x1 number of pitsburge x2 number of IMB
max 30 x1 + 15 x2
st
40 x1 + 30 x2 < 6000
40x1 >500
30x2>500
x2 750
x1,x2> 0
Question is Formulate this LP Model write in standared form
To formulate the given Linear Programming (LP) model and write it in standard form, we need to follow these steps:
Step 1: Define the decision variables:
Let x1 be the number of pitsburge.
Let x2 be the number of IMB.
Step 2: Define the objective function:
The objective function is to maximize profit, which is represented by:
Maximize 30x1 + 15x2.
Step 3: Define the constraints:
The given constraints are as follows:
40x1 + 30x2 < 6000,
40x1 > 500,
30x2 > 500,
x1, x2 > 0.
Step 4: Transform the constraints into standard form:
To convert the constraints into standard form, we need to introduce slack variables (s1, s2) for the less than or equal to constraints and surplus variables (s3, s4) for the greater than or equal to constraints.
Converting the first constraint:
40x1 + 30x2 + s1 = 6000, where s1 ≥ 0.
Converting the second constraint:
40x1 − s2 = 500, where s2 ≥ 0.
Converting the third constraint:
−30x2 + s3 = −500, where s3 ≥ 0.
Adding the non-negativity constraints:
x1, x2, s1, s2, s3 ≥ 0.
Step 5: Write the LP model in standard form:
The standard form of the LP model is as follows:
Maximize 30x1 + 15x2 + 0s1 + 0s2 + 0s3 (objective function)
subject to:
40x1 + 30x2 + s1 + 0s2 + 0s3 = 6000
40x1 - s2 + 0s1 + 0s3 = 500
0x1 - 30x2 + 0s1 + s3 = -500
x1, x2, s1, s2, s3 ≥ 0
This is the formulated LP model in standard form for the given problem.