Editing Multidisciplinary design optimization (section)

==Problem formulation==
Problem formulation is normally the most difficult part of the process. It is the selection of design variables, constraints, objectives, and models of the disciplines. A further consideration is the strength and breadth of the interdisciplinary coupling in the problem.<ref name="edo2021">{{Cite book|url=https://www.researchgate.net/publication/352413464|title=Engineering Design Optimization|last1=Martins|first1=Joaquim R. R. A.|last2=Ning|first2=Andrew|date=2021-10-01|publisher=Cambridge University Press|isbn=978-1108833417|language=en}}</ref>

===Design variables===
A design variable is a specification that is controllable from the point of view of the designer. For instance, the thickness of a structural member can be considered a design variable. Another might be the choice of material. Design variables can be continuous (such as a wing span), discrete (such as the number of ribs in a wing), or Boolean (such as whether to build a monoplane or a [[biplane]]). Design problems with continuous variables are normally solved more easily.

Design variables are often bounded, that is, they often have maximum and minimum values. Depending on the solution method, these bounds can be treated as constraints or separately.

One of the important variables that needs to be accounted is an uncertainty. Uncertainty, often referred to as epistemic uncertainty, arises due to lack of knowledge or incomplete information. Uncertainty is essentially unknown variable but it may causes the failure of system.

===Constraints===
A constraint is a condition that must be satisfied in order for the design to be feasible. An example of a constraint in aircraft design is that the [[lift (force)|lift]] generated by a wing must be equal to the weight of the aircraft. In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using [[Lagrange multiplier]]s.

===Objectives===
An objective is a numerical value that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other methods allow multiobjective optimization, such as the calculation of a [[Pareto efficiency|Pareto front]].

===Models===
The designer must also choose models to relate the constraints and the objectives to the design variables. These models are dependent on the discipline involved. They may be empirical models, such as a [[regression analysis]] of aircraft prices, theoretical models, such as from [[computational fluid dynamics]], or reduced-order models of either of these. In choosing the models the designer must trade off fidelity with analysis time.

The multidisciplinary nature of most design problems complicates model choice and implementation. Often several iterations are necessary between the disciplines in order to find the values of the objectives and constraints. As an example, the aerodynamic loads on a wing affect the structural deformation of the wing. The structural deformation in turn changes the shape of the wing and the aerodynamic loads. Therefore, in analysing a wing, the aerodynamic and structural analyses must be run a number of times in turn until the loads and deformation converge.

===Standard form===
Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following form:

: find <math>\mathbf{x}</math> that minimizes <math>J(\mathbf{x})</math> subject to <math>\mathbf{g}(\mathbf{x})\leq\mathbf{0} </math>, <math>\mathbf{h}(\mathbf{x}) = \mathbf{0} </math> and <math>\mathbf{x}_{lb}\leq \mathbf{x} \leq \mathbf{x}_{ub} </math>

where <math>J</math> is an objective, <math>\mathbf{x}</math> is a [[Vector (geometric)|vector]] of design variables, <math>\mathbf{g}</math> is a vector of inequality constraints, <math>\mathbf{h}</math> is a vector of equality constraints, and <math>\mathbf{x}_{lb}</math> and <math>\mathbf{x}_{ub}</math> are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by -1. Constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints.