technische universiteit eindhoven Analysis of one product /one location inventory control models prof.dr. A.G. de Kok 1
|
|
- Earl Reed
- 5 years ago
- Views:
Transcription
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
JSM Survey Research Methods Section. Is it MAR or NMAR? Michail Sverchkov
JSM 2013 - Survey Researh Methods Seton Is t MAR or NMAR? Mhal Sverhkov Bureau of Labor Statsts 2 Massahusetts Avenue, NE, Sute 1950, Washngton, DC. 20212, Sverhkov.Mhael@bls.gov Abstrat Most methods that
More informationA Simple Inventory System
A Smple Inventory System Lawrence M. Leems and Stephen K. Park, Dscrete-Event Smulaton: A Frst Course, Prentce Hall, 2006 Hu Chen Computer Scence Vrgna State Unversty Petersburg, Vrgna February 8, 2017
More informationPHYSICS 212 MIDTERM II 19 February 2003
PHYSICS 1 MIDERM II 19 Feruary 003 Exam s losed ook, losed notes. Use only your formula sheet. Wrte all work and answers n exam ooklets. he aks of pages wll not e graded unless you so request on the front
More informationSTK4900/ Lecture 4 Program. Counterfactuals and causal effects. Example (cf. practical exercise 10)
STK4900/9900 - Leture 4 Program 1. Counterfatuals and ausal effets 2. Confoundng 3. Interaton 4. More on ANOVA Setons 4.1, 4.4, 4.6 Supplementary materal on ANOVA Example (f. pratal exerse 10) How does
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationThe corresponding link function is the complementary log-log link The logistic model is comparable with the probit model if
SK300 and SK400 Lnk funtons for bnomal GLMs Autumn 08 We motvate the dsusson by the beetle eample GLMs for bnomal and multnomal data Covers the followng materal from hapters 5 and 6: Seton 5.6., 5.6.3,
More informationCharged Particle in a Magnetic Field
Charged Partle n a Magnet Feld Mhael Fowler 1/16/08 Introduton Classall, the fore on a harged partle n eletr and magnet felds s gven b the Lorentz fore law: v B F = q E+ Ths velot-dependent fore s qute
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationBrander and Lewis (1986) Link the relationship between financial and product sides of a firm.
Brander and Lews (1986) Lnk the relatonshp between fnanal and produt sdes of a frm. The way a frm fnanes ts nvestment: (1) Debt: Borrowng from banks, n bond market, et. Debt holders have prorty over a
More informationMachine Learning: and 15781, 2003 Assignment 4
ahne Learnng: 070 and 578, 003 Assgnment 4. VC Dmenson 30 onts Consder the spae of nstane X orrespondng to all ponts n the D x, plane. Gve the VC dmenson of the followng hpothess spaes. No explanaton requred.
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More informationController Design for Networked Control Systems in Multiple-packet Transmission with Random Delays
Appled Mehans and Materals Onlne: 03-0- ISSN: 66-748, Vols. 78-80, pp 60-604 do:0.408/www.sentf.net/amm.78-80.60 03 rans eh Publatons, Swtzerland H Controller Desgn for Networed Control Systems n Multple-paet
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationQueueing Networks II Network Performance
Queueng Networks II Network Performance Davd Tpper Assocate Professor Graduate Telecommuncatons and Networkng Program Unversty of Pttsburgh Sldes 6 Networks of Queues Many communcaton systems must be modeled
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More informationEquilibrium Analysis of the M/G/1 Queue
Eulbrum nalyss of the M/G/ Queue Copyrght, Sanay K. ose. Mean nalyss usng Resdual Lfe rguments Secton 3.. nalyss usng an Imbedded Marov Chan pproach Secton 3. 3. Method of Supplementary Varables done later!
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationComplement of an Extended Fuzzy Set
Internatonal Journal of Computer pplatons (0975 8887) Complement of an Extended Fuzzy Set Trdv Jyot Neog Researh Sholar epartment of Mathemats CMJ Unversty, Shllong, Meghalaya usmanta Kumar Sut ssstant
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationExact Inference: Introduction. Exact Inference: Introduction. Exact Inference: Introduction. Exact Inference: Introduction.
Exat nferene: ntroduton Exat nferene: ntroduton Usng a ayesan network to ompute probabltes s alled nferene n general nferene nvolves queres of the form: E=e E = The evdene varables = The query varables
More informationOrder Full Rate, Leadtime Variability, and Advance Demand Information in an Assemble- To-Order System
Order Full Rate, Leadtme Varablty, and Advance Demand Informaton n an Assemble- To-Order System by Lu, Song, and Yao (2002) Presented by Png Xu Ths summary presentaton s based on: Lu, Yngdong, and Jng-Sheng
More informationNewsvendor Bounds and Heuristics for Serial Supply Chains with Regular and Expedited Shipping
Newsvendor Bounds and Heursts for Seral Supply Chans wth egular and xpedted Shppng Sean X. Zhou, Xul Chao 2 Department of Systems ngneerng and ngneerng Management, The Chnese Unversty of Hong Kong, Shatn,
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationLab 2e Thermal System Response and Effective Heat Transfer Coefficient
58:080 Expermental Engneerng 1 OBJECTIVE Lab 2e Thermal System Response and Effectve Heat Transfer Coeffcent Warnng: though the experment has educatonal objectves (to learn about bolng heat transfer, etc.),
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationIndeterminate pin-jointed frames (trusses)
Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationAn (almost) unbiased estimator for the S-Gini index
An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for
More informationPhase Transition in Collective Motion
Phase Transton n Colletve Moton Hefe Hu May 4, 2008 Abstrat There has been a hgh nterest n studyng the olletve behavor of organsms n reent years. When the densty of lvng systems s nreased, a phase transton
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationVoltammetry. Bulk electrolysis: relatively large electrodes (on the order of cm 2 ) Voltammetry:
Voltammetry varety of eletroanalytal methods rely on the applaton of a potental funton to an eletrode wth the measurement of the resultng urrent n the ell. In ontrast wth bul eletrolyss methods, the objetve
More informationProbability and Random Variable Primer
B. Maddah ENMG 622 Smulaton 2/22/ Probablty and Random Varable Prmer Sample space and Events Suppose that an eperment wth an uncertan outcome s performed (e.g., rollng a de). Whle the outcome of the eperment
More informationMAE140 - Linear Circuits - Fall 13 Midterm, October 31
Instructons ME140 - Lnear Crcuts - Fall 13 Mdterm, October 31 () Ths exam s open book. You may use whatever wrtten materals you choose, ncludng your class notes and textbook. You may use a hand calculator
More informationApplied Stochastic Processes
STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationHorizontal mergers for buyer power. Abstract
Horzontal mergers for buyer power Ramon Faul-Oller Unverstat d'alaant Llus Bru Unverstat de les Illes Balears Abstrat Salant et al. (1983) showed n a Cournot settng that horzontal mergers are unproftable
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationSELECTED PROOFS. DeMorgan s formulas: The first one is clear from Venn diagram, or the following truth table:
SELECTED PROOFS DeMorgan s formulas: The frst one s clear from Venn dagram, or the followng truth table: A B A B A B Ā B Ā B T T T F F F F T F T F F T F F T T F T F F F F F T T T T The second one can be
More informationExercise 10: Theory of mass transfer coefficient at boundary
Partle Tehnology Laboratory Prof. Sotrs E. Pratsns Sonneggstrasse, ML F, ETH Zentrum Tel.: +--6 5 http://www.ptl.ethz.h 5-97- U Stoffaustaush HS 7 Exerse : Theory of mass transfer oeffent at boundary Chapter,
More information11 Tail Inequalities Markov s Inequality. Lecture 11: Tail Inequalities [Fa 13]
Algorthms Lecture 11: Tal Inequaltes [Fa 13] If you hold a cat by the tal you learn thngs you cannot learn any other way. Mark Twan 11 Tal Inequaltes The smple recursve structure of skp lsts made t relatvely
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationExercises of Chapter 2
Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard
More informationAPPENDIX 2 FITTING A STRAIGHT LINE TO OBSERVATIONS
Unversty of Oulu Student Laboratory n Physcs Laboratory Exercses n Physcs 1 1 APPEDIX FITTIG A STRAIGHT LIE TO OBSERVATIOS In the physcal measurements we often make a seres of measurements of the dependent
More informationSome Results on the Counterfeit Coins Problem. Li An-Ping. Beijing , P.R.China Abstract
Some Results on the Counterfet Cons Problem L An-Png Bejng 100085, P.R.Chna apl0001@sna.om Abstrat We wll present some results on the ounterfet ons problem n the ase of mult-sets. Keywords: ombnatoral
More informationThermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)
hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In
More informationOnline Appendix. t=1 (p t w)q t. Then the first order condition shows that
Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationSTOCHASTIC INVENTORY MODELS INVOLVING VARIABLE LEAD TIME WITH A SERVICE LEVEL CONSTRAINT * Liang-Yuh OUYANG, Bor-Ren CHUANG 1.
Yugoslav Journal of Operatons Research 10 (000), Number 1, 81-98 STOCHASTIC INVENTORY MODELS INVOLVING VARIABLE LEAD TIME WITH A SERVICE LEVEL CONSTRAINT Lang-Yuh OUYANG, Bor-Ren CHUANG Department of Management
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationMAE140 - Linear Circuits - Winter 16 Final, March 16, 2016
ME140 - Lnear rcuts - Wnter 16 Fnal, March 16, 2016 Instructons () The exam s open book. You may use your class notes and textbook. You may use a hand calculator wth no communcaton capabltes. () You have
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationGeneral Tips on How to Do Well in Physics Exams. 1. Establish a good habit in keeping track of your steps. For example, when you use the equation
General Tps on How to Do Well n Physcs Exams 1. Establsh a good habt n keepng track o your steps. For example when you use the equaton 1 1 1 + = d d to solve or d o you should rst rewrte t as 1 1 1 = d
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationPhysics 2B Chapter 17 Notes - Calorimetry Spring 2018
Physs 2B Chapter 17 Notes - Calormetry Sprng 2018 hermal Energy and Heat Heat Capaty and Spe Heat Capaty Phase Change and Latent Heat Rules or Calormetry Problems hermal Energy and Heat Calormetry lterally
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More informationLecture 6/7 (February 10/12, 2014) DIRAC EQUATION. The non-relativistic Schrödinger equation was obtained by noting that the Hamiltonian 2
P470 Lecture 6/7 (February 10/1, 014) DIRAC EQUATION The non-relatvstc Schrödnger equaton was obtaned by notng that the Hamltonan H = P (1) m can be transformed nto an operator form wth the substtutons
More information18.1 Introduction and Recap
CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng
More informationNotes prepared by Prof Mrs) M.J. Gholba Class M.Sc Part(I) Information Technology
Inverse transformatons Generaton of random observatons from gven dstrbutons Assume that random numbers,,, are readly avalable, where each tself s a random varable whch s unformly dstrbuted over the range(,).
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationLecture 2: Gram-Schmidt Vectors and the LLL Algorithm
NYU, Fall 2016 Lattces Mn Course Lecture 2: Gram-Schmdt Vectors and the LLL Algorthm Lecturer: Noah Stephens-Davdowtz 2.1 The Shortest Vector Problem In our last lecture, we consdered short solutons to
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationA NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011
A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,
More informationLinear Feature Engineering 11
Lnear Feature Engneerng 11 2 Least-Squares 2.1 Smple least-squares Consder the followng dataset. We have a bunch of nputs x and correspondng outputs y. The partcular values n ths dataset are x y 0.23 0.19
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationLecture 4. Instructor: Haipeng Luo
Lecture 4 Instructor: Hapeng Luo In the followng lectures, we focus on the expert problem and study more adaptve algorthms. Although Hedge s proven to be worst-case optmal, one may wonder how well t would
More informationIntroduction to Continuous-Time Markov Chains and Queueing Theory
Introducton to Contnuous-Tme Markov Chans and Queueng Theory From DTMC to CTMC p p 1 p 12 1 2 k-1 k p k-1,k p k-1,k k+1 p 1 p 21 p k,k-1 p k,k-1 DTMC 1. Transtons at dscrete tme steps n=,1,2, 2. Past doesn
More information