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Issues with Perfectly Correlated Demand

We have examined the beneficial effect in aggregating independent demands. But what if the demand is correlated—in fact, what if it is perfectly correlated? Rather than the normalized compression toward the center, the distribution would remain proportionally similar.

What happens to the coefficient of variation metric? Let a demand curve D(t) have σ = σ(D(t)) and μ = μ(D(t)). Therefore, the coefficient of variation of this demand function is cv = σ/μ. However, since D+(t) = D1(t) + D2(t) + . . . + Dm(t), now D+(t) = m × D(t). The mean is μ(D+(t)) =m × μ(D(t)). However, the variance σ2(c × X) of a constant times a random variable is the square of the constant times the variance of the random variable: σ2(c × X) = c2 × σ2(X). Therefore, σ2(D+(t)) = σ2(m × D(t)) = m2 × σ2(D(t)) and the standard deviation is σ(D+(t)) = m × σ(D(t)). Rather than the factor that we discovered earlier, the new coefficient of variation is . In other words, there is no change to the coefficient of variation. Put another way, there is no smoothing effect through aggregation.

In most cases, there is likely to be a mix of correlated demand and independent demand, so the penalty cost reduction benefit of demand aggregation will be somewhere between none and the reduction ...