Maximize ƒ(x) = 5x1₁ − x² + 8x2 − 2x²2 3x₁ + 2x₂ ≤ 6 x1 ≥ 0, subject to and x₂ > 0 x2
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- 2. Apply the first phase of the 2-phase simplex algorithm to the following linear pro- gramme giving the initial tableau and each further tableau produced. Give the starting tableau for the second phase if there is one. maximize 2x1 x23x3 subject to X2 - x3 ≤ 2, x13x2+2x3 ≥ 3, 2x12x2 x3 = 4, x1, x2, x3 0.. Use the simplex algorithm to solve the following linear optimisation problem: maximise f(x1, x2, x3) = 3x1 + x2+ 2x3 subject to: Зх1 + 2л2 + з 0.Use Simplex Algorithm to determine the optimal solution of this LP problem. Maximize z = 2x1 − x2 + 2x3subject to:2x1 + x2 ≤ 10x1 + 2x2 − 2x3 ≤ 20x2 + 2x3 ≤ 5x1, x2, x3 ≥ 0
- 4. Consider the following LP: max z = 6x1 + x2 X1 + x2 5 5 2x, + x2 < 6 st. X1,x2 2 0 (a) Solve this problem using the Simplex Algorithm and obtain the optimal tableau for this problem. (b) If we add a new variable x3 such that: max z = 6x1 + x2 + x3 X1 + x2 + ax3 S 5 2x, + x2 + bx356 X1, X2, X3 20 where a and b are scalar values, define the conditions under which the current basis stay st. optimal.Suppose that we are carrying out the simplex algorithm on a linear program in standard inequality form (with 3 variables and 4 constraints) and suppose that we have reached a point where we have obtained the following tableau. Apply one more pivot operation, indicating the highlighted row and column and the row operations you carry out. What can you conclude from your updated tableau? I1 I2 23 S1 S2 $3 SA $1 -2 0 1 1 0 0 0 3 82 3 0 -2 0 1 2 0 6 12 1 1 -3 0 0 1 0 2 SA -3 0 2 0 0 -1 1 4 -2 -2 0 11 0 0 -4 0-8Apply the Simplex Algorithm using Tucker tableau to the following LP problem. Use x₁ the first entering variable. Maximize: f(x₁, X2, X3, X4) = 5X1 + 6x2 + 9x3 + 8x4 ubject to x₁ + 2x2 + 3x3 + x4 ≤ 5 x1 + x2 + 2x3 + 3x4≤ 3 X1, X2, X3, X4 ≥0
- Use the simplex algorithm to solve the following linear optimisation problem: maximise f(x1, x2, x3) = 3x1+ x2 + 2x3 subject to: 3x1 + 2x2 + x3 0. You must address each of the 5 steps of the algorithm as presented in the course notes and videos in your answer.Use the simplex algorithm to solve the following linear optimisation problem: maximisef(x1,x2,x3)=3x1+x2+2x3 subjectto: 3x1 +2x2 +x3 ≤8 x1 +2x2 +2x3 ≤4, x1,x2,x3 ≥0. You must address each of the 5 steps of the algorithm as presented in the course notes and videos in your answer.Solve the following problem using the tableau implementation of the simplex algorithm. minimize - 6x - 4x2 + 2x3 subject to x, + x2 + 4x3 < 20 - 5x2 + 5x3 < 100 X1 + 3x2 + x3 s 400 *1 20 *3 20
- Solve the following linear programs (LPs) using the simplex algorithm. • For each tableau you obtain, write the associated BFS and its objective value. • For each LP, find the complete optimal solution set and write it using appropriate mathematical notation. Use only the information in the tableaux to justify your answer. minimize z(x1, x2) subject to -x1 X2 + x2 0A workshop that produces two types of chemicals, Article (1) needs 2 km of material B and 3 working hours, and Article (2) provides 4 kg of B and 5 hours, knowing that the owner of 30 units of raw material B in that workshop employs 7 workers. The profit of Article (1) is 3 dinars per unit, and the profit of the second chemical is 2 dinars per unit The problem is required to be represented in a linear programming mod if you know that the number of working hours per worker is 8 hours ?