Onlne Supplemen for Dynamc Mul-Technology Producon-Invenory Problem wh Emssons Tradng by We Zhang Zhongsheng Hua Yu Xa and Baofeng Huo Proof of Lemma For any ( qr ) Θ s easy o verfy ha he lnear programmng problem n equaon (5) s feasble and does no have he unbound opmal value. I s clear ha he rank of he coeffcen marx for (5) s. In oher words each basc feasble soluon has no more han non-zero elemens. Therefore s opmal o produce wh no more han wo echnologes for any ( qr ) Θ. Proof of Lemma From he dscusson n subsecon 4. and he algorhm ha deermnes he effecve echnology se S we know ha f r μ q r μ ( = L m ) hen s opmal o produce wh echnologes and n S. The producon quanes of echnologes and are q = ( r μq) ( μ μ) and q = ( μ q r) ( μ μ) respecvely and he mnmum producon cos s ha F( qr ) = ( μ q r)( c c ) ( μ μ ) + c q= ω q vr = L m. When fndng echnology + n S Sep 3 n he algorhm ha deermnes S mples Consequenly for = L m v ( c c ) ( μ μ ) > ( c c ) ( μ μ ) = L m. + + + + c c c c ( c c ) ( c c ) c c + + + + + + + v+ = = > μ+ μ+ μ μ+ ( μ μ+ ) ( μ μ+ ) μ μ+ I follows ha ω+ ω+ = ( c+ + μ+ v+ ) ( c+ + μ+ v+ ) = μ+ v+ μ+ v+ >. Snce μ > L > μ m and < c < L < cm we have < v < L < vm and < ω < L < ω m.
Proof of Lemma 3 We frs show he convexy of F( qr ). For any r and r le q [ r μ r μ ] and q [ r μ r μ ]. Lemma mples ha where F( q r ) = c q + cq and F( q r ) = c q + c q q + q = q μ q + μq = r q + q = q and μ q + μq = r Therefore for any λ []. λf( q r ) + ( λ) F( q r ) = c ( λq ) + c ( λq ) + c [( λ) q ] + c [( λ) q ] F[ λq + ( λ) q λr + ( λ) r ]. The nequaly holds because λq λ q ( λ) q and ( ) ha mnmze he producon cos under gven ( λq + ( λ) q λr + ( λ) r ). Consequenly F( qr ) s convex. r λ q are he feasble soluons Nex we prove he submodulary of F( qr ). We only need o show ha for all r q q and F( q r ) F( q r ) F( q r ) F( q r ). (A.) We verfy wheher equaon (A.) holds or no n wo cases: () here exss such ha r μ q q r μ r μ q r μ < r μ ; and () here exs and such ha +. q r μ + + Case In hs case r μ q q r μ ( ) ( ) ω ω ω( ). r μ q q r μ ; r μ. q r μ < r μ + q r μ + + F q r F q r = q vr q + vr = q q We dscuss wo subcases: () here exss such ha such ha Snce Subcase In hs subcase r r r μ q q r μ and () here exs and F( q r ) F( q r ) = ω q v r ω q + v r = ω ( q q ). s clear ha ω ω. Consequenly Subcase I s clear ha μ μ or equvalenly. From Lemma hs mples ha F( q r) Fq ( r) = ω ( q q) ω ( q q) = Fq ( r) Fq ( r). r μ q r μ < r μ q r μ + + + r q r < L < r + q r μ μ μ μ. From Subcase we have F( r μ r) Fq ( r) Fr ( μ r) Fq ( r)
F( r μ r ) F( r μ r ) F( r μ r ) F( r μ r ) = L + + + F( q r ) F( r μ r ) F( q r ) F( r μ r ). Combnng he above equaons we have Case + + F q r F q r F q r F q r r μ q r μ < r μ q r μ ( ) ( ) ( ) ( ). + + + From Case we know ha F( r μ r ) F( q r ) F( r μ r ) F( q r ) F( r μ r ) F( r μ r ) F( r μ r ) F( r μ r ) = L + + + Combnng he above equaons we have F( q r ) F( r μ r ) F( q r ) F( r μ r ). + + F q r F q r F q r F q r ( ) ( ) ( ) ( ). Equaon (A.) holds n boh Case and Case. Therefore F( qr ) s submodular. In summary F( qr ) s convex and submodular. Proof of Lemma 5 Le W % ( x z) = W( x z) and H% ( y w ) = H ( y w ). Then W% ( x z) = mn { F( q r) + H% ( x q z r)} ( qr ) Θ = mn q+ y= x r+ w= z ( q r y w) Ω{ F( qr ) + H % ( yw )} where Ω= {( qryw ) : r rμ q rμm yw }. I can be verfed ha Ω s a nonempy closed convex sublace. The supermodulary and convexy of H mply ha H % s submodular and convex. From Theorem n Chen e al. (3) we have ha W% ( x z) s submodular and convex n. I follows ha W( x z ) s supermodular and convex. Proof of Proposon The supermodulary and convexy of H ( y w K ) W ( x z K ) and V ( x z K ) can be proved by nducon. I s clear ha V ( ) T + x z s supermodular and convex. Suppose ha V ( ) + x z K s supermodular and convex. Ths mples ha H ( y w K) = G ( y) + E [ V ( y D w K% )] s also supermodular and convex because () γ + + G ( y) s convex and () boh supermodulary and convexy are preserved under expecaon and 3
lnear operaon. As shown n Lemma F( qr ) s submodular and convex. Lemma 5 mples ha W ( x z K) = mn{ F( q r) + H ( x+ q z r K ):( q r) Θ} s supermodular and convex so W ( x z K ) s submodular and convex n ( x z ). Snce C s a convex funcon C ( z z) s submodular and convex n ( zz ) (Topks 998). I s well known ha submodulary and convexy are preserved under mnmzaon. Accordngly we have he submodulary and convexy of V ( x z K) = mn{ C ( z z K) + W ( x z K): z } n ( x z ). Ths mples ha V ( x z K ) s supermodular and convex. In summary H ( y w K ) W ( x z K ) and V ( x z K ) are boh supermodular and convex under any gven K for all perods. Proof of Theorem Before provng Theorem we rewre problem (3) based on he formula of F( qr ). For = L m we defne W ( x z K) mn μ μ { ω y+ vw+ H ( y w K)} ω x v z w z( z w) y x ( z w) V ( x z K) mn{mn { K ( z z) + W ( x z K)}mn { k ( z z) + W ( x z K)}}. z z z z Then problem (3) can be rewren as V ( x z K) = mn{ V ( x z K) L V ( x z K)}. m Here V ( x z K ) s he opmal cos for he problem usng us echnologes and. Based on Lemma 5 s easy o verfy he supermodulary and convexy of W ( x z K ) and V ( x z K ). Defne he followng hresholds for he allowance level: L ( x K) argmn { K z W ( x z K)} z + K z + z + z + U ( x ) argmn { k z W ( x z K)} = L m l ( x K) sup{ z :( W ( x z K)) < v} u ( x K) nf{ z :( W ( x z K)) > v} = L m. I s easy o verfy ha () L ( x K) U ( x K ) ( k K ) and () L ( x K) l ( x K) u ( x K) U ( x K ) f k < v < K from he convexy of W ( x z K ). Smlar o he proof of Theorems -4 n Gong and Zhou (3) can be proved from s supermodulary and convexy ha he opmal soluons o V ( x z K ) have he followng properes leng y and z be he opmal soluons o V ( x z K ): 4
LEMMA A.. The followng resuls hold: () If v K hen produce exclusvely wh echnology : when z U ( x K ) sell allowances down o z = U ( x K ) and hen produce up o ˆU y = max{ S ( K ) x}; when L ( x K) < z< U ( x K ) do no rade allowances and hen produce up o y = max{ s ( μ x+ z K ) x}; and when z L ( x K ) purchase allowances up o z = L ( x K ) and hen produce up o ˆ L y = max{ S ( K ) x}. () If v k hen produce exclusvely wh echnology : when z U ( x K ) sell allowances down o z = U ( x K ) and hen produce up o max{ ˆU y = S ( K ) x}; when L ( x K) < z< U ( x K ) do no rade allowances and hen produce up o y = max{ s ( μ x+ z K ) x}; and when z L ( x K ) purchase allowances up o z = L ( x K ) and hen produce up o y = max{ S ( K ) x}. 5 ˆ L () If k < v < K hen when z U ( x K ) sell allowances down o z = U ( x K ) and hen produce exclusvely wh echnology up o ˆU y = max{ S ( K ) x}; when u ( x K) z< U ( x K ) do no rade allowances and hen produce exclusvely wh echnology up o y = max{ s ( μ x+ z K ) x}; when l ( x K) < z< u ( x K ) do no rade allowances and hen produce [ μx+ z μs( K) w( K )] ( μ μ) uns wh echnology and [ μ S( K) + w( K ) μ x z] ( μ μ) uns wh echnology up o y = S ( K ) ; when L ( x K) < z l ( x K ) do no rade allowances and hen produce exclusvely wh echnology up o purchase allowances up o o y = max{ S ( K ) x}. ˆ L y = max{ s ( μ x+ z K ) x}; and when z L ( x K ) z = L ( x K ) and hen produce exclusvely wh echnology up The defnons of ˆ U S ( K ) ˆ L S ( K ) s ( μ x+ z K ) S ( K ) and w ( K ) can be found n Secon 5. In Lemma A. L ( x K ) U ( x K ) l ( x K ) and u ( x K ) have he followng specfc formulas: ˆL + L ( ) ( ( ) ) (max{ ˆL L x K = μ S K x + w S ( K) x}) ˆU + U ( ) ( ( ) ) (max{ ˆU U x K = μ S K x + w S ( K ) x}) = L m + l ( x K) = μ ( S ( K) x) + w (max{ S ( K) x}) u ( x K) = μ ( S ( K) x) + w (max{ S ( K ) x}) = L m. + Noe ha x< S ( K ) when l ( x K) < z< u ( x K ). From Lemma A. we know ha he opmal soluons o V ( x z K ) have he followng properes: () If v K hen produce
exclusvely wh echnology () If v k hen produce exclusvely wh echnology () If k < v < K hen (a) produce exclusvely wh echnology when z u ( x K ) (b) produce wh echnologes and smulaneously when l ( x K) < z< u ( x K ) and (c) produce exclusvely wh echnology when z l ( x K ). Lemma A. shows he opmal soluons y z and he correspondng echnology selecon for V ( x z K ). Please noe ha producng wh echnology ( = L m ) s feasble o boh V ( x z K ) and V+ ( x z K ). Consequenly he opmal soluons o V ( x z K) = mn{ V ( x z K ) L V ( x z K )} can be characerzed by dscussng he opmal m soluons o V ( x z K) L V ( x z K) as shown n Lemma A.. For example f he opmal m soluon o V ( x z K ) s also feasble o V+ ( x z K ) hen mn{ V ( x z K) V + ( x z K)} = V+ ( x z K ). We nex characerze he srucures of he opmal polces when v K vm k and v k K v + respecvely. Case K v < L < vm From Lemma A.() we know ha when v K he opmal soluon o V ( x z K ) s producng exclusvely wh echnology. Snce producng exclusvely wh echnology ( = 3 L m ) s feasble o boh V ( x z K ) and V ( x z K ) we have V ( x z K ) = mn{ V ( x z K) L V ( x z K)} = V ( x z K). Consequenly he opmal polces have he m followng srucure: when z U ( ) x K sell allowances down o z ( ) = U x K and hen produce exclusvely wh echnology up o ˆ U y = max{ S ( K ) x}; when L ( x K ) < z < U ( ) x K do no rade allowances and hen produce exclusvely wh echnology up o y = max{ s ( μx+ z K ) x}; and when z L ( ) x K purchase allowances up o ( ) z = L x K and hen produce exclusvely wh echnology up o ˆ L y = max{ S ( K ) x}. Case v < L < vm k From Lemma A.() we know ha when v k he opmal soluon o V ( x z K ) s producng exclusvely wh echnology. Snce producng exclusvely wh echnology ( = L m ) s feasble o boh V ( x z K ) and V+ ( x z K ) we have m m V ( x z K) = mn{ V ( x z K) L V ( x z K)} = V ( x z K). Consequenly he opmal polces have he followng srucure: when z U ( x K ) sell allowances down o z = U ( x K ) and hen produce exclusvely wh echnology m up o m m 6
y = max{ S ( K ) x}; when L ( x K) < z< U ( x K ) do no rade allowances and hen ˆU m m m produce exclusvely wh echnology m up o y = max{ s ( μ x+ z K ) x}; and when m m z Lm ( x K ) purchase allowances up o z = Lm( x K ) and hen produce exclusvely wh echnology m up o max{ ˆ L y = S ( K ) x}. Case 3 v < L< v k K v < L < v + m Accordng o he analyses n Cases and we have m V( xz K) = mn{ V ( xz K) L V ( xz K)} = mn{ V ( xz K) V ( xz K)}. m + From Lemma A.( ) we know ha he opmal soluons o V ( ) x z K and V + ( x z K ) are he same. Thus he srucure of he opmal polces s: when z U ( ) x K sell allowances down o z = U ( x K ) and hen produce exclusvely wh echnology up o y = max{ S ( K ) x}; when ˆU L ( x K) < z< U ( x K ) do no rade allowances and hen produce exclusvely wh echnology up o y = max{ s ( ) }; μ x+ z K x and when ( ) z L x K purchase allowances up o z ( ) = L x K and hen produce exclusvely wh echnology up o ˆ L y = max{ S ( ) }. K x Proof of Theorem Recall ha v L v k v L v K v L v Smlar o Cases and n he < < < + < < < + < < m. proof of Theorem we have We frs consder v mn{ V ( x z K) L V ( x z K)} = V ( x z K) (A.9) mn{ V ( x z K) L V ( x z K)} = V ( x z K). (A.) + m + mn{ V ( x z K) V ( x z K )} and + mn{ V ( x z K) V ( x z K )}. Snce + k Lemma A.() ndcaes ha he opmal soluon o V ( x z K ) s producng exclusvely wh echnology. I s clear ha hs soluon s also feasble o Consequenly Smlarly from + + V ( ). + x z K mn{ V ( x z K) V ( x z K)} = V ( x z K ). (A.) v < K v + and Lemma A.() we have Concludng equaons (A.9-) we have mn{ V ( x z K) V ( x z K)} V ( x z K ). (A.) + = V ( x z K) = mn{ V ( x z K) L V ( x z K)} = mn{ V ( x z K) L V ( x z K)}. (A.3) m + If = + Lemma A.() ndcaes ha he opmal polces have he same srucure as 7
shown n Theorem. We nex consder he case wh = +. Snce k < v + ( v + < K) we have c + + kμ + > c + k μ ( c + + μ + K > c + + μ K + ). From he defnons of ˆ U S ( ) K and ˆ L S ( K ) we have ˆU ˆU S + ( K) S ( K ) and ˆL ( ) ˆL S K S ( K ). Therefore + + Defne U + ( x K) U ( ) x K and L + ( x K) L + ( x K ) for all x. (A.4) y% max{ s + ( μ + x+ z K ) x}. Now we dscuss he relaonshp beween u + ( x K ) and l + ( x K ). From Lemma A. we have W + ( x z K) = c ( y% + x) + H( y% μ + x+ z μ + y% K) when u + ( x K) z< U + ( x K ) W + ( x z K) = c ( y% + x) + H( y% μ + x+ z μ + y% K) when L + ( x K) < z l + ( x K ). I s clear ha u + ( x K ) l ( ) + x K and L ( ) + x K can be rewren as: u ( x K) = nf{ z :( W ( x z K)) > v u ( x K) z< U ( x K)} Snce + + + z + + + l ( x K) = sup{ z :( W ( x z K)) < v L ( x K) < z l ( x K)} + + + z + + + L ( x K) = nf{ z : ( W ( x z K)) > K L ( x K) < z l ( x K)}. + + + z + + + < + < he convexes of W + ( x z K ) and W ( x z ) v v K L ( x K) u ( x K) l ( x K ) for all x. + + + + K mply ha Recall ha Ln ( x K) ln ( x K) un ( x K) Un ( x K ) n= + +. We nex characerze he srucure of he opmal polces when = + n fve cases. (A.5) Case z U ( ) x K Lemma A.() ndcaes ha when z U ( x K ) s opmal o produce exclusvely wh echnology ( = + + ) and sell allowances. From (A.4)) we have mn{ V ( x z K) V ( x z K)} = V ( x z K ) when + + + U ( x ) U ( x K ) (equaon + K z U ( ) x K (A.6) Because producng exclusvely wh echnology + wh sellng allowances s feasble o V ( ) + x z K when z U ( ). x K Equaon (A.6) ndcaes ha when z U ( ) x K s opmal o sell allowances down o level z = U ( x K ) and hen produce exclusvely wh echnology up o U y = max{ S ( ) } K x. Case l + ( x K) < z< U ( ) x K Snce z l ( x K) u ( x K ) (equaon (A.5)) Lemma A.() ndcaes ha he ˆ + + opmal soluon o V + ( x z K ) s producng exclusvely wh echnology +. Ths soluon s clearly feasble o V ( ) + x z K. Hence 8
mn{ V + ( x z K) V + ( x z K)} = V + ( x z K ) when l + ( x K) < z< U ( x K ). Ths mples ha: () when u ( ) ( ) + x K z< U x K do no rade allowances and hen produce exclusvely wh echnology up o y = max{ s ( ) }; μ x+ z K x and () when l ( x K) < z< u ( x K) do no rade allowances and hen produce + + ( μ + x+ z μ + S+ ( K) w + ( K )) ( μ μ + ) uns wh echnology and ( μ S ( K) + w ( K ) μ x z) ( μ μ ) uns wh echnology +. + + + Case 3 u ( x K) z l ( x K ) + + Snce L + ( x K) u + ( x K ) (equaon (A.5)) and l + ( x K) U + ( x K ) Lemma A.() mples ha V + ( x z K ) and V + ( x z K ) have he same opmal soluon when u ( x K) z l ( x K ). Therefore when u ( x K) z l ( x K ) do no rade + + + + allowances and hen produce exclusvely wh echnology + up o y = max{ s ( μ x+ z K ) x}. + + Case 4 Snce L ( x K) < z< u ( x K ) + + z u ( x K) l ( x K ) (equaon (A.5)) Lemma A.() suggess ha he + + opmal soluon o V ( ) + x z K s producng exclusvely wh echnology +. Ths soluon s also feasble o V + ( x z K ). Consequenly mn{ V + ( x z K) V + ( x z K)} = V + ( x z K ) when L + ( x K) < z< u + ( x K ). Ths mples ha: () when L + ( x K) z< l + ( x K ) do no rade allowances and hen produce exclusvely wh echnology + up o y = max{ s ( μ x+ z K ) x}; and () when + + + + l ( x K) < z< u ( x K) do no rade allowances and hen produce ( μ x+ z μ S ( K) w ( K)) ( μ μ ) uns wh echnology + and + + + + + + ( μ S ( K) + w ( K ) μ x z) ( μ μ ) uns wh echnology +. + + + + + + Case 5 z L + ( x) Lemma A.() ndcaes ha s opmal o produce exclusvely wh echnology ( = + + ) and purchase allowances when z L ( x K ). From L + ( x K) L + ( x K ) (equaon (A.4)) we have mn{ V ( x z K) V ( x z K)} = V ( x z K ) when + + + z L + ( x K ) (A.7) because producng exclusvely wh echnology + wh purchasng allowances s feasble o V + ( x z K ) when z L + ( x K ). Equaon (A.7) ndcaes ha: when z L + ( x K ) s opmal o purchase allowances up o z = L ( x K ) and hen produce exclusvely wh + 9
echnology + up o y = max{ Sˆ ( K ) x}. L + So far we have fully characerzed he opmal polces for he case wh = + whch has he same srucure as shown n Theorem. Smlarly for > + can be proved ha he opmal polcy has he same srucure as shown n Theorem. Deals are omed for brevy.