5 Optimization and Decision Models Online Quiz Questions

A company chooses production quantities to make its total manufacturing cost as small as possible while meeting demand and capacity restrictions. Which expression represents the objective of the model?

True or false: For a differentiable function on a closed interval, checking only interior points where the derivative is zero is sufficient to find the absolute maximum and minimum.

Three sides of a rectangular garden are fenced with 100 metres of fencing, while the fourth side borders a wall. If $x$ is the width perpendicular to the wall, the area is $A(x)=100x-2x^2$ for $0\leq x\leq 50$. What is the maximum area, in square metres? Enter the exact numerical value.

A decision model has objective function $P(x)=40x-x^2$, the constraint $x\geq 0$, and no requirement that $x$ be an integer. How should this model be classified based on the form of its functions?

True or false: For an equality-constrained optimization problem, the Lagrange multiplier condition states that the gradients of the objective and constraint are parallel at a candidate optimum.

What is the term for a constraint that holds with equality at an optimal solution?

For $f(x,y)=x+y$ on the region $x\geq 0$, $y\geq 0$, and $x+y\leq 10$, which statement is correct?

Formulate an optimization model for a manufacturer that chooses quantities $x$ and $y$ of two products. Profit is $\$40$ per unit of the first product and $\$30$ per unit of the second, while machine time is limited by $2x+y\leq 100$. State the variables, objective, constraints, and any appropriate variable-domain restriction. Briefly explain what must be done after formulation to identify and interpret the optimum.

A transportation model decides how many buses to use. A continuous relaxation produces $x=3.6$. What is the appropriate modeling response?

A manufacturer chooses quantities $x$ and $y$ of two products. Profit is $P(x,y)=40x+30y$, subject to $2x+y\leq 100$ and nonnegativity restrictions. Which part of the model is the objective function?

For a differentiable function optimized on a closed interval $[a,b]$, which candidates must be evaluated to find an absolute maximum or minimum?

A transit model uses $x$ to represent the number of buses purchased. Which modeling change is necessary before solving the optimization problem?

At a critical point of a twice-differentiable function of two variables, suppose $D=f_{xx}f_{yy}-(f_{xy})^2>0$ and $f_{xx}>0$. What does the second-derivative test conclude?

True or false: Checking only interior critical points is sufficient to find the absolute maximum and minimum of an objective on a closed, bounded feasible region.

A rectangular garden has three fenced sides and a wall as the fourth side. With 100 metres of fencing, the area is $A(x)=100x-2x^2$ for $0\leq x\leq 50$, where $x$ is the width perpendicular to the wall. What width $x$, in metres, gives the maximum area?

What term names the set of all choices that satisfy every constraint in an optimization model?

An optimization model recommends a decision using a particular objective, set of constraints, data values, and assumptions. Select all conclusions that are justified by the model's result.

An optimization procedure identifies an interior point as a local maximum of an objective function. Select all statements that are logically justified by this information alone.