Consider the LP with n variables and 2n linear constraints, henceforth de- noted CUBER: max S = xn -10x1 10x1 ≤ 9 -1 - xi 10x+1 ≤ 0 i = 1,...,n-1 Ꮖ 1, . xi+10x+1 ≤ 10 i = 1,..., n − 1 ,...,xn≥0 a) (No written answer to this question is required.) Let n = 4. Introduce slack variables w₁,..., wg and an artificial variable xo representing con- straint tolerance. Apply the online pivot tool https://vanderbei.princeton.edu/JAVA/pivot/simple.html in order to find a feasible solution to CUBE by means of Phase 1 of the Sim- plex algorithm. Verify that the dictionary after Phase 1 (the column corresponding to x deleted and restored in row 0) is: 10000 + 10000 + 1000 W3 + 110 5 + 17 1 S = 1 x1 = - 10 աշ = x2 = - 100 W4 = 95 49 - 1 x3 - = 1000 W6 = 499 50 - 1 = W8 = 10000 4999 500 - - 100 15 -1 1000 -1 W1 W1 w₁ W1 10000 W1 1 500 W₁ - W3 - - - - - 15 1 1000 W3 W3 w3 W3 W3 - - - - ws W5 TW5 100 W5 15 w5 - - W7 W7

Consider the LP with n variables and 2n linear constraints, henceforth de- noted CUBER: max S = xn -10x1 10x1 ≤ 9 -1 - xi 10x+1 ≤ 0 i = 1,...,n-1 Ꮖ 1, . xi+10x+1 ≤ 10 i = 1,..., n − 1 ,...,xn≥0 a) (No written answer to this question is required.) Let n = 4. Introduce slack variables w₁,..., wg and an artificial variable xo representing con- straint tolerance. Apply the online pivot tool https://vanderbei.princeton.edu/JAVA/pivot/simple.html in order to find a feasible solution to CUBE by means of Phase 1 of the Sim- plex algorithm. Verify that the dictionary after Phase 1 (the column corresponding to x deleted and restored in row 0) is: 10000 + 10000 + 1000 W3 + 110 5 + 17 1 S = 1 x1 = - 10 աշ = x2 = - 100 W4 = 95 49 - 1 x3 - = 1000 W6 = 499 50 - 1 = W8 = 10000 4999 500 - - 100 15 -1 1000 -1 W1 W1 w₁ W1 10000 W1 1 500 W₁ - W3 - - - - - 15 1 1000 W3 W3 w3 W3 W3 - - - - ws W5 TW5 100 W5 15 w5 - - W7 W7