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■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s

❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s

▼árt♦♥ ❉rót♦s ❛✱ Pét❡r ●②ör❣②✐❜✱ ❚❛♠ás ❑✐s❛❜

❛■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛♥❞ ❈♦♥tr♦❧✱ ❇✉❞❛♣❡st ❜❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❉❡♣❛rt♠❡♥t✱ ❊öt✈ös ▲♦rá♥❞ ❯♥✐✈❡rs✐t②✱ ❇✉❞❛♣❡st

❆✉ss♦✐s ❈♦♠❜✳ ❖♣t✳ ✇♦r❦s❤♦♣✱ ✷✵✶✹✳✵✶✳✵✻✲✶✵✳

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s

❈♦♥t❡♥t

✶

■♥tr♦❞✉❝t✐♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s

✷

Pr♦❞✉❝❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

✸

❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s

❇❛s✐❝ ♥♦t✐♦♥s

✶ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐❤♠s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s

ρ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❆ ❢♦r ❛ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ Π ❆ ❤❛s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t②✱ ❛♥❞ ❆(■) ≤ ρ❖P❚(■), ∀■ ∈ Π P♦❧②♥♦♠✐❛❧ ❚✐♠❡ ❆♣♣r♦①✐♠❛t✐♦♥ ❙❝❤❡♠❡ ❢♦r Π✿

✐✮ ❋❛♠✐❧② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s {❆ε}ε>✵ ✐✐✮ ❆ε ✐s ❛♥ (✶ + ε)✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r Π ✐✐✐✮ ❆ε ❤❛s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t

❋✉❧❧② P♦❧②♥♦♠✐❛❧ ❚✐♠❡ ❆♣♣r♦①✐♠❛t✐♦♥ ❙❝❤❡♠❡ ❢♦r Π✿

✐✮ ❋❛♠✐❧② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s {❆ε}ε>✵ ✐✐✮ ❆ε ✐s ❛♥ (✶ + ε)✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r Π ✐✐✐✮ ❆ε ❤❛s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣❧❡①✐t② ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t ❛♥❞ ✐♥ ✶/ε

✷ ❝❧❛ss ❆P❳ = {Π | Π ❛❞♠✐ts ❛ ρ✲❛♣♣r♦①✳ ❛❧❣✳ ❢♦r s♦♠❡ ρ > ✶} ✸ ❯♥❧❡ss P = ◆P✱ ❝❧❛ss ❆P❳ = ❝❧❛ss P❚❆❙ ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s

■♥tr♦❞✉❝t✐♦♥ t♦ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s

❊①t❡♥❞ ❜❛s✐❝ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ♥♦♥✲r❡♥❡✇❛♥❧❡ r❡s♦✉r❝❡s✳ ✸ ❝❛s❡s✿ ❖♥❧② ♣r♦❞✉❝❡r ❥♦❜s ❖♥❧② ❝♦♥s✉♠❡r ❥♦❜s ❇♦t❤ ♣r♦❞✉❝❡r ❛♥❞ ❝♦♥s✉♠❡r ❥♦❜s ✭♥♦t ✐♥ t❤✐s t❛❧❦✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

Pr♦❞✉❝❡r ❏♦❜s

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

Producer jobs

5 10 S t 15

u1 u3 u2 u4

Cumulative demand Cumulative production

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

❊❛r❧✐❡r ✇♦r❦ ♦♥ ♣r♦❞✉❝❡r ❥♦❜s

Pr♦❜❧❡♠ ❞❛t❛ ❙✐♥❣❧❡ ♠❛❝❤✐♥❡ ❙❡t ♦❢ ♣r♦❞✉❝ts t②♣❡s ❙✳ ❏♦❜s✿ ❏❥✱ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♣❥✱ t②♣❡ s❥ ∈ ❙✱ ❛♥❞ ❛ q✉❛♥t✐t② ❛s

❥

♣r♦❞✉❝❡❞ ❢r♦♠ ♣r♦❞✉❝t s❥✳ ❊①t❡r♥❛❧ ❞❡♠❛♥❞ r❡q✉❡sts ✇✐t❤ ❞❡❛❞❧✐♥❡s ✉✶ ≤ · · · ≤ ✉q✱ ♣r♦❞✉❝t t②♣❡s ρℓ ∈ ❙✱ ❛♥❞ q✉❛♥t✐t✐❡s r❡q✉❡st❡❞ ❜ρ

ℓ ✳

❘❡s✉❧ts ♦❢ ❇♦②s❡♥ ❡t ❛❧ ✭✷✵✶✷✮ ❉❡❝✐❞✐♥❣ ✇❤❡t❤❡r ❛ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ ❡①✐sts ✐s ◆P ❝♦♠♣❧❡t❡ ❖❜❥❡❝t✐✈❡✿ ♠✐♥✐♠✐③❡ ♠❛①✐♠✉♠ st♦❝❦ s✐③❡ ✭❛ss✉♠✐♥❣ t❤❛t ✇❡ ❦♥♦✇ t❤❛t ❛ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ ❡①✐sts✮ ❙❡✈❡r❛❧ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♣❡❝✐❛❧ ❝❛s❡s

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

◆❡✇ r❡s✉❧ts ♦♥ ♣r♦❞✉❝❡r ❥♦❜s

❉✉❡ ❞❛t❡s ✐♥st❡❛❞ ♦❢ ❞❡❛❞❧✐♥❡s✱ s❤✐♣♠❡♥ts ♠❛② ❜❡ ❧❛t❡ ❆ s✐♥❣❧❡ ❥♦❜ ♠❛② ♣r♦❞✉❝❡ s❡✈❡r❛❧ t②♣❡s ♦❢ ♣r♦❞✉❝ts ❖❜❥❡❝t✐✈❡✿ ♠✐♥✐♠✐③❡ ❛ r❡❣✉❧❛r ❢✉♥❝t✐♦♥ ♦❢ s❤✐♣♠❡♥t ❞❛t❡s✱ ❡✳❣✳✱ t❛r❞✐♥❡ss ▼❛✐♥ r❡s✉❧ts✿ Pr♦❜❧❡♠ ❈♦♠♣❧❡①✐t② ✶| ♣❥ = ♣, |❙| = ✶ | γ ❖(♥ ❧♦❣ ♥) ✶| ❛s

❥ = ❛s | γ

❖(♥ ❧♦❣ ♥) ✶| ♣❥ = ✶, |❙| = ✷, q = ✷ | γ❚ ♦r❞✐♥❛r② ◆P❤ ✶| ♣❥ = ✶, |❙| = ✷ |γ❚ str♦♥❣❧② ◆P❤ ✶| |❙| = ✶|γ❚ str♦♥❣❧② ◆P❤ ✶| ♣❥ = ✶, q = ✷ | γ❚ str♦♥❣❧② ◆P❤ ✶| |❙| = ❝♦♥st, q = ❝♦♥st | γ ♣s❡✉❞♦✲♣♦❧②t✐♠❡ ✶| |❙| = ✶, q = ✷ | ♣s✉♠ + ❚♠❛① ❋P❚❆❙

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

Producer jobs

5 10 S t 15

u1 u3 u2 u4

Cumulative demand Cumulative production

• 1. shipment
• 2. shipment
• 3. shipment
• 4. shipment

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

P♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s

✶ ❙♦❧✈✐♥❣ ✬✶| ♣❥ = ♣, |❙| = ✶ | γ✬ ✐♥ ❖(♥ ❧♦❣ ♥) t✐♠❡

❙❝❤❡❞✉❧❡ t❤❡ ❥♦❜s ✐♥ ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❛s

❥ ♦r❞❡r✳

✷ ❙♦❧✈✐♥❣ ✬✶| ❛s

❥ = ❛s | γ✬ ✐♥ ❖(♥ ❧♦❣ ♥) t✐♠❡

❙❝❤❡❞✉❧❡ t❤❡ ❥♦❜s ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♦r❞❡r ✭❙P❚ r✉❧❡✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

Ps❡✉❞♦✲♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ✜①❡❞ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s ❛♥❞ ❞❡❧✐✈❡r✐❡s

❋✐①❡❞ |❙| ✭♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s✮✱ q ✭♥✉♠❜❡r ♦❢ ❞❡❧✐✈❡r✐❡s✮ t✐♠❡ ✐♥t❡r✈❛❧ ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ❞❡❧✐✈❡r✐❡s✱ ❛♥❞ ❜❡t✇❡❡♥ ✵ ❛♥❞ t❤❡ ✜rst ❞❡❧✐✈❡r② ❉❡✜♥❡ ❛♥ ❛❝②❝❧✐❝ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ♥❏ + ✶ ❧❡✈❡❧s✱ ✵, ✶, . . . , ♥ ◆♦❞❡s✿ ◆(❥, P✶, . . . , Pq, ∆✶

✶, . . . , ∆♥❙ q )✱ ✇❤❡r❡

✐✮ ❥ ✐s t❤❡ ❥♦❜ ❥✉st s❝❤❡❞✉❧❡❞ ✐✐✮ P✐ ✐s t❤❡ t♦t❛❧ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♦❢ t❤♦s❡ ❥♦❜s s❝❤❡❞✉❧❡❞ ✐♥ s❧♦t ✐ ✐✐✐✮ ∆s

✐ ✐s t❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ r❡s♦✉r❝❡ s s❝❤❡❞✉❧❡❞ ✐♥ s❧♦t ✐

❆r❝s✿ ❢♦r♠ ◆(❥, P✶, . . . , Pq, ∆✶

✶, . . . , ∆♥❙ q ) t♦

◆(❥ + ✶, P✶, . . . , P✐ + ♣❥, Pq, ∆✶

✶, . . . , ∆s ✐ + ❛s ❥ , . . . , ∆♥❙ q )

❆♥② ❞✐r❡❝t❡❞ ♣❛t❤ ❢r♦♠ t❤❡ s♦✉r❝❡ ♥♦❞❡ ❛t ❧❡✈❡❧ ✵ t♦ ❛ t❡r♠✐♥❛❧ ♥♦❞❡ ❛t ❧❡✈❡❧ ♥❏ r❡♣r❡s❡♥ts ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥✳ Pr♦✈✐❞❡s t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ ❢♦r ❛♥② r❡❣✉❛❧ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❧✐✈❡r② ❝♦♠♣❧❡t✐♦♥ t✐♠❡s

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

■❧❧✉str❛t✐♦♥

(✵, ✵, ✵, ✵, ✵) (✶, ✹, ✵, ✻, ✵) (✶, ✵, ✹, ✵, ✻) (✷, ✼, ✵, ✶✶, ✵) (✷, ✵, ✼, ✵, ✶✶) · · · · · · · · · · · · · · · ✹(✻) ✹(✻) ✸(✺) ✸(✺) ✸(✺) ✸(✺) ✹(✶) ✹(✶) ✹(✶) ✹(✶)

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts

❋P❚❆❙ ❛❧❣♦r✐t❤♠s ❢♦r ✶|q = ✷, |❙| = ✶|❚♠❛①

✶ ❚♠❛① ❝❛♥ ❜❡ ✵✱ t❤❡r❡❢♦r❡✱ ✇❡ s❤✐❢t ✐t ❜② ❛ ♣r♦❜❧❡♠ ✐♥st❛♥❝❡

s♣❡❝✐✜❝ ✈❛❧✉❡✱ ❡✳❣✳✱ ✉q✱ t❤❡ ❧❛st ❞❡❧✐✈❡r② ❞✉❡✲❞❛t❡✳

✷ ❋P❚❆❙✲✶✿ ♦♥❡ ❝❛♥ t✉r♥ t❤❡ ❛❧r❡❛❞② s❡❡♥ ♣s❡✉❞♦✲♣♦❧②♥♦♠✐❛❧

t✐♠❡ ❛❧❣♦r✐t❤♠ ✐♥t♦ ❛♥ ❋P❚❆❙ ♦❢ t✐♠❡ ❝♦♠♣❧❡①✐t② (♥✼/ε✹)

✸ ❋P❚❆❙✲✷✿ ✉s❡ ❛♥ ❋P❚❆❙ ❢♦r ❜✐♥❛r② ❦♥❛♣s❛❝❦ t♦ s♦❧✈❡ ✐t ✐♥

❡✳❣✳✱ ❖(♥✸/ε) t✐♠❡ ✭st❛t❡ ♦❢ t❤❡ ❛rt ✐s ♠✉❝❤ ❜❡tt❡r✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥s✉♠❡r ❏♦❜s

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

Pr♦❜❧❡♠ ❞❛t❛ ❙✐♥❣❧❡ ♠❛❝❤✐♥❡✱ s❡t ♦❢ r❡s♦✉r❝❡ t②♣❡s ❘ ❏♦❜s✿ ❏❥✱ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♣❥✱ ❛♥❞ q✉❛♥t✐t✐❡s ❛ρ

❥ r❡q✉✐r❡❞ ❢r♦♠

❡❛❝❤ r❡s♦✉r❝❡ ρ ∈ ❘ ❊①t❡r♥❛❧ s✉♣♣❧② ❛t t✐♠❡ ♣♦✐♥ts ✵ = ✉✶ < · · · < ✉q✱ ❛♥❞ q✉❛♥t✐t✐❡s s✉♣♣❧✐❡❞ ❜ρ

ℓ ❢r♦♠ ❡❛❝❤ ρ ∈ ❘✱ ℓ = ✶, . . . , q

✉q+✶ = ∞✱ ❛♥❞

❥ ❛❥ = ℓ ❜ℓ

❙❝❤❡❞✉❧❡s ❙t❛rt✐♥❣ t✐♠❡ ❙❥ ❢♦r ❡❛❝❤ ❥♦❜ ❏❥✱ ❋❡❛s✐❜❧❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❥♦❜s ❞♦ ♥♦t ♦✈❡r❧❛♣ ✐♥ t✐♠❡✱ ❛♥❞ ❙❥ < ✉ℓ+✶ ✐♠♣❧✐❡s

• ✐ : ❙✐≤❙❥

❛ρ

✐ ≤ ℓ

• ❦=✶

❜ρ

❦,

∀ρ ∈ ❘

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

5 10 S t 15

Consumer jobs

u1=0 u3 u2 u4=uq

Cumulative supply Cumulative demand

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❊❛r❧✐❡r ✇♦r❦ ♦♥ ❝♦♥s✉♠❡r ❥♦❜s

❈❛r❧✐❡r ✭✽✹✮✱ ●r✐❣♦r✐❡✈ ❡t ❛❧ ✭✷✵✵✺✮✿ ✶|r♠ = ✶|❈♠❛①✱ ✶|r♠ = ✶|▲♠❛① ❛r❡ ◆P✲❤❛r❞ ✶|r♠ = ✶, ♣❥ = ♣|❈♠❛①✱ ✶|r♠ = ✶, ♣❥ = ✶|▲♠❛① ♣♦❧②♥♦♠✐❛❧❧② s♦❧✈❛❜❧❡ ✶|r♠ = ✶|❈♠❛① r❡❞✉❝❡s t♦ ❋✷||❈♠❛① ✐♥ ❝❛s❡ ♦❢ ✉♥✐t s✉♣♣❧② ✐♥ ❡❛❝❤ t✐♠❡ ✉♥✐t✳ ✶|r♠ = ✷, ♣❥ = ✶|❈♠❛① ◆P✲❤❛r❞✱ t❤❡r❡ ❡①✐st ✷✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✶|r♠ = ✷, ♣❥ = ♣|▲♠❛① ◆P✲❤❛r❞✱ t❤❡r❡ ❡①✐st ✷✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❙✉♠♠❛r② ♦❢ ♥❡✇ r❡s✉❧ts

✶ ❋P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ✷|❈♠❛① ✷ P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①✿

❯s❡ s♦♠❡ ✐❞❡❛s ♦❢ ❈❤❡❦✉r✐ ❛♥❞ ❑❤❛♥♥❛ ✭✷✵✵✻✮ ❢♦r ▼✉❧t✐♣❧❡ ❑♥❛♣s❛❝❦ Pr♦❜❧❡♠ ✭▼❑P✮

✸ P❚❆❙ ❢♦r ✶|r❥, r♠ = ✶, q = ❝♦♥st|❈♠❛① ✹ P❚❆❙ ❢♦r ✶|r❥, r♠ = ❝♦♥st, r − ❛❣r, q = ❝♦♥st|❈♠❛①

r✲❛❣r✿ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ π ♦❢ ❥♦❜s s✉❝❤ t❤❛t ❛π(✶) ≤ ❛π(✷) ≤ · · · ≤ ❛π(♥) ✭❝♦♠♣♦♥❡♥t✇✐s❡✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥

①t❥ = ✶ ⇐ ⇒ ❥♦❜ ❥ st❛rts ❜❡t✇❡❡♥ ✉t ❛♥❞ ✉t+✶✱ ♦t❤❡r✇✐s❡ ✵✳ ❈t✿ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤♦s❡ ❥♦❜s st❛rt✐♥❣ ✐♥ [✉t, ✉t+✶)✳ ❈t > ✉t+✶ ✐s ♣♦ss✐❜❧❡✳ ❜′

✐ := ✐ ℓ=✶ ❜ℓ

0=u1 u2 u3 uq

...

• 1. period

C1 C2

• 2. period

...

J 1 J 2 J 3 J 4 J 5

x13=x12=x25=x31=x34=1

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

■P ♠♦❞❡❧

♠✐♥

q

♠❛①

ℓ=✶

 ✉ℓ +

q

• t=ℓ
• ❥∈J

♣❥①t❥   ✭✶✮ s.t.

• ❥∈J

❛❥ ℓ

• t=✶

①t❥

• ≤ ❜′

ℓ,

ℓ = ✶, . . . , q ✭✷✮

q

• t=✶

①t❥ = ✶, ❥ ∈ J ✭✸✮ ①t❥ ∈ {✵, ✶}, t = ✶, . . . , q; ❥ ∈ J ✭✹✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

P❚❆❙✿ Pr❡❧✐♠✐♥❛r✐❡s

❋♦r ❛♥② ε✱ ✇❡ ❛✐♠ ❛t (✶ + ❖(ε)) ❛♣♣r♦①✐♠❛t✐♦♥ ◆♦t✐❝❡ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❥♦❜s ❧♦♥❣❡r t❤❛♥ ε♣s✉♠ ✐s ❛t ♠♦st ✶/ε✳ ❲❡ s❝❤❡❞✉❧❡ t❤❡♠ ✐♥ ❛❧❧ ♣♦ss✐❜❧❡ ✇❛②s✳ ❋♦r ❛♥② ❛ss✐❣♥❡♥t ¯ ①❇

t❥ ♦❢ ✧❜✐❣ ❥♦❜s✧ t♦ s✉♣♣❧② ♣❡r✐♦❞s✱ ✇❡

s❝❤❡❞✉❧❡ t❤❡ r❡♠❛✐♥✐♥❣ ✧s♠❛❧❧ ❥♦❜s✧ ✇✐t❤ ♠✐♥✐♠✉♠ ♠❛❦❡s♣❛♥ ❆❢t❡r ✉q t❤❡ s❝❤❡❞✉❧❡ ✐s tr✐✈✐❛❧ → s❝❤❡❞✉❧❡ ❛s ♠❛♥② s♠❛❧❧ ❥♦❜s ❛s ✇❡ ❝❛♥ ❜❡❢♦r❡ ✉q ✭r❡s♣❡❝t r❡s♦✉r❝❡ ❝♦♥str❛✐♥ts✦✮

0=u1 u2 u3 u4=uq

• 1. period
• 2. period

J 1 J 2 J 3 J 4 J 5 J 6 J 7

• 3. period
• 4. period

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

▼♦❞❡❧ ❢♦r s❝❤❡❞✉❧✐♥❣ t❤❡ s♠❛❧❧ ❥♦❜s

❖P❚¯

①❇ := ♠❛①

• t<q,❥∈S

♣❥①t❥ ✭✼✮ s.t.

• ❥∈S

❛❥ ℓ

• t=✶

①t❥

• ≤ ˜

❜ℓ(¯ ①❇), ℓ ≤ q − ✶ ✭✽✮

• ❥∈S

♣❥①t❥ ≤ ♠❛①{✵, ✉t+✶ − ❈t(¯ ①❇)} + ε♣s✉♠, t ≤ q − ✶ ✭✾✮

q−✶

• t=✶

①t❥ ≤ ✶, ❥ ∈ S ✭✶✵✮ ①t❥ ∈ {✵, ✶}, t ≤ q − ✶; ❥ ∈ S ✭✶✶✮

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥♥❡❝t✐♦♥ ✇✐t❤ t❤❡ ▼✉❧t✐♣❧❡ ❑♥❛♣s❛❝❦ Pr♦❜❧❡♠

❚❤❡ ▼❑P ♣r♦❜❧❡♠ ❚❤❡r❡ ❛r❡ ♠ ❦♥❛♣s❛❝❦s✱ ❡❛❝❤ ❤❛✈✐♥❣ ✐ts ♦✇♥ s✐③❡ ❚❤❡r❡ ❛r❡ ♥ ✐t❡♠s ❡❛❝❤ ❤❛✈✐♥❣ ❛ s✐③❡ ❛♥❞ ❛ ♣r♦✜t ■t❡♠s ♠✉st ❜❡ ❛❧❧♦❝❛t❡❞ t♦ ❦♥❛♣s❛❝❦s s✉❝❤ t❤❛t

◆♦ ❦♥❛♣s❛❝❦ ✐s ♦✈❡r✜❧❧❡❞ ❚❤❡ t♦t❛❧ ♣r♦✜t ♦❢ s❡❧❡❝t❡❞ ✐t❡♠s ✐s ♠❛①✐♠✐③❡❞

❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ▼❑P✿ ❚✇♦ t②♣❡s ♦❢ ❦♥❛♣s❛❝❦ ❝♦♥str❛✐♥ts✱ ♦♥❡ ✐s ♥❡st❡❞✱ t❤❡ ♦t❤❡r ✐s ✧st❛♥❞❛r❞✧ ❚❤❡ ✐t❡♠s ❛r❡ t❤❡ s♠❛❧❧ ❥♦❜s✱ ❛♥❞ t❤❡ ❦♥❛♣s❛❝❦s ❝♦♥str❛✐♥ts ❛r❡ ✭✽✮ ❛♥❞ ✭✾✮ ❘❡❝②❝❧❡ s♦♠❡ ✐❞❡❛s ❢r♦♠ t❤❡ P❚❆❙ ♦❢ ❈❤❡❦✉r✐ ❛♥❞ ❑❤❛♥♥❛ ✭✷✵✵✻✮ ❢♦r ▼❑P

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❍♦✇ t♦ s❝❤❡❞✉❧❡ s♠❛❧❧ ❥♦❜s❄

✶ ✬●✉❡ss✬ t❤❡ ♦♣t✐♠✉♠ ❖P❚¯

①❇

❚❡st ❡❛❝❤ ♠❡♠❜❡r ♦❢ ● := {♣❙(¯

①❇) ♠❛① (✶ + ε)✐ | ✵ ≤ ✐ ≤ ❣}

✇❤❡r❡ ❣ ≤ ⌊✷ε−✶ ❧♥ ♥⌋ ✭♣❙(¯

①❇) ♠❛① (✶ + ε)❣+✶ > ♥ · ♣❙(¯ ①❇) ♠❛①

≥ ❖P❚¯

①❇✮

✷ ❋♦r ❡❛❝❤ ❣✉❡ss O ∈ ●✱ ♣❛rt✐t✐♦♥ t❤❡ s❡t ♦❢ s♠❛❧❧ ❥♦❜s ✐♥t♦

s✉❜s❡ts ❙✶, . . . , ❙❤✿ ❙ℓ := {❥ ∈ ❙(¯ ①❇) | (✶ + ε)ℓ(εO/♥) ≤ ♣❥ < (✶ + ε)ℓ+✶(εO/♥)} ✇❤❡r❡ ❤ ≤ ⌊✹ε−✶ ❧♥ ♥⌋ + ✶✱ ❢♦r (εO/♥)(✶ + ε)❤ > O

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

S1

Sl

S3 S2

0=u1 u2 u3 u4

• 1. period
• 2. period

J 1 J 2 J 3 J 4 J 5

• 3. period
• 4. period

V 1

3

… p1' =(1+ε)(εO /n') p3' =(1+ε)

3(εO /n')

p2' =(1+ε)

2(εO/n' )

Schedule of big jobs Partition the small jobs into sets ; each set is ordered in increasing resource req.

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

S1

… …

S3 S2

0=u1 u2 u3 u4

• 1. period
• 2. period

J 3 J 4 J 1 J 5 J 2

• 3. period
• 4. period

U 1

1

U 1

2

U 1

3

… U 2

1 U 3 1

U 3

2

U 3

1

U 2

2 U 2 3

U 1

1 U 1 2

U 2

3

U 3

2

U 2

2

U 2

1

U 3

3

… U 1

3

0=u1 u2 u3 u4

• 1. period
• 2. period

J 3 J 4 J 1 J 5 J 2

• 3. period
• 4. period

… U 3

3

❈♦♠♣❧❡①✐t②✿ ❖(q✶/ε · (✷ε−✶ ❧♥ ♥) · (♥✷ε−✶ ❧♦❣✷ ♥ + (q♥) · ♥❖(ε✶−q+ε−✷)))

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❞❡❧✐✈❡r② t❛r❞✐♥❡ss ✭♣r♦❞✉❝❡r ❥♦❜s✮✱ ❛♥❞ t❤❡ ♠❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ✭❝♦♥s✉♠❡r ❥♦❜s✮ ♣r♦❜❧❡♠s

▲❡♠♠❛

• ✐✈❡♥ ❛♥ ✐♥st❛♥❝❡ ■ = {♥, q, (♣❥, ❛❥)♥

❥=✶, (✉ℓ, ❜ℓ)q ℓ=✶} ♦❢ t❤❡ ❉❡❧✐✈❡r②

t❛r❞✐♥❡ss ♣r♦❜❧❡♠✳ ❉❡✜♥❡ ❛♥ ✐♥st❛♥❝❡ ■ ′ = {♥, q, (♣❥, ❛❥)♥

❥=✶, (✉′ ℓ, ❜′ ℓ)q ℓ=✶} ♦❢ t❤❡ ▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥

♣r♦❜❧❡♠✿ ✉′

ℓ = ✉q − ✉q+✶−ℓ

❜′

ℓ = ❜q+✶−ℓ

ℓ = ✶, . . . , q. ❚❤❡♥✱ ✐❢ σ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❥♦❜s ❣✐✈✐♥❣ ❛ ♠❛①✐♠✉♠ ❞❡❧✐✈❡r② t❛r❞✐♥❡ss ♦❢ ❚ σ

♠❛① ❢♦r ■✱ t❤❡♥ s❝❤❡❞✉❧✐♥❣ t❤❡ ❥♦❜s ✐♥ r❡✈❡rs❡ σ ♦r❞❡r

❣✐✈❡s ❛ s❝❤❡❞✉❧❡ ♦❢ ♠❛❦❡s♣❛♥ ✉q + ❚ σ

♠❛① ❢♦r ✐♥st❛♥❝❡ ■ ′ ♦❢ t❤❡

▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥♥❡❝t✐♦♥ ✳✳✳

▲❡♠♠❛

• ✐✈❡♥ ❛♥ ✐♥st❛♥❝❡ ■ = {♥, q, (♣❥, ❛❥)♥

❥=✶, (✉ℓ, ❜ℓ)q ℓ=✶} ♦❢ t❤❡

▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✳ ❉❡✜♥❡ ❛♥ ✐♥st❛♥❝❡ ■ ′ = {♥, q, (♣❥, ❛❥)♥

❥=✶, (✉′ ℓ, ❜′ ℓ)q ℓ=✶} ♦❢ t❤❡ ❉❡❧✐✈❡r② t❛r❞✐♥❡ss ♣r♦❜❧❡♠✿

✉′

ℓ = ✉q − ✉q+✶−ℓ

❜′

ℓ = ❜q+✶−ℓ

ℓ = ✶, . . . , q. ❚❤❡♥✱ ✐❢ ❙ ✐s ❛ s❝❤❡❞✉❧❡ ✇✐t❤ ❛ ♠❛❦❡s♣❛♥ ♦❢ ❈ ❙

♠❛① ❢♦r ■✱ t❤❡♥

s❝❤❡❞✉❧✐♥❣ t❤❡ ❥♦❜s ✐♥ r❡✈❡rs❡ ♦r❞❡r ✭✇✐t❤♦✉t ❛♥② ❞❡❧❛②s ❛♠♦♥❣ t❤❡♠✮ ❣✐✈❡s ❛ s❝❤❡❞✉❧❡ ♦❢ ♠❛①✐♠✉♠ t❛r❞✐♥❡ss ❛t ♠♦st ❈ ❙

♠❛① − ✉q

❢♦r ✐♥st❛♥❝❡ ■ ′ ♦❢ t❤❡ ❉❡❧✐✈❡r② t❛r❞✐♥❡ss ♣r♦❜❧❡♠✳

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

5 10 S t 15

u1 u3 u2 u4

Cumulative demand Cumulative production

• 1. shipment
• 2. shipment
• 3. shipment
• 4. shipment

5 10 S t 15

u' 1 u' 3 u' 2 u' 4

Cumulative supply

• 1. supply
• 2. supply
• 3. supply
• 4. supply

Cumulative consumption

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ❝♦♥♥❡❝t✐♦♥

❈♦r♦❧❧❛r② ▲❡t (■❉, ■▼) ❜❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥st❛♥❝❡s ♦❢ t❤❡ ❉❡❧✐✈❡r② t❛r❞✐♥❡ss ❛♥❞ t❤❡ ▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s✳ ❚❤❡♥ t❤❡ ♦♣t✐♠✉♠ ✈❛❧✉❡ ❚ ∗

♠❛①(■❉) ♦❢ t❤❡ ❉❡❧✐✈❡r② t❛r❞✐♥❡ss ♣r♦❜❧❡♠ ❡q✉❛❧s ❈ ∗ ♠❛①(■❉) − ✉q✱

t❤❡ ♦♣t✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ ▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ♠✐♥✉s ✉q✳ ❚❤❡♦r❡♠ ❚❤❡ ❉❡❧✐✈❡r② t❛r❞✐♥❡ss ♣r♦❜❧❡♠ ❛❞♠✐ts ❛ ❋P❚❆❙ ✴ P❚❆❙ ❢♦r s♦♠❡ q ❛♥❞ r ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ▼❛❦❡s♣❛♥ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❛❞♠✐ts ❛♥ ❋P❚❆❙ ✴ P❚❆❙ ❢♦r t❤❡ s❛♠❡ q ❛♥❞ r✳

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ Pr♦❞✉❝❡r ❥♦❜s ❈♦♥s✉♠❡r ❥♦❜s ❊❛r❧✐❡r ✇♦r❦ ◆❡✇ r❡s✉❧ts P❚❆❙ ❢♦r ✶|r♠ = ✶, q = ❝♦♥st|❈♠❛①

❈♦♥❝❧✉❞✐♥❣ r❡♠❛r❦s

❘❡s✉❧ts ♦♥ t❤❡ ✇❛② ❆♣♣r♦①✐♠❛t✐♦♥ r❡❞✉❝t✐♦♥s ❜❡t✇❡❡♥ ❑♥❛♣s❛❝❦ ❛♥❞ ❙❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s ✭✐♥ ❜♦t❤ ❞✐r❡❝t✐♦♥s✦✮ P❚❆❙ ❢♦r ✜①❡❞ ♥✉♠❜❡r ♦❢ s✉♣♣❧✐❡s ❛♥❞ ✜①❡❞ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s P❚❆❙ ❢♦r ♣r♦❜❧❡♠s ✇✐t❤ r❡❧❡❛s❡ ❞❛t❡s✱ ❛♥❞ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ✜①❡❞ ♥✉♠❜❡r ♦❢ s✉♣♣❧✐❡s ❛♥❞ ✜①❡❞ ♥✉♠❜❡r ♦❢ r❡s♦✉r❝❡s ❖♣❡♥ ♣r♦❜❧❡♠s ❉♦❡s t❤❡r❡ ❡①✐st ❛ P❚❆❙ ✇❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ s✉♣♣❧✐❡s ✴ ❞❡❧✐✈❡r② ❞✉❡ ❞❛t❡s ✐s ♣❛rt ♦❢ t❤❡ ✐♥♣✉t❄ ❊①t❡♥s✐♦♥✿ ▼✉❧t✐♣❧❡ ♠❛❝❤✐♥❡s❄ ❊①t❡♥s✐♦♥✿ Pr❡❝❡❞❡♥❝❡ ❝♦♥str❛✐♥ts❄

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

❘❡❢❡r❡♥❝❡s

❘❡❢❡r❡♥❝❡s

◆✳ ❇♦②s❡♥✱ ❙✳ ❇♦❝❦✱ ▼✳ ❋❧✐❡❞♥❡r✱ ❙❝❤❡❞✉❧✐♥❣ ♦❢ ✐♥✈❡♥t♦r② r❡❧❡❛s✐♥❣ ❥♦❜s t♦ s❛t✐s❢② t✐♠❡✲✈❛r②✐♥❣ ❞❡♠❛♥❞✿ ❛♥ ❛♥❛❧②s✐s ♦❢ ❝♦♠♣❧❡①✐t②✱ ❏♦✉r♥❛❧ ♦❢ ❙❝❤❡❞✉❧✐♥❣✱ ❱♦❧✉♠❡ ✶✻ ✭✷✵✶✸✮✱ ✶✽✺✕✶✾✽✳ ❈✳ ❈❤❡❦✉r✐✱ ❙✳ ❑❤❛♥♥❛✿ ❆ P♦❧✐♥♦♠✐❛❧ ❚✐♠❡ ❆♣♣r♦①✐♠❛t✐♦♥ ❙❝❤❡♠❡ ❢♦r t❤❡ ▼✉❧t✐♣❧❡ ❑♥❛♣s❛❝❦ Pr♦❜❧❡♠✱ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ❱♦❧✉♠❡ ✸✺✱ ◆♦✳ ✸ ✭✷✵✵✻✮✱ ✼✶✸✕✼✷✽✳ ❏✳ ❈❛r❧✐❡r✱ Pr♦❜❧è♠❡s ❞✬♦r❞♦♥♥❛♥❝❡♠❡♥ts á ❝♦♥tr❛✐♥t❡s ❞❡ r❡ss♦✉r❝❡s✿ ❛❧❣♦r✐t❤♠❡s ❡t ❝♦♠♣❧❡①✐té❡✱ ❚❤ès❡ ❞✬ét❛t✱ ✶✾✽✹✳ ❆✳ ●r✐❣♦r✐❡✈✱ ▼✳ ❍♦❧t❤✉✐❥s❡♥✱ ❏✳ ✈❛♥ ❞❡ ❑❧✉♥❞❡rt✿ ❇❛s✐❝ ❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠s ✇✐t❤ ❘❛✇ ▼❛t❡r✐❛❧ ❈♦♥str❛✐♥ts✱ ◆❛✈❛❧ ❘❡s❡❛r❝❤ ▲♦❣✐st✐❝s✱ ❱♦❧✳ ✺✷ ✭✷✵✵✺✮✱ ✺✷✼✕✺✸✺✳ ▼✳ ❉rót♦s✱ ❚✳ ❑✐s✱ ❙❝❤❡❞✉❧✐♥❣ ♦❢ ✐♥✈❡♥t♦r② r❡❧❡❛s✐♥❣ ❥♦❜s t♦ ♠✐♥✐♠✐③❡ ❛ r❡❣✉❧❛r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ ❞❡❧✐✈❡r② t✐♠❡s✱ ❏♦✉r♥❛❧ ♦❢ ❙❝❤❡❞✉❧✐♥❣✱ ❱♦❧✉♠❡ ✶✻ ✭✷✵✶✸✮✱ ✸✸✼✕✸✹✻✳ P✳ ●②ör❣②✐✱ ❚✳ ❑✐s✿ ❆♣♣r♦①✐♠❛t✐♦♥ s❝❤❡♠❡s ❢♦r s✐♥❣❧❡ ♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s✱ ✐♥ ♣r✐♥t ✭✷✵✶✸✮ ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s