1. True or false: If all the coefficients ,,…, in the objective function P = + + … +arenonpositive, then the only solution of the problem is = = … = and P =0.
Let us havefollowing equation.
-2x1-3x2-4x3 = 0
We see that onlysolution for above equation is x 1=0, x2=0 and x3 =0 as variable values have tobe ≥0. Hence, given statement is true.
2. Trueor false: The pivot column of a simplex tableau identifies the variable whosevalue is to be decreased in order to increase the value of the objectivefunction (or at least keep it unchanged).
The simplex method is an iterative process. Starting atsome initial feasible solution (a corner point – usually the origin), eachiteration moves to another corner point with an improved (or at least notworse) value of the objective function. Iteration stops when an optimal solution(if it exists) is found. The value of the variable can beincreaesd also. Hence, given statement is false.
3. Trueor false: The ratio associated with the pivot row tells us by how much thevariable associated with the pivot column can be increased while thecorresponding point still lies in the feasible set.
This statement is true as we consider the lowest ratio for further iterations.
4. Trueor false: At any iteration of the simplex procedure, if it is not possible tocompute the ratios or the ratios are negative, then one can conclude that thelinear programming problem has no solution.
A linear programming will have no solution if the simplex method breaks down at somestage. For example, at some stage, there are no non-negative ratios in ourcomputations, and then the linear programming problem has no solution. Hence,statement is true.
5. Trueor false: If the last row to the left of the vertical line of the final simplextableau has a zero in a column that is not a unit column, then the linear programmingproblem has infinitely many solutions.
A linear programming problem will have infinitely manysolutions if, for example, the last row to the left of the vertical line of thefinal simplex tableau has a zero in a column that is not a unit column, or ifthe final tableau contains two or more indentical unit columns. Hence, givenstatement is true.
6. Trueor false: Suppose you are given a linear programming problem satisfying theconditions-
· Theobjective function is to be minimized.
· All thevariables involved are nonnegative, and
· Eachlinear constraint may be written so that the expression involving the variablesis greater than or equal to a negative constant.
Then the problem can be solved using thesimplex method to maximize the objective function P = -C.
These are the requirements of a linear programming problem to be solvedby simplex method. Hence, it is true.
7. Trueor false: The objective function of the primal problem can attain an optimalvalue that is different from the optimal value attained by the dual problem.
Primal and dual have samesolution; hence, given statement is false.