In LPP, both the objective function and all constraints are linear expressions in decision variables. No products of variables or powers appear, so the feasible region becomes polygonal.
In Z = ax + by, “a” and “b” are known as
A Decision limits
B Corner values
C Objective coefficients
D Constraint signs
Objective coefficients show contribution of each variable to Z, such as profit per unit or cost per unit. They control the slope and direction of improvement of the objective line.
The line 2x + 3y = 12 has x-intercept
A 4
B 3
C 2
D 6
For x-intercept, set y = 0. Then 2x = 12, so x = 6. Thus the line meets the x-axis at (6,0). This helps in quick graphing.
The line 2x + 3y = 12 has y-intercept
A 4
B 6
C 3
D 2
For y-intercept, set x = 0. Then 3y = 12, so y = 4. The line meets the y-axis at (0,4), useful for the intercept method.
If a point makes one constraint false, that point is
A Optimal point
B Binding point
C Infeasible point
D Redundant point
Feasibility requires satisfying every constraint and non-negativity. If even one inequality fails, the point lies outside the feasible region and cannot be considered for optimization.
A feasible region created by x ≥ 0, y ≥ 0 is restricted to
A Second quadrant
B First quadrant
C Third quadrant
D Fourth quadrant
x ≥ 0 means right side of y-axis, and y ≥ 0 means above x-axis. Their intersection is the first quadrant, which matches real-world nonnegative decision variables.
The feasible region is found by taking intersection of
A All circles
B All parabolas
C All tangents
D All half-planes
Each linear inequality gives a half-plane. The feasible set is the common area satisfying all inequalities together. This intersection produces a convex polygonal region in 2D.
A point is a vertex of feasible region if it is intersection of
A Two objective lines
B Two axes only
C Two boundary lines
D Two midpoints
Vertices occur where boundary lines of constraints meet (including axes). These intersection points form the corners of the feasible region and are candidates for optimal solutions.
The corner point method requires calculating Z at
A All feasible vertices
B All infeasible points
C All interior points
D Only origin
After identifying feasible vertices, we compute the objective value at each. The best value gives the optimum because linear objectives achieve optimum at vertices when feasible.
If objective line overlaps a feasible boundary segment, optimal solutions are
A Exactly one
B Infinitely many
C Always none
D Always two
When the objective function is parallel and coincides with a boundary edge at the optimum, every point on that feasible segment gives the same objective value, producing infinite optima.
A constraint is called binding at a point when
A Slack is maximum
B It is removed
C Slack is zero
D It is violated
For a ≤ constraint, binding means the left side equals the right side, leaving no unused resource. Such constraints often determine the position of the optimal corner point.
Slack value for constraint x + y ≤ 10 at (3,4) is
A 3
B 7
C 1
D 0
Compute left side: 3 + 4 = 7. Slack = 10 − 7 = 3. This means 3 units of the resource limit remain unused at that feasible point.
Surplus value for constraint x + y ≥ 10 at (6,6) is
A 10
B 0
C 2
D 6
Left side is 6 + 6 = 12. Surplus = 12 − 10 = 2. This shows the requirement is exceeded by 2 units at that feasible point.
When converting x + y ≤ 8 to equality, we add
A Surplus variable
B Artificial variable
C Objective variable
D Slack variable
A ≤ constraint becomes equality by adding a nonnegative slack variable: x + y + s = 8. The slack represents unused capacity in the constraint.
When converting x + y ≥ 8 to equality, we subtract
A Slack variable
B Surplus variable
C Decision variable
D Corner variable
A ≥ constraint becomes equality by subtracting surplus: x + y − s = 8, s ≥ 0. Surplus shows how much the left side exceeds the minimum requirement.
The equation of objective line for Z = 5x + 2y at Z = 20 is
A 5x − 2y = 20
B 5x + 2y ≤ 20
C 5x + 2y = 20
D 5x + 2y ≥ 20
Iso-objective lines are formed by fixing Z to a constant. So Z = 20 gives 5x + 2y = 20. Different constants give parallel lines used in shifting method.
Slope of objective line 4x + y = Z is
A -4
B 4
C -1/4
D 1/4
Write y = -4x + Z. The slope is -4. Objective coefficients determine slope, so changing coefficients changes direction of improvement and which vertex becomes optimal.
If objective line is parallel to a constraint boundary, then their slopes are
A Opposite
B Multiplicative
C Undefined
D Equal
Two lines are parallel when their slopes match. In LPP, if objective line becomes parallel to a boundary edge at optimum, it can create multiple optimal solutions along that edge.
A bounded optimum is guaranteed only when feasible region is
A Empty and bounded
B Nonempty and unbounded
C Nonempty and bounded
D Empty and unbounded
If the feasible region exists and is bounded, Z will attain a maximum and minimum for any linear objective. Unbounded feasible regions may still have optimum, but not guaranteed.
“Unbounded solution” refers to
A Objective unlimited
B Constraints inconsistent
C Only one vertex
D Only integer points
Unbounded solution means objective value can increase (or decrease) without limit while staying feasible. It occurs when constraints do not restrict movement in the improving direction.
For inequality 2x + y ≤ 6, the origin (0,0) gives
A False
B True
C Undefined
D Zero only
Substitute (0,0): 2·0 + 0 = 0 ≤ 6 is true. So the shaded side for this constraint includes the origin, helping choose correct half-plane quickly.
A quick feasibility check for a point uses
A Substitution method
B Differentiation method
C Integration method
D Factor method
To test whether a point is feasible, substitute its coordinates into every inequality and non-negativity condition. If all statements are true, the point is feasible.
If two constraints give same boundary line, one becomes
A Binding constraint
B Objective constraint
C Redundant constraint
D Artificial constraint
When constraints are identical or one is always implied by the other, one adds no new restriction. Removing it does not change feasible region, so it is redundant.
If constraints x ≥ 5 and x ≤ 2 are together, the LPP is
A Unbounded
B Infeasible
C Alternate
D Degenerate
No real number can satisfy x ≥ 5 and x ≤ 2 simultaneously. Hence no feasible point exists, so the feasible region is empty and the LPP is infeasible.
A feasible region is convex mainly because it is
A Union of half-planes
B Product of lines
C Set of circles
D Intersection of half-planes
Each linear inequality gives a convex half-plane. Intersection of convex sets remains convex, so the feasible region is convex. This supports the theorem that optimum occurs at vertices.
In a maximization problem, the optimal vertex is generally the one giving
A Lowest Z value
B Zero Z value
C Highest Z value
D Random Z value
For maximization, we compute Z at all feasible vertices. The vertex with maximum Z is optimal. If a tie occurs along an edge, there are multiple optimal solutions.
In a minimization problem, the optimal vertex is generally the one giving
A Lowest Z value
B Highest Z value
C Largest x value
D Largest y value
For minimization, we select the feasible vertex with the smallest objective value. Graphically, iso-cost lines are shifted toward decreasing Z until they first touch the feasible region.
A common graphical mistake that changes solution is
A Neat axis labels
B Clear line drawing
C Wrong shading side
D Correct intercepts
If one inequality is shaded on the wrong side, the feasible region becomes incorrect. Then the computed vertices and the chosen optimum can be completely wrong even with correct arithmetic.
If feasible region exists but objective line never stops improving, the model shows
A Redundant objective
B Inconsistent constraints
C Alternate optimum
D Unbounded optimum
A feasible region may be unbounded, and if objective increases along an unbounded feasible direction, Z has no finite maximum (or minimum). This indicates missing limiting constraints.
The “corner point theorem” applies when feasible region is
A Curved ellipse
B Convex polygon
C Random cloud
D Disconnected arcs
For linear objective over a convex polygonal feasible region, an optimum (if it exists) occurs at some vertex. Hence evaluating vertices is sufficient in two-variable LPP.
If three constraints meet at one feasible vertex, it indicates
A Degeneracy possible
B Always infeasible
C Always unbounded
D Always alternate
Degeneracy can occur when more than two constraints are active at a vertex, causing repeated basic feasible solutions in simplex. In 2D, it appears as multiple lines meeting at one corner.
A “feasible solution” differs from “optimal solution” because optimal is
A Any feasible point
B Always interior point
C Best feasible value
D Always origin
Feasible solutions only satisfy constraints. Optimal solution is the feasible one that maximizes or minimizes the objective function. Many feasible points exist, but optimal gives best objective value.
If constraint line passes through origin, intercept method gives
A Both intercepts zero
B One intercept zero
C Both intercepts same
D No intercepts
If a line passes through origin, then either x-intercept or y-intercept (or both) involves zero. Intercept method still works, but another point is needed to draw the line correctly.
The intersection of x + y = 8 and x = 3 is
A (3,5)
B (5,3)
C (3,8)
D (8,3)
Substitute x = 3 into x + y = 8 gives 3 + y = 8, so y = 5. The intersection point is (3,5), often a candidate vertex.
To check whether (3,5) satisfies x + 2y ≤ 14, compute
A 14 ≤ 13
B 15 ≤ 14
C 13 ≤ 14
D 14 ≤ 15
Substitute x = 3, y = 5: x + 2y = 3 + 10 = 13. Since 13 ≤ 14 is true, the point satisfies this constraint and may remain feasible.
If objective function is Z = 0x + 5y, Z depends only on
A x only
B x and y
C y only
D y only
Coefficient of x is zero, so x does not affect Z. Objective becomes Z = 5y. Graphically, iso-objective lines are horizontal, and optimum depends on extreme feasible y-values.
A zero coefficient in a constraint like 0x + y ≤ 6 means the line is
A Vertical line
B Horizontal line
C Slant line
D Curved line
0x + y ≤ 6 is simply y ≤ 6, whose boundary is y = 6, a horizontal line. Such constraints directly limit y regardless of x.
A vertical boundary line occurs when constraint is of form
A y = constant
B x + y = 0
C x = constant
D xy = constant
x = k represents a vertical line parallel to y-axis. In LPP, constraints like x ≤ k or x ≥ k create vertical half-planes, limiting x directly.
A two-variable LPP with equality constraint often makes feasible region
A A line segment
B A full plane
C A circle region
D A parabola region
An equality restricts solutions to lie on a line. When combined with other inequalities and non-negativity, the feasible set may become a segment or a point on that line.
A feasible region can be a single point when
A Objective is constant
B Coefficients are zero
C Lines are parallel
D Constraints meet at one
If constraints intersect at exactly one common point and no other point satisfies all inequalities, the feasible set is a single point. That point becomes automatically optimal if feasible.
In maximization, if Z values at vertices are 12, 18, 18, the optimum is
A Z = 12
B Z = 0
C Z = 18
D Z = 6
The maximum objective value among feasible vertices is 18. Because it occurs at more than one vertex, the LPP may have alternate optimal solutions along the edge between those vertices.
“Sensitivity idea” in LPP mainly refers to
A Drawing neat graphs
B Effect of changes
C Solving faster
D Choosing integers
Sensitivity (intro) studies how small changes in coefficients or resource limits can affect feasibility and optimal solution. It helps understand stability of the chosen plan under changes.
“Shadow price” in LPP is closest to
A Value of resource
B Number of vertices
C Length of edge
D Slope of axis
Shadow price (intro) measures how much the optimal objective value would improve if a resource limit increases by one unit, assuming the same constraints remain binding in that range.
Duality in LPP (intro) connects
A Lines and circles
B Graph and table
C Slope and intercept
D Max and min forms
Duality links a primal LPP (often maximization) to a dual LPP (often minimization) with related constraints and variables. Solutions give consistent bounds and economic interpretation.
Artificial variables are mainly introduced in simplex for
A Changing objective slope
B Drawing feasible region
C Starting feasible basis
D Removing vertices
In simplex, some constraints (≥ or =) do not directly give an initial basic feasible solution. Artificial variables help start the algorithm, later removed using Big-M or two-phase method.
In a diet LPP, decision variables usually represent
A Food quantities
B Profit margins
C Vertex numbers
D Constraint slopes
Diet problems choose amounts of different foods. Constraints ensure minimum nutrients like protein or calories. Objective commonly minimizes total cost while meeting all nutritional requirements.
In a transportation LPP, constraints often represent
A Angle and length
B Supply and demand
C Roots and powers
D Area and volume
Transportation models distribute goods from sources to destinations. Supply constraints limit shipping from each source, demand constraints ensure required amounts reach each destination, usually minimizing total cost.
In an assignment LPP, each worker typically is assigned
A Many jobs
B No jobs
C One job only
D Two jobs
Assignment problems impose one-to-one restrictions: each worker gets exactly one job and each job gets exactly one worker. Objective minimizes cost/time or maximizes efficiency under these rules.
A three-variable LPP is harder graphically because it needs
A 3D plotting
B 1D plotting
C Only shading
D Only intercepts
With three variables, constraints form planes in 3D and feasible region becomes a polyhedron. Graphical approach becomes difficult, so simplex and computational methods are preferred.
Converting word problem to LPP begins by
A Drawing objective lines
B Finding vertices
C Choosing shading
D Defining decision variables
First identify what quantities must be decided, such as units produced or amounts shipped. Then write the objective function and constraints from given conditions, including non-negativity.