I have question&its solutiion but how?in LP

[emadhamdy2002]

Blending problem:
Decision variables
x1: tons of ore from mine 1 in each ton of the blend
x2: tons of ore from mine 2 in each ton of the blend
x3: tons of ore from mine 3 in each ton of the blend
x4: tons of ore from mine 4 in each ton of the blend
Objective function
Min cost
Min 800x1+400x2+600x3+500x4
Constraints
Element A: 10x1+3x2+8x3+2x4>=5
Element B: 90x1+250x2+75x3+175x4>=100
Element C: 45x1+25x2+20x3+37x4>=30
Weight: x1+x2+x3+x4=1

If you know that Element B has slack of 31.666
x1=0.259259
x2=0.703705
x3=0.037037
x4=0.00000
there is a question
what is the amount used of element B?compute it

i found a solution please if it is right can anyone explain it to me

90*0.259259+250*0.703705+75*0.037037+175*0-31.66=100

therefore the amount used =131.66

Can anyone Explain it to me?

 

[mmwave]

Sure! Let's break down the solution step by step. First, we need to understand the constraints and what they mean. The first three constraints (Element A, B, and C) represent the minimum amount of each element that needs to be present in the blend. The fourth constraint (Weight) ensures that the total weight of the blend is equal to 1 ton. This means that the blend is made up of 100% of the four different types of ore, with each type contributing a certain amount (represented by the decision variables x1, x2, x3, and x4).

Now, let's look at the objective function. The goal is to minimize the cost of the blend, which is represented by the expression 800x1+400x2+600x3+500x4. This means that we want to find the values of x1, x2, x3, and x4 that will result in the lowest cost for the blend.

Next, we are given the information that Element B has a slack of 31.666. This means that the amount of Element B used in the blend is 31.666 less than the minimum amount required (represented by the coefficient of x4 in the constraint for Element B). We can use this information to solve for the values of x1, x2, and x3.

To do this, we can set x4 equal to 0 in the constraint for Element B and solve for the remaining variables. This gives us the following equations:

10x1+3x2+8x3 >= 5
90x1+250x2+75x3 >= 131.666 (since we know that the slack is 31.666)
45x1+25x2+20x3 >= 30

We can then solve these equations using a method called linear programming. This involves graphing the equations and finding the point where they intersect, which represents the optimal solution. In this case, the optimal solution is x1=0.259259, x2=0.703705, and x3=0.037037.

Finally, we can use these values to calculate the amount of Element B used in the blend. We know that x4=0, so we can plug in the values for x1, x2, and x3 into the constraint for Element B:

90*0.259259+250*0.703705+