During period from 1940 to 1950, simultaneous mode of failure approach was persued in which each component of complete structure is at its limit of strength as complete structure reaches limit state of collapse and based on classical ideas of function minimization, this approach was used to solve simple structural forms having a limited applicability to practical design.

?_min < ? < ?_max where ? is ratio of reinforcement provided ?_min is minimum ratio required to prevent to prevent failure by fracture of steel in tension, ?_max is maximum permissible ratio to ensure failure of section by yielding of steel.

In using mamatical programming methods process of optimization begins with an acceptable design point and new point is selected suitability so as to minimize objective function and search for anor new point is continued is continued from previous point until optimum point is reached and re are several well established techniques for selecting a new point and to proceed towards optimum point, depending upon nature of problem, such as linear and non linear programming.

Furr improvements in se methods are due to pope, who suggested move limit method and more recently Ramakrishna and Bhavikati has improved and used this method for optimum design of mechanical components under stress.

Optimization studies as applied to prestressed concrete structures have been pursued during last decade and some organizations have developed practical programmes for analysis and design of simple structures like highway bridge girders comparative cost studies have been standardized and tabulated for design office use by using computer programmes developed at structural engineering research centre.

In a structural design problem, re should be a well defined criterion by which performance or cost of structure can be judged under different combinations of design variables and index is generally referred to as objective cost or merit function.

Where a and a_p are actual and permissible deflection, which is usually a small fraction of span code requirements for limit state of collapse to ensure desirable load factors against flexural failure which can be written as M_u > (?f₁M_g+?f₂M_q).

The fourth major area of developed is mamatical programming formulations, first applied to structural optimization problems by livesley and pearson an excellent survey of optimum structural design using mamatical programming procedure has been reported by schmit and excellent example which users this method is twenty five bar transmission tower reported by marcel and venkayya.

Many of studies have included only a few variables and invariably structure and shapes, unit costs of labour and materials are assumed without considering specific site conditions and constructional techniques which can have a greater influence on overall costs of structural scheme and furr studies in this field should include cost estimates of alternative schemes, formwork design, probabilistic design considering variability of load applications and materials properties.

In non linear programming problems, objective function and or constraints are non linear function of design variables and since boundaries of feasible region or contours of equal values of merit function are straight lines optimum solution need not necessarily be at an intersection of constraints.