Effective Dynamic Pricing Strategies with Stochastic Demand

A research and education initiative at the MIT Sloan School of Management Effective Dynamic Pricing Strategies with Stochastic Demand Paper 158 Lap ...
Author: Stuart Adams
0 downloads 3 Views 460KB Size
A research and education initiative at the MIT Sloan School of Management

Effective Dynamic Pricing Strategies with Stochastic Demand Paper 158

Lap Mui Ann Chan David Simchi-Levi Julie Swann

July 2001

For more information,

please visit our website at http://ebusiness.mit.edu or contact the Center directly at [email protected]

or 617-253-7054

e&‚{$%‚ *kts8${ _„${$t 7„s‚$‚R b$Y 7B{YsR${ *‚8stI Ns} Z$ Vtt ]YstG *s%$I 7$8{Y$TN‚%$ FZj$‚ 7bstt FZjk 1((d

VAR„s{ G‚ {BtR$I‚„ `B$tjk I‚‚„8$t$t }„${‚R stI }„BIZ{$Bt %sjZ‚R $t s 8Zj$}j‚ }‚„$BI YB„$9Bt ZtI‚„ s ‚t‚„sj tBtTRs$Bts„k RB{YsR${ I‚8stI PZt{$Bt

VII$$Btsj ,‚k

sRRZ8}$BtR s„‚ Ys Y‚ s%s$jsAj‚ }„BIZ{$Bt {s}s{$k $R j$8$‚I Zt8‚ I‚8stI $R jBR stI Rsj‚R s„‚ I$R{„‚$Bts„k G‚ stsjk9‚ stI {B8}s„‚ }s„$sj }jstt$t stI I‚jsk‚I R„s‚$‚R $t bY${Y s I‚{$R$Bt 8sk A‚ I‚jsk‚I Zt$j I‚8stI $R „‚sj$9‚I $t }„‚%$BZR }‚„$BIR 3B„ ‚-s8}j‚ $t *‚jsk‚I _„BIZ{$Bt w*‚jsk‚I _„${$tD }„${$t w}„BIZ{$BtD $R I‚‚„8$t‚I s Y‚ A‚$tt$t BP Y‚ YB„$9Bt stI Y‚ }„BIZ{$Bt w}„${$tD I‚{$R$Bt $R 8sI‚ s Y‚ A‚$tt$t BP ‚s{Y }‚„$BI A‚PB„‚ {ZRB8‚„ B„I‚„R s„‚ „‚{‚$%‚I {sR‚ BP BZ„ 8BI‚j $R bY‚„‚ s R$tj‚ }„${‚ $R {YBR‚t B%‚„ Y‚ YB„$9Bt

V R}‚{$sj

G‚ I‚%‚jB}

Y‚Z„$R${R PB„ Y‚ R„s‚$‚R AsR‚I Bt I‚‚„8$t$R${ s}}„B-$8s$BtR stI stsjk9‚ Y‚ bB„RT{sR‚ }‚„PB„8st{‚ ]B8}Zs$Btsj stsjkR$R $R }‚„PB„8‚I B I‚%‚jB} $tR$YR sABZ Y‚ }‚„PB„8st{‚ BP Y‚ Y‚Z„$R${R

i‚R‚s„{Y

RZ}}B„‚I $t }s„ Ak !Wi ]Bt„s{R W(((d;T(TFTdE; stI W(((d;TxTdT(1n1 W73 ]Bt„s{R

**Tn11M1M stI *yTn1x s „‚R‚s„{Y „st P„B8 Y‚ WsZ„sj 7{$‚t{‚R stI i‚R‚s„{Y ]BZt{$j BP ]stsIs wW7ei]D s „st P„B8 p‚t‚„sj BB„R stI s „st P„B8 Y‚ ‚

G 7{YBBj  *‚}  7{YBBj

ZR$t‚RR ]‚t‚„ s yq

BP sts‚8‚t =t$%‚„R$k BP qB„BtB qB„BtB !ts„$B BP ]$%$j stI et%$„Bt8‚tsj et$t‚‚„$t stI Y‚ et$t‚‚„$t 7kR‚8R *$%$R$Bt yq BP ytIZR„$sj stI 7kR‚8R et$t‚‚„$t p‚B„$s ytR$Z‚ BP q‚{YtBjBk

d

1

]Bt‚tR d yt„BIZ{$Bt dd N$‚„sZ„‚ i‚%$‚b                                  ddd _„${$t stI yt%‚tB„k ]Bt„Bj                       dd1 7}‚{$h{ _„${$t 7„s‚$‚R                        

n x E 

1 qY‚ 7B{YsR${ _„${$t _„BAj‚8R 1d WBs$Bt stI VRRZ8}$BtR                        11 *‚jsk‚I _„BIZ{$Bt 7„s‚k                       11d VtsjkR$R BP *‚jsk‚I _„BIZ{$Bt PB„ s p$%‚t _„${‚ z‚{B„   111 d  hx wV  dDH L hx wV  dD qYZR $ $R {j‚s„ Ys Ptx $R tBtT$t{„‚sR$t R$t{‚ Ptx wV D  Ptx wV L dD ) hx xh wV D  8xh wV D k ( x

x

x

x

1 p$%‚t 0 0 ) d, 3 3 3 , x  sRRZ8‚ Ys Pt0 $R tBtT$t{„‚sR$t G‚ RYBb Ys O0 $R {Bt{s%‚ T *‚ht‚ V0 sR ( $P >0 k Pt0 wdD ‚jR‚ sR 8s- \V U V k ( stI >0  Pt0 wV Di 3 T 0R0 L t0 w{ L R0 D O0 w{D ) € >0 wV0  {D L t0 wV0 D Z t0 w{D

$P {  V0  R0 $P V0  R0 c {  V0 $P V0 c {

dd T 7$t{‚ Pt0 $R tBtT$t{„‚sR$t stI YZR t0 $R {Bt{s%‚ stI Ak Y‚ {YB${‚ BP V0  Ey?w(, E7zwR0 , V0  {0 DD $R Y‚ V0 Ys 8s-$8$9‚R O0 w{0 D T G‚ {BtR$I‚„ Y‚R‚ {sR‚R stI RYBb Ys PO0 $R tBtT$t{„‚sR$t b$Y$t ‚s{Y BP Y‚8U wsD ]sR‚ dU {  V0  R0 U O0 w{D  O0 w{  dD ) t0 w{ L R0 D  t0 w{ L R0  dD  >0 A‚{sZR‚ BP Y‚ {YB${‚ BP V0  7$t{‚ Pt0 $R tBtT$t{„‚sR$t RB $R PO0 w{D PB„ Y$R {sR‚ wAD ]sR‚ 1U V0  R0 c {  V0 U O0 w{D  O0 w{  dD ) >0  PO0 w{D $R {BtRst stI YZR tBtT$t{„‚sR$t $t Y$R {sR‚ sR b‚jj w{D ]sR‚ nU { k V0 U O0 w{D  O0 w{  dD ) t0 w{D  t0 w{  dD  >0 A‚{sZR‚ BP Y‚ {YB${‚ BP V0  Vs$t R$t{‚ Pt0 $R tBtT$t{„‚sR$t RB PO0 w{D $R tBtT$t{„‚sR$t T qB R‚‚ Ys PO0 $R tBtT$t{„‚sR$t PB„ sjj { b‚ {B8}s„‚ Y‚ Y„‚‚ {sR‚R sR { $t{„‚sR‚R G‚ Ys%‚ Ys PO0 $R A$‚R bY‚t { $R R8sjj w >0 D $ I‚{„‚sR‚R B >0 $t Y‚ R‚{BtI {sR‚ stI bY‚t { $R js„‚R PO0 $R R8sjj‚R w >0 D qYZR PO0 $R tBtT$t{„‚sR$t stI O0 $R {Bt{s%‚ n p$%‚t Ys Pt0 $R tBtT$t{„‚sR$t stI O0 $R {Bt{s%‚ PB„ 0 ) 0, 3 3 3 , x  b‚ tBb RYBb Ys Pt0 $R tBtT$t{„‚sR$t d

T *‚ht‚ C0  sR ( $P h0 k PO0 wdD_0 ‚jR‚ sR 8s- \{ U { k ( stI h0  PO0 w{D  _0 i d

d

T PO wV >D  _0 H 0 w>D3 0 Z >cV C 0

d

d

$P V  C0  $P V k C0 

d

d

d

d

0

d

d

d

d

0

T 7$t{‚ O0 $R {Bt{s%‚ stI PO0 $R tBtT$t{„‚sR$t Ey?wC0  , V0 D $R Y‚ C Ys 8s-T $8$9‚R t0 wV0 D d

d

d

d

T G‚ {BtR$I‚„ Pt0 b$Y$t R‚%‚„sj {sR‚RU d

wsD ]sR‚ dU V c C0  U Pt0 wV D  Pt0 wV L dD ) PO0 wV D  PO0 wV L dD  ( IZ‚ B Y‚ {Bt{s%$k BP O0  wAD ]sR‚ 1U V k C0  U Pt0 wV D  Pt0 wV L dD ) = h h h0 0 wV  C0  D  PO0 wC0  L dD  _0 0 wV  C0  D d

d

d

d

d

d

0

d

L



0

d

d

>cV C0 d

d

d

d

d

>PO0 wV  >D  PO0 wV  > L

d

h dDH 00d d w>D

 (

qY$R $R „Z‚ R$t{‚  = h h $ h0 0 wV  C0  D  PO0 wC0  L dD  _0 0 wV  C0  D  ( A‚{sZR‚ BP Y‚ {YB${‚ BP C0   stI 0

d

0

d

d

d

d

d

d

d

d

d1 $$



h0

>cV C0 d

>PO0 wV  >D  PO0 wV  > L dDH 0 w>D  ( PB„ V k C0   ( d

d

d

IZ‚ B Y‚ {Bt{s%$k BP O0  w{D ]sR‚ nU V ) C0  U Pt0 wC0  D  Pt0 wC0  L dD  =  = h h ) PO0 wC0  D  _0  h0 d  0 w(D  PO0 wC0  L dD  _0 0 w(D d

d

d

d



d

) PO0 wC0  D  _0  h0



d

0

=

d

d

d

d

d

h0

=

0

d

d

d

d  0 w(D

=

d

d

h0

L PO0 wC0  D  PO0 wC0  L dD 0 w(D  ( IZ‚ B Y‚ {YB${‚ BP C0  stI Y‚ {Bt{s%$k BP O0  d

d

d

d

d

T 7$t{‚ Pt0 wV D  Pt0 wV L dD  ( PB„ sjj V  b‚ Ys%‚ Ys Pt0 $R tBtT $t{„‚sR$t stI t0 $R {Bt{s%‚ d

d

d

d

qY‚ j‚88s YZR $8}j$‚R Y‚ B}$8sj }Bj${k PB„ Y‚ *‚jsk‚I _„BIZ{$Bt 7„s‚k ]B„Bjjs„k 11 p$%‚t s %‚{B„ BP }„${‚R h  Y‚„‚ ‚-$RR st B}$8sj }Bj${k PB„ Y‚ *‚jsk‚I _„BIZ{$Bt 7„s‚k b$Y st B}$8sj B„I‚„TZ}TB j‚%‚j wV0 D stI st B}$8sj Rs%‚TZ}TB wC0 D qYZR Y‚ R„Z{Z„‚ BP Y‚ *‚jsk‚I _„BIZ{$Bt }„BAj‚8 {B8A$t‚I b$Y Y‚ j$8$R Bt }„BIZ{$Bt {s}s{$k j‚sIR B s 8BI$h‚I B„I‚„TZ}TB }Bj${k yP Y‚„‚ $R RZ^{$‚t {s}s{$k Y‚t }„BIZ{‚ ‚tBZY B A„$t Y‚ $t%‚tB„k j‚%‚j Z} B Y‚ B„I‚„TZ}TB j‚%‚j BY‚„b$R‚ }„BIZ{‚ B 8s-$8Z8 {s}s{$k 7$8$js„jk Y‚ B}$8sj }Bj${k PB„ $t%‚tB„k B Rs%‚ PB„ Y‚ PZZ„‚ $R B R‚ sR$I‚ Y‚ Rs%‚TZ}TB s8BZt $P }BRR$Aj‚ BY‚„b$R‚ R‚ sR$I‚ Y‚ 8s-$8Z8 s%s$jsAj‚ B„‚ $8}B„stjk Y‚ j‚88s sjRB $8}j$‚R Ys Y‚ sRRZ8}$Bt BP bY‚t B „‚`‚{ {ZRT B8‚„R ‚$Y‚„ A‚PB„‚ I‚8stI $R BAR‚„%‚I B„ sP‚„ $ $R BAR‚„%‚I YsR tB $8}s{ Bt Y‚ B}$8sj $t%‚tB„k }Bj${k ytI‚‚I Y‚ {Bt{s%$k BP O0 w{D $8}j$‚R Y‚ PBjjBb$t ]B„Bjjs„k 1n p$%‚t s %‚{B„ BP }„${‚R h  sRRZ8‚ Ys Y‚ I‚{$R$Bt B „‚`‚{ B„ s{{‚} {ZRB8‚„R $R 8sI‚ sP‚„ Y‚ }‚„$BI I‚8stI YsR A‚‚t BAR‚„%‚I qY‚„‚ ‚-$RR st B}$8sj }Bj${k PB„ Y‚ *‚jsk‚I _„BIZ{$Bt 7„s‚k {Ys„s{‚„$9‚I Ak st B„I‚„TZ}TB j‚%‚j wV0 D stI st B}$8sj Rs%‚TZ}TB wC0 D qYZR ,tBb$t Y‚ s8BZt BP BAR‚„%‚I I‚8stI IB‚R tB Ys%‚ stk $8}s{ Bt YBb 8Z{Y $t%‚tB„k Y‚ h„8 RYBZjI „stRP‚„ PB„ PZZ„‚ }‚„$BIR 111

0 …0 D RZA`‚{ B {L ) ( {0 ) {0 L …0  ;0 , 0 ) d, 1, 3 3 3 , x …0  R0 , 0 ) d, 1, 3 3 3 , x {0 , …0 , ;0 $t‚‚„  (, 0 ) d, 1, 3 3 3 , x3 )d

Ld

qY‚ BA`‚{$%‚ $R B I‚‚„8$t‚ }„${‚R $t ‚s{Y }‚„$BI stI Y‚ }„BIZ{$Bt R{Y‚IZj‚ PB„ Y‚ YB„$9Bt $t B„I‚„ B 8s-$8$9‚ }„Bh qY‚ h„R {BtR„s$t „‚JZ$„‚R Ys Y‚ $t$$sj $t%‚tB„k PB„ Y‚ h„R }‚„$BI $R 9‚„B qY‚ R‚{BtI {BtR„s$t ‚tRZ„‚R Ys $t%‚tB„k $R Asjst{‚I stI ‚tBZY Zt$R s„‚ }„BIZ{‚I B 8‚‚ {ZRB8‚„ Rsj‚R _„BIZ{$Bt $R j$8$‚I Ak Y‚ {s}s{$k BP Y‚ RkR‚8 $t Y‚ Y$„I {BtR„s$t stI Y‚ jsR „‚JZ$„‚8‚t $R Ys }„BIZ{ A‚ I‚8stI‚I B„ }„BIZ{‚I $t $t‚‚„ Zt$R G‚ {BtR$I‚„ tBb Y‚ }„BAj‚8 BP R‚$t Y‚ }„${‚ %‚{B„ PB„ Y‚ *‚jsk‚I _„BIZ{$Bt sjB„$Y8 ZR$t Y‚ I‚‚„8$t$R${ }„BAj‚8 3B„ stk }„${‚ h  Q0 wh D $R Y‚ {B„„‚R}BtI$t ‚-}‚{‚I I‚8stI s }‚„$BI 0 stI Qwh D $R Y‚ %‚{B„ BP ‚-}‚{‚I I‚8stIR PB„ }„${‚ h  qB s}}jk _„BAj‚8 h h  b‚ 8ZR „‚js‚ Y‚ „‚%‚tZ‚ PZt{$Bt p0 w;0 D B Y‚ ‚-}‚{‚I I‚T 8stI {Z„%‚ Q0 wh D 7$t{‚ ‚%‚t $t Y‚ I‚‚„8$t$R${ {sR‚ „‚sj$9‚I I‚8stI + 8sk A‚ j‚RR Yst  BAR‚„%‚I I‚8stI sR ‚t‚„s‚I Ak Y‚ I‚8stI {Z„%‚ b‚ j‚ p0 w;0 D ) 8s- ;0 h0 U Q0 wh0 D  ;0  h0

G‚ „‚P‚„ B _„BAj‚8 h h b$Y Y$R „‚%‚tZ‚ PZt{$Bt sR h h wQD T

Suggest Documents