Modeling Demand Propagation in Guaranteed Service Models

Transcription

1 Modelng Demand Propagaton n Guaranteed Servce Models Madelene Löhnert Jörg Rambau DATEV eg, Nürnberg, (emal: madelene.loehnert@gmx.de) Unversty of Bayreuth, Bayreuth, Germany (e-mal: joerg.rambau@un-bayreuth.de) Abstract: In ths paper, the class of guaranteed servce models for mult-echelon nventory management s enhanced wth explct demand propagaton. More specfcally, the known mxed nteger lnear programmng formulaton for the guaranteed servce model s refned by new varables and restrctons so that t descrbes the nternal demand propagaton exactly for lnear demand bound functons. Wth ths feature, aspects lke outsourcng as well as decsondependent stochastc demands at nternal stock ponts can be expressed exactly. The relevance of the new model s shown n an llustratve example, where the new model s able to fnd a soluton wth almost 40% lower actual cost compared to the exstng approxmatve model wthout explct demand propagaton. Keywords: mult-echelon nventory management, mxed-nteger lnear programmng, demand propagaton, outsourcng, lost demand 1. INTRODUCTION Inventory management n ts basc form poses the followng problem: Gven a stock pont wth uncertan demand, when and how much supply has to be replenshed n order to serve all customers at the lowest possble expected cost (nventory holdng cost plus backorderng cost)? If several stock ponts form an nventory network, the problem s called mult-echelon nventory management. Specal cases occur when each stock pont has a unque predecessor (dvergent systems), a unque successor (convergent systems), or both (seral systems). Moreover, the backorderng case (unsatsfed demand s served later) s dstngushed from the lost-sales case (unsatsfed demand s lost). For more detaled background on mult-echelon nventory management, see de Kok and Fransoo (2003). It s known that usng ndvdually optmal polces at all stock ponts rarely consttutes an optmal mult-echelon polcy. However, the computaton of an optmal multechelon polcy s dffcult n all but the most basc (famous) cases lke the seral case wth backorderng and lnearplus-fxed orderng cost (Clark and Scarf, 1960). Over all, n the backorderng case optmal polces could be derved more often than n the lost-sales case. The reason s as follows: Snce the order volumes of downstream stock ponts depend on ther orderng decsons to be optmzed, demands at all upstream stock ponts are endogenous. Ths makes optmzaton more complcated. In the backorderng case and dvergent systems t can be assumed that all demands are propagated upstream n an unchanged manner. In the lost-demands case ths s not true anymore, and the upstream demands can have ntractable dstrbutons, dependng on the downstream Project of the Bayreuth research center for modelng and smulaton (MODUS). orderng polces. Smlar problems arse n the case of outsourcng opportuntes or any other decson opton that satsfes part of the demand outsde the mult-echelon system to be optmzed. Varous approaches to approxmate optmal polces have been suggested. They can be dvded nto the Stochastc Servce Model paradgm (SSM) and the Guaranteed Servce Model paradgm (GSM), see Graves and Wllems (2003) for a comparson and (Magnant et al., 2006) for an approxmate mxed-nteger lnear programmng formulaton (MILP) for the GSM. Ths paper deals wth the GSM branch of research. In the GSM, bounded demand s assumed n the sense that a demand bound functon s known,.e., for each node and for each duraton there s a maxmal value for the total demand presented to the node durng any tme nterval wth that duraton. Based on ths nformaton, each stock pont decdes for each of ts successors the maxmal tme between recevng an order and sendng off the supply to ts successor. Gven ths guaranteed servce tme, each successor stock pont can predct the latest pont n tme when ts order arrves at ts nventory, gven that the (transportaton) delay between the stock ponts s determnstc (or bounded). From the demand bound functon, mnmal nventory levels can be computed that ensure that the guaranteed servce tmes can be obeyed. Ths approach uses two debatable assumptons: bounded demand and constant delays. The most common nterpretaton s: f anythng goes wrong n practce (demand bounds are exceeded, unforeseen transportaton delays occur), then some unmodeled operatonal flexblty s appled so that the servce tmes can be guaranteed anyway. And here s a serous gap n the justfcaton of the GSM paradgm: f that operatonal flexblty s, e.g., out-

2 sourcng demand to an external emergency suppler, then the demand propagaton nsde the nventory network s changed, and the nternal demand bound functons derved from smple upstream propagaton would overestmate the upstream demand. In ths paper, the GSM s enhanced wth exact demand propagaton. Ths can not only be appled to the GSM wth explct outsourcng or lost demands but also to the recent Stochastc GSM (SGSM), that was developed n order to explctly account for outsourcng n scenaros wth dfferent demand bound functons and expedtng n scenaros wth dfferent delays. The correspondng twostage stochastc MILP (2SMILP) was ntroduced by Rambau and Schade (2010) and further studed by Schade (2012) and Rambau and Schade (2014). The man contrbuton of ths paper s a new model, a 2SMILP, denoted by SGSM-DP, n whch the demand propagaton n the presence of outsourcng s captured exactly by explct computaton of demand bounds from the outsourcng decsons nsde the model. In a small threeechelon example wth fve stock-ponts (three of whch face exogenous demands) and three demand scenaros the new model mproves the total expected cost by 40%. The downsde s that the CPU tme ncreases from around 10ms (SGSM) to slghtly over 1s (SGSM-DP). Remark: The problem of explct demand propagaton n the SGSM was prelmnary nvestgated n the master s thess by Löhnert (2016); the results n ths paper are based on that thess. 2. PROBLEM STATEMENT Let G = ( N, A ) be a dvergent nventory network,.e., N s a set of nventory nodes and A N N s a set of supply relatons so that for each node j N there s at most one N wth (, j) A. Moreover, let D be the leaves of the network, whch correspond to the nodes wth exogenous demands, called demand nodes. There are gven determnstc demand bound functons φ (x ) at all nodes that specfy the maxmal cumulatve demands arrvng n any tme nterval of length x. At the demand nodes these functons are exogenously gven, and at upstream nodes they are accumulated from downstream demands. Snce the network s dvergent, ths yelds unque demand bound functons throughout. Moreover, there are delays L for the transports of supples to nodes. There are margnal holdng costs of h > 0 for nventory at node. At each demand node a duraton specfes how long ts end customers are prepared to wat (wthout ncurrng a backorderng cost!) for an order to be fulflled. The dea of a guaranteed servce model s to mpose that the whole system delvers to all end customers n tme. Then, no backorderng cost s ncurred, nether n demand nodes nor n nternal nodes. Among all systems that can guarantee the servce tmes for the end customers, a system s sought wth mnmal nventory holdng costs for safety stock. The ndependent decsons are guaranteed (outgong) servce tmes for all nodes. The meanng s that node guarantees ts successors that the tme between recevng and delverng an order s at most. The guaranteed ngong servce tmes are dependent auxlary varables used to smplfy the formulaton of some constrants. They mean that node need not wat any longer than +L for an order to arrve. The dependent varable x denotes maxmal duratons for whch delveres are taken from node s nventory. The materal quantty y s the nventory needed to be able to delver from stock all orders that arrve durng x tme unts. The orgnal GSM MILP (Magnant et al., 2006) reads as follows: mn N h y (1) 0 D (2) j 0 (j, ) A (3) + L 0 N (4) y φ (x ) 0 N (5),, x, y, q 0 N (6),, x, y Z N (7) The model s brefly explaned n the followng. The objectve (1) measures the total holdng cost per tme unt for safety stock over all nodes. In partcular, backorderng costs are not consdered. Constrant (2) bounds the guaranteed servce tmes for each demand node by ts customers acceptable servce tmes. Constrant (3) ensures that the ngong servce tmes at a node cannot be earler than the outgong servce tmes of the suppler. Constrant (4) computes, from ts gven n- and outgong guaranteed servce tmes and the delays, the mnmal amount of tme unts x that node has to be able to delver from ts on-hand nventory. Constrant (5) transforms ths quantty, by means of the demand bound functons, nto a mnmal necessary nventory level. Moreover, all varables are nonnegatve, and all tme unts and quanttes are consdered ntegers. Ths corresponds to nventory networks storng expensve materals whose nventory levels are checked perodcally, e.g., on a daly bass. Note that ths model s always feasble. For example, set := 0. Ths has the meanng that each node must delver the full demand to downstream nodes mmedately. Then, settng x := L and y := φ (x ) for all leads to a feasble, yet most probably expensve soluton. Ths soluton s called the all -soluton, because each node guarantees to delver all orders mmedately from stock. Gven a feasble GSM soluton, a correspondng replenshment polcy s straght-forward: just order n each node exactly the observed demand mmedately. Ths order wll arrve at most x tme unts later. Untl then, the avalable on-hand nventory y s suffcent to keep the promsed outgong servce tme, as specfed by. Thus, each feasble soluton leads to a feasble nventory management polcy, and an optmal soluton yelds a polcy that has mnmal nventory holdng costs for safety stock among all polces based on fxed guaranteed servce tmes. As dscussed n the ntroducton, the bounded-demand assumpton s very often nterpreted as usually fulflled so that n all remanng cases operatonal flexblty can be used to ensure the guaranteed servce tmes. But what happens f for some tme the demands exceed the demand

3 bound functons and outsourcng s used as operatonal flexblty? Assume, the shortage n Constrant (5) s flled by outsourcng, and t shall be explctly accounted for. Then t has to be specfed how many peces are ordered at a suppler outsde the system. Denote by q the outsourcng quantty at node at a margnal cost of c > 0. Then, ths outsourcng decson can be ncorporated nto Constrant (5) as y + q φ (x ) 0., (5 ) (Throughout the paper, we wll use prmed equaton numbers for constrants modfed from earler constrants.) The total cost s then ncreased by N c q. The resultng model s denoted by GSM-o: mn ( ) h y + c q (1 ) N 0 D (2) j 0 (j, ) A (3) + L 0 N (4) y + q φ (x ) 0 N (5 ),, x, y, q 0 N (6 ),, x, y, q Z N (7 ) Ths model s always feasble by gnorng the outsourcng opportunty and usng the all -soluton. Note, however, that now there s no clear way how to set the modfed demand bound functons φ at nternal nodes. Formally, all demand bound functons are gven as exogenous data, whch does not match realty anymore, snce the demand propagated upstream s possbly reduced by the downstream outsourcng decsons q. Thus, n ts current form, the GSM-o (and for that matter, also the orgnal GSM) s only an approxmate model that can be used f there s no frequent use of operatonal flexblty n form of outsourcng. Ths s usually ensured by settng the demand bound functon to a large quantle (> 90%) of some normally dstrbuted stochastc demand. Ths quantle s called the servce level and s the fracton of the actual demand that has to be fulflled from nventory. A large servce level can be justfed at end customer nodes. However, at nternal nodes t s not so clear why mantanng a large servce level should be the most effcent polcy. The stuaton s worse n the SGSM. The SGSM consders several demand scenaros characterzed by ndvdual demand bound functons and adapts the outsourcng quantty to the observed demand scenaro. Thus, large outsourcng quanttes occur naturally n the hgh-demand scenaros of the SGSM. And n these scenaros, the demand propagaton of the model s not exact. Stock Pont 1 (Master) h 1 = 1 c 1 = 2 L 1 = 1 Stock Pont 2 2 = 0 h 2 = 2 c 2 = 1 L 2 = 1 Fg. 1. An example network exhbtng the problem wth demand propagaton Consder the followng lttle example. The network together wth the data except the demand bound functons s shown n Fgure 1. Let the exogenous demand bound functon at the demand node be φ 2 (x) = x. Accordng to the usual a-pror demand propagaton, ths results n φ 1 (x) = x as well. Consder a choce of y 2 = 0, q 1 = 0, and 1 = 0,.e., evaluate the cost of the good-lookng decson where nothng s stored at the demand node (expensve margnal holdng cost), where nothng s outsourced at the upstream node (expensve margnal outsourcng cost), and where all delveres from the upstream nventory are mmedate. In that case, Constrant (2) mples 2 = 2 = 0, and, by Constrant (3), 2 = 0 holds as well. Ths results, by Constrant (4) n x 2 = L 2 = 1 and x 1 = L 1 = 1. (Note that all varables are requred to be non-negatve.) Snce y 2 = q 1 = 0, t follows, by Constrant (5 ), that q 2 = y 1 = 1. The ncurred cost of ths soluton s then c 2 + h 1 = 2. In realty, however, the upstream demand bound functon n the presence of complete outsourcng at the demand node s actually dentcal to zero. Thus, n realty, whenever q 2 = 1 we can afford y 1 = 0, leadng to a cost of c 2 = 1 for the same ndependent decsons, only half as much as predcted by the model. Wth lttle more effort examples can be constructed where the optmalty of decsons s assessed ncorrectly by the GSM-o. Summary: For scenaros n whch frequent outsourcng s preferable a more explct, consstent model s needed. And ths s the modelng problem posed n ths paper. In the followng, the GSM-o s enhanced wth outsourcng wth explct demand propagaton for lnear demand bound functons of the form φ (x ) := α x. The exact handlng of more general demand bound functons s subject of research n progress. 3. A GSM WITH OUTSOURCING AND DEMAND PROPAGATION The constructon works n two steps. Frst, a model wth non-lnear restrctons (GSM-NL) s derved, n whch the number of peces n ordered at node per tme unt by downstream nodes s explctly computed. Second, the non-lnear restrctons are lnearzed by means of extra bnary varables wth the so-called bg-m-method (GSM- DP). For ths secton t s assumed that x = 0 mples q = 0, whch s wth no loss of generalty because each feasble soluton wth x = 0 and q > 0 can be modfed by settng q = 0 mantanng feasblty at a reduced cost. For the frst step, consder the suffcent nventory nequalty (5 ) y + q φ (x ) := α x for some α 0. For nternal nodes, the demand shall be endogenously defned by propagaton from downstream nodes. Thus, for nternal nodes the exogenous demand rate per tme unt α (data) must be replaced by an endogenous demand rate n (a varable). Ths yelds the non-lnear suffcent-nventory nequalty y + q n x (5 ) for the requred nventory at nternal nodes. In order to compute an upstream demand rate n from downstream outsourcng quanttes q j wth (, j) A, two cases are dstngushed. If x j = 0, then, by assumpton,

4 no outsourcng happens,.e., q j = 0, and, therefore, the unreduced demand rate s propagated upstream. Otherwse, the outsourcng quantty q j s used to cover a part of the demand at j over a perod of x j tme unts. Thus, because of lnear demand bound functons, the demand per tme unt upstream at s reduced by qj x j. Summarzng ths concept over all successors of node, the demand-rate balance equatons are obtaned that are non-lnear for all but the demand nodes: n = α D, (8) n = n j q j N\D. (9) x j j:(,j) A j:(,j) A x j 0 The followng non-lnear GSM extenson summarzes the model, whch s denoted by GSM-o-NL: mn ( ) h y + c q (1 ) N n 0 D (2) j 0 (j, ) A (3) + L 0 N (4) y + q n x 0 N (5 ) j:(,j) A n j + j:(,j) A x j 0 n = α D (8) q j = 0 x j N\D (9),, x, y, q, n 0 N (6 ),, x, y, q Z N (7 ) Agan, gnorng the outsourcng opportunty,.e., mposng q = 0 combned wth the all -soluton (always delver the full demand from nventory mmedately) s a feasble soluton for ths model. Next, the GSM-o-NL s lnearzed. The man dea s as follows: f x were a constant n all restrctons, then the system of restrctons would be lnear. Snce x s an nteger, all possble cases (x = k, k K) can be classfed by the possble values of x. The respectve cases are then ndcated by addtonal bnary multple-choce varables z k, exactly one of whch has to be set to one, formally: K z k = 1 N. (10) If z k = 1, then node s prepared to cover demand for x = k tme unts from ts nventory y. The maxmal k needed,.e., the maxmal tme to wat for an order to arrve, can be estmated by the sum of all delays L j on a drected path to n the nventory network. Denote by k max ths upper bound on k, and let K := {0, 1,..., k max } be the set of all k to be consdered n the model. We can recover x from the z k formally as follows: K kz k = x N. (11) Note that k max multpled by the sum of all end customer demand rates downstream of s an estmate for the maxmal possble shortage that can be experenced n node. Let M bg denote ths maxmal possble shortage. Then n x (y + q ) M bg n any optmal soluton. Constrant (5 ) can therefore be expressed usng the constant k and z k nstead of the x by the lnear constrant n k (y + q ) (1 z k )M bg. (5 ) Indeed: Whenever x = k, then z k = 1, and thus the rght-hand sde evaluates to zero. The quantty n k s the maxmal requred amount of materal n node durng k tme unts. Ths should not exceed the avalable materal n node, whch s y + q. Whenever x k, then z k = 0, and the rght hand sde evaluates to M bg, whch s so large that the nequalty becomes redundant. Contrant (9) can be lnearzed usng an addtonal varable n j for the demand rate propagated from j to. The total demand arrvng at node s consequently n = n j N\D. (9 a) j:(,j) A It s requred that the outsourcng quanttes q j are exactly the dfference between downstream demand at j and the propagated demand from j to accumulated over k tme unts. Ths can be ensured by the condtonal equaton q j = (n j n j )k whenever z k = 1. Ths condtonal equaton can be reformulated by usng the same bg-m technque as above separately for the mpled condtonal nequaltes q j (n j n j )k and q j (n j n j )k whenever z k = 1. Ths results n the two lnear nequaltes q j (n j n j )k (1 z jk )M bg j, (9 b) q j + (n j n j )k (1 z jk )M bg j. (9 c) If k = 0, the propagated demand rates have yet to be ted to the downstream demand rates. In that case q j = 0, and n j n j q j (, j) A, (9 d) enforces that the demand rate s propagated completely upstream. Note that for k 1 ths constrant s redundant, n partcular vald, so that no bg-m s needed. The complete resultng model wth demand propagaton, denoted by GSM-o-DP, reads as follows: mn ( ) h y + c q (1 ) N 0 D (2) j 0 (j, ) A (3) + L 0 N (4) n k (y + q ) (1 z k )M bg 0 N, k K (5 ) n = α D (8) n n j = 0 N\D (9 a) j:(,j) A q j (n j n j )k (1 z jk )M bg j 0 (, j) A, k K (9 b) q j + (n j n j )k (1 z jk )M bg j 0 (, j) A,

5 k K (9 c) n j n j q j 0 (, j) A (9 d) K z k = 1 N (10) x K kz k = 0 N (11),, x, y, q, n, n j 0 N (6 ),, x, y, q Z N (7 ) z k {0, 1} N k K (12) In the same way as above, the all -soluton wthout outsourcng s feasble for ths model. Ths model can be extended to the stochastc model SGSM n the same way as the GSM-o was extended to the orgnal SGSM. To ths end, consder a fnte set of demand scenaros ω Ω wth probabltes p ω. The demand bound functons at the demand nodes are now scenarodependent lnear functons and are gven by α ωx. The guaranteed servce tmes and, the tmes to delver from nventory x and the nventory level decsons y n the SGSM-DP are consdered as here-and-now decsons,.e., they have to be taken wthout knowng the realzed scenaro. The outsourcng quanttes q ω are consdered as wat-and-see decsons that can be taken as soon as the scenaro has realzed. Consequently, also the propagated demand quanttes n ω, nω j are both scenaro dependent. The resultng two-stage stochastc mxed-nteger lnear program wth recourse, whch s denoted by SGSM-DP, that mnmzes expected total costs s obtaned from the GSM-o-DP as follows: for each wat-and-see decson varable ntroduce a copy for each scenaro ω Ω. All constrants that contan a wat-and-see decson varable are then requred to hold for each scenaro ω Ω. The result s summarzed n the followng model, where the equaton numbers wth a superscrpt ω reflect a constrant that now has to hold n each scenaro: mn ( h y + p ω c q ω ) N ω Ω (1 ) 0 D (2) j 0 (j, ) A (3) + L 0 N (4) n ω k (y + q ω ) (1 z k )M bg 0 N, k K, ω Ω (5 ω ) n ω = α ω D, ω Ω (8 ω ) q ω j (n ω j n ω j)k (1 z jk )M bg j 0 (, j) A, k K, ω Ω (9 b ω ) q ω j + (n ω j n ω j)k (1 z jk )M bg j 0 (, j) A, k K, ω Ω (9 c ω ) n ω j n ω j q ω j 0 (, j) A, n ω j:(,j) A x n ω j = 0 ω Ω (9 d ω ) N\D, ω Ω (9 a ω ) K z k = 1 N (10) K kz k = 0 N (11),, x, y 0 N (6 ) q ω, n ω, n ω j 0 N, ω Ω (6 ω ),, x, y Z N (7 ) q ω Z N ω Ω (7 ω ) z k {0, 1} N, k K (12) Once more, the all -soluton wth no outsourcng s feasble; ths tme each node must set y to the maxmal demand over x = L tme unts among all scenaros n order to be able to delver from stock under all crcumstances. Now, n each scenaro the ndvdually specfed outsourcng decsons lead to the exact endogenously derved upstream demand rates. 4. EXAMPLE In the followng the mpact of the enhanced model SGSM- DP s llustrated for a stll small example by Löhnert (2016). The example network together wth all nput data s dsplayed n Fgure 2. For the SGSM, the nternal demands n each scenaro are approxmated by the complete accumulated downstream demand. For the SGSM-DP nternal demands are computed by the model. In the Table 1, the resultng cost of the orgnal SGSM model by Rambau and Schade (2014) s compared to the cost of the new SGSM-DP model. Note that the SGSM s suggestons for outsourcng need only be mplemented (and payed) n realty f there really s a shortage gven the nventory levels. Snce the SGSM overestmates demands n the presence of outsourcng downstream, some upstream outsourcng mght turn out to be superfluous. Thus, two cost values are presented for the SGSM soluton: the model cost and the actual cost. The actual cost of the orgnal SGSM s computed by fxng the here-and-now decsons (servce tmes s, duratons x, and nventores y ) of the SGSM n the SGSM-DP. The model cost and the actual cost of the SGSM s then compared wth the model cost of the SGSM- DP. The computatons were performed on a Dell VOSTRO 3450 Laptop wth (Intel M, 64bt, 2.30GHz, 4GB RAM, Wndows 7 Professonal) usng the free MILP solver SCIP (Gamrath et al., 2016). It can be seen that the orgnal SGSM s model cost (optmal objectve functon value of the SGSM) overestmates the actual cost of ts strategc nventory decsons (objectve functon value of

6 Stock Pont 2 2 = 1 h 2 = 3 c 2 = 8 L 2 = 5 α 1 2 = 5 α 2 2 = 2 α 3 2 = 25 Stock Pont 1 (Master) h 1 = 2 c 1 = 4 L 1 = 3 α 1 1 = 15 α 2 1 = 7 α 3 1 = 50 Stock Pont 4 4 = 6 h 4 = 5 c 4 = 8 L 4 = 5 α 1 4 = 5 α 2 4 = 2 α 3 4 = 15 Stock Pont 3 (Internal) h 3 = 4 c 3 = 6 L 3 = 5 α 1 3 = 10 α 2 3 = 5 α 3 3 = 25 Stock Pont 5 5 = 7 h 5 = 5 c 5 = 8 L 5 = 4 α 1 5 = 5 α 2 5 = 3 α 3 5 = 10 Fg. 2. An example network wth hgh cost dfferences between SGSM and SGSM-DP SGSM SGSM-DP # var s # cons s CPU tme/s < cost (actual cost) 747 (567) 410 Table 1. Results for the example nstance (rounded) the SGSM n the SGSM-DP) by over 30%. Moreover, because of the approxmaton of the nternal demands, the SGSM s nventory decson s almost another 40% more expensve than the SGSM-DP s optmal soluton wth explct demand propagaton n each scenaro. Admttedly, ths example was constructed n such a way that dfferences become vsble. However, there s no reason to assume that n more complcated real-lfe nstances the dfference s less pronounced. 5. CONCLUSIONS The SGSM-DP can mprove strategc nventory decsons: explct accountng for outsourcng and demand propagaton are advantages over all exstng models n the GSM branch of research. By presentng an easy example wth three echelons, three demand nodes, two other nodes, and three scenaros, we have shown evdence for the fact that the problems wth demand propagaton n GSM models are not neglectable. The advantages come at a cost: The SGSM-DP s computatonally more demandng (by a factor of over 100 n the example!), and ths wll become more serous of an ssue for real-lfe scale problems. Consderng the fact that the model usually has to be solved separately for each product class handled by a network, short computaton tmes can become vtal. Ths paper showed a logcally sound way to exactly model demand propagaton n GSM-type models for lnear demand bound functons. It has yet to be nvestgated how the modelng approach can be generalzed to, say, pecewse lnear demand bound functons, and how large the computaton tmes grow for more complcated supply chans n practce. Some mportant dstrbuton systems, lke the real-world US spare-part dstrbuton system of a large German automoble manufacturer (whch motvated the SGSM research n the frst place), are of moderate sze (two echelons, a sngle root node, less than ten demand nodes) so that the SGSM-DP can probably be solved for them by drectly feedng the model and the data to a solver. For more complcated mult-company producton supply chans, taylormade algorthmc methods from mxed-nteger lnear programmng mght be needed and should therefore be nvestgated. REFERENCES Clark, A.J. and Scarf, H. (1960). Optmal polces for a mult-echelon nventory problem. Management Scence, 6(4), de Kok, T.G. and Fransoo, J.C. (2003). Plannng supply chan operatons: defnton and comparson of plannng concepts. Handbooks n operatons research and management scence, 11, Gamrath, G., Fscher, T., Gally, T., Glexner, A.M., Hendel, G., Koch, T., Maher, S.J., Mltenberger, M., Müller, B., Pfetsch, M.E., Puchert, C., Rehfeldt, D., Schenker, S., Schwarz, R., Serrano, F., Shnano, Y., Vgerske, S., Wennger, D., Wnkler, M., Wtt, J.T., and Wtzg, J. (2016). The SCIP Optmzaton Sute 3.2. Techncal Report 15-60, ZIB, Takustr.7, Berln. Graves, S.C. and Wllems, S.P. (2003). Supply chan desgn: safety stock placement and supply chan confguraton. Handbooks n operatons research and management scence, 11, Löhnert, M. (2016). Das Stochastc Guaranteed Servce Model n velstufgen Lagernetzen. Masterarbet, Unverstät Bayreuth. Magnant, T.L., Shen, Z.J.M., Shu, J., Smch-Lev, D., and Teo, C.P. (2006). Inventory placement n acyclc supply chan networks. Operatons Research Letters, 34(2), Rambau, J. and Schade, K. (2010). The stochastc guaranteed servce model wth recourse for mult-echelon warehouse management. In Proceedngs of the Internatonal Symposum on Combnatoral Optmzaton (ISCO 2010), volume 36 of Electronc Notes n Dscrete Mathematcs, Elsever. Rambau, J. and Schade, K. (2014). The stochastc guaranteed servce model wth recourse for mult-echelon warehouse management. Mathematcal Methods of Operatons Research, 79(3), Schade, K. (2012). Bestandsoptmerung n mehrstufgen Lagernetzwerken. Stochastsche Optmerung. Sprnger.