Which one is correct?
We introduce the xx model, which exploits the lack of consistency.
We introduce the xx model, which exploits lack of consistency.
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If you ask me as a reader, both are grammatical and I just have to say that "(a) lack of consistency" stands for generality, and "the lack" for specificity" which is preferable in your sentence. But in the non-affirmative form "lack" is viewed as a mass noun such as:
"There was no lack of amusement at the theater."
Also, if you are using "lack" as a general reference [for its meaning], you could point out that if in singular, it really depends on your writing style, interpretation, or even approach.