S​ 1​ : TRUE: The dual of the dual linear programming problem is again the primal problem
S​ 2​ : TRUE: If either the primal or the dual problem has an unbounded objective function value, the other problem has no feasible solution.
S​ 3​ : TRUE: If either the primal or dual problem has a finite optimal solution, the other one also possesses the same, and the optimal value of the objective functions of the two problems are equal.

Step1: In vogel’s approximation method first find out the row difference(by calculating the difference between the two smallest values of that row) and column difference(by calculating the difference between the two smallest values of that row)
Then select the row having highest row difference and after selecting row choose the column/cell of that row having minimum value.
After that fulfill the demand by using supply from the selected row and column.
After that if demand becomes zero then delete that column and if the row become zero then delete that row and then repeat step 1.

In the Hungarian method for solving assignment problem, an optimal assignment requires that the maximum number of lines that can be drawn through squares with zero opportunity cost is equal to the number of rows or columns.

The initial solution of a transportation problem is said to be non-degenerate basic feasible solution if it satisfies:
→The solution must be feasible, i.e. it must satisfy all the supply and demand constraints.
→The number of positive allocations must be equal to m+n-1, where m is the number of rows and n is the number of columns.
→All the positive allocations must be in independent positions
A few terms used in connection with transportation models are defined below.
1. Feasible solution: A feasible solution to a transportation problem is a set of non-negative allocations, xij that satisfies the rim (row and column) restrictions.
2. Basic feasible solution: A feasible solution to a transportation problem is said to be a basic feasible solution if it contains no more than m + n – 1 non – negative allocations, where m is the number of rows and n is the number of columns of the transportation problem.
3. Optimal solution: A feasible solution (not necessarily basic) that minimizes (maximizes) the transportation cost (profit) is called an optimal solution.
4. Non-degenerate basic feasible solution: A basic feasible solution to a (m x n) transportation problem is said to be non – degenerate if, the total number of non-negative allocations is exactly m + n – 1 (i.e., number of independent constraint equations), and these m + n – 1 allocations are in independent positions.
5. Degenerate basic feasible solution: A basic feasible solution in which the total number of non-negative allocations is less than m + n – 1 is called degenerate basic feasible solution.
Note: It is very standard and regular question and asked in UGC-NET Dec-2015 paper-3
Ref-https://solutionsadda.in/ugc-net-cs-2015-dec-paper-3/

Duality in LPP:
TRUE: The final simplex table giving optimal solution of the primal also contains optimal solution of its dual in itself.
TRUE: If either the primal or the dual problem has a finite optimal solution, then the other problem also has a finite optimal solution.
TRUE: If either problem has an unbounded optimal solution, then the other problem has no feasible solution at all.

The occurrence of degeneracy while solving a transportation problem means that the solution so obtained is not feasible.

The initial solution of a transportation problem is said to be non-degenerate basic feasible solution if it satisfies:
→The solution must be feasible, i.e. it must satisfy all the supply and demand constraints.
→The number of positive allocations must be equal to m+n-1, where m is the number of rows and n is the number of columns.
→All the positive allocations must be in independent positions.

Feasible Solution: The set of values of decision variables X_j (j = 1, 2, ... n) which satisfy all the constraints and non-negativity conditions of a linear programming problem simultaneously is said to constitute the feasible solution to that problem.
Basic Solution: For a set of m simultaneous equations in n variables (n > m), a solution obtained by setting (n – m) variables equal to zero and solving for remaining m variables is called a “basic feasible solution”.
Basic Feasible Solution: A feasible solution to LP problem which is also the basic solution is called the “basic feasible solution”. Basic feasible solutions are of two types;
(a) Degenerate: A basic feasible solution is called degenerate if value of at least one basic variable is zero.
(b) Non-degenerate: A basic feasible solution is called ‘non-degenerate’ if all values of m basic variables are non-zero and positive.
Hence the correct option is option(A)

→ The region of feasible solution of an LP problem has a convexity property in geometry, provided the feasible solution of the problem exists.

→ The basic feasible solutions for a standard form LP are vertices for the convex set of feasible solution for the LP.

→ If an LP has a finite optimal objective function value, then the optimal value occurs at at least one basic feasible solution.

Properties of convex optimization problems:
1. Every local minimum is a global minimum
2. The optimal set is convex
3. If the objective function is strictly convex, then the problem has at most one optimal point.
These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma.

→ Demand-Sensitive Pricing model is designed to bring prices down by increasing the number of customers who buy a particular product at once.
→ Demand sensitivity is referred to the change in demand for a product when its price is changed by a small amount.
Price Elasticity of demand = ((%change in quantity demanded) / (% change in price of a particular product))