Exam 10: Nonlinear Programming

Lush Lawns, Inc. provides a lawn fertilizer and weed control service. They are adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A budget of $2000 is to be used on this promotional campaign over the next quarter. Based on past experience in promoting its other services, Lush Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media: s = 2x1² - 10x2² - 2_x1x2 + 18_x1 + 34x2 s.t. x₁ + x₂ = 2 Solve.

If a firm's profit is Z = 12x - 6x² + 30, and their minimum production level of x is equal to 0.5, then the level of x that maximizes profit is:

If a nonlinear program has been correctly formulated, procedures guarantee a solution.

If a nonlinear programming problem results in profit (Z) of $50, and the Lagrange multiplier for a constraint is -2, the new profit will be ________ if the right-hand side of the constraint is increased by 1 unit.

The Lagrange multiplier is analogous to the dual variables in a linear programming problem.

If a firm's profit is Z = 100p -8p² +16, then the maximum profit occurs where p = ________.

If a nonlinear programming model consists of a single nonlinear objective function and a single linear constraint, it is called a(n) ________ optimization problem.

The Lagrange multiplier is ________ to the dual variables in a linear programming problem.

Assume a nonlinear programming problem with a single constraint has been solved. The value of the Lagrange multiplier is $0.75 and the value of the optimal profit (Z) is $25. If the right-hand side of the constraint is increased from 38 to 42, the new value of Z will be ________.

The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value.

The slope of a curve at its highest point equals 1.

Sara's Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is: x₁ = 1800 - 150p₁ x₂ = 1500 - 300p₂ The cost for a catnip ball is $2 and for the mouse, $3. Sara has only 200 ounces of catnip on hand. A ball uses a tenth of an ounce and a toy mouse uses one-quarter of an ounce. -Determine the prices that Sara should charge to maximize profit.