Answer:
z=10
Explanation:
To write the given linear program in standard form, we need to convert the inequalities into equalities and introduce slack variables.
The original linear program:
Maximize: Z = x1 + 2x2
Subject to:
x1 + 2x2 ≤ 10
2x1 + 6x2 ≤ 16
x1, x2 ≥ 0
To convert the inequalities into equalities, we introduce slack variables:
Let s1 and s2 be the slack variables for the first and second constraints, respectively.
The linear program in standard form becomes:
Maximize: Z = x1 + 2x2
Subject to:
x1 + 2x2 + s1 = 10
2x1 + 6x2 + s2 = 16
x1, x2, s1, s2 ≥ 0
1.1 In a basic solution, the number of variables set equal to zero is equal to the number of constraints. In this case, there are 2 constraints, so 2 variables will be set equal to zero in a basic solution.
1.2 To find all the basic solutions, we need to set the slack variables (s1 and s2) equal to zero and solve the system of equations.
Setting s1 = 0 and s2 = 0, we have:
x1 + 2x2 = 10
2x1 + 6x2 = 16
Solving these equations, we can use row operations to eliminate x1:
Row 2 - 2 * Row 1:
0x1 + 2x2 = 16 - 2 * 10
2x2 = -4
x2 = -2
Substituting x2 = -2 into the first equation:
x1 + 2(-2) = 10
x1 - 4 = 10
x1 = 14
So, the basic solution is (x1, x2) = (14, -2).
1.3 To find the optimal solution, we need to compute the value of each basic feasible solution.
Substituting the basic solution (14, -2) into the objective function:
Z = 14 + 2(-2)
Z = 14 - 4
Z = 10
Therefore, the optimal solution is Z = 10.
Note: This is just the first iteration. If there are more variables and constraints, the process may involve multiple iterations using the simplex method to find the optimal solution