Chapter 19: Linear Programming

Click and drag on elements in order Put the steps in order to graphically solve a linear programming word problem.

1. set up 2. plot the constraints 3. identify 4. plot the objective . determine

Which of the following expresses the constraint, "the ratio of x1 to x2 must be at least 5 to 3," in a mathematically suitable format for use in linear programming?

3x1 - 5x2 ≥ 0

Which of the following is a correct definition of graphical linear programming?

A method for finding optimal solutions to two-variable problems.

Select the best definition of sensitivity analysis.

Assessing the impact of potential changes to the numerical values of an LP model.

Linear programming problems involve limitations or restrictions on multiple variables. What are some of those constraints? (Select all that apply.)

Budgets Materials Machine time Labor

What is an alternate term for linear programming in operations management?

Constrained optimization

Which ribbon in Excel contains Solver?

Data

Linear programming requires a single overriding goal or objective be chosen. What are the two choices generally used?

Either max or min

True or false: In a graphical minimization problem, slide the objective function isocost line away from the origin point to find the optimal solution.

F

In linear programming, what is the set of combinations of the decision variables that satisfy all constraints?

Feasible solution space

Using graphical linear programming, the set of all points that satisfies all constraints is called what?

Feasible solution space

What is a necessary condition of a redundant constraint?

Its removal does not alter the feasible solution space.

Match the linear programming model assumption with its definition.

Linearity: The impact - Divisibility: Noninteger values -Certainty: Values of - Nonnegativity: Negative vales

What is the main advantage of using a computer program to do linear programming versus graphing the equations manually?

More than two variables can be analyzed using a computer program.

Select all that apply What are the three types of potential changes used in sensitivity analysis? (Select three.)

Objective function coefficients Constraint coefficients Right-hand values of constraints

What are the four components of a linear programming model which must be specified in order to solve the mathematical problem? (Select four.)

Parameters Decision variables Constraints Objective function

The feasible solution space forms what type of shape typically?

Polygon

The range of coefficient values over which the solution quantities of all the decision values remains the same is called what?

Range of optimality

A constraint that does not form a unique boundary of the feasible solution space.

Redundant constraint

Non-integer values for quantities need to be acceptable for LP models to work effectively; however, when maximizing is the goal, what should typically be done with a non-integer solution value for products made in units?

Round it down to the nearest whole number

What are the first steps used to plot the optimal objective function line?

Set it equal to an arbitrary amount to find the slope. Plot the parallel line in the feasible space that either maximizes or minimizes.

What is the first step in the graphical approach to linear programming?

Set up the objective function and constraints in mathematical format.

The amount by which the value of the objective function would change by a one-unit change in the right hand value of a constraint is called what?

Shadow price

The simplex method can be used in a spreadsheet software such as Microsoft Excel. The routine needed to calculate the solution is called what?

Solver

True or false: In linear programming the solution is always at one of the corner points of the feasible solution space polygon.

T

Select all that apply What are the key differences in graphically solving minimization problems (versus maximization ones)?

The feasible solution space will not contain the origin point. The optimal point is the one closest to the origin. One of constraints must be of the ≥ or = variety.

Select all that apply Which of the following are examples of constraints on decision variables?

The ratio of product #1 to product #2 must be 3 to 2. Make at least 10 units of product #1.

Graphical linear programming is a method for finding optimal solutions _____-variable problems.

Two

Given a constraint of 4x1 + 10x2 ≤ 100, what are the axes intercepts when plotting it on a graph?

X1 = 25 when X2 = 0; X2 = 10 when X1 = 0.

The simplex method uses a(n) _______________ to solve a linear programming problem.

algorithm

A constraint that forms the optimal corner point of the feasible solution space is called a ____________ constraint.

binding

The limited range of values over which the value of the shadow price remains constant is known as the range of

feasibility

What are the two general-purpose linear programming solution techniques covered in this chapter?

graphical linear programming and computer solutions

If a constraint is _______________, its shadow price is zero.

nonbinding

The simplex method of linear programming is typically chosen over the graphical method because it can solve problems with more than ______ decision variables.

two

A binding constraint limits the value of the objective function. If it could be ______________, an improved solution would be possible.

relaxed

In the range of feasibility,how does the shadow price change?

remains constant

True or false: A change in the value of a constraint will cause a change in the optimal solution of a problem.

F

Match the linear programming component with its definition.

Objective function: A mathematical expression that can be used to determine the profit or cost, e.g., for a given solution Decision variables: Choices available to the decision maker in terms of amounts of either inputs or outputs Parameters: Numerical values representing the fixed amount of quantity available or required Constraints: Limitations that restrict the alternatives available to decision makers in the form of ≥, =, ≤.

Given a constraint of 4x1 + 10x2 ≤ 100, what is the step-by-step procedure for plotting it on a graph?

Change the inequality sign to an equal sign and set each variable alternately to zero to solve for the other.

Select all that apply What are some applications of linear programming in operations that involve problems of resource limitations?

Determining optimal schedules for airlines for planes, pilots, and ground personnel Developing shipping plans that will minimize shipping costs Determining optimal blends of animal feed mixes given available inputs

What are some applications of linear programming in operations that involve problems of resource limitations?

Developing optimal production schedules Identifying the optimal mix of products in a factory Locating warehouses to minimize transportation costs

What are some applications of linear programming in operations that involve problems of resource limitations?

Developing optimal production schedules Locating warehouses to minimize transportation costs Identifying the optimal mix of products in a factory

The technique of substituting the coordinates of each corner point into the objective function to determine which corner point is optimal is called what?

Enumeration approach

In graphical linear programming how do you solve for x1 and x2 decision variables where two constraint lines cross at the objective function line?

Use algebra to solve for simultaneous equations using the constraint lines