In an LPP, the variables representing choices are called
A Random variables
B Dependent variables
C Decision variables
D Constant terms
Decision variables represent quantities we can choose, like units produced or hours used. They are the unknowns whose values must satisfy constraints and optimize the objective function.
The expression to maximize or minimize in LPP is the
A Constraint equation
B Feasible region
C Intercept form
D Objective function
The objective function is a linear expression of decision variables representing profit, cost, time, etc. The goal is to maximize or minimize this value while meeting all constraints.
Restrictions written as inequalities/equations are called
A Constraints
B Objective terms
C Vertices
D Iso-lines
Constraints are conditions like resource limits or requirements. They are linear inequalities or equalities that restrict the possible values of decision variables in the model.
The set of all points satisfying every constraint is the
A Objective line
B Inconsistent set
C Feasible region
D Slack set
The feasible region contains all points that satisfy all constraints simultaneously, including non-negativity. Solutions must come from this region; outside points violate at least one constraint.
A point inside the feasible region is a
A Optimal solution
B Feasible solution
C Redundant solution
D Artificial solution
Any point that satisfies all constraints and non-negativity is feasible. It may not be best, but it is allowed. Optimal solutions are special feasible solutions giving best objective value.
The best feasible solution is called
A Optimal solution
B Bounded solution
C Slack solution
D Basic variable
An optimal solution is a feasible solution that gives the maximum or minimum value of the objective function. Graphically, it occurs at a suitable corner point in two-variable LPP.
In graphical method, optimal value usually occurs at
A Any interior point
B Midpoint of edge
C Corner point
D Origin only
By the corner point (vertex) principle for linear problems, if an optimum exists, at least one optimal solution occurs at a vertex of the feasible region polygon.
Constraints like x ≥ 0, y ≥ 0 are
A Binding constraints
B Redundant constraints
C Artificial constraints
D Non-negativity constraints
Non-negativity constraints restrict decision variables to be zero or positive, reflecting real quantities like production and time. They keep the feasible region in the first quadrant.
A solution set forming a closed polygon is a
A Bounded region
B Unbounded region
C Infeasible region
D Parallel region
A bounded feasible region is enclosed and finite. All feasible points lie within a closed boundary, meaning the variables cannot grow without limit while satisfying constraints.
If feasible region extends infinitely in some direction, it is
A Bounded
B Redundant
C Unbounded
D Degenerate
An unbounded feasible region is not closed and extends infinitely. This does not always mean unbounded optimum, but it can lead to objective increasing without limit in some cases.
If no point satisfies all constraints together, the LPP is
A Optimal
B Bounded
C Alternate
D Infeasible
An infeasible LPP has an empty feasible set. Graphically, constraint half-planes do not overlap in a common region, so no solution can satisfy all conditions at once.
A constraint that does not change the feasible region is
A Redundant constraint
B Binding constraint
C Active constraint
D Standard constraint
A redundant constraint is automatically satisfied by all feasible points due to other constraints. Removing it does not affect the feasible region or the optimal solution.
Constraints that contradict each other lead to
A Feasible region
B Alternate optimum
C Inconsistent constraints
D Slack variables
Inconsistent constraints cannot be satisfied together. They produce no feasible region. For example, x ≥ 5 and x ≤ 2 simultaneously makes the feasible set empty.
The graph of a linear inequality represents a
A Circle region
B Half-plane
C Parabola region
D Ellipse region
A linear inequality divides the plane by a line and includes one side of it. That side is a half-plane, often chosen using a test point method.
Shading in graphical method shows the
A Objective values
B Vertex label
C Feasible side
D Slope sign
Shading indicates which side of each constraint line satisfies the inequality. The feasible region is the intersection of all shaded half-planes along with x ≥ 0, y ≥ 0.
An objective line for Z = ax + by is also called
A Binding line
B Redundant line
C Constraint line
D Iso-profit line
For maximization (profit) or minimization (cost), Z = constant gives a family of parallel lines. These are iso-profit or iso-cost lines used to locate the best feasible point.
The slope of objective line Z = ax + by is
A -a/b
B a/b
C -b/a
D ab
Writing ax + by = Z gives y = -(a/b)x + Z/b. Hence the slope is -a/b. Moving this line parallel improves Z depending on maximization or minimization.
The “moving line” method means
A Rotating objective line
B Changing constraints
C Shifting objective line
D Deleting vertices
In graphical optimization, the objective line is moved parallel in the direction of increasing (or decreasing) Z until it last touches the feasible region, giving the optimal point.
If objective line is parallel to an edge of feasible region, it may give
A No solution
B Multiple optima
C Infeasible set
D Redundant region
When an objective line coincides with a boundary segment at the optimum, every point on that segment gives the same objective value, producing alternative (multiple) optimal solutions.
A constraint is “binding” at a point when
A It is violated
B It is redundant
C It holds as equality
D It is removed
A binding constraint is active at the solution, meaning the left side equals the right side. It typically limits the optimum; non-binding constraints have slack (extra allowance).
The extra unused amount in a ≤ type constraint is called
A Slack
B Surplus
C Artificial
D Degeneracy
Slack is the difference between the right-hand side and the left-hand side for a ≤ constraint at a feasible point. It measures unused resources like leftover time or material.
For a ≥ constraint, the extra above minimum is
A Slack
B Intercept
C Surplus
D Corner value
Surplus is the amount by which the left-hand side exceeds the right-hand side for a ≥ constraint. It shows how much the requirement is exceeded beyond the minimum level.
In standard form (intro), most constraints are written as
A ≥ type
B quadratic type
C logarithmic type
D ≤ type
In introductory standardization for simplex, constraints are commonly converted to ≤ form by adding slack variables. Equalities and ≥ constraints need different handling in advanced methods.
A slack variable is added to convert
A ≥ to =
B = to ≤
C ≤ to =
D = to ≥
For a ≤ constraint, adding a nonnegative slack variable converts it into an equality, representing unused resource. Example: x + y ≤ 10 becomes x + y + s = 10.
A surplus variable is used to convert
A ≥ to =
B ≤ to =
C = to ≤
D = to ≥
For a ≥ constraint, subtracting a nonnegative surplus variable converts it into equality. Example: x + y ≥ 10 becomes x + y − s = 10, where s measures excess.
The feasible region of linear constraints is always
A Non-convex
B Circular
C Convex
D Random
Intersection of half-planes is convex. So for any two feasible points, the line segment joining them lies entirely in the feasible region. This property supports corner point optimization.
The method of checking which side of a line satisfies inequality uses
A Derivative test
B Test point
C Mean value
D L’Hospital
To decide shading, choose a simple test point (often origin if not on the line). Substitute into inequality. If true, shade the side containing that point; otherwise shade opposite.
The corner points are found mainly by
A Intersections of lines
B Random selection
C Only x-axis cuts
D Only y-axis cuts
Corner points (vertices) occur where boundary lines intersect each other or axes. Solving pairs of constraint equations gives intersection points, then we check which are feasible.
The “corner point evaluation” step means
A Shade half-planes
B Delete constraints
C Compute Z at vertices
D Draw circles
After listing feasible vertices, evaluate the objective function value at each corner point. The best value among them gives the optimum for two-variable LPP.
A feasible region in first quadrant implies
A x ≤ 0, y ≤ 0
B x + y ≤ 0
C xy ≤ 0
D x ≥ 0, y ≥ 0
First quadrant means both coordinates are nonnegative. In LPP, this matches non-negativity constraints. Graphical solutions usually consider only this region for practical variables.
If the objective value can increase forever in feasible region, solution is
A Bounded optimum
B Alternate optimum
C Unbounded optimum
D Degenerate optimum
If feasible region allows moving in a direction that keeps satisfying constraints while increasing Z without limit, then maximum does not exist. This is an unbounded solution case.
If the objective line never touches feasible region, the problem is
A Infeasible
B Feasible
C Bounded
D Optimal
If there is no feasible region, objective lines cannot touch it. Graphically, constraints do not overlap in a common area, so there is no feasible point to optimize.
A two-variable LPP can be solved mainly by
A Taylor series
B Laplace method
C Graphical method
D Newton’s method
With two decision variables, constraints can be drawn on a plane and the feasible region identified. Then objective is optimized using corner point method or moving line method.
Graphical method becomes difficult mainly when variables are
A One variable
B Two variables
C Exactly two
D More than two
Graphing inequalities is practical in 2D. With three or more variables, visualization and vertex finding become hard, so algebraic methods like simplex are preferred.
The line ax + by = c crosses axes at
A (a/c, 0) and (0, b/c)
B (c/a, 0) and (0, c/b)
C (c, 0) and (0, c)
D (ab, 0) and (0, ab)
Setting y = 0 gives x = c/a, so x-intercept is (c/a,0). Setting x = 0 gives y = c/b, so y-intercept is (0,c/b), assuming a,b ≠ 0.
A “feasible corner point” means
A Any intersection point
B Only x-axis intercept
C Intersection inside feasible region
D Only y-axis intercept
Many line intersections exist, but only those satisfying all constraints are feasible. Such feasible intersections are corner points of the feasible polygon and candidates for optimum.
If two constraints are parallel and separate, feasible region may be
A Always bounded
B Always triangle
C Always circle
D Empty
Two parallel constraints like x ≥ 5 and x ≤ 2 never overlap, making feasibility impossible. Such contradictions create an empty feasible set and the LPP becomes infeasible.
The coefficients in objective function represent
A Random errors
B Vertex numbers
C Profit or cost rates
D Inequality signs
Objective coefficients show contribution per unit of each decision variable. In production, they often represent profit per item; in minimization, they represent cost per unit.
Maximization problems usually model
A Highest profit
B Lowest profit
C Constant output
D Random demand
In a maximization LPP, the objective is to get the largest possible value, commonly profit, output, or utility, while respecting all resource and requirement constraints.
Minimization problems usually model
A Highest cost
B Highest constraints
C Lowest cost
D Lowest variables
In minimization LPP, the goal is smallest possible objective value, commonly cost, time, or wastage. Feasible solutions are compared to find the minimum while satisfying constraints.
A “diet problem” in LPP usually aims to
A Maximize taste
B Minimize cost
C Maximize weight
D Minimize nutrients
Diet-type LPPs typically choose amounts of foods to meet nutrition constraints at minimum cost. Decision variables are food quantities, constraints are nutrient requirements, objective is total cost.
Transportation problems mainly focus on
A Drawing graphs
B Solving quadratics
C Finding derivatives
D Shipping cost planning
Transportation-type models allocate shipments from sources to destinations to minimize total transportation cost under supply and demand constraints. It is a special class within linear programming.
Assignment problems mainly focus on
A One-to-one allocation
B One person many jobs
C Random selection
D Infinite matching
Assignment problems allocate tasks to agents so each task and each agent is used exactly once, usually minimizing cost or time. It is a special structured LPP type.
A constraint written with “=” is called
A Slack constraint
B Surplus constraint
C Equality constraint
D Random constraint
Equality constraints require exact satisfaction, like meeting a fixed demand. In graphical solutions, they appear as a line boundary, and feasible points must lie on that line.
A feasible region formed by linear inequalities is generally a
A Polygon region
B Circle
C Parabola
D Spiral
In two variables, each linear inequality gives a half-plane, and their intersection is typically a polygon (possibly unbounded). Its vertices are key points for finding optimum.
A “corner point theorem” says optimum occurs at
A Any feasible point
B Only origin
C Some corner point
D Only midpoint
The theorem states that if an optimal value exists for a linear objective over a convex polygonal feasible region, then at least one optimal solution is at a vertex.
Alternative optimum occurs when objective line
A Cuts feasible region
B Overlaps an edge
C Is perpendicular edge
D Passes through origin
If the objective line at optimal value coincides with a boundary segment, then every point on that segment gives the same objective value, creating infinitely many optimal solutions.
A constraint that is active at optimum typically has
A Positive slack
B Negative slack
C Zero slack
D Infinite slack
Active or binding constraints are satisfied exactly at the optimal point, so slack is zero for ≤ constraints. They usually determine the location of the optimal corner point.
“Feasibility” in LPP means
A Best objective value
B Maximizing variables
C Minimizing vertices
D Satisfying all constraints
Feasibility checks whether a point meets every constraint including non-negativity. Only feasible points are allowed solutions; optimality is decided later by comparing objective values.
The first step in solving a graphical LPP is usually
A Compute derivatives
B Guess optimum
C Plot constraints
D Add artificial vars
In graphical method, begin by drawing each constraint line and shading the satisfying half-plane. Then find the intersection (feasible region), list its vertices, and evaluate the objective function.