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For the following LP Maximize Z = 3x1 + 7x2 + 5x3 subject to: x1 + x2 + x3 ≤ 50 2x1 + 3x2 + x3 ≤ 100 x1 , x2 , x3 ≥ 0. The final simplex tableau is given below, where s1 and s2 are the slack variables for constraint 1 and 2. The optimal solution given by this tableau is: x1 = 0, x2 = 25, x3 = 25, s1 = 0, s2 = 0, Z = 300. Basic variable | Eq. | Coefficient of: Z x1 x2 x3 s1 s2 | Right Side Z | (0) | 1 3 0 0 4 1 | 300 x3 | (1) | 0 1/2 0 1 3/2 -1/2 | 25 x2 | (2) | 0 1/2 1 0 -1/2 1/2 | 25 Using the information from the optimal simplex tableau: a. Determine the range of c1 (coefficient of x1 in the objective function) within which current basis remains optimal. b. Determine the range of c2 (coefficient of x2 in the objective function) within which current basis remains optimal. c. Determine the range of b1 (right hand side of the first constraint) within which current basis remains optimal. d. If the RHS of the first constraint (b1) is changed from 50 to 60, find out the new objective function value for Z from the information given in the optimal simplex tableau (without re-solving).
Sri K.
The following are simplex tables corresponding to a particular iteration of the simplex method for a maximization problem. For this problem assume that xi and si are decision and slack variables that are constrained to be non-negative and Ri is an artificial variable added for the two-phase method. Interpret these tables to comment on: i) whether the simplex method has converged, if so, state the solution. ii) whether there is any inconsistency in the table iii) whether the optimal solution is unbounded, unique (i.e., there are no alternative optima), non-existent or degenerate. If none of them is true, say so. a) [10pt] Table 1 b) [10pt] Table 2 c) [10pt] Table 3
Problem 2. (20%) Consider the following linear program: min y1,y2,y3,y4 9y1 + 3y2 + 3y3 + y4 subject to y1 ≥ 0, y2 ≥ 0, y3 ≥ 0, y4 ≥ 0 -3y1 - 2y2 + y3 - y4 ≤ -2, -y1 - y2 - y3 ≤ -1, (b) (5%) Write down the augmented form of the above LP. Also write down the corresponding basic feasible (BF) solution to (y1, y2, y3, y4) = (0, 1, 0, 0). (notice that we should be maximizing an objective in the augmented form) (c) (10%) Show that (y1, y2, y3, y4) = (0, 1, 0, 0) is an optimal solution to the LP using the simplex method (with the tableau form). Precisely, identify a basis that is associated with the BF solution you have derived in part (b), then apply row operations to the simplex table such that it is valid (i.e., containing only 1 basic variable per row, etc.). Depending on the basis you have selected initially, you may need to run the simplex method for one or two iterations before the optimality test is passed. Optimality can then be guaranteed by observing the signs of the coefficients in the 0th row.
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