For constraint 3x + 2y ≤ 18, point (4,4) gives LHS value
A 20
B 18
C 16
D 14
Substitute x = 4, y = 4 into 3x + 2y: 3(4) + 2(4) = 12 + 8 = 20. Since 20 exceeds 18, the point violates this constraint.
For LPP with x ≥ 0, y ≥ 0, x + y ≥ 6, the point (2,3) is
A Feasible point
B Boundary point
C Infeasible point
D Optimal point
Check x + y at (2,3): 2 + 3 = 5, which is not ≥ 6, so it violates the constraint. Even though it satisfies non-negativity, it is infeasible.
If constraints are x ≥ 0, y ≥ 0, x + y ≥ 2, the feasible region is
A Bounded region
B Empty region
C Single point
D Unbounded region
x + y ≥ 2 is the half-plane above the line x + y = 2 in the first quadrant. This region extends infinitely, so it is unbounded, though still feasible.
In Z = 5x + 4y, if Z is fixed at 40, the objective line is
A 5x + 4y = 40
B 5x + 4y ≤ 40
C 5x + 4y ≥ 40
D 5x − 4y = 40
Iso-profit or iso-cost lines are formed by setting the objective equal to a constant value. Thus fixing Z = 40 gives the line 5x + 4y = 40.
Slope of objective line for Z = 5x + 4y is
A -4/5
B -5/4
C 5/4
D 4/5
Write 5x + 4y = Z ⇒ y = -(5/4)x + Z/4. Therefore slope is -5/4. Objective slope helps compare with feasible edges for alternate optimum.
If objective slope equals slope of a feasible boundary edge, then optimum is likely
A No solution
B Unique solution
C Multiple solutions
D Infeasible region
When the objective line is parallel to and coincides with a feasible edge at optimal level, every point along that segment gives the same objective value, producing alternate optimal solutions.
Consider constraints x + y ≤ 6, x + y ≤ 10, x ≥ 0, y ≥ 0. The constraint x + y ≤ 10 is
A Redundant
B Binding always
C Inconsistent
D Artificial
Any point satisfying x + y ≤ 6 automatically satisfies x + y ≤ 10. Hence the second constraint does not shrink the feasible region and is redundant.
For constraints x ≥ 0, y ≥ 0, x + y ≤ 0, the feasible region is
A Empty set
B Unbounded strip
C Full quadrant
D Only (0,0)
In first quadrant, x and y are nonnegative. x + y ≤ 0 forces x = 0 and y = 0. So the feasible region reduces to the single point (0,0).
If feasible region is nonempty but objective can increase without limit, the LPP has
A Unique optimum
B No feasible point
C Unbounded optimum
D Redundant objective
Unbounded optimum occurs when the feasible region allows moving in an improving direction indefinitely while remaining feasible. Then the maximum (or minimum) is not finite.
Which equality gives boundary for inequality 2x − y ≥ 4?
A 2x − y = 4
B 2x − y = -4
C 2x + y = 4
D 2x + y = -4
To draw an inequality, first draw its boundary line by replacing ≥ with equality. Then choose the correct side (half-plane) by testing a point like the origin.
Convert 2x − y ≥ 4 into ≤ form by multiplying -1 gives
A -2x − y ≤ -4
B 2x + y ≤ 4
C -2x + y ≤ -4
D 2x − y ≤ 4
Multiply both sides by -1 reverses inequality: -(2x − y) ≤ -4 ⇒ -2x + y ≤ -4. This is commonly used while standardizing constraints.
For constraint y ≥ 3, which point satisfies it?
A (4,2)
B (2,4)
C (0,1)
D (3,0)
y ≥ 3 requires y-coordinate at least 3. Among choices, only (2,4) has y = 4 ≥ 3. Others have y < 3, so they violate the constraint.
For constraint x ≤ 5, the point (6,1) is
A Infeasible point
B Feasible point
C Boundary point
D Optimal point
x ≤ 5 means x cannot exceed 5. The point (6,1) has x = 6, so it violates the constraint and is infeasible regardless of other constraints.
Intersection of lines x + y = 7 and 2x + y = 10 is
A (4,3)
B (2,5)
C (3,4)
D (5,2)
Subtract first from second: (2x+y)−(x+y)=10−7 ⇒ x=3. Substitute into x+y=7 gives y=4. So intersection is (3,4), a candidate vertex.
If (3,4) is tested in constraint x + 2y ≤ 10, result is
A Feasible
B Infeasible
C On boundary
D Always optimal
Substitute x=3, y=4: x + 2y = 3 + 8 = 11, which is not ≤ 10. So the point violates this constraint and cannot be feasible.
A feasible solution that makes two constraints equalities simultaneously is typically a
A Interior point
B Random point
C Midpoint only
D Vertex point
In 2D LPP, a vertex occurs where two boundary lines intersect and both constraints are tight. If that intersection also satisfies remaining constraints, it becomes a feasible vertex.
A degenerate vertex in simplex corresponds to
A Negative basic variable
B Infinite objective
C Zero basic variable
D No constraints
Degeneracy occurs when a basic feasible solution has one or more basic variables equal to zero. It often happens when more than required constraints meet at a vertex.
If three constraints are binding at a feasible point in 2D, the point indicates
A Degeneracy chance
B Unboundedness sure
C Infeasibility sure
D Redundancy sure
In two variables, usually two constraints define a vertex. If three constraints bind at the same feasible point, it suggests overlapping boundaries and may produce a degenerate basic feasible solution.
A bounded feasible region guarantees existence of
A Only max
B Both max and min
C Only min
D Neither
For a linear objective over a nonempty bounded feasible region, both maximum and minimum values exist and occur at feasible points, often at vertices due to linearity.
If objective function is constant on an edge, then along that edge Z is
A Increasing only
B Decreasing only
C Same value
D Undefined
If the objective line coincides with a feasible edge, every point on that segment yields identical Z. This is the geometric reason behind alternate optimal solutions in LPP.
A constraint is redundant if removing it
A Keeps region same
B Makes region empty
C Makes region bounded
D Changes objective
Redundant constraints do not affect feasible region. They are implied by other constraints. Removing them keeps the set of feasible points unchanged and does not alter optimum.
If feasible region is unbounded, optimum must be unbounded?
A Yes, always
B Only in minimization
C No, not always
D Only in maximization
An unbounded feasible region can still have a finite optimum if the objective improves toward a direction blocked by constraints. Unbounded region only means feasible points extend infinitely, not objective.
If Z = 3x + 2y is maximized and feasible set is x ≥ 0, y ≥ 0, x − y ≥ 1, maximum is
A Unbounded
B 1
C 3
D 2
Constraint x − y ≥ 1 allows moving to larger x and y while keeping x at least y+1. Since coefficients are positive, Z grows without limit, so maximum is unbounded.
If Z = 3x + 2y is minimized with x ≥ 0, y ≥ 0, x − y ≥ 1, minimum occurs at
A (0,1)
B (1,0)
C (2,2)
D (0,0)
Feasible points must satisfy x ≥ y + 1 with nonnegativity. Smallest values occur when y is minimal (0) and x is minimal allowed (1). Then Z = 3(1)+0 = 3.
Slack in constraint 2x + y ≤ 9 at (4,1) is
A 1
B 2
C 3
D 0
Compute LHS: 2(4)+1 = 9. Slack = 9−9 = 0, so actually slack is zero. That means the constraint is binding at (4,1), not slack 1.
If a ≤ constraint has slack zero at a feasible point, that constraint is
A Redundant
B Inconsistent
C Binding
D Optional
Slack zero means the inequality is satisfied exactly as equality, so the constraint is active or binding. Binding constraints typically form the boundary edges passing through the point.
For minimization, if the objective line touches feasible region at an interior point only, then
A Not possible
B Unique optimum
C Alternate optimum
D Unbounded optimum
A linear objective cannot have a unique optimum strictly in the interior of a polygonal feasible region. If optimum exists, it occurs on boundary, typically at a vertex or along an edge.
A convex set property used in LPP means
A Segment goes outside
B Segment stays inside
C Only circles exist
D Only triangles exist
Convexity means for any two feasible points, the entire line segment joining them is also feasible. Feasible regions of linear inequalities are convex, supporting vertex-based optimization.
If objective function coefficients are both negative in maximization, optimum tends to occur near
A Smallest feasible values
B Largest feasible values
C Random values
D Unbounded edge
With negative coefficients, increasing variables decreases Z. So to maximize Z, we prefer smaller variable values within feasible region, often near origin or smallest boundary point.
In a diet problem, increasing a minimum nutrient requirement usually makes feasible region
A Always larger
B Always same
C Smaller or shift
D Always empty
A higher “at least” requirement tightens constraints (moves boundary outward), reducing or shifting the feasible region. It can even make the problem infeasible if requirements exceed possible combinations.
A constraint with “=” can make feasible region
A Always empty
B Always unbounded
C Always bounded
D Lower dimensional
Equality constraints force solutions to lie exactly on a line in 2D, reducing the feasible set from an area to a line segment or point when combined with other constraints.
If objective line slope is steeper than all feasible boundary edges, optimum likely lies on
A An extreme vertex
B Any interior point
C All points
D No point
The direction of objective improvement depends on slope. If objective slope is beyond the range of edge slopes, the last touching point occurs at a single extreme vertex, giving a unique optimum.
In LPP, “feasibility” means
A Objective highest
B Z equals zero
C Constraints satisfied
D Graph is neat
Feasibility is about satisfying all constraints, including non-negativity. It is different from optimality, which compares objective values among feasible points to find best solution.
If two feasible vertices give same maximum, objective line is
A Perpendicular to edge
B Parallel to edge
C Randomly tilted
D Circular
Equal best values at two vertices typically means the objective line at optimum overlaps the boundary segment joining them. This happens when objective slope matches slope of that edge.
In simplex, a basic feasible solution corresponds geometrically to
A A vertex
B An interior point
C A circle point
D A midpoint
In linear programming geometry, basic feasible solutions correspond to corner points of the feasible region. Simplex moves from one vertex to another, improving objective until optimum.
If a constraint is multiplied by -1, the inequality sign must
A Stay same
B Become equality
C Reverse direction
D Disappear
Multiplying an inequality by a negative number flips its direction. For example, x ≥ 2 becomes -x ≤ -2. This is important in standard form conversions.
A “shadow price” is linked to
A Objective slope only
B Binding constraint change
C Redundant constraint only
D Graph scaling only
Shadow price measures how optimal objective value changes with a small change in the right-hand side of a binding constraint. It reflects the value of an additional unit of that resource.
If a constraint is non-binding at optimum, its shadow price (intro) is usually
A Zero
B Positive
C Negative
D Infinite
Non-binding constraints have slack, meaning the resource is not fully used. Small changes in that constraint’s limit do not affect the optimal solution locally, so shadow price is zero.
Dual variables in LPP (intro) are associated with
A Decision variables
B Objective constants
C Constraints
D Graph axes
In duality, each primal constraint corresponds to a dual variable (often interpreted as a shadow price). Dual variables reflect value of resources and connect to sensitivity analysis ideas.
A common reason for unbounded optimum is
A Too many constraints
B Extra redundant constraint
C Objective is constant
D Missing limiting constraint
If constraints do not restrict movement in the direction that improves objective, the objective can increase indefinitely. This typically happens when an important upper-bound constraint is missing.
If feasible region is a line segment and objective is not parallel to it, optimum occurs at
A One endpoint
B Midpoint only
C Every point
D No point
On a line segment, a linear objective changes monotonically unless it is constant along the segment. So optimum occurs at an endpoint vertex unless objective is parallel giving same value everywhere.
If objective is parallel to feasible segment, then optimum on that segment is
A Only one endpoint
B Every point optimal
C No feasible
D Always unbounded
Parallel objective means objective has same value along that feasible segment at optimal level. Therefore all points on that segment provide the same optimum value, creating infinite optimal solutions.
A feasibility region that is “strip” between two parallel lines is
A Bounded
B Empty
C Unbounded
D Single point
A strip extends infinitely in at least one direction even though it is bounded between parallel constraints. Hence it is unbounded as a feasible set, though objective may still have optimum.
For maximization, if objective coefficients are (a,b) and both positive, increasing both variables generally
A Increases Z
B Decreases Z
C Keeps Z same
D Makes Z zero
With positive coefficients, Z = ax + by increases when x or y increases. Therefore, if feasible region allows large values in improving direction, it may cause unbounded optimum.
If constraint is y ≥ 2 and nonnegativity exists, y ≥ 0 becomes
A Inconsistent constraint
B Binding always
C Artificial constraint
D Redundant constraint
If y must be at least 2, then y is automatically nonnegative. So y ≥ 0 adds no new restriction and is redundant, though it is still often written for standard form.
When formulating from word problem, “at least 10 units” translates to
A ≤ 10 constraint
B ≥ 10 constraint
C = 10 constraint
D ≠ 10 constraint
“At least” means minimum requirement, so the expression must be greater than or equal to that number. This becomes a ≥ type linear constraint in the LPP.
Transportation and assignment problems are solved efficiently using
A Only graphical method
B Only derivatives
C Special algorithms
D Only integration
These are structured LPPs. Transportation uses methods like Vogel’s approximation and MODI, while assignment often uses Hungarian method. They exploit structure for faster solutions than general simplex.
A feasible solution where objective is best is called
A Optimal solution
B Redundant solution
C Inconsistent solution
D Artificial solution
Optimal solution is the feasible point giving maximum or minimum objective value. It must satisfy all constraints and provide the best Z among all feasible solutions.
“Corner point method” fails directly when feasible region has
A Infinitely many vertices
B Exactly four vertices
C No vertices
D Exactly three vertices
Corner point method relies on evaluating objective at vertices. If feasible region is empty or has no corner points (rare cases like a line without endpoints in feasible set), method cannot apply.
If constraints allow feasible points but none satisfy x ≥ 0, y ≥ 0, then issue is
A Objective wrong
B Nonnegativity violated
C Redundant constraints
D Alternate optimum
Many real-world LPPs require nonnegative decision variables. If the region that satisfies other constraints lies entirely in negative coordinates, the model becomes infeasible after applying nonnegativity conditions.