Minimize [tex]z=6 x_1+8 x_2[/tex]
subject to
[tex]
2 x_1+3 x_2 \geq 7
[/tex]
[tex]
4 x_1+5 x_2 \geq 9
[/tex]
[tex]x_1, x_2 \geq 0[/tex]
Solve by the simplex method.
Answer (1)
Convert inequalities to equalities using surplus and artificial variables.
Apply the simplex method to find the optimal values of x 1 and x 2 .
Substitute the optimal values into the objective function to find the minimum value of z.
The minimum value of the objective function is 3 56 .
Explanation
Problem Setup We are given a linear programming problem and asked to solve it using the simplex method. The problem is to minimize the objective function z = 6 x 1 + 8 x 2 subject to the constraints 2 x 1 + 3 x 2 ≥ 7 , 4 x 1 + 5 x 2 ≥ 9 , and x 1 , x 2 ≥ 0 .
Converting to Standard Form To solve this using the simplex method, we first convert the inequalities to equalities by introducing surplus variables s 1 and s 2 . This gives us 2 x 1 + 3 x 2 − s 1 = 7 and 4 x 1 + 5 x 2 − s 2 = 9 . We also introduce artificial variables a 1 and a 2 to obtain an initial basic feasible solution: 2 x 1 + 3 x 2 − s 1 + a 1 = 7 and 4 x 1 + 5 x 2 − s 2 + a 2 = 9 . The objective function becomes z = 6 x 1 + 8 x 2 + M a 1 + M a 2 , where M is a large positive number.
Simplex Method Solution After performing the simplex method (which involves setting up the initial tableau, selecting entering and leaving variables, and performing row operations), we arrive at the optimal solution. The result of this process is: x 1 = 0 and x 2 = 3 7 .
Calculating the Minimum Value Substituting these values into the objective function, we get: z = 6 ( 0 ) + 8 ( 3 7 ) = 3 56 = 18.666... .
Final Answer Therefore, the minimum value of the objective function is 3 56 .
Examples
Linear programming is used in many real-world applications, such as optimizing resource allocation, production planning, and transportation logistics. For example, a company might use linear programming to determine the optimal production levels for different products to maximize profit, subject to constraints on available resources such as labor, materials, and equipment. Another application is in transportation, where linear programming can be used to find the most efficient routes for delivering goods, minimizing transportation costs and delivery times. These optimization techniques are crucial for businesses to make informed decisions and improve their efficiency and profitability.
