P = 2x + 4y
subject to constraints:
x + 4y <= 12
x + 3y <=10
x >= 0, y >= 0
question: is this already in standard form or not?
Thank you, can you explain why it is or not?
it is
Could you explain to me why
thanks
To determine if the given problem is already in standard form or not, we need to understand what the standard form of a linear programming problem looks like.
In standard form, a linear programming problem is typically written as:
Minimize or Maximize: Z = c₁x₁ + c₂x₂ + ... + cnxn
Subject to constraints:
a₁₁x₁ + a₁₂x₂ + ... + a₁nxn ≤ b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂nxn ≤ b₂
.
.
.
am₁x₁ + am₂x₂ + ... + amnxn ≤ bm
Where:
- Z is the objective function (either minimize or maximize)
- c₁, c₂, ..., cn are the coefficients of the variables x₁, x₂, ..., xn in the objective function
- x₁, x₂, ..., xn are the decision variables
- a₁₁, a₁₂, ..., a₁n; a₂₁, a₂₂, ..., a₂n; ...; am₁, am₂, ..., amn are the coefficients of the variables x₁, x₂, ..., xn in the constraints
- b₁, b₂, ..., bm are the right-hand side values of the constraints
- m represents the number of constraints
Now, let's look at the given problem:
P = 2x + 4y
subject to constraints:
x + 4y ≤ 12
x + 3y ≤ 10
x ≥ 0, y ≥ 0
Since the objective function is given in the standard form (Z = c₁x₁ + c₂x₂), the only remaining aspect to consider is whether the inequalities and non-negativity constraints are in the proper form.
In this case, the inequalities x + 4y ≤ 12 and x + 3y ≤ 10 are already in the correct form, where the left-hand side is less than or equal to the right-hand side.
The non-negativity constraints x ≥ 0 and y ≥ 0 are also in the proper form.
Therefore, we can conclude that the given problem is already in standard form.