An optimisation-based approach for process plant layout - Industrial

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Process Systems Engineering

An optimisation-based approach for process plant layout Jude Ejeh, Songsong Liu, Mazaher Molaei Chalchooghi, and Lazaros G Papageorgiou Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00260 • Publication Date (Web): 22 Jun 2018 Downloaded from http://pubs.acs.org on June 24, 2018

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Industrial & Engineering Chemistry Research

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†

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❙♦♥❣s♦♥❣ ▲✐✉✱

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▼❛③❛❤❡r ▼✳ ❈❤❛❧❝❤♦♦❣❤✐✱

❛♥❞ ▲❛③❛r♦s ●✳

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P❛♣❛❣❡♦r❣✐♦✉

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❙❝❤♦♦❧ ♦❢ ▼❛♥❛❣❡♠❡♥t✱ ❙✇❛♥s❡❛ ❯♥✐✈❡rs✐t②✱ ❇❛② ❈❛♠♣✉s✱ ❋❛❜✐❛♥ ❲❛②✱ ❙✇❛♥s❡❛ ❙❆✶ ✽❊◆✱ ❯♥✐t❡❞ ❑✐♥❣❞♦♠❀ ❊✲♠❛✐❧✿ ❧✳♣❛♣❛❣❡♦r❣✐♦✉❅✉❝❧✳❛❝✳✉❦

P❤♦♥❡✿ ✰✹✹ ✭✵✮✷✵ ✼✻✼✾ ✷✺✻✸

❆❜str❛❝t ❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛♥ ♦♣t✐♠✐s❛t✐♦♥✲❜❛s❡❞ ❛♣♣r♦❛❝❤ ❢♦r t❤❡ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠✳ ❚❤❡ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠ ✐♥✈♦❧✈❡s ❞❡t❡r♠✐♥✐♥❣ t❤❡ ♠♦st ❡✣❝✐❡♥t ✲ ❜❛s❡❞ ♦♥ ♣r❡❞❡✜♥❡❞ ❝r✐t❡r✐❛ ✲ s♣❛t✐❛❧ ❛rr❛♥❣❡♠❡♥t ♦❢ ❛ s❡t ♦❢ ♣r♦✲ ❝❡ss ♣❧❛♥t ❡q✉✐♣♠❡♥t ✇✐t❤ ❛ss♦❝✐❛t❡❞ ❝♦♥♥❡❝t✐✈✐t②✳ ❆ ♥✉♠❜❡r ♦❢ ❝♦st ❛♥❞ ♠❛♥❛❣❡✲ ♠❡♥t✴❡♥❣✐♥❡❡r✐♥❣ ❞r✐✈❡rs ✭❡✳❣✳ ❝♦♥♥❡❝t✐✈✐t②✱ ♦♣❡r❛t✐♦♥s✱ ❧❛♥❞ ❛r❡❛✱ s❛❢❡t②✱ ❝♦♥str✉❝t✐♦♥✱ r❡tr♦✜t✱ ♠❛✐♥t❡♥❛♥❝❡✱ ♣r♦❞✉❝t✐♦♥ ♦r❣❛♥✐s❛t✐♦♥✮ ❤❛✈❡ ❜❡❡♥ ❝♦♥s✐❞❡r❡❞ ♦✈❡r t❤❡ ❧❛st t✇♦ ❞❡❝❛❞❡s ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ ♣♦t❡♥t✐❛❧ s❛✈✐♥❣s ✐♥ t❤❡ ♦✈❡r❛❧❧ ♣❧❛♥t ❞❡s✐❣♥ ♣r♦❝❡ss✳ ❚❤✐s ✇♦r❦ ❝♦♥st✐t✉t❡s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ✇♦r❦ ❜② P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r✲ ❣✐♦✉ ✶ t♦ ❛❞❞r❡ss t❤❡ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠✳ ◆❡✇ ❢❡❛t✉r❡s ✐♥tr♦❞✉❝❡❞ ♠♦❞❡❧❧❡❞ t❛❧❧ ❡q✉✐♣♠❡♥t ✇✐t❤ ❤❡✐❣❤ts ❣r❡❛t❡r t❤❛♥ t❤❡ t②♣✐❝❛❧ ✢♦♦r ❤❡✐❣❤t ✐♥ ❝❤❡♠✐❝❛❧ ♣r♦❝❡ss ♣❧❛♥ts✱ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ ♣♦✐♥t ❛t ❛ ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❤❡✐❣❤t ❢♦r ❡❛❝❤ ❡q✉✐♣♠❡♥t✳ ✶

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❚❤❡ ♥✉♠❜❡r ♦❢ ✢♦♦rs✱ ❧❛♥❞ ❛r❡❛✱ ❛❧❧♦❝❛t✐♦♥ ♦❢ ❡❛❝❤ ❡q✉✐♣♠❡♥t t♦ ❛ ✢♦♦r ❛♥❞ t❤❡ ♦✈❡r✲ ❛❧❧ ❧❛②♦✉t ♦❢ ❡❛❝❤ ✢♦♦r ✇❡r❡ ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ♦♣t✐♠✐s❛t✐♦♥ ♠♦❞❡❧ ✇❤✐❧st ♣r❡✈❡♥t✐♥❣ ♦✈❡r❧❛♣ ♦❢ ❡q✉✐♣♠❡♥t✳ ❚❤❡ ❝♦♥♥❡❝t✐♦♥ ❝♦sts✱ ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧✱ ❛s ✇❡❧❧ t❤❡ ❝♦♥✲ str✉❝t✐♦♥ ❝♦sts ✇❡r❡ ❛❝❝♦✉♥t❡❞ ❢♦r ✇✐t❤ ❛♥ ♦✈❡r❛❧❧ ♦❜❥❡❝t✐✈❡ t♦ ♠✐♥✐♠✐③❡ t❤❡ t♦t❛❧ ❝♦st✳ ❚❤❡ ♦✈❡r❛❧❧ ♣r♦❜❧❡♠ ✇❛s ❢♦r♠✉❧❛t❡❞ ❛s ❛ ♠✐①❡❞✲✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✭▼■▲P✮ ♠♦❞❡❧ ❜❛s❡❞ ♦♥ ❛ ❝♦♥t✐♥✉♦✉s ❞♦♠❛✐♥ r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ✐ts ❛♣♣❧✐❝❛❜✐❧✐t② ❞❡♠♦♥str❛t❡❞ ❜② ❛ ♥✉♠❜❡r ♦❢ ✐❧❧✉str❛t✐✈❡ ❡①❛♠♣❧❡s✳ ❘❡s✉❧ts s❤♦✇❡❞ ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ❡q✉✐♣♠❡♥t ❤❛♥❞❧❡❞ ❜② t❤❡ ♣r♦♣♦s❡❞ ♠♦❞❡❧s ✐♥ ♠♦❞❡st ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡s✳ ❋✐♥❛❧❧②✱ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ✇❡r❡ ✐♥❝❧✉❞❡❞ t♦ ✐♥❝r❡❛s❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝②✱ ❛♥❞ t❤❡✐r ♣❡r❢♦r♠❛♥❝❡ t❡st❡❞ ✇✐t❤ t❤❡ ❧❛st ❡①❛♠♣❧❡✳

❑❡②✇♦r❞s ▼✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t❀ ▼✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✭▼■▲P✮❀ ❖♣t✐♠✐s❛t✐♦♥


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✶ ■◆❚❘❖❉❯❈❚■❖◆

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✶ ■♥tr♦❞✉❝t✐♦♥ ❖✈❡r t❤❡ ♣❛st t✇♦ ❞❡❝❛❞❡s✱ t❤❡ ❧❛②♦✉t ♦❢ ❡q✉✐♣♠❡♥t ✐♥ ♣r♦❝❡ss ♣❧❛♥ts ❤❛✈❡ ❣❛✐♥❡❞ ✐♥❝r❡❛s✐♥❣ r❡❧❡✈❛♥❝❡✱ ❢r♦♠ ❛♥ ❡❝♦♥♦♠✐❝✱ s❛❢❡t②✱ ♦♣❡r❛t✐♦♥❛❧ ❛♥❞ ♠❛✐♥t❡♥❛♥❝❡✲r❡❧❛t❡❞ ♣♦✐♥t ♦❢ ✈✐❡✇✳ ✷✱✸ ▲❛②♦✉t ❞❡s✐❣♥ ❣❡♥❡r❛❧❧② ✐♥✈♦❧✈❡s ♦❜t❛✐♥✐♥❣ ❛ s✉✐t❛❜❧❡ ❛rr❛♥❣❡♠❡♥t ♦❢ ✐t❡♠s ❛♥❞ t❤❡✐r ❝♦♥♥❡❝✲ t✐♦♥s ✇✐t❤✐♥ ❛ ♣r❡❞❡✜♥❡❞ ❛r❡❛ ❜❛s❡❞ ♦♥ ♣r❡✲s♣❡❝✐✜❡❞ ❝r✐t❡r✐❛✳ ❚❤❡s❡ ✐t❡♠s ♠❛② ❜❡ ♣r♦❝❡ss ✈❡ss❡❧s ❛♥❞ ❡q✉✐♣♠❡♥t✱ ✇♦r❦ ❝❡♥tr❡s ♦r ❞❡♣❛rt♠❡♥ts ✇✐t❤✐♥ ❛♥ ♦r❣❛♥✐s❛t✐♦♥❀ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥s ❜② ♣✐♣❡s✱ ❝♦♥✈❡②♦rs✱ ✈❡❤✐❝✉❧❛r tr❛♥s♣♦rt ♦r ❛♥② s✉✐t❛❜❧❡ ♠❛t❡r✐❛❧ ❤❛♥❞❧✐♥❣ ❡q✉✐♣♠❡♥t✳ ✸✱✹ ▲❛②♦✉t ❞❡s✐❣♥s ❤❛✈❡ ❜❡❡♥ ❢♦✉♥❞ t♦ ❤❛✈❡ ❛ tr❡♠❡♥❞♦✉s ✐♠♣❛❝t ♦♥ t❤❡ ❝♦sts ❛♥❞ ♣r♦❞✉❝t✐✈✐t② ♦❢ ❛♥② s②st❡♠ ✐♥✈♦❧✈❡❞ ✷ ✱ ❛s s✉❝❤ r❡s❡❛r❝❤❡rs ❤❛✈❡ ❝♦♥st❛♥t❧② s♦✉❣❤t ♦✉t ♠❡t❤♦❞s t♦ ✜♥❞ t❤❡ ❜❡st s♦❧✉t✐♦♥ t♦ s✉❝❤ ❞❡s✐❣♥ ♣r♦❜❧❡♠s ✲ ❣❡♥❡r❛❧❧② r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❧❛②♦✉t ♣r♦❜❧❡♠✳ ❲❤❡♥ ❝♦♥❝❡r♥❡❞ ✇✐t❤ ✐t❡♠s t❤❛t ❛r❡ ♥❡❡❞❡❞ ❢♦r t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ ❣♦♦❞s ♦r ❞❡❧✐✈❡r② ♦❢ s❡r✈✐❝❡s✱ ✐t ❤❛s ❜❡❡♥ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❢❛❝✐❧✐t② ❧❛②♦✉t ♣r♦❜❧❡♠✳ ✹ ❆♠✐❞st t❤❡ ❝❧❛ss❡s ♦❢ t❤❡ ❢❛❝✐❧✐t② ❧❛②♦✉t ♣r♦❜❧❡♠ ✭s❡❡ ❉r✐r❛ ❡t ❛❧✳ ❢♦r ❛ ❞❡t❛✐❧❡❞ r❡✈✐❡✇✮ ✐s t❤❡ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠✱ ♦♥ ✇❤✐❝❤ t❤✐s ✇♦r❦ ♣r✐♠❛r✐❧② ❢♦❝✉s❡s✳ Pr♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠ ❛❞❞r❡ss❡s t❤❡ ❧❛②♦✉t ♣r♦❜❧❡♠ ❢♦r ✐t❡♠s ✐♥ ❛ t②♣✐❝❛❧ ❝❤❡♠✐❝❛❧ ♣r♦❝❡ss ♣❧❛♥t✳ ✺ Pr❡✈✐♦✉s ❛♣♣r♦❛❝❤❡s t♦ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ❞❡s✐❣♥ ✇❡r❡ ❢r♦♠ ❛ ♣r❛❝t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✇❤❡r❡ ❞❡❝✐s✐♦♥s ✇❡r❡ ♠❛❞❡ ❜❛s❡❞ ♦♥ ❡♥❣✐♥❡❡r✐♥❣ ❥✉❞❣❡♠❡♥t ✻ ✱ ❜✉t ❡s♣❡❝✐❛❧❧② ❢r♦♠ t❤❡ ❧❛t❡ ✾✵s✱ ♠♦r❡ s❝✐❡♥t✐✜❝ ❛♣♣r♦❛❝❤❡s ✐♥ t❡r♠s ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧❧✐♥❣ ❤❛✈❡ ❜❡❡♥ ❛❞♦♣t❡❞ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ✶✱✼✕✶✷ ✳ ▼♦st ♦❢ t❤❡ ♣r❡✈✐♦✉s r❡s❡❛r❝❤ ❤❛✈❡ ❢♦❝✉s❡❞ ♦♥ ❛ ❝♦st✲r❡❞✉❝✐♥❣ ♦❜❥❡❝t✐✈❡✱ ✇✐t❤ ❛ ✈❛r②✐♥❣ ♥✉♠❜❡r ♦❢ s❝❡♥❛r✐♦s✳ ●❡♦r❣✐❛❞✐s ❛♥❞ ▼❛❝❝❤✐❡tt♦ ✾ ♣r♦♣♦s❡❞ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤ ✇✐t❤ ❛♥ ♦❜❥❡❝t✐✈❡ t♦ ♠✐♥✐♠✐③❡ ❝♦st ✲ ♣✉♠♣✐♥❣✱ ♣✐♣❡ ❝♦♥♥❡❝t✐♦♥ ❛♥❞ ✢♦♦r ❝♦♥str✉❝t✐♦♥ ✲ ❛♥❞ ✇❛s ❢♦r♠✉❧❛t❡❞ ❛s ❛ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✭▼■▲P✮ ♣r♦❜✲ ❧❡♠✳ ❘❡s✉❧ts s❤♦✇❡❞ t❤❡ ♠♦❞❡❧ ✇❛s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t ❢♦r ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ✉♥✐ts✳ P❛♣❛❣❡♦r❣✐♦✉ ❛♥❞ ❘♦tst❡✐♥ ✶✷ ❢✉rt❤❡r s♦❧✈❡❞ t❤❡ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠ ♦✈❡r ❛ ❝♦♥t✐♥✉✲ ♦✉s s♣❛t✐❛❧ ❞♦♠❛✐♥ ✇❤✐❧st ✐♥tr♦❞✉❝✐♥❣ ♠♦r❡ r❡❛❧✐st✐❝ ❝♦♥str❛✐♥ts ❢♦r s✐♥❣❧❡✲✢♦♦r ❝❛s❡ st✉❞✐❡s✳ ❇❛r❜♦s❛✲P♦✈♦❛ ❡t ❛❧✳ ✼✱✽ ❛❝❝♦✉♥t❡❞ ❢♦r ❞✐✛❡r❡♥t ✐♥♣✉ts ❛♥❞ ♦✉t♣✉ts ❢♦r ❡❛❝❤ ❡q✉✐♣♠❡♥t✱ ✐rr❡❣✲ ✉❧❛r s❤❛♣❡s✱ ❛ ✸✲❞✐♠❡♥s✐♦♥❛❧ r❡♣r❡s❡♥t❛t✐♦♥✱ ❛❧❧♦✇❡❞ ❢♦r ❞✐✛❡r❡♥t ♣r♦❞✉❝t✐♦♥ s❡❝t✐♦♥s✱ ❛♥❞ ACS Paragon Plus Environment


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s❛❢❡t② ❛♥❞ ♦♣❡r❛❜✐❧✐t② ❢❛❝t♦rs✳ ❚❤❡s❡ ♣r♦❜❧❡♠s ✇❡r❡ ❛❧❧ ❢♦r♠✉❧❛t❡❞ ❛s ❛♥ ▼■▲P ❣✐✈✐♥❣ ❝♦♠✲ ♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t s♦❧✉t✐♦♥s ❢♦r ❛ s♠❛❧❧ ♥✉♠❜❡r ♦❢ ✉♥✐ts✳ ❋♦r♠✉❧❛t✐♦♥s t♦ ❛❞❞r❡ss s❝❡♥❛r✐♦s ✇✐t❤ ♠✉❧t✐♣❧❡ ✢♦♦rs ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❝♦♥s✐❞❡r❡❞✳ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶✵ ❞❡✈❡❧♦♣❡❞ ❛♥ ▼■▲P ❢♦r♠✉❧❛t✐♦♥ t♦ ❛❧❧♦✇ ❢♦r ❛ ♠✉❧t✐✲✢♦♦r ❧❛②♦✉t ✇✐t❤ ✢♦♦r ❝♦♥str✉❝t✐♦♥ ❝♦sts ✭✜①❡❞ ❛♥❞ ❛r❡❛✲❞❡♣❡♥❞❡♥t✮ ❛♥❞ ✈❡rt✐❝❛❧ ♣✉♠♣✐♥❣ ❝♦sts✳ ❚❤❡ ♠♦❞❡❧ s✉❝❝❡ss❢✉❧❧② ❤❛♥❞❧❡❞ ✉♣ t♦ ✶✶ ✉♥✐ts ✇✐t❤✐♥ ❛ ♠♦❞❡st ❈P❯ t✐♠❡✳ ❋♦r ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ✉♥✐ts✱ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ♣r♦♣♦s❡❞ ❛ r✐❣♦r♦✉s ❞❡❝♦♠♣♦s✐t✐♦♥ ❛♣♣r♦❛❝❤ ❛♥❞ ❛♥ ✐t❡r❛t✐✈❡ s♦❧✉t✐♦♥ s❝❤❡♠❡✱ ✇✐t❤ t❤❡ ❧❛t❡r ♣r♦❞✉❝✐♥❣ ❛ s♦❧✉t✐♦♥ ✇✐t❤ ❛ ❤✐❣❤❡r ❞❡❣r❡❡ ♦❢ ♦♣t✐♠❛❧✐t②✳ ❆♥ ✐♠♣r♦✈❡♠❡♥t✲t②♣❡ ❛❧❣♦r✐t❤♠ ✇❛s ♣r♦♣♦s❡❞ ❜② ❳✉ ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶✸ ❛♣♣❧✐❝❛❜❧❡ t♦ ❛ s✐♥❣❧❡ ✢♦♦r✳ ❆♣❛rt ❢r♦♠ ❛ ❝♦st ♠✐♥✐♠✐s❛t✐♦♥ ♦❜❥❡❝t✐✈❡✱ ♦t❤❡r ❛✉t❤♦rs ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ♦t❤❡r ❝r✐t❡r✐❛ t♦ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳ P❡♥t❡❛❞♦ ❛♥❞ ❈✐r✐❝ ✺ ♣r❡s❡♥t❡❞ ❛ ♠✐①❡❞ ✐♥t❡❣❡r ♥♦♥✲❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✭▼■◆▲P✮ ♣r♦❜❧❡♠ ♥♦t ♦♥❧② t♦ ♠✐♥✐♠✐③❡ ♣✐♣✐♥❣ ❝♦st✱ ❧❛♥❞ ❛❧❧♦❝❛t✐♦♥ ❝♦st✱ ❜✉t ❛❧s♦ ✜♥❛♥❝✐❛❧ r✐s❦ ❛♥❞ s❛❢❡t② ❝♦sts✳ ❖t❤❡r ❢❛❝t♦rs ❤❛✈❡ ❜❡❡♥ t❤❡ r♦✉t✐♥❣ ❛♥❞ ❧❛②♦✉t ♦❢ ♣✐♣❡s ✻ ✱ s❛❢❡t② ❛♥❞ r✐s❦ ❛ss❡ss♠❡♥t ✶✶✱✶✹✕✶✼ ✱ ❧♦♥❣✲❧❡♥❣t❤ ❡q✉✐♣♠❡♥t ✶✼✕✶✾ ✱ ❛r❡❛ ♠✐♥✐♠✐s❛t✐♦♥ ✷✵ ❀ ✇✐t❤ ✈❛r②✐♥❣ s♦❧✉✲ t✐♦♥ t❡❝❤♥✐q✉❡s ✶✼✱✷✶✕✷✺ ✳ ❚❤✐s ♣❛♣❡r ♣r♦♣♦s❡s ❛ ❣❡♥❡r✐❝ ♠♦❞❡❧ t♦ t❤❡ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠✱ ✐♥✐t✐❛❧❧② ♣r♦♣♦s❡❞ ❜② P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✭❛♥ ✐♠♣r♦✈❡♠❡♥t ♦♥ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r✲ ❣✐♦✉ ✶✵ ✮ ❜✉t ❡①t❡♥❞❡❞ t♦ ❛❝❝♦✉♥t ❢♦r ❧♦♥❣✲❧❡♥❣t❤ ❡q✉✐♣♠❡♥t✱ ✐✳❡✳ ❡q✉✐♣♠❡♥t t②♣✐❝❛❧❧② s♣❛♥✲ ♥✐♥❣ ♠✉❧t✐♣❧❡ ✢♦♦rs✱ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ ♣♦✐♥ts ❛t ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❤❡✐❣❤ts ❢♦r ❡❛❝❤ ❡q✉✐♣♠❡♥t✳ ❑✉ ❡t ❛❧✳ ✶✽ ♣r❡s❡♥t❡❞ ❛ ▼■◆▲P ♠♦❞❡❧ ❛♥❞ ❛❝❝♦✉♥t❡❞ ❢♦r t❛❧❧ ❡q✉✐♣♠❡♥t✱ ♠✐♥✐♠✐③✐♥❣ t♦t❛❧ ❧❛②♦✉t ❛r❡❛ ✇✐t❤ ❛ ❤②❜r✐❞ ♦♣t✐♠✐s❛t✐♦♥ ♠❡t❤♦❞✳ ❚❤✐s ✇♦r❦ s❡❡❦s t♦ ♣r♦♣♦s❡ ❛ ♠♦r❡ ❡✣❝✐❡♥t ▼■▲P ♠♦❞❡❧ t♦ ❤❛♥❞❧❡ ❛ ❧❛r❣❡r ♥✉♠❜❡r ♦❢ ✉♥✐ts t❤❛♥ ♣r❡✈✐♦✉s❧② r❡❝♦r❞❡❞✳ ❙②♠♠❡tr② ❜r❡❛❦✲ ✐♥❣ ❝♦♥str❛✐♥ts ❛r❡ ❛❧s♦ ✐♥tr♦❞✉❝❡❞ t♦ r❡❞✉❝❡ s♦❧✉t✐♦♥ s❡❛r❝❤ ♦✈❡r ♠✉❧t✐♣❧❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s✳ ❚❤❡ r❡st ♦❢ t❤✐s ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✿ t❤❡ ♣r♦❜❧❡♠ ✐s ❞❡s❝r✐❜❡❞ ✐♥ s❡❝t✐♦♥ ✷ ✇✐t❤ ✉♥❞❡r❧②✐♥❣ ❛ss✉♠♣t✐♦♥s❀ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❛r❡ t❤❡♥ ♣r♦♣♦s❡❞ ✐♥ s❡❝t✐♦♥ ✸ t♦ ❛❞❞r❡ss t❤❡ ♣r♦❜❧❡♠❀ ❝❛s❡ st✉❞✐❡s t♦ s❤♦✇ ♠♦❞❡❧ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❛♣♣❧✐❝❛❜✐❧✐t② ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ s❡❝t✐♦♥


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✷ P❘❖❇▲❊▼ ❉❊❙❈❘■P❚■❖◆

✺

✹✱ ✇❤❡r❡ ❛❞❞✐t✐♦♥❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ❛r❡ ❛❧s♦ ✐♥tr♦❞✉❝❡❞ ❛♥❞ t❡st❡❞✳ ❋✐♥❛❧❧②✱ ❝♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s ❛r❡ ❣✐✈❡♥ ✐♥ s❡❝t✐♦♥ ✺✳

✷ Pr♦❜❧❡♠ ❉❡s❝r✐♣t✐♦♥ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s ❛r❡ ♠❛❞❡ ✐♥ t❤❡ ♠♦❞❡❧ ❢♦r♠✉❧❛t✐♦♥✳ ❊q✉✐♣♠❡♥t ✐t❡♠s ❛r❡ ❞❡✲ s❝r✐❜❡❞ ❜② r❡❝t❛♥❣✉❧❛r s❤❛♣❡s✳ ■♥ ❛❧❧ ❝❛s❡s✱ r❡❝t✐❧✐♥❡❛r ❞✐st❛♥❝❡s ❛r❡ ❛ss✉♠❡❞✱ ✇✐t❤ ❝♦♥♥❡❝✲ t✐♦♥s ❜❡t✇❡❡♥ ❡q✉✐♣♠❡♥t t❛❦❡♥ ❢r♦♠ t❤❡✐r r❡s♣❡❝t✐✈❡ ❣❡♦♠❡tr✐❝❛❧ ❝❡♥tr❡s ✐♥ t❤❡ ①✲② ♣❧❛♥❡✳ ❱❡rt✐❝❛❧ ❞✐st❛♥❝❡s ❛r❡ ♠❡❛s✉r❡❞ ❢r♦♠ ❛ ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❤❡✐❣❤t ♦❢ ❡❛❝❤ ❡q✉✐♣♠❡♥t✳ ❊❛❝❤ ❡q✉✐♣♠❡♥t ✐s ❛❧❧♦✇❡❞ t♦ r♦t❛t❡ ✐♥ ✾✵♦ ❛♥❣❧❡s ✐♥ t❤❡ ①✲② ♣❧❛♥❡✱ ❛♥❞ ♠✉st st❛rt ❢r♦♠ t❤❡ ❜❛s❡ ♦❢ ❡❛❝❤ ❛✈❛✐❧❛❜❧❡ ✢♦♦r✳ ❚❤❡ ✢♦♦r ❤❡✐❣❤t ✐s ✜①❡❞ ❛❝r♦ss ❛❧❧ ✢♦♦rs ❛♥❞ ✐❢ ❛♥ ❡q✉✐♣♠❡♥t ❡①❝❡❡❞s s✉❝❤ ❤❡✐❣❤t✱ ✐t ✐s ❛❧❧♦✇❡❞ t♦ ❡①t❡♥❞ t❤r♦✉❣❤ ❛✈❛✐❧❛❜❧❡ ✢♦♦rs✳ ❚❤❡ ♣r♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥ ❢♦r t❤❡ ♣r♦❝❡ss ❧❛②♦✉t ♣r♦❜❧❡♠ ✐s ❛s ❢♦❧❧♦✇s✳

●✐✈❡♥ ✿ • ❛ s❡t ♦❢ ◆ ❡q✉✐♣♠❡♥t ✐t❡♠s ❛♥❞ t❤❡✐r ❞✐♠❡♥s✐♦♥s ✭❧❡♥❣t❤✱ ❜r❡❛❞t❤ ❛♥❞ ❤❡✐❣❤t✮❀ • ❛ s❡t ♦❢ ❑ ♣♦t❡♥t✐❛❧ ✢♦♦rs ❢♦r ❧❛②♦✉t❀ • ❝♦♥♥❡❝t✐✈✐t② ♥❡t✇♦r❦ ❛♠♦♥❣st ❡q✉✐♣♠❡♥t❀ • ❝♦st ❞❛t❛ ✭❝♦♥♥❡❝t✐♦♥✱ ♣✉♠♣✐♥❣✱ ❧❛♥❞✱ ❛♥❞ ❝♦♥str✉❝t✐♦♥✮❀ • ✢♦♦r ❤❡✐❣❤t❀ • s♣❛❝❡ ❛♥❞ ❡q✉✐♣♠❡♥t ❛❧❧♦❝❛t✐♦♥ ❧✐♠✐t❛t✐♦♥s❀ • ♠✐♥✐♠✉♠ s❛❢❡t② ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ ❡q✉✐♣♠❡♥t ✐t❡♠s❀

❞❡t❡r♠✐♥❡ ✿ • t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ r❡q✉✐r❡❞ ✢♦♦rs ❢♦r t❤❡ ❧❛②♦✉t❀ • t❤❡ ❜❛s❡ ❧❛♥❞ ❛r❡❛ ♦❝❝✉♣✐❡❞❀ ACS Paragon Plus Environment


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

Page 6 of 33

✻

• t❤❡ ❛r❡❛ ♦❢ ✢♦♦rs❀ • s♣❛t✐❛❧ ❡q✉✐♣♠❡♥t ❛❧❧♦❝❛t✐♦♥ t♦ ✢♦♦rs❀

s♦ ❛s t♦ ✿ ♠✐♥✐♠✐s❡ t❤❡ t♦t❛❧ ♣❧❛♥t ❧❛②♦✉t ❝♦st✳

✸

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛t✐♦♥

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ❝♦♥st✐t✉t❡s ❛♥ ❡①t❡♥s✐♦♥ t♦ t❤❡ ✇♦r❦ ♦❢ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r✲ ❣✐♦✉ ✶ ✳ ❆ ❢✉❧❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ♣r♦♣♦s❡❞ ❜② P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✐s ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳ ❚✇♦ ❢♦r♠✉❧❛t✐♦♥s ✭❆ ✫ ❇✮ ❛r❡ ♣r♦♣♦s❡❞✱ ♣r✐♠❛r✐❧② ❞✐✛❡r✐♥❣ ✐♥ t❤❡ ♠❛♥♥❡r t❛❧❧ ❡q✉✐♣♠❡♥t ✭✇❤✐❝❤ s♣❛♥ ♠✉❧t✐♣❧❡ ✢♦♦rs✮ ❛r❡ ❤❛♥❞❧❡❞✳ ❚❤❡ ✜rst ❤❛♥❞❧❡s ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❛s ✐s✱ ❜② ❛ s❡t ♦❢ ❝♦♥str❛✐♥ts ❛s s✉♠♠❛r✐s❡❞ ✐♥ ❚❛❜❧❡ ✶ ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♠♦❞❡❧s ♣r♦♣♦s❡❞✳ ❚❤❡ s❡❝♦♥❞ ❢♦r♠✉❧❛t✐♦♥ ✐s ❛ ❧✐♥❡❛r✐s❡❞ ❢♦r♠ ♦❢ t❤❡ ♠♦❞❡❧ ♣r♦♣♦s❡❞ ❜② ❑✉ ❡t ❛❧✳ ✶✽ ✭❛ ❞✐r❡❝t ❡①t❡♥s✐♦♥ ♦❢ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✮✱ ✇✐t❤ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❞✐✈✐❞❡❞ ✐♥t♦ ♣s❡✉❞♦✲s✐♥❣❧❡ ✢♦♦r ✉♥✐ts ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥s❡❝✉t✐✈❡ ✢♦♦rs t❤❡② r❡q✉✐r❡✳

❚❛❜❧❡ ✶✿ ❙✉♠♠❛r② ♦❢ ❡q✉❛t✐♦♥s ❛❞♦♣t❡❞ ✐♥ ♠♦❞❡❧s ❆✳✶ ✲ ❆✳✸ ❈♦♥str❛✐♥ts ❋❧♦♦r ❊q✉✐♣♠❡♥t ♦r✐❡♥t❛t✐♦♥ ▼✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❉✐st❛♥❝❡ ◆♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❆r❡❛ ▲❛②♦✉t ❞❡s✐❣♥

❋♦r♠✉❧❛t✐♦♥ ❆ ♠♦❞❡❧ ❆✳✶ ❆✳✷ ❆✳✸ ✶ ✲ ✹✱ ✺ ✲ ✻ ✶ ✲ ✹✱ ✺ ✲ ✻ ✶ ✲ ✹✱ ✺ ✲ ✻ ✾✱ ✼ ✲ ✽ ✾✱ ✼ ✲ ✽ ✾✱ ✼ ✲ ✽ ✶✵✲✶✹ ✶✺ ✶✷✱ ✶✸ ❛♥❞ ✶✻ ✷✹✱✷✶✱ ✷✷✱ ✶✵✱ ✶✷✱ ✷✹ ❛♥❞ ✷✹✱✷✶✱ ✷✷✱ ✷✶✱ ✷✷✱ ✷✸✱ ❛♥❞ ✷✸ ❛♥❞ ✷✸ ✶✼ ✲ ✷✵ ✶✼ ✲ ✷✵ ✶✼ ✲ ✷✵ ✸✶ ✲ ✸✻ ✸✶ ✲ ✸✻ ✸✶ ✲ ✸✻ ✸✵✱ ✷✺ ✲ ✷✾ ✸✵✱ ✷✺ ✲ ✷✾ ✸✵✱ ✷✺ ✲ ✷✾


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✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆ ◆♦♠❡♥❝❧❛t✉r❡

■♥❞✐❝❡s i,j

❡q✉✐♣♠❡♥t ✐t❡♠ ✐♥ ♠♦❞❡❧s ❆✳✶ ✲ ❆✳✸

i′ ,j ′

❡q✉✐♣♠❡♥t ✐t❡♠ ✐♥ ♠♦❞❡❧ ❇

k, k ′

✢♦♦r ♥✉♠❜❡r

p

♣s❡✉❞♦ ✉♥✐ts

s

r❡❝t❛♥❣✉❧❛r ❛r❡❛ s✐③❡s

❙❡ts I

s❡t ♦❢ ❡q✉✐♣♠❡♥t ✐t❡♠ ❢♦r ♠♦❞❡❧s ❆✳✶✲ ❆✳✸

I′

s❡t ♦❢ ❡q✉✐♣♠❡♥t ✐t❡♠ ❢♦r ♠♦❞❡❧ ❇

MF

s❡t ♦❢ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥ts

Pi

s❡ts ♦❢ ♣s❡✉❞♦ ✉♥✐ts ❢♦r ❡q✉✐♣♠❡♥t i

P❛r❛♠❡t❡rs αi ✱ βi ,γi

❞✐♠❡♥s✐♦♥s ♦❢ ✐t❡♠ i

Mi

♥✉♠❜❡r ♦❢ ✢♦♦rs r❡q✉✐r❡❞ ❜② ✐t❡♠ i

fij

✶ ✐❢ ✢♦✇ ❞✐r❡❝t✐♦♥ ❜❡t✇❡❡♥ i ❛♥❞ j ✐s ♣♦s✐t✐✈❡❀ ✵✱ ♦t❤❡r✇✐s❡

OPij

❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❜❛s❡ ❛♥❞ ♦✉t♣✉t ♣♦✐♥t ♦♥ ❡q✉✐♣♠❡♥t i

IPij

❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❜❛s❡ ❛♥❞ ✐♥♣✉t ♣♦✐♥t ♦♥ ❡q✉✐♣♠❡♥t j

BM

❛ ❧❛r❣❡ ♥✉♠❜❡r

FH

✢♦♦r ❤❡✐❣❤t

LC

❧❛♥❞ ❝♦st

F C1

✜①❡❞ ✢♦♦r ❝♦♥str✉❝t✐♦♥ ❝♦st

F C2

❛r❡❛✲❞❡♣❡♥❞❡♥t ✢♦♦r ❝♦♥str✉❝t✐♦♥ ❝♦st

Cijc

❝♦♥♥❡❝t✐♦♥✴♣✐♣✐♥❣ ❝♦sts ACS Paragon Plus Environment

✼


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

Cijh

❤♦r✐③♦♥t❛❧ ♣✉♠♣✐♥❣ ❝♦sts

Cijv

✈❡rt✐❝❛❧ ♣✉♠♣✐♥❣ ❝♦sts

■♥t❡❣❡r ✈❛r✐❛❜❧❡s NF

♥✉♠❜❡r ♦❢ ✢♦♦rs r❡q✉✐r❡❞ ❢♦r ❧❛②♦✉t

❇✐♥❛r② ✈❛r✐❛❜❧❡s Oi

✶ ✐❢ ❧❡♥❣t❤ ♦❢ ✐t❡♠ i ✐s ❡q✉❛❧ t♦ αi ❀ ✵✱ ♦t❤❡r✇✐s❡

Wk

✶ ✐❢ ✢♦♦r k ✐s ♦❝❝✉♣✐❡❞❀ ✵✱ ♦t❤❡r✇✐s❡

nijk

✶ ✐❢ ✐t❡♠s i ❛♥❞ j ❛r❡ ❛ss✐❣♥❡❞ t♦ ✢♦♦r k ❀ ✵✱ ♦t❤❡r✇✐s❡

Nij

✶ ✐❢ ✐t❡♠s i ❛♥❞ j ❛r❡ ❛ss✐❣♥❡❞ t♦ t❤❡ s❛♠❡ ✢♦♦r❀ ✵✱ ♦t❤❡r✇✐s❡

Vik

✶ ✐❢ ✐t❡♠ i ✐s ❛ss✐❣♥❡❞ t♦ ✢♦♦r k

Qs

✶ ✐❢ r❡❝t❛♥❣✉❧❛r ❛r❡❛ s ✐s s❡❧❡❝t❡❞ ❢♦r t❤❡ ❧❛②♦✉t❀ ✵✱ ♦t❤❡r✇✐s❡

E1ij ✱ E2ij

♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❜✐♥❛r②✱ ❛ s❡t ♦❢ ✈❛❧✉❡s ♦❢ ✇❤✐❝❤ ♣r❡✈❡♥ts ❡q✉✐♣♠❡♥t ♦✈❡r❧❛♣ ✐♥ ♦♥❡ ❞✐r❡❝t✐♦♥ ✐♥ t❤❡ ①✲② ♣❧❛♥❡

s Sik

✶ ✐❢ ✐t❡♠ i s♣❛♥s ♠♦r❡ t❤❛♥ ♦♥❡ ✢♦♦r ❛♥❞ ❜❡❣✐♥s ❛t ✢♦♦r k ❀ ✵✱ ♦t❤❡r✇✐s❡

f Sik

✶ ✐❢ ✐t❡♠ i s♣❛♥s ♠♦r❡ t❤❛♥ ♦♥❡ ✢♦♦r ❛♥❞ t❡r♠✐♥❛t❡s ❛t ✢♦♦r k ❀ ✵✱ ♦t❤❡r✇✐s❡

❈♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s li

❧❡♥❣t❤ ♦❢ ✐t❡♠ i

di

❜r❡❛❞t❤ ♦❢ ✐t❡♠ i

hi

❤❡✐❣❤t ♦❢ ✐t❡♠ i

xi ✱ yi ✱ z i

❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝❛❧ ❝❡♥tr❡ ♦❢ ✐t❡♠ i

Rij

r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ① ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱


Page 8 of 33

✽

Page 9 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

✾

✐❢ i ✐s t♦ t❤❡ r✐❣❤t ♦❢ j r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ① ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱

Lij

✐❢ i ✐s t♦ t❤❡ ❧❡❢t ♦❢ j r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ② ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱

Aij

✐❢ i ✐s ❛❜♦✈❡ j r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ② ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱

Bij

✐❢ i ✐s ❜❡❧♦✇ j r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ③ ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱

Uij

✐❢ i ✐s ❤✐❣❤❡r t❤❛♥ j r❡❧❛t✐✈❡ ❞✐st❛♥❝❡ ✐♥ ③ ❝♦♦r❞✐♥❛t❡s ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j ✱

Dij

✐❢ i ✐s ❧♦✇❡r t❤❛♥ j

T Dij

t♦t❛❧ r❡❝t✐❧✐♥❡❛r ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✐t❡♠s i ❛♥❞ j

N Qs

❧✐♥❡❛r✐s❛t✐♦♥ ✈❛r✐❛❜❧❡ ❡①♣r❡ss✐♥❣ t❤❡ ♣r♦❞✉❝t ♦❢ N F ❛♥❞ Qs

ARs

♣r❡❞❡✜♥❡❞ r❡❝t❛♥❣✉❧❛r ✢♦♦r ❛r❡❛ s

FA

✢♦♦r ❛r❡❛

X max , Y max

❞✐♠❡♥s✐♦♥s ♦❢ ✢♦♦r ❛r❡❛

✸✳✶

❋♦r♠✉❧❛t✐♦♥ ❆

■♥ ❢♦r♠✉❧❛t✐♦♥ ❆✱ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥ts ❛r❡ tr❡❛t❡❞ ❛s ❛ s✐♥❣❧❡ ✉♥✐t st❛rt✐♥❣ ♦♥ ♦♥❡ ✢♦♦r

k ✱ s♣❛♥♥✐♥❣ t❤r♦✉❣❤ ❝♦♥s❡❝✉t✐✈❡ ✢♦♦rs ❛♥❞ t❡r♠✐♥❛t✐♥❣ ❛t ❛ ❤✐❣❤❡r ✢♦♦r k ′ ✳ ❋❧♦♦rs ❛r❡ ♥✉♠❜❡r❡❞ ❢r♦♠ ❜♦tt♦♠ t♦ t♦♣✳ ❚❤r❡❡ ❡q✉✐✈❛❧❡♥t ♠♦❞❡❧s ✲ ❆✳✶✱ ❆✳✷ ❛♥❞ ❆✳✸ ❛r❡ ❞❡s❝r✐❜❡❞ ✇❤✐❝❤ ♦♥❧② ❞✐✛❡r ❜② t❤❡ s❡ts ♦❢ ❝♦♥str❛✐♥ts ✉s❡❞ t♦ ♠♦❞❡❧ t❤❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t✳

✸✳✶✳✶ ❋❧♦♦r ❝♦♥str❛✐♥ts ❊✈❡r② ❡q✉✐♣♠❡♥t i ❛✈❛✐❧❛❜❧❡ ✐s ❛ss✐❣♥❡❞ t♦ ❛♥ ❡q✉✐✈❛❧❡♥t ♥✉♠❜❡r ♦❢ ✢♦♦rs✱ Mi ✳



✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

X

Page 10 of 33

✶✵

Vik = Mi

∀i

✭✶✮

k

✇❤❡r❡ Vik ✐s ❛ ❜✐♥❛r② ✈❛r✐❛❜❧❡ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s ✐❢ ❛♥ ❡q✉✐♣♠❡♥t i ✐s ❛ss✐❣♥❡❞ t♦ ✢♦♦r k. ❆ ✈❛r✐❛❜❧❡✱ nijk ✱ ✐s ✐♥tr♦❞✉❝❡❞ t♦ ❞❡t❡r♠✐♥❡ ✐❢ ❡q✉✐♣♠❡♥t i ❛♥❞ j ❛r❡ ❛ss✐❣♥❡❞ t♦ ✢♦♦r k ✱ ❣✐✈❡♥ ❜②✿

nijk ≥ Vik + Vjk − 1

∀ i = 1, ..., N − 1, j 6= i, k = 1, ..., K ∀ i = 1, ..., N − 1, j 6= i, k = 1, ..., K

Nij ≥ nijk

✭✷✮ ✭✸✮

❚❤❡ ✈❛r✐❛❜❧❡ Nij t❛❦❡s t❤❡ ✈❛❧✉❡ ♦❢ ✶ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t❡♠s i ❛♥❞ j ❛r❡ ♦♥ ❛♥② s❛♠❡ ✢♦♦r✳ s ❋✉rt❤❡r♠♦r❡✱ ❛ ✢♦♦r ✇✐❧❧ ♦♥❧② ❡①✐st ✐❢ ❛♥ ❡q✉✐♣♠❡♥t st❛rts ♦♥ ✐t✳ ❋♦r t❤✐s✱ ♥❡✇ ✈❛r✐❛❜❧❡s Sik f ❛♥❞ Sik ′ ❛r❡ ✐♥tr♦❞✉❝❡❞ t❛❦✐♥❣ ✈❛❧✉❡s ♦❢ ✶ ✐❢ ❛♥ ❡q✉✐♣♠❡♥t i st❛rts ❛t ✢♦♦r k ❛♥❞ t❡r♠✐♥❛t❡s

❛t ✢♦♦r k ′ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✉s✿ s ≤ Wk Sik

∀ i, k

✭✹✮

❆❧s♦✱ ❛ ✢♦♦r s❤♦✉❧❞ ❡①✐st ♦♥❧② ✐❢ ❛♥ ❡q✉✐♣♠❡♥t ✐s ❛ss✐❣♥❡❞ t♦ ✐t✱ ❛♥❞ ♥♦t ❥✉st ♣❛s✐♥❣ t❤r♦✉❣❤ ✐t ✭❢♦r ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t✮✱ ❛♥❞ t❤❛t t❤❡ ♣r❡❝❡❞✐♥❣ ✢♦♦r ✐s ❛❧s♦ ♦❝❝✉♣✐❡❞✳ ❚❤✐s ❝♦♥str❛✐♥t ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✿

∀ k = 2, ..., K

Wk ≤ Wk−1

✭✺✮

❚❤❡ ♥✉♠❜❡r ♦❢ ✢♦♦rs r❡q✉✐r❡❞ ✐s t❤❡♥ ❞❡t❡r♠✐♥❡❞ ❜② ❡q✉❛t✐♦♥ ✻ ✶ ✳

NF ≥

X

Wk

k


✭✻✮

Page 11 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

✶✶

✸✳✶✳✷ ❊q✉✐♣♠❡♥t ♦r✐❡♥t❛t✐♦♥ ❝♦♥str❛✐♥ts ❆ ✾✵◦ r♦t❛t✐♦♥ ♦❢ ❡q✉✐♣♠❡♥t ♦r✐❡♥t❛t✐♦♥ ✐s ❛❧❧♦✇❡❞ ✐♥ t❤❡ ①✲② ♣❧❛♥❡✳ ❚❤✐s ✐s r❡♣r❡s❡♥t❡❞ ❜② ❡q✉❛t✐♦♥s ✼ ❛♥❞ ✽ ✶ ✳

li = αi Oi + βi (1 − Oi )

∀i

di = αi + βi − li

∀i

✭✼✮ ✭✽✮

❘♦t❛t✐♦♥ ✐♥ t❤❡ ③✲❛①✐s ✐s ❞❡❡♠❡❞ ✉♥r❡❛❧✐st✐❝ ❛s s✉❝❤ ❡q✉✐♣♠❡♥t ♦r✐❡♥t❛t✐♦♥ ✐s ✜①❡❞ ❢♦r ❝♦♥✲ str✉❝t✐♦♥ ✐♥ ✈✐rt✉❛❧❧② ❛❧❧ ❝❛s❡s✳

hi = γ i

✭✾✮

∀i

✸✳✶✳✸ ▼✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❝♦♥str❛✐♥ts ■♥ ♦r❞❡r ❢♦r ❡q✉✐♣♠❡♥t ✐t❡♠s r❡q✉✐r✐♥❣ ♠♦r❡ t❤❛♥ ♦♥❡ ✢♦♦r t♦ s♣❛♥ ❛❝r♦ss s✉❝❝❡ss✐✈❡ ✢♦♦rs✱ t❤❡ ❝♦♥str❛✐♥ts ❜❡❧♦✇ ❛r❡ ✐♥tr♦❞✉❝❡❞✳ ❚❤r❡❡ ❛❧t❡r♥❛t✐✈❡ ♠♦❞❡❧s ❛r❡ s❤♦✇♥ ✭▼♦❞❡❧s ❆✳✶✱ ❆✳✷ ❛♥❞ ❆✳✸✮✳

▼♦❞❡❧ ❆✳✶ ✐♥❝❧✉❞❡s ❡q✉❛t✐♦♥s ✶✵ ✲ ✶✹✳ s −Vik + Vi,k−1 + Sik ≥0

∀ i, k

✭✶✵✮

f ≥0 −Vik + Vi,k+1 + Sik

∀ i, k

✭✶✶✮

f s ❊q✉❛t✐♦♥s ✶✵ ❛♥❞ ✶✶ ❡♥s✉r❡s t❤❛t t❤❡ ❜✐♥❛r② ✈❛r✐❛❜❧❡s Sik t❛❦❡ ❛ ✈❛❧✉❡ ♦❢ ✶ ✐❢ ❛♥ ❛♥❞ Sik

❡q✉✐♣♠❡♥t st❛rts ❛♥❞ ❡♥❞s ♦♥ ❛ ♣❛rt✐❝✉❧❛r ✢♦♦r✱ ✇✐t❤ ❡❛❝❤ ❡q✉✐♣♠❡♥t ❜❡✐♥❣ r❡str✐❝t❡❞ t♦ st❛rt ❛♥❞ ❡♥❞ ♦♥ ♦♥❧② ♦♥❡ ✢♦♦r ❜②✿

X

s Sik =1

∀i

✭✶✷✮

X

f =1 Sik

∀i

✭✶✸✮

k

k



Page 12 of 33

✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

✶✷

❚❤❡ ❝♦♥str❛✐♥t t♦ r❡str✐❝t ❡q✉✐♣♠❡♥t t♦ ♦❝❝✉♣② s✉❝❝❡ss✐✈❡ ✢♦♦rs ✐s t❤❡♥ ❣✐✈❡♥ ❛s✿ k+M Xi

s Vik′ ≥ Mi .Sik

✭✶✹✮

∀ i, k

k′ =k

▼♦❞❡❧ ❆✳✷ ♠♦❞❡❧s t❤❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❝♦♥str❛✐♥t ❜② t❤❡ ❡q✉❛t✐♦♥ ❜❡❧♦✇✿ M i −1 X

Vi,k+θ ≥ (M i − 1).(Vik − Vi,k−1 )

∀ i, k

✭✶✺✮

θ=1

❊q✉❛t✐♦♥ ✶✺ ❡♥s✉r❡s t❤❛t ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ♦❝❝✉♣② ❝♦♥s❡❝✉t✐✈❡ ✢♦♦rs✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ❡q✉❛❧❧✐♥❣ t❤❡ r❡q✉✐r❡❞ ♥✉♠❜❡r ♦❢ ✢♦♦rs Mi ✳

▼♦❞❡❧ ❆✳✸ ♠❛❦❡s ✉s❡ ♦❢ t❤❡ ❜✐♥❛r② ✈❛r✐❛❜❧❡s Siks ❛♥❞ Sikf ❛♥❞ ✐s ❞❡s❝r✐❜❡❞ ✉s✐♥❣ ❡q✉❛t✐♦♥s ✶✷✱ ✶✸ ❛♥❞ ✶✻ ❜❡❧♦✇✳ f s Vik − Vi,k−1 = Sik − Si,k−1

∀ i, k

✭✶✻✮

■♥ ❛❧❧ ❝❛s❡s ✭♠♦❞❡❧s ❆✳✶✱ ❆✳✷✱ ❛♥❞ ❆✳✸✮ ❢♦r t❤❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t✱ ❡✈❡r② ❡q✉✐♣♠❡♥t s ✐t❡♠ i s♣❛♥♥✐♥❣ ♠♦r❡ t❤❛♥ ♦♥❡ ✢♦♦r st❛rts ♦♥ ❛ s✐♥❣❧❡ ✢♦♦r ✭Sik = 1✮ ❛♥❞ t❡r♠✐♥❛t❡s ♦♥ f ❛♥♦t❤❡r ✢♦♦r ✭Sik ′ = 1✮✱ ♦❝❝✉♣②✐♥❣ s✉❝❝❡ss✐✈❡ ✢♦♦rs✱ t❤❡ s✉♠ ♦❢ ✇❤✐❝❤ ✐s ❡q✉❛❧ t♦ ✐ts r❡q✉✐r❡❞

♥✉♠❜❡r ♦❢ ✢♦♦rs ✭Mi ✮✳


Page 13 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

✶✸

✸✳✶✳✹ ◆♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❝♦♥str❛✐♥ts ❚♦ ♣r❡✈❡♥t t✇♦ ♦r ♠♦r❡ ❡q✉✐♣♠❡♥t t♦ ♦❝❝✉♣② t❤❡ s❛♠❡ s♣❛❝❡ ✇✐t❤✐♥ ❛ ✢♦♦r✱ ❝♦♥str❛✐♥ts ✶✼ ✲ ✷✵ ❛r❡ ✐♥tr♦❞✉❝❡❞ ✶ ✿

li + lj 2

xi − xj + BM (1 − Ni,j + E1ij + E2ij ) ≥

∀ i = 1, ..., N − 1, j = 2, ..., N ✭✶✼✮

xj − xi + BM (2 − Ni,j − E1ij + E2ij ) ≥

li + lj 2

∀ i = 1, ..., N − 1, j = 2, ..., N ✭✶✽✮

yi − yj + BM (2 − Ni,j + E1ij − E2ij ) ≥

di + dj 2

∀ i = 1, ..., N − 1, j = 2, ..., N ✭✶✾✮

yj − yi + BM (3 − Nij − E1ij − E2ij ) ≥

di + dj 2

∀ i = 1, ..., N − 1, j = 2, ..., N ✭✷✵✮

✸✳✶✳✺ ❉✐st❛♥❝❡ ❝♦♥str❛✐♥ts ❉✐st❛♥❝❡ ❝♦♥str❛✐♥ts ❛r❡ ❞❡s❝r✐❜❡❞ ❜② ❡q✉❛t✐♦♥s ✷✶✱ ✷✷ ❛♥❞ ✷✸ ✶ ✱ t♦ ❞❡t❡r♠✐♥❡ t❤❡ r❡❧❛t✐✈❡ ❞✐st❛♥❝❡s ✐♥ t❤❡ ① ❛♥❞ ② ❝♦♦r❞✐♥❛t❡s r❡s♣❡❝t✐✈❡❧②✳

Rij − Lij = xi − xj

∀(i, j) : fij = 1

✭✷✶✮

Aij − Bij = yi − yj

∀(i, j) : fij = 1

✭✷✷✮

T Dij = Rij + Lij + Aij + Bij + Uij + Dij

∀(i, j) : fij = 1

✭✷✸✮

Pr♦✈✐s✐♦♥ ✐s ♠❛❞❡ ❢♦r ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❡q✉✐♣♠❡♥t i ❛♥❞ j ❛t ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❤❡✐❣❤ts ♦❢ ❡✐t❤❡r ❡q✉✐♣♠❡♥t ❛s ❞❡s❝r✐❜❡❞ ❜② ❡q✉❛t✐♦♥ ✷✹❀

Uij − Dij = F H

X

s s (k − 1)(Sik − Sjk ) + OPij − IPij

∀(i, j) : fij = 1

✭✷✹✮

k

✇❤❡r❡ OPij r❡♣r❡s❡♥ts t❤❡ ✈❡rt✐❝❛❧ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❜❛s❡ ♦❢ ❡q✉✐♣♠❡♥t i t♦ ✐ts ♦✉t♣✉t ♣♦✐♥t✱




Page 14 of 33

Page 15 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

✶✺

❚❤❡ ❧❛♥❞ ❛r❡❛ ✐s t❤❡♥ ❝❛❧❝✉❧❛t❡❞ ❛s✿

F A = X max Y max

✭✷✾✮

❆♥ ❛❞❞✐t✐♦♥❛❧ ❧❛②♦✉t ❞❡s✐❣♥ ❝♦♥str❛✐♥t ✐s ✐♥❝❧✉❞❡❞ ❢♦r t❤❡ ③✲❝♦♦r❞✐♥❛t❡s t♦ ❞❡✜♥❡ ✐ts ❣❡♦♠❡t✲ r✐❝❛❧ ❝❡♥tr❡✱ ❣✐✈❡♥ ❜②✿

zi =

hi 2

∀i

✭✸✵✮

❊q✉❛t✐♦♥ ✸✵ ❛❧s♦ ❡♥s✉r❡s ❡q✉✐♣♠❡♥t i st❛rts ❢r♦♠ t❤❡ ❜❛s❡ ♦❢ t❤❡ ✢♦♦r k ✇❤❡r❡ ✐t ✐s ♣❧❛❝❡❞✳

✸✳✶✳✼ ❆r❡❛ ❈♦♥str❛✐♥ts ■♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❜✐❧✐♥❡❛r t❡r♠s ✐♥ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ✢♦♦r ❛r❡❛✱ F A ✭❡q✳ ✷✾✮✱ ❡q✉❛t✐♦♥s ✸✶ ✲ ✸✻ ✶ ❛r❡ ✐♥tr♦❞✉❝❡❞✳ ❚❤❡ ❛r❡❛ ♦❢ ❡❛❝❤ ✢♦♦r ✐s ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❛ s❡t ♦❢ S ♣r❡❞❡✜♥❡❞ r❡❝t❛♥❣✉❧❛r ❛r❡❛ s✐③❡s✱ ARs ✱ ✇✐t❤ ❞✐♠❡♥s✐♦♥s (X s , Y s )✳

FA =

X

ARs Qs

✭✸✶✮

s

X

Qs = 1

✭✸✷✮

s

❚❤❡ ✢♦♦r ❧❡♥❣t❤ ❛♥❞ ❜r❡❛❞t❤ ✐s s❡❧❡❝t❡❞ ❢r♦♠ t❤❡ ❝❤♦s❡♥ r❡❝t❛♥❣✉❧❛r ❛r❡❛ s✐③❡ ❞✐♠❡♥s✐♦♥s✿

X max =

X

X s Qs

✭✸✸✮

X

Y s Qs

✭✸✹✮

s

Y max =

s



✸ ▼❆❚❍❊▼❆❚■❈❆▲ ❋❖❘▼❯▲❆❚■❖◆

Page 16 of 33

✶✻

❆❧s♦✱ ❛ ♥❡✇ t❡r♠ N Qs ✐s ✐♥tr♦❞✉❝❡❞ ✐♥ ♦r❞❡r t♦ ❧✐♥❡❛r✐s❡ t❤❡ ❝♦st t❡r♠ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ✢♦♦rs✿

N Qs ≤ K Qs ∀s X NF = N Qs

✭✸✺✮ ✭✸✻✮

s

✸✳✶✳✽ ❖❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❚❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❢♦r ❡❛❝❤ ♠♦❞❡❧ ✲ t♦ ♠✐♥✐♠✐③❡ t❤❡ t♦t❛❧ ❝♦st ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❝♦♥♥❡❝t✐♦♥ ❝♦st✱ ♣✉♠♣✐♥❣ ❝♦st✱ ❧❛♥❞ ❛r❡❛ ❝♦st✱ ✢♦♦r ❝♦♥str✉❝t✐♦♥ ❝♦st ❛♥❞ ✢♦♦r✲❛r❡❛ ❞❡♣❡♥❞❡♥t ❝♦st✳ ❚❤✐s ✐s ❣✐✈❡♥ ❛s✿

min

X X

[Cijc T Dij + Cijv Dij + Cijh (Rij + Lij + Aij + Bij )]

i j6=i:fij =1

✭✸✼✮

+F C1 · N F + F C2

X

ARs · N Qs + LC · F A

s

✸✳✷

❋♦r♠✉❧❛t✐♦♥ ❇

❋♦r♠✉❧❛t✐♦♥ ❇ ✐s ❛ ❧✐♥❡❛r✐s❡❞ ♠♦❞❡❧ ♦❢ ❑✉ ❡t ❛❧✳ ✶✽ ✇✐t❤ ❛❞❛♣t❛t✐♦♥s ❢r♦♠ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ❛s ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛❞❞✐t✐♦♥❛❧ ❝♦♥✲ str❛✐♥ts ❛r❡ ✐♥❝❧✉❞❡❞ t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❛♥❞ ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❝♦♥♥❡❝t✐♦♥ ❤❡✐❣❤t✳

✸✳✷✳✶ ▼✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❝♦♥str❛✐♥t ▼✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❛r❡ s♣❧✐t ✐♥t♦ ♣s❡✉❞♦ ✉♥✐ts ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♥✉♠❜❡r ♦❢ ✢♦♦rs t❤❡② ♦❝❝✉♣② ✭❋✐❣✉r❡ ✷✮✳ ❚❤✐s ❛♣♣r♦❛❝❤ ♣r❡s❡♥ts ❛ ♠♦r❡ ❛❝❝✉r❛t❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t t❤❛t ❤❛✈❡ ✈❛r②✐♥❣ s✐③❡s ❛❧♦♥❣ ✐ts ❤❡✐❣❤t ✲ ❞✉❡ t♦ ❛ r❛♥❣❡ ✭✐♥ t②♣❡ ❛♥❞ s✐③❡✮ ♦❢ ❛✉①✐❧✐❛r② ✉♥✐ts ♣❧❛❝❡❞ ❜② s✉❝❤ ❡q✉✐♣♠❡♥t✳ ❚❤❡ ❞r❛✇❜❛❝❦✱ ❤♦✇❡✈❡r✱ ✐s t❤❛t ❛♥ ✐♥❝r❡❛s❡ ✐♥ t❤❡ ♥✉♠❜❡r ♦❢ ✉♥✐ts ✐♥ t❤❡ ▼■▲P ♠♦❞❡❧ ✐s r❡❛❧✐s❡❞ ✇❤✐❝❤ ❝❛♥ ❧❡❛❞ t♦ ❧❛r❣❡r ❝♦♠♣✉t❛t✐♦♥❛❧ ACS Paragon Plus Environment

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Page 18 of 33

✹ ❈❆❙❊ ❙❚❯❉■❊❙

✶✽

❆♥❞ ❢♦r ♣s❡✉❞♦ ✉♥✐ts t♦ ❜❡ ♣❧❛❝❡❞ ❞✐r❡❝t❧② ❛❜♦✈❡ ✐t✬s ♣r❡❝❡❞✐♥❣ ❝♦✉♥t❡r♣❛rt✿

xp = xp+1

∀ pǫPi

✭✸✾✮

yp = yp+1

∀ pǫPi

✭✹✵✮

❛♥❞ ✜♥❛❧❧②✱ ❝♦♥s✐st❡♥❝② ✐♥ ✾✵♦ r♦t❛t✐♦♥ ✐s ❡♥s✉r❡❞ ❢♦r ❛❧❧ ♣s❡✉❞♦ ✉♥✐ts ♦❢ ❛ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣✲ ♠❡♥t✿

Op = Op+1

✭✹✶✮

∀ pǫPi

✸✳✷✳✷ ❉✐st❛♥❝❡ ❝♦♥str❛✐♥ts ❋♦r ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❡q✉✐♣♠❡♥t i′ ❛♥❞ j ′ ❛t ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ❤❡✐❣❤ts ♦❢ ❡✐t❤❡r ❡q✉✐♣♠❡♥t✱ t❤❡ ❝♦♥str❛✐♥t ❜❡❧♦✇ ✐s ✉s❡❞❀

Ui′ j ′ − Di′ j ′ = F H

X

∀(i′ , j ′ )ǫI ′ : fi′ j ′ = 1 ✭✹✷✮

(k − 1)(Vi′ k − Vj ′ k ) + OPi′ j ′ − IPi′ j ′

k

❚❤❡ r❡s✉❧t✐♥❣ ♠♦❞❡❧ ✭❇✮ t❤✉s ❝♦♥s✐sts ♦❢ ❝♦♥str❛✐♥ts ✸✽ ✲ ✹✷❀ ❛♥❞ ❡q✉❛t✐♦♥s ❙✶✲❙✶✺❀ ❙✶✼✲❙✷✽ ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ♠♦❞❡❧ ♦❢ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ❞❡s❝r✐❜❡❞ ✐♥ t❤❡ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ∀ i′ , j ′ ǫ I ′ ❀ ❛♥❞ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✿

min

X

X

[Cic′ j ′ T Di′ j ′ + Civ′ j ′ Di′ j ′ + Cih′ j ′ (Ri′ j ′ + Li′ j ′ + Ai′ j ′ + Bi′ j ′ )]

i′ j ′ 6=i′ :fi′ j ′ =1

✭✹✸✮

+F C1 · N F + F C2

X

ARs · N Qs + LC · F A

s

✹

❈❛s❡ ❙t✉❞✐❡s

■♥ t❤✐s s❡❝t✐♦♥✱ ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝❛s❡ st✉❞✐❡s ❛♣♣❧✐❡❞ t♦ ❡❛❝❤ ❢♦r♠✉❧❛t✐♦♥ ✐s s❤♦✇♥✳ ❊❛❝❤ ❡①❛♠♣❧❡ ✇❛s ♠♦❞❡❧❧❡❞ ✉s✐♥❣ ●❆▼❙ ✷✻ ♠♦❞❡❧❧✐♥❣ s②st❡♠ ✈✷✹✳✼✳✶ ✇✐t❤ t❤❡ ❈P▲❊❳ ✈✶✷✳✻ ❘ ❊✺✲✶✻✺✵ ❈P❯ ✇✐t❤ ✸✷●❇ ❘❆▼✱ ❛♥❞ t❤❡ ❧❛②♦✉t ♣❧♦tt❡❞ ✉s✐♥❣ ❘ ❳❡♦♥ s♦❧✈❡r ♦♥ ❛♥ ■♥t❡❧ ❘ ✷✵✶✼✳ ❊❛❝❤ r✉♥ ✇❛s s♦❧✈❡❞ t♦ ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t②✳ ❋♦r t❤❡ ✢♦♦r ❛r❡❛✱ ❆✉t♦❞❡s❦ ❆✉t♦❈❆❉


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✹ ❈❆❙❊ ❙❚❯❉■❊❙

✶✾

✜✈❡ ❛❧t❡r♥❛t✐✈❡ s✐③❡s ✭✶✵♠✱ ✷✵♠✱ ✸✵♠✱ ✹✵♠ ❛♥❞ ✺✵♠✮ ✇❡r❡ ✉s❡❞ ✐♥ ❡①❛♠♣❧❡s ✶ ❛♥❞ ✷✱ s❡✈❡♥ ❛❧t❡r♥❛t✐✈❡ s✐③❡s ✭✶✵♠✱ ✷✵♠✱ ✸✵♠✱ ✹✵♠✱ ✺✵♠✱ ✻✵♠ ❛♥❞ ✼✵♠✮ ✐♥ ❡①❛♠♣❧❡ ✸✱ ❣✐✈✐♥❣ ❛ t♦t❛❧ ♦❢ ✷✺ ❛♥❞ ✹✾ ♣♦ss✐❜❧❡ ❛r❡❛ s✐③❡s r❡s♣❡❝t✐✈❡❧②✱ ❡①❝❡♣t ♦t❤❡r✇✐s❡ st❛t❡❞✳ ❉❛t❛ ♦♥ ❝♦♥♥❡❝t✐✈✐t② ❝♦sts ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ♣r♦❝❡ss s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ♦❢ ❛ ❝❛s❡ st✉❞②✳ P✐♣❡ s✐③✐♥❣ ✈❛❧✉❡s ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❞❡s✐❣♥ ❡q✉❛t✐♦♥s✱ ♠❛t❝❤❡❞ ✇✐t❤ ❝♦♠♠❡r❝✐❛❧❧② ❛✈❛✐❧❛❜❧❡ ♣✐♣❡ s✐③❡s ❛♥❞ t❤❡ ❝♦sts ❡st✐♠❛t❡❞ ♣❡r ✉♥✐t ❧❡♥❣t❤ ❢♦r ❛ s❡❧❡❝t❡❞ ♠❛t❡r✐❛❧ ♦❢ ❝♦♥str✉❝t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡ ❝♦♠♣♦♥❡♥ts ✐♥ ❡❛❝❤ str❡❛♠✳ P✉♠♣✐♥❣ ❝♦sts ❛r❡ ❡st✐♠❛t❡❞ ❜❛s❡❞ ♦♥ ❝✉rr❡♥t ❡❧❡❝tr✐❝✐t② ♣r✐❝❡s ❛♥❞ ♠❛t❡r✐❛❧ ✢♦✇ r❛t❡ ✐♥ ❡❛❝❤ str❡❛♠✱ ❛♥❞ ❝♦♥str✉❝t✐♦♥ ❝♦sts ❛r❡ ❡①tr❛♣♦❧❛t❡❞ ❢r♦♠ ♣❛st ❞❛t❛ ♦♥ s✐♠✐❧❛r ♣❧❛♥t ❞❡s✐❣♥ ❜❛s❡❞ ♦♥ str✉❝t✉r❛❧ ❡♥❣✐♥❡❡r✐♥❣ ❝❛❧❝✉❧❛t✐♦♥s✳ ❋♦r ❛❧❧ ❝❛s❡ st✉❞✐❡s✱ ❞❛t❛ ♦♥ ❡q✉✐♣♠❡♥t ❞✐♠❡♥s✐♦♥s✱ ❝♦♥♥❡❝t✐✈✐t② ❛♥❞ ❝♦♥str✉❝t✐♦♥ ❝♦sts✱ ❛♥❞ ✈❡rt✐❝❛❧ ❝♦♥♥❡❝t✐♦♥ ♣♦✐♥ts ❛r❡ ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳ ❋♦r ❛❧❧ t❤r❡❡ ❡①❛♠♣❧❡s ✐♥✈❡st✐❣❛t❡❞ ✐♥ t❤✐s ✇♦r❦✱ ❜♦t❤ ❢♦r♠✉❧❛t✐♦♥s ❆ ❛♥❞ ❇ ②✐❡❧❞ ❡q✉✐✈❛❧❡♥t ♦♣t✐♠❛❧ ♣❧❛♥t ❧❛②♦✉ts ❛♥❞ t❤❡ s❛♠❡ ♦❜❥❡❝t✐✈❡ ✈❛❧✉❡✱ ✇✐t❤ ♠❛✐♥ ❞✐✛❡r❡♥❝❡s ✐♥ ❧❛②♦✉t ♦r✐❡♥t❛t✐♦♥ ❛♥❞ r❡✢❡❝t✐♦♥✳ ❚❤✉s✱ ♦♥❧② t❤❡ ♦♣t✐♠❛❧ ❧❛②♦✉t r❡s✉❧t ❢♦r ♠♦❞❡❧ ❆✳✶ ✐s ♣r❡s❡♥t❡❞ ❢♦r ❡❛❝❤ ❡①❛♠♣❧❡✳ ▲❛②♦✉t r❡s✉❧ts ❢♦r t❤❡ ♦t❤❡r ♠♦❞❡❧s ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ t❤❡ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳

✹✳✶

❊①❛♠♣❧❡ ✶

❊①❛♠♣❧❡ ✶ ✐s ❛♥ ❊t❤②❧❡♥❡ ♦①✐❞❡ ♣❧❛♥t ✉s❡❞ ✐♥ t❤❡ ✇♦r❦ ♦❢ P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶✵ ✱ ♦r✐❣✐♥❛❧❧② ♣r❡s❡♥t❡❞ ❜② P❡♥t❡❛❞♦ ❛♥❞ ❈✐r✐❝ ✺ ✳ ❚❤❡ ♣❧❛♥t ❝♦♥s✐sts ♦❢ ✼ ✉♥✐ts ✇✐t❤ ♣r♦❝❡ss ✢♦✇ ❞✐❛❣r❛♠ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳ ❚❤❡ s✉♠♠❛r② ♦❢ t❤❡ ♠♦❞❡❧ st❛t✐st✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❡r❢♦r♠❛♥❝❡ ✐s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✷ ❛♥❞ t❤❡ ♦♣t✐♠❛❧ ❧❛②♦✉t ✐♥ ❋✐❣✉r❡ ✹✳ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❣❛✈❡ ❛ t♦t❛❧ ❝♦st ♦❢ ✻✻✱✷✻✷✳✵ r♠✉ ✲ ✷✷✪ ❝♦♥♥❡❝t✐♦♥ ❝♦sts✱ ✹✹✪ ♣✉♠♣✐♥❣ ❝♦sts ❛♥❞ ✸✹✪ ❝♦♥str✉❝t✐♦♥ ❝♦sts✳ ❊❛❝❤ ✢♦♦r ❤❛❞ ❛♥ ❛r❡❛ ♠❡❛s✉r✐♥❣ ✷✵♠①✷✵♠✱ t♦t❛❧❧✐♥❣ t✇♦ ✢♦♦rs s❡❧❡❝t❡❞ ❢♦r ❝♦♥str✉❝t✐♦♥ ♦✉t ♦❢ ❢♦✉r ♣r♦✈✐❞❡❞ t♦ t❤❡ ♠♦❞❡❧✳ ❚❛❜❧❡ ✷ s❤♦✇s t❤❛t t❤❡ ♠♦❞❡❧s ✐♥ ❢♦r♠✉❧❛t✐♦♥ ❆ ❛r❡ ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t✳ ❋♦r♠✉❧❛t✐♦♥ ❇ ✇❛s ✐♥❤❡r❡♥t❧② ❧❛r❣❡r ✐♥ s✐③❡ ❢♦r ❡❛❝❤ ❝❛s❡ st✉❞②✱ ❛s ❛ ❣r❡❛t❡r ♥✉♠❜❡r ♦❢ ✉♥✐ts ♥❡❡❞❡❞ t♦ ❜❡ ❤❛♥❞❧❡❞ t❤❛♥ ✐♥ ❢♦r♠✉❧❛t✐♦♥s ❆✳




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✹ ❈❆❙❊ ❙❚❯❉■❊❙

✷✶

❋✐❣✉r❡ ✹✿ ❊①❛♠♣❧❡ ✶ ❧❛②♦✉t r❡s✉❧ts ✹✳✷

❊①❛♠♣❧❡ ✷

❘ ✈✽✳✵✳ ■t ❝♦♥s✐sts ♦❢ ❊①❛♠♣❧❡ ✷ ✐s ❛♥ ❯r❡❛ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥t✱ s✐♠✉❧❛t❡❞ ✇✐t❤ ❆s♣❡♥ P▲❯❙

✽ ✉♥✐ts✱ ✷ ♦❢ ✇❤✐❝❤ ❡①❝❡❡❞ t❤❡ ✢♦♦r ❤❡✐❣❤t ♦❢ ✽♠✳ ❚❤❡ ♣r♦❝❡ss ✢♦✇ ❞✐❛❣r❛♠ ♦❢ t❤❡ ♣❧❛♥t ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺✳ ❆ t♦t❛❧ ♦❢ ✷✺ ♣♦ss✐❜❧❡ ❛r❡❛ s✐③❡s ✇❡r❡ ✉s❡❞ r❛♥❣✐♥❣ ❢r♦♠ ✺♠ ✲ ✹✺♠✳ ❊❛❝❤ ✉♥✐t ❤❛❞ t♦ ❜❡ ♣❧❛❝❡❞ ❛ ♠✐♥✐♠✉♠ ♦❢ ✹♠ ❢r♦♠ ❛♥♦t❤❡r ✐♥ ❡✐t❤❡r ❞✐r❡❝t✐♦♥✳ ❚❤❡ s✉♠♠❛r② ♦❢ t❤❡ ♠♦❞❡❧ st❛t✐st✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❡r❢♦r♠❛♥❝❡ ✐s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✸✳ ❚❤❡ ❧❛②♦✉t r❡s✉❧t ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✻✳ ❆❧❧ ✹ ✢♦♦rs ✇❡r❡ ❛ss✐❣♥❡❞ ✉♥✐ts✱ ✇✐t❤ ❡❛❝❤ ✢♦♦r ❤❛✈✐♥❣ ❛♥ ❛r❡❛ ♠❡❛s✉r✐♥❣ ✶✺♠①✺♠✱ ❛♥❞ ❛ t♦t❛❧ ❝♦st ♦❢ ✶✶✼✱✹✸✶✳✵ r♠✉ ✲ ✻✪ ❝♦♥♥❡❝t✐♦♥✱ ✷✻✪ ♣✉♠♣✐♥❣ ❛♥❞ ✻✽✪ ❝♦♥str✉❝t✐♦♥ ❝♦sts✳ ❚❤❡ t✇♦ ♠✉❧t✐✲✢♦♦r ✉♥✐ts ✭✉♥✐ts ✷ ❛♥❞ ✹✮ ♦❝❝✉♣✐❡❞ s✉❝❝❡ss✐✈❡ ✢♦♦rs ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡✐r ❤❡✐❣❤t ✭✢♦♦rs ✶✲✹ ❛♥❞ ✷✲✸ r❡s♣❡❝t✐✈❡❧②✮✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts s❤♦✇❡❞ ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ✇❛s ❛❝❤✐❡✈❡❞ ❜② ❛❧❧ ♠♦❞❡❧s ✐♥ ❛ r❡❧❛t✐✈❡❧② s❤♦rt t✐♠❡ ✭✉♥❞❡r ✺s✮✳ ❖✈❡r❛❧❧ r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ♠♦❞❡❧ ✐s ❝❛♣❛❜❧❡ ♦❢ ❤❛♥❞❧✐♥❣ ♣r♦❝❡ss✲s♣❡❝✐✜❝ ❝♦♥❞✐t✐♦♥s ✇❤✐❧st ❞❡❝✐❞✐♥❣ ✢♦♦r ♣❧❛❝❡♠❡♥t ❡✈❡♥ ❢♦r ♠✉❧t✐✲✢♦♦r ✉♥✐ts✳



✹ ❈❆❙❊ ❙❚❯❉■❊❙

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✷✷

❋✐❣✉r❡ ✺✿ ❋❧♦✇ ❞✐❛❣r❛♠ ♦❢ ❯r❡❛ Pr♦❞✉❝t✐♦♥

❚❛❜❧❡ ✸✿ ❙✉♠♠❛r② ♦❢ ♠♦❞❡❧ st❛t✐st✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❡r❢♦r♠❛♥❝❡ ❢♦r ❊①❛♠♣❧❡ ✷

❚♦t❛❧ ❈♦st ❈P❯ ✭s✮ ◆✉♠❜❡r ♦❢ ◆✉♠❜❡r ♦❢ ◆✉♠❜❡r ♦❢

✭r♠✉✮ ❜✐♥❛r② ✈❛r✐❛❜❧❡s ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s ❡q✉❛t✐♦♥s

❯r❡❛ Pr♦❞✉❝t✐♦♥ ✭✽ ✉♥✐ts✮ ❆✳✶ ❆✳✷ ❆✳✸ ❇ ✶✶✼✱✹✸✶✳✵ ✹✳✽ ✹✳✾ ✷✳✾ ✹✳✵ ✶✹✺ ✶✹✺ ✶✹✺✸ ✷✽✶ ✸✶✺ ✷✽✸ ✸✶✺ ✶✺✾ ✻✶✶ ✺✹✼ ✺✹✼ ✶✱✷✼✾


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✹ ❈❆❙❊ ❙❚❯❉■❊❙

✷✸

❋✐❣✉r❡ ✻✿ ❊①❛♠♣❧❡ ✷ ❧❛②♦✉t r❡s✉❧ts




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✹ ❈❆❙❊ ❙❚❯❉■❊❙

✷✺

✇✐t❤ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡s ✐s ❛ttr✐❜✉t❡❞ t♦ t❤❡ ❣r❡❛t❡r ♥✉♠❜❡r ♦❢ ✉♥✐ts ❝♦♥s✐❞❡r❡❞✳ ❈♦♥♥❡❝t✐♦♥ ❛♥❞ ❝♦♥str✉❝t✐♦♥ ❝♦sts ❝♦♥st✐t✉t❡❞ ✻✪ ❛♥❞ ✹✹✪ ♦❢ t❤❡ t♦t❛❧ ❝♦st r❡s♣❡❝t✐✈❡❧②✳ ❚❛❜❧❡ ✹✿ ❙✉♠♠❛r② ♦❢ ♠♦❞❡❧ st❛t✐st✐❝s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❡r❢♦r♠❛♥❝❡ ❢♦r ❊①❛♠♣❧❡ ✸

❚♦t❛❧ ❈♦st ❈P❯ ✭s✮ ◆✉♠❜❡r ♦❢ ◆✉♠❜❡r ♦❢ ◆✉♠❜❡r ♦❢

✭r♠✉✮ ❜✐♥❛r② ✈❛r✐❛❜❧❡s ❝♦♥t✐♥✉♦✉s ✈❛r✐❛❜❧❡s ❡q✉❛t✐♦♥s

❈❉❯ P❧❛♥t ✭✶✼ ✉♥✐ts✮ ❆✳✶ ❆✳✷ ❆✳✸ ❇ ✼✹✾✱✻✾✶✳✹ ✷✾✷✳✸ ✶✵✺✳✵ ✹✹✳✶ ✶✸✶✳✻ ✺✻✽ ✺✻✽ ✺✻✽ ✶✱✺✻✾ ✶✱✺✵✼ ✶✱✸✽✽ ✶✱✺✵✼ ✸✽✼ ✸✱✷✶✸ ✷✱✾✾✸ ✷✱✾✼✺ ✶✶✱✹✽✸

❚❤❡ ❧❛②♦✉t ♦❢ ❡q✉✐♣♠❡♥t ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✽✳ ❆s ❛❧❧ t❤❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ✭✺✱ ✻✱ ✼✱ ✶✷ ❛♥❞ ✶✺✮ st❛rt✐♥❣ ✢♦♦rs ✇❡r❡ ♣r❡❞❡✜♥❡❞✱ t❤❡② ❛❧❧ st❛rt❡❞ ❢r♦♠ t❤❡ ✜rst ✢♦♦r t❤r♦✉❣❤ t❤❡ ♥✉♠❜❡r ♦❢ ✢♦♦rs r❡q✉✐r❡❞ ❜❛s❡❞ ♦♥ t❤❡✐r ❤❡✐❣❤t✳ ■t ✐s ✇♦rt❤② ♦❢ ♥♦t✐♥❣ t❤❛t ❛❧t❤♦✉❣❤ ❡q✉✐♣♠❡♥t ✶✹ r❡q✉✐r❡❞ ✺ ✢♦♦rs ❜❛s❡❞ ♦♥ ✐ts ❤❡✐❣❤t✱ ❛ t♦t❛❧ ♦❢ ✹ ✢♦♦rs ✇❛s ❞❡❝✐❞❡❞ ❜② ❛❧❧ ❢♦r♠✉❧❛t✐♦♥s✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ✜❢t❤ ✢♦♦r ✇❛s ❞❡❡♠❡❞ ✉♥♥❡❝❡ss❛r② ❛s ♥♦ ♦t❤❡r ❡q✉✐♣♠❡♥t ♦t❤❡r t❤❛♥ ❡q✉✐♣♠❡♥t ✶✹ ✇❛s t♦ ❜❡ ♣❧❛❝❡❞ ♦♥ s✉❝❤ ✢♦♦r ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❛♥ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥✱ ❛♥❞ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str✉❝t✐♦♥ ❝♦st ❢♦r ❛ ✜❢t❤ ✢♦♦r ✇❛s ❡❧✐♠✐♥❛t❡❞✳ ❙♦✱ ❛❧t❤♦✉❣❤ ❛ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❝❛♥ s♣❛♥ ❛ s♣❡❝✐✜❡❞ ♥✉♠❜❡r ♦❢ ✢♦♦rs ❜❛s❡❞ ♦♥ ✐ts ❤❡✐❣❤t✱ ♥♦t ❡✈❡r② ♦♥❡ ♦❢ t❤♦s❡ ✢♦♦rs ♥❡❡❞ t♦ ❜❡ ❝♦♥str✉❝t❡❞✳ Pr❛❝t✐❝❛❧ ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡ ♠✉❧t✐✲✢♦♦r ❧❛②♦✉ts ❛❜♦✉t ✜r❡❞ ❤❡❛t❡rs ✇✐t❤ ❧♦♥❣ st❛❝❦s✱ ❞✐st✐❧❧❛t✐♦♥ ❝♦❧✉♠♥s ❛♥❞ ✢❛r❡ st❛❝❦s✳

❋✐❣✉r❡ ✽✿ ❊①❛♠♣❧❡ ✸ ❧❛②♦✉t r❡s✉❧ts



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✷✻

✹✳✸✳✶ ❙②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❝❛s❡ st✉❞✐❡s s♦❧✈❡❞✱ ✐t ❜❡❝♦♠❡s ❡✈✐❞❡♥t ❢r♦♠ t❤❡ ❧❛②♦✉t r❡s✉❧ts ✭♣r❡s❡♥t❡❞ ❛❜♦✈❡ ❛♥❞ ✐♥ s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✮ t❤❛t t❤❡r❡ ❝❛♥ ❡①✐st ♠✉❧t✐♣❧❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❢♦r ❡❛❝❤ ♣r♦❜❧❡♠✳ ❚❤❡s❡ ♠✉❧t✐♣❧❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❞✐✛❡r ♦♥❧② ✐♥ ❧❛②♦✉t ♦r✐❡♥t❛t✐♦♥✴r❡✢❡❝t✐♦♥ ❛♥❞✴♦r tr❛♥s❧❛t✐♦♥ ♦❢ ❡q✉✐♣♠❡♥t ♦♥ t❤❡ ❧❛②♦✉t ❛r❡❛✱ r❡s✉❧t✐♥❣ ✐♥ ✐♥❡✣❝✐❡♥t ❈P❯ ✉s❛❣❡ ✷✼ ✱ ❡s♣❡❝✐❛❧❧② ❢♦r ❧❛r❣❡r ♣r♦❜❧❡♠ ✐♥st❛♥❝❡s✳ ■♥ ♦r❞❡r t♦ ✐♥❝r❡❛s❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝② ❛♥❞ r❡❞✉❝❡ s②♠♠❡tr②✱ ❛❞❞✐t✐♦♥❛❧ ❝♦♥str❛✐♥ts✱ ❛❞❛♣t❡❞ ❢r♦♠ ❲❡st❡r❧✉♥❞ ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✷✼ ❛r❡ ✐♥tr♦❞✉❝❡❞✿

✇❤❡r❡ α = min

li d i , 2 2

+ min

xi + yi − xj − yj ≥ α · Nij

✭✹✹✮

E1ij = 0

✭✹✺✮

lj d j , 2 2

✳ ❚❤❡s❡ ✜① t❤❡ r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥ ♦❢ i ❛♥❞ j ❛s ✇❡❧❧ ❛s ♦♥❡

♦❢ t❤❡ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ❜✐♥❛r② ✈❛r✐❛❜❧❡s✱ E1ij = 0✳ ❚❤❡ ❧❛tt❡r ❡♥❢♦r❝❡s ❡✐t❤❡r ♦❢ xi −xj ≥ ♦r yi − yj ≥

di +dj 2

li +lj 2

❛❝❝♦r❞✐♥❣ t♦ ❡q✉❛t✐♦♥s ✶✼ ✲ ✷✵✱ ✐✳❡✳✱ ✉♥✐t i ✐s r❡❧❛t✐✈❡❧② ❧♦❝❦❡❞ ✐♥ ❛ ❝♦♠♣❛ss

♣♦✐♥t ◆♦rt❤ ❊❛st ♦❢ ✉♥✐t j ❤❡♥❝❡ ❜r❡❛❦✐♥❣ t❤❡ s②♠♠❡tr②✳ ❚❤❡ ❛❜♦✈❡ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ✇❡r❡ ❛♣♣❧✐❡❞ t♦ ❡①❛♠♣❧❡ ✸ ✲ ❈r✉❞❡ ❞✐st✐❧❧❛t✐♦♥ ♣❧❛♥t ✇✐t❤ ♣r❡✲❤❡❛t✐♥❣ tr❛✐♥ ✲ ❜② ✐♥❝♦r♣♦r❛t✐♥❣ ❡q✉❛t✐♦♥s ✹✹ ❛♥❞ ✹✺ ✐♥ ▼♦❞❡❧ ❆✳✶ ✇❤❡r❡ t❤❡ ❝❤♦✐❝❡ ♦❢ ❡q✉✐♣♠❡♥t i ❛♥❞ j ❛r❡ ❛♠♦♥❣ ♠✉❧t✐✲✢♦♦r ✉♥✐ts ♦♥❧②✱ ❜❛s❡❞ ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝r✐t❡r✐❛ ✷✼ ✿ ❛ ✲ ❊q✉✐♣♠❡♥t i ❛♥❞ j ❤❛✈✐♥❣ t❤❡ ❤✐❣❤❡st ❝♦♥♥❡❝t✐♦♥ ❝♦st✿ ✉♥✐ts ✺ ❛♥❞ ✻❀ ❜ ✲ ❚✇♦ ❡q✉✐♣♠❡♥t ✇✐t❤ t❤❡ ❧❛r❣❡st ❛r❡❛✿ ✉♥✐ts ✼ ❛♥❞ ✶✺❀ ❝ ✲ ❚✇♦ ❡q✉✐♣♠❡♥t ✇✐t❤ t❤❡ s♠❛❧❧❡st ❛r❡❛✿ ✉♥✐ts ✻ ❛♥❞ ✶✷✳ ❚❤❡ r❡s✉❧t✐♥❣ ♠♦❞❡❧s ✇❡r❡ s♦❧✈❡❞ t♦ ❣❧♦❜❛❧ ♦♣t✐♠❛❧✐t② ❛♥❞ ❛ s♦❧✉t✐♦♥ ♦❢ ✼✹✾✱✻✾✶✳✹ r♠✉ ✇❛s ♦❜t❛✐♥❡❞ ✐♥ ❛❧❧ ❛❧t❡r♥❛t✐✈❡ ❝❤♦✐❝❡s ♦❢ ❡q✉✐♣♠❡♥t✱ t❤❡ s❛♠❡ ❛s ♠♦❞❡❧ ❆✳✶ ✇✐t❤♦✉t t❤❡s❡ ❝♦♥✲ str❛✐♥ts✳ ❚❤❡✐r ❝♦♠♣✉t❛t✐♦♥❛❧ ❈P❯ t✐♠❡s ❛r❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✺ ✲ ✇❤❡r❡ ❢♦r ❡❛❝❤ ✉♥✐t s❡❧❡❝t✐♦♥ ACS Paragon Plus Environment

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✺ ❈❖◆❈▲❯❉■◆● ❘❊▼❆❘❑❙

✷✼

❛❧t❡r♥❛t✐✈❡✱ t❤❡ r❡s✉❧t✐♥❣ ♠♦❞❡❧ ✐s ♥❛♠❡❞ ❛♣♣r♦♣r✐❛t❡❧② ✲ ♠♦❞❡❧ ❆✳✶❛ ❢♦r ♠♦❞❡❧ ❆✳✶ ✇✐t❤ s②♠✲ ♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ✇✐t❤ ❡q✉✐♣♠❡♥t i ❛♥❞ j ❝❤♦s❡♥ ❜② ❝r✐t❡r✐♦♥ ❛✱ ❤❛✈✐♥❣ t❤❡ ❤✐❣❤❡st ❝♦♥♥❡❝t✐♦♥ ❝♦st✱ ❛♥❞ s♦ ♦♥✳ ❚❤❡ r❡s✉❧ts s❤♦✇ t❤❛t t❤❡ ♥❡✇ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ❤❛s ✐♠♣r♦✈❡❞ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✣❝✐❡♥❝② ❜② ♦♥❡ ♦r❞❡r ♦❢ ♠❛❣♥✐t✉❞❡✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ✐♠♣r♦✈❡♠❡♥ts ✇❡r❡ ❛❧s♦ ❢♦✉♥❞ ❢♦r t❤❡ ♦t❤❡r ♠♦❞❡❧s ❞❡✈❡❧♦♣❡❞ ❛♥❞ ❡①❛♠♣❧❡s ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ✇♦r❦✳ ❚❛❜❧❡ ✺✿ ❙✉♠♠❛r② ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣❡r❢♦r♠❛♥❝❡ ❢♦r ❊①❛♠♣❧❡ ✸❀ ▼♦❞❡❧ ❆✳✶ ✇✐t❤ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥tr❛✐♥ts ▼♦❞❡❧ ❆✳✶ ❆✳✶❛ ❆✳✶❜ ❆✳✶❝

✺

❈P❯ ✭s✮ ✷✾✷✳✸ ✸✶✳✷ ✸✸✳✼ ✼✶✳✷

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s

❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ▼■▲P ♠♦❞❡❧ ❜② P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✇❛s ♣r♦♣♦s❡❞ t♦ ❛❞❞r❡ss ♥❡✇ ❝♦♥❝❡r♥s ✐♥ t❤❡ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ♣r♦❜❧❡♠✳ ❚❤❡s❡ ❝♦♥❝❡r♥s ✐♥❝❧✉❞❡❞ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ✐♥ ❝❡rt❛✐♥ ♣r♦❝❡ss ♣❧❛♥ts ✇✐t❤ ❞❡s✐❣♥✲s♣❡❝✐✜❡❞ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t ❝♦♥♥❡❝t✐♦♥ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡q✉✐♣♠❡♥t✳ ❆ t♦t❛❧ ♦❢ ❢♦✉r ✭✹✮ ♠♦❞❡❧s ✇❡r❡ ♣r♦♣♦s❡❞✱ ❣r♦✉♣❡❞ ✐♥ t✇♦ ❜r♦❛❞ ❝❧❛ss❡s✳ ❚❤❡ ✜rst ❝❧❛ss ❤❛♥❞❧❡❞ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ❛s t❤❡② ✇❡r❡ ✭s✐♥❣❧❡ t❛❧❧ ✉♥✐ts✮ ❤❛✈✐♥❣ ❝♦♥str❛✐♥ts t♦ ❡♥s✉r❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ✇❡r❡ ❛ss✐❣♥❡❞ t♦ ❝♦♥s❡❝✉t✐✈❡ ✢♦♦rs✱ ✇✐t❤ t❤r❡❡ ✭✸✮ ❛❧t❡r♥❛t✐✈❡ ♠♦❞❡❧s r❡s✉❧t✐♥❣✳ ❋♦r t❤❡ s❡❝♦♥❞ ❝❧❛ss✱ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ✇❡r❡ ❜r♦❦❡♥ ✉♣ t♦ s✐♥❣❧❡✲✢♦♦r✲♣s❡✉❞♦✲ ✉♥✐ts ❛♥❞ t❤❡ ❧❛②♦✉t s♦❧✈❡❞ ❜❛s❡❞ ♦♥ ❛ ❧✐♥❡❛r✐s❡❞ ♠♦❞❡❧ ♦❢ ❑✉ ❡t ❛❧✳ ✶✽ ✳ ❊❛❝❤ ♦❢ t❤❡ ❢♦✉r ✭✹✮ ♠♦❞❡❧s ✇❡r❡ ✈❛❧✐❞❛t❡❞ ✇✐t❤ ❝❛s❡ st✉❞✐❡s ❛♥❞ ♠♦❞❡❧ ♣❡r❢♦r♠❛♥❝❡ ✇❡r❡ ❛s❝❡rt❛✐♥❡❞ ❛♥❞ ❝♦♠♣❛r❡❞✳ ❆❧❧ ♠♦❞❡❧s s♦❧✈❡❞ ♣r♦❜❧❡♠s ♦❢ ✉♣ t♦ ✶✼ ✉♥✐ts ✇❡❧❧ ✉♥❞❡r ✼ ♠✐♥✉t❡s✱ ❛s ❝♦♠♣❛r❡❞ t♦ ♣r❡✈✐♦✉s ✇♦r❦ ✶ ✇❤✐❝❤ ♦♥❧② ❤❛♥❞❧❡❞ ❛ ♠❛①✐♠✉♠ ♦❢ ✶✶ ✉♥✐ts s✐♠✉❧t❛♥❡♦✉s❧②✳ ❚♦t❛❧ ❝♦st ❞✐str✐❜✉t✐♦♥ ✇❡r❡ ❝♦♥s✐st❡♥t ✇✐t❤ ❡①♣❡❝t❡❞ ✈❛❧✉❡s ✲ ❝♦♥str✉❝t✐♦♥ ❛♥❞ ♣✉♠♣✐♥❣ ❝♦sts t❛❦❡♥ ✉♣ t❤❡ ❧❛r❣❡r ♣♦rt✐♦♥✱ ✇✐t❤ ❝♦♥♥❡❝t✐♦♥ ❝♦sts ❢♦❧❧♦✇✐♥❣✳ ❆❧❧ ♠♦❞❡❧s ✇❡r❡ ❛❜❧❡ t♦ ACS Paragon Plus Environment


✺ ❆❈❑◆❖❲▲❊❉●❊▼❊◆❚

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✷✽

❤❛♥❞❧❡ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t✱ ❛♥❞ ❞❡❝✐❞❡ ✇❤❡t❤❡r ❛ ✢♦♦r ❜❡ ❝♦♥str✉❝t❡❞ ❛♥❞ ✉s❡❞ ❡✈❡♥ ✐❢ ❛ ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t ✇❡r❡ ❛ss✐❣♥❡❞ t♦ ✐t✳ ■♥ ♠♦st ❝❛s❡s t❤♦✉❣❤✱ ❢♦r♠✉❧❛t✐♦♥s ❆ ✇❛s ❢♦✉♥❞ t♦ ❜❡ ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t t❤❛♥ ❇✳ ❋✐♥❛❧❧②✱ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts ✇❡r❡ ✐♥tr♦❞✉❝❡❞ r❡❞✉❝✐♥❣ t❤❡ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ♠✉❧t✐♣❧❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s t❤❛t ❧❡❛❞ t♦ ❣r❡❛t❡r ❈P❯ ✉s❛❣❡✳ ❚❤❡s❡ ❝♦♥str❛✐♥ts ✜①❡❞ t❤❡ r❡❧❛t✐✈❡ ♣♦s✐t✐♦♥s ♦❢ t✇♦ ✉♥✐ts i ❛♥❞ j ✳ ❚❤❡ ❝❤♦✐❝❡ ♦❢ ✉♥✐ts i ❛♥❞ j ✇❡r❡ ❜❛s❡❞ ♦♥ t❤r❡❡ ❝r✐t❡r✐❛✿ ❧❛r❣❡st ❝♦♥♥❡❝t✐♦♥✱ ❧❛r❣❡st ❛r❡❛s ♦r s♠❛❧❧❡st ❛r❡❛ ❛♠♦♥❣st ♠✉❧t✐✲✢♦♦r ❡q✉✐♣♠❡♥t✳ ❆ r❡❞✉❝t✐♦♥ ✐♥ ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✉♣ t♦ ✸✶✳✷s ✇❛s ♦❜t❛✐♥❡❞ ❢♦r ❡①❛♠♣❧❡ ✸✱ ❛s ❝♦♠♣❛r❡❞ t♦ ✷✾✷✳✸s ✇✐t❤♦✉t s②♠♠❡tr② ❜r❡❛❦✐♥❣✳ ❋✉rt❤❡r ✇♦r❦ ✇✐❧❧ ❡♥t❛✐❧ ♠♦❞❡❧ ✈❛❧✐❞❛t✐♦♥ ✇✐t❤ ❧❛r❣❡r ❝❛s❡ st✉❞✐❡s ❛♥❞ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥❛❧ ♠❡t❤♦❞s✱ ❡✳❣✳ ❞❡❝♦♠♣♦s✐t✐♦♥ t❡❝❤♥✐q✉❡s✳

❆❝❦♥♦✇❧❡❞❣❡♠❡♥t ❏❖❊ ❛❝❦♥♦✇❧❡❞❣❡s t❤❡ ✜♥❛♥❝✐❛❧ s✉♣♣♦rt ♦❢ t❤❡ P❡tr♦❧❡✉♠ ❚❡❝❤♥♦❧♦❣② ❉❡✈❡❧♦♣♠❡♥t ❋✉♥❞ ✭P❚❉❋✮✱ ◆✐❣❡r✐❛✳

❙✉♣♣♦rt✐♥❣ ■♥❢♦r♠❛t✐♦♥ ❆✈❛✐❧❛❜❧❡ ❆ ❢✉❧❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ♣r♦♣♦s❡❞ ❜② P❛ts✐❛t③✐s ❛♥❞ P❛♣❛❣❡♦r❣✐♦✉ ✶ ✱ ❞❛t❛ ❢♦r t❤❡ ❝❛s❡ st✉❞✐❡s ♣r❡s❡♥t❡❞✱ ❛♥❞ ❧❛②♦✉t r❡s✉❧ts ❢♦r ♠♦❞❡❧s ❆✳✷✱ ❆✳✸ ❛♥❞ ❇ ✐s ❛✈❛✐❧❛❜❧❡ ❛s s✉♣♣♦rt✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳

❆✉t❤♦r ■♥❢♦r♠❛t✐♦♥ ❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r

✯❊♠❛✐❧✿ ❧✳♣❛♣❛❣❡♦r❣✐♦✉❅✉❝❧✳❛❝✳✉❦❀ P❤♦♥❡✿ ✰✹✹ ✭✵✮✷✵ ✼✻✼✾ ✷✺✻✸✳


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✺ ❆❯❚❍❖❘ ■◆❋❖❘▼❆❚■❖◆

✷✾

❖❘❈■❉

❙♦♥❣s♦♥❣ ▲✐✉✿ ✵✵✵✵✲✵✵✵✶✲✽✹✶✷✲✷✼✹❳ ▲❛③❛r♦s ●✳ P❛♣❛❣❡♦r❣✐♦✉✿ ✵✵✵✵✲✵✵✵✸✲✹✻✺✷✲✻✵✽✻

◆♦t❡s

❚❤❡ ❛✉t❤♦rs ❞❡❝❧❛r❡ ♥♦ ❝♦♠♣❡t✐♥❣ ✜♥❛♥❝✐❛❧ ✐♥t❡r❡st✳



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✺ ❘❊❋❊❘❊◆❈❊❙

✸✵

❘❡❢❡r❡♥❝❡s ✭✶✮ P❛ts✐❛t③✐s✱ ❉✳ ■✳❀ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳ ❊✣❝✐❡♥t ❙♦❧✉t✐♦♥ ❆♣♣r♦❛❝❤❡s ❢♦r t❤❡ ▼✉❧t✐✢♦♦r Pr♦❝❡ss P❧❛♥t ▲❛②♦✉t Pr♦❜❧❡♠✳

■♥❞✉str✐❛❧ ✫ ❊♥❣✐♥❡❡r✐♥❣ ❈❤❡♠✐str② ❘❡s❡❛r❝❤ ✷✵✵✸✱

✹✷✱ ✽✶✶✕✽✷✹✳ ✭✷✮ ❆♥❥♦s✱ ▼✳ ❋✳❀ ❱✐❡✐r❛✱ ▼✳ ❱✳ ❈✳ ▼❛t❤❡♠❛t✐❝❛❧ ❖♣t✐♠✐③❛t✐♦♥ ❆♣♣r♦❛❝❤❡s ❢♦r ❋❛❝✐❧✐t② ▲❛②✲ ♦✉t Pr♦❜❧❡♠s✿ ❚❤❡ ❙t❛t❡✲♦❢✲t❤❡✲❆rt ❛♥❞ ❋✉t✉r❡ ❘❡s❡❛r❝❤ ❉✐r❡❝t✐♦♥s✳ ❊✉r♦♣❡❛♥

❏♦✉r♥❛❧

♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ✷✵✶✼✱ ✷✻✶✱ ✶✕✶✻✳ ✭✸✮ ▼❡❝❦❧❡♥❜✉r❣❤✱ ❏✳

Pr♦❝❡ss ♣❧❛♥t ❧❛②♦✉t✱ ✷♥❞ ❡❞✳❀ ▲♦♥❣♠❛♥✿ ◆❡✇ ❨♦r❦✱ ✶✾✽✺❀ ♣ ✻✷✺✳

✭✹✮ ❉r✐r❛✱ ❆✳❀ P✐❡rr❡✈❛❧✱ ❍✳❀ ❍❛❥r✐✲●❛❜♦✉❥✱ ❙✳ ❋❛❝✐❧✐t② ❧❛②♦✉t ♣r♦❜❧❡♠s✿ ❆ s✉r✈❡②✳

❆♥♥✉❛❧

❘❡✈✐❡✇s ✐♥ ❈♦♥tr♦❧ ✷✵✵✼✱ ✸✶✱ ✷✺✺✕✷✻✼✳ ✭✺✮ P❡♥t❡❛❞♦✱ ❋✳ ❉✳❀ ❈✐r✐❝✱ ❆✳ ❘✳ ❆♥ ▼■◆▲P ❆♣♣r♦❛❝❤ ❢♦r ❙❛❢❡ Pr♦❝❡ss P❧❛♥t ▲❛②♦✉t✳

■♥❞✉str✐❛❧ ✫ ❊♥❣✐♥❡❡r✐♥❣ ❈❤❡♠✐str② ❘❡s❡❛r❝❤ ✶✾✾✻✱ ✸✺✱ ✶✸✺✹✕✶✸✻✶✳ ✭✻✮ ●✉✐r❛r❞❡❧❧♦✱ ❘✳❀ ❙✇❛♥❡②✱ ❘✳ ❊✳ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ✇✐t❤ ♣✐♣❡ r♦✉t✐♥❣✳

❈♦♠♣✉t❡rs ✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✵✺✱ ✸✵✱ ✾✾✕✶✶✹✳ ✭✼✮ ❇❛r❜♦s❛✲P♦✈♦❛✱ ❆✳ P✳❀ ▼❛t❡✉s③✱ ❘✳❀ ◆♦✈❛✐s③✱ ❆✳ ◗✳ ❖♣t✐♠❛❧ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ❧❛②♦✉t ♦❢ ✐♥❞✉str✐❛❧ ❢❛❝✐❧✐t✐❡s✳

■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❘❡s❡❛r❝❤ ✷✵✵✶✱ ✸✾✶✷✱ ✷✺✻✼✕

✷✺✾✸✳ ✭✽✮ ❇❛r❜♦s❛✲Pó✈♦❛✱ ❆✳ P✳❀ ▼❛t❡✉s✱ ❘✳❀ ◆♦✈❛✐s✱ ❆✳ ◗✳ ❖♣t✐♠❛❧ ✸❉ ❧❛②♦✉t ♦❢ ✐♥❞✉str✐❛❧ ❢❛❝✐❧✐t✐❡s✳

■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❘❡s❡❛r❝❤ ✷✵✵✷✱ ✹✵✱ ✶✻✻✾✕✶✻✾✽✳

✭✾✮ ●❡♦r❣✐❛❞✐s✱ ▼✳❀ ▼❛❝❝❤✐❡tt♦✱ ❙✳ ▲❛②♦✉t ♦❢ ♣r♦❝❡ss ♣❧❛♥ts✿ ❆ ♥♦✈❡❧ ❛♣♣r♦❛❝❤✳ ❈♦♠♣✉t❡rs

✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✶✾✾✼✱ ✷✶✱ ❙✸✸✼✕❙✸✹✷✳ ✭✶✵✮ P❛ts✐❛t③✐s✱ ❉✳ ■✳❀ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳ ❖♣t✐♠❛❧ ♠✉❧t✐✲✢♦♦r ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t✳

♣✉t❡rs ✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✵✷✱ ✷✻✱ ✺✼✺✕✺✽✸✳ ACS Paragon Plus Environment

❈♦♠✲

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✺ ❘❊❋❊❘❊◆❈❊❙

✸✶

✭✶✶✮ P❛ts✐❛t③✐s✱ ❉✳ ■✳❀ ❑♥✐❣❤t✱ ●✳❀ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳ ❆♥ ▼■▲P ❛♣♣r♦❛❝❤ t♦ s❛❢❡ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t✳

❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❘❡s❡❛r❝❤ ❛♥❞ ❉❡s✐❣♥ ✷✵✵✹✱ ✽✷✱ ✺✼✾✕✺✽✻✳

✭✶✷✮ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳❀ ❘♦tst❡✐♥✱ ●✳ ❊✳ ❈♦♥t✐♥✉♦✉s✲❞♦♠❛✐♥ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧s ❢♦r ♦♣✲ t✐♠❛❧ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉t✳

■♥❞✉str✐❛❧ ✫ ❊♥❣✐♥❡❡r✐♥❣ ❈❤❡♠✐str② ❘❡s❡❛r❝❤ ✶✾✾✽✱ ✺✽✽✺✱

✸✻✸✶✕✸✻✸✾✳ ✭✶✸✮ ❳✉✱ ●✳❀ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳ Pr♦❝❡ss ♣❧❛♥t ❧❛②♦✉t ✉s✐♥❣ ❛♥ ✐♠♣r♦✈❡♠❡♥t✲t②♣❡ ❛❧❣♦r✐t❤♠✳

❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❘❡s❡❛r❝❤ ❛♥❞ ❉❡s✐❣♥ ✷✵✵✾✱ ✽✼✱ ✼✽✵✕✼✽✽✳ ✭✶✹✮ ▼❡❞✐♥❛✲❍❡rr❡r❛✱ ◆✳❀ ❏✐♠é♥❡③✲●✉t✐érr❡③✱ ❆✳❀ ●r♦ss♠❛♥♥✱ ■✳ ❊✳ ❆ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❣r❛♠✲ ♠✐♥❣ ♠♦❞❡❧ ❢♦r ♦♣t✐♠❛❧ ❧❛②♦✉t ❝♦♥s✐❞❡r✐♥❣ q✉❛♥t✐t❛t✐✈❡ r✐s❦ ❛♥❛❧②s✐s✳

❈♦♠♣✉t❡rs ❛♥❞

❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✶✹✱ ✻✽✱ ✶✻✺✕✶✽✶✳ ✭✶✺✮ ❍✉❛♥❣✱ ❈✳❀ ❲♦♥❣✱ ❈✳ ❑✳ ❖♣t✐♠✐s❛t✐♦♥ ♦❢ s✐t❡ ❧❛②♦✉t ♣❧❛♥♥✐♥❣ ❢♦r ♠✉❧t✐♣❧❡ ❝♦♥str✉❝t✐♦♥ st❛❣❡s ✇✐t❤ s❛❢❡t② ❝♦♥s✐❞❡r❛t✐♦♥s ❛♥❞ r❡q✉✐r❡♠❡♥ts✳ ❆✉t♦♠❛t✐♦♥

✐♥ ❈♦♥str✉❝t✐♦♥ ✷✵✶✺✱

✺✸✱ ✺✽✕✻✽✳ ✭✶✻✮ ❏✉♥❣✱ ❙✳ ❋❛❝✐❧✐t② s✐t✐♥❣ ❛♥❞ ♣❧❛♥t ❧❛②♦✉t ♦♣t✐♠✐③❛t✐♦♥ ❢♦r ❝❤❡♠✐❝❛❧ ♣r♦❝❡ss s❛❢❡t②✳ ❑♦r❡❛♥

❏♦✉r♥❛❧ ♦❢ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✶✻✱ ✸✸✱ ✶✕✼✳ ✭✶✼✮ ❳✐♥✱ P✳❀ ❑❤❛♥✱ ❋✳❀ ❆❤♠❡❞✱ ❙✳ ▲❛②♦✉t ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❛ ❋❧♦❛t✐♥❣ ▲✐q✉❡✜❡❞ ◆❛t✉r❛❧ ●❛s ❋❛❝✐❧✐t② ❯s✐♥❣ ■♥❤❡r❡♥t ❙❛❢❡t② Pr✐♥❝✐♣❧❡s✳

❏♦✉r♥❛❧ ♦❢ ❖✛s❤♦r❡ ▼❡❝❤❛♥✐❝s ❛♥❞ ❆r❝t✐❝

❊♥❣✐♥❡❡r✐♥❣ ✷✵✶✻✱ ✶✸✽✱ ✵✹✶✻✵✷✳ ✭✶✽✮ ❑✉✱ ◆✳✲❑✳❀ ❍✇❛♥❣✱ ❏✳✲❍✳❀ ▲❡❡✱ ❏✳✲❈✳❀ ❘♦❤✱ ▼✳✲■✳❀ ▲❡❡✱ ❑✳✲❨✳ ❖♣t✐♠❛❧ ♠♦❞✉❧❡ ❧❛②♦✉t ❢♦r ❛ ❣❡♥❡r✐❝ ♦✛s❤♦r❡ ▲◆● ❧✐q✉❡❢❛❝t✐♦♥ ♣r♦❝❡ss ♦❢ ▲◆●✲❋P❙❖✳ ❙❤✐♣s ❛♥❞ ❖✛s❤♦r❡ ❙tr✉❝t✉r❡s

✷✵✶✸✱ ✾✱ ✸✶✶✕✸✸✷✳ ✭✶✾✮ ❍✇❛♥❣✱ ❏✳❀ ▲❡❡✱ ❑✳ ❨✳ ❖♣t✐♠❛❧ ❧✐q✉❡❢❛❝t✐♦♥ ♣r♦❝❡ss ❝②❝❧❡ ❝♦♥s✐❞❡r✐♥❣ s✐♠♣❧✐❝✐t② ❛♥❞ ❡✣❝✐❡♥❝② ❢♦r ▲◆● ❋P❙❖ ❛t ❋❊❊❉ st❛❣❡✳

❈♦♠♣✉t❡rs ❛♥❞ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✷✵✶✹✱

✻✸✱ ✶✕✸✸✳ ACS Paragon Plus Environment


Page 32 of 33

✺ ❘❊❋❊❘❊◆❈❊❙

✸✷

✭✷✵✮ ❑✉✱ ◆✳❀ ❏❡♦♥❣✱ ❙✳✲❨✳❀ ❘♦❤✱ ▼✳✲■✳❀ ❙❤✐♥✱ ❍✳✲❑✳❀ ❍❛✱ ❙✳❀ ❍♦♥❣✱ ❏✳✲✇✳ ▲❛②♦✉t ▼❡t❤♦❞ ♦❢ ❛ ❋P❙❖ ✭❋❧♦❛t✐♥❣✱ Pr♦❞✉❝t✐♦♥✱ ❙t♦r❛❣❡✱ ❛♥❞ ❖✛✲▲♦❛❞✐♥❣ ❯♥✐t✮ ❯s✐♥❣ t❤❡ ❖♣t✐♠✐③❛t✐♦♥ ❚❡❝❤♥✐q✉❡✳ ❱♦❧✉♠❡ ✶❇✿ ❖✛s❤♦r❡ ❚❡❝❤♥♦❧♦❣②✳ ✷✵✶✹❀ ♣♣ ✶✕✶✶✳ ✭✷✶✮ P❛r❦✱ P✳ ❏✳❀ ▲❡❡✱ ❈✳ ❏✳ ❚❤❡ ❘❡s❡❛r❝❤ ♦❢ ❖♣t✐♠❛❧ P❧❛♥t ▲❛②♦✉t ❖♣t✐♠✐③❛t✐♦♥ ❜❛s❡❞ ♦♥ P❛rt✐❝❧❡ ❙✇❛r♠ ❖♣t✐♠✐③❛t✐♦♥ ❢♦r ❊t❤②❧❡♥❡ ❖①✐❞❡ P❧❛♥t✳

❏♦✉r♥❛❧ ♦❢ t❤❡ ❑♦r❡❛♥ ❙♦❝✐❡t②

♦❢ ❙❛❢❡t② ✷✵✶✺✱ ✸✵✱ ✸✷✕✸✼✳ ✭✷✷✮ ❑❤❡✐r❦❤❛❤✱ ❆✳❀ ◆❛✈✐❞✐✱ ❍✳❀ ▼❡ss✐ ❇✐❞❣♦❧✐✱ ▼✳ ❉②♥❛♠✐❝ ❢❛❝✐❧✐t② ❧❛②♦✉t ♣r♦❜❧❡♠✿ ❛ ♥❡✇ ❜✐❧❡✈❡❧ ❢♦r♠✉❧❛t✐♦♥ ❛♥❞ s♦♠❡ ♠❡t❛❤❡✉r✐st✐❝ s♦❧✉t✐♦♥ ♠❡t❤♦❞s✳

■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥

❊♥❣✐♥❡❡r✐♥❣ ▼❛♥❛❣❡♠❡♥t ✷✵✶✺✱ ✻✷✱ ✸✾✻✕✹✶✵✳ ✭✷✸✮ ◆❛✈✐❞✐✱ ❍✳❀ ❇❛s❤✐r✐✱ ▼✳❀ ❇✐❞❣♦❧✐✱ ▼✳ ▼✳ ❆ ❤❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ ♦♥ t❤❡ ❢❛❝✐❧✐t② ❧❛②♦✉t ♣r♦❜❧❡♠ ❜❛s❡❞ ♦♥ ❣❛♠❡ t❤❡♦r②✳

■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ Pr♦❞✉❝t✐♦♥ ❘❡s❡❛r❝❤ ✷✵✶✷✱

✺✵✱ ✶✺✶✷✕✶✺✷✼✳ ✭✷✹✮ ◆❛❜❛✈✐✱ ❙✳ ❘✳❀ ❚❛❣❤✐♣♦✉r✱ ❆✳ ❍✳❀ ▼♦❤❛♠♠❛❞♣♦✉r ●♦r❥✐✱ ❆✳ ❖♣t✐♠✐③❛t✐♦♥ ♦❢ ❋❛❝✐❧✐t② ▲❛②♦✉t ♦❢ ❚❛♥❦ ❢❛r♠s ✉s✐♥❣ ●❡♥❡t✐❝ ❆❧❣♦r✐t❤♠ ❛♥❞ ❋✐r❡❜❛❧❧ ❙❝❡♥❛r✐♦✳ ❈❤❡♠✐❝❛❧ Pr♦❞✉❝t

❛♥❞ Pr♦❝❡ss ▼♦❞❡❧✐♥❣ ✷✵✶✻✱ ✶✶ ✳ ✭✷✺✮ ❋✉r✉❤♦❧♠❡♥✱ ▼✳❀ ●❧❡tt❡✱ ❑✳❀ ❍♦✈✐♥✱ ▼✳❀ ❚♦rr❡s❡♥✱ ❏✳ ❆ ❝♦❡✈♦❧✉t✐♦♥❛r②✱ ❤②♣❡r ❤❡✉r✐st✐❝ ❛♣♣r♦❛❝❤ t♦ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ♦❢ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ ♣r♦❝❡ss ♣❧❛♥t ❧❛②♦✉ts ✲ ❆ ❝♦♠♣❛r❛t✐✈❡ st✉❞②✳

✷✵✶✵ ■❊❊❊ ❲♦r❧❞ ❈♦♥❣r❡ss ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ■♥t❡❧❧✐❣❡♥❝❡✱ ❲❈❈■ ✷✵✶✵ ✲ ✷✵✶✵

■❊❊❊ ❈♦♥❣r❡ss ♦♥ ❊✈♦❧✉t✐♦♥❛r② ❈♦♠♣✉t❛t✐♦♥✱ ❈❊❈ ✷✵✶✵ ✷✵✶✵✱ ✭✷✻✮ ❘♦s❡♥t❤❛❧✱ ❊✳ ●❆▼❙✲❆ ✉s❡r✬s ❣✉✐❞❡✳ ●❆▼❙ ❉❡✈❡❧♦♣♠❡♥t ❈♦r♣♦r❛t✐♦♥✳ ❲❛s❤✐♥❣t♦♥✱ ❉❈✱ ❯❙❆✱ ✷✵✵✽✳ ✭✷✼✮ ❲❡st❡r❧✉♥❞✱ ❏✳❀ P❛♣❛❣❡♦r❣✐♦✉✱ ▲✳ ●✳ ■♠♣r♦✈❡❞ P❡r❢♦r♠❛♥❝❡ ✐♥ Pr♦❝❡ss P❧❛♥t ▲❛②♦✉t Pr♦❜❧❡♠s ❯s✐♥❣ ❙②♠♠❡tr②✲❜r❡❛❦✐♥❣ ❝♦♥str❛✐♥ts✳

❉❡s✐❣♥ ✷✵✵✷✱ ✹✽✺✕✹✽✽✳


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An optimisation-based approach for process plant layout - Industrial

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