Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts
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1 Onlne Appendx for Sraegc safey socs n supply chans wh evolvng forecass Tor Schoenmeyr Sephen C. Graves Opsolar, Inc. 332 Hunwood Avenue Hayward, CA A. P. Sloan School of Managemen Massachuses Insue of Technology Cambrdge MA Augus 28 Ths onlne appendx accompanes Schoenmeyr and Graves (28. In hs Appendx, we provde proofs and dealed dervaons for some of he clams made hroughou he ex of Schoenmeyr and Graves (28. The order s he same as ha used n he ex. Dervaon of (3 Δ f ( + s E[ D Ψ, Ψ,...] E[ D Ψ, Ψ,...] + s + s 2 E[ μ+ RHX Ψ, Ψ,...] E[ μ+ RHX Ψ, Ψ,...] + s + s 2 RHE[ X Ψ, Ψ,...] RHE[ X Ψ, Ψ,...] + s + s 2 RHEFX [ + V Ψ, Ψ,...] RHEFX [ + V Ψ + s + s + s + s 2 RHFE[ X Ψ, Ψ,...] RHFE[ X Ψ, Ψ,...] + s + s 2 s... RHF E[ X Ψ, Ψ,...] RHF E[ X Ψ, Ψ,...] s 2 ( [, 2,...] ( [, 2,...] ( [ X + V Ψ, Ψ,...] s RHF X E X Ψ Ψ s RHF FX V E FX V + + Ψ Ψ s RHF FX V E F + RHF V s+ 2, Ψ,...] (A28 35
2 Proof ha (7 holds f and only f [ ( ] E I T S consan In hs proof, and he ones o follow, we wll use he noaon E[.] o ndcae expecaon condonal on all evens (specfcally, all realzaons of he forecas revson process up o me, nclusve. Frs we noe ha for n T S + < < + + s mpossble o hold [ (] snce any conrol only affecs I ( afer a lead-met + S +. Furhermore, f E I for > + T + S + wll auo- E[ I( + T + S + ] s held consan every perod, hen [ ( ] macally be held consan as well. Hence, eepng [ ( ] E I consan E I T S consan s n some sense he bes we can do n erms of eepng expeced fuure nvenory consan. We now proceed wh provng he clam. If: Ths wll be shown n he dervaon of ( below. Only f: We need o assume ha [ ( ] E I + T + S I, and show ha hs mples he polcy + (7. In wha follows we use he deny E[ P ] P for s: s s 36
3 [ ( + ] [ ( ] ( [ ( ] [ ( ] I E I + T + S E I + T + S+ + E I + T + S+ E I + T + S+ + T+ S+ S + T+ S+ S I + I( E[ P ] + P I( E [ P ] + P S T S+ S TS+ + T+ S+ S + T+ S+ S I + E[ P+ T ] [ ] [ ] + S S E P P P E P P S T S+ S TS+ + T+ S+ S [ + T ] + S+ S ( [ P ] E[ P ] + P S + T+ S+ S S + T+ S+ S ( E [ P ] E[ P ] P + I E P + E ( ( I E[ P ] + E [ P ] E[ P ] + E [ P ] E[ P ] + + [ + T ] + S+ S + E [ P ] P S + T+ S+ S ( E [ P ] E[ P ] P + + T+ S+ S [ + T ] ( ( [ ] [ ] [ ] + S+ S + I E P + P P + + ( ( I E P + E P P + E P E P + P Comparng he frs and las lne, and rewrng n erms of P, we have + T+ S+ S + T+ S+ S ( P E[ P ] P E [ P ] E[ P ] E [ P ] (A29 For he specal case, we have ha P j D and E [ D + ] f ( + j. Inserng hese denes n (A29 gves us (7. Hence he clam s rue for, we now proceed o prove for greaer, by nducon. Suppose ha he polcy (7 s used for some -, and ha [ ( ] I E I + T + S + for all. Then we have: + T+ S+ S + T+ S+ S ( P E[ P ] P E [ P ] E[ P ] E [ P ] + T+ S+ S + + ( f ( + + T + S S + P f ( + + f ( + f ( f( + + f( + + Δf( f ( + + Δ f( f ( + + Δf ( (A3 37
4 Snce we showed ha he clam was rue for, we have ha [ ( ] mples polcy (7 for all. I E I + T + S + for all Proof ha (7 and (8 are equvalen From he nvenory balance equaon (9 we can descrbe I ( + T + SI n erms of I ( and ncomng and ougong orders. T SI T SI + + (A3 I ( + T + SI I ( P ( + S + P ( + T SI Tang he expecaon a me gves us T + SI T + SI [ ] [ ]. E [ I ( + T + SI ] I ( E P ( + S + E P ( + T SI We noe ha n he rghmos sum + T SI and so he orders P( + T SI have already been realzed and he expecaon operaor s redundan. If we furhermore smplfy he las summaon ndex and separae ou he las erm we ge T + SI T + SI [ ] [ ] I ( E P ( + S + E P ( + T SI T + SI T + SI [ ] I ( E P ( + S + P ( + P ( So far we have no used any properes of he polcy self, only of he generc nvenory balance equaons. Now suppose ha he order P ( s deermned by (8. Then: T + SI T + SI [ ] I ( E P ( + S + P ( + P ( T+ SI T+ SI T+ SI T+ SI I( E[ P ( S ] P( E[ P ( S ] P( I( I I Thus when (8 s used E [ I ( + T + SI ] I and s hus consan. Bu we have already shown ha hs propery unquely characerzes he polcy (7, and so (7 and (8 mus be equvalen. 38
5 Dervaon of ( + T+ S+ S + ( T+ S+ ( T+ S+ ( ( S S ++ S ( T + S I T S I S P P + τ ( + + τ I + S P + P + + τ + + I( + S Δ f( j + f( + + Δf( j + f( + + j + τ j + I( + S + τ + Δ f( j + f τ ( + + Δ fl( + + Δ f( j + f( + + j l τ + + τ j + τ + + I( + S Δ f( j + Δ fl( + + Δf( j + j l τ+ + τ j + τ τ I( + S Δ f( j + Δ fl( + + Δf( j Δ fl( + + j l+ + τ j + l τ + + τ + + τ j + + τ I( + S Δf( j Δ f( j+ + Δf( j Δ f( j+ + j j τ j j + j τ+ + τ + + τ + τ + + τ I( + S Δf( j Δ f( j+ + Δf( j Δ f( j+ + j + j + τ j + τ j+ + τ + + τ I( + S Δf( j Δ f( j + Δf( j Δf( j + j + j+ + τ j + τ j++ + τ + + I( + S Δ f( j + Δf( j + j + τ j 39
6 We noe ha + τ + Δf ( j s he change of nvenory caused by new forecas revsons n he + j me wndow [ +, + τ ], and + τ j + Δf ( j s he replenshmen for he forecass ha happened durng [ + τ, ]. These expressons are dencal excep for a ranslaon of τ. If here were no change of forecass durng [ +, + τ ], + τ + + j brng he safey soc bac o s defaul value whch we denoe Δ f ( j, hen he replenshmen would I. Mahemacally, we have ha I S f j I + ( + + Δ ( + τ j. Hence + τ + ( Δ ( + j I T S I f j. Noe also ha I s he average nvenory level: + τ + EI [ ( + T + S+ ] EI [ ] E Δ f( j I (A32 + j Dervaon of (5 For he bound o be vald, we need o show ha We have ha ( 2 2 zσ I ( F ( F (. (A33 4
7 var[ I ( ] + τ + var Δf( j + j + τ + var Δf ( j (A34 + j var Δf( j var Δf( j + j + τ + j var Δf( j var Δf( j. + j + τ + j We noe ha snce Δf ( j are..d., we can add or deduc any consan from and j, and sll preserve he varance: var Δf( j var Δf( j + j + τ + j var Δf( j var Δfτ ( jτ + j + τ + j Because + τ, we have (A35 4
8 Thus var Δf( j var Δfτ ( jτ + j + τ + j τ var Δf( j var Δf τ ( j + j + τ+ j τ τ + τ var Δf( j var Δf( j + j + τ τ+ j var Δf( j var Δf( j + j + j + j + j var Δf( j var Δf( j j+ + j var ( ( var f ( j f ( j ( f j j f j ( j j+ j+ + + var ( var ( ( Dj f j ( Dj f j (A36 j+ j+ + + var[ I( ] var Dj f( j var Dj f( j j+ j+ ( (, (A37 whch combned wh he defnon (4 of F, gves he clamed relaon. 42
9 Dervaon of (25 var( D f ( var( Δf ( j+ var( Δ f ( + var( f ( var( f ( j+ var( Δ f ( + f ( var( f ( j+ var( D var( f ( var( f ( var( D var( D Fnally, usng (24, we have: j j j (A38 var( f ( var( D var( D 2 ( ρ ( D, f ( var( D (A39 Dervaon of (26 under addonal ndependence assumpon We have from (4 and (5 ha: + + (, var ( j ( var ( j ( (A4 B z D f j D f j j+ j+ If we now mae he addonal ndependence assumpon (as saed, hen ( Dj f( j are ndependen for dfferen j. Then z var D f ( j var D f ( j z var D f ( j j+ j+ j+ + ( j ( j j (A4 43
10 And fnally, usng (25 + z var Dj f( j j ( ρ zσ( D ( D, f ( j j j ( ( zσ( D + ρ ( D, f ( j ρ ( D, f ( j j j j+ + j+ j+ (A ( + ρ ( j, ( j+ + zσ D T + S S D f j 44
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