. Let S be a reverse cver fr SPMCLSSP and U asubsetfi\s. Theinequality v i (es i + er i) U is valid fr SPMCLSSP. Prf. Accrding t prpsitin 8, the inequality ψ U f i + v i e di y i+ i U v idi e ψ U f i + v idi e (1 y i)+ v i e di + (f i µ S) y i (52) i U i U f i + v idi e µ S y i ψ U f i + v idi e + v i esi + er i + diy e i di e U is valid fr SPMCLSSP. We btain the inequatin (52) by simplificatin. Prpsitin 12. Let S be a reverse cver f the SPMCLSSP prblem and U I\S such that f i + v i e di µ S i U. We cnsider an rder [1],...,[ U ] such that f [1] + v [1] e d[1] f [ U ] + v [ U ] d[ U ] e. Let T = I\{S U} and (T,T ) any partitin f T. The inequality (ψ U is defined by (46)): U v i (es i + er i) is valid fr SPMCLSSP. + min v idi e ψ U f i + v idi e (1 y i)+ (f i µ S) y i + v idi e i U i U 8 < ψ U f i + v idi e : f i + v i e di 9 = ; i T (v ix i (µ S f i) y i) (53) Prf. The prf f this prpsitin is similar t ne described fr prpsitin 4. 15

16 7 Separatin heuristic fr cver inequalities In this sectin, we present a fast cmbinatrial separatin heuristic t create cver inequalities fr the SPMCLSSP prblem, which are als valid fr the MCLSSP. Indeed, fr the latter prblem, we generate the cver inequalities fr each perid f the planning hrizn which crrespnds t the cver inequalities f the SPMCLSSP. In rder t build a cver inequality fr the SPMCLSSP prblem, the first step is t define acversets, then we cmpute λ S and µ 1 (see (25) and prpsitin 5). The secnd step is t examine all the elements i I\S t create the sets U and T. We use a greedy algrithm t create the set S. We srt the elements i Iaccrding t the descending rder f the value: max n f i,v idi e λ S (1 yi ) v i (es i + er i ) (54) The frmula (54) is btained frm the inequality (51) by cnsidering the nly terms in relatin with S. The value f (54) represents the cntributin f the vilatin f the inequality (51) by the item i S and depends n λ S. Hwever, λ S is nt knwn in advance. Therefre, we estimate the value f λ S. T d that, we srt the items i Iaccrding t the descending rder f their resurce cnsumptin by using the frmula (55). Frmula (54) wuld give a better set S but cnnt be used since λ S is nt knwn, s in practice frmula (55) is used. f i + v i e di y i (55) We can ntice that the frmula (55) represents the resurce cnsumptin f the item i if the ttal requirement e di is prduced. In rder t create a cver set S, we greedily add the srted elements accrding t the frmula (55) until we get a cver set. Fr the design f the set U (respectively T ), we examine all the elements i I\S and check if the crrespnding value f the expressin btained by summing up the terms f the inequality (51) in relatin with U (respectively t T ) is psitive. In this case, we add the elements i t U (respectively t T ). We derive a valid inequality using (51) with the sets S, U and T btained. If the value f the inequality is psitive, we get a cut. The basic principle previusly described is captured in the fllwing algrithm. Algrithm 2 Separatin heuristic fr cver inequalities 1: Order the elements f I in descending rder accrding t the frmula (55). 2: S,U,T,i 0 np 3: i arg min i i=1,...,n k=1 f [k] + v [k] d[k] e >c 4: S {[1],...,[i ]} n 5: i 1 arg max 6: i 2 arg max \{i1 } 7: λ S P i f i + v idi e nf i + v idi e k=1 f [k] + v [k] e d[k] c 8: if ( S 2) and (f i2 + v i2 e di2 λ S) then 9: µ 1 nf i1 + v i1 e di1 λ S 10: i i +1 11: while (i N) d 12: if (φ S f [i] + v [i] d[i] e y [i] + v [i] d[i] e 1 y [i] v[i] es [i] + er [i] > 0) then 13: U U {[i]} 14: else if (v [i] x [i] > µ 1 f [i] y [i] ) then 15: T T {[i]} 16

17 16: end if 17: i i +1 18: end while 19: end if 20: if (The inequality (51) based n S, U and T is vilated) then 21: Add the inequality (51) at the current nde. 22: end if We recall that the evaluatin f the superadditive functins φ S is in O(N) fr each item. The separatin heuristic fr generating the cver inequalities fr the SPMCLSSP prblem is bviusly in O(N 2 ). Mrever, since each perid is examined separately t create valid inequalities fr the MCLSSP prblem, the separatin heuristic is then in O(N 2 T ) fr the latter prblem. 8 Separatin heuristic fr reverse cver inequalities The idea f the separatin heuristic fr reverse cver inequalities is similar t the previus ne. The first step is t create a reverse cver set S in rder t define µ S (see (40)). The secnd step is t examine all the elements i I\S t create the sets U and T. We use a greedy algrithm t create S by srting the elements i I accrding t the descending rder f the value: v i e di ψ U f i + v i e di (1 y i ) v i (es i + er i ) (56) In the same way, the frmula (56) is btained frm the inequality (53) by cnsidering the nly terms in relatin with S. The value f (56) represents the cntributin f the vilatin f the inequality (53) by the item i S and depends n µ S. Hwever, µ S is nt knwn in advance. Therefre, we estimate the value f µ S. T d that, we srt the items i Iaccrding t the descending rder f their resurce cnsumptin by using the frmula (55). Frmula (56) wuld give a better set S but cnnt be used since λ S is nt knwn, s in practice frmula (55) is used. We illustrate this principle in the fllwing algrithm: Algrithm 3 Separatin heuristic fr reverse cver inequalities 1: Order the elements f I in descending rder accrding t the frmula (55) 2: S, i 1 3: while (i N) d 4: S S {i P } i 5: µ S = c j=1 f [j] + v [j] d[j] e 6: if (µ S < 0) then 7: T, U, i i +1 8: while (i N) d 9: if ( f [i] µ S y [i] + v [i] e d [i] v [i] es [i] + er [i] > 0) then 10: U U {[i]} 11: else if (v [i] x [i] µ S f [i] y [i] > 0) then 12: T T {[i]} 13: end if 14: i i +1 15: end while 16: if (The inequality (53) based n S, U and T is vilated) then 17: Add the inequality (53) at the current nde. 18: i N +1 19: else 20: i i +1 21: end if 22: else 23: i N +1 17

18 24: end if 25: end while We recall that the evaluatin f the superadditive functins ψ S is in O(N) fr each item. The separatin heuristic fr cnstructing reverse cver inequalities fr the SPMCLSSP prblem is bviusly in O(N 2 ). Since each perid is examined separately t create valid inequalities fr the MCLSSP prblem, the separatin heuristic is then in O(N 2 T ) fr the MCLSSP. 9 Cmputatinal issues and results In this sectin, we discuss cmputatinal issues that arise in using the classes f inequalities previusly identified. We reprt cmputatinal results frm cut-and-branch and branch-and-cut framewrks. The cut-and-branch methd cnsists in adding cuts nly at the first nde (r rt) f the branch-and-bund tree in rder t imprve the lwer bund. The branch-and-cut methd cnsists in adding cuts nt nly at the first nde but at ther ndes f the branch-and-bund tree. Usually, cuts are nt added at all the ndes f the branch-and-bund tree in rder nt t slw dwn the ttal CPU time while slving the prblem. Our algrithm is implemented in the C++ prgramming language and it is intergrated in an APS sftware. It uses the callable CPLE 9.0 library [20] that prvides callback functins that allw the user t implement his wn branch-and-cut algrithm. We have perfrmed cmputatinal tests n a series f extended instances frm the lt-sizing library LOTSIZELIB [22], initially described in Trigeir et al. [36] and als used by Miller [25]. Trigeir et al. [36] instances are dented tr N T,whereN is the number f items and T is the number f perids. These are characterized by a variable resurce cnsumptin equal t ne, and enugh capacity t satisfy all the requirement ver the planning hrizn. They are als characterized, by an imprtant setup cst, a small fixed resurce requirement (setup time) and n it which dentes the last perid at which an item i prduced at perid t can be cnsumed. The characteristics f Trigeir et al. [36] prblems are presented in table 1. Instance N T tr tr tr tr tr tr Table 1: Instances f Trigeir et al. [36]. Since these instances have enugh capacity t satisfy all the requirements ver the planning hrizn, we make sme mdificatins t induce shrtages. We have derived 24 new benchmarks 2 frm the tr N T instances by augmenting the fixed resurce requirements (setup times), the variable resurce requirements and by adding it. We have als generated shrtage csts. Mre details are given belw. These new benchmarks fall int 4 classes f 6 instances each: The first class was btained by increasing the variable resurce requirements and adding it. Variable resurce requirements are multipled by a cefficient (1 + ρ) such that 0 ρ c t, c t represents the available resurce capacity at perid t. it are generated such that we cannt anticipate prductin mre than 1 T perids, T dentes the number f 3 perids. 2 Test prblems can be btained frm the crrespnding authr. 18

19 The secnd class is btained by carring ut the same mdificatins n the variable resurce requirements than the first class. it are generated such that we cannt anticipate prductin mre than 2 T perids. 3 The third class is based n the first ne. In fact, we carried ut sme mdificatins n fixed resurce requirements which are increased by multiplying them by a cefficient (1 + τ) such that τ 0.1 c t. The last class is btained by carring ut the same mdificatins n the variable and fixed resurce requirements than the third class. it are generated such that we cannt anticipate prductin mre than 2 T perids. 3 Shrtage csts are cnsidered as penalty csts and their values must be higher than ther cst cmpnents. Therefre, ϕ it arefixedsuchthatϕ it >> max i,t {α i t ; β i t ; γ i t }. Mrever, shrtage csts have the feature that they decrease ver the hrizn. In fact, demands in the first perids f the hrizn crrespnd t real rders and nt frecasts by ppsitin t the demands in the last perids that are usually nly predictins. They are generated in the same way fr all the described instances. We carried ut a cmparisn between the fllwing methds: An algrithm based n the standard branch-and-cut f CPLE slver that we dente by BC. An algrithm based n the standard branch-and-cut f CPLE slver including all the cuts presented in this paper dented by BC+. In bth algrithms BC and BC+, we used the aggregated mdel defined in sectin 1 by the set f cnstraints (1)-(7). Tw kinds f specialized cuts in mixed linear prblems are used in bth methds, the flw cver cuts and the Mixed Integer Runding (MIR) cuts frm CPLE 9.0 slver. The flw cver cuts are accurate regarding the MCLSSP prblem because f the flw structure induced by the flw cnservatin cnstraints (2). Fr a cmplete descriptin f these flw cver cuts, the reader can refer t Gmry [16], Nemhauser and Wlsey [27] and Wlsey [40]. The MIR cuts were primarily applied t bth capacity and maximum prductin cnstraints. Fr mre details n the MIR cuts, we can refer t Padberg et al. [30] and Van Ry and Wlsey [34]. Fr all the algrithms, LB and UB represent respectively the lwer bund and the upper bund values at the terminatin f the algrithm. NB Ndes is the number f the ndes explred in the branch-and-bund tree, U Cuts is the number f the cuts added during the branch-and-cut algrithm (cver, reverse cver and (l, S) inequalities). F Cuts and MIR Cuts represents respectively the number f flw cver and MIR cuts added by the slver during the branch-and-cut algrithm. All the algrithm cmparisns are based n the fllwing criteria. The first ne called GAP is equal t UB LB / UB, and the secnd ne is a CPU time dented Time. The cmputatins are perfrmed n a Pentium IV 2.66 Ghz PC. At the rt nde f the BC+ methd, we use algrithms 1, 2 and 3 (see sectins 4, 7 and 8) until we d nt find any mre vilated inequalities. The same prcedure is fllwed in the branch-and-bund tree. The branching strategy in bth algrithms is depth-first search t find a feasible slutin. Upper bunds are either btained when LP slutins are integral r by the LP based heuristics f the slver. Generally, ur cmputatinal results shw that adding inequalities at the rt nde imprves cnsiderably the lwer bunds. The average imprvement f the lwer bund at the rt nde f BC+ is 80% fr the first class, 53% fr the secnd class, 73% fr the third class and 48% fr the last class. This rate is the percentage btained between the best lwer bund bserved at the first nde f BC and the best ne fund at the end f the BC+ methd. Table 2 summarizes the cmputatinal behaviur based n a time-limit criterin. We allw a maximum f 600 secnds CPU time fr all the algrithms. Frm table 2, we can easily ntice that using the valid inequalities described in this paper imprves the perfrmance f the branch-and-cut algrithm. Clearly BC+ slves the test prblems mre effectively than BC. The valid inequalities that we have prpsed are interesting since all the lwer bunds given by BC+ are better than thse given by BC. 19

20 N T Methd UB LB NB Ndes U Cuts MIR Cuts F Cuts GAP Class BC ,46% 6 15 BC ,77% 6 30 BC ,09% 6 30 BC ,75% BC ,28% BC ,53% BC ,91% BC ,07% BC ,39% BC ,62% BC ,13% BC ,60% Class BC ,35% 6 15 BC ,17% 6 30 BC ,06% 6 30 BC ,55% BC ,43% BC ,80% BC ,16% BC ,89% BC ,43% BC ,86% BC ,21% BC ,73% Class BC ,52% 6 15 BC ,76% 6 30 BC ,73% 6 30 BC ,77% BC ,77% BC ,91% BC ,19% BC ,14% BC ,73% BC ,63% BC ,15% BC ,08% Class BC ,37% 6 15 BC ,03% 6 30 BC ,62% 6 30 BC ,58% BC ,43% BC ,06% BC ,24% BC ,36% BC ,32% BC ,71% BC ,92% BC ,44% Table 2: Cmputatinal results: time-limit criterin. 20

21 We als can ntice that the upper bunds btained by BC+ are better than thse btained by BC. In fact, usually a gd lwer bund implies a better upper bund. One reasn is that a better lwer bund can be used t prune dminated ndes f the research tree and allw the branch-andbund algrithm t visit mre interesting branches t find better slutins. An additinal reasn is that when the LP relaxatin is tighter, it is easier t find integer slutins, either because the LP slutin is mre ften integral r because LP based heuristics f the slver are mre successful. Mrever, the number f ndes explred by the BC methd is much mre higher than the ne explred by BC+. The rati between these tw numbers varies frm 150% t 500% fr the small instances and 300% t 36000% fr the bigger instances. In fact, generating cuts takes time and having many cuts slws dwn the LP reslutin at each nde. Thus, when we enable the cuts that we have develped, the slver generates a number f cuts lwer than the number btained when they are disabled. Since we generate cuts befre the slver, the previus bservatin can be explained by the fact that these cuts dminate part f standard cuts generated by the slver. Anther remark that we can make by visualizing table 2 is that the third and frth classes are mre difficult than the first and the secnd nes. Indeed, the third and frth class prblems are characterized by a higher fixed resurce cnsumptin values than the first and the secnd nes. We can als ntice that prblems with a small it have a higher GAP than the nes with a big ne. In fact, instances f class 1 and class 3 have reach a smaller GAP than respectively instances frm class 2 and class 4 when using bth algrithms BC and BC+. Table 3 shws the percentages f generated cuts by BC+ fr all instances. %ls cuts, %C cuts and %RC cuts are respectively the percentage f (l, S), cver and reverse cver cuts generated by BC+. We use a time-limit criterin f 600 secnds fr BC+. N T %ls cuts %C cuts %RC cuts N T %ls cuts %C cuts %RC cuts Class 1 Class ,08% 95,84% 0,08% ,48% 77,20% 1,32% ,62% 85,32% 0,05% ,93% 64,89% 1,17% ,16% 86,80% 0,04% ,19% 34,24% 14,57% ,96% 55,93% 0,11% ,64% 21,44% 6,92% ,78% 81,16% 0,06% ,00% 27,43% 1,56% ,16% 28,80% 0,04% ,49% 10,18% 2,34% Class 2 Class ,58% 97,35% 0,06% ,30% 90,18% 0,52% ,60% 93,30% 0,11% ,69% 83,11% 1,20% ,97% 77,03% 0,00% ,11% 45,11% 9,78% ,31% 72,68% 0,01% ,13% 38,36% 2,51% ,41% 68,56% 0,04% ,78% 27,98% 2,23% ,46% 39,49% 0,05% ,05% 17,05% 2,90% Table 3: Cmputatinal results: cuts percentages. Frm table 3, we can ntice that the percentage f reverse cver cuts generated by BC+ is very small. We can als ntice that BC+ generates mre (l, S) cuts than cver cuts fr class 1 and class 2 benchmarks except fr instances with 24 items and 30 perids. Fr class 3 and class 4 instances, BC+ generates mre cver cuts than (l, S) cuts except fr instances with 6 items. We have als tested BC+ using each family f valid inequalities separately ((l, S), cver and reverse cver inequalities). Table 4 summarizes the cmputatinal results n class 1 and class 3 f instances. GAP ls, GAP C and GAP RC represents respectively the GAP when nly the family f (l, S) inequalities is used, the GAP when nly the family f cver inequalities is used and the GAP when nly the family f reverse cver inequalities is used. These tests shw that the family f (l, S) inequalities is the mst effective. The family f cver inequalities is less effective than the family f (l, S) inequalities. Using the family f reverse cver 21

22 N T GAP ls GAP C GAP CR Class ,56% 2,45% 2,40% ,93% 11,03% 10,02% ,55% 2,50% 2,54% ,87% 3,37% 3,24% ,68% 1,40% 1,39% ,33% 9,44% 9,10% Class ,48% 3,16% 4,45% ,17% 14,71% 19,40% ,43% 4,28% 5,32% ,65% 12,40% 14,63% ,19% 2,20% 2,60% ,88% 7,27% 7,71% Table 4: Cmputatinal results by cuts family. inequalities alne is nt really effective. T give a relevant cmparisn cncerning the number f added cuts and explred ndes fr bth algrithms BC and BC+, we slve sme prblems t a given GAP. Since we cannt slve these prblems t ptimality in a reasnable CPU time, we use a time-limit criterin f 1800 secnds fr BC. We use the GAP btained at the end f BC as a stpping criterin fr BC+. We als use a time-limit f 1800 secnds fr BC+. Table 5 summarizes the cmputatinal behaviur f the instances f class 1 based n a minimum GAP criterin. N T Methd NB Ndes U Cuts MIR Cuts F Cuts GAP Time Class BC ,03% BC ,94% BC ,51% BC ,65% BC ,18% BC ,05% BC ,60% BC ,66% BC ,14% BC ,88% BC ,00% BC ,00% 175 Table 5: Cmputatinal results: minimum GAP criterin. Frm table 5, we can easily ntice that t reach the same GAP, BC+ des nt need a significant number f ndes and time cmparing t BC. We remark that withut branching, BC+ gets alwergap than BC s ne fr the instances with 12 items and 30 perids and the ne with 30 items and 24 perids. The cuts imprve cnsiderably the GAP at the rt nde. Many authrs reprted experimental results shwing that using the facility lcatin-based frmulatin prvide a better LP relaxatin based lwer bund than the ne btained by the aggregated frmulatin. (see [8], [31]). We carried ut sme cmputatinal experiments using the 22