A Markov chain is irreducible if the state space S is a single communicating class. If a Markov chain has more than one class, it is called reducible.

Pivot number is 2 in this tableau.

n = 7 ; m = 4 ; x = 7 – 4 = 3 . pg. 87. Correct answer is E.

Some of the properties of duality contribute solving of the primal problem by considering primal-dual model relationships. Before getting into these, recall that x’s are the components of a feasible solution for the primal model and, whereas the corresponding dual solution comprises of y’s. For a basic feasible solution of the primal model, the objective value, Z is equal or greater than W, the objective value of the corresponding dual solution. This is called weak duality property: the researcher begins at any point, then reaches the optimum at a level by lowering the costs and thus increasing the revenues simultaneously. In this sense the correct answer is E.

The value at the intersection of the pivotal column and row is called the pivotal number. It is alternatively called as pivot number or pivot element 

The correct answer is D. 

-3 p + 1 (1 – p) = -4 p + 1 . pg. 164. Correct answer is B.

In symmetric games, all players have the same actions and even in case of interchanging players, the actions of players remain the same. The other words,
the actions in a symmetric game depend on the strategies used, not on the players of the game.

A linear program can be related to another one reflecting the opposite of itself. These two programs have the same conditions (cost and constraint coefficients), yet they are modeled from counter-viewpoints. Consequently, they reach the same optimal solution mutually. Here, the inversed pair of the original model is called the dual of the original (or primal) model. The dual model interchanges costs and constraints of the primal model, and converts the objective from maximization to minimization.

A slack or surplus is utilized to reconstruct a constraint to obtain equation format. A slack variable represents the remainder of an expendable resource and applies to less than or equal constraints. 

The correct answer is A.

A basic solution—whether it is feasible or not—is a corner point of the solution space. The corner points of system of equations are obtained by setting n – m number of variables equal to zero and solving the equations for the remaining m number of variables.

The correct answer is D.

The solution of a transportation model follows the steps that the simplex method offers:

1. Determine a basic feasible solution to initiate,
2. Check the optimality of the basic feasible solution as the simplex method does. If it is not, optimal, then move to the next step. If optimal, stop.
3. Iterate to the next basic feasible solution by determining the entering and leaving variable using the rate of improvement on the objective value. Return to step 2.

(7 !) / ((4 !) ((7 – 4) !)) = (7 6 5) / (3 2) = 35 . pg. 87. Correct answer is A.

In a degenerate basic feasible solution, a degenerate basic variable acts as a non-basic variable, however, it does not prevent reaching to the optimal.

The assignment model has a tailored solution method such as the transportation model does. The solution method of the assignment model is called the Hungarian Method, named by the nationality of its developers. So, the correct answer is option B.

greatest in columns : -1 1 -2 2 -5 ; min : -5 . pg. 158 . Correct answer is E.

The assignment model has a tailored solution method such as the transportation model does. The solution method of the assignment model is called the Hungarian Method, named by the nationality of its developers. The steps of Hungarian Method is given below. 1. Identify the smallest value of each row for the cost matrix of the assignment problem. Subtract each row’s smallest value from all the costs in the respective row. 2. Identify the smallest value of each column for this altered matrix. Subtract each column’s smallest value from all the costs in the respective column. 3. Mask the columns and rows out that have a zero value. The number of masked out rows and columns must be at a minimum. 4. If the number of masked out rows and columns is equal to n, then the optimum can be obtained from the present matrix; move on to the next step. If not, skip to Step 6. 5. Identify the optimal solution by the coordinates of the zero-valued elements in the present matrix. 6. Identify the smallest value except for the ones in masked out rows and columns. This value is then subtracted from the values of unmasked rows and columns and, added to the intersections of masked out rows and columns. Return to Step 3. The correct answer is D.

then the following explanation is obtained 

and

Because of the

the result is