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- • Jt(x) — Jt(x − 1) is non-decreasing in t, Vx. - - • Jt(x) — Jt(x − 1) is non-increasing in x, Vt. - Hint: The second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x): = max0

- • Jt(x) — Jt(x − 1) is non-decreasing in t, Vx. - - • Jt(x) — Jt(x − 1) is non-increasing in x, Vt. - Hint: The second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x): = max0

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Consider the Dynamic Program formulation of the n-class optimal booking problem with sequential arrivals. In this
problem, show some monotonicities of the value function.
• Jt(x) − Jt(x − 1) is non-decreasing in t, ∀x.
• Jt(x) − Jt(x − 1) is non-increasing in x, ∀t.
Consider that the second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x) = max0≤u≤min{Dt,x} (pt)u + Jt−1(x − u) in x, i.e., Ht(Dt, x + 1) − Ht(Dt, x) ≤ Ht(Dt, x) − Ht(Dt, x − 1) for any x ∈ Z+ for any t. Image attatched for clarity.

- • Jt(x) — Jt(x − 1) is non-decreasing in t, Vx. - - • Jt(x) — Jt(x − 1) is non-increasing in x, Vt. - Hint: The second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x): = max0<u<min{D₁,x} Ptu + Jt-1(x - u) in x, i.e., H₁(Dt, x + 1) − H₁(Dt, x) ≤ Ht(Dt, x) — Ht(Dt, x-1) for any x € Z+ for any t. - -
Transcribed Image Text:- • Jt(x) — Jt(x − 1) is non-decreasing in t, Vx. - - • Jt(x) — Jt(x − 1) is non-increasing in x, Vt. - Hint: The second property implies the first property. To prove the second property, given Dt, consider the discrete concavity of Ht(Dt, x): = max0<u<min{D₁,x} Ptu + Jt-1(x - u) in x, i.e., H₁(Dt, x + 1) − H₁(Dt, x) ≤ Ht(Dt, x) — Ht(Dt, x-1) for any x € Z+ for any t. - -
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