technische universiteit eindhoven Analysis of one product /one location inventory control models prof.dr. A.G. de Kok 1

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1 TU/e tehnshe unverstet endhoven Analyss of one produt /one loaton nventory ontrol models prof.dr. A.G. de Kok Aknowledgements: I would lke to thank Leonard Fortun for translatng ths ourse materal nto Englsh and hm and Karel van Donselaar for rtal remarks.

2 Analyss of stok ontrol models for one loaton wth one produt. Introduton Ths handout deals wth stok ontrol models for one produt n one stok loaton, the (s,nq) model. We wll derve expressons for performane measures most frequently used n prate. Thereby we observe the framework of onepts and notatons defned by Slver, Pyke & Peterson [998], SPP for short. Why a handout n addton to SPP? The reason s smple to explan: the formulae derved n SPP have a too lmted valdty. Extensons or adaptatons of these formulae, n SPP presented va footnotes, appeared n prate essental for ther applablty. SPP s makng assumptons, mpltly or expltly, about the behavour of the demand proess, the order proess and the delvery proess. In ths handout these assumptons frst are made explt, n order to determne the valdty of the SPP formulae. Next we wll replae the dervatons of SPP by an analyss that s vald wthout restrtve assumptons. In ths way we obtan formulae wth a general applablty. Furthermore, we dstngush between the dervaton of formulae on the one hand and the numeral proessng of the formulae on the other. In our opnon ndustral engneers should be able to derve these formulae themselves. Suh a dervaton gves them nsght n the mehansms behnd the ontrol rules. One the formulae have been derved, for an ndustral engneer numeral proessng beomes less nterestng. For ths reason the handout s aompaned by a spreadsheet. It enables numeral analyss of stok ontrol models on the bass of the formulae derved. By analysng varous operatonal stuatons, we obtan nsght n the effets of unertanty n demand proess, supply performane, order osts, stok-keepng osts and the flexblty or speed of the delvery proesses. The struture of ths handout s as follows. Frst we defne the relevant parameters and varables, n Seton. Next, n Seton 3, we lst the assumpton made by SPP. In Seton 4 we gve a detaled analyss of the ( s, Q) model. Here the assumptons of SPP stll play an mportant part. But n Seton 5 we drop these assumptons one after the other, whh leads to results that have a wde applablty to real-lfe problems. Then we analyse the (R,S) model n a smlar way (Seton 6). Fnally, n Seton 7, we present the essental formulae of related models, namely the (s,s), (R,s,S) and (R,s,Q) model, wthout gvng the analyss.

3 Analyss of stok ontrol models for one loaton wth one produt 3. Defntons X (t) : = net stok at tme t X ( t ) : = net stok just before tme t ( t ) = lm X t ; X () t t Y () t : = nventory poston at tmet D : = demand per ustomer or demand per perod ( ] t,t D : = demand durng the nterval ( t ], wth ( t, t ] = { x t < t } x s : = reorder pont,t Q : = order quantty th τ : = replenshment order moment after tme t = ( =,, ) τ : =, tme orgn at whh the frst replenshment order s plaed L : = delvery tme of the replenshment order plaed at tme t = τ ( =,, ) υ : = expeted net stok mmedately before the arrval of a replenshment order (safety stok) B ( t,t ] : = demand bakordered n ( t,t ] + x : = max (, x) P {...} : = Probablty { } E [...] : = Expetaton [ ] σ (...) : = Varane [ ]

4 Analyss of stok ontrol models for one loaton wth one produt 4 k Φ (k) : = exp( z ) dz. π ( k) { } + G : = ( Z k ) k E = ( z k) exp z dz π P P : probablty of not beng out-of-stok just before a replenshment order arrves : long-run fraton of total demand, whh s beng delvered from stok on hand (also known as fll-rate)

5 Analyss of stok ontrol models for one loaton wth one produt 5 3. Assumptons In SPP the followng assumptons le at the bass of all formulae derved n Chapters 7, 8 and 9: () ( ] t,t D has a normal dstrbuton wth expetaton ( t t ) µ and varane t t σ. () At the moment of orderng the stok poston s exatly equal to s. () Subsequent orders annot overtake eah other; so: an order plaed later annot arrve earler. (v) Delvery tmes are onstant and equal to L. (v) The net nventory after arrval of an order s postve. Impltly there s another assumpton that we are gong to use too: (v) The reorder quantty s onstant and equal to Q. (v) All demand whh annot be met mmedately from stok s bakordered. In our dervatons of expressons for P, P and other performane haratersts we wll only need assumpton () and (v). For the manual omputaton of values for these expressons the other assumptons are very useful, sne they enable the use of tables for funtons assoated wth the normal dstrbuton funton. However, n most pratal stuatons one or more assumptons are volated, so that we need a omputer for the omputaton of P and P. Fortunately, the expressons derved are relatvely smple and are easly mplemented n e.g. an Exel spreadsheet (f. De Kok []).

6 Analyss of stok ontrol models for one loaton wth one produt 6 4. Analyss of the (s,q) model 4. Performane measurement wth P (SPP, pages 66-68) We start by defnng the performane measure P : () P : = probablty of no stok out just before the arrval of an order. If we onsder an order plaed at t = τ,.e. the frst order after t =, we an wrte { } () P = P X ( τ L ) +. Ths expresson s easy to understand: at tme t = ( + L ) before that arrval, the net stok s X ( τ ) τ the order arrves. Just + L, whereas P s the probablty that ths quantty s not negatve. Equaton () gves us an expresson for P. It s not very X τ. But useful yet, beause we do not know the probablty dstrbuton of ( + L ) we an rewrte ( + L ) ) dstrbuton of the demand durng the nterval ( ] ( ] X τ n terms of a known dstrbuton, namely the probablty t,t. That demand equals D t,t, wth t en t known onstants, and aordng to assumpton () t has a normal dstrbuton. 4. The probablty dstrbuton of X ( τ ) + L To derve the probablty densty funton of X τ + L we wll study an nventory system durng two replenshment yles. We wll start analysng the nventory system at a moment n tme at whh the frst replenshment order s beng plaed. Ths moment n tme s taken as the tme orgn. In other words, we start our tme-sale at ths pont n tme: tme. The other moments n tme, whh wll be used n our analyss, are: - L, the delvery tme of the order plaed at tme τ, the moment the seond replenshment order s plaed - - τ + L, the moment the seond replenshment order s delvered These moments n tme an be reognsed n Fgure, whh shows the net nventory X (t) as well as the nventory poston Y ( t) as a funton of tme for a (s,q)- ontrolled nventory system.

7 Analyss of stok ontrol models for one loaton wth one produt 7 Inventory s+ Q Net stok X t () Inventory poston () Y t X( τ + L ) s L L L τ τ + L B ( τ, τ + L ] X (( τ+ L) ) Tme Fgure - The nventory n a (s,q)-system as a funton of tme. In order to derve the probablty dstrbuton funton of X ( τ ) the general nventory balane equaton. For any (3) t X plus all orders arrvng n t t X = ( ] t + L we have:, we frst state ( ] t,t mnus all demand n t,t. Next we note that all orders plaed after t = wll arrve after tme t = ()). Hene: ( ] (4) all orders arrvng n, L = all orders outstandng at tme, and so L (assumpton ( (5) X L ) = plus all outstandng orders at tme mnus all demand n = Y D, L. X ( L ) s Q D (, L ] ( L X (, L ] ( ] Wth assumpton () ths yelds: (6a) = +. Ths s the nventory level after the arrval of the order quantty Q. So just before that arrval we have:

8 Analyss of stok ontrol models for one loaton wth one produt 8 (6b) X ( L ) = s D( L ] Intermezzo, Apparently, X ( L ) and ( L ). X have a normal dstrbuton f the demand D s normally (, ] dstrbuted. Then ther mean value and varane are easly derved from those of D L X τ + ) the stuaton s more omplated. ( L We return to the performane measure P. Between L and τ + L no orders arrve, beause of assumpton (). So we have: = X ( L ) D( L τ ] (7) ( + L ) X τ +., L Usng expresson (6a) for X ( L ) we get: (8) X ( τ ) = s + Q D(, L ] D( L τ + L ] + L, = s Q D(, + L ] + τ. Here a new dffulty arses: τ and L are random varables and so τ + L s a random varable too, and we do not know the probablty dstrbuton of D (, τ + L ]. Ths problem an be solved n the followng way. We rewrte D (, τ + L ]: (9) D (, + L ] = D(, τ ] + D( τ τ + L ] τ., The frst term on the rght hand sde s the demand durng a replenshment yle: D = Y () Y τ ). () (, τ ] ( Assumpton () tells us that Y = s+ Q and Y ( τ ) = s () (, τ ] D = Q. Substtuton of (9) and () nto (8) yelds: = s D( τ τ ] () ( + L ) X τ +., L Returnng to () and substtutng () we obtan: { } (3) P = (, ] P D τ τ + L s., and so Note that ths formula has been derved wthout any assumpton on the number of outstandng orders at the moment an order s plaed,.e. at n τ, wth n =,,... In ase there s no outstandng order at the moment an order s plaed, equaton (3) an easly be derved from a smple drawng: see Fgure.. For

9 Analyss of stok ontrol models for one loaton wth one produt 9 s+q Inventory s L. µ p.d.f. of demand durng L perods L L τ L τ + L v Tme out-of-stok probablty (= P ) F gure The nventory level durng the nterval from t = τ untl t = τ + L. The serve measure P s equal to the probablty that deman d D( τ, τ + L) does not exeed the reorder level s. More llustratons, showng how nventory level and number of outstandng orders, develop n tme an be found n Appendes A and B. 4.3 Calulatons for demand wth a normal dstrbuton T he reorder level s For onvenene we now drop the ndex of τ and L. Aeptng assumptons () and D τ, τ + L has a normal dstrbuton wth mean Lµ and varane (v) we know that ( ] Lσ. Therefore we an wrte (3) as: ( ] D ττ, + L Lµ s Lµ (4) P = P σ L σ L s Lµ = Φ, σ L w th Φ(.) the standard-normal probablty dstrbuton funton. Intermezzo The funton s related to expressons used by SPP by a smple equaton: Φ (k) : = ( k). If we requre that P = α, then we have p u

10 Analyss of stok ontrol models for one loaton wth one produt s Lµ (5) Φ = α. σ L If we wrte ths as Φ( k ) α α =, we fnd s Lµ (6) = kα. σ L Ths leads us an expresson for the reorder level s (.f. SPP, page 55): (7) s = L + k σ L. µ α So f P = α s gven, we an determne the orrespondng value for k α. Ths number follows from α = Φ k ), by usng a table or a omputer program suh as Exel, n ( α whh the funton Φ s avalable. Fnally, f L, µ and σ are known, we an substtute all these values nto (7) and obtan a value for the reorder level s. The safety stok We defne the safety stok ν as the expeted net stok just before the arrval of an order. We have: (8) ν : = E X ( τ + L) Usng () we get: or. ν = E s D( ττ, + L] = s E D( ττ, + L] (9) ν = s Lµ = kασ L. Beause of ths result, k α s alled the safety fator. Numeral example (SPP, page 67) Suppose we know that µ L = 58.3 unts and σ L = 3. unts. We requre a performane level P =.9. From a table for k α Φ, or one for pu k k α = Φ, we fnd k α =.8 so that ν =.8x3. = unts and s = = unts.

11 Analyss of stok ontrol models for one loaton wth one produt Average nventory Consder the order plaed at tme and the next one, plaed at tme τ. They wll arrve respetvely at t = L and t = τ + L. In between no orders arrve, beause of assumpton (). So for the average nventory durng the replenshment yle we an wrte: ( ) () E[ X] = E[ X( L )] + E X ( τ + L ). We substtute (6a) and () nto () and fnd: ( ) () E[ X] = s+ Q E D(, L ] + s E D( τ, τ + L ] = Q + s µ L. Usng (9) we get: () [ ] E X = Q+ k α σ L. So on average the nventory equals Q plus the safety stok ν. Numeral example (SPP, page 67), ontnued Suppose agan that µ L = 58.3 unts, σ L = 3. unts, and the requred performane level P =.9. Then E[ X] = Q+ 7 unts. 4.4 Performane measurement wth P (SSP, page 68-69) We start wth two defntons: (3) P : = fraton of demand satsfed dretly from the shelf, and (4) P : = fraton of demand delvered as a bakorder. Note the dfferene between E[ X] and E X ( t) ; E X ( t) partular perod t, whle E[ X] s the average net stok durng the entre replenshment yle. s the expeted net stok n a

12 Analyss of stok ontrol models for one loaton wth one produt In (3) and (4) the word fraton refers to the long-term behavour of the stohast proesses nvolved,.e. the demand proess and the net nventory poston. Mathematally we have then: (5) P = (, t] ( ] t B lm. t D, Sne we are onsderng nfnte tme, B (, L ] and (, L ] long-term fraton. So we an also wrte: D are not relevant for the (6) P = B lm D ( L, t] ( L t] t,. Now we take t very large and equal to τ + L wth N very large too. Then we get: (7) B D wth τ = ( L, t] ( L, t] B ( L, τ N + LN ] ( L, τ + L ] = = = N D N N N = B D N N ( τ + L, τ + L ] ( τ + L, τ + L ]. Substtuton of (7) nto (6) gves, (8) P = lm N lm = lm N = N N N = N N B D B D ( τ + L, τ + L ] ( τ + L, τ + L ] ( τ + L, τ + L ] ( τ + L, τ + L ]. We are allowed to make ths step, beause the seres n (8) are bounded. P an be rewrtten as: N lm B( τ + L, τ + L ] N N = (9) P =. N lm D( τ + L, τ + L ] N N The argument ( τ L, τ + L ] th + represents the replenshment yle after t =. All replenshment yles are stohastally dental. They start wth the arrval of a replenshment quantty Q. Just before that arrval the net stok equals X τ L = ( τ L ] ( ) s D τ, +, n aordane wth equaton (6b) f = and wth () f =. X denotes the stok mmedately after the arrval of the replenshment quantty Q. So the yle begns wth a net stok of sze +

13 Analyss of stok ontrol models for one loaton wth one produt 3 X ( τ + L ) = s + Q D( τ, τ + L ] X ( + L ) = s D τ τ + L ( ], and ends wth a net stok of sze τ,. Aordng to the Law of Large Numbers, f Z are dentally dstrbuted stohast varables, then N lm Z N N = [ ] = E Z So we have E B( L, τ+ L] (3) P =. E D( L, τ + L] Ths means that (3) P = (the expeted quantty baklogged n a replenshment yle) (the expeted demand n a replenshment yle) It s easy to see that: (3) E D( L, τ+ L] = E D (, τ ] ( ] + L D, L Usng assumpton () we an now wrte: (33) E [ D( L, τ + L ] = E D(, τ ] = E D( (, ] + D(, + L ] D(, L ] τ τ τ = E D(, ] E D(, L ] E D(, L ] τ + τ τ+. = Q. Note that ths result has a general valdty, beause (33) desrbes an nput-output balane equaton: the average demand durng a replenshment yle should equal the average amount replenshed durng a replenshment yle. Fnally, we have to fnd an expresson for E B( L, τ+ L], the expeted quantty n baklog durng the nterval ( L, τ + L ]. We onsder three stuatons, namely: () X ( τ + L ) ) ( () () X ( τ + L ) ) and X ( L ) ( < X ( τ + L ) ) ( < and X L <.

14 Analyss of stok ontrol models for one loaton wth one produt 4 Note that at tme L we onsder the net stok after arrval of the order plaed at t =, whereas at tme τ + L the order plaed at t = τ has not yet arrved. In stuaton () there s no baklog, n stuaton () there s a baklog of sze X (( τ + L ) ), and n stuaton () all demand n ( L, τ + L ] s baklogged,.e. D ( L, τ + L ]. These three stuatons an be expressed n one formula: (34) ( L, + L ] = ( D( τ, τ + L ] s) + ( D( L ] ( s + Q) ) + τ. B, Intermezzo: Before provng ths formula, we generalse t (ths generalsed result wll be used n paragraph 5). To do so, we opy two formulae from Seton 4.: = s D( τ τ ] () ( + L ) X τ +, (, L (6a) L = s + Q D, L. X ) ( ] Wth these formulae, we an also wrte (34) as (34b) (, τ ] ( τ ) + ( ) B L + L = X + L X L We proof formula (34) by usng equaton () and (6a), mentoned n the Intermezzo above, and by lookng at eah of the three stuatons. Stuaton (): X ( τ + L ) ) Sne ( L ) X X ( τ + L ) ( (35a) D (, L ] s + Q (35b) D( L ] s and so τ,,τ + + (36a) ( D(, L ] ( s + Q) ) = + (36b) ( D( τ τ + L ] s)., =, we have Formula (34) then yelds ( L + L ] B, τ =, as t should beause n stuaton () there s no baklog. Conluson: formula (34) orret for Stuaton (). Stuaton (): X ( τ + L ) < and X ( L ) In equaton (34) we substtute () and (6a). Then we get: (37) B ( L τ L ] = X ( τ + L ) + ( X + +, L +

15 Analyss of stok ontrol models for one loaton wth one produt 5 As X (( τ + L ) ) < we have X ( τ + L ) follows from X that ( ) ( L ) (38) B ( L + L ] = X ( τ + ) τ., L + ( = X ( τ + ) ) L + X L =. And so. In the same way t Ths ndeed s the baklog for stuaton (). Conluson: equaton (34) s orret for stuaton (). Stuaton (): X ( τ + L ) < and X ( L ) <. We return to (37). Now ths an be wrtten as: (39) B ( L, τ + L ] = X ( τ + L ) X L = X ( L ) X ( τ + ) L = D ( L, + L ] τ. In stuaton () all demand n ( L τ + ] s baklogged,.e. equal to D ( L + ], L Conluson: equaton (34) s orret for stuaton (). τ., L After ths analyss, we an state that (34) s an expresson for the baklog n ( L, τ + L ] vald n all relevant stuatons. Now we an return to (3), the expresson for P. Substtuton of (33) and (34) gves: + + ( E ( D L s) E ( D( L] ( s Q) ) ), +, + Q (4) P = ( τ τ ] Wth ths expresson we an alulate P, f we know s, Q and the dstrbuton of two stohast varables: D (, L ] and D ( τ, τ + L ]. These varables are dstrbuted dentally. Now usng assumpton (v),.e. the net nventory after arrval of an order s postve, we have (4) P { (, L ] s + Q} = D. Therefore, the seond expetaton n (4) s zero and so + E ( D L s). Q (4) P = ( τ, τ + ]. 4.5 Calulatons for demand wth a normal dstrbuton Wth assumptons () and (v), D ( τ + ] τ s normally dstrbuted wth mean µ L and, L varane σ L. So t makes sense to modfy P nto:

16 Analyss of stok ontrol models for one loaton wth one produt 6 ( ] σ L D τ, τ + L µ L s µ L (43) P = E Q σ L σ L + σ L s µ L = E Z. Q σ L + Here Z has a standard-normal dstrbuton. Therefore, Z s onneted wth two probablty funtons: (44) Φ( x) : = P{ Z x} and (45) k : = = π x exp G E ( Z k ) + = ( y k) y exp dy π k y dy. Next we defne a quantty k β : (46) k β : s µ L =. σ L Then, also usng (45), we an wrte (43) as: σ L (47) P = G( k β ). Q If we are lookng for a value of s suh that P = β, then σ L (48) β = G( k ) or Q (49) ( β ) G k β β Q =. σ L Now, f β, µ, σ, L and Q are known, we an alulate the rght-hand sde of (49). It s a numeral value, say C. Then we have G ( k β ) = C and from a table we obtan a numeral value for k β. Fnally, wth (46) we get the re-order level s: (5) s = L + k σ L. µ β

17 Analyss of stok ontrol models for one loaton wth one produt 7 There s an mportant dfferene wth the safety fator k α for the P measure: depends for a gven β not only on L, µ and σ, but also on Q. If Q nreases, k β dereases and so do s and the safety stok. Numeral example Lke n Seton 4.3 we suppose that µ L = 58.3 unts and σ L = 3. unts. We requre a performane level P =.9 and we take Q = pees. Then (48) gves G = k β From a table for k β G we fnd by nterpolaton =.45, so that s = = unts. Apparently, n ths stuaton the P-measure produes values for s and the safety stok slghtly lower than wth the P -measure. Numeral example (SPP, page 69) Now we have µ L = 5 gallons and σ L =.4 gallons. Management requres a P - =.75. From the table level of.99, whereas Q = gallons. Now (49) gves G ( k β ) we obtan =.58 so that s = (.4) = gallons. k β k β k β

18 Analyss of stok ontrol models for one loaton wth one produt 8 5. Elmnaton of assumptons In the prevous hapters, our analyss was based on fve assumptons, lsted n Chapter 3. They made t easy to obtan results that have proven to be useful n pratal stuatons untl 98. However, owng to hangng ondtons, n partular smaller order quanttes and hgher demand varablty than n the past, the assumptons tend to lose ther valdty n modern prate. In ths hapter we wll remove them one after the other, n order to obtan results wth a broader valdty than those of SPP. 5. Assumpton () Assumpton () says that demand has a normal dstrbuton. It an easly be elmnated, for nstane by assumng that D (, L ] and D ( τ, τ + L ] have the same gamma dstrbuton. In many ases ths turns out to be realst. Then the problem s redued E D, L σ D, L, beause they to fndng an expresson for ( ] and one for ( ] ( ) determne the parameters of the orret gamma dstrbuton. Suh expressons depend on the assumptons made about the demand proess. In prate we an estmate the E D, L σ D, L by measurng demand durng value of the varables ( ] and ( ] the delvery tmes of orders and subsequently alulatng mean value and varane of the sample data. Ths approah has an mportant onsequene: the expressons are only vald for the measured delvery tmes, so that these measurements lose ther value as soon as the measurements hange. We return to ths subjet n Seton 5.4, where we elaborate the expressons for E D(, L ] and D (, L ] σ under dfferent assumptons on the demand proess and lead tme dstrbutons. 5. Assumpton () Aordng to assumpton (), the stok poston at the moment of orderng s exatly equal to s. Ths assumpton s only vald f all ustomers order the same quantty, say unts. Then t s obvous to take Q as a multple of, so that at the moment of orderng the stok poston ndeed always equals s. In all more realst ases, however, the stok poston at the moment of orderng wll be s U, wth U a nonnegatve stohast varable alled the undershoot. See Fgure 3.

19 Analyss of stok ontrol models for one loaton wth one produt 9 s+ Q Net stok X t () Inventory poston Y t () Inventory X( τ + L ) s undershoot U L τ τ + L X (( τ + L) ) Tme Fgure 3 - The undershoot D If demand per ustomer s equal to we an fnd good approxmatons for EU [ ] and E U, the mean value and the varane of the undershoot (see Tjms [994]): (5) EU [ ] σ D E D + E D (5) E U ( D ) 3 E 3E D. In the formulae derved n the prevous setons, we now need to aount for the fat that the stok poston equals s U at the moment of orderng. We an do so by means of the relatons (f. (6a) and ()): (5a) X ( L ) s + Q U D( L ] (5b) ( τ ) =,, ( + ) = ( τ, +L] X L s U D τ, where U and U are the undershoots at tmes and τ. After substtuton of these relatons the analyss an proeed n the same way as before. It starts by makng an D, L + U, and takng the same assumpton about the probablty dstrbuton for ( ] one for D( τ, τ + L ] + U. For a normal dstrbuton of D,.e. D N ( µ, σ ) get, wth v = σ µ the oeffent of varaton: we

20 Analyss of stok ontrol models for one loaton wth one produt EU v µ 3 E U = + v 3 µ. (53a) [ ] = ( + ) (53b) However, a gamma dstrbuton s preferable beause t s more realst. So f wth ( λ ) Γ( α ) x α α exp x λ x P{ D x} = dx, E D α σ α = and λ =, ( D ) E D equatons (5) and (5) produe: (54a) ( α + ) EU [ ] = λ (54b) ( α + )( α + ) E U =. 3λ One we have determned the frst two moments for U, we know the mean and varane of (note that var U = E U E U ). Sne we also know the mean U and varane of D(, L] the mean and varane of (, ] τ τ +, we an add them to the mean and varane of to get D τ τ + L + U. Next, we hoose whh pdf (normal, gamma or...) we onsder most approprate for D, L U D τ, τ + L + U. ( τ τ + ] + and we ft ths pdf to the mean and varane of ( ] U Next, we have to ombne the general formulas for P (formula ()), the safety stok (8), the average nventory (), the bakorders (34) and P (4) wth formulas (5a) and (5b) to take nto aount the undershoot. Ths results n the followng formulas: { } ( ' ) P = P D( τ, τ+ L] + U s ( 8' ) v= s E D( τ, τ + L ] E[ U ] Q ( ' ) E[ X] = s+ E D( τ, τ + L ] E[ U ] ( 34b' ) B( L, τ+ L] = ( D( τ, τ+ L] + Us) ( D(, L] + U sq) ( τ τ ] ' P = E ( D, + L + Us) E ( D(, L] + U sq) Q Gven the pdf for (, ] relevant logsts varables. D τ τ + L + U, these formulas enable us to determne all

21 Analyss of stok ontrol models for one loaton wth one produt Numeral analyss has revealed that t s ndeed rual to take U nto aount. See, for nstane, the Exel spreadsheet Classal Inventory Models (De Kok []). 5.3 Assumpton () Orders annot overtake eah other, aordng to assumpton (). Elmnaton of ths assumpton s not sensble n the framework of our present model. So assumpton () has a general valdty. Indeed, we order one produt at one suppler. There s no reason whatsoever for the suppler to hange the sequene n whh he arres out the delveres of the same produt for the same ustomer. At most wll the suppler ombne orders to mprove effeny durng produton or transport. 5.4 Assumpton (v) Ths assumpton says that delvery tmes are onstant and equal to L. If we assume furthermore that L equals an nteger number of perods, say K, we an fnd expressons for E D, L σ D, L n the followng way. Defne: ( ] and ( ] (55) D : = demand n perod k, k ( ] K D k k = then D, L =. Note that L has the dmenson tme whereas K s a dmensonless number. Suppose that { D k } dstrbuted stohast varables. Then: (56a) E D(, L] = K E[ D] (56b) σ (, ], ( D L ) Kσ ( D) =. For onvenene we defne (57a) E [ D(, L ] = L E[D], (57b) ( D(, L] ) L σ (D Intermezzo L σ = ). are mutually ndependent and dentally too as a number of perods. Then In ase assumpton () s true, demand has a normal dstrbuton wth [ ] σ ( D) = σ, so that equatons (57) hange nto: (, ] ( (, ]) E D L = L µ σ D L = L σ. L = K and E D µ = and

22 Analyss of stok ontrol models for one loaton wth one produt Next we drop the assumpton that L s onstant, but we ontnue to assume that t equals an nteger number of perods, say K. Then we an wrte (58) E D(, L] = K E D k k = n = E D P{ K = n} n= k= k = ne [ D] P{ K = n} n= = E [ D] np{ K = n} n= = E[ D] E[ K ] and [ ] (59) E D (, L] = K E k = D k n = E D P{ K = n} n= k= k n n k k n= k= k= = σ D + E D P{ K = n} ( n D + n E D ) P{ K = n} = σ [ ] n= = σ ( D) n P{ K n} + E [ D] n P{ K = n} = n= n= D E K + E D E K. = σ [ ] [ ] Sne ( (, L] ) = E D (, L] [ ] E D( L] [ ] σ D, we fnd after substtuton of (57) and (58): (6) ( D (, L] ) σ = E[ K ] σ ( D) + E[ K ] E [ D] E [ K ] E [ D] = E[ K] σ ( D) σ [ K] E [ D] +.

23 Analyss of stok ontrol models for one loaton wth one produt 3 Ths formula an also be found n SPP, on page 83, whereby K has been replaed by L. So n order to keep dmensons orret, t s essental to nterpret L as a number of perods and not as delvery tme. In ase of stohast delvery tmes, we frst have to alulate E [ D(, L ] and σ ( D (, L] ). Next we have to make an assumpton about the probablty dstrbuton of D (, L]. For normal dstrbutons and usng the P measure, we obtan for the re-order level (6) s E[ D(, L ] + kσ ( D(, L] ) =. Instead of departng from nformaton about demand per perod, we also an base our analyss on D, demand per ustomer, and A, the tme between the arrvals of ustomers. An often used assumpton s P{ A x} = e λx, x. Ths means that A has an exponental dstrbuton. That s equal to sayng that the arrval proess of ustomers s a Posson proess. For suh an arrval proess, the followng expressons for E D(, L] and σ D (, L ] an be derved: = λ (6a) (, ] [ ] E D L E L E D (6b) (, ] [ ] D L = E L E D + L E D σ λ λ σ. It should be noted that these expressons have to be ombned wth formulae for EU [ ] and E U. 5.5 Assumpton (v) The net nventory after arrval of an order s postve, aordng to ths assumpton. Q>> D, L. It only regards the expresson for P. But an Ths s realst f ( ] expresson for P has been derved already n Seton 4.4: (4) P = ( τ τ ] + + ( E ( D L s) E ( D( L] ( s Q) ) ), +, + Q Wth formula (4), P an be determned numerally under the assumpton that D ( τ, τ + L ] and D(, L ] have a normal or gamma dstrbuton. Calulaton of the reorder pont s for gven P = β s also done numerally, by means of bseton or a smlar method. Here we use the fat that P s strtly asendng n s. So assumpton (v) s only needed n order to use tables for the alulaton of the reorder pont. It should be emphaszed here that assumpton (v) s nowadays not vald due to the reduton of the replenshment bath sze Q. It an easly be seen that formula (4) beomes negatve as Q dereases, whh of ourse should not be the ase for a vald expresson for a serve measure lke P..

24 Analyss of stok ontrol models for one loaton wth one produt 4 Agan we are able to take the undershoot nto aount n a smple way through the fat that the undershoot s ndependent of the tme durng the subsequent lead tme. We only have to add the mean and varane of the undershoot to the mean and varane, respetvely, of the demand durng the lead tme. Thereafter the expressons (4) and (4) an be appled.

25 Analyss of stok ontrol models for one loaton wth one produt 5 6. Analyss of the (R,S) model 6. The P measure The ( R, S ) model mples that after eah perod we reorder suh a quantty that the nventory poston beomes S. Then we fnd, analogous to the analyss of the (s,q) model (see Seton 4., equaton ()) that { } (63) P = P X ( τ L ) But now we have τ = R and +. (64a) X ( L) = S D(, L] (64b) X ( R L ) S D(, R L ] so that + = +, { } (65) P = (, ] P D R+ L S. Wth assumptons () and (v) we fnd that S = µ R+ L + kσ R+L. (66) In general we have: (67) S = E D(, R+ L ] + kσ D(, R+ L ] If L= K*,.e. f L equals an nteger and possbly stohast number of perods, we get for demand that s mutually ndependent and dentally dstrbuted: (68a) E D(, R+ L ] = ( R+ E[ K] ) E[ D] (68b) ( ] ( D R+ L ) = ( E[ K] + R) ( D) + ( K) E [ D] σ σ σ,, where R s the number of perods that expresses the length of the revew perod. Lke before, we an replae K by L, f we gnore the dmenson of L.

26 Analyss of stok ontrol models for one loaton wth one produt 6 6. The P measure In the same way as for the P measure, and usng the expressons (64), we fnd + + (69) P = E ( D( R + L ] S ) E ( D, L S ). E D, R [ ( ] ( [ ] [ ( ] ]) The problem now s that assumpton (v) s mostly not vald any more, unless R >> L or S >> D(, L ]. In many ases ths s not true. Ths means that n fat we annot use the approxmaton that follows from SPP, equaton (7.4), whh gnores the last expetaton n (69) and yelds: + [ ]. (7) P E ( D[ R + L ] S ) E [ D ( R ], Nevertheless, f we do gnore ths restrton, then the assumptons () and (v) lead to σ L+ R S µ L+ R (7) P = G. µ R σ L+ R If we use ths formula to fnd S for a gven β va (7) G( k) ( ) µ R β =, σ L + R we get a value for S that s too hgh. It s also possble that we do not fnd a soluton at all, sne (7) beomes negatve for small values of R. One more we an say that the problem to use the orret expressons s only of a numeral nature. Suh problems an easly be solved by omputer software. We refer to the aforementoned Exel spreadsheet Classal Inventory Models (De Kok []). If we apply assumptons () and (v) to the orret formula (69), we obtan S L+ R µ S Lµ (73) P = σ L+ RG σ LG Rµ. σ L+ R σ L It s lear that formula (73) an easly be appled usng the tables for the funton... n SPP f S s gven. The opposte,.e. omputng G. gven a target value for G P, requres a omputerzed algorthm lke bseton. Fnally we note that assumpton () s not relevant for the (, ) RS model.

27 Analyss of stok ontrol models for one loaton wth one produt 7 7. Essental elements from the analyses of the (s,s), (R,s,Q) and (R,s,S) models The essene of the analyss s ontaned n the expressons for X ( L ) and ( X τ + L ). Gven these expressons and e.g. assumptons () and (v), we an derve expressons for P, P and [ ] E X. Ths approah has been mplemented n the spreadsheet Classal Inventory Models (De Kok []). The (s,s) model For ths model we have,.f. equatons (64a) and (5b): ( ] (74a) X L = S D L,, (74b) X ( + L ) = s U D( τ τ + ] τ,, L where U s the undershoot of the reorder level s. The (R,s,Q) model Smlarly,.f. equaton (5): (75a) X ( L ) s + Q U D( L ], R, =, (75b) X ( + L ) = s U R D( τ τ + ] τ.,, L The(R,s,S) model Fnally, for ths model we have: ( ] (76a) X L = S D L,, (76b) X ( + L ) = s U R D( τ τ + ] τ.,, L In expressons (75) and (76), and U are the undershoots n the perod U,R,R reorder models that are derved from the demands per revew perod. Under assumpton (),.e. demand durng ntervals s normally dstrbuted, we fnd: E U = + Rµ, (77a) [ R ] (77b) 3 E U = + 3 R R R µ. Agan a gamma dstrbuton s preferable beause t s more realst. So f

28 Analyss of stok ontrol models for one loaton wth one produt 8 wth ( λx) Γ( α ) x α α exp λ x P{ D(, R] x} = dx, (, ] (, R] E D R α σ = and λ = ( D ) α E D (, R], we fnd (78a) [ ] (78b) ( α + ) EU R = λ ( α + )( α + ) E U R =. 3λ

29 Analyss of stok ontrol models for one loaton wth one produt 9 Referenes Tjms, H.C., 994, Stohast Models: An Algorthm Approah, Wley, Chhester. De Kok, A.G.,, Classal Inventory Models: Student Verson, Exel spreadsheet, Studyweb, C3.

30 Analyss of stok ontrol models for one loaton wth one produt 3 Appendx A Sample paths for the ase of a sngle outstandng order R E[D] σ(d) 5 E[L] 4 Q s Y(t) X(t) s Number of outstandng orders 3 3 4

31 Analyss of stok ontrol models for one loaton wth one produt 3 Appendx B Sample paths for the ase of multple outstandng orders R E[D] σ(d) 5 E[L] 4 Q 5 s Y(t) X(t) s Number of outstandng orders 3 3 4