Linear Programming

Step 1: Understanding the Concept:
This is a linear programming problem (LPP). The goal is to find the maximum value of a linear objective function given a set of linear inequality constraints, which define a feasible region. The maximum value of will occur at one of the vertices (corner points) of the feasible region.

Step 2: Key Formula or Approach:
1. Graph the inequalities to identify the feasible region. 2. Determine the coordinates of the vertices of this region. 3. Evaluate the objective function at each vertex. 4. The largest value of found is the maximum value.

Step 3: Detailed Explanation:
The constraints are: 1. (First quadrant) 2. (Region below the line ) 3. (Region below the line ) Let's find the vertices of the feasible region:
Vertex 1 (Origin): Intersection of and , which is .
Vertex 2 (x-intercept): Intersection of and . This gives . The point is . (Check other constraint: , which is true).
Vertex 3 (y-intercept): Intersection of and . This gives . The point is . (Check other constraint: , which is true).
Vertex 4 (Intersection of lines): Intersection of and . Subtracting the first equation from the second: . Substitute into : . The point is . The vertices of the feasible region are (0,0), (1.5, 0), (0,2), and (1,1). Now, evaluate at each vertex:
At (0,0):
At (1.5, 0):
At (0,2):
At (1,1): The values of Z are 0, 3, 5, and 6. The maximum value is 6.
Step 4: Final Answer:
The maximum value of Z is 6, which occurs at the point (0,2).

Step 1: Understanding the Concept:
A redundant constraint is a constraint in a linear programming problem that does not affect the feasible region. In other words, if we remove a redundant constraint, the set of feasible solutions remains the same. We can identify redundant constraints by checking if they are automatically satisfied whenever the other constraints are met.

Step 2: Detailed Explanation:
Let's analyze the given constraints: 1. 2. 3. 4. 5. Let's check for redundancy: - Consider constraint . If is satisfied, then is certainly greater than 0. So, is redundant. - Consider constraint . If is satisfied, then is certainly greater than 0. So, is redundant. - Consider constraint . Let's see if the other "stronger" constraints and imply this one. If and , then the smallest value the expression can take is when and are at their minimums: Since , any point that satisfies and will automatically satisfy , which means it will definitely satisfy . Therefore, the constraint is made redundant by the constraints and . The set of non-redundant constraints is just and . The constraints , , and are all redundant. The question asks for "the redundant constraints" from the options. Let's evaluate the options: (A) : is a defining constraint of the final feasible region, not redundant. (B) : As shown above, this constraint is redundant. (C) : and are not redundant. (D) : is not redundant. The question is slightly ambiguous. The full set of redundant constraints is . Option (B) lists one of these. It is the best choice among the given options.

Step 3: Final Answer:
The constraints and are made redundant by and . The constraint is also made redundant by and . Therefore, is a redundant constraint.

Step 1: Understanding the Concept:
This question tests fundamental theorems and definitions related to Linear Programming Problems (LPP). We need to evaluate the correctness of four key statements.

Step 2: Detailed Explanation:
Let's analyze each statement:
A. There exists only finite number of basic feasible solutions to LPP.
A basic solution is found by choosing basic variables from total variables, where is the number of constraints. The maximum number of ways to do this is , which is a finite number. Since the number of basic solutions is finite, the number of basic feasible solutions (which is a subset of the basic solutions) must also be finite. This statement is true.
B. Any convex combination of k - different optimum solution to a LPP is again an optimum solution to the problem.
The set of all optimal solutions to an LPP is a convex set. By definition of a convex set, any convex combination of points within the set is also in the set. Therefore, any convex combination of optimal solutions is also an optimal solution. This statement is true.
C. If a LPP has feasible solution, then it also has a basic feasible solution.
This is a statement of the Fundamental Theorem of Linear Programming. It guarantees that if a feasible solution exists (and the feasible region is non-empty), then at least one vertex (a basic feasible solution) must exist. This statement is true.
D. A basic solution to AX = b is degenerate if one or more of the basic variables vanish.
This is the precise definition of a degenerate basic solution. In a non-degenerate basic solution, all basic variables are strictly positive. If at least one of these basic variables is zero, the solution is called degenerate. This statement is true.
Step 3: Final Answer:
All four statements A, B, C, and D are correct principles in the theory of Linear Programming.