We generalize the idea of (${in}$, ${in}{vee}q_k$)-fuzzy ordered semi-group and give the concept of (${in}$, ${in}{vee}q_k$)-fuzzy ordered $mathcal{LA}$-semigroup. We show that (${in}$, ${in}{vee}q_k$)-fuzzy left (right, two-sided) ideals, (${in}$, ${in}{vee}q_k$)-fuzzy (generalized) bi-ideals, (${in}$, ${in}{vee}q_k$)-fuzzy interior ideals and (${in}$, ${in}{vee}q_k$)-fuzzy (1, 2)-ideals need not to be coincide in an ordered $mathcal{LA}$-semigroup but on the other hand, we prove that all these (${in}$, ${in}{vee}q_k$)-fuzzy ideals coincide in a left regular class of an ordered $mathcal{LA}$-semigroup. Further we investigate some useful conditions for an ordered $mathcal{LA}$-semigroup to become a left regular ordered $mathcal{LA}$-semigroup and characterize a left regular ordered $mathcal{LA}$-semigroup in terms of (${in}$, ${in}{vee}q_k$)-fuzzy one-sided ideals. Finally we connect an ideal theory with an (${in}$, ${in}{vee}q_k$)-fuzzy ideal theory by using the notions of duo and (${in}{vee}q_k$)-fuzzy duo.