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The impact of dampening demand variability in a production/inventory system with multiple retailers (abstract only)

We study a supply chain consisting of a single manufacturer and two retailers. The manufacturer produces goods on a make-to-order basis, while both retailers maintain an inventory and use a periodic replenishment rule. As opposed to the traditional
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  The impact of dampening demand variability in aproduction/inventory system with multiple retailers B. Van Houdt Department of Mathematics and ComputerScienceUniversity of Antwerp - IBBTAntwerpen, Belgium P´erez Department of Electrical and ElectronicsEngineeringUniversidad de los AndesBogot´a, Colombia ABSTRACT We study a supply chain consisting of a single man-ufacturer and two retailers. The manufacturer pro-duces goods on a make-to-order basis, while both re-tailers maintain an inventory and use a periodic re-plenishment rule. As opposed to the traditional ( r,S  )policy, where a retailer at the end of each period or-ders the demand seen during the previous period, weassume that the retailers dampen their demand vari-ability by smoothing the order size. More specifically,the order placed at the end of a period is equal to  β  times the demand seen during the last period plus(1  −  β  ) times the previous order size, with  β   ∈  (0 , 1]the smoothing parameter.We develop a GI/M/1-type Markov chain with onlytwo nonzero blocks  A 0  and  A d  to analyze this sup-ply chain. The dimension of these blocks prohibitsus from computing its rate matrix  R  in order to ob-tain the steady state probabilities. Instead we rely onfast numerical methods that exploit the structure of the matrices  A 0  and  A d , i.e., the power method, theGauss-Seidel iteration and GMRES, to approximatethe steady state probabilities.Finally, we provide various numerical examples thatindicate that the smoothing parameters can be setin such a manner that all the involved parties bene-fit from smoothing. We consider both homogeneousand heterogeneous settings for the smoothing param-eters. 1. INTRODUCTION Consider a two-echelon supply chain consisting of a single retailer and a single manufacturer, where theretailer places an order for a batch of items withthe manufacturer at regular time instants, i.e., thetime between two orders is fixed and denoted as  r .The manufacturer may be regarded as a single serverqueue that produces these items and delivers them tothe retailer as soon as a complete order is finished.The retailer sells these items and maintains an in-ventory on hand to meet customer demands. Whenthe customer demand exceeds the current inventoryon hand, only part of the demand is immediately ful-filled and the remaining items are delivered as soonas new items become available at the retailer. Hence,items are backlogged instead of being lost (i.e., thereare no  lost sales  ). We assume that the manufacturerdoes not maintain an inventory, but simply producesitems whenever an order arrives, i.e., it operates on a make-to-order   basis.A key performance measure in such a system is the  fill-rate  , which is a measure for the proportion of cus-tomer demands that can be met without any delay.In order to guarantee a certain fill-rate it is impor-tant to determine the size of the orders placed at theregular time instants. This size will depend on thecurrent  inventory position  , defined as the inventoryon hand plus the number of items on order minusthe number of backlogged items. The rule that de-termines the order size is termed the  replenishment  rule. A well-studied replenishment rule exists in or-dering an amount such that the inventory position israised after each order to some fixed position  S  , calledthe  base-stock level  . This basically means that at theregular time instants, you simply order the amount of items sold since the last order instant. As a result, theorder policy of the retailer is called an ( r,S  ) policy.A common approach in the analysis of such a pol-icy is to assume an  exogenous   lead time, which meansthat the time required to  deliver   an order is indepen-dent of the size of the current order and independentof the lead time of previous orders. In [4] the ( R,S  )policy was studied with  endogenous   lead times, mean-ing the lead times depend on the order size and con-secutive lead times are correlated. The results in [4]indicate that exogenous lead times result in a severeunderestimation of the required inventory on hand,as expected.When the lead times are endogenous, it is clear thata high variability in the order sizes comes at a cost,as this increases the variability of the arrival processat the manufacturer and therefore increases the leadtimes. As a result, replenishment rules that smooththe order pattern at the retailer were studied in [3]and it was shown that the retailer can reduce the up-stream demand variability without having to increasehis safety stock (much) to maintain customer serviceat the same target level. Moreover, on many occa-sions the retailer can even decrease his safety stocksomewhat when he smooths his orders. This is clearlyadvantageous for both the retailer and the manufac-  turer. The manufacturer receives a less variable orderpattern and the retailer can decrease his safety stockwhile maintaining the same fill rate, so that a coop-erative surplus is realized.In this paper we analyze the same set of replenish-ment rules as in [3], but now we look at a two-echelonsupply chain consisting of one manufacturer and tworetailers, where either both, one or neither of theretailers uses a smoothing rule. The main questionthat we wish to address therefore exists in studyingwhether all parties can still benefit when the ordersare smoothed and moreover who benefits most.As in [3], one of the key steps in the analysis of this supply chain system will exist in setting up aGI/M/1-type Markov chain [7], that has only two non-zero blocks, denoted as  A 0  and  A d . However,as opposed to [3], the size of these blocks often pro-hibits us from storing them into main (or secondary)memory. This implies that iteratively computing thedense  R  matrix, used to express the matrix geomet-ric steady state vector of the GI/M/1-type Markovchain, by one of the existing methods such as func-tional iterations or cyclic reduction [1], is no longerpossible/efficient. Instead, we will rely on the spe-cific structure of the matrices  A 0  and  A d  and willmake use of numerical methods typically used to solvelarge finite Markov chains, such as the shuffling algo-rithm [5], Kronecker products, the power method, theGauss-Seidel iteration and GMRES [9]. 2. MODEL DESCRIPTION We consider a two-echelon supply chain with tworetailers and a single manufacturer, where both retail-ers maintain their own inventory. Every period, bothretailers observe their customer demand. If there isenough on-hand inventory available at a retailer, thedemand is immediately satisfied. If not, the shortageis backlogged. To maintain an appropriate amount of inventory on hand, both retailers place a replenish-ment order with the manufacturer at the end of everyperiod. The manufacturer does not hold a finishedgoods inventory but produces the orders on a make-to-order basis. The manufacturers production sys-tem is characterized by a single server queueing modelthat sequentially processes the orders, which requirestochastic processing times. Once the complete re-plenishment order of both retailers is produced, themanufacturer replenishes both inventories. Hence,the order in which the two orders are produced isirrelevant, as shipping only occurs when both ordersare ready.The time from the moment an order is placed tothe moment that it replenishes the retailers inven-tory, is the  replenishment   lead time  T  r . The queueingprocess at the manufacturer clearly implies that theretailers replenishment lead times are stochastic andcorrelated with the order quantity. The sequence of events in a period is as follows. The retailer first re-ceives goods from the manufacturer, then he observesand satisfies customer demand and finally, he placesa replenishment order with the manufacturer. Thefollowing additional assumptions are made:1. Customer demand during a period for retailer i  is independently and identically distributed(i.i.d.) over time according to an arbitrary,finite, discrete distribution  D ( i ) with a maxi-mum of   m ( i ) D  , for  i  = 1 and 2. The demandat the retailers is also assumed to be indepen-dent of each other. For further use, denote m D  =  m (1) D  + m (2) D  .2. The order quantity  O ( i ) t  of retailer  i  during pe-riod  t  is determined by the retailers replenish-ment rule and influences the variability in theorders placed on the manufacturer. Possible re-plenishment rules are discussed in the next sec-tion.3. The replenishment orders are processed by a sin-gle FIFO server. This excludes the possibility of order crossovers. When the server is busy, neworders join a queue of unprocessed orders.4. The orders placed during period  t  are deliveredwhen both orders have been produced.5. Orders consist of multiple items and the pro-duction time of a single item is i.i.d. accordingto a discrete-time phase type (PH) distributionwith representation ( α,U  ). For further use, wedefine  u ∗ =  e  −  Ue , with  e  a column vector of ones.The PH distribution is determined using the match-ing procedure presented in [3], that matches the firsttwo moments of the production time using an order 2representation, meaning the matrix  U   is a 2  ×  2 ma-trix and  α  a size 2 row vector, even if the squaredcoefficient of variation is small by exploiting the scal-ing factor as in [2]. This implies that the length of a time slot is chosen as half of the mean productiontime of an item. In other words, the mean productiontime of an item is two time slots, while the length of aperiod is denoted as  d  time slots, where  d  is assumedto be an integer.The time from the moment the order arrives at theproduction queue to the point that the production of the entire batch is finished, is the  production   lead timeor response time, denoted by  T  p . Note that the pro-duction lead time is not necessarily an integer numberof periods. Since in our inventory model events occuron a discrete time basis with a time unit equal to oneperiod, the replenishment lead time  T  r  is expressed interms of an integer number of periods. For instance,suppose that the retailer places an order at the  end  of period  t , and it turns out that the production leadtime is 1 . 4 periods. This order quantity will be addedto the inventory in period  t  + 2, and due to our se-quence of events, can be used to satisfy demand inperiod  t  + 2. As such, we state that the replenish-ment lead time  T  r  is  ⌊ T  p ⌋  periods, i.e., 1 period inour example. 3. REPLENISHMENT RULES  The retailers considered in this paper apply an( r,S  ) policy with or without smoothing, meaningamongst others they place an order at the end of each period. Without smoothing, the order size issuch that the inventory position  IP  , defined as theon-hand inventory plus the number of items on orderminus the backlogged items, equals some fixed  S   af-ter the order is placed. In other words, the size of the order  O t  at the end of period  t  simply equals thedemand  D t  observed during period  t .If smoothing is applied with parameter 0  < β <  1,we do not order the difference between  S   and  IP  , butinstead only order  β   times  S   −  IP  . As will becomeclear below, this does not imply that fewer items areordered in the long run, it simply means that someitems will be ordered at a later time. As shown in [3],this rule is equivalent to stating that the size of theorder at the end of period  t , denoted  O t , is given by O t  = (1  − β  ) O t − 1  + βD t , where  D t  is the demand observed by a retailer in pe-riod  t . Hence, setting  β   = 1 implies that we do notsmooth. This equation also shows that the mean or-der size is still equal to the mean demand size  E  [ D ].It is also easy to show [3] that the variance of theorder size  Var [ O ] equals β  (2  − β  ) Var [ D ] , meaning the variance decreases to zero as  β   ap-proaches zero, where  Var [ D ] is the variance in thedemand. It is also possible to consider  β   values be-tween 1 and 2, but this would amplify the variabilityinstead of dampening it.The key question that our analytical model will an-swer is how to select the base-stock level  S   such thatthe fill-rate, a measure for the proportion of demandsthat can be immediately delivered from the inventoryon hand, defined as1  −  expected number of backlogged itemsexpected demand  , is sufficiently high. The level  S   is typically expressedusing the  safety stock   SS  , defined as the average netstock just before a replenishment arrives (where thenet stock equals the inventory on hand minus thenumber of backlogged items). For a retailer thatsmooths with parameter  β  ,  S   and  SS   are related asfollows [3] S   =  SS   + ( E  [ T  r ] + 1) E  [ D ] + 1  − β β  E  [ D ] , where  E  [ T  r ] is the mean replenishment lead time.Thus, a good policy will result in a smaller safetystock  SS  , which implies a lower average storage costfor the retailer. 4. THE MARKOV CHAIN Both Markov chains developed in this section are ageneralization of the Markov chain introduced in [3],for the system with a single retailer. The numericalmethod to attain their stationary probability vector,discussed in Section 5, is however very different.From now on we will express all our variables intime slots, where the length of a single slot equals half of the mean production time, i.e.,  α ( I  − U  ) − 1 e/ 2, andorders are placed by both retailers every  d  time slots.Hence, the order size of retailer  i  at the end of period t  is now written as  O ( i ) td  and O ( i ) td  = (1  − β  i ) O ( i )( t − 1) d  + β  i D ( i ) , where  β  i  is the smoothing parameter of retailer  i , for i  = 1 , 2. As the order size must be an integer, theinteger amount ordered  O ( i ∗ ) td  will equal  ⌈ O ( i ) td  ⌉  withprobability  O ( i ) td  − ⌊ O ( i ) td  ⌋  and  ⌊ O ( i ) td  ⌋  with probabil-ity  ⌈ O ( i ) td  ⌉ −  O ( i ) td  in case  O ( i ) td  is not an integer. Thisguarantees that  E  [ O ( i ∗ ) td  ] =  E  [ O ( i ) td  ] =  E  [ D ( i ) ].The joint order  O ∗ td  of both retailers placed at time td  equals  O (1 ∗ ) td  + O (2 ∗ ) td  . Recall, both these orders areonly delivered by the manufacturer when the jointorder has been produced. Next, define the followingrandom variables: •  t n : the time of the n -th observation point, whichwe define as the  n -th time slot during which theserver is busy, •  a ( n ): the arrival time of the joint order in ser-vice at time  t n , •  B n : the age of the joint order in service at time t n , expressed in time slots, i.e.,  B n  =  t n  − a ( n ), •  C  n : the number of items part of the joint orderin service that still need to start or completeservice at time  t n , •  S  n : the service phase at time  t n .All events, such as arrivals, transfers from the wait-ing line to the server, and service completions areassumed to occur at instants immediately after thediscrete time epochs. This implies that the age of anorder in service at some time epoch  t n  is at least 1.We start by introducing the Markov chain for the casewhere both retailers smooth. 4.1 Both retailers smooth It is clear that the stochastic process( B n ,C  n ,O (1) a ( n ) ,O (2) a ( n ) ,S  n ) n ≥ 0  forms a discrete timeMarkov process on the state space  N 0 ×{ ( c,x 1 ,x 2 ) | c  ∈{ 1 ,...,m D } , 1  ≤  x i  ≤  m ( i ) D  ,i  ∈ { 1 , 2 }} × { 1 , 2 } , asthe PH service requires only two phases. Notethat the process makes use of the order quantities O ( i ) a ( n )  instead of the integer values  O ( i ∗ ) a ( n ) . Since thisorder quantity is a real number, the Markov process( B n ,C  n ,O (1) a ( n ) ,O (2) a ( n ) ,S  n ) n ≥ 0  has a continuous statespace which makes it very hard to find its steadystate vector.Therefore, instead of keeping track of   O ( i ) a ( n )  in anexact manner, we will round it in a probabilistic wayto the nearest multiple of 1 /g , where  g  ≥  1 is an inte-ger termed the  granularity   of the system. Clearly, the  larger  g , the better the approximation. Hence, we ap-proximate the Markov process above by the Markovchain ( B n ,C  n ,O g, (1) a ( n ) ,O g, (2) a ( n ) ,S  n ) n ≥ 0  on the discretestate space  N 0  × { ( c,x 1 ,x 2 ) | c  ∈ { 1 ,...,m D } ,x i  ∈ S ( i ) g  ,i  ∈ { 1 , 2 }} × { 1 , 2 } , where  S ( i ) g  =  { 1 , 1 + 1 /g, 1 +2 /g,...,m ( i ) D  }  and the quantity  O g, ( i ) td  evolves as fol-lows. Let x  = (1  − β  i ) O g, ( i )( t − 1) d  + β  i D ( i ) , then  O g, ( i ) td  =  x  if   x  ∈  S ( i ) , otherwise it equals  ⌈ x ⌉ g with probability  g ( x  − ⌊ x ⌋ g ), or  ⌊ x ⌋ g  with probabil-ity  g ( ⌈ x ⌉ g  − x ), where  ⌈ x ⌉ g  ( ⌊ x ⌋ g ) rounds up (down)to the nearest element in  S ( i ) g  . Notice, by induction,we have  E  [ O g, ( i ) td  ] =  E  [ D ( i ) ]. Using this probabilis-tic rounding, we can easily compute the conditionalprobabilities  P  [ O g, ( i ) td  =  q  ′ | O g, ( i )( t − 1) d  =  q  ], which we de-note as  p ( i ) g  ( q,q  ′ ), from  D ( i ) (see [3, Eqn. (12)] fordetails).The transition matrix  P  g  of the Markov chain( B n ,C  n ,O (1) a ( n ) ,O (2) a ( n ) ,S  n ) n ≥ 0  is a GI/M/1-typeMarkov chain [7] with the following structure, P  g  =  A d  A 0 ...... A d  A 0 A d  A 0 ... ...  , as  B n  either increases by one if the same joint or-der remains in service, or decreases by  d  −  1 if a joint order is completed. Hence, there are  d  occur-rences of   A d  on the first block column. The size  m of the square matrices  A 0  and  A d  is 2 m D m g , with m g  =   2 i =1 ( m ( i ) D  g  −  g  + 1), which is typically suchthat we cannot store the matrices  A 0  and  A d  inmemory. Although we can eliminate close to 50%of the states by removing the transient states with C  n  >  ⌈ O (1) a ( n ) ⌉  +  ⌈ O (2) a ( n ) ⌉ , the size  m  remains prob-lematic and this would slow down the numerical solu-tion method presented in Section 5. A more detaileddiscussion of the structure of   A 0  and  A d  is given inSection 5.1. 4.2 One retailer smooths Assume without loss of generality that retailer onesmooths, while retailer two does not, i.e.,  β  1  <  1 and β  2  = 1. In this case we can also rely on the Markovchain defined above, but now there is no longer a needto keep track of   O g, (2) a ( n ) , as the orders of retailer two aredistributed according to  D (2) . This not only simpli-fies the transition probabilities, but also considerablyreduces the time and memory requirements of the nu-merical solution method introduced in Section 5. Al-though storing the matrices  A 0  and  A d  in memorymay no longer be problematic, a numerical approachas presented in the next section outperforms the moretraditional approach that relies on computing the ratematrix  R  [7] by a considerable margin. 5. NUMERICAL SOLUTION The objective of this section is to introduce a nu-merical method to compute the steady state distribu-tion of the Markov chain introduced in Section 4.1 byavoiding the need to store the matrices  A 0  and  A d . 5.1 Fast multiplication In order to multiply the vector  x  = ( x 0 ,x 1 ,... ) with P  g , where  x i  is a length  m  = 2 m D m g  vector, withoutstoring the matrices  A 0  or  A d , we will write  P  g  as thesum of   P  (0) g  + P  ( d ) g  =  A 0 ... A 0 ...  +  A d ... A d ...  , and compute  xP  g  as  xP  (0) g  +  xP  ( d ) g  . To express thetime complexity of these multiplications, assume  x i  =0 for  i  ≥  n  for some  n  (as will be the case in the nextsubsection).The matrix  A 0  corresponds to the case where thesame joint order remains in service, meaning  C  n  ei-ther remains the same or decreases by one. Due tothe order of the random variables, the matrix  A 0  isa bi-diagonal block Toeplitz matrix, with blocks of size 2 m g . The block appearing on the main diago-nal equals  I   ⊗ U  , as the production of the same itemcontinues in this case. The block below the main di-agonal is  I   ⊗ u ∗ α , as the item is finished, but at leastone item of the joint order still needs to be produced.Hence, A 0  =  I   ⊗ U I   ⊗ u ∗ α I   ⊗ U  ... ... I   ⊗ u ∗ α I   ⊗ U   , where  I   is the size  m g  unity matrix and we have  m D blocks  I   ⊗ U   on the main diagonal. As the PH repre-sentation is of order 2 (even in case of low variability),we can multiply  x  with  P  (0) g  in  O ( mn ) time.When multiplying with  A d , we first argue that  A d can be written as A d  = ( e 1  ⊗  ( I   ⊗ u ∗ ))( W  1  ⊗ W  2 )( Y   ⊗ α ) , where  e 1  is a size  m D  column vector which equals onein its first entry and zero elsewhere,  W  i  is a squarematrix of size  m ( i ) D  g  −  g  + 1 and  Y   is a  m g  ×  m g m D matrix. To understand this decomposition we splitthe transition in four steps. First, a service comple-tion of an order must occur, meaning  C  n  must equalone and the item in service must be completed. Thus,the matrix ( e 1  ⊗  ( I   ⊗  u ∗ ) describes this step. Next,in step 2, we determine the new order size for eachretailer based on the previous order size (using thegranularity  g ). Let the ( q,q  ′ )-th entry of   W  i  equal  p ( i ) g  ( q,q  ′ ) (as defined in Section 4.1), for  i  = 1 , 2. Aseach retailer determines its next order size indepen-dently,  W  1  ⊗ W  2  captures to step 2. To complete the  transition we need to determine the joint  integer   ordersize given the individual  granularity   g  order sizes of both retailers (in step 3) and the initial service phaseof the first item part of the joint order (in step 4).Step 4 is clearly determined by  α , while step 3 cor-responds to the matrix  Y  . A row of the matrix  Y  contains either 1, 2 or 4 non-zero entries (dependingon whether the row corresponds to a case where both,one or none of the granularity  g  orders are integers).Thus, when multiplying  x  = ( x 0 ,x 1 ,... ) with  P  ( d ) g  ,each of the vectors  x i  is first reduced to a length  m g vector in  O ( nm g ) time, because of ( e 1  ⊗  ( I   ⊗  u ∗ )).A multiplication with  W  1  ⊗ W  2  is done in two steps.First we multiply with ( I  ⊗ W  2 ), which can be triviallydone in  O (( m (2) D  g ) 2 m (1) D  g ) =  O ( m g m (2) D  g ) for eachvector, followed by the multiplication with ( W  1  ⊗ I  ).This latter multiplication can be rewritten as a multi-plication with ( I  ⊗ W  1 ) using the shuffle algorithm[5].Hence, it can also be done in  O ( m g m (1) D  g ). Due to itssparse structure, a multiplication with  Y   can be im-plemented in  O ( m g ) time. In conclusion, the overalltime required to multiply  x  with  P  ( d ) g  can be writ-ten as  O ( nm g ( m (1) D  +  m (2) D  ) g ) =  O ( nmg ) and thetime needed to multiply  x  with  P  g  is therefore also O ( nmg ). In practice, for  g  small, the multiplicationwith  P  (0) g  is more time demanding than the multipli-cation with  P  ( d ) g  and a considerable percentage of thetime is also spend on allocating memory. 5.2 The power method, the Gauss-Seideliteration and GMRES To determine the steady state probability vector of the transition matrix  P  g  we rely on the fast matrixmultiplication between a vector  x  and  P  g  introducedabove.When combined with the power method, we ba-sically start with some initial vector  x (0) and de-fine  x ( k  + 1) =  x ( k ) P  g  until the infinity norm of  x ( k  + 1)  −  x ( k ) is smaller than some predefined  ǫ 1 (e.g.,  ǫ 1  = 10 − 8 ). If we start from an empty system, x (0) has only one nonzero component  x 0 (0) of length m  and  x ( k ) has  k  + 1 nonzero components  x 0 ( k ) to x k ( k ). Whenever some of the last components aresmaller than some predefined  ǫ 2 , we reduce the lengthof   x ( k ) (by adding these components to the last com-ponent larger than  ǫ 2 ). Notice, introducing  ǫ 2  is notexactly equivalentto a truncation of the Markov chainat some predefined level  N  . Instead we dynamicallytruncate the vector  x  during the computation and itslength may still vary over time. The impact of both ǫ 1  used by the stopping criteria and  ǫ 2  used by thedynamic truncation will be examined in Section 7.1.Both these parameters will be used in a similar man-ner for the other iterative schemes as well.When applying the  forward   Gauss-Seidel iteration[8], we compute  x ( k  + 1) from  x ( k ) by solving thelinear system x ( k  + 1)( I   − P  (0) g  ) =  x ( k ) P  ( d ) g  , which can be done efficiently using forward substitu-tion as ( I   − P  (0) g  ) is upper triangular. If   x  is an arbi-trary stochastic vector, we initialize  x (0) such thatit solves  x (0)( I   −  P  (0) g  ) =  x . As indicated in [8],this Gauss-Seidel iteration is equivalent to a precon-ditioned power method if we use ( I  − P  (0) g  ) as the pre-conditioning matrix  M  . Notice, we can benefit fromthe fast multiplications discussed in the previous sec-tion when computing  x ( k ) P  ( d ) g  as well as during theforward substitution phase.The GMRES method [9] computes an approximatesolution of the linear system ( I   −  P  ′ g ) x  = 0, by find-ing a vector  x (1) that minimizes  ( I   − P  ′ g ) x  2  overthe set  x (0) +  K ( I   −  P  ′ g ,r 0 ,n ). Here  r 0  is the resid-ual of an initial solution  x (0):  r 0  =  − ( I   −  P  ′ g ) x (0); K ( I   − P  ′ g ,r 0 ,n ) is the Krylov subspace, i.e., the sub-space spanned by the vectors  { r 0 , ( I  − P  ′ g ) r 0 ,..., ( I  − P  ′ g ) n − 1 r 0 } ; and  n  is the dimension of the Krylov sub-space [6]. To do this GMRES relies on the Arnoldi it-eration to find an orthonormal basis  V  n  for the Krylovsubspace, such that  V  ′ n ( I   −  P  ′ g ) V  n  =  H  n , where  H  n is an upper Hessenberg matrix of size  n . Once  V  n and  H  n  have been obtained, a vector  y n  is found suchthat  J  ( y ) =  βe 1  −  ˜ H  n y  2 is minimized. Here  β   isthe 2-norm of   r 0 ,  e 1  is the first column of the identitymatrix, and ˜ H  n  is an ( n  + 1)  ×  n  matrix whose first n  rows are identical to  H  n , and its last row has onenonzero element that also results from the Arnoldiiteration. A new approximate solution  x (1) is com-puted as  x (1) =  x (0)+ V  n y n . The process is then re-peated with  x (1) as  x (0) until the difference betweentwo consecutive solutions is less than some predefined ǫ . Although this algorithm is defined to solve linearsystems of the type  Ax  =  b , with  A  nonsingular, itcan also be used to solve homogeneous systems with A  singular, as is the case with Markov chains [10].The GMRES algorithm also benefits from the fastmultiplication discussed in the previous section. Tofind the residual  r 0  at each iteration we need tocompute the product ( I   −  P  ′ g ) x (0) =  x (0)  −  P  ′ g x (0).Also, for the Arnoldi process we need to determinethe vectors  v j  = ( I   −  P  ′ g ) j − 1 r 0 , which are computediteratively, and require  n  −  1 products of the type( I   −  P  ′ g ) v j − 1  =  v j − 1  −  P  ′ g v j − 1 . As with the powermethod, when analyzing several scenarios we can usethe final approximate solution of one scenario as thestarting solution for the next one to speed up conver-gence. 6. THE SAFETY STOCK The required safety stock  SS  i  for each retailer toguarantee a certain fill rate is one of the main perfor-mance measures of this supply chain problem. Thederivation for the case where both retailers smoothis nearly identical to the one presented in [3] and ismainly included for reasons of completeness. As in-dicated in Section 3, computing  SS  i  is equivalent todetermining the base-stock  S  i  provided that we knowthe mean replenishment lead time  E  [ T  r ] (which equalsthe floor of the production lead time  T  p ). The pro-duction lead time distribution  T  p  is easy to obtain
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