With upsizing

complexifying and unifying in modern petro-chemical systems

the combined complexity among streams of mass

energy and information as well as among unit operations keeps increasing exponentially. P-graph theory has found widespread applications by generation of rigorous superstructures as a result of axiom constraints which may reduce creation of redundant structures. This review on P-graph theory began with the mathematical definition

fundamental axioms

solution algorithms and workflow

and a case study to show the modeling framework and graphic representation. A literature study of past 20 years' publications systemically summarized application of P-graph theory in traditional process system engineering of separation network synthesis

reaction path synthesis and heat exchange synthesis

and recent expansion in new areas of process technology selection

supply chain and process optimization. The advantages and disadvantages of P-graph theory and mathematical programming were compared

and improving ideas were proposed to solve nonlinear problems with P-graph theory. The prospective research and application of P-graph theory were forecasted

including multi-objective optimization in consideration of economic and environmental factors

effective solution of large complex nonlinear programming problems by the combination of both P-graph and mathematical programming

and other potential applications.