Transportation of fluid is a very important aspect of process industries, especially for oil and gas industries. Pipelines have been considered as the most effective and safest way of transporting fluids. Transportation of gas through a pipeline is an energy-intensive process; hence, energy optimization in gas transportation networks is an important issue to be considered while framing the environmental policies. In this paper, a novel graphical methodology based on pinch analysis approach for simultaneous minimization of capital investment and compression energy requirement in gas allocation network with the aid of thermodynamic relations is developed. The results of the proposed methodology are expressed as a Pareto optimal front. The e-constraint method is used to generate Pareto optimal front for the two objectives. Identifying the relationship between capital investment and energy requirement gives the opportunity to the decision-maker for choosing the suitable optimal operating point based on the operating and economic conditions of the process. This result allows the planner to calculate the effects via increasing or decreasing energy requirement or capital investment. The applicability of the proposed methodology is demonstrated through two illustrative examples.