Continuous


Linear Nonlinear
ILP INLP
LP NLP
MILP MINLP

troubles because the selected algorithm could perhaps not provide a feasible solu tion or in several cases it shows error messages. In addition, in some cases, how ever, a valid solution seems to be achieved, but after inspecting the results, it wa$ detected that some constraints have not been satisfied [26].

In general, three options exist to solve mathematical models. Researchers try to choose the best one based on the model's complexity and solution time limitations These options are exact methods, heuristic methods, and metaheuristic methods.

Exact Techniques

The problem featuring in the natural gas transmission and distribution networks because of its nonlinear and nonconvex nature cannot be solved using classical techniques like exact methods from mathematical programming because these methods are usually time consuming and unable to solve NP-hard problems even on a small scale. A number of researchers have tried to solve the developed models by exact techniques, but they had to oversimplify their approaches and compressor station models, which in practice may be inaccurate.

Heuristic Technique

The heuristic methods give the final solution in shorter time in comparison with exact methods, but there is a risk of trapping in the first local optimality. Therefore, achieving a global optimal solution is not guaranteed.

Metaheuristic Techniques

The best choice to solve NP-hard problems, that their solution time is dependent on the problem size exponentially, is the metaheuristic method, which guarantees find­ing the global optimum solution through decreasing the problem complexity with­out any limitations regarding the problem size. Some of the common effective methods, which researchers in different fields are interested in and in natural gas network planning also achieved many successes, are GA, SA, and TS. Chung et al. [17] used a GA for the problem of transmission networks planning to avoid arriving at local optimality and utilized a fuzzy decision analysis to select the best possible planning scheme. Mahlke et al. [20] exploited an SA to find a feasible solution in a reliable short time because of its simplicity to apply. TS allows designers to take advantages of the previous information in the selection of algorithms and subalgo- rithms. In optimization problems dealing with natural gas networks, the high non- convexity of objective functions and the capability of TS to escape from local optimality have made it very efficient with an appropriate discrete solution space. Borraz-Sanchez and Rfos-Mercado |4] combined TS with nonsequential DP lor the fuel cost minimization in the natural gas transmission network.