All questions of Linear Programming for Mechanical Engineering Exam

Explanation:

Objective Function:
- Max z = 40x1 + 60x2

Constraints:
1. 2x + y ≤ 90
2. x + y ≤ 50
3. x + 2y ≤ 80
4. x, y ≥ 0

Analysis:
- To find the maximum value of z, we need to optimize the objective function while satisfying all the given constraints.
- The given constraints form a feasible region in the x-y plane.
- By solving the system of inequalities, we can identify the feasible region and determine the corner points.

Corner Points:
- A: (0, 0)
- B: (40, 0)
- C: (20, 30)
- D: (0, 40)

Optimal Solution:
- By evaluating the objective function at each corner point, we find that the maximum value of z (zmax) occurs at point C (20, 30).
- Substitute x = 20 and y = 30 into the objective function: zmax = 40(20) + 60(30) = 2400
Therefore, the correct answer is 2400.

When one of the constraint is parallel to objective function i.e., if slopes are equal then the LPP may contain multiple optimal solutions.

Hungarian method is used in assignment problem (but remember assignment problem is a special case of transportation problem).

Shadow price in Linear Programming

Shadow price in linear programming refers to the value assigned to one unit of a particular resource or constraint. It is the change in the objective function value per unit increase in the availability of a resource or constraint.

Significance of Shadow Price

The shadow price provides valuable information about the problem and helps in decision-making. It indicates the opportunity cost of using a resource or constraint. It helps in determining the economic feasibility of the problem and identifying the critical resources.

Calculation of Shadow Price

The shadow price can be calculated using the dual simplex method or sensitivity analysis. In the dual simplex method, the shadow price is the negative of the reduced cost of the corresponding dual variable. In sensitivity analysis, the shadow price is the change in the objective function value due to a unit increase in the availability of a resource or constraint.

Interpretation of Shadow Price

A positive shadow price indicates that an increase in the availability of a resource or constraint will increase the objective function value. A negative shadow price indicates that a decrease in the availability of a resource or constraint will increase the objective function value. A zero shadow price indicates that the resource or constraint is not binding.

Conclusion

In conclusion, the shadow price in linear programming refers to the value assigned to one unit of a particular resource or constraint. It provides valuable information about the problem and helps in decision-making. The shadow price can be calculated using the dual simplex method or sensitivity analysis.

Linear Programming Model Development Process

The correct order of steps followed for the development of a linear programming model is 4, 1, 2, 3. Let's discuss each step in detail:

Step 4: Develop Inequations (or Equations) for the Constraints
- This step involves formulating the constraints that restrict the feasible region of the problem.
- Constraints are the limitations or restrictions that must be adhered to in order to find the optimal solution.
- Constraints can be represented as inequalities (inequations) or equalities (equations).
- These constraints can include limitations on resources, capacity, demand, time, etc.

Step 1: State the Problem in the Form of a Linear Programming Model
- In this step, the problem is transformed into a mathematical model based on linear relationships.
- The decision variables, objective function, and constraints are defined.
- The problem is analyzed to identify the goal or objective that needs to be maximized or minimized.

Step 2: Determine the Decision Variables
- Decision variables are the unknown quantities that need to be optimized in the problem.
- They represent the choices or decisions that the decision-maker can make.
- Decision variables are usually denoted by symbols and can be continuous or discrete.

Step 3: Write the Objective Function
- The objective function represents the goal or objective that needs to be optimized.
- It is a mathematical representation of the quantity that needs to be maximized or minimized.
- The objective function is typically a linear equation, although it can also be nonlinear in some cases.

Correct Order: 4, 1, 2, 3
- The correct order of steps for the development of a linear programming model is 4, 1, 2, 3.
- First, we need to formulate the constraints (inequations or equations) that define the limitations of the problem.
- Then, we state the problem in the form of a linear programming model by defining the objective function and decision variables.
- Finally, we determine the decision variables and write the objective function based on the problem requirements.

Summary:
The correct order for the development of a linear programming model is 4, 1, 2, 3. Constraints are formulated first, followed by stating the problem in the form of a linear programming model, determining the decision variables, and writing the objective function. This sequential process allows for the systematic formulation and optimization of the problem.

Explanation:
Supply and Demand Discrepancy in Transportation Problems
Supply and demand in a transportation problem are represented by rows (supply points) and columns (demand points) respectively. When the total supply is less than the total demand, it creates a discrepancy that needs to be addressed.

Adding a Dummy Row
- When the total supply is less than the total demand in a transportation problem, a dummy row should be added.
- This dummy row helps to balance out the shortage in supply by introducing additional supply points to meet the demand requirements.
- By adding a dummy row, the transportation problem can be solved effectively, ensuring that all demand points are met.

Importance of Adding a Dummy Row
- Adding a dummy row helps in maintaining the balance between supply and demand in the transportation problem.
- It ensures that all demand points are satisfied, even when there is a shortage of supply.
- The dummy row acts as a placeholder for additional supply points, helping to optimize the transportation process.

Conclusion
In conclusion, when the total supply is less than the total demand in a transportation problem, adding a dummy row is the most appropriate solution. This approach ensures that the transportation problem can be solved efficiently, meeting all demand requirements and maintaining a balance between supply and demand.

Simplex Method Constraints
- Statement I is true, as the Simplex method can handle only ≤ type constraints. This is because the method is designed to work with linear programming problems where the constraints are in the form of inequalities.

Number of Decision Variables
- Statement II is false. The Simplex method can be applied to problems with any number of decision variables, not just when the number is greater than or equal to 3. The method is a powerful tool for solving linear programming problems regardless of the number of variables involved.

Objective Function Type
- Statement III is false. The Simplex method can be applied to both maximization and minimization problems. It is not limited to maximization problems only. The method can optimize the objective function by moving through feasible solutions to reach the optimal solution.
Therefore, the correct statements are I and II. The Simplex method is a widely used algorithm for solving linear programming problems with constraints in the form of ≤, =, or ≥ and can handle problems with any number of decision variables. Additionally, it can optimize both maximization and minimization objective functions.

Solution:

When solving a two-variable linear programming problem by graphical method, if one of the constraint lines is parallel to the objective function line, then the problem will have an infinite number of optimal solutions.

Explanation:

To understand why this is the case, let's consider the following example of a two-variable linear programming problem:

Maximize Z = 3x + 2y

Subject to:

2x + y ≤ 10

3x + y ≥ 12

x, y ≥ 0

We can graph the two constraint lines and the objective function line on the same coordinate plane as shown below:


As we can see, the constraint line 3x + y = 12 is parallel to the objective function line Z = 3x + 2y. This means that any point on the constraint line will have the same objective function value of Z = 12.

Since the feasible region of the problem is bounded (i.e., it is a polygon), there must be at least one corner point that is optimal. However, any corner point that lies on the constraint line 3x + y = 12 will be optimal since it has the same objective function value as any other point on that line.

Therefore, the problem has an infinite number of optimal solutions, all of which lie on the constraint line 3x + y = 12.

Conclusion:

When one of the constraint lines is parallel to the objective function line in a two-variable linear programming problem solved by graphical method, the problem will have an infinite number of optimal solutions.

The correct answer is option 'D': The simplex algorithm is an iterative procedure.

The simplex method is a widely used algorithm for solving linear programming problems. It was developed by George Dantzig in the late 1940s and has since become a fundamental tool in optimization theory and operations research.

Iterative Procedure:
The simplex algorithm is an iterative procedure, meaning it applies a set of steps repeatedly until a solution is found or a specified stopping criterion is met. It starts with an initial feasible solution and then iteratively improves the solution until an optimal solution is reached.

Steps of the Simplex Method:
1. Formulate the linear programming problem in standard form, which involves writing the objective function and the constraints as a system of linear equations or inequalities.
2. Identify an initial feasible solution by introducing slack or surplus variables to convert the inequalities into equations.
3. Choose an entering variable, which is a non-basic variable that can be increased to improve the objective function value.
4. Choose a leaving variable, which is a basic variable that can be decreased to maintain feasibility.
5. Update the basic and non-basic variables using a pivot operation, which essentially swaps the entering and leaving variables.
6. Repeat steps 3-5 until an optimal solution is reached, i.e., no non-basic variable can be increased to further improve the objective function value.
7. The final solution will have all non-basic variables set to zero, and the values of the basic variables represent the optimal solution.

Advantages of the Simplex Method:
- The simplex algorithm is efficient and can solve linear programming problems with a large number of variables and constraints.
- It guarantees convergence to an optimal solution if one exists.
- It can handle problems with both equality and inequality constraints.
- It provides sensitivity analysis, allowing decision-makers to understand how changes in the problem parameters affect the optimal solution.

Limitations of the Simplex Method:
- The simplex method may not terminate if the problem is unbounded, meaning there is no upper limit on the objective function value.
- It may also fail to converge if the problem is degenerate, meaning there are multiple optimal solutions.
- The simplex method can become computationally expensive for problems with a large number of variables and constraints.

In conclusion, the simplex algorithm is an iterative procedure used to solve linear programming problems. It is a powerful tool for optimization and can handle problems with both equality and inequality constraints. However, it may have limitations in terms of termination and computational complexity for certain problem types.

Linear programming is a mathematical technique used to determine the best possible utilization of limited resources to achieve a specific objective. It is widely used in various fields including business, economics, engineering, and manufacturing.



In linear programming, a set of mathematical equations are formulated to represent the constraints and objectives of the problem. The objective function is then optimized subject to these constraints to find the best possible solution. This allows businesses to make informed decisions on how to allocate resources efficiently.



For example, a manufacturing company may use linear programming to determine the optimal production mix of different products given limited resources such as labor, materials, and equipment. By solving the linear programming model, the company can maximize its profits or minimize costs while meeting demand and capacity constraints.



Linear programming helps businesses make better decisions by providing a systematic approach to resource allocation. It allows companies to analyze various scenarios and choose the most cost-effective and efficient solution. This can lead to increased productivity, profitability, and overall performance.

Overall, linear programming is a powerful tool for optimizing resources and making informed decisions in a wide range of applications. By utilizing this mathematical technique, businesses can achieve their goals while making the most effective use of limited resources.

Linear Programming in Problem Solving
Linear programming is a mathematical method used to determine the best possible outcome in a given mathematical model for a set of constraints. It is widely used in various fields to optimize resource allocation and decision-making processes. Two common types of problems that linear programming can solve are assignment problems and transportation problems.

Assignment Problems
Assignment problems involve assigning a set of tasks to a set of resources in the most efficient way possible. Linear programming can be used to optimize this assignment process by minimizing costs or maximizing efficiency. This can be applied in various scenarios such as job assignments, project scheduling, or machine assignments in manufacturing.

Transportation Problems
Transportation problems involve determining the most cost-effective way to transport goods from a set of sources to a set of destinations. Linear programming can be used to optimize transportation routes, minimize transportation costs, and maximize the efficiency of the transportation network. This is commonly used in logistics, supply chain management, and distribution planning.

Both Assignment and Transportation Problems
Linear programming can be used to solve both assignment problems and transportation problems efficiently. By formulating these problems as linear programming models, optimal solutions can be found to allocate resources, assign tasks, or plan transportation routes effectively. This makes linear programming a versatile tool for solving a wide range of optimization problems in different industries.
In conclusion, linear programming can be effectively used to solve both assignment problems and transportation problems by formulating them as mathematical models and applying optimization techniques to find the best possible solutions.

There are certain implications when one of the decision variables in a Linear Programming Problem (LPP) is zero. Let's break down the answer in detail:

When a decision variable is zero in an LPP, it may lead to a degenerate solution. A degenerate solution occurs when the redundant constraint is fully consumed, meaning that the constraint involving the zero decision variable no longer plays a role in determining the feasible region. This can result in the degeneracy of the solution.

The redundant constraint is the one that does not affect the feasible region when removed. When a decision variable is zero, it implies that the corresponding constraint is fully utilized and no longer contributes to defining the feasible region. This can lead to degeneracy in the solution.

Degenerate solutions can complicate the optimization process as they may require additional iterations to reach the optimal solution. They can also affect the efficiency of the algorithm used to solve the LPP. Therefore, it is essential to be aware of the presence of degenerate solutions in order to address them effectively.
In conclusion, when a decision variable is zero in an LPP, it can lead to a degenerate solution if the redundant constraint is fully consumed. Understanding the implications of degeneracy is crucial in optimizing the solution to linear programming problems.

Basic Feasible Solution in Linear Programming

Basic feasible solutions are important concepts in linear programming as they help in finding optimal solutions to optimization problems. Here's an explanation of what constitutes a basic feasible solution:

Constraints and Non-negativity Restrictions

- A basic feasible solution must satisfy all the constraints of the linear programming problem. This means that the solution must adhere to all the limitations and requirements set by the problem.
- In addition to satisfying the constraints, a basic feasible solution must also satisfy the non-negativity restrictions. This means that all decision variables in the solution must be greater than or equal to zero.

Optimizing the Objective Function

- While a basic feasible solution does not necessarily optimize the objective function, it provides a starting point for finding the optimal solution. By satisfying all constraints and non-negativity restrictions, a basic feasible solution narrows down the search space for the optimal solution.
- The optimal solution is found by further refining the basic feasible solution through techniques such as the simplex method or graphical method. These methods iteratively improve the solution until the optimal values of the decision variables are obtained.

In conclusion, a basic feasible solution in linear programming satisfies both constraints and non-negativity restrictions, setting the foundation for finding the optimal solution to the optimization problem.

Explanation:

Simplex procedure is an iterative method used to solve linear programming problems. In each iteration of the simplex procedure, a basic feasible solution (BFS) is obtained by setting non-basic variables to zero and solving for basic variables. The BFS is then used to determine the direction of movement towards the optimal solution.

When there are multiple BFS that provide the same optimal solution, the problem is said to be degenerate. In such cases, the simplex procedure may encounter tie situations where two or more non-basic variables have the same reduced cost. This means that the objective function value remains the same as we increase or decrease the value of any of these variables.

Tie situations in the simplex procedure can lead to cycling, where the algorithm keeps revisiting the same BFS without making any progress towards the optimal solution. This can result in an infinite loop, making it impossible to find the optimal solution.

However, leaving variables in the basis during the simplex procedure can help to break ties and prevent cycling. By leaving non-basic variables in the basis, we can introduce more constraints and reduce the number of possible BFS. This can help to ensure that the simplex procedure converges to the optimal solution without cycling.

Therefore, tie situations in the simplex procedure do not necessarily imply optimality, cycling, or no solution. Instead, they indicate degeneracy, which can be resolved by leaving variables in the basis.

Explanation:

Transportation problems:
- In transportation problems, the nonsquare matrix is made square by adding a dummy row or a dummy column.
- This process is known as matrix balancing, where dummy rows or columns are added to make the matrix square.
- The addition of dummy rows or columns ensures that the total supply is equal to the total demand, making it easier to solve the transportation problem.

Correct statement:
- In transportation problems, the nonsquare matrix is made square by adding a dummy row or a dummy column.
Therefore, the statement "In transportation problems, the nonsquare matrix is made square by adding a dummy row or a dummy column" is correct, and not incorrect as mentioned in the question.

Explanation:

Number of Basic Variables in Simplex Method
In the simplex method, the number of basic variables is equal to the number of constraints in the linear programming problem. This is because each constraint in the problem corresponds to a basic variable in the simplex method.

Basic Variables vs. Non-Basic Variables
- Basic variables are the variables that are chosen as the basic variables at each iteration of the simplex method.
- Non-basic variables are the remaining variables in the problem that are not chosen as basic variables.

Role of Basic Variables
- Basic variables play a crucial role in the simplex method as they form the basis of the solution.
- The values of the basic variables determine the values of the non-basic variables through the constraints of the linear programming problem.

Number of Decision Variables vs. Number of Constraints
- The number of decision variables in a linear programming problem is not necessarily equal to the number of basic variables in the simplex method.
- The number of constraints, on the other hand, directly corresponds to the number of basic variables in the simplex method.
Therefore, the correct answer to the question is option 'B', as the number of basic variables in the simplex method is equal to the number of constraints in the linear programming problem.

Explanation:

Given Linear Programming Problem (LPP):
- Objective function: Zmin = 2A + 3B
- Constraints:
- 0.6A + 0.3B ≥ 45
- 0.1A + 0.5B ≥ 25
- A, B ≥ 0

Analysis:
- To find the correct statement, we need to analyze the coefficients of A and B in the objective function and constraints.

Objective Function:
- Coefficient of A: 2
- Coefficient of B: 3

First Constraint:
- Coefficients of A and B are 0.6 and 0.3 respectively.
- To maintain the feasibility of the constraint, an increase in A should be accompanied by a greater increase in B.

Second Constraint:
- Coefficients of A and B are 0.1 and 0.5 respectively.
- To maintain the feasibility of the constraint, an increase in A should be accompanied by a greater increase in B.

Conclusion:
- Since the coefficients of B are greater in both constraints as well as the objective function, a decrease in A will lead to an increase in B.
- Therefore, the correct statement is: Decrease of 3 units in A will cause increase of 2 units in B.

Number of Possible Ways of Making Assignments in an Assignment Problem

An assignment problem is a special type of linear programming problem where we have n facilities and n jobs to be assigned to these facilities. The objective is to minimize the total cost of assignments while ensuring that each facility is assigned to only one job and each job is assigned to only one facility. In such a problem, the number of possible ways of making assignments can be calculated as follows:

1. Understanding the Problem Requirements

To calculate the number of possible ways of making assignments, we need to first understand the problem requirements. In an assignment problem, we have n facilities and n jobs, and we need to assign each job to one facility and each facility to one job. Therefore, we need to find the total number of ways in which we can make such assignments.

2. Using the Fundamental Principle of Counting

The fundamental principle of counting states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together. Using this principle, we can calculate the number of possible ways of making assignments in an assignment problem.

Since we have n facilities and n jobs, there are n ways to assign the first job to a facility. Once the first job is assigned to a facility, there are n - 1 ways to assign the second job to a facility (since we cannot assign it to the same facility as the first job). Similarly, there are n - 2 ways to assign the third job to a facility (since we cannot assign it to the same facility as the first two jobs), and so on.

Therefore, the total number of possible ways of making assignments is:

n x (n - 1) x (n - 2) x ... x 2 x 1 = n!

This means that there are n! (n factorial) possible ways of making assignments in an assignment problem, where n is the number of facilities and jobs.

3. Choosing the Correct Option

Out of the given options, option A (n!) is the correct answer since it represents the total number of possible ways of making assignments in an assignment problem with n facilities and n jobs. The other options (n^2, 2n, 2^n) do not provide the correct answer since they do not consider the fundamental principle of counting and the specific requirements of an assignment problem.

Feasible Region in LP Model

The feasible region is a term used in linear programming to describe the set of all possible solutions to an optimization problem. When solving a linear programming problem graphically, the feasible region is the area bounded by the constraints.

Features of Feasible Region:

1. It is a convex polygon.
2. Every point inside the feasible region satisfies all the constraints.
3. The objective function is optimized at one of the corners of the feasible region.
4. The feasible region can be either bounded or unbounded.

Infeasible Region and Unbounded Solution

An infeasible region is a region that does not contain any feasible solutions. It is the region outside the feasible region, where at least one of the constraints is violated.

An unbounded solution is a solution that can be increased or decreased without limit. It occurs when the objective function can be improved indefinitely in a particular direction. In other words, the feasible region extends infinitely in one or more directions.

Conclusion

In conclusion, the feasible region is the area bounded by the constraints in a linear programming problem. It contains all possible solutions that satisfy the constraints. An infeasible region is a region that does not contain any feasible solutions, while an unbounded solution occurs when the feasible region extends infinitely in one or more directions.

Explanation:

Dual of the LP problem:
- The dual of the LP problem involves switching the roles of variables and constraints.
- In this case, since the primal problem has 6 variables and 2 constraints, the dual will have 2 variables and 6 constraints.

Correspondence to the dual:
- Option 1 correctly corresponds to the dual with 2 variables and 6 constraints.
- Option 4 corresponds to the minimization of the objective function in the dual problem.
Therefore, the correct answer is option B which includes both 1 and 4.

Objective Function in Linear Programming:
The objective function in a linear programming problem represents the quantity that needs to be maximized or minimized. It is typically denoted by 'z' and is a linear combination of decision variables with coefficients representing the contribution of each variable to the overall objective.

Correct Expression for Objective Function:
The correct expression for an objective function in a Linear Programming Problem is typically a linear combination of decision variables. In this case, the correct expression given is:

z = 3x1 + 10x2
Here, 'z' represents the objective function to be optimized, 'x1' and 'x2' are the decision variables, and 3 and 10 are the coefficients of these variables respectively. This expression indicates that the objective function is a linear combination of 'x1' and 'x2' with specific coefficients attached to each variable.

Explanation:
- The objective function in a Linear Programming Problem is crucial as it defines the goal of the optimization process.
- In the given expression, 'z = 3x1 + 10x2', the coefficients 3 and 10 indicate the contribution or weight of each decision variable towards the objective.
- By maximizing or minimizing this objective function subject to certain constraints, optimal values of the decision variables 'x1' and 'x2' can be determined.
- The objective function plays a key role in guiding the optimization algorithm towards finding the best solution that satisfies all constraints while optimizing the objective.
In conclusion, the correct expression for an objective function in a Linear Programming Problem is a linear combination of decision variables, with specific coefficients representing the contribution of each variable towards the overall objective.

Simplex Method in Linear Programming

Iterative Solution
- The simplex method is an iterative procedure where the steps are repeated until an optimal solution is reached.
- At each iteration, the algorithm moves from one basic feasible solution to another, improving the objective function value each time.

Constants of Constraints Equation
- The constants of the constraints equation may be positive or negative, as the simplex method can handle both types of coefficients.

Inequalities to Equations
- In the simplex method, inequalities are typically converted into equations by introducing slack, surplus, or artificial variables.
- This transformation allows the algorithm to work with equations instead of inequalities.

Two-Variable Problems
- The simplex method can be used for problems with any number of variables, including two-variable problems.
- It is a powerful tool for solving linear programming problems with multiple decision variables.

Conclusion
- The simplex algorithm is an iterative procedure that can handle both positive and negative constants in constraints equations, convert inequalities into equations, and be used for problems with any number of variables.

Corner Points

- The simplex method of solving linear programming problems involves determining the optimal solution by moving from one corner point (or vertex) of the feasible region to another.
- The corner points are the intersection points of the constraint lines that define the feasible region.

Feasible Region

- The feasible region is the area in which all constraints of the linear programming problem are satisfied.
- The corner points represent the extreme points of the feasible region.

Optimal Solution

- By evaluating the objective function at each corner point, the simplex method allows us to find the optimal solution.
- The optimal solution is the corner point that maximizes or minimizes the objective function, depending on the problem.

Advantages of Using Corner Points

- The simplex method is efficient because it only considers a finite number of corner points instead of all the points in the feasible region.
- By focusing on the corner points, the simplex method simplifies the problem-solving process and reduces computational complexity.

Therefore, the correct answer is option 'b) only the corner points of the feasible region', as the simplex method specifically utilizes these corner points to find the optimal solution for linear programming problems.

Linear Programming Conditions for Optimization
1. Objective function as a linear function of variables: In linear programming, the objective function is to be maximized or minimized, and it must be expressed as a linear combination of decision variables. This means that the coefficients of the variables are constants and the variables appear in a linear form.
2. Resources are unlimited: Linear programming assumes that resources such as raw materials, labor, and budget are unlimited. This simplifies the optimization process as there are no constraints on the availability of resources.
3. Inter-related and non-negative decision variables: The decision variables in linear programming are inter-related, meaning that they are connected in a linear manner. Additionally, these variables must be non-negative, which means they cannot take on negative values in the optimization process.
Therefore, the correct statement is option 'D' - 1 and 3 only, as linear programming involves an objective function expressed linearly and non-negative inter-related decision variables, while assuming unlimited resources.

**Correct Statement:**
The correct statement is option 'C': In the standard form of a linear programming problem, all constraints are in the form of equations.

**Explanation:**
Linear programming is a mathematical optimization technique used to determine the best possible outcome in a given mathematical model, given a set of constraints. The standard form of a linear programming problem involves maximizing or minimizing a linear objective function, subject to a set of linear constraints.

In the standard form, the objective function and all constraints must be in the form of equations rather than inequalities. This is because the objective function represents the quantity to be maximized or minimized, and the constraints represent the limitations or restrictions on the variables.

**Reason for Incorrect Options:**

a) Option 'A': Of less than or equal to type
This option is incorrect because linear programming problems may involve constraints of both less than or equal to and greater than or equal to type. However, in the standard form, all constraints are represented as equations.

b) Option 'B': Of greater or equal to type
This option is also incorrect for the same reason as option 'A'. Linear programming problems can have constraints of both types, but in the standard form, they are all represented as equations.

d) Option 'D': Some constraints are of less than equal to type and some are of greater than equal to type
This option is incorrect because, as mentioned earlier, in the standard form, all constraints are represented as equations and not as inequalities of any type.

**Conclusion:**
In the standard form of a linear programming problem, all constraints are represented as equations. This is because the standard form requires the objective function and constraints to be in the form of equations to ensure consistency and compatibility within the mathematical model.

Removing Infeasibility in LP Problem

In linear programming (LP) problems, infeasibility means that there is no feasible solution that satisfies all the constraints. Infeasibility can occur due to various reasons such as conflicting constraints, inconsistent data, or incorrect formulation of the problem. In such cases, the LP problem cannot be solved using standard optimization techniques.

To remove infeasibility, one of the following approaches can be taken:

1. Removing a Constraint:

Removing a constraint can make the problem feasible by relaxing the system of constraints. This approach is usually taken when the infeasibility is caused by a single constraint that is too restrictive. By removing this constraint, the feasible region is expanded, and a solution may become feasible. However, this approach should be taken with caution as removing a constraint may change the nature of the problem and the optimal solution.

2. Adding a Variable:

Adding a variable can also make the problem feasible by introducing more degrees of freedom into the system. This approach is usually taken when the infeasibility is caused by the lack of variables to satisfy all the constraints. By adding a variable, the feasible region is expanded, and a solution may become feasible. However, this approach should be taken with caution as adding a variable may increase the complexity of the problem and the computation time.

3. Adding a Constraint:

Adding a constraint can make the problem feasible by restricting the feasible region. This approach is usually taken when the infeasibility is caused by the lack of constraints to satisfy all the variables. By adding a constraint, the feasible region is reduced, and a solution may become feasible. However, this approach should be taken with caution as adding a constraint may make the problem more restrictive and reduce the feasible region further.

Conclusion:

In conclusion, to remove infeasibility in LP problems, one can remove a constraint, add a variable, or add a constraint. However, each approach should be taken with caution as it may change the nature of the problem and the optimal solution.