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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✐♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s Pr♦❞✉❝❡r ❥♦❜s ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s ❈♦♥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✐♦♥ ❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s Pr♦❞✉❝❡r ❥♦❜s ❙❝❤❡❞✉❧✐♥❣ ✇✐t❤ ♥♦♥✲r❡♥❡✇❛❜❧❡ r❡s♦✉r❝❡s ❈♦♥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✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s Pr♦❞✉❝❡r ❏♦❜s ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s Producer jobs Cumulative production S 15 Cumulative demand 10 5 u 1 t u 2 u 3 u 4 ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s ❊❛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 ρ ℓ ∈ ❙ ✱ ❛♥❞ 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✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s ◆❡✇ 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 | γ ✶ | ❛ s ❖ ( ♥ ❧♦❣ ♥ ) ✶ | ♣ ❥ = ✶ , | ❙ | = ✷ , 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✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s Producer jobs Cumulative production S 15 Cumulative demand 10 5 t u 1 u 2 u 3 u 4 1. shipment 2. shipment 3. shipment 4. shipment ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s

■♥tr♦❞✉❝t✐♦♥ ❊❛r❧✐❡r ✇♦r❦ Pr♦❞✉❝❡r ❥♦❜s ◆❡✇ r❡s✉❧ts ❈♦♥s✉♠❡r ❥♦❜s P♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✶ ❙♦❧✈✐♥❣ ✬✶ | ♣ ❥ = ♣ , | ❙ | = ✶ | γ ✬ ✐♥ ❖ ( ♥ ❧♦❣ ♥ ) t✐♠❡ ❙❝❤❡❞✉❧❡ t❤❡ ❥♦❜s ✐♥ ♥♦♥✲✐♥❝r❡❛s✐♥❣ ❛ s ❥ ♦r❞❡r✳ ❥ = ❛ s | γ ✬ ✐♥ ❖ ( ♥ ❧♦❣ ♥ ) t✐♠❡ ✷ ❙♦❧✈✐♥❣ ✬✶ | ❛ s ❙❝❤❡❞✉❧❡ t❤❡ ❥♦❜s ✐♥ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ♣r♦❝❡ss✐♥❣ t✐♠❡ ♦r❞❡r ✭❙P❚ r✉❧❡✮ ❉rót♦s✱ ●②ör❣②✐✱ ❑✐s