100+ Linear Programming (LPP) MCQs with Answers| Operations Research Exam Preparation

Section 1: Fundamentals of Linear Programming (MCQ 1–20)

Q1. What is the primary objective of Linear Programming (LPP)?

A. To eliminate uncertainty in decision-making

B. To optimize an objective function subject to constraints

C. To forecast future demand

D. To perform statistical analysis

Correct Answer: B

Short Explanation: Linear Programming is a mathematical optimization technique used to maximize or minimize an objective function while satisfying a set of linear constraints.

Exam Tip: This is the most frequently asked conceptual question in university examinations.

Q2. Linear Programming is mainly used for:

A. Optimization

B. Probability analysis

C. Sampling

D. Accounting

Correct Answer: A

Short Explanation: The main purpose of LPP is to determine the best possible solution from available alternatives while considering limited resources.

Q3. Which branch of mathematics forms the basis of Linear Programming?

A. Geometry and Algebra

B. Trigonometry

C. Calculus only

D. Number Theory

Correct Answer: A

Short Explanation: LPP combines algebraic equations with geometric concepts, especially in the graphical solution method.

Q4. Which of the following is NOT a component of a Linear Programming Problem?

A. Decision Variables

B. Objective Function

C. Constraints

D. Probability Distribution

Correct Answer: D

Short Explanation: Decision variables, objective function, and constraints are the fundamental components of every LPP. Probability distributions are not essential components.

Exam Tip: "Components of LPP" is a repeatedly asked topic in BBA, MBA, and Engineering examinations.

Q5. In a Linear Programming Problem, the objective function represents:

A. Available resources

B. Mathematical expression to maximize or minimize

C. Decision variables only

D. Constraints only

Correct Answer: B

Short Explanation: The objective function defines the goal of optimization, such as maximizing profit or minimizing cost.

Q6. Which of the following is an example of a decision variable?

A. Profit

B. Production quantity of Product X

C. Budget

D. Machine capacity

Correct Answer: B

Short Explanation: Decision variables represent the unknown quantities whose values are determined through optimization.

Q7. Constraints in Linear Programming represent:

A. Goals

B. Resource limitations

C. Forecast values

D. Random variables

Correct Answer: B

Short Explanation: Constraints define the limitations within which the objective function must be optimized.

Q8. Which symbol generally represents a decision variable?

A. x

B. P

C. M

D. R

Correct Answer: A

Short Explanation: Decision variables are commonly denoted by x₁, x₂, x₃, etc.

Q9. Which of the following is TRUE regarding Linear Programming?

A. All relationships must be linear.

B. Relationships can be exponential.

C. Relationships can be logarithmic.

D. Relationships may be nonlinear.

Correct Answer: A

Short Explanation: The term "Linear" means both the objective function and constraints must be linear expressions.

Exam Tip: Questions on the meaning of "linear" appear frequently in objective examinations.

Q10. Which of the following is NOT generally solved using Linear Programming?

A. Product Mix

B. Transportation Planning

C. Resource Allocation

D. DNA Sequencing

Correct Answer: D

Short Explanation: Linear Programming is commonly applied to optimization problems like production planning and resource allocation, whereas DNA sequencing is not a standard LPP application.

Q11. Which of the following best defines a Linear Programming Problem (LPP)?

A. A mathematical technique for solving nonlinear equations

B. A statistical method for data analysis

C. A mathematical optimization technique for maximizing or minimizing a linear objective function subject to linear constraints

D. A forecasting technique

Correct Answer: C

Short Explanation: Linear Programming is a mathematical optimization technique used to obtain the best possible value of an objective function while satisfying all linear constraints.

Exam Tip: This is one of the most frequently asked definition-based questions in university and competitive examinations.

Q12. The objective function in an LPP may be used to:

A. Only maximize profit

B. Only minimize cost

C. Either maximize or minimize a measurable quantity

D. Calculate probability

Correct Answer: C

Short Explanation: The objective function can represent profit, cost, time, distance, or any measurable quantity that needs optimization.

Q13. The constraints in an LPP are generally expressed as:

A. Linear equations or inequalities

B. Quadratic equations

C. Exponential equations

D. Differential equations

Correct Answer: A

Short Explanation: Constraints define the limitations on available resources and are expressed using linear equations or inequalities.

Q14. Which of the following is NOT an objective of Linear Programming?

A. Efficient utilization of resources

B. Optimization of profit or cost

C. Determining the best feasible solution

D. Eliminating all business risks

Correct Answer: D

Short Explanation: Linear Programming helps optimize decisions but cannot eliminate uncertainty or business risks.

Q15. Linear Programming belongs to which area of study?

A. Marketing

B. Operations Research

C. Financial Accounting

D. Organizational Behaviour

Correct Answer: B

Short Explanation: Linear Programming is one of the most important quantitative techniques used in Operations Research.

Exam Tip: Many exams ask the direct question: "Linear Programming is a technique of ______."

Q16. Which of the following is NOT normally treated as a decision variable?

A. Quantity of Product A to produce

B. Number of workers to assign

C. Selling price fixed by the government

D. Number of trucks to dispatch

Correct Answer: C

Short Explanation: Decision variables are controllable by the decision-maker. A government-fixed selling price is not under managerial control.

Q17. In an LPP, resource limitations are represented by:

A. Objective function

B. Decision variables

C. Constraints

D. Constants only

Correct Answer: C

Short Explanation: Constraints mathematically represent limited resources such as labor, raw materials, machine hours, or budget.

Q18. Which of the following is an example of an objective function?

A. x₁ + x₂ ≤ 50

B. x₁ ≥ 0

C. Maximize Z = 5x₁ + 3x₂

D. x₂ ≤ 25

Correct Answer: C

Short Explanation: An objective function specifies what needs to be optimized, such as maximizing profit or minimizing cost.

Q19. Which of the following statements is correct?

A. Every optimization problem is an LPP.

B. Every LPP is an optimization problem.

C. Every statistical problem is an LPP.

D. Every optimization problem has nonlinear constraints.

Correct Answer: B

Short Explanation: All Linear Programming Problems are optimization problems, but many optimization problems are nonlinear and therefore are not LPPs.

Exam Tip: This is a common conceptual MCQ designed to test the difference between optimization and Linear Programming.

Q20. The success of Linear Programming mainly depends upon:

A. Random guessing

B. Proper mathematical formulation of the problem

C. Trial and error only

D. Market competition

Correct Answer: B

Short Explanation: Even the best solution methods cannot produce correct results if the problem is formulated incorrectly. Proper identification of decision variables, objective function, and constraints is essential.

Section 2: Characteristics and Assumptions of Linear Programming (MCQ 21–30)

Q21. Which of the following is NOT a basic assumption of Linear Programming?

A. Linearity

B. Certainty

C. Probability

D. Divisibility

Correct Answer: C

Short Explanation: The fundamental assumptions of LPP are Linearity (Proportionality), Additivity, Divisibility, Certainty, and Non-negativity. Probability is not one of these assumptions.

Exam Tip: This is among the most repeated conceptual MCQs in university examinations.

Q22. The assumption of linearity implies that:

A. Relationships between variables are linear.

B. Resources are unlimited.

C. Variables are random.

D. Costs always remain constant.

Correct Answer: A

Short Explanation: Both the objective function and all constraints must be linear mathematical expressions.

Q23. Which assumption states that the contribution of each decision variable is directly proportional to its value?

A. Additivity

B. Proportionality

C. Certainty

D. Feasibility

Correct Answer: B

Short Explanation: The proportionality assumption means doubling a decision variable doubles its contribution to the objective function and resource usage.

Exam Tip: Many exams use the terms Linearity and Proportionality interchangeably. Understand the concept rather than memorizing the wording.

Q24. The assumption of additivity means:

A. Total effect equals the sum of individual effects.

B. Variables cannot be added together.

C. Only one decision variable exists.

D. Constraints are nonlinear.

Correct Answer: A

Short Explanation: There are no interaction effects among decision variables. The total profit or resource usage is simply the sum of individual contributions.

Q25. Which assumption allows decision variables to take fractional values?

A. Certainty

B. Divisibility

C. Additivity

D. Feasibility

Correct Answer: B

Short Explanation: The divisibility assumption allows variables to take decimal or fractional values unless the problem specifically requires integer solutions.

Exam Tip: If variables must be whole numbers, the problem becomes an Integer Programming Problem, not a standard LPP.

Q26. The assumption of certainty means that:

A. All model coefficients are known and constant.

B. Future demand is unknown.

C. Probabilities are assigned to all events.

D. Decision variables are random.

Correct Answer: A

Short Explanation: Profits, costs, resource requirements, and availability are assumed to be known with certainty during the planning period.

Q27. The non-negativity restriction requires that:

A. Decision variables must be greater than or equal to zero.

B. Constraints must be negative.

C. Profit cannot be positive.

D. Resources must be unlimited.

Correct Answer: A

Short Explanation: Negative production, negative labor, or negative quantities usually have no practical meaning, so decision variables are restricted to zero or positive values.

Q28. Which assumption ignores interaction among decision variables?

A. Additivity

B. Divisibility

C. Certainty

D. Feasibility

Correct Answer: A

Short Explanation: Each variable contributes independently. Their combined effect is obtained by simple addition.

Q29. Which assumption is violated if the profit per unit changes with the production level?

A. Proportionality

B. Divisibility

C. Non-negativity

D. Feasibility

Correct Answer: A

Short Explanation: If profit per unit changes as production increases, the relationship is no longer linear, violating the proportionality assumption.

Exam Tip: Questions based on identifying the violated assumption are very common in MBA and Engineering entrance exams.

Q30. Standard Linear Programming models assume that all coefficients are:

A. Random and changing

B. Unknown

C. Fixed and known

D. Estimated after obtaining the solution

Correct Answer: C

Short Explanation: LPP assumes complete certainty. Therefore, coefficients in the objective function and constraints remain constant throughout the analysis.

Section 3: Decision Variables, Objective Function, Constraints & Mathematical Formulation (MCQ 31–40)

Q31. The first step in formulating a Linear Programming Problem is to:

A. Draw the feasible region

B. Identify the decision variables

C. Solve the simplex table

D. Calculate the optimal solution

Correct Answer: B

Short Explanation: Every LPP begins by identifying the unknown quantities (decision variables) that must be determined.

Exam Tip: Questions on the steps of LPP formulation are repeatedly asked in university examinations.

Q32. Decision variables represent:

A. Known constants

B. Unknown quantities whose values are to be determined

C. Available resources

D. Market demand only

Correct Answer: B

Short Explanation: Decision variables are controllable quantities such as units to produce, workers to assign, or products to transport.

Q33. The objective function is always expressed in terms of:

A. Decision variables

B. Constraints only

C. Constants only

D. Random variables

Correct Answer: A

Short Explanation: The objective function is a linear expression involving decision variables and their coefficients.

Q34. Which of the following is a correctly written objective function?

A. Maximize Z = 8x₁ + 5x₂

B. x₁ + x₂ ≤ 20

C. x₁ ≥ 0

D. x₂ ≤ 15

Correct Answer: A

Short Explanation: An objective function specifies the quantity to maximize or minimize, whereas the others are constraints.

Q35. Constraints in an LPP describe:

A. Managerial goals

B. Resource limitations and requirements

C. Future uncertainty

D. Market competition

Correct Answer: B

Short Explanation: Constraints restrict the values that decision variables can take because of limited resources.

Q36. Which mathematical symbol is commonly used for a "less than or equal to" constraint?

A. ≥

B. =

C. ≤

D. ≠

Correct Answer: C

Short Explanation: Most resource constraints are represented using the ≤ sign because resource usage cannot exceed availability.

Q37. A production problem has two products. The appropriate decision variables are generally:

A. Total profit and total cost

B. Quantity of Product 1 and Quantity of Product 2

C. Machine hours and labour hours

D. Market demand and selling price

Correct Answer: B

Short Explanation: Decision variables represent the quantities that management must decide.

Q38. In the objective function

Maximize Z = 40x₁ + 60x₂

the numbers 40 and 60 represent:

A. Decision variables

B. Resource limits

C. Objective function coefficients

D. Constraint coefficients

Correct Answer: C

Short Explanation: These coefficients indicate the contribution (e.g., profit or cost) of each unit of the corresponding decision variable.

Exam Tip: Many exams ask students to identify coefficients, decision variables, and objective function from a given mathematical model.

Q39. Which of the following is NOT part of a complete mathematical formulation of an LPP?

A. Decision variables

B. Objective function

C. Constraints

D. Histogram

Correct Answer: D

Short Explanation: A complete LPP consists of decision variables, an objective function, constraints, and non-negativity restrictions. A histogram is unrelated.

Q40. Which of the following represents a complete Linear Programming model?

A. Maximize Z = 5x₁ + 3x₂

B. Maximize Z = 5x₁ + 3x₂ subject to constraints and x₁, x₂ ≥ 0

C. x₁ + x₂ ≤ 20

D. x₁ ≥ 0

Correct Answer: B

Short Explanation: A valid LPP must include:

Decision variables
Objective function
Constraints
Non-negativity restrictions

Exam Tip: If any of these four elements is missing, the mathematical model is considered incomplete.

Section 4: Feasible Solution, Basic Solution, Basic Feasible Solution, Degeneracy, Optimal Solution, Multiple Solutions, Unbounded Solution, and Infeasible Solution (MCQ 41–50)

(These are among the most frequently tested concepts in university exams, GATE-level OR courses, and management programs.)

Q41. A feasible solution of an LPP is one that:

A. Maximizes the objective function only

B. Satisfies all the constraints and non-negativity restrictions

C. Has the highest profit

D. Contains only integer values

Correct Answer: B

Short Explanation: A feasible solution is any solution that satisfies every constraint, including the non-negativity restrictions. It may or may not be the optimal solution.

Exam Tip: Remember: Every optimal solution is feasible, but every feasible solution is not necessarily optimal.

Q42. The collection of all feasible solutions is called the:

A. Objective function

B. Feasible region

C. Constraint equation

D. Decision space

Correct Answer: B

Short Explanation: The feasible region consists of all points that satisfy every constraint simultaneously.

Q43. An optimal solution is a feasible solution that:

A. Violates one constraint

B. Gives the best value of the objective function

C. Has the largest number of variables

D. Always contains integers

Correct Answer: B

Short Explanation: Among all feasible solutions, the optimal solution provides the maximum or minimum objective value, depending on the problem.

Q44. A solution that does not satisfy one or more constraints is called:

A. Basic solution

B. Feasible solution

C. Infeasible solution

D. Optimal solution

Correct Answer: C

Short Explanation: Any solution violating at least one constraint is infeasible and cannot be considered for optimization.

Q45. A basic solution is obtained by:

A. Setting all variables equal to zero

B. Setting a suitable number of variables equal to zero and solving for the remaining variables

C. Maximizing the objective function directly

D. Ignoring all constraints

Correct Answer: B

Short Explanation: For a system with n variables, a basic solution is obtained by selecting basic variables and setting the remaining non-basic variables to zero.

Exam Tip: Questions on basic vs basic feasible solutions are commonly asked in simplex method topics.

Q46. A basic feasible solution is:

A. A basic solution that satisfies all constraints

B. Any feasible solution

C. A solution with negative variables

D. A solution outside the feasible region

Correct Answer: A

Short Explanation: A basic feasible solution (BFS) is both a basic solution and a feasible solution. The simplex method moves from one BFS to another.

Q47. A degenerate basic feasible solution occurs when:

A. All variables are positive

B. One or more basic variables are equal to zero

C. All constraints are violated

D. The objective function becomes nonlinear

Correct Answer: B

Short Explanation: Degeneracy occurs when at least one basic variable has a value of zero. It may lead to cycling in the simplex method.

Exam Tip: The phrase "basic variable equals zero" is the key identifier of degeneracy.

Q48. Which of the following statements is correct?

A. Every basic solution is feasible.

B. Every feasible solution is basic.

C. Every basic feasible solution is feasible.

D. Every infeasible solution is optimal.

Correct Answer: C

Short Explanation: A basic feasible solution always satisfies all constraints. However, a basic solution need not be feasible.

Q49. A Linear Programming Problem has multiple (alternate) optimal solutions when:

A. The feasible region is empty.

B. More than one feasible solution gives the same optimal value of the objective function.

C. All variables are zero.

D. Constraints are inconsistent.

Correct Answer: B

Short Explanation: Alternate optimal solutions occur when two or more feasible solutions produce the same maximum or minimum objective value.

Exam Tip: In the graphical method, alternate optimal solutions occur when the objective function is parallel to a binding constraint at the optimum.

Q50. An unbounded solution means:

A. There is no feasible solution.

B. The objective function can be improved indefinitely without violating any constraint.

C. All decision variables are zero.

D. The problem has multiple optimal solutions.

Correct Answer: B

Short Explanation: An unbounded solution indicates that the objective value can increase (or decrease, in a minimization problem) without limit because the constraints do not sufficiently restrict the feasible region.

Exam Tip: Do not confuse:

Infeasible Problem → No feasible solution exists.
Unbounded Problem → Feasible solutions exist, but no finite optimum exists.

Section 5: Graphical Method of Solving Linear Programming Problems (MCQ 51–60)

(These topics are among the most frequently tested in university examinations and competitive OR courses.)

Q51. The graphical method can be directly used when an LPP has:

A. One decision variable

B. Two decision variables

C. Three decision variables

D. Four or more decision variables

Correct Answer: B

Short Explanation: The graphical method is practical only for problems with two decision variables, as they can be represented on a two-dimensional graph.

Exam Tip: Some books mention that one-variable problems can also be solved graphically on a number line, but in Operations Research, the standard answer is two decision variables.

Q52. In the graphical method, each constraint is represented by:

A. A point

B. A straight line

C. A circle

D. A parabola

Correct Answer: B

Short Explanation: Every linear constraint is first plotted as a straight line, and then the appropriate half-plane is identified.

Q53. The feasible region in the graphical method is obtained by:

A. Taking the union of all half-planes

B. Finding the common intersection of all constraint half-planes

C. Drawing only the objective function

D. Ignoring non-negativity restrictions

Correct Answer: B

Short Explanation: The feasible region consists of all points that satisfy every constraint simultaneously.

Q54. The Corner Point (Extreme Point) Theorem states that:

A. The optimal solution, if it exists, occurs at one or more corner points of the feasible region.

B. The optimal solution always lies at the center of the feasible region.

C. Every feasible point is optimal.

D. The optimal solution always occurs on the x-axis.

Correct Answer: A

Short Explanation: For Linear Programming Problems, if an optimal solution exists, at least one corner (extreme) point will be optimal.

Exam Tip: This is one of the most frequently asked theoretical questions in LPP.

Q55. The intersection of two constraint lines generally represents:

A. A corner (extreme) point

B. A decision variable

C. The objective function

D. An infeasible solution

Correct Answer: A

Short Explanation: Corner points are formed where two or more constraint boundaries intersect.

Q56. In the graphical method, the objective function is evaluated at:

A. Every point in the feasible region

B. Only the corner (extreme) points

C. Only the origin

D. Only the midpoint of the feasible region

Correct Answer: B

Short Explanation: According to the Corner Point Theorem, it is sufficient to evaluate the objective function at all corner points.

Q57. Which of the following is NOT a step in the graphical method?

A. Plot the constraints

B. Identify the feasible region

C. Evaluate the objective function at corner points

D. Differentiate the objective function

Correct Answer: D

Short Explanation: Differentiation is not used in the graphical solution of Linear Programming.

Q58. A constraint is called binding at the optimal solution if:

A. It is satisfied as a strict inequality.

B. It holds as an equality at the optimal solution.

C. It is ignored during optimization.

D. It has no effect on the solution.

Correct Answer: B

Short Explanation: A binding constraint is active at the optimum and directly limits the objective function.

Exam Tip: Remember:

Binding Constraint → Equality
Non-binding Constraint → Strict Inequality

This distinction is asked frequently in exams.

Q59. A non-binding constraint is one that:

A. Is satisfied as an equality.

B. Is violated by the optimal solution.

C. Is satisfied with some slack at the optimal solution.

D. Makes the problem infeasible.

Correct Answer: C

Short Explanation: A non-binding constraint has unused resources (positive slack) at the optimal solution and does not directly determine the optimum.

Q60. If the feasible region is empty, then the Linear Programming Problem is:

A. Degenerate

B. Unbounded

C. Infeasible

D. Alternate Optimal

Correct Answer: C

Short Explanation: If no point satisfies all constraints simultaneously, there is no feasible region, and hence the problem is infeasible.

Exam Tip: A common exam trick is to confuse empty feasible region with unbounded feasible region:

Empty Feasible Region → Infeasible Problem
Infinite Feasible Region → May still have a finite optimal solution

Section 6: Standard Form, Canonical Form, Model Conversion, Slack, Surplus & Artificial Variables (MCQ 61–70)

(These topics are extremely important because they are the foundation of the Simplex Method and are asked repeatedly in BBA, MBA, B.Com, Engineering, GATE, UGC NET, and other university examinations.)

Important Note: Different books sometimes use the terms Standard Form and Canonical Form differently. The following MCQs use the definitions that are most widely accepted in Operations Research textbooks and university syllabi. Where terminology varies, the explanation clarifies the distinction.

Q61. Before applying the Simplex Method, a Linear Programming Problem is generally converted into:

A. Differential form

B. Standard form

C. Quadratic form

D. Matrix inversion form

Correct Answer: B

Short Explanation: The Simplex Method requires the LPP to be expressed in standard form so that the initial simplex tableau can be constructed.

Exam Tip: This is one of the most frequently repeated introductory questions on the Simplex Method.

Q62. A slack variable is added to a constraint of the type:

A. ≥

B. =

C. ≤

D. ≠

Correct Answer: C

Short Explanation: A slack variable converts a "less than or equal to (≤)" constraint into an equality by representing the unused portion of a resource.

Example:


x_1+x_2\le10


x_1+x_2+s_1=10

Q63. A slack variable represents:

A. Excess usage of resources

B. Unused or idle resources

C. Decision variables

D. Profit

Correct Answer: B

Short Explanation: Slack indicates the amount of resource left unused after satisfying the solution.

Q64. A surplus variable is introduced for a constraint of the type:

A. ≤

B. ≥

C. =

D. <

Correct Answer: B

Short Explanation: A surplus variable is subtracted from a "greater than or equal to (≥)" constraint to convert it into an equality.

Example:


x_1+x_2\ge12

becomes


x_1+x_2-s_1=12

Q65. A surplus variable indicates:

A. Unused resources

B. Excess above the minimum required level

C. Profit earned

D. Cost incurred

Correct Answer: B

Short Explanation: Surplus measures how much the left-hand side exceeds the required minimum level.

Q66. Why is an artificial variable introduced?

A. To increase profit

B. To obtain an initial basic feasible solution when one is not readily available

C. To replace decision variables

D. To reduce the number of constraints

Correct Answer: B

Short Explanation: Artificial variables are temporary variables introduced to start the Simplex Method in problems involving "=" or "≥" constraints.

Exam Tip: Artificial variables have no physical meaning. They are computational devices used only during the solution process.

Q67. Artificial variables are generally used in:

A. Graphical Method only

B. Big M Method and Two-Phase Method

C. Transportation Problem only

D. Assignment Problem only

Correct Answer: B

Short Explanation: Both the Big M Method and the Two-Phase Method use artificial variables to establish an initial basic feasible solution.

Q68. Which variable is subtracted from a constraint?

A. Slack Variable

B. Artificial Variable

C. Surplus Variable

D. Decision Variable

Correct Answer: C

Short Explanation: Only a surplus variable is subtracted while converting a "≥" constraint into an equality.

Q69. Which of the following variables is always non-negative in Linear Programming?

A. Slack Variable

B. Surplus Variable

C. Artificial Variable

D. All of the above

Correct Answer: D

Short Explanation: Like decision variables, slack, surplus, and artificial variables are also subject to the non-negativity restriction.

Q70. Which statement is correct?

A. Slack variables are added to "≥" constraints.

B. Surplus variables are added to "≤" constraints.

C. Artificial variables may be required for "=" and "≥" constraints.

D. Slack variables are used only in the graphical method.

Correct Answer: C

Short Explanation: Artificial variables are required whenever an initial basic feasible solution cannot be obtained directly, which commonly occurs with equality and greater-than-or-equal-to constraints.

Exam Tip (Highly Repeated):

Remember this exam shortcut:

Constraint	Variable Introduced
≤	Add Slack Variable (+S)
≥	Subtract Surplus Variable (−S) + Add Artificial Variable (+A)
=	Add Artificial Variable (+A)

This table alone answers many university objective questions.

Section 7: Simplex Method (MCQ 71–80)

(This is one of the highest-weightage topics in Linear Programming examinations. This section covers the Simplex Method in depth: Initial Simplex Tableau, Entering Variable, Leaving Variable, Pivot Element,Iteration, Optimality Test,Minimization vs. Maximization, Common Simplex Concepts)

Quality Note: The Simplex Method is one of the most frequently tested topics in BBA, MBA, B.Com, Engineering, GATE, UGC NET, and university examinations. These MCQs focus on concepts rather than lengthy calculations.

Q71. The Simplex Method is primarily used to solve Linear Programming Problems involving:

A. One decision variable

B. Two decision variables only

C. More than two decision variables

D. No decision variables

Correct Answer: C

Short Explanation: Although it can solve smaller problems, the Simplex Method is especially useful for LPPs with more than two decision variables, where the graphical method is impractical.

Exam Tip: Graphical Method → Usually 2 variables
Simplex Method → 2 or more variables, especially more than 2.

Q72. The Simplex Method starts with:

A. Any feasible solution

B. An initial basic feasible solution (IBFS)

C. The optimal solution

D. An infeasible solution

Correct Answer: B

Short Explanation: The Simplex Method begins with an Initial Basic Feasible Solution and moves systematically toward the optimal solution.

Q73. In the Simplex Method, the variable entering the basis is called the:

A. Leaving variable

B. Incoming (Entering) variable

C. Artificial variable

D. Slack variable

Correct Answer: B

Short Explanation: The entering variable is the non-basic variable selected to improve the objective function value.

Q74. The variable that leaves the basis during an iteration is determined using the:

A. Largest coefficient rule

B. Minimum Ratio Test

C. Maximum Ratio Test

D. Corner Point Rule

Correct Answer: B

Short Explanation: The Minimum Ratio Test identifies the leaving variable while maintaining feasibility.

Exam Tip: Minimum Ratio Test = Leaving Variable
This question appears repeatedly in university exams.

Q75. The element at the intersection of the entering variable column and leaving variable row is called the:

A. Objective coefficient

B. Pivot element

C. Slack value

D. Ratio value

Correct Answer: B

Short Explanation: The pivot element is used to perform row operations and generate the next simplex tableau.

Q76. The purpose of pivot operations is to:

A. Increase the number of constraints

B. Obtain a new basic feasible solution

C. Remove the objective function

D. Eliminate decision variables

Correct Answer: B

Short Explanation: Each pivot operation transforms the current tableau into another tableau representing a new basic feasible solution.

Q77. The Simplex Method continues until:

A. All constraints disappear

B. The optimality condition is satisfied

C. All variables become zero

D. Every variable enters the basis

Correct Answer: B

Short Explanation: Iterations continue until no further improvement in the objective function is possible according to the optimality criterion.

Q78. During the Simplex Method, each iteration generally produces:

A. A new infeasible solution

B. A new basic feasible solution

C. A nonlinear solution

D. A random solution

Correct Answer: B

Short Explanation: Each iteration moves from one Basic Feasible Solution (BFS) to another while improving (or at least not worsening) the objective function.

Q79. Which statement about the Simplex Method is correct?

A. It examines every feasible solution.

B. It examines only basic feasible solutions.

C. It ignores constraints.

D. It always requires the graphical method first.

Correct Answer: B

Short Explanation: The Simplex Method is efficient because it evaluates only basic feasible solutions rather than every feasible point.

Exam Tip: This is a classic conceptual question that tests understanding of why the Simplex Method is computationally efficient.

Q80. The main objective of each Simplex iteration is to:

A. Make all variables equal to zero.

B. Improve the value of the objective function while maintaining feasibility.

C. Increase the number of constraints.

D. Eliminate decision variables.

Correct Answer: B

Short Explanation: Each iteration aims to move toward the optimal solution without violating any constraints.

Section 8: Big M Method, Two-Phase Method & Special Cases in the Simplex Method (MCQ 81–90)

(These topics are among the most challenging and highest-scoring areas in university and competitive examinations. This section covers Big M Method,Two-Phase Method, Artificial Variables in Detail, Special Cases in the Simplex Method (Degeneracy, Cycling, Multiple Optima, Unboundedness, Infeasibility))

Quality Note: This section has been carefully worded because some universities use slightly different terminology. The concepts and correct answers are consistent with standard OR textbooks.

Q81. The primary purpose of the Big M Method is to:

A. Reduce the number of decision variables

B. Handle LPPs containing artificial variables

C. Eliminate slack variables

D. Draw the feasible region

Correct Answer: B

Short Explanation: The Big M Method is used when artificial variables are required. A very large penalty (M) is assigned to artificial variables so that they are forced out of the final solution.

Exam Tip: Remember: Big M = Large Penalty Method.

Q82. In the Big M Method for a maximization problem, artificial variables are assigned:

A. A large positive coefficient

B. A large negative coefficient

C. Zero coefficient

D. Unit coefficient

Correct Answer: B

Short Explanation: For maximization problems, artificial variables receive a coefficient of −M in the objective function so that the algorithm avoids keeping them in the final solution.

Exam Tip: For minimization problems, many textbooks assign +M. Always check the sign convention used in your syllabus.

Q83. The symbol M in the Big M Method represents:

A. Market demand

B. Machine capacity

C. A very large positive constant

D. Minimum value

Correct Answer: C

Short Explanation: The value M is assumed to be extremely large compared to ordinary coefficients in the objective function.

Q84. The Two-Phase Method is mainly used to:

A. Solve nonlinear programming problems

B. Eliminate the need for constraints

C. Obtain an initial basic feasible solution before optimization

D. Replace the graphical method

Correct Answer: C

Short Explanation: The Two-Phase Method first removes artificial variables (Phase I) and then optimizes the original objective function (Phase II).

Q85. In the Two-Phase Method, Phase I aims to:

A. Maximize profit

B. Minimize cost

C. Eliminate artificial variables by obtaining a feasible solution

D. Solve the dual problem

Correct Answer: C

Short Explanation: Phase I minimizes the sum of artificial variables (or an equivalent auxiliary objective) to obtain a feasible starting solution.

Q86. Phase II of the Two-Phase Method begins after:

A. Drawing the graph

B. Obtaining a feasible basic solution without artificial variables

C. Adding more constraints

D. Eliminating slack variables

Correct Answer: B

Short Explanation: Once a feasible basis is obtained in Phase I, the original objective function is restored and optimized in Phase II.

Q87. If an artificial variable remains positive in the final solution of Phase I, the original LPP is:

A. Degenerate

B. Unbounded

C. Infeasible

D. Optimal

Correct Answer: C

Short Explanation: A positive artificial variable at the end of Phase I indicates that no feasible solution exists for the original Linear Programming Problem.

Exam Tip: This is one of the most repeated conceptual questions in the Two-Phase Method.

Q88. Degeneracy in the Simplex Method occurs when:

A. The feasible region is empty

B. At least one basic variable becomes zero

C. The objective function is nonlinear

D. There are no constraints

Correct Answer: B

Short Explanation: Degeneracy occurs when one or more basic variables have a value of zero in a basic feasible solution.

Q89. Cycling in the Simplex Method is associated with:

A. Unbounded solutions only

B. Degeneracy

C. Graphical Method

D. Duality

Correct Answer: B

Short Explanation: Cycling is a rare phenomenon in which the Simplex Method revisits the same basic feasible solutions repeatedly due to degeneracy.

Exam Tip: Although cycling is rarely encountered in practice, the concept is frequently tested in theory exams.

Q90. Which statement is correct?

A. Every degenerate solution causes cycling.

B. Cycling may occur because of degeneracy, but degeneracy does not always cause cycling.

C. Cycling occurs in every Simplex problem.

D. Degeneracy and cycling are unrelated concepts.

Correct Answer: B

Short Explanation: Degeneracy is necessary for cycling to occur, but most degenerate problems do not cycle.

Section 9: Duality in Linear Programming (MCQ 91–100)

This is one of the most important advanced topics in Linear Programming and appears regularly in MBA, Engineering, GATE, UGC NET, and university examinations. This section cover Duality, including Primal vs. Dual, Dual Variables, Duality Theorems, Complementary Slackness, Economic Interpretation (Shadow Prices), Relationship between Primal and Dual.

Quality Note: Duality is one of the most frequently tested advanced topics in Operations Research. The following MCQs focus on concepts that repeatedly appear in university, engineering, management, and competitive examinations.

Q91. In Linear Programming, every primal problem has:

A. No corresponding problem

B. Exactly one dual problem

C. Two dual problems

D. Infinite dual problems

Correct Answer: B

Short Explanation: Every Linear Programming Problem (Primal) has one corresponding Dual Problem constructed according to standard duality rules.

Exam Tip: Remember: One Primal ⇄ One Dual.

Q92. The objective of the dual problem is determined by the objective of the primal problem. If the primal is a maximization problem, the dual is generally a:

A. Maximization problem

B. Minimization problem

C. Quadratic problem

D. Nonlinear problem

Correct Answer: B

Short Explanation: For standard LPPs, a maximization primal corresponds to a minimization dual, and vice versa.

Q93. In a standard maximization primal problem with m constraints and n decision variables, the dual problem has:

A. m variables and n constraints

B. n variables and m constraints

C. m variables and m constraints

D. n variables and n constraints

Correct Answer: A

Short Explanation: The number of constraints and variables are interchanged when forming the dual:

Primal: m constraints, n variables
Dual: m variables, n constraints

Exam Tip: A favorite exam question: Variables and constraints interchange between primal and dual.

Q94. In constructing the dual, the coefficient matrix of the primal is:

A. Squared

B. Transposed

C. Differentiated

D. Integrated

Correct Answer: B

Short Explanation: The constraint coefficient matrix of the primal becomes its transpose in the dual formulation.

Q95. If the primal problem has an optimal solution, then the dual problem:

A. Has no solution

B. Also has an optimal solution with the same objective value

C. Is always infeasible

D. Is always unbounded

Correct Answer: B

Short Explanation: According to the Strong Duality Theorem, if both problems have feasible optimal solutions, their optimal objective values are equal.

Exam Tip: This is one of the most repeated theory questions in OR examinations.

Q96. The Weak Duality Theorem states that:

A. Any feasible solution of the primal equals any feasible solution of the dual.

B. The objective value of a feasible maximization primal cannot exceed that of any feasible dual minimization solution.

C. Primal and dual always have identical variables.

D. Every primal problem is unbounded.

Correct Answer: B

Short Explanation: Weak Duality provides a bound: the objective value of any feasible primal solution cannot exceed that of any feasible dual solution (for the standard max–min pair).

Q97. Which of the following best describes the Strong Duality Theorem?

A. Every feasible solution is optimal.

B. If optimal solutions exist for both primal and dual, their objective values are equal.

C. Primal and dual have the same number of variables.

D. The dual always has more constraints.

Correct Answer: B

Short Explanation: Strong Duality guarantees equality of the optimal objective values for the primal and dual when optimal solutions exist.

Q98. Complementary Slackness establishes the relationship between:

A. Decision variables and objective coefficients only

B. Optimal solutions of the primal and dual problems

C. Graphical method and simplex method

D. Slack variables only

Correct Answer: B

Short Explanation: Complementary Slackness provides conditions linking the optimal primal variables with the optimal dual variables.

Exam Tip: If the syllabus includes Duality, Complementary Slackness is an important theoretical topic.

Q99. In Linear Programming, a shadow price is another name for:

A. Decision variable

B. Dual variable (dual value)

C. Slack variable

D. Artificial variable

Correct Answer: B

Short Explanation: A shadow price (also called a dual value or marginal value) measures how much the optimal objective value changes when the right-hand side of a constraint changes by one unit, within the allowable range.

Exam Tip: The terms Shadow Price, Dual Value, and Marginal Value are often used interchangeably in OR.

Q100. Which statement about the relationship between the primal and dual is correct?

A. If one problem is unbounded, the other must have a feasible optimal solution.

B. If one problem is unbounded and feasible, the other is infeasible.

C. Both problems are always unbounded.

D. Both problems are always infeasible.

Correct Answer: B

Short Explanation: A fundamental result of duality is:

If one problem is feasible and unbounded, the corresponding dual is infeasible.
Conversely, if one problem is infeasible, the other may be infeasible or unbounded, depending on the situation.

Exam Tip (Highly Repeated): Remember these standard results:

✅ Both feasible → Equal optimal objective values.
✅ One feasible & unbounded → The other is infeasible.

Section 10: Sensitivity Analysis (Post-Optimality Analysis) (MCQ 101–110)

Q101. Sensitivity Analysis is also known as:

A. Probability Analysis

B. Post-Optimality Analysis

C. Regression Analysis

D. Simulation Analysis

Correct Answer: B

Short Explanation: Sensitivity Analysis examines how changes in model parameters affect the optimal solution after an LPP has already been solved.

Exam Tip: Sensitivity Analysis = Post-Optimality Analysis. Both terms are used interchangeably in most OR textbooks.

Q102. The primary purpose of Sensitivity Analysis is to study:

A. Past business performance

B. The effect of changes in model parameters on the optimal solution

C. Market demand forecasting

D. Employee productivity

Correct Answer: B

Short Explanation: It helps decision-makers understand whether the current optimal solution remains valid when coefficients or resource levels change.

Q103. Which of the following is NOT normally studied in Sensitivity Analysis?

A. Changes in objective function coefficients

B. Changes in resource availability

C. Changes in constraint coefficients

D. Changing a linear model into a nonlinear model

Correct Answer: D

Short Explanation: Sensitivity Analysis assumes the problem remains a Linear Programming Problem. It does not convert the model into a nonlinear one.

Q104. The value that indicates the improvement in the objective function resulting from one additional unit of a resource is called:

A. Pivot Value

B. Shadow Price

C. Slack Variable

D. Artificial Variable

Correct Answer: B

Short Explanation: The shadow price measures the marginal value of an additional unit of a constrained resource within the allowable range.

Exam Tip: Shadow Price is one of the most frequently asked theory questions in OR.

Q105. A shadow price of zero generally indicates that:

A. The corresponding constraint is non-binding.

B. The problem is infeasible.

C. The objective function is zero.

D. The solution is unbounded.

Correct Answer: A

Short Explanation: If a constraint is non-binding, extra units of that resource do not improve the objective value; therefore, its shadow price is typically zero.

Q106. Sensitivity Analysis helps managers primarily by:

A. Eliminating uncertainty completely

B. Evaluating the impact of changes without resolving the entire problem

C. Replacing the Simplex Method

D. Removing constraints

Correct Answer: B

Short Explanation: It allows managers to assess the effect of parameter changes efficiently without solving the LPP from scratch, provided the changes remain within allowable limits.

Q107. The allowable increase and allowable decrease refer to changes in:

A. Decision variable names

B. Model parameters while preserving the current optimal basis

C. Number of constraints only

D. Number of objective functions

Correct Answer: B

Short Explanation: Within the allowable range, the current optimal basis remains unchanged even though parameter values vary.

Q108. Which of the following is most closely associated with Sensitivity Analysis?

A. "What-if" Analysis

B. Network Analysis

C. Inventory Analysis

D. Queuing Analysis

Correct Answer: A

Short Explanation: Sensitivity Analysis is often called What-if Analysis because it answers questions such as, "What if the profit changes?" or "What if additional resources become available?"

Q109. If a change remains within the allowable range, then generally:

A. The current optimal basis remains unchanged.

B. The problem becomes infeasible.

C. The objective function disappears.

D. All decision variables become zero.

Correct Answer: A

Short Explanation: Small parameter changes within the allowable limits usually do not change the current optimal basis, although the objective value may change.

Q110. Sensitivity Analysis is most useful because real-world business conditions:

A. Never change

B. Frequently change after an optimal solution has been obtained

C. Are always uncertain in exactly the same way

D. Always satisfy all assumptions perfectly

Correct Answer: B

Short Explanation: Business environments are dynamic. Sensitivity Analysis helps determine whether the current optimal solution remains effective when profits, costs, or resources change.

Exam Tip: A common theory question is: "Why is Sensitivity Analysis important in Operations Research?" The key answer is decision-making under changing business conditions.

Section 11: Frequently Asked Conceptual & Exam-Oriented MCQs (111–120)

This section focus on the kinds of questions that students commonly find tricky in actual examinations while remaining fully aligned with standard OR textbooks. These are based on concepts that repeatedly appear in university papers and competitive exams. The wording is original, but the concepts are taken from the standard OR textbooks already cited.

Q111. Which of the following is NOT an application of Linear Programming?

A. Product Mix Problem

B. Production Planning

C. Media Selection

D. Random Sampling

Correct Answer: D

Short Explanation: Linear Programming is widely used for optimization problems such as production planning, transportation, blending, scheduling, media selection, and resource allocation. Random sampling belongs to statistics.

Exam Tip: Questions asking "Which is NOT an application of LPP?" appear very frequently.

Q112. Which of the following is essential for applying Linear Programming?

A. At least one objective function

B. At least one constraint

C. Linear relationships among variables

D. All of the above

Correct Answer: D

Short Explanation: A valid LPP requires an objective function, one or more constraints, and linear relationships between variables.

Q113. If all constraints of an LPP are satisfied but the objective function is not optimal, the solution is:

A. Optimal Solution

B. Basic Solution

C. Feasible Solution

D. Infeasible Solution

Correct Answer: C

Short Explanation: A solution satisfying all constraints is feasible. It becomes optimal only when it provides the best objective value.

Q114. Which of the following methods is NOT used to solve Linear Programming Problems?

A. Graphical Method

B. Simplex Method

C. Big M Method

D. Newton-Raphson Method

Correct Answer: D

Short Explanation: Newton-Raphson is a numerical method for solving equations, not a standard method for solving Linear Programming Problems.

Q115. The feasible region of a Linear Programming Problem is always:

A. Determined by the intersection of all constraints

B. Determined by the objective function only

C. Independent of constraints

D. Always unbounded

Correct Answer: A

Short Explanation: The feasible region is the common region satisfying all constraints and the non-negativity conditions.

Q116. The objective function is evaluated at the corner points because:

A. Every feasible point is optimal.

B. According to the Corner Point Theorem, an optimal solution (if it exists) occurs at a corner point.

C. The graphical method requires differentiation.

D. Corner points always have integer coordinates.

Correct Answer: B

Short Explanation: The Corner Point Theorem states that if an optimal solution exists, at least one corner point will be optimal.

Q117. Which statement about slack variables is correct?

A. They are added only to equality constraints.

B. They measure unused resources.

C. They represent excess production.

D. They are always removed before solving.

Correct Answer: B

Short Explanation: Slack variables indicate the unused portion of resources in "≤" constraints.

Q118. Which of the following statements about artificial variables is correct?

A. They have physical meaning in the original problem.

B. They become decision variables in the final solution.

C. They are temporary variables introduced only for computational purposes.

D. They are used in every Linear Programming Problem.

Correct Answer: C

Short Explanation: Artificial variables have no practical interpretation. They are introduced only to obtain an initial basic feasible solution.

Exam Tip: If an artificial variable remains positive in the final solution, the original LPP is infeasible.

Q119. In the Simplex Method, the entering variable is selected because it:

A. Improves the objective function.

B. Is always the largest variable.

C. Violates a constraint.

D. Is an artificial variable.

Correct Answer: A

Short Explanation: The entering variable is chosen based on the optimality criterion to improve the objective function value.

Q120. Which statement is TRUE?

A. Every optimal solution is feasible.

B. Every feasible solution is optimal.

C. Every infeasible solution is basic.

D. Every basic solution is optimal.

Correct Answer: A

Short Explanation: Optimality always requires feasibility. However, many feasible solutions may exist that are not optimal.

Exam Tip (Very High Frequency): Remember these relationships:

✔ Every Optimal Solution is Feasible.
✔ Every Basic Feasible Solution is Feasible.
✘ Every Feasible Solution is not necessarily Optimal.
✘ Every Basic Solution is not necessarily Feasible.

Section 12: Advanced Conceptual & Statement-Based MCQs (121–130)

This section includes higher-level conceptual MCQs that test understanding rather than simple memorization. These are common in MBA, Engineering, GATE, NET, and international university examinations.

Q121. Which of the following conditions is necessary for a Linear Programming Problem?

A. All variables must be integers.

B. Relationships among variables must be linear.

C. Every constraint must be an equality.

D. The objective function must always maximize profit.

Correct Answer: B

Short Explanation: The defining characteristic of an LPP is linearity. Decision variables may be fractional, constraints may be inequalities or equalities, and the objective may be maximization or minimization.

Exam Tip: Whenever you see Linear Programming, think first of linear objective function + linear constraints.

Q122. If the feasible region consists of only one point, then:

A. The problem is infeasible.

B. That point is the only feasible solution and therefore the optimal solution.

C. The problem is unbounded.

D. Multiple optimal solutions always exist.

Correct Answer: B

Short Explanation: If only one feasible point exists, it is automatically the optimal solution because there are no alternative feasible solutions.

Q123. Which statement about binding constraints is correct?

A. They have positive slack.

B. They are satisfied as equalities at the optimal solution.

C. They can be ignored during optimization.

D. They never affect the objective function.

Correct Answer: B

Short Explanation: A binding constraint is active at the optimal solution and directly limits the value of the objective function.

Q124. If a resource has unused capacity at the optimal solution, then the corresponding constraint is generally:

A. Binding

B. Non-binding

C. Artificial

D. Degenerate

Correct Answer: B

Short Explanation: Unused capacity means the slack is positive; therefore, the constraint is non-binding.

Q125. Which of the following statements is correct regarding slack variables?

A. Slack variables can take negative values.

B. Slack variables are added to ≤ constraints and represent unused resources.

C. Slack variables are used only in the graphical method.

D. Slack variables are decision variables of the original problem.

Correct Answer: B

Short Explanation: Slack variables are introduced during model conversion and represent unused resources. They are not original decision variables.

Q126. Which statement correctly distinguishes slack and surplus variables?

A. Both are added to ≤ constraints.

B. Slack is added to ≤ constraints, while surplus is subtracted from ≥ constraints.

C. Slack and surplus are both artificial variables.

D. There is no difference between them.

Correct Answer: B

Short Explanation: Slack variables convert ≤ constraints into equalities, whereas surplus variables convert ≥ constraints into equalities by subtraction.

Exam Tip: Remember the rule:

≤ → + Slack
≥ → − Surplus

Q127. Which of the following situations requires the introduction of an artificial variable?

A. Every ≤ constraint

B. Every non-negativity restriction

C. Equality constraints and most ≥ constraints when an initial basic feasible solution is not directly available

D. Every objective function

Correct Answer: C

Short Explanation: Artificial variables are introduced only when necessary to construct an initial basic feasible solution.

Q128. Which of the following best describes the movement of the Simplex Method?

A. Random movement through the feasible region

B. Movement from one basic feasible solution to another until optimality is reached

C. Direct movement to the center of the feasible region

D. Examination of every feasible point

Correct Answer: B

Short Explanation: The Simplex Method efficiently moves along the edges of the feasible region, visiting basic feasible solutions that improve the objective function.

Q129. Which statement regarding multiple optimal solutions is correct?

A. They occur only when the feasible region is empty.

B. They occur when more than one feasible solution yields the same optimal objective value.

C. They occur only in minimization problems.

D. They indicate that the problem is infeasible.

Correct Answer: B

Short Explanation: Multiple (alternate) optimal solutions exist when two or more feasible solutions produce the same best objective value.

Q130. Which of the following statements is TRUE?

A. Every Linear Programming Problem has a feasible solution.

B. Every feasible solution is unique.

C. Every optimal solution satisfies all constraints.

D. Every unbounded problem is infeasible.

Correct Answer: C

Short Explanation: An optimal solution must always satisfy all constraints and non-negativity restrictions. However, not every LPP has a feasible solution, and unboundedness is different from infeasibility.

Exam Tip (Very High Frequency): Never confuse:

Feasible → Constraints are satisfied.
Optimal → Feasible + Best objective value.
Infeasible → No feasible solution exists.
Unbounded → Feasible solutions exist, but no finite optimum exists.

Section 13: Application-Based & Exam-Trap MCQs (131–140)

In this section we cover application-based and exam-trap MCQs. These questions are commonly seen in university papers, engineering exams, and management entrance tests.

Q131. A factory manufactures two products using limited labor and raw materials. Which Operations Research technique is most appropriate to determine the production quantities that maximize profit?

A. Queuing Theory

B. Linear Programming

C. Game Theory

D. Markov Analysis

Correct Answer: B

Short Explanation: When the objective is to maximize profit or minimize cost under limited resources, Linear Programming is the appropriate optimization technique.

Exam Tip: Whenever you see limited resources + maximize/minimize, think Linear Programming.

Q132. A company wants to minimize transportation cost from several warehouses to different markets. Which OR technique is most suitable?

A. Inventory Model

B. Transportation Model

C. Replacement Model

D. Sequencing Model

Correct Answer: B

Short Explanation: Although based on Linear Programming, the Transportation Model is a specialized optimization technique designed specifically for transportation problems.

Q133. Which of the following real-life problems is least likely to be solved using Linear Programming?

A. Product mix optimization

B. Diet planning

C. Machine scheduling with linear constraints

D. Predicting tomorrow's stock market price

Correct Answer: D

Short Explanation: Linear Programming is an optimization technique, not a forecasting method.

Q134. If all coefficients in the objective function are multiplied by the same positive constant, the optimal solution generally:

A. Changes completely.

B. Remains the same, although the objective value is scaled.

C. Becomes infeasible.

D. Becomes unbounded.

Correct Answer: B

Short Explanation: Multiplying the objective function by a positive constant changes only the scale of the objective value, not the optimal decision variables.

Exam Tip: A positive scaling of the objective function does not change the optimal solution.

Q135. If one additional unit of a scarce resource increases the maximum profit by ₹25, then the shadow price of that resource is:

A. ₹0

B. ₹1

C. ₹25

D. Cannot be determined

Correct Answer: C

Short Explanation: The shadow price measures the increase in the optimal objective value resulting from one additional unit of a resource, within the allowable range.

Q136. Which statement correctly distinguishes a feasible solution from an optimal solution?

A. Both terms have the same meaning.

B. Every feasible solution gives the maximum profit.

C. Every optimal solution is feasible, but not every feasible solution is optimal.

D. Every feasible solution is basic.

Correct Answer: C

Short Explanation: Feasibility means satisfying all constraints; optimality means selecting the best feasible solution.

Q137. In a maximization problem, if all constraints are removed, the objective function is most likely to become:

A. Infeasible

B. Unbounded

C. Degenerate

D. Redundant

Correct Answer: B

Short Explanation: Without constraints, there is nothing to restrict the decision variables, so the objective function can often increase indefinitely.

Exam Tip: This question tests the conceptual difference between constraints and the objective function.

Q138. Which of the following situations most likely violates the certainty assumption of Linear Programming?

A. Machine hours are known exactly.

B. Raw material availability is fixed.

C. Profit per unit changes unpredictably every day.

D. Decision variables are non-negative.

Correct Answer: C

Short Explanation: The certainty assumption requires all coefficients in the model to be known and constant during the planning period.

Q139. Which statement about the objective function is correct?

A. It represents the limitations on resources.

B. It expresses the goal of optimization.

C. It defines the feasible region.

D. It always contains slack variables.

Correct Answer: B

Short Explanation: The objective function mathematically represents the goal, such as maximizing profit or minimizing cost.

Q140. Which of the following statements is FALSE?

A. Linear Programming is an optimization technique.

B. Constraints represent limitations.

C. The graphical method is suitable for problems with two decision variables.

D. Artificial variables represent actual physical resources.

Correct Answer: D

Short Explanation: Artificial variables are introduced only for computational convenience. They have no physical or economic meaning in the original problem.

Exam Tip (Very High Frequency): A favorite objective question is: "Which variable has no physical meaning?" ✔ Artificial Variable

Section 14: Assertion–Reason MCQs (141–150)

In this section we include Assertion–Reason (A–R) MCQs, a format frequently used in university examinations, engineering entrance tests, and management exams. These questions test conceptual understanding more deeply than standard MCQs.

Directions: Choose the correct answer using the following codes:

A. Both Assertion (A) and Reason (R) are true, and R is the correct explanation of A.

B. Both Assertion (A) and Reason (R) are true, but R is not the correct explanation of A.

C. Assertion (A) is true, but Reason (R) is false.

D. Assertion (A) is false, but Reason (R) is true.

Q141.

Assertion (A): Linear Programming is an optimization technique.

Reason (R): It determines the best feasible solution while satisfying linear constraints.

A. Both A and R are true, and R correctly explains A.

B. Both A and R are true, but R is not the correct explanation.

C. A is true, but R is false.

D. A is false, but R is true.

Correct Answer: A

Short Explanation: Linear Programming optimizes an objective function subject to linear constraints. The reason correctly explains why it is an optimization technique.

Q142.

Assertion (A): Every optimal solution is feasible.

Reason (R): An optimal solution must satisfy all constraints.

Correct Answer: A

Short Explanation: Optimality is meaningful only among feasible solutions. Therefore, every optimal solution must satisfy all constraints.

Exam Tip: One of the most repeated conceptual relationships in LPP.

Q143.

Assertion (A): Every feasible solution is optimal.

Reason (R): Every feasible solution satisfies all constraints.

Correct Answer: D

Short Explanation: The assertion is false because a feasible solution may not maximize or minimize the objective function. The reason is true because feasible solutions do satisfy all constraints.

Q144.

Assertion (A): Slack variables are added to ≤ constraints.

Reason (R): Slack variables represent unused resources.

Correct Answer: A

Short Explanation: Both statements are true, and the reason explains the purpose of adding slack variables.

Q145.

Assertion (A): Artificial variables have economic significance.

Reason (R): Artificial variables are introduced only to obtain an initial basic feasible solution.

Correct Answer: D

Short Explanation: Artificial variables do not have physical or economic meaning. The reason correctly states why they are introduced.

Q146.

Assertion (A): The graphical method is suitable for solving every Linear Programming Problem.

Reason (R): It is practical only for problems with two decision variables.

Correct Answer: D

Short Explanation: The assertion is false because the graphical method is generally limited to two decision variables. The reason is true.

Q147.

Assertion (A): The Simplex Method moves from one basic feasible solution to another.

Reason (R): Each iteration seeks to improve the objective function while maintaining feasibility.

Correct Answer: A

Short Explanation: The reason correctly explains the working principle of the Simplex Method.

Q148.

Assertion (A): A binding constraint has zero slack at the optimal solution.

Reason (R): A binding constraint is satisfied as an equality.

Correct Answer: A

Short Explanation: A binding constraint is active at the optimum; therefore, its slack is zero.

Q149.

Assertion (A): If a maximization problem is unbounded, a finite optimal solution does not exist.

Reason (R): The objective function can increase indefinitely without violating the constraints.

Correct Answer: A

Short Explanation: This is the definition of an unbounded maximization problem.

Q150.

Assertion (A): Every Linear Programming Problem has an optimal solution.

Reason (R): Some Linear Programming Problems may be infeasible or unbounded.

Correct Answer: D

Short Explanation: The assertion is false because an LPP may be infeasible or unbounded. The reason is true and explains why an optimal solution is not guaranteed.

Exam Tip (Very High Frequency): An LPP can have:

A unique optimal solution
Multiple optimal solutions
No feasible solution (infeasible)
No finite optimal solution (unbounded)

Section 15: Numerical Concept & Problem-Based MCQs (151–160)

In this section we include numerical concept-based MCQs. These are not lengthy calculations, but they test the concepts behind numerical problems. Such questions are extremely common in BBA, MBA, B.Com, Engineering, GATE, and university examinations.

Q151. Consider the following objective function:

Maximize

$$ Z = 8x_1 + 5x_2 $$

The numbers 8 and 5 are called:

Decision variables
Objective function coefficients
Constraint coefficients
Slack variables

Correct Answer: B. Objective function coefficients

Short Explanation:
The numerical values multiplying the decision variables in the objective function are called objective function coefficients.

Q152. Given the constraint

$$ 3x_1 + 2x_2 \le 18 $$

After converting it into an equality, the equation becomes:

$$3x_1 + 2x_2 - s_1 = 18$$
$$3x_1 + 2x_2 + s_1 = 18$$
$$3x_1 + 2x_2 + a_1 = 18$$
$$3x_1 + 2x_2 = 18$$

Correct Answer: B

Short Explanation:
A slack variable is added to every ≤ constraint to convert it into an equality.

Q153. Given the constraint

$$ 2x_1 + x_2 \ge 10 $$

Which variable is introduced first?

Slack Variable
Surplus Variable
Decision Variable
Dual Variable

Correct Answer: B

Short Explanation:
A surplus variable is subtracted from every ≥ constraint. An artificial variable is then added if required to obtain an initial basic feasible solution.

Exam Tip:
For ≥ constraints:
• Subtract Surplus Variable
• Add Artificial Variable

Q154. If the optimal solution of an LPP is

$$ x_1 = 5,\qquad x_2 = 3 $$

and both satisfy every constraint, then this solution is:

Infeasible
Feasible
Degenerate
Artificial

Correct Answer: B

Short Explanation:
A solution satisfying all constraints and non-negativity restrictions is a feasible solution.

Q155. A maximization problem has five corner points in its feasible region.

How many objective function values are normally evaluated using the Corner Point Method?

2
3
5
Depends on the number of constraints

Correct Answer: C

Short Explanation:
The Corner Point Method requires evaluating the objective function at every corner point.

Q156. If one corner point gives

$$ Z = 120 $$

and another gives

$$ Z = 145 $$

for a maximization problem, the better solution is:

120
145
Both are optimal
Cannot be determined

Correct Answer: B

Short Explanation:
For maximization problems, the largest objective value is selected.

Q157. Suppose a constraint has

$$ \text{Slack} = 0 $$

This means:

The constraint is non-binding.
The constraint is binding.
The problem is infeasible.
The solution is degenerate.

Correct Answer: B

Short Explanation:
Zero slack indicates that the entire available resource is utilized.

Exam Tip:
Zero Slack = Binding Constraint

Q158. Suppose a constraint has

$$ \text{Slack} = 8 $$

This indicates:

Eight units of the resource remain unused.
Eight constraints are violated.
Eight artificial variables are required.
The solution is infeasible.

Correct Answer: A

Short Explanation:
Slack measures the unused portion of available resources.

Q159. During the Simplex Method, the minimum ratio test is used to determine the:

Objective value
Entering variable
Leaving variable
Artificial variable

Correct Answer: C

Short Explanation:
The minimum ratio test identifies which basic variable leaves the basis while maintaining feasibility.

Q160. A manager receives one additional hour of machine time.

Sensitivity analysis shows a shadow price of ₹40.

This means:

Machine cost increases by ₹40.
Maximum profit increases by ₹40 (within the allowable range).
Production cost decreases by ₹40.
Slack increases by ₹40.

Correct Answer: B

Short Explanation:
The shadow price measures the improvement in the objective function resulting from one additional unit of a constrained resource, provided the change is within the allowable range.

Exam Tip:
Shadow Price = Marginal Value of a Resource

Section 16: Advanced Mixed Concept MCQs (161–170)

Q161. Which of the following is NOT a characteristic of a standard Linear Programming Problem?

A. Linear objective function

B. Linear constraints

C. Non-negativity restrictions

D. Exponential objective function

Correct Answer: D

Short Explanation: A standard Linear Programming Problem requires both the objective function and all constraints to be linear. An exponential objective function makes the problem nonlinear.

Exam Tip: Whenever the question mentions nonlinear, quadratic, or exponential, it is generally not a standard LPP.

Q162. A company wants to determine the best combination of products to manufacture using limited labor and machine hours. This is an example of:

A. Assignment Problem

B. Product Mix Problem

C. Sequencing Problem

D. Inventory Problem

Correct Answer: B

Short Explanation: The Product Mix Problem is one of the most common applications of Linear Programming, where limited resources are allocated among competing products.

Q163. Which of the following assumptions ensures that the total contribution of all decision variables equals the sum of their individual contributions?

A. Certainty

B. Divisibility

C. Additivity

D. Proportionality

Correct Answer: C

Short Explanation: The additivity assumption states that the total effect is simply the sum of the individual effects of each decision variable.

Q164. If an LPP has three decision variables, the graphical method is generally:

A. The preferred solution method

B. Not practical

C. Always more accurate than the Simplex Method

D. Applicable without modification

Correct Answer: B

Short Explanation: The graphical method is practical only for problems involving two decision variables. Problems with three or more variables are typically solved using the Simplex Method or computer software.

Exam Tip: Graphical Method → Maximum practical limit = Two decision variables.

Q165. The feasible region of a Linear Programming Problem is always:

A. Determined by the objective function

B. Determined by the intersection of all constraints

C. Independent of non-negativity restrictions

D. Always bounded

Correct Answer: B

Short Explanation: The feasible region is the common area satisfying every constraint and all non-negativity restrictions.

Q166. A constraint that does not affect the location of the optimal solution is generally called:

A. Binding Constraint

B. Non-binding Constraint

C. Artificial Constraint

D. Objective Constraint

Correct Answer: B

Short Explanation: A non-binding constraint has positive slack at the optimal solution and therefore does not determine the optimum.

Q167. Which of the following statements about the Simplex Method is correct?

A. It always examines every feasible solution.

B. It moves from one basic feasible solution to another.

C. It ignores the objective function.

D. It can solve only minimization problems.

Correct Answer: B

Short Explanation: The Simplex Method systematically moves from one basic feasible solution to another, improving the objective value until no further improvement is possible.

Q168. The dual variable associated with a resource constraint is commonly interpreted as:

A. Production quantity

B. Shadow Price

C. Slack Variable

D. Decision Variable

Correct Answer: B

Short Explanation: The dual variable measures the marginal value of an additional unit of the corresponding resource and is commonly known as the shadow price.

Q169. If every constraint in an LPP is satisfied and the objective function cannot be further improved, the solution is:

A. Infeasible

B. Degenerate

C. Optimal

D. Artificial

Correct Answer: C

Short Explanation: An optimal solution is the best feasible solution with respect to the objective function.

Q170. Which of the following statements is TRUE?

A. Every binding constraint has positive slack.

B. Every non-binding constraint has zero slack.

C. A binding constraint has zero slack at the optimal solution.

D. Slack variables may take negative values in a standard LPP.

Correct Answer: C

Short Explanation: A binding constraint is active at the optimal solution and therefore has zero slack. In a standard LPP, slack variables are non-negative.

Exam Tip (Frequently Asked):

Remember these relationships:

Binding Constraint → Slack = 0
Non-binding Constraint → Slack > 0
Shadow Price > 0 → Usually indicates a binding resource (subject to standard assumptions)
Shadow Price = 0 → Often indicates a non-binding constraint

Section 17: Advanced LPP Concepts & Previous Exam Pattern MCQs (171–180)

These are integrated concept questions that combine multiple LPP topics. Similar concepts frequently appear in university semester examinations and competitive tests.

Q171. Which of the following best describes the purpose of an objective function in Linear Programming?

A. To define the feasible region

B. To express the goal of optimization

C. To introduce slack variables

D. To determine the number of constraints

Correct Answer: B

Short Explanation: The objective function represents the goal of the decision-maker, such as maximizing profit or minimizing cost.

Q172. Which statement regarding decision variables is correct?

A. They represent unknown quantities whose values are determined by solving the model.

B. They always represent available resources.

C. They are introduced only during the Simplex Method.

D. They cannot take fractional values.

Correct Answer: A

Short Explanation: Decision variables are the unknown quantities that the Linear Programming model determines to optimize the objective function.

Exam Tip: Do not confuse decision variables with slack or artificial variables.

Q173. If all constraints are satisfied but one decision variable is negative, the solution is:

A. Optimal

B. Feasible

C. Infeasible

D. Degenerate

Correct Answer: C

Short Explanation: A feasible solution must satisfy both the constraints and the non-negativity restrictions.

Q174. Which of the following assumptions allows decision variables to take fractional values?

A. Certainty

B. Additivity

C. Divisibility

D. Proportionality

Correct Answer: C

Short Explanation: The divisibility assumption permits decision variables to have fractional values unless additional restrictions (such as integer programming) are imposed.

Q175. In a maximization problem, if increasing a resource does not improve the optimal objective value, the corresponding shadow price is generally:

A. Positive

B. Negative

C. Zero

D. Infinite

Correct Answer: C

Short Explanation: A shadow price of zero generally indicates that the resource is not fully utilized and is therefore non-binding.

Q176. Which statement about redundant constraints is correct?

A. They change the feasible region.

B. They do not affect the feasible region.

C. They always create infeasible solutions.

D. They always require artificial variables.

Correct Answer: B

Short Explanation: A redundant constraint is already implied by other constraints and therefore does not alter the feasible region.

Exam Tip: Redundant constraints increase model size but generally do not change the optimal solution.

Q177. The main advantage of the Simplex Method over the Graphical Method is that it:

A. Eliminates constraints.

B. Can efficiently solve problems involving many decision variables.

C. Does not require an objective function.

D. Guarantees integer solutions.

Correct Answer: B

Short Explanation: The Simplex Method is computationally efficient and suitable for solving Linear Programming Problems with many decision variables.

Q178. Which statement is correct regarding alternate (multiple) optimal solutions?

A. They indicate that the problem is infeasible.

B. They indicate that more than one optimal solution exists with the same objective value.

C. They occur only in minimization problems.

D. They imply degeneracy.

Correct Answer: B

Short Explanation: Alternate optimal solutions occur when two or more feasible solutions produce the same optimal objective value.

Q179. In Linear Programming, constraints primarily represent:

A. Business objectives

B. Resource limitations or requirements

C. Decision variables

D. Market demand forecasts

Correct Answer: B

Short Explanation: Constraints describe the limitations, requirements, or restrictions under which decisions must be made.

Q180. Which one of the following statements is correct?

A. Every feasible solution is optimal.

B. Every optimal solution satisfies all constraints.

C. Every Linear Programming Problem has a unique optimal solution.

D. Every binding constraint has positive slack.

Correct Answer: B

Short Explanation: An optimal solution must always satisfy all constraints and non-negativity restrictions. However, an LPP may have multiple optimal solutions, no feasible solution, or be unbounded.

Exam Tip (Very Important Revision):

Remember these fundamental facts:

✔ Optimal ⇒ Feasible
✔ Binding Constraint ⇒ Slack = 0
✔ Non-binding Constraint ⇒ Slack > 0
✔ Alternate Optimum ⇒ Same objective value at multiple feasible solutions
✔ Unbounded ≠ Infeasible

Section 18: Master Revision MCQs (181–200)

These questions are designed as a Master Revision Set. Rather than introducing new theory, they integrate multiple LPP concepts and target common exam mistakes. They are based on the same standard Operations Research textbooks already used throughout this collection.

Q181. Which combination is essential for a Linear Programming Problem?

A. Linear objective function, linear constraints, and non-negativity restrictions

B. Nonlinear objective function and equality constraints only

C. Integer decision variables only

D. Probability distributions

Correct Answer: A

Short Explanation: A standard LPP requires a linear objective function, linear constraints, and non-negativity restrictions.

Q182. The primary purpose of constraints in an LPP is to:

A. Maximize profit

B. Represent limitations or requirements

C. Introduce decision variables

D. Eliminate uncertainty

Correct Answer: B

Short Explanation: Constraints define the limitations (such as labor, material, or machine time) within which decisions must be made.

Q183. Which assumption states that all model coefficients are known with certainty?

A. Additivity

B. Divisibility

C. Certainty

D. Proportionality

Correct Answer: C

Short Explanation: The certainty assumption requires all coefficients in the objective function and constraints to remain known and constant during the planning period.

Q184. Which assumption allows production of 2.5 units if meaningful in the application?

A. Proportionality

B. Divisibility

C. Additivity

D. Certainty

Correct Answer: B

Short Explanation: The divisibility assumption allows fractional values for decision variables.

Q185. Which variable is added to a ≤ constraint to convert it into an equality?

A. Surplus Variable

B. Artificial Variable

C. Slack Variable

D. Dual Variable

Correct Answer: C

Short Explanation: Slack variables represent unused resources and convert ≤ constraints into equalities.

Q186. Which variable is subtracted from a ≥ constraint?

A. Slack Variable

B. Surplus Variable

C. Artificial Variable

D. Dummy Variable

Correct Answer: B

Short Explanation: A surplus variable is subtracted from a ≥ constraint before introducing an artificial variable if required.

Q187. The Graphical Method is generally suitable for:

A. Any number of decision variables

B. Two decision variables

C. Five decision variables

D. Ten decision variables

Correct Answer: B

Short Explanation: The Graphical Method is practical only for problems involving two decision variables.

Q188. The Simplex Method mainly improves the objective function by moving between:

A. Random points

B. Basic feasible solutions

C. Infeasible solutions

D. Midpoints of constraints

Correct Answer: B

Short Explanation: The Simplex Method progresses from one basic feasible solution to another until no further improvement is possible.

Q189. A binding constraint at the optimal solution has:

A. Positive slack

B. Zero slack

C. Negative slack

D. Infinite slack

Correct Answer: B

Short Explanation: A binding constraint is active at the optimum; therefore, no unused resource remains.

Q190. Which variable has no physical or economic meaning in the original problem?

A. Decision Variable

B. Slack Variable

C. Artificial Variable

D. Dual Variable

Correct Answer: C

Short Explanation: Artificial variables are introduced only for computational purposes and are not part of the original model.

Exam Tip: One of the most frequently asked conceptual questions.

Q191. If an LPP has no feasible solution, it is called:

A. Degenerate

B. Infeasible

C. Unbounded

D. Redundant

Correct Answer: B

Short Explanation: An infeasible problem has no solution satisfying all constraints simultaneously.

Q192. If the objective function can increase indefinitely without violating any constraint, the problem is:

A. Degenerate

B. Feasible

C. Unbounded

D. Alternate

Correct Answer: C

Short Explanation: An unbounded maximization problem has no finite optimal solution.

Q193. A shadow price measures:

A. Market selling price

B. Marginal value of an additional unit of a resource

C. Production cost

D. Inventory holding cost

Correct Answer: B

Short Explanation: The shadow price indicates how much the optimal objective value changes when one additional unit of a constrained resource becomes available (within the allowable range).

Q194. The Big M Method is primarily used when:

A. No constraints exist

B. Artificial variables are required

C. The graphical method fails

D. Integer variables are present

Correct Answer: B

Short Explanation: The Big M Method penalizes artificial variables so that they leave the basis if a feasible solution exists.

Q195. The Two-Phase Method begins by:

A. Solving the dual problem

B. Finding an initial feasible basis

C. Drawing the feasible region

D. Maximizing the original objective function

Correct Answer: B

Short Explanation: Phase I is used to obtain a feasible basic solution by eliminating artificial variables.

Q196. According to the Strong Duality Theorem, if both the primal and dual have optimal solutions:

A. Their objective values are equal.

B. Their variables are equal.

C. Their constraints are identical.

D. Both become infeasible.

Correct Answer: A

Short Explanation: Strong Duality states that the optimal objective values of the primal and dual are equal.

Q197. Which statement about Sensitivity Analysis is correct?

A. It is performed before formulating the LPP.

B. It studies the effect of changes in model parameters after obtaining an optimal solution.

C. It converts an LPP into a nonlinear model.

D. It replaces the Simplex Method.

Correct Answer: B

Short Explanation: Sensitivity Analysis examines the stability of the optimal solution when model parameters change.

Q198. Which of the following is NOT a standard assumption of Linear Programming?

A. Proportionality

B. Additivity

C. Certainty

D. Randomness

Correct Answer: D

Short Explanation: Randomness is not a standard assumption of classical Linear Programming.

Q199. Which statement is always true?

A. Every feasible solution is optimal.

B. Every optimal solution is feasible.

C. Every basic solution is optimal.

D. Every LPP has a unique solution.

Correct Answer: B

Short Explanation: Optimal solutions must satisfy all constraints, but many feasible solutions are not optimal.

Q200. Which statement best summarizes the purpose of Linear Programming?

A. To predict future demand

B. To optimize the use of limited resources subject to linear constraints

C. To eliminate business risk completely

D. To forecast market prices

Correct Answer: B

Short Explanation: Linear Programming provides a mathematical framework for allocating limited resources efficiently while satisfying linear constraints.

Exam Tip (Final Revision): Remember these five fundamentals:

✔ Objective Function → What to optimize
✔ Decision Variables → Unknown quantities to determine
✔ Constraints → Resource limitations/requirements
✔ Feasible Solution → Satisfies all constraints
✔ Optimal Solution → Best feasible solution

The above set of MCQs is a complete set of 200 original, textbook-aligned Linear Programming MCQs covering:

Fundamentals and assumptions
Mathematical formulation
Graphical Method
Feasible, basic, and optimal solutions
Slack, surplus, and artificial variables
Standard and canonical forms
Simplex Method
Big M Method
Two-Phase Method
Degeneracy, cycling, alternate optima, infeasibility, and unboundedness
Duality and Complementary Slackness
Sensitivity (Post-Optimality) Analysis
Application-based questions
Assertion–Reason questions
Numerical concept questions
Integrated master revision questions

This structure is suitable for BBA, B.Com, MBA, M.Com, Engineering, university semester exams, and many competitive examinations that include Linear Programming in their Operations Research syllabus.

About the Author
Lata Agarwal
Mathematics, Science and Astronomy professional, M.Sc. and M.Phil. in Maths with 10+ years of experience as Assistant Professor and Subject Matter Expert.
Author at Prinsli.com

References

These MCQs are based on concepts presented in Hillier & Lieberman, Hamdy A. Taha, Kanti Swarup, P.K. Gupta & Man Mohan, and S.D. Sharma, and

Hamdy A. Taha, Operations Research: An Introduction, Pearson.
Kanti Swarup, P. K. Gupta & Man Mohan, Operations Research, Sultan Chand & Sons.
J. K. Sharma, Operations Research: Theory and Applications, Macmillan India.
Frederick S. Hillier & Gerald J. Lieberman, Introduction to Operations Research, McGraw-Hill.
H. M. Wagner, Principles of Operations Research, Prentice Hall.