2. For the following linear program maximize x18x23x3 subject to 2x1 +8x2+2x3 ≤ 4, 2x14x2 3x3 ≤ 4, x12x2 x3 ≤ 1, x1, x2, x30 determine whether x = (0, ½, 0) is an optimal solution using the principle of com- plementary slackness.

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Solve the linear programming problem in the attached image (excel solver or simplex tableau). Here x3 is unrestricted, so we have to take: x3=(x’3-x”3), also introduce slack and artificial variables.

Q6. Consider the following linear program in standard maximization form: -5 max{r1 – x2} subject to -4 X1, 12 a) Use the Simplex Method (as seen in class) to solve the above linear program. Be sure to: o Indicate all slack and artificial variables you introduce. o Indicate the objective S of the extended tableau. o Whenever your tableau is in RRREF, indicate the current solution's values for all vari- ables. o Before pivoting, indicate the entry you plan to convert to the next pivot. o Clearly indicate why you should stop the algorithm and give a precise conclusion. b) Draw the feasible region in a fairly large and well-scaled diagram. Determine the coordi- nates of each vertex and label the vertices of the feasible region in your diagram. c) Based on the graphical approach, describe the optimal solutions of this LP (if any). VI VI AI

Convert the following linear program to standard form: Maximize z = 3x₁ + 5x2 - 4x3 ubject to 7x1 - 2x2 - 3x3 ≥4 - 2x1 + 4x2 + 8x3 = -3 5x13x2 - 2x3 ≤9 x1 ≥ 1, X₂ ≤7, X3 ≥ 0.