Negative Questions MCQs for Sub-Topics of Topic 13: Linear Programming

Question 1. Which of the following is NOT a fundamental component of a Linear Programming Problem?

(A) Objective function.

(B) Constraints.

(C) Decision variables.

(D) Non-linear relationships.

Question 2. Which of the following is NOT a valid type of objective for a Linear Programming Problem?

(A) Maximize profit.

(B) Minimize cost.

(C) Maximize the square of production quantity.

(D) Minimize transportation distance.

Question 3. In a manufacturing LPP, which of the following would typically NOT be a constraint?

(A) Limited labour hours.

(B) Minimum production quantity required.

(C) Raw material availability.

(D) The total profit earned.

Question 4. Which of the following is NOT true about decision variables in an LPP?

(A) They represent the quantities to be determined.

(B) They are typically non-negative.

(C) They represent the limitations in the problem.

(D) They are variables that influence the objective function and constraints.

Question 5. The 'linear' aspect of Linear Programming means which of the following is NOT allowed?

(A) An objective function like $Z = 5x + 2y$.

(B) A constraint like $3x + 4y \leq 100$.

(C) A constraint like $x^2 + y^2 \leq 50$.

(D) A constraint like $x \geq 0, y \geq 0$.

Question 6. Which of the following is NOT a typical type of constraint inequality used in LPP?

(A) Less than or equal to ($\leq$).

(B) Greater than or equal to ($\geq$).

(C) Not equal to ($\neq$).

(D) Equal to ($=$) (can be used, although often converted).

Question 7. If a problem involves maximizing profit from producing two items, X and Y, which of the following is NOT a necessary element for it to be an LPP?

(A) Profit per unit for X and Y are constants.

(B) Production resource usage per unit for X and Y are constants.

(C) Total available resources are specified limits.

(D) Profit per unit varies depending on the quantity produced.

Question 8. Which of the following statements about the non-negativity restriction ($x_i \geq 0$) is NOT true?

(A) It is a common constraint in most real-world LPPs.

(B) It ensures that quantities are not negative.

(C) It guarantees that the feasible region is always bounded.

(D) It restricts the feasible region to certain quadrants (e.g., first quadrant in 2D).

Question 9. Which of the following is NOT a goal that can be directly addressed by a single LPP formulation?

(A) Maximize total revenue.

(B) Minimize total production cost.

(C) Simultaneously maximize profit and minimize environmental impact as separate objectives.

(D) Minimize the deviation from production targets.

Question 10. In a diet problem, which of the following would NOT be represented as a constraint?

(A) Minimum daily requirement of Vitamin C.

(B) Maximum allowable intake of calories.

(C) The cost per unit of a food item.

(D) Minimum amount of fibre needed.

Question 1. Which of the following is NOT a correct representation of a linear constraint in two variables $x_1$ and $x_2$?

(A) $5x_1 + 2x_2 \leq 50$.

(B) $x_1 - x_2 = 10$.

(C) $4x_1 \geq 0$.

(D) $x_1^2 + x_2^2 = 100$.

Question 2. When formulating an LPP, which of the following steps is NOT essential?

(A) Identifying the decision variables.

(B) Defining the objective function mathematically.

(C) Graphing the feasible region.

(D) Expressing all limitations as linear constraints.

Question 3. In the standard mathematical formulation of an LPP, Optimize $Z = \mathbf{c}^T \mathbf{x}$ subject to $A\mathbf{x} \leq \mathbf{b}, \mathbf{x} \geq \mathbf{0}$, which of the following is NOT always true?

(A) $\mathbf{x}$ is a vector of variables.

(B) $\mathbf{c}$ is a vector of constants.

(C) $\mathbf{b}$ is a vector of constants.

(D) The matrix $A$ must be square.

Question 4. A constraint like "The total production of item A and item B must be less than or equal to 200" (where $x_A, x_B$ are quantities) is formulated. Which of the following is NOT a correct formulation?

(A) $x_A + x_B \leq 200$.

(B) $x_A + x_B \leq 200$, $x_A \geq 0$, $x_B \geq 0$.

(C) $x_A + x_B = 200$.

(D) $x_A + x_B \leq 200$ (assuming non-negativity is handled separately).

Question 5. Which of the following is NOT a valid linear objective function?

(A) Maximize $Z = 10x_1 + 15x_2 - 5x_3$.

(B) Minimize $C = 7y_1 + 3y_2$.

(C) Optimize $Z = x_1/x_2$.

(D) Maximize Profit = $\sum_{i=1}^n p_i x_i$.

Question 6. When translating a real-world problem into an LPP, which of the following is NOT a typical type of information needed?

(A) Profit/cost per unit of each activity.

(B) Amount of each resource consumed/required per unit of each activity.

(C) Total availability of each resource or minimum requirement levels.

(D) Historical data on market price fluctuations.

Question 7. Consider a problem where a factory has two types of machines, M1 and M2, and produces a single product. The constraint for total machine hours available is formulated. Which information is NOT directly used in forming this constraint inequality?

(A) Number of units of the product made.

(B) Machine M1 hours required per unit.

(C) Machine M2 hours required per unit.

(D) Profit per unit of the product.

Question 8. In the formulation of an LPP with two variables $x$ and $y$, which of the following is NOT a valid type of linear constraint?

(A) $x \leq 10$.

(B) $y \geq 5$.

(C) $x + y = 15$.

(D) $x/y \leq 2$.

Question 9. Which of the following is NOT a characteristic of the constraints in a correctly formulated LPP?

(A) They involve linear combinations of decision variables.

(B) They represent limitations or requirements.

(C) The right-hand side values are constants.

(D) They must all be inequalities of the same type (e.g., all $\leq$).

Question 10. When formulating an LPP from a scenario, which action is NOT typically performed first?

(A) Understanding the problem and the goal.

(B) Identifying the decision variables.

(C) Defining the objective function.

(D) Calculating the optimal solution values.

Question 1. Which of the following is NOT a standard type of Linear Programming Problem application?

(A) Manufacturing.

(B) Diet planning.

(C) Transportation.

(D) Forecasting stock prices.

Question 2. In a typical Manufacturing Problem, which of the following is NOT a common constraint?

(A) Maximum machine hours available.

(B) Minimum number of units to produce.

(C) Total profit must be a specific value.

(D) Availability of raw materials.

Question 3. Which of the following objectives is NOT typical for a Transportation Problem?

(A) Minimize total shipping cost.

(B) Maximize the total quantity transported.

(C) Meet demand at destinations from available supply at sources.

(D) Balance supply and demand.

Question 4. In a Diet Problem aimed at minimizing cost, which of the following is NOT a type of constraint?

(A) Minimum intake of a vitamin.

(B) Maximum intake of fat.

(C) Total cost must be exactly $\textsf{₹}500$.

(D) Minimum calorie requirement.

Question 5. Which of the following is NOT a characteristic of a Blending Problem?

(A) Involves mixing different ingredients.

(B) Constraints relate to the properties of the final mixture.

(C) Objective is usually to minimize the cost of the blend.

(D) Decision variables represent which machine to use for processing.

Question 6. A problem involves assigning workers to tasks to maximize total efficiency. This is an Assignment Problem, which is a special case of LPP. Which of the following is NOT a typical aspect of such a problem?

(A) Each worker is assigned to exactly one task.

(B) Each task is assigned to exactly one worker.

(C) There is a measure of efficiency/cost for each worker-task assignment.

(D) The constraints involve minimum nutrient requirements.

Question 7. Which of the following types of data is NOT commonly needed for a Transportation Problem formulation?

(A) Supply quantity at each source.

(B) Demand quantity at each destination.

(C) Unit transportation cost between each source-destination pair.

(D) Production cost at each source.

Question 8. In a Production Problem, which of the following might NOT be a limiting resource represented by a constraint?

(A) Labour hours.

(B) Machine capacity.

(C) Raw material supply.

(D) Customer preference.

Question 9. Which of the following is NOT a valid decision variable in a Diet Problem?

(A) Quantity (in kg) of a specific food item.

(B) Number of servings of a food item.

(C) Total daily intake of Vitamin A.

(D) Quantity (in grams) of a food ingredient.

Question 10. Which type of LPP is NOT primarily focused on the physical movement of goods between locations?

(A) Transportation Problem.

(B) Transshipment Problem.

(C) Manufacturing Problem.

(D) Vehicle Routing Problem (a more complex variant, but related).

Question 1. Which of the following statements about the feasible region of an LPP is NOT true?

(A) It is the set of all points that satisfy all constraints.

(B) It is always a convex set.

(C) It can be empty.

(D) It is always a bounded polygon.

Question 2. When graphing linear inequalities in two variables to find the feasible region, which of the following is NOT a correct step?

(A) Graph the boundary line for each inequality.

(B) Use a test point (like the origin) to check which side of the line satisfies the inequality.

(C) Shade the region that violates the inequality.

(D) The feasible region is the intersection of the satisfied regions for all constraints.

Question 3. Which of the following points is NOT necessarily on the boundary of the feasible region defined by $x+y \leq 5$, $x \geq 0$, $y \geq 0$?

(A) $(0,0)$.

(B) $(5,0)$.

(C) $(2.5, 2.5)$.

(D) $(1,1)$.

Question 4. If the feasible region of an LPP is unbounded, which of the following is NOT a possible outcome?

(A) A unique optimal solution exists at a corner point.

(B) Multiple optimal solutions exist along an edge.

(C) The objective function is unbounded over the region.

(D) There is no feasible solution.

Question 5. A point is considered infeasible if it does NOT satisfy which condition(s)?

(A) It does not satisfy the objective function.

(B) It does not satisfy at least one of the constraints.

(C) It lies outside the feasible region.

(D) It may satisfy some constraints but not all.

Question 6. Which of the following sets of constraints for $x, y \geq 0$ would NOT result in an empty feasible region?

(A) $x+y \leq 2$ and $x+y \geq 3$.

(B) $x \geq 5$ and $x \leq 4$.

(C) $x+y \leq 5$ and $2x+2y \geq 12$.

(D) $x \geq 2$ and $y \geq 3$ and $x+y \leq 10$.

Question 7. The feasible region is the intersection of half-planes. Which of the following is NOT true about a half-plane defined by $ax + by \leq c$?

(A) It includes the boundary line $ax + by = c$.

(B) It is a convex set.

(C) It is always bounded.

(D) It contains all points $(x,y)$ satisfying the inequality.

Question 8. Which of the following is NOT a vertex (corner point) of the feasible region defined by $x \leq 5$, $y \leq 4$, $x \geq 0$, $y \geq 0$?

(A) $(0,0)$.

(B) $(5,0)$.

(C) $(5,4)$.

(D) $(3,2)$.

Question 9. If the feasible region consists of a single point, which of the following is NOT true?

(A) The problem has a unique feasible solution.

(B) The problem is infeasible.

(C) The single point is the optimal solution.

(D) The feasible region is bounded.

Question 10. Which of the following is NOT a correct interpretation of a feasible region being "bounded"?

(A) It can be enclosed within a finite circle.

(B) Its area is finite.

(C) Its boundaries extend infinitely.

(D) It has a finite number of corner points (unless it's a point or line segment).

Question 1. Which of the following is NOT a type of solution for an LPP?

(A) Feasible solution.

(B) Infeasible solution.

(C) Optimal feasible solution.

(D) Non-linear solution.

Question 2. Which of the following is NOT true about a feasible solution?

(A) It satisfies all the constraints.

(B) It lies within or on the boundary of the feasible region.

(C) It must yield the maximum value of the objective function.

(D) It satisfies the non-negativity restrictions.

Question 3. An infeasible solution is NOT characterized by which of the following?

(A) It violates at least one constraint.

(B) It lies outside the feasible region.

(C) It can be an optimal solution if the feasible region is unbounded.

(D) It fails to satisfy all restrictions simultaneously.

Question 4. Which of the following is NOT true about an optimal feasible solution?

(A) It must be a feasible solution.

(B) It yields the best objective function value among all feasible solutions.

(C) It always occurs at a unique point.

(D) For a bounded feasible region, it occurs at a corner point.

Question 5. Which of the following is NOT necessarily a property of a corner point of the feasible region?

(A) It is a feasible solution.

(B) It is an intersection of boundary lines.

(C) It is always the unique optimal solution.

(D) For a bounded region, the optimal solution is found among the corner points.

Question 6. If an LPP has no feasible region, which of the following is NOT true?

(A) The constraints are contradictory.

(B) There is no point satisfying all constraints.

(C) There exists an optimal solution at infinity.

(D) There are no feasible solutions.

Question 7. Which of the following points would NOT be a feasible solution for constraints $x \geq 1$, $y \geq 1$, $x+y \leq 5$?

(A) $(1,1)$.

(B) $(2,2)$.

(C) $(0,0)$.

(D) $(1,3)$.

Question 8. If an LPP has multiple optimal solutions, which of the following is NOT true?

(A) The optimal value of the objective function is attained at more than one feasible point.

(B) The optimal solutions lie on a line segment connecting two corner points.

(C) The optimal solution is unique.

(D) The objective function is parallel to an edge of the feasible region at the optimum.

Question 9. Which of the following points is NOT an infeasible solution for the constraint $2x+3y \leq 12$ (assuming $x,y \geq 0$)?

(A) $(6,1)$.

(B) $(0,5)$.

(C) $(3,2)$.

(D) $(7,0)$.

Question 10. If the feasible region is bounded and non-empty, which of the following is NOT guaranteed?

(A) Existence of at least one feasible solution.

(B) Existence of an optimal solution.

(C) The optimal solution occurs at a corner point.

(D) The optimal solution is unique.

Question 1. Which of the following is NOT a step in the Corner Point Method for solving LPPs with a bounded feasible region?

(A) Find the feasible region.

(B) Identify the corner points of the feasible region.

(C) Evaluate the objective function at the origin.

(D) Compare the objective function values at the corner points.

Question 2. According to the Corner Point Theorem, for a bounded, non-empty feasible region, which of the following is NOT true?

(A) An optimal solution exists.

(B) The optimal solution occurs at at least one corner point.

(C) The optimal solution occurs at every point in the feasible region.

(D) The maximum and minimum values of the objective function are attained at corner points.

Question 3. If a maximization problem has an unbounded feasible region, which of the following is NOT a possible outcome regarding the optimal solution?

(A) A unique optimal solution exists at a corner point.

(B) Multiple optimal solutions exist along an edge.

(C) The objective function has an unbounded maximum value.

(D) The problem has no feasible solution.

Question 4. If the maximum value of the objective function $Z=ax+by$ for a bounded feasible region is attained at two adjacent corner points, P and Q, which of the following is NOT true?

(A) The value of Z is the same at P and Q.

(B) Every point on the line segment PQ is an optimal solution.

(C) The objective function line is parallel to the edge PQ at the optimal value.

(D) The problem has a unique optimal solution.

Question 5. For a minimization problem with an unbounded feasible region, if the objective function values decrease indefinitely, which of the following is NOT true?

(A) The problem has an unbounded solution.

(B) No minimum value exists.

(C) An optimal solution exists at a corner point.

(D) The objective function can be made arbitrarily small within the feasible region.

Question 6. Evaluating the objective function $Z = 10x + 5y$ at the corner points (0,0), (5,0), and (0,8) of a feasible region. Which of the following values of Z is NOT obtained?

(A) 0.

(B) 50.

(C) 40.

(D) 90.

Question 7. If the feasible region of an LPP is empty, the Corner Point Method would NOT lead to which conclusion?

(A) The problem has no feasible solution.

(B) The problem has no optimal solution.

(C) The objective function is unbounded.

(D) The constraints are contradictory.

Question 8. Which of the following situations would NOT lead to a unique optimal solution for a bounded feasible region?

(A) The optimal value is attained at only one corner point.

(B) The objective function is parallel to an edge, and the optimal value is attained along that edge.

(C) The feasible region is a single point.

(D) The optimal value is the same at several non-adjacent corner points.

Question 9. The Corner Point Method is based on the fact that the feasible region is a convex set. Which of the following is NOT a property of a convex set in this context?

(A) The feasible region can be empty, a point, a line segment, or a polygon/unbounded region.

(B) The optimal value of a linear function occurs at its extreme points.

(C) Any point connecting two points within the set lies outside the set.

(D) The intersection of convex sets is convex.

Question 10. If a minimization problem has an unbounded feasible region but the objective function $Z=ax+by$ (with $a,b > 0$) is bounded below, which of the following is NOT true?

(A) A minimum value exists.

(B) The minimum value is attained at a corner point.

(C) The objective function can be made arbitrarily small.

(D) The minimum value is the smallest value of Z among the corner points.

Question 1. The graphical method for solving LPPs is generally NOT suitable for problems with:

(A) Exactly two decision variables.

(B) Linear constraints.

(C) Non-negativity restrictions.

(D) Three or more decision variables.

Question 2. When using the graphical method, which of the following is NOT a correct way to determine the feasible region for an inequality constraint?

(A) Plot the boundary line (equality).

(B) Use a test point (like the origin) to check which side of the line satisfies the inequality.

(C) Shade the region that violates the inequality.

(D) The feasible region is the intersection of the satisfied regions for all constraints.

Question 3. If the feasible region obtained by the graphical method is empty, which of the following is NOT true?

(A) The LPP has no feasible solution.

(B) The LPP has no optimal solution.

(C) The constraints are inconsistent.

(D) The objective function is unbounded.

Question 4. Using the Iso-profit/Iso-cost line method, which of the following is NOT a correct step to find the optimal solution?

(A) Draw a line representing the objective function for an arbitrary value.

(B) Move this line parallel to itself.

(C) For maximization, move it towards the origin to find the maximum value.

(D) The optimal solution is the last point(s) the line touches the feasible region.

Question 5. If a maximization problem with an unbounded feasible region has an unbounded solution, which of the following is NOT observed graphically?

(A) The feasible region extends infinitely.

(B) The objective function line can be moved indefinitely far from the origin while still intersecting the feasible region.

(C) The objective function values increase without limit within the feasible region.

(D) The optimal solution is found at the corner point furthest from the origin.

Question 6. When solving a minimization problem graphically, which of the following is NOT a key difference compared to solving a maximization problem?

(A) The goal is to find the smallest value of the objective function.

(B) When evaluating corner points, you select the point yielding the minimum Z.

(C) The feasible region is always bounded for minimization problems.

(D) Using the Iso-cost line, you move it towards the origin (usually) to find the minimum Z.

Question 7. If, during the graphical method, the objective function line is parallel to one edge of the bounded feasible region, and the optimal value is attained along this edge, which of the following is NOT true?

(A) There are multiple optimal solutions.

(B) Any point on that edge segment is an optimal solution.

(C) The value of the objective function is constant along that edge.

(D) The problem has an unbounded solution.

Question 8. Which of the following is NOT a typical special case encountered and identified using the graphical method?

(A) No feasible solution.

(B) Unbounded solution.

(C) Multiple optimal solutions.

(D) Integer optimal solution (unless specifically an integer LPP).

Question 9. For the constraint $y \geq 2$ with $x \geq 0$, which of the following points is NOT in the feasible region according to the graphical method?

(A) $(0,2)$.

(B) $(3,2)$.

(C) $(5,5)$.

(D) $(4,1)$.

Question 10. The graphical method is primarily useful for teaching and understanding LPP concepts because:

(A) It can solve problems of any size.

(B) It provides a visual representation of the feasible region and the optimization process.

(C) It clearly shows how the optimal solution is found at the boundary/corners.

(D) It is the most efficient method for large-scale industrial problems.