Arkadiusz Kijek – Optymalizacja struktury branżowej portfela inwestycyjnego na przykładzie gałęzi polskiego przemysłu przetwórczego

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤  ļGODGHVW\PXODQW } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = K k k k k V V 1 t p t p t t t t A D Az A z A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A₀D A₁z h A₂z h A z h p zt = t+ t + t +…+ p t j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P2= ujul jl l=1 m j=1 m = u_j2 2_j j=1 m + u_ju_{l jl} l= ≠1 l j ≠ l j m j=1 m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P2 = ' u u u ' min 1 'u= (1) P Z u'μ = 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 aμ b b ac u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b= ' 1μ oraz c=μ' 1μ ) ( )' ( ) ( 1 2 1 2 2 _{Z Z u u Z u u} P P P P = + + 1 0 uj , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤  ļGODQRPLQDQW]ZDUWRĂFLÈQRPLQDOQÈNk } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = _K k k k k V V 1 t p t p t t t t A D Az A z A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A0D A1z h A2z h A z h p zt = t+ t + t +…+ p t j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P2= ujul jl l=1 m j=1 m = u_j2 2_j j=1 m + u_ju_{l jl} l= ≠1 l j ≠ l j m j=₁ m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P= ' 2 u u u ' min 1 'u= (1) P Z u'μ = 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 _{ac b} aμ b u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b= ' 1μ oraz c=μ' 1μ ) ( )' ( ) ( _{1 2 1 2} 2 _{Z Z u u Z u u} P P P P = + + 1 0 u_j , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤  ļGODQRPLQDQW]SU]HG]LDïHPQRPLQDOQ\P>N1k, N2k@ } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = K k k k k V V 1 t p t p t t t t A D Az Az A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A₀D A₁z h A₂z h A z h p zt = t+ t + t +…+ p t j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P2= ujul jl l=1 m j=1 m = u_j2 _j2 j=1 m + u_ju_{l jl} l= ≠1 l j ≠ l j m j=1 m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P2 = ' u u u ' min 1 'u= (1) P Z u'μ = 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 _{ac b} aμ b u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b=' 1μ oraz c=μ' 1μ ) ( )' ( ) ( 1 2 1 2 2 _{Z Z u u Z u u} P P P P = + + 1 0 uj , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤  JG]LH z_jkļ]QRUPDOL]RZDQDZDUWRĂÊkWHM]PLHQQHMZjW\PRELHNFLH x_jkļSLHUZRWQDZDUWRĂÊkWHM]PLHQQHMZjW\PRELHNFLH =SRZRGXZ\VWÚSRZDQLDREVHUZDFMLQLHW\SRZ\FKZUR]NïDGDFKQLHNWöU\FK]PLHQQ\FKL]ZLÈ ]DQHJR]W\P]DJURĝHQLD]QLHNV]WDïFHQLHPZ\QLNöZSRMDZLïVLÚSUREOHP]ZïDĂFLZ\PGRVWRVRZDQLHP

A. Kijek

84

IRUPXïXQLWDU\]DF\MQ\FK5R]ZLÈ]DQLHPE\ïR]DVWÈSLHQLHZHZ]RUDFKZDUWRĂFLPLQLPDOQHMLPDN V\PDOQHM ]PLHQQHM RGSRZLDGDMÈF\PL LP ZLHONRĂFLDPL ] UR]NïDGöZ SRZVWDï\PL SR Z\HOLPLQR ZDQLXREVHUZDFMLVNUDMQ\FKRUD]SU]\XZ]JOÚGQLHQLXVSHF\ILNLZVNDěQLNöZ2EVHUZDFMHVNUDMQH ]LGHQW\ILNRZDQRQDSRGVWDZLHJUDILF]QHMDQDOL]\UR]NïDGöZZVNDěQLNöZ:SU]\SDGNXZ\VWÚSR ZDQLDZDUWRĂFLPQLHMV]\FKRGQRZHMZDUWRĂFLPLQLPDOQHMRUD]ZLÚNV]\FKRGQRZHMZDUWRĂFLPDN V\PDOQHM]DVWÚSRZDQRMHZ\]QDF]RQ\PLZLHONRĂFLDPLHNVWUHPDOQ\PL

1DVWÚSQLHGODZ\EUDQHJR]HVWDZX]PLHQQ\FKEXGXMHVLÚV\QWHW\F]Q\PLHUQLNNRQG\FMLEUDQĝ RNUHĂODQ\ MDNR LQGHNV NRQG\FML $JUHJDFMÚ SU]HSURZDG]RQR ]D SRPRFÈ IRUPXï EH]Z]RUFRZ\FK NWöUHXĂUHGQLDMÈ]QRUPDOL]RZDQHZDUWRĂFL]PLHQQ\FKSU]\Z\NRU]\VWDQLXV\VWHPXZDJ:EDGD QLXZDJLREOLF]RQRZHGïXJQDVWÚSXMÈFHJRZ]RUX*UDELñVNLV  } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = K k k k k V V 1 t p t p t t t t A D Az A z A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A₀D A₁z h A₂z h A z h p z_t = _t+ _t + _t +…+ _{p t} j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P2= ujul jl l=₁ m j=₁ m = u_j2 2_j j=₁ m + ujul jl l= ≠1 l j ≠ l j m j=1 m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P2= ' u u u ' min 1 'u= (1) P Z u'μ = 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 _{ac b} aμ b u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b= ' 1μ oraz c=μ' 1μ ) ( )' ( ) ( 1 2 1 2 2 _{Z Z u u Z u u} P P P P = + + 1 0 uj , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤  JG]LH z_jļZDUWRĂÊ]PLHQQHMV\QWHW\F]QHMGODjWHJRRELHNWX ƠkļZDJDQDGDQDkWHM]PLHQQHM]JRGQLH]SRQLĝV]ÈIRUPXïÈ*UDELñVNLV  } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = K k k k k V V 1 t p t p t t t t A D Az Az A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A₀D A₁z h A₂z h A z h p zt = t+ t + t +…+ p t j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P 2 _{= u} jul jl l=1 m j=1 m = u2_{j j}2 j=1 m + u_ju_{l jl} l= ≠1 l j ≠ l j m j=1 m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P2= ' u u u ' min 1 'u= (1) P Z u'μ= 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 _{ac b} aμ b u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b= ' 1μ oraz c=μ' 1μ ) ( )' ( ) ( 1 2 1 2 2 _{Z Z u u Z u u} P P P P = + + 1 0 u_j , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤ JG]LHVkR]QDF]DZVSöïF]\QQLN]PLHQQRĂFLkWHM]PLHQQHM 7DNVNRQVWUXRZDQ\PLHUQLNV\QWHW\F]Q\SU]\MPXMHZDUWRĂFL]SU]HG]LDïX!-HĝHOLMHJR ZDUWRĂÊ]EOLĝRQDMHVWGRMHGQRĂFLR]QDF]DWRZ\VRNLSR]LRPNRQG\FMLDMHĂOLZDUWRĂÊMHVWEOLVND ]HUDĂZLDGF]\WRRVïDEHMNRQG\FML:]ZLÈ]NX]W\PZDUWRĂÊPLHUQLNDPRĝHRG]ZLHUFLHGODÊVWR SLHñQDWÚĝHQLDEDGDQHJR]MDZLVNDDZSURZDG]RQ\PEDGDQLXŘSR]LRPNRQG\FMLVHNWRUöZ2WU]\ PDQ\LQGHNVNRQG\FMLEUDQĝZGDOV]\PSRVWÚSRZDQLX]RVWDï]DVWRVRZDQ\MDNR]PLHQQDU\]\ND

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=DVWRVRZDQLH LQGHNVX NRQG\FML EUDQĝ MDNR ]PLHQQHM U\]\ND SR]ZDOD QD SRPLDU PRGHORZDQLH RUD]SURJQR]RZDQLHSR]LRPXU\]\NDVHNWRURZHJR6HNWRU\JRVSRGDUF]HWZRU]ÈV\VWHPRELHNWöZ PLÚG]\NWöU\PL]DFKRG]ÈUöĝQRNLHUXQNRZHSRZLÈ]DQLD:]ZLÈ]NX]W\PGREDGDQLD]DOHĝQRĂFL V\WXDFMLHNRQRPLF]QRILQDQVRZHMEUDQĝPRĝQDZ\NRU]\VWDÊPRGHOZHNWRURZHMDXWRUHJUHVML9$5 0RGHO9$5]RVWDïVIRUPXïRZDQ\SU]H]6LPVDMDNRPRGHOZLHORUöZQDQLRZ\ZNWöU\PNDĝ GD]H]PLHQQ\FKHQGRJHQLF]Q\FKMHVWZ\MDĂQLDQDSU]H]VZRMHZïDVQHRSöěQLHQLDRUD]RSöěQLHQLD SR]RVWDï\FK]PLHQQ\FKHQGRJHQLF]Q\FK0DRQ]DVWRVRZDQLHZPRGHORZDQLX]PLHQQ\FKVWDFMR QDUQ\FK:SU]\SDGNXJG\PRGHO9$5Z\NRU]\VWXMHVLÚGRSURJQR]RZDQLDWDNMDNZSURSRQRZD QHMSU]H]DXWRUDSURFHGXU]HMHVWRQUöZQLHĝRGSRZLHGQLGOD]PLHQQ\FKQLHVWDFMRQDUQ\FK/đWNH SRKOV 0RGHO9$5pX]XSHïQLRQ\]PLHQQ\PLGHWHUPLQLVW\F]Q\PLSU]HGVWDZLDVLÚQDVWÚSXMÈFR } { min } { max } { min jk j jk j jk j jk jk _{x x} x x z = – – – – – – – – – – – – } { min } { max } { max jk j jk j jk jk j jk _{x x} x x z = = ≤ ≤ ≤ > > < k jk k jk j jk jk j k jk jk j k jk j jk jk N x N x x x N x x N x x z gdy , } { max } { max gdy , } { min } { min = k jk k jk j jk jk j k jk k k jk jk j k jk j jk jk N x N x x x N x N N x x N x x z 2 2 2 1 1 1 gdy , } { max } { max gdy ,1 gdy , } { min } { min = – – – = = K k k jk j z z 1 = K k k k k V V 1 t p t p t t t t A D Az A z A z e z = ₀ + _{1 1}+ _{2 2}+…+ + ]' [ 1t 2t mt t z z z z = … ]' ' [ 1t 2t mt t e e e e = … ) ( ˆ ˆ ) 2 ( ˆ ˆ ) 1 ( ˆ ˆ ˆ ) ( ˆ h A₀D A₁z h A₂z h A z h p zt = t+ t + t +…+ p t j t t j z zˆ ( )= + ) ( ˆ h zt = = = = 1 0 ˆ ˆ ˆ ) ( ˆ h i i e i z h ) (h z i ˆ e ˆ ˆ m j j j P u z Z 1 P2 = ujul jl l=₁ m j=₁ m = uj2 2_j j=₁ m + ujul jl l= ≠1 l j ≠ l j m j=1 m = = = + = m j m l j l j l jl m j j j P u u u 1 1 1 2 2 2 l j jl jl = μ ' u ZP= u u P2= ' u u u ' min 1 'u= (1) P Z u'μ = 2 1 opt_{(Z ) Z u u} u P = P + ) ( 1 1 2 1 _{ac b} aμ b u = ) ( 1 1 2 2 _{ac b} bμ c u = + 1 ' = a , b= ' 1μ oraz c=μ' 1μ ) ( )' ( ) ( 1 2 1 2 2 _{Z Z u u Z u u} P P P P = + + 1 0 uj , j = 1, 2, …, m.

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∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ α α – – – – φ σ σ σ σ σ σ σ ι ι ι ι ι ι σ σ σ σ σ ρ ρ φ φ – – – – – – – – – σ ≤ ≤