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▼♦❞❡❧✐♥❣ ❛♥❞ ❆♥❛❧②s✐s ♦❢ ❙✐♠♣❧❡ ▼❛r❦❡t ❙tr❛t❡❣✐❡s ❆❧❡①❛♥❞❡r ❘✳ ❇❧♦❝❦ ❏✉❧② ✶✼✱ ✷✵✶✼ ✶ ✴ ✷✹

■♥tr♦❞✉❝t✐♦♥ ❚❤✐s ♣r♦❥❡❝t ❛✐♠s t♦ ♦❜s❡r✈❡ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ s✐♠♣❧❡ ♠❛r❦❡t str❛t❡❣✐❡s ❛r❡ ✉s❡❞ ✐♥ ❛ ♣❛rt✐❝✉❧❛r ♠❛r❦❡t ♣❧❛❝❡✳ ▼♦❞❡❧✐♥❣ ❜❛s❡❞ ♦♥ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥s✳ ❙✐♠♣❧❡ s♦❢t✇❛r❡ ❞❡✈❡❧♦♣❡❞ ✐♥ ❙❛❣❡▼❛t❤ ❬❙ + ✶✼❪ ✭❛r❜✐tr❛r② ♣r❡❝✐s✐♦♥ ❝♦♠♣✉t✐♥❣✮ t♦ tr❛❝❦ st❛t✐st✐❝s ❛♥❞ ♣❧♦t ❞❛t❛ ❢♦r ❛♥❛❧②s✐s✳ ✷ ✴ ✷✹

▼♦❞❡❧ ❲❡ ❛♥❛❧②③❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧✳ ◮ ❚❤❡r❡ ✐s ❛ ♠❛r❦❡t ♣❧❛❝❡ ❢♦r ♦♥❡ ♣r♦❞✉❝t✱ ✇✐t❤ n s❡❧❧❡rs ❛♥❞ ❛ ♠❛r❦❡t ♣r✐❝❡ mp ✳ ◮ ❊❛❝❤ s❡❧❧❡r ❤❛s ❛ ♣r✐❝✐♥❣ str❛t❡❣② ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ ❛t ♠♦st ( n − 1) ♦t❤❡r s❡❧❧❡rs ❛♥❞✴♦r t❤❡ ❝✉rr❡♥t ♦r st❛rt✐♥❣ ♠❛r❦❡t ♣r✐❝❡✳ ◮ ❊❛❝❤ s❡❧❧❡r ❤❛s ❛♥ ✐♥✐t✐❛❧ ♣r✐❝❡ ❢♦r t❤❡ ♠❛r❦❡t✳ ◮ ❚❤❡ ♠❛r❦❡t ♣r✐❝❡ ❛❧s♦ ❤❛s ❛ str❛t❡❣②✳ ❚❤✐s ♠♦❞❡❧ ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ s❡❧❧❡r✬s ♦♥❧② ♠❛r❦❡t ✱ ✇❤❡r❡ t❤❡ ♣r✐❝✐♥❣ str❛t❡❣✐❡s ♦❢ t❤❡ s❡❧❧❡rs ✐s t❤❡ s♦❧❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ♠❛r❦❡t ❛s ❛ ✇❤♦❧❡✳ ✸ ✴ ✷✹

▼♦❞❡❧ ❊①♣❡r✐♠❡♥ts ✐♥ t❤✐s ♠♦❞❡❧ ❢♦❧❧♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦✉t❧✐♥❡✳ ◮ ❚❤❡ ♠❛r❦❡t s♣❛♥s ♦✈❡r t r♦✉♥❞s ♦r t✐♠❡ ✐♥st❛♥❝❡s✳ ◮ ❉✉r✐♥❣ ❛ r♦✉♥❞ t i ✱ ✇❡ ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ s❡❧❧❡r✭s✮ ❛♣♣❧② t❤❡✐r ♣r✐❝✐♥❣ str❛t❡❣②✭✲✐❡s✮ ❛♥❞ ✉♣❞❛t❡ t❤❡✐r ♣r✐❝❡✭s✮✳ ◮ ❆t t❤❡ ❡♥❞ ♦❢ ❛ r♦✉♥❞ t i ✱ t❤❡ ♠❛r❦❡t ♣r✐❝❡ ✐s ✉♣❞❛t❡❞✳ ✹ ✴ ✷✹

▼♦❞❡❧ ▼♦❞❡❧ ✐s ❜r♦❛❞ ✐♥ ♦r❞❡r t♦ ❛❧❧♦✇ ❛ ✇✐❞❡r r❛♥❣❡ ♦❢ ♣♦ss✐❜❧❡ r❡s✉❧ts ❛♥❞ ♦❜s❡r✈❛t✐♦♥s✱ ❜✉t ❛t t❤❡ s❛♠❡ t✐♠❡ ✐s ❧✐♠✐t❡❞✳ ◮ ◆♦ ❝❧❛ss✐❝❛❧ s✉♣♣❧② ❛♥❞ ❞❡♠❛♥❞ ❢♦r ❡①❛♠♣❧❡✳ ▼❛♥② ♣❛r❛♠❡t❡rs ❛r❡ ✜①❡❞ t♦ ❛❧❧♦✇ ❝♦♠♣❛r❛❜❧❡ r❡s✉❧ts t♦ ❜❡ ❣❡♥❡r❛t❡❞✳ ●❡♥❡r❛❧ ❛ss✉♠♣t✐♦♥s ◮ ❙❡❧❧❡rs ❝❛♥♥♦t ♣r✐❝❡ ❜❡❧♦✇ ✩✵✳ ◮ ❙❡❧❧❡rs ♠❛② ♥♦t ✉♣❞❛t❡ t❤❡✐r ♣r✐❝❡ ♠♦r❡ t❤❛♥ ♦♥❝❡ ♣❡r r♦✉♥❞✳ ◮ ❊❛❝❤ s❡❧❧❡r str❛t❡❣② ✐s r♦✉♥❞ ✐♥❞❡♣❡♥❞❡♥t ✳ ❋✐①❡❞ ▼❛r❦❡t Pr✐❝❡ ❙tr❛t❡❣② ◮ t❤❡ ❛✈❡r❛❣❡ ♦❢ ❛❧❧ s❡❧❧❡r ♣r✐❝❡s ❛t t❤❡ ❡♥❞ ♦❢ r♦✉♥❞ t i ✳ ✺ ✴ ✷✹

❙❡❧❧❡r ❙tr❛t❡❣✐❡s ●✉✐❞❡❧✐♥❡ ❢♦r s❡❧❧❡r str❛t❡❣✐❡s ◮ ❉❡t❡r♠✐♥✐st✐❝ ✈s✳ ◆♦♥❞❡t❡r♠✐♥✐st✐❝ ◮ ❯♥❜♦✉♥❞❡❞ ✈s✳ ❇♦✉♥❞❡❞ ❊①♣❡r✐♠❡♥ts ❛r❡ ♣❡r❢♦r♠❡❞ ✇✐t❤ ❡❛❝❤ ♦❢ t❤❡ ❛❜♦✈❡ ❝♦♠❜✐♥❛t✐♦♥s✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❛ ♠✐① ♦❢ ❜♦✉♥❞❡❞ ❛♥❞ ✉♥❜♦✉♥❞❡❞ ❝♦♥str❛✐♥ts✳ ◮ ❋♦r ❞❡t❡r♠✐♥✐st✐❝ str❛t❡❣② ❡①♣❡r✐♠❡♥ts✱ ❛❧❧ s❡❧❧❡rs ❤❛✈❡ ❞❡t❡r♠✐♥✐st✐❝ str❛t❡❣✐❡s✳ ◮ ❋♦r ♥♦♥❞❡t❡r♠✐♥✐st✐❝✱ ❛t ❧❡❛st ♦♥❡ s❡❧❧❡r ♠✉st ❤❛✈❡ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ str❛t❡❣②✳ ✻ ✴ ✷✹

❙❡❧❧❡r ❙tr❛t❡❣✐❡s ❛♥❞ ▼❛r❦❡t ❙❦❡✇ ❲❤❛t ✐s ▼❛r❦❡t ❙❦❡✇❄ ◮ ❲❤❡♥ t❤❡ str❛t❡❣✐❡s ♦❢ t❤❡ s❡❧❧❡r ♣✉❧❧ t❤❡ ♠❛r❦❡t ✉♣✇❛r❞s ♦r ❞♦✇♥✇❛r❞s ✭✐♥ t❡r♠s ♦❢ ♣r✐❝❡✮✳ ❚❤r❡❡ ♣♦ss✐❜❧❡✿ ▲♦✇✲❙❦❡✇✱ ◆♦✲❙❦❡✇✱ ❛♥❞ ❍✐❣❤✲❙❦❡✇ ◮ ▲♦✇✲❙❦❡✇ ✐♥❤❡r❡♥t❧② ✢♦♦r❡❞ ❛t ✩✵ ❜② ♦✉r ❛ss✉♠♣t✐♦♥s✱ ❜✉t ✐s ♦t❤❡r✇✐s❡ s②♠♠❡tr✐❝ t♦ ❍✐❣❤✲❙❦❡✇✳ ◮ ◆♦✲❙❦❡✇ ♠❡❛♥s r♦✉❣❤❧② ♦♥ ❛✈❡r❛❣❡ ✱ t❤❡ ♠❛r❦❡t ❞♦❡s♥✬t s❦❡✇ ✉♣ ♦r ❞♦✇♥ ✏t♦♦ ♠✉❝❤✑✳ ◮ ❍✐❣❤✲❙❦❡✇ ♥♦t ❝♦♥s✐❞❡r❡❞✱ ❛s ✐t ✐s s②♠♠❡tr✐❝ t♦ ▲♦✇✲❙❦❡✇ ✭❡s♣❡❝✐❛❧❧② ✇❤❡♥ ❛ ♠❛r❦❡t ❝❛♣ ✐s ♣r♦✈✐❞❡❞✮✳ ❇r♦❛❞ ❝❛t❡❣♦r✐③❛t✐♦♥✱ ❞♦❡s ♥♦t ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ❛ ♠❛r❦❡t s❦❡✇s ✐♥ ❛ ♣❛rt✐❝✉❧❛r ❞✐r❡❝t✐♦♥✳ ✼ ✴ ✷✹

❊①♣❡r✐♠❡♥t P❛r❛♠❡t❡rs ❍♦✇ t♦ ❝❤♦♦s❡ ❛ s❡❧❧❡r t♦ ✉♣❞❛t❡ t❤❡✐r ♣r✐❝❡ ✐♥ ❛ r♦✉♥❞❄ ◮ ❖♥❡ ❘❛♥❞♦♠ ❙❡❧❧❡r✿ ♦♥❡ s❡❧❧❡r ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠ ♦✉t ♦❢ n ✳ ◮ ❚✇♦ ❘❛♥❞♦♠ ❙❡❧❧❡rs✿ t✇♦ ❞✐st✐♥❝t s❡❧❧❡rs ❝❤♦s❡♥ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ◮ ❆❧❧ ❙❡❧❧❡rs✿ ❡❛❝❤ s❡❧❧❡r ✉♣❞❛t❡s t❤❡✐r ♣r✐❝❡ ▼♦r❡ ♣❛r❛♠❡t❡r ✜①✐♥❣ ◮ ◆✉♠❜❡r ♦❢ r♦✉♥❞s✿ t = 500 ✳ ◮ ◆✉♠❜❡r ♦❢ s❡❧❧❡r✿ n = 5 ❛♥❞ n = 10 ✳ ◮ ❙t❛rt✐♥❣ ♠❛r❦❡t ♣r✐❝❡✿ smp = $2500 ✳ ❙t✐❧❧ ❛ ❣r❡❛t ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ❧❡❢t ✐♥ t❤❡ ♠♦❞❡❧ ❜❡❝❛✉s❡ ♦❢ s❡❧❧❡r str❛t❡❣✐❡s✳ ❲❡ r✉♥ ❡❛❝❤ ❡①♣❡r✐♠❡♥t ❢♦r ✷✵ tr✐❛❧s❀ ✐✳❡✳✱ ❛t t❤❡ ❡♥❞ ♦❢ ✺✵✵ r♦✉♥❞s✱ st❛t✐st✐❝s ❛r❡ ❝♦❧❧❡❝t❡❞✱ t❤❡ ♠❛r❦❡t ✐s r❡s❡t✱ ❛♥❞ t❤❡ ❡①♣❡r✐♠❡♥t ✐s r✉♥ ❛❣❛✐♥✳ ✽ ✴ ✷✹

❘❡s✉❧ts ❢r♦♠ ❛ ▲♦✇✲❙❦❡✇ ▼❛r❦❡t ◆♦♥❞❡t❡r♠✐♥✐st✐❝ ◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❇♦✉♥❞❡❞ ◆♦♥❞❡t❡r♠✐♥✐st✐❝ ▼✐① ❯♥❜♦✉♥❞❡❞ f 1 ( x [ n ] ) = g 1 ( x [ n ] ) = ❙❡❧❧❡r ✶ h 1 ( x [ n ] ) = g 1 ( x [ n ] ) � � ave { x i } − rr (5 , 16) max 100 , f 1 ( x [ n ] ) i f 2 ( x [ n ] ) = g 2 ( x [ n ] ) = h 2 ( x [ n ] ) = f 2 ( x [ n ] ) ❙❡❧❧❡r ✷ rr (145 , 161) � � ave { x i } min 5( mp ) , f 2 ( x [ n ] ) 100 i f 3 ( x [ n ] ) = g 3 ( x [ n ] ) = h 3 ( x [ n ] ) = f 3 ( x [ n ] ) ❙❡❧❧❡r ✸ rr (67 , 74) � � ave { x i } max 0 . 15( mp ) , f 3 ( x [ n ] ) 100 i g 4 ( x [ n ] ) = f 4 ( x [ n ] ) = h 4 ( x [ n ] ) = g 4 ( x [ n ] ) ❙❡❧❧❡r ✹ rr (75 , 86) � � max 75 , f 4 ( x [ n ] ) min i { x i } 100 f 5 ( x [ n ] ) = g 5 ( x [ n ] ) = h 5 ( x [ n ] ) = f 5 ( x [ n ] ) ❙❡❧❧❡r ✺ xi<mp { x i } − rr (20 , 31) ave � � max 0 . 1( smp ) , f 5 ( x [ n ] ) f 6 ( x [ n ] ) = g 6 ( x [ n ] ) = ❙❡❧❧❡r ✻ h 6 ( x [ n ] ) = f 6 ( x [ n ] ) � � rr (107 , 116) min 3( smp ) , f 6 ( x [ n ] ) max i { x i } 100 f 7 ( x [ n ] ) = g 7 ( x [ n ] ) = xi � 0 . 75( mp ) { x i } − min ❙❡❧❧❡r ✼ h 7 ( x [ n ] ) = g 7 ( x [ n ] ) � � max 50 , f 7 ( x [ n ] ) rr (9 , 21) g 8 ( x [ n ] ) = 75 f 8 ( x [ n ] ) = i ❡✈❡♥ { x i } ave h 8 ( x [ n ] ) = g 8 ( x [ n ] ) ❙❡❧❧❡r ✽ � � 100 max 100 , f 8 ( x [ n ] ) f 9 ( x [ n ] ) = g 9 ( x [ n ] ) = ❙❡❧❧❡r ✾ h 9 ( x [ n ] ) = f 9 ( x [ n ] ) rr (120 , 136) � � i ♦❞❞ { x i } ave min 500 + ( mp ) , f 9 ( x [ n ] ) 100 g 10 ( x [ n ] ) = f 10 ( x [ n ] ) = ave { x i } h 10 ( x [ n ] ) = g 10 ( x [ n ] ) ❙❡❧❧❡r ✶✵ � � i max 60 , f 10 ( x [ n ] ) ✾ ✴ ✷✹