5 Optimization and Decision Models Free Online FlashCards

What are the four components of an optimization model?

The four parts are decision variables, an objective function, constraints, and the feasible region.

What makes a solution feasible?

A feasible solution satisfies every constraint, including restrictions such as nonnegativity.

What is the objective function?

The objective function is the quantity being maximized or minimized, such as profit, cost, time, area, or risk.

What is the correct order for formulating an applied model?

A valid model defines variables and units, states the objective, translates limitations into constraints, determines the domain, solves candidates, and interprets the result.

What dimensions maximize the three-sided garden’s area?

For the garden, $A(x)=100x-2x^2$ on $[0,50]$. The maximum is $1250\text{ m}^2$ at $x=25\text{ m}$ and $y=50\text{ m}$.

How is an interior critical point identified?

An interior critical point occurs where $f_x=f_y=0$, or where at least one partial derivative does not exist.

What does the second-derivative test conclude when $D<0$?

With $D=f_{xx}f_{yy}-(f_{xy})^2$, $D<0$ indicates a saddle point; $D>0$ uses the sign of $f_{xx}$ to distinguish a local minimum or maximum.

What must boundary analysis include?

For a closed, bounded region, check interior critical points, every boundary component, boundary critical points, endpoints, and corners.

Where are the extrema of $x+y$ on the triangular region?

For $f(x,y)=x+y$ with $x\geq0$, $y\geq0$, and $x+y\leq10$, the maximum is $10$ on the boundary $x+y=10$, and the minimum is $0$ at $(0,0)$.

What equations define Lagrange-multiplier candidates?

For an equality constraint $g(x,y)=c$, solve $\nabla f=\lambda\nabla g$ together with $g(x,y)=c$.

What is the geometric meaning of Lagrange multipliers?

At a constrained optimum, the objective’s level curve is tangent to the constraint curve, so their gradients are parallel.

What rectangle maximizes area for fixed perimeter?

With fixed perimeter $P$, the rectangle of greatest area is a square: $x=y=P/4$.