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The present work is devoted to study the high-energy $QCD$ events, such asthe di-jet productions from proton-proton inelastic collisions at the $LHC$ inthe forward-center and the forward-forward configurations, using the$unintegrated$ parton distribution functions ($UPDF$) in the$k_t$-factorization framework. The $UPDF$ of $Kimber$ et. al. ($KMR$) and$Martin$ et.al. ($MRW$) are generated in the leading order ($LO$) andnext-to-leading order ($NLO$), using the $Harland-Lang$ et al. ($MMHT2014$)$PDF$ libraries. While working in the forward-center and the forward-forwardrapidity sectors, one can probe the parton densities at very low longitudinalmomentum fractions ($x$). Therefore, such a computation can provide a valuabletest-field for these $UPDF$. We find very good agreement with the correspondingdi-jet production data available from $LHC$ experiments. On the other hand, aswe have also stated in our previous works, (i.e. the protons longitudinal andtransverse structure function as well as hadron-hadron $LHC$ $W/Z$ production),the present calculations based on the $KMR$ prescriptions show a betteragreement with the corresponding experimental data. This conclusion isachieved, due to the particular visualization of the angular orderingconstraint ($AOC$), despite the fact that the $LO-MRW$ and the $NLO-MRW$formalisms both employ better theoretical descriptions of the$Dokshitzer$-$Gribov$-$Lipatov$ -$Altarelli$-$Parisi$ ($DGLAP$) evolutionequation, and hence are expected to produce better results. The form of the$AOC$ in the $KMR$ prescription automatically includes the re-summation of thehigher-order $ln({1/x})$ type contributions, i.e. the$Balitski$-$Fadin$-$Kuraev$-$Lipatov$ ($BFKL$) logarithms, in the $LO$-$DGLAP$evolution equation.

LHC production of forward-center and forward-forward di-jets in the $k_t$-factorization $unintegrated$ parton distribution frameworks