Consider the following linear programming problem (P) in standard form: (P) min{c"x : Ax = b, x > 0}, where A E R"mxn, b E R™, and c e R". Let ī e R" be a vertex of the feasible region of (P). (4.2) Let d e R" be a feasible direction at ī such that d # 0. Prove that -d e R" is not a feasible direction at . (Hint: Consider the effect of the hypothesis d # 0 on the subvector dN, where BC {1,.,n} and N C {1,.., n} denote the corresponding index sets of the vertex .)

c) A student wishes to allocate her available study time of 60 hours per week between two subjects in such a way as to maximize her grade average. She formulates two functions governing the grades of each module which can be taken as: 9₁ (t₁) = 20 + 20√₁, with t₁ > 0 and 9₂ (t₂) = -80 + 3t2, with t₂ > 80/3 maximize the grade average subject to the time constraint t₁ + t₂ = 60. You do not need to show that you have obtained a maximum.

1. Write the following linear program into standard form, then write the linear program corresponding to the first phase simplex: min x1,X2 ,X3ER 2x1 + x2 + X3 X1 – x2 + 2x3 < (1) subject to x1 + x2 – 3 -5 X1, X2, X3 T 2. Determine whether x = (0 2 0 1 1)ʻ is a vertex of max x1 + 2x2 – X4 + x5 X1,X2,X3,x4,X5ER xi + x2 -x2 + 3x3 + x4 -x2 + x3 – x4 + 4x5 (2) -1 subject to 1 X1; X2; X3, X4, X5 3. Determine whether x= (0 3 0 9 2)' is a solution to max 3x2 + 2x5 x1,x2,x3,x4,X5ER X1 + x2 – x5 X2 + x3 + x5 -2x2 + x4 + x5 (3) 5 subject to 5 X1, X2, X3, X4, X5 || || || A VI A