- [[#Appendix A.0 Step 0 — Classic Newsvendor Model|Appendix A.0 Step 0 — Classic Newsvendor Model]]
- [[#Appendix A.0 Step 0 — Classic Newsvendor Model#A.0.1 Expected Cost|A.0.1 Expected Cost]]
- [[#Appendix A.0 Step 0 — Classic Newsvendor Model#A.0.2 Optimal Quantity (Critical Fractile)|A.0.2 Optimal Quantity (Critical Fractile)]]
- [[#Appendix A.0 Step 0 — Classic Newsvendor Model#A.0.3 Business Intuition|A.0.3 Business Intuition]]
- [[#Appendix A.1 Quality-Linear Commitments|Appendix A.1 Quality-Linear Commitments]]
- [[#Appendix A.1 Quality-Linear Commitments#Appendix A.1.1 Symmetric Commitments|Appendix A.1.1 Symmetric Commitments]]
- [[#Appendix A.1.1 Symmetric Commitments#A.1.1 Expected Loss|A.1.1 Expected Loss]]
- [[#Appendix A.1.1 Symmetric Commitments#A.1.2 Optimal Quality|A.1.2 Optimal Quality]]
- [[#Appendix A.1 Quality-Linear Commitments#Appendix A.1.2 Asymmetric Commitments|Appendix A.1.2 Asymmetric Commitments]]
- [[#Appendix A.1.2 Asymmetric Commitments#A.1.2.1 Expected Loss|A.1.2.1 Expected Loss]]
- [[#Appendix A.1.2 Asymmetric Commitments#A.1.2.2 Expanded Form|A.1.2.2 Expanded Form]]
- [[#Appendix A.1.2 Asymmetric Commitments#A.1.2.3 Optimal Quality (Primal Problem)|A.1.2.3 Optimal Quality (Primal Problem)]]
- [[#Appendix A.1.2 Asymmetric Commitments#A.1.2.4 Optimal Responsiveness (Dual Problem)|A.1.2.4 Optimal Responsiveness (Dual Problem)]]
- [[#Appendix A.1.2 Asymmetric Commitments#A.1.2.5 Verification|A.1.2.5 Verification]]
- [[#Appendix A.2 Quality-Non-Linear Commitments|Appendix A.2 Quality-Non-Linear Commitments]]
- [[#Appendix A.2 Quality-Non-Linear Commitments#Appendix A.2.1 Non-Linear Symmetric Commitments|Appendix A.2.1 Non-Linear Symmetric Commitments]]
- [[#Appendix A.2.1 Non-Linear Symmetric Commitments#A.2.1.1 Expected Loss|A.2.1.1 Expected Loss]]
- [[#Appendix A.2.1 Non-Linear Symmetric Commitments#A.2.1.2 Optimal Quality|A.2.1.2 Optimal Quality]]
- [[#Appendix A.2 Quality-Non-Linear Commitments#Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters|Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters]]
- [[#Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters#A.2.2.1 Expected Loss|A.2.2.1 Expected Loss]]
- [[#Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters#A.2.2.2 First-Order Condition|A.2.2.2 First-Order Condition]]
- [[#Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters#A.2.2.3 Special Limits — Closed-Form Solutions|A.2.2.3 Special Limits — Closed-Form Solutions]]
- [[#A.2.2.3 Special Limits — Closed-Form Solutions#1. Symmetric Responsiveness ($β_c=β_r=β$)|1. Symmetric Responsiveness ($β_c=β_r=β$)]]
- [[#A.2.2.3 Special Limits — Closed-Form Solutions#2. Customer-Dominant Responsiveness ($β_c\gg β_r$)|2. Customer-Dominant Responsiveness ($β_c\gg β_r$)]]
- [[#A.2.2.3 Special Limits — Closed-Form Solutions#3. Partner-Dominant Responsiveness ($β_r\gg β_c$)|3. Partner-Dominant Responsiveness ($β_r\gg β_c$)]]
- [[#A.2.2.3 Special Limits — Closed-Form Solutions#4. High-Match-Value Limit ($V\gg C_u,C_o$)|4. High-Match-Value Limit ($V\gg C_u,C_o$)]]
## Appendix A.0 Step 0 — Classic Newsvendor Model
| Symbol | Interpretation |
| ------------------- | ------------------------------------- |
| $q\in\mathbb R_{+}$ | order quantity (decision) |
| $D$ | stochastic demand with CDF $F(\cdot)$ |
| $p,c$ | selling price / purchase cost |
### A.0.1 Expected Cost
$\operatorname{EC}(q)=c\int_{0}^{q}(q-x)\,f(x)\,dx \;+\;(p-c)\int_{q}^{\infty}(x-q)\,f(x)\,dx .$
### A.0.2 Optimal Quantity (Critical Fractile)
$q^∗=\frac{p−c}{p}$
### A.0.3 Business Intuition
_Balance_ overage and opportunity loss. A higher gross margin $(p-c)$ shifts the critical fractile upward, justifying more aggressive stocking.
---
## Appendix A.1 Quality-Linear Commitments
### Appendix A.1.1 Symmetric Commitments
|Symbol|Interpretation|
|---|---|
|$q!\in![0,1]$|product quality (decision)|
|$P_c(q)=q$|customer commitment probability|
|$P_r(q)=1-q$|resource‑partner commitment probability|
|$C_u,C_o$|under‑ / over‑mismatch unit costs|
|$V$|match bonus when both commit|
#### A.1.1 Expected Loss
| | $\color{skyblue}{Q}=0$ (partner absent) | $\color{skyblue}{Q}=1$ (partner present) |
| --------------------------------------- | ------------------------------------------------------------------------ | ---------------------------------------------------------------------- |
| $\color{skyblue}{D}=1$ (customer shows) | $\color{red}{q}^{2}
amp;nbsp;⟶ **shortage** incurs *opportunity* cost $C_u$ | $(1-\color{red}{q})\color{red}{q}amp;nbsp;⟶ **match** gains value $+V$ |
| $\color{skyblue}{D}=0$ (no customer) | $\color{red}{q}(1-\color{red}{q})amp;nbsp;⟶ **match** (no cost) | $(1-\color{red}{q})^{2}amp;nbsp;⟶ **excess** incurs *overage* cost $C_o$ |
$L(q)=C_uq^{2}+C_o(1-q)^{2}-Vq(1-q).$
#### A.1.2 Optimal Quality
$
q^{*}=\frac{V+2C_o}{2\,(C_u+C_o+V)},
$
### Appendix A.1.2 Asymmetric Commitments
|Symbol|Interpretation|
|---|---|
|$q \in [0,1]$|product quality (decision)|
|$\beta_r, \beta_c$|resource partner and customer responsiveness parameters|
|$P_c(q) = \beta_c \cdot q$|customer commitment probability|
|$P_r(q) = 1 - \beta_r \cdot q$|resource partner commitment probability|
|$C_u, C_o$|under‑ / over‑mismatch unit costs|
|$V$|match bonus when both commit|
#### A.1.2.1 Expected Loss
The 2×2 outcome matrix becomes:
| | $\color{skyblue}{Q}=0$ (partner absent) | $\color{skyblue}{Q}=1$ (partner present) |
|---|---|---|
| $\color{skyblue}{D}=1$ (customer shows) | $(\beta_c q)^2$ ⟶ **shortage** incurs _opportunity_ cost $C_u$ | $(\beta_c q)(1-\beta_r q)$ ⟶ **match** gains value $+V$ |
| $\color{skyblue}{D}=0$ (no customer) | $(1-\beta_c q)(\beta_r q)$ ⟶ **no match** (no cost) | $(1-\beta_c q)(1-\beta_r q)$ ⟶ **excess** incurs _overage_ cost $C_o$ |
$L(q,\beta_r,\beta_c) = C_u(\beta_c q)^2 + C_o(1-\beta_c q)(1-\beta_r q) - V(\beta_c q)(1-\beta_r q)$
#### A.1.2.2 Expanded Form
$L(q,\beta_r,\beta_c) = C_u\beta_c^2 q^2 + C_o(1-\beta_c q-\beta_r q+\beta_r\beta_c q^2) - V\beta_c q(1-\beta_r q)$
$= q^2(C_u\beta_c^2 + C_o\beta_r\beta_c + V\beta_r\beta_c) + q(-C_o\beta_c - C_o\beta_r - V\beta_c) + C_o$
#### A.1.2.3 Optimal Quality (Primal Problem)
Given $\beta_r, \beta_c$, minimize with respect to $q$:
$\frac{\partial L}{\partial q} = 2q(C_u\beta_c^2 + C_o\beta_r\beta_c + V\beta_r\beta_c) + (-C_o\beta_c - C_o\beta_r - V\beta_c) = 0$
$q^* = \frac{C_o(\beta_c + \beta_r) + V\beta_c}{2(C_u\beta_c^2 + C_o\beta_r\beta_c + V\beta_r\beta_c)}$
#### A.1.2.4 Optimal Responsiveness (Dual Problem)
Given $q$, the first-order conditions for $\beta_r, \beta_c$ are:
$\frac{\partial L}{\partial \beta_r} = C_o q(1-\beta_c q) + V\beta_c q^2 = 0$
$\frac{\partial L}{\partial \beta_c} = 2C_u\beta_c q^2 - C_o q(1-\beta_r q) - Vq(1-\beta_r q) = 0$
#### A.1.2.5 Verification
When $\beta_r = \beta_c = 1$, this reduces to the symmetric case:
$L(q,1,1) = C_u q^2 + C_o(1-q)^2 - Vq(1-q)$
---
## Appendix A.2 Quality-Non-Linear Commitments
### Appendix A.2.1 Non-Linear Symmetric Commitments
| Symbol | Meaning |
| ----------------------------- | ------------------------------- |
| $q \in \mathbb{R}$ | quality level (decision) |
| $P_c(q) = \frac{1}{1+e^{-q}}$ | customer commitment probability |
| $P_r(q) = \frac{1}{1+e^{q}}$ | partner commitment probability |
| $C_u, C_o$ | under-/over-mismatch unit costs |
| $V$ | match bonus when both commit |
#### A.2.1.1 Expected Loss
$L(q) = C_u P_c(q)[1-P_r(q)] + C_o P_r(q)[1-P_c(q)] - V P_c(q)P_r(q)$
#### A.2.1.2 Optimal Quality
The first-order condition $L'(q) = 0$ yields:
$C_u P_c(1-P_c)(1-P_r) - C_u P_c P_r(1-P_r) + C_o P_r(1-P_r)(1-P_c) - C_o P_r P_c(1-P_c) - V P_c(1-P_c)P_r - V P_c P_r(1-P_r) = 0$
Factoring and simplifying with $P_c = \frac{e^{-q}}{1+e^{-q}}$ and $P_r = \frac{1}{1+e^{-q}}$:
$P_c(V + 2C_o) = P_r(V + 2C_u)$
Substituting the sigmoid forms and solving:
$q^* = \ln\left(\frac{2C_o + V}{2C_u + V}\right)$
This closed-form solution exists due to the symmetric responsiveness parameters ($\beta_c = 1, \beta_r = -1$).
----
### Appendix A.2.2 General Non-Linear with Asymmtric $β$ Parameters
| Symbol | Meaning |
|---------------------------------------------|---------------------------------------------------|
| $q\in\mathbb R$ | quality level (decision) |
| $P_c(q)=\dfrac1{1+e^{-β_c q}}$ | customer commitment probability |
| $P_r(q)=\dfrac1{1+e^{+β_r q}}$ | partner commitment probability |
| $β_c,β_r>0$ | steepness of each sigmoid |
| $C_u,C_o,V>0$ | under-commit cost; over-commit cost; match bonus |
#### A.2.2.1 Expected Loss
We define
$
L(q)
= C_u\,P_c(q)\bigl[1 - P_r(q)\bigr]
+ C_o\,P_r(q)\bigl[1 - P_c(q)\bigr]
- V\,P_c(q)\,P_r(q).
$
#### A.2.2.2 First-Order Condition
1. Compute the derivatives
$
P_c'(q) = β_c\,P_c(1-P_c),
\quad
P_r'(q) = -β_r\,P_r(1-P_r).
$
2. Differentiate $L(q)$ term-by-term:
$
\frac{d}{dq}\bigl[C_uP_c(1-P_r)\bigr]
= C_u\bigl[P_c'(1-P_r) - P_c\,P_r'\bigr],
$
$
\frac{d}{dq}\bigl[C_oP_r(1-P_c)\bigr]
= C_o\bigl[P_r'(1-P_c) - P_r\,P_c'\bigr],
$
$
\frac{d}{dq}\bigl[-V\,P_cP_r\bigr]
= -V\bigl[P_c'P_r + P_c\,P_r'\bigr].
$
3. Assemble $L'(q)$:
$
L'(q)
= P_c'\Bigl[C_u(1-P_r) - (C_o+V)P_r\Bigr]
+ P_r'\Bigl[C_o(1-P_c) - (C_u+V)P_c\Bigr].
$
4. Substitute $P_c',P_r'$ and set $L'(q)=0$:
$
β_cP_c(1-P_c)\bigl[C_u(1-P_r) - (C_o+V)P_r\bigr]
-β_rP_r(1-P_r)\bigl[(C_u+V)P_c - C_o(1-P_c)\bigr]
=0.
$
5. Rearranged into the implicit equation
$
\boxed{
\frac{β_c\,P_c(1-P_c)}{β_r\,P_r(1-P_r)}
= \frac{\,C_o(1-P_c)\;-\;(C_u+V)P_c\,}
{\,C_u(1-P_r)\;-\;(C_o+V)P_r\,}
}.
$
#### A.2.2.3 Special Limits — Closed-Form Solutions
We now derive the four tractable cases step by step.
---
##### 1. Symmetric Responsiveness ($β_c=β_r=β$)
- Let $x = e^{βq}$. Then
$
P_c = \frac{x}{1+x},\quad P_r = \frac{1}{1+x}.
$
- Implicit FOC becomes
$
(1-P_c)(C_u+V) \;=\; P_c\,(C_o+V).
$
- Substitute $1-P_c = 1/(1+x)$ and $P_c=x/(1+x)$:
$
\frac{1}{1+x}(C_u+V)
= \frac{x}{1+x}(C_o+V)
\;\Longrightarrow\;
x = \frac{C_u+V}{C_o+V}.
$
- Hence
$
\boxed{
q^* = \frac{1}{β}\,
\ln\!\Bigl(\frac{C_u+V}{C_o+V}\Bigr).
}
$
---
##### 2. Customer-Dominant Responsiveness ($β_c\gg β_r$)
from
$\boxed{\; \beta_c P_c(1-P_c)\,\bigl[\,C_u(1-P_r)-(C_o+V)P_r\bigr] = \beta_r P_r(1-P_r)\,\bigl[(C_u+V)P_c-C_o(1-P_c)\bigr] \; } \tag{FOC}$
1. Divide the FOC by $\beta_r$. As $\beta_c/\beta_r\to\infty$, the left-hand factor $\beta_cP_c(1-P_c)$ explodes **unless** its square-bracket term is driven to 0. Hence to leading order
$C_u(1-P_r)-(C_o+V)P_r = 0.$
2. Solve for $P_r$:
$P_r^\star = \frac{C_u}{C_u+C_o+V},\qquad \frac{1-P_r^\star}{P_r^\star}=\frac{C_o+V}{C_u}.$
3. Invert the (decreasing) logistic for $P_r$:
$P_r(q)=\frac{1}{1+e^{\beta_r q}} \;\Longrightarrow\; q^\star =\frac{1}{\beta_r}\, \ln\left(\frac{C_o+V}{C_u}\right).$
$\boxed{\; q^{*}_{\text{cust-dom}} =\frac{1}{\beta_r}\, \ln\left(\frac{C_o+V}{C_u}\right)\;}$
---
##### 3. Partner-Dominant Responsiveness ($β_r\gg β_c$)
- By symmetry, swap roles:
$
C_o(1-P_c) - (C_u+V)P_c = 0
\;\Longrightarrow\;
P_c = \frac{C_o}{C_u+C_o+V}.
$
- With $P_c = 1/(1+e^{-β_c q})$, we get
$
e^{-β_c q} = \frac{1-P_c}{P_c}
= \frac{C_u+V}{C_o}
\;\Longrightarrow\;
q^* = \frac{1}{β_c}\,
\ln\!\Bigl(\frac{C_o}{C_u+V}\Bigr).
$
- Hence
$
\boxed{
q^* = \frac{1}{β_c}\,
\ln\!\Bigl(\frac{C_o}{C_u+V}\Bigr).
}
$
---
##### 4. High-Match-Value Limit ($V\gg C_u,C_o$)
- Approximate $L(q)\approx -V\,P_cP_r$, so FOC becomes
$
β_c P_c(1-P_c) = β_r P_r(1-P_r).
$
- Solving this logistic marginal equality yields
$
\boxed{
q^* = \frac{1}{β_c + β_r}\,
\ln\!\Bigl(\frac{β_r}{β_c}\Bigr).
}
$