In this work, we present the benchmark generator PUBOi, Polynomial Unconstrained Binary Optimization, that combines subproblems to create instances of pseudo-boolean optimization problems. Any mono-objective pseudoboolean functions including existing classical optimization problems can be expressed with Walsh functions. The benchmark generator can tune main features of problems such as problem dimension, non-linearity degree, and neutrality. Additionally, to be able to create instances with properties similar to those of real-like combinatorial optimization problems, the goal of PUBOi is to introduce the notion of variable importance. Indeed, the importance of decision variables can be tuned using three benchmark parameters. In the version presented here, we consider four subproblems already used in Chook generator for benchmarking quantum computers and algorithms as a basis. We also present the impact of benchmark parameters using a fitness landscape analysis that empirically shows these parameters to significantly impact the variable importance.