(12)–(13) — the SD problem decomposes into N single-product newsvendor problems

Proved

MixFlex.RiskNeutral.sd_optimal_investment

by mikedeng1 · Sep 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

flexibilitynewsvendorp2o-batch-p200ap2o-gran-per-chapterp2o-plan-paperp2o-v1unreliable-supply

In the SD network of the mission's model, suppose $\theta>0$ and let $K^*=(K^*_1,\dots,K^*_N)\ge 0$ satisfy, for every product $n$ ,

F_{X_n}(K^*_n)=1-\frac{(\lambda+(1-\lambda)\theta)\,c}{\theta p}.

Then $K^*$ maximizes $V^{SD}_{RN}$ over all $K\ge 0$ , and the optimal SD value is

V^{SD,*}_{RN}=V^{SD}_{RN}(K^*)=\theta p\sum_{n=1}^N\mathbb E\big[\tilde X_n\,\mathbf 1\{\tilde X_n\le K^*_n\}\big].

These are equations (12)–(13): the SD investment problem splits into $N$ independent single-product problems, each solved by the same critical fractile.

Formalization Note As in (10)–(11), $F^{-1}_{X_n}$ is replaced by the hypothesis $F_{X_n}(K^*_n)=$ fractile, and $\int_0^{K^*_n}x f_{X_n}(x)\,dx$ by $\mathbb E[\tilde X_n\mathbf 1\{\tilde X_n\le K^*_n\}]$ . $\theta>0$ is needed for the fractile to be defined.

Preamble

import Mathlib
import Definitions.Def_MixFlex_RiskNeutral_Model

Formal statement

namespace MixFlex.RiskNeutral

open MeasureTheory ProbabilityTheory

/-- (12)–(13), p. 42: the SD problem decomposes into `N` single-product problems; any `K* ≥ 0`
with `F_{X_n}(K*_n) = 1 − (λ + (1 − λ)θ) c / (θ p)` for every `n` is an optimal SD investment,
and the optimal SD value is `θ p Σ_n E[X_n · 1{X_n ≤ K*_n}]`. -/
theorem sd_optimal_investment (P : Params) (hP : P.Standing) {N : ℕ} {Ω : Type*}
    [MeasurableSpace Ω] (μ : Measure Ω) (X : Ω → Fin N → ℝ) (Y : Ω → Fin (N + 1) → ℝ)
    (hS : Setting P μ X Y) (htheta : 0 < P.theta) (Kstar : Fin N → ℝ) (hK : ∀ n, 0 ≤ Kstar n)
    (hq : ∀ n, μ.real {ω | X ω n ≤ Kstar n}
      = 1 - (P.lam + (1 - P.lam) * P.theta) * P.c / (P.theta * P.p)) :
    (∀ K : Fin N → ℝ, (∀ n, 0 ≤ K n) → vSD P μ X Y K ≤ vSD P μ X Y Kstar) ∧
    vSD P μ X Y Kstar
      = P.theta * P.p * ∑ n, ∫ ω, X ω n * (if X ω n ≤ Kstar n then 1 else 0) ∂μ := by sorry

end MixFlex.RiskNeutral

Source

Tomlin and Wang, On the value of mix flexibility and dual sourcing in unreliable newsvendor networks, Manufacturing Service Oper. Management 7(1), 2005, p. 42, §3.1, (12)–(13)

Human review

Endorsed by Shuze Chen · Oct 1, 2026
Confirmed by the moderator at approval.
Endorsed by mikedeng1 · Oct 1, 2026
Confirmed by the mission captain (proposal self-audit).

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