Brand Selection and its Matrix Structure -Expansion to the Second Order Lag-

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1 Journl of Compuons & Modellng vol.3 no ISS: (prn) (onlne) Scenpress Ld 3 Brnd Selecon nd s Mrx Srucure -Expnson o he Second Order Lg- Kuhro Tkeysu nd Yuk Hguch src Focusng h consumers re p o uy superor rnd when hey re ccusomed or ored o use curren rnd new nlyss mehod s nroduced. Before uyng d nd fer uyng d s sed usng lner model. When ove sed evens occur rnson mrx ecomes upper rngulr mrx. In hs pper equon usng rnson mrx s exended o he second order lg nd he mehod s newly re-ul. These re confrmed y numercl exmples. S-sep forecsng model s lso nroduced. Ths pproch mkes possle o denfy rnd poson n he mrke nd cn e uled for uldng useful nd effecve mrkeng pln. Keywords: rnd selecon; mrx srucure; rnd poson Tokoh Unversy. Sesunn Unversy. rcle Info: Receved : July 3. Revsed : ugus 5 3 Pulshed onlne : Sepemer 5 3

2 88 Brnd Selecon nd s Mrx Srucure Inroducon I s ofen oserved h consumers selec upper clss rnd when hey uy nex me fer hey re ored o use curren rnd. Suppose h former uyng d nd curren uyng d re ghered. lso suppose h upper rnd s loced upper n he vrle rry. Then rnson mrx ecomes upper rngulr mrx under he supposon h former uyng vrles re se npu nd curren uyng vrles re se oupu. If he op rnd were seleced from lower rnd skppng nermede rnds correspondng pr n upper rngulr mrx would e. These re verfed n numercl exmples wh smple models. If rnson mrx s denfed s-sep forecsng cn e execued. Generled forecsng mrx componens equons re nroduced. Unless plnners for producs noce s rnd poson wheher s upper or lower hn oher producs mrx srucure mkes possle o denfy hose y clculng consumers cves for rnd selecon. Thus hs proposed pproch mkes effecve o execue mrkeng pln nd/or eslsh new rnd. Qunve nlyss concernng rnd selecon hs een execued y Ymnk[5] Tkhsh e l.[4]. Ymnk[5] exmned purchsng process y Mrkov Trnson Proly wh he npu of dversng expense. Tkhsh e l.[4] mde nlyss y he Brnd Selecon Proly model usng logscs dsruon. In Tkeysu e l. (7) mrx srucure ws nlyed for he cse rnd selecon ws execued for upper clss. In hs pper equon usng rnson mrx s exended o he second order lg nd he mehod s newly re-ul. Such reserch s que new one. Herenfer mrx srucure s clrfed for he selecon of rnd n secon. Exenson of he model o he second order lg s mde n secon 3. umercl clculon s execued n secon 4. Secon 5 s summry.

3 Kuhro Tkeysu nd Yuk Hguch 89 Brnd Selecon nd Is Mrx Srucure. Upper shf of rnd selecon ow suppose h x s he mos upper clss rnd y s he second upper clss rnd nd s he lowes clss rnd. Consumer s ehvor of selecng rnd mgh e y y x x ec. x mgh e few. Suppose h x s curren uyng vrle nd x s prevous uyng vrle. Shf o x s execued from x y or. Therefore x s sed n he followng equon. j represens rnson proly from j -h o -h rnd. x x y 3 Smlrly nd y y 3 33 These re re-wren s follows. Se x 3 x y 3 y () 33 x y x y hen s represened s follows. Here ()

4 9 Brnd Selecon nd s Mrx Srucure R R R s n upper rngulr mrx. To exmne hs generng followng d whch re ll conssed y he d n whch rnson s mde from lower rnd o upper rnd (3) (4) prmeer cn e esmed usng les squre mehod. Suppose where (5) 3 nd mnme followng J J T Mn (6)  whch s n esmed vlue of s oned s follows. T T (7) In he d group whch re ll conssed y he d n whch rnson s mde from lower rnd o upper rnd esmed vlue  should e upper rngulr mrx. If followng d whch shf o lower rnd re dded only few n

5 Kuhro Tkeysu nd Yuk Hguch 9 equon (3) nd (4) Â would conn mnue ems n he lower pr rngle.. Sorng rnd rnkng y re-rrngng row In generl d vrles my no e n order s y x. In h cse lrge nd smll vlue le scered n Â. Bu re-rrngng hs we cn se n order y shfng row. The lrge vlue prs re ghered n upper rngulr mrx nd he smll vlue prs re ghered n lower rngulr mrx. Â Â y x y x (8).3 Mrx srucure under he cse skppng nermede clss rnd s skpped I s ofen oserved h some consumers selec he mos upper clss rnd from he mos lower clss rnd nd skp selecng he nermede clss rnd. We suppose y x w v rnds (suppose hey re ld from upper poson o lower poson s y x w v > > > > ). In he ove cse selecon shfs would e v y v Shfng row

6 9 Brnd Selecon nd s Mrx Srucure Suppose hey do no shf o y x w from o x w from y nd o w from x hen Mrx srucure would e s follows. v w x y v w x y 55 (9) 3 Exenson of The Model o The Second Order Lg We exend Eq.() o he second order lg n hs secon. We hve nlyed he uomole purchsng cse (Tkeysu e l. (7)). In h cse we could on he d (curren uyng d former uyng d efore former uyng d). We hve nlyed hem y dvdng he d (curren uyng d former uyng d) nd (former uyng d efore former uyng d) nd pu hem o Eq.(5) o pply he model. Bu hs s knd of smplfed mehod o pply o he model. If we hve furher me lg model nd we cn uled he d s s he esmon ccurcy of prmeer would e more ccure nd he forecsng would e more precse. Therefore we nroduce new model whch exends Eq.() o he second order lg model s follows. () Where x x x p

7 Kuhro Tkeysu nd Yuk Hguch 93 () () () p () () () p () () () p () () () p p R ( ) () p () p () pp () p () p () pp p p p p R R In order o esme we se he followng equon n he sme wy s efore. ( ) () J T Mn () Eq.() s expressed s follows. ( ) ( ) (3) whch s n esmed vlue of ( ) he sme wy s Eq.(7). s oned s follows n ( T T T T ) ( ) ( ) (4) Ths s re-wren s :

8 94 Brnd Selecon nd s Mrx Srucure ( ) T T T T T T (5) We se hs s : ( ) ( ) F E E D C B T (6) In he d group of upper shf rnd Ê ecomes n upper rngulr mrx. Whle D nd F re dgonl mrx n ny cse. Ths wll e mde cler n he numercl clculon ler. 4 Forecsng fer rnson mrx s esmed we cn mke forecsng. We show some of hem n he followng equons. (7) ( ) (8) ( ) ( ) 3 3 (9) ( ) ( ) { } 4 4 () 5 umercl Exmple In hs secon we consder he cse here s no shf o lower rnd. We consder he cse h rnd selecon shfs o he sme clss or upper clsses. s

9 Kuhro Tkeysu nd Yuk Hguch 95 ove referenced correspondng pr of rnson mrx mus e n upper rngulr mrx. Suppose followng evens occur. Here we se p 3 n Eq.(). o > < Shf from lower rnd o mddle rnd : 5 evens Shf from lower rnd o lower rnd : 3 evens 3 Shf from lower rnd o lower rnd : 4 evens 4 Shf from lower rnd o upper rnd : 3 evens 5 Shf from lower rnd o mddle rnd : 4 evens 6 Shf from lower rnd o lower rnd : 3 evens 7 Shf from mddle rnd o mddle rnd : evens 8 Shf from mddle rnd o mddle rnd : 3 evens 9 Shf from upper rnd o upper rnd : even Shf from mddle rnd o upper rnd : evens < o > Shf from mddle rnd o upper rnd : 5 evens Shf from lower rnd o upper rnd : 3 evens 3 Shf from lower rnd o mddle rnd : 4 evens 4 Shf from upper rnd o upper rnd : 3 evens 5 Shf from mddle rnd o mddle rnd : 4 evens 6 Shf from lower rnd o lower rnd : 3 evens 7 Shf from mddle rnd o upper rnd : evens

10 96 Brnd Selecon nd s Mrx Srucure 8 Shf from mddle rnd o mddle rnd : 3 evens 9 Shf from upper rnd o upper rnd : even Shf from upper rnd o upper rnd : evens Shf from lower rnd o mddle rnd : evens Shf from mddle rnd o mddle rnd : even 3 Shf from mddle rnd o upper rnd : evens Vecor n hese cses re expressed s follows

11 Kuhro Tkeysu nd Yuk Hguch Susung hese o Eq.(4) we on he followng equon.

12 98 Brnd Selecon nd s Mrx Srucure () ( ) s we hve seen efore we cn confrm h Ê pr n Eq.(7) s n upper rngulr mrx nd D F pr n Eq.(7) re dgonl mrces. T Ê pr s here y lower rngulr mrx. We cn fnd h f Ê pr ecomes n upper rngulr mrx hen he ems compose upper shf or he sme level shf. Clculon resuls of ( ) ecome s follows. ( ) () 6 Concluson Consumers ofen uy hgher grde rnd producs s hey re ccusomed or ored o use curren rnd producs hey hve. Formerly we hve presened he pper nd mrx srucure ws clrfed when rnd selecon ws execued owrd hgher grde rnd. In Tkeysu e l. (7) mrx srucure ws nlyed for he cse rnd selecon ws execued for upper clss. In hs pper equon usng rnson mrx ws exended o he second order lg nd he mehod ws newly re-ul. In he numercl exmple mrx

13 Kuhro Tkeysu nd Yuk Hguch 99 srucure s hypohess ws verfed. We cn uled he d s s for he d n whch me lg exs y hs new model nd esmon ccurcy of prmeer ecomes more ccure nd forecsng ecomes more precse. Such reserch s quesonnre nvesgon of consumers cvy n uomole purchsng should e execued n he ner fuure o verfy oned resuls. References [] D.. ker Mngemen Brnds Equy Smon & Schuser US99. [] H. Khr Mrkeng Scence (In Jpnese) Tokyo Unversy Press 987. [3] H. Khr nd Y. Sug Curren movemen of Mrkeng Scence (In Jpnese) Operons Reserch 4 (994) [4] Y. Tkhsh nd T. Tkhsh Buldng Brnd Selecon Model Consderng Consumers Royly o Brnd (In Jpnese) Jpn Indusrl Mngemen ssocon 53(5) () [5] H. Ymnk Qunve Reserch Concernng dversng nd Brnd Shf (In Jpnese) Mrkeng Scence Chkr-Shoo Pulshng 98. [6] K. Tkeysu nd Y. Hguch nlyss of he Preference Shf of Cusomer Brnd Selecon Inernonl Journl of Compuonl Scence (4) (7)

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