By A. Kaveh

The booklet provides lately constructed effective metaheuristic optimization algorithms and their purposes for fixing numerous optimization difficulties in civil engineering. The ideas is additionally used for optimizing difficulties in mechanical and electric engineering.

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**Sample text**

A typical section for composite steel–concrete box girder is shown in Fig. 1. As it is depicted, the design variables in each section are slab thickness (tc), top flange width (bf), top flange thickness (tf), web depth (Dw), web thickness (tw), and bottom flange thickness (tb). The center to center distance of the top flanges and the inclination angle of web from the vertical direction are fixed to 160 cm and 100 , respectively, for the entire girder because of fabrication conditions. As a result, the width of bottom flange is a function of other variables.

50) induces randomness into the algorithm. This term can be interpreted as the random portion of the search space traveled by team i before it stops after the applied force is removed. Here, α is a constant chosen from the interval [0,1]; Xmax and Xmin are the vectors containing the upper and lower 20 2 Optimum Design of Castellated Beams Using the Tug of War Algorithm bounds of the permissible ranges of the design variables, respectively; ∘ denotes element by element multiplication; and rand(1, n) is a vector of uniformly distributed random numbers.

The objective function can be expressed as Fcost ¼ ρAinitial ðL0 Þ Â p1 þ Lcut Â p2 þ Lweld Â p3 ð2:10Þ In practice, in order to support high shear forces close to the connection or for reasons of fire safety, sometimes it becomes necessary to fill certain openings using steel plates. In this case, the price of plates is added to the total cost. Therefore, the objective function can be expressed as Fcost-filled ¼ ρðAinitial ðL0 Þ þ 2Ahole Â tw Þ Â p1 þ Lcut Â p2 þ ðLweld Þ Â p3 ð2:11Þ where p1, p2, and p3 are the price of the weight of the beam per unit weight, length of cutting, and welding per unit length, L0 is the initial length of the beam before castellation process, ρ is the density of steel, Ainitial is the area of the selected universal beam section, Ahole is the area of a hole, and Lcut and Lweld are the cutting length and welding length, respectively.