Discussion
If the demand distribution is exponential with mean \(1/\alpha\), then the optimal inventory level \(S^{*}\) is :
Answer: C
The optimal level is \(S^{*} = \dfrac{1}{\alpha}\log\dfrac{h + p}{h + c}\).
With purchase cost \(c\), holding cost \(h\) and shortage cost \(p\), the single-period optimum satisfies \(F(S^{*}) = \frac{p - c}{p + h}\). For exponential demand, \(F(S) = 1 - e^{-\alpha S}\), so
\[ \begin{aligned} e^{-\alpha S^{*}} &= 1 - \frac{p - c}{p + h} = \frac{h + c}{h + p} \\ S^{*} &= \frac{1}{\alpha}\log\frac{h + p}{h + c} \end{aligned} \]
Since \(p \gt c\), the logarithm is positive, as a stock level must be. Option B has the fraction inverted, giving a negative level, and options with the factor \(\alpha\) instead of \(1/\alpha\) have the wrong units (the mean demand is \(1/\alpha\)).
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