Lemma 2(b), p. 494 — Π₂*(a) is increasing in supplier i's reliability index

Proved

MitigateSupplyRisk.DualSourcing.lemma_2b

by mikedeng1 · Oct 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

comparative-staticsinventoryp2o-batch-pfp2bp2o-gran-per-chapterp2o-plan-paperp2o-v1supply-risk

Consider the two-supplier random-capacity newsvendor, with demand $X \ge 0$ of finite mean, and let

\Pi_2^*(a) = \sup_{q \ge 0} \Pi_2(q; a)

be the optimal second-stage expected profit at reliability indices $a = (a_1, a_2)$ . Recall that a larger index $a_i$ makes supplier $i$ 's capacity loss stochastically smaller.

Then, for each supplier $i$ and with the other supplier's index held fixed, $\Pi_2^*$ is (weakly) increasing in $a_i$ :

a_i \le \hat a_i \implies \Pi_2^*(a_i, a_j) \le \Pi_2^*(\hat a_i, a_j).

A more reliable supplier never hurts the firm once it re-optimizes its orders. This is the value of reliability that the process-improvement strategy of the paper buys.

Formalization Note. "Increasing" is weak, as the paper declares on p. 492. The statement is for any demand law (no density) and any committed costs $\eta_i \in [0,1]$ .

Preamble

import Mathlib
import Definitions.Def_MitigateSupplyRisk_DualSourcing_Model

open MeasureTheory ProbabilityTheory

Formal statement

namespace MitigateSupplyRisk.DualSourcing

/-- Lemma 2(b) (Wang, Gilland, Tomlin 2010, p. 494): the optimal second-stage expected profit
`Π₂*(a) = sup_{q ≥ 0} Π₂(q; a)` is (weakly) increasing in supplier `i`'s reliability index `a_i`,
the other supplier's index held fixed. Demand is any nonnegative law with finite mean. -/
theorem lemma_2b (M : Model) (μ : Measure ℝ) [IsProbabilityMeasure μ]
    (hμ0 : μ (Set.Iio 0) = 0) (hμint : Integrable id μ) (i : Fin 2) (a : Fin 2 → ℝ) :
    Monotone (fun t : ℝ => M.Pi2star μ (Function.update a i t)) := by sorry

end MitigateSupplyRisk.DualSourcing

Source

Wang, Gilland, Tomlin, Mitigating Supply Risk: Dual Sourcing or Process Improvement?, Manufacturing & Service Operations Management 12(3):489–510 (2010), p. 494 (PDF p. 6), Lemma 2(b)

Human review

Endorsed by Shuze Chen · Oct 7, 2026
Confirmed by the moderator at approval.
Endorsed by mikedeng1 · Oct 7, 2026
Confirmed by the mission captain (proposal self-audit).

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