Aa Pacc Maageme Revew 4) 009) 5-5 Compeve Facl Locao Problem wh Demad Depedg o he Facle Shogo Shode a* Kuag-Yh Yeh b Hao-Chg Ha c a Facul of Bue Admrao Kobe Gau Uver Japa bc Urba Plag Deparme Naoal Cheg Kug Uver Tawa Acceped 3 Jue 008 wwwapmrmaagemecueduw Abrac I h paper we deal wh compeve facl locao problem whch he demad of each cuomer deped o he facle I our model wo compeve compae ed o locae her facle o he mare order o mamze he oal demad acqured from he mare The dace bewee he facle ad he demad po are meaured b recagular dace A fr we coder he lear model ad derve he opmal polc for locag he facle We he eed he model o plaar cae ad alo derve he opmal polc Keword: Compeve facl locao recagular dace Sacelberg Equlbrum Iroduco There are ma paper o compeve facl locao problem Drezer ad Hamacher ed 00) I ma of her paper here are ome demad po o he mare ad ome muuall compeve compae eer he mare o coruc her facle Drezer 98; Ham 983; Hoellg 99) The obecve of her compae o mamze he oal demad acqured from he mare I her paper he auhor aume ha he demad of each cuomer doe o deped o he facle I h paper we coder wo compeve facl locao problem whch here are demad po o he mare ad wo compeve compae ed o coruc her facle o h mare equece I our model we aume ha he demad of each demad po deped o he facle Th becaue he prce of good boh facle are o alwa he ame ad herefore he amou bough b cuomer deped o her prce Ad we aume ha each cuomer ulze he earer oe amog wo facle A fr we coder he problem o he lear mare I he cae where he demad of all demad po are decal o boh facle he model ver mple ad we ca eal fd he opmal oluo o ha boh facle are locaed adacel o he meda po I h paper however we aume ha he demad are o decal o boh facle Therefore he meda po for boh facle are o he ame locao I our model we aume ha he deco of boh compae are o doe a he ame me Drezer 98; Ham 983; Shode ad Drezer 003) Therefore he leader compa locae a facl ad he he follower compa locae a facl o a o mamze ow oal demad acqured from he mare Our obecve o derve he opmal polce for boh compae Ne we coder he locao problem o he plaar mare I he opmal locao boh * Correpodg auhor E-mal: hode@baobegauacp 5
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 compae mu ae he locao of aoher compa o accou I our model he dace bewee wo po are meaured b recagular dace We eed our lear model o he plaar cae aurall I h cae we alo coruc he opmal polc o locae her facle for wo compeve compae Model formulao There are demad po o a mare Two compeve compae eer h mare order o coruc her facle ad ed o mamze oal demad acqured from he mare I h model we aume he demad are o decal for boh facle If a cuomer bu he good a oe facl he he cuomer doe o ecearl bu he ame amou of he good a aoher facl A fr we defe he followg oao X : Y : A : locao of he facl for he leader compa locao of he facl for he follower compa locao of he -h demad po w : demad of v : demad of A for he facl X A for he facl Y I h paper we coder wo pe of dace d A B) bewee wo po A ad B I he lear cae meaured b uual dace bewee wo po a ad b o he le ha d a b) = a b ) I he plaar cae meaured b recagular dace bewee wo po A = a a ) ad B = b b ) o he plae ha d A B) = a b a b ) Each cuomer ulze earer oe of wo facle If he dace o boh facle are he ame he cuomer ulze boh facle half ad half Th rae o eeal becaue we ca eal mod our model for he dfere rae Ne we coder he followg area domaed b each facl a) he area domaed b he facl X 3) R X X Y ) = A d X A) d Y A) { } b) he area domaed b he facl Y { A d Y A) d X )} R Y X Y ) = A c) he equdace area o boh facle { A d X A) d Y )} R XY X Y ) = = A Therefore he demad acqured b he facl X w w A ) RX X Y A RXY X Y ) 4) 5) 6
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 ad he demad acqured b he facl Y v v A ) RY X Y A RXY X Y ) The each compa locae a facl o a o mamze he demad acqured b elf Therefore he leader compa coder he followg fr age problem Ad he follower compa coder he followg ecod age problem The Fr Sage Problem) mamze w w X A RX X Y* X )) A RXY X Y* X )) where Y * X ) he opmal oluo of he followg ecod age problem The Secod Sage Problem) For gve X mamze v v Y A RY X Y )) A RXY X Y )) The wo problem are he opmal locao problem for he leader compa ad he follower compa repecvel I he fr age problem he leader compa mae a deco b coderg he opmal locao of he follower facl ha he opmal oluo of he ecod age problem) Ad he ecod age problem he follower compa locae a facl order o mamze ow demad acqured b he mare afer owg he locao of he leader compa I our model we aume he follower compa cao locae he facl a he ame locao a he leader facl 3 Locao model o he lear mare I h eco we coder he lear mare There are demad po a a L a o a lear mare where a a L a ) I h model he leader compa ad he follower compa locae wo facle ad repecvel Ad le w w L w ad v v L v deoe he demad of he demad po for he facle ad repecvel If w = v for a = L h model a mple model The he opmal oluo for he leader compa o locae he meda po Ad he opmal oluo for he follower compa o locae a he adace po o he leader compa Tha w w ad w w 6) he leader compa locae a facl a he a For he follower compa he opmal locao * deermed a follow: a) v > v [ a a ) * 7
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 b) v v a a ] * c) v = v [ a a ) a a ] * I h paper he demad are o commo for boh facle Therefore w w ad w w 7) a a meda po for he facl ad v v ad v v 8) a a meda po for he facl The he followg proper hold for he leader compa [Proper ] The opmal locao for he facl of he leader compa e o he demad po [Proof] We aume ha he opmal locao o o a demad po ha * = b a b a ) The he follower compa locae oe of adace po ha he followg wo cae mu be codered a) If v v b) If v v he * = a he * = a w I cae he amou demaded of he leader compa Th qua alo me cae * = a Ad cae he amou demaded of he leader compa w whch alo me cae * = a Therefore he are coradcg he aumpo From h proper eal ow ha we ca ge he mamum demad b locag o oe of demad po Therefore he followg we have ol o coder he demad po a a caddae for he opmal locao of he leader compa We mu coder he followg hree cae abou he locao of wo meda po a ad a for he leader compa ad he follower compa repecvel a) a a b) a a > c) a = a 8
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 Two cae a) ad b) are mmerc herefore we coder wo cae a) ad c) h eco A fr we coder he cae where a a I h cae he leader compa locae a facl a he po a he leader compa ca acqure more ha half of oal demad Bu we had beer o move he facl cloe o a If v v he opmal oluo for he leader compa o locae a facl a he po a I h cae eal how ha he follower compa locae a facl a he po a Therefore he leader ad he follower acqure he amou w ad v repecvel The oluo ) = a a ) o ol a Sacelberg equlbrum oluo bu alo a Nah equlbrum oluo Ad v v he leader compa locae a facl a a ad he follower compa locae a facl a a A a reul he leader compa ad he follower compa acqure he amou w ad v repecvel Ne we coder he cae where a = a I h cae he leader compa locae a facl a a becaue o he follower compa ca locae a facl a a ha a ) ad he leader compa cao acqure more ha half of oal demad Therefore h cae he leader compa locae a facl a a The a) v v he he follower locae a facl a a b) v > v 4 Locao model o he plaar mare he he follower locae a facl a a There are demad po A A L A o a plaar mare Le a b ) deoe he coordae of he -h demad po A The leader compa ad he follower compa locae he facle X ad Y repecvel Le ) ad ) deoe he coordae of he wo muual compeve facle X ad Y repecvel Ad le w w L w ad v v L v deoe he demad of he demad po for he facle X ad Y repecvel I h plaar model he dace bewee wo po are meaured b recagular dace The coderg he locao of he follower compa mlarl a he lear cae he leader compa mu locae a facl Now we coder he area domaed b he leader compa We coder he cae where hereafer We ca mlarl coder he cae where b replacg X ad Y I he followg we coder he followg cae Cae) I h cae he area domaed b he leader compa 9
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 0 > 9) Cae ) > The area domaed b he leader compa > 0) Cae 3) = The area domaed b he leader compa ) Ad he area ol domaed b boh wo compeve compae } : ) { ad } : ) { Cae 4) > The area domaed b he leader compa > )
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 Cae 5) > > The area domaed b he leader compa ) ) ) > 3) Cae 6) > = The area domaed b he leader compa 4) Ad he area ol domaed b boh wo compeve compae { ) : } ad { ) : } For he facl coruced b he leader compa he follower compa locae a facl o a o mamze he oal demad acqured b ow facl Coderg he above cae he opmal locao of he follower compa bouded he followg pe a) wo adace po o he facl of he leader compa alog -a b) wo adace po o he facl of he leader compa alog -a c) for each of four quadra here are hree wa of adace mehod for eample for he fr quadra from Cae ) o Cae 3) I pe c) Cae ) equvale o pe b) ad Cae ) equvale o pe a) I Cae 3) le V ad V deoe he um of he wegh v for he demad po he ol domaed area { ) : } ad { ) : } b boh wo compeve compae repecvel Ad le V 3 deoe he um of he wegh v for he demad po he domaed area b he follower compa The he oal demad acqured b he follower compa V V ) / V3 Bu he followg equal hold geerall ma V V3 V V3 ) V V ) V3 5) Therefore we do o coder Cae 3) hereafer Now reumber b b L accordg o he creag order of magude ha b a a L a ad a ) L m) ad b ) L p)
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 The he followg proper hold [Proper ] The opmal locao for he facl of he leader compa e o he lace po coruced b horzoal ad vercal le gog hrough he demad po [Proof] We aume ha he locao ) of he leader compa o a lace po The accordg o he above dcuo he follower compa locae a facl he horzoal or vercal dreco of h po The he follower compa ed o ge he mamum of A a reul a) b) c) d) v v v a > a b > b v v greae he he demad acqured b he leader compa w a > a v greae he he demad acqured b he leader compa w a a v greae he he demad acqured b he leader compa w b > b v greae he he demad acqured b he leader compa w b We aume a ) a ) ad b b ) The he leader compa ca ge he mamum demad b locag a facl a oe of four po a b ) a b ) a ) ad a b ) b Le a ) ) ad a ) ) deoe he meda po weghed demad po for he leader compa ad he follower compa repecvel The he followg e cae are poble a) a ) ) b) a ) ) = b c) a ) ) d) a = a ) ) e) a = a ) ) = b f) a = a ) ) g) a ) ) h) a ) ) = b b
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 ) a ) ) I hee cae a) c) g) ad h) are fudameall mlar Ad b) d) f) ad h) are alo muuall mlar Afer all we have ol o coder hree cae a) b) ad e) Fr we coder he cae a ) ) If he leader compa locae a facl a he meda po a ) ) he compa ca ge a lea half of he demad The compa however poble o ge more demad b acceg o he meda po a ) ) of he follower compa If he leader compa locae a facl a a ) ) effecve o he followg wo cae a) If v > ma v v v hold he he leader compa ca a ) a ) b b acqure he demad v a a ) b) If v > ma v v v hold he he leader compa ca b a ) a b acqure he demad v b b l ) If eher of hee wo codo hold he leader compa doe o locae a facl a he lace po a ) ) The he leader compa locae a facl a a ) ) he leader compa ge he demad V v Ad he leader compa locae a facl a ) X = a ) l a ) l ) he leader compa ge he demad V Y = v Therefore he opmal polc of he leader compa a follow: a) If V V l he he leader compa locae a facl a a b ) b) X Y b b ) If V V l he he leader compa locae a facl a a ) X Y ) b The ecod cae a ) ) = b Th cae mlar o he lear model Therefore he followg polc opmal Cae ) v > ma v v v a ) a ) b b The leader compa locae a facl a he lace po a ) ) ad ge he demad v a a ) If Cae ) v > ma v v v b a ) a b w > w he leader compa locae a facl a he lace po a b ) ad b b a a ) ) ge he demad w If oherwe he leader compa locae a facl a he lace po b b a ) ) ad ge he demad w a a ) 3
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 If Cae 3) v > ma v v v b a ) a b w > w he leader compa locae a facl a he lace po a b ) ad b b a a ) ) ge he demad w If oherwe he leader compa locae a facl a he lace po b b a ) ) ad ge he demad w a a ) Cae 4) v ma v v v a ) a ) b b The leader compa locae a facl a he lace po a b ) ad ge he demad a a ) w ) Ad he la cae a = a ) ) = b I h cae he leader compa clearl locae a facl a he po a ) ) The he opmal locao for he facl of he follower compa rerced amog he four po a ) ) a ) ) a ) ) ad a ) l ) ) a) If he follower compa locae a facl a he po a ) ) he he leader compa ge he oal demad w a a ) b) If he follower compa locae a facl a he po a ) ) he he leader compa ge he oal demad w b b c) If he follower compa locae a facl a he po a ) ) he he leader compa ge he oal demad w a a ) d) If he follower compa locae a facl a he po a ) l ) ) he he leader compa ge he oal demad w b b 5 Cocluo I h paper we have codered compeve facl locao problem whch he demad of each cuomer wa dfere bewee he facle of he leader compa ad he follower compa A fr we have codered a lear mare ad each of wo compeve compae ed o locae a facl o h mare order o mamze oal demad acqured from h mare ad he we have eeded he lear model o he plaar pe Referece Drezer Z 98) Compeve locao raege for wo facle Regoal Scece ad Urba Ecoomc 4) 485-493 4
S Shode e al / Aa Pacc Maageme Revew 4) 009) 5-5 Drezer Z Hamacher HW ed 00) Facl Locao: Applcao ad Theor Sprger New Yor Ham SL 983) O locag ew facle a compeve evrome Europea Joural of Operaoal Reearch ) 9-35 Hoellg H 99) Sabl compeo The Ecoomc Joural 3953) 4-57 Shode S Drezer Z 003) A compeve facl locao problem o a ree ewor wh ochac wegh Europea Joural of Operaoal Reearch 49) 47-5 5