# 🟧G12: Mathematical Methods Framework ## 🟧G1: Linear Quality Model The linear model transforms the classical newsvendor by making quality $q \in [0,1]$ the decision variable. Stakeholder responses are linear and opposing: - Customer commitment probability: $P_c(q) = q$ (increases with quality) - Resource partner commitment probability: $P_r(q) = 1-q$ (decreases with quality) This creates four outcomes with probabilities: - Both commit: $P_c \cdot P_r = q(1-q)$ → Value $V$ - Customer only: $P_c(1-P_r) = q^2$ → Underage cost $C_u$ - Partner only: $P_r(1-P_c) = (1-q)^2$ → Overage cost $C_o$ - Neither: $(1-P_c)(1-P_r) = 0$ → No cost **Optimal quality:** $q^* = \frac{V+2C_o}{2(C_u+C_o+V)}$ **Key insight:** The "cost-priority principle" - quality adjusts to avoid the more expensive mismatch. When overage is costly ($C_o > C_u$), increase quality. When underage is costly ($C_u > C_o$), decrease quality. ## 🟧G2: Sigmoid Quality Model The sigmoid model captures realistic S-shaped stakeholder responses: - Customer: $P_c(q) = \frac{1}{1+e^{-\beta_c q}}$ - Partner: $P_r(q) = \frac{1}{1+e^{\beta_r q}}$ For the symmetric case ($\beta_c = 1, \beta_r = -1$): **Optimal quality:** $q^* = \ln\left(\frac{2C_o + V}{2C_u + V}\right)$ **Key insights:** 1. **Symmetric responsiveness:** When both stakeholders respond equally, balance net penalties 2. **Asymmetric responsiveness:** When one stakeholder is more sensitive ($\beta_c \gg \beta_r$), their preferences dominate 3. **High match value:** When $V \gg C_u, C_o$, maximize joint commitment probability ## Comparison: 🟧G1 vs 🟧G2 |Aspect|Linear (G1)|Sigmoid (G2)| |---|---|---| |Response curves|Linear opposing|S-shaped realistic| |Solution|Always closed-form|Closed-form for special cases| |Cost-priority|Pure cost ratios|Moderated by responsiveness β| |Business insight|Simple trade-offs|Behavioral steepness matters| Both models demonstrate how quality decisions must balance opposing stakeholder preferences, but G2's sigmoid functions capture the diminishing returns and saturation effects observed in real markets.