An individual wishes to invest $5000 over the next year in two types of investment: investment A yields 5% and investment B yields 8%. market research recommends an allocation of at least 25% in A and at most 50% in B. Moreover, investment in A should be at least half the investment in B. Formulate this as a linear programming problem.
Solve the following LP problem using Simplex method
Maximizez=3x1+2x2+x3Maximize \quad z = 3x_1 + 2x_2 + x_3Maximizez=3x1+2x2+x3 Subject tox1+x2+x3≤9Subject\ to \quad x_1 + x_2 + x_3 \le 9Subject tox1+x2+x3≤9 2x1+3x2+5x3≤302x_1 + 3x_2 + 5x_3 \le 302x1+3x2+5x3≤30 2x1−x2−x3≤82x_1 - x_2 - x_3 \le 82x1−x2−x3≤8 x1,x2,x3≥0x_1, x_2, x_3 \ge 0x1,x2,x3≥0
Solve the LP model using Big-M
Minimizez=2x1+3x2Minimize \quad z = 2x_1 + 3x_2Minimizez=2x1+3x2 Subject tox1+x2≥6Subject\ to \quad x_1 + x_2 \ge 6Subject tox1+x2≥6 7x1+x2≥147x_1 + x_2 \ge 147x1+x2≥14 x1,x2≥0x_1, x_2 \ge 0x1,x2≥0
Solve the following LPP using dual simplex method:
Minimizez=2x1+4x2Minimize \quad z = 2x_1 + 4x_2Minimizez=2x1+4x2 Subject to2x1+x2≥4Subject\ to \quad 2x_1 + x_2 \ge 4Subject to2x1+x2≥4 x1+2x2≥3x_1 + 2x_2 \ge 3x1+2x2≥3 2x1+2x2≤122x_1 + 2x_2 \le 122x1+2x2≤12 x1,x2≥0x_1, x_2 \ge 0x1,x2≥0
What is operations research? Why operations research is important in the field of Mechanical Engineering?
Alumco manufacturers Aluminum sheets and Aluminum bars. The maximum production capacity is estimated at either 800 sheets or 600 bars per day. The maximum daily demand is 550 sheets and 560 bars. The profit per ton is $40 per sheet and $35 per bar. Formulate the problem as a linear programming problem.
Use graphical method to solve the following problem
Maximizez=2x1+3x2Maximize \quad z = 2x_1 + 3x_2Maximizez=2x1+3x2
Subject tox1+x2≤1Subject \ to \quad x_1 + x_2 \le 1Subject tox1+x2≤1
3x1+x2≤43x_1 + x_2 \le 43x1+x2≤4
x1,x2≥0x_1, x_2 \ge 0x1,x2≥0
Solve the following LP problem using simplex method Minimize z=10x1+15x2+20x3z = 10x_1 + 15x_2 + 20x_3z=10x1+15x2+20x3 Subject to 2x1+4x2+6x3≤242x_1 + 4x_2 + 6x_3 \leq 242x1+4x2+6x3≤24 3x1+9x2+6x3≤303x_1 + 9x_2 + 6x_3 \leq 303x1+9x2+6x3≤30 x1,x2,x3≥0x_1, x_2, x_3 \geq 0x1,x2,x3≥0
Solve the following transportation problem using Northwest corner cell method and Least cost cell method.
Solve the LPP using Big-m method Minimize z=10x1+15x2+20x3z = 10x_1 + 15x_2 + 20x_3z=10x1+15x2+20x3 Subject to 2x1+4x2+6x3≥242x_1 + 4x_2 + 6x_3 \geq 242x1+4x2+6x3≥24 3x1+9x2+6x3≥303x_1 + 9x_2 + 6x_3 \geq 303x1+9x2+6x3≥30 x1,x2,x3≥0x_1, x_2, x_3 \geq 0x1,x2,x3≥0
Form the dual of the following primal problem Minimize z=20x1+40x2z = 20x_1 + 40x_2z=20x1+40x2 Subject to 2x1+20x2=102x_1 + 20x_2 = 102x1+20x2=10 20x1+3x2=2020x_1 + 3x_2 = 2020x1+3x2=20 4x1+15x2=304x_1 + 15x_2 = 304x1+15x2=30 x1 and x2≥0x_1 \text{ and } x_2 \geq 0x1 and x2≥0
Use duality to solve the following LP problem Minimize z=2x1+x2z = 2x_1 + x_2z=2x1+x2 Subject to x1+2x2≤10x_1 + 2x_2 \leq 10x1+2x2≤10 x1+x2≤6x_1 + x_2 \leq 6x1+x2≤6 x1−x2≤2x_1 - x_2 \leq 2x1−x2≤2 x1−2x2≤1x_1 - 2x_2 \leq 1x1−2x2≤1 x1 and x2≥0x_1 \text{ and } x_2 \geq 0x1 and x2≥0