# Regression model when under-estimations costs us more than over-estimations

We have a factory and we are planning how many items produce in 2014. During the learning process we minimize the mean squared error. But, under-estimations costs us more than over-estimations. Let's say we estimated and produced $p$ items, but we have to sell $d$ items.

• In case of $p > d$, getting rid of 1 surplussed item costs us 30\$. • In case of$p < d$, buying 1 item (from friendly factory) costs us 200\$.

Can this additional data can be used to formulate better objective function? If yes, what it will be?

Here's how. Suppose your regression model takes the form $y_i=x_i'\beta+u_i$. Then let $\hat\beta$ be the minimizer of

$$\Biggl[200 \sum_{i=1}^n \max(y_i-x_i'\tilde\beta,0) - 30 \sum_{i=1}^n \min(y_i-x_i'\tilde\beta,0)\Biggr]$$

with respect to $\tilde\beta$.

• Shouldn't it be $\Biggl[200 \sum_{i=1}^n \max(y_i-x_i'\tilde\beta,0) - 30 \sum_{i=1}^n \min(y_i-x_i'\tilde\beta,0)\Biggr]$? Feb 1, 2014 at 18:41
• You're right; I'm editing my answer.
– JPi
Feb 1, 2014 at 18:48

Your function is telling you that if you underestimate, it is costlier to buy than overestimating and get rid of surplus. You need to consider this in your decision. You need some certainty that you overproduce. This is different than utility. You need to add an optimum based on utility. In both cases maybe you gain money but less utility if you underestimate or grossly overestimate.

• But how to add this optimum based on utility? I mean formulas, for example our starting objective function is $min_{a,b}[(a+b-10)^2 + (2a+b-20)^2]$, how to improve it? Feb 1, 2014 at 17:26
• Dynamic weights could be a solution but the optimum will be hard to obtain. Feb 1, 2014 at 17:38