In this video, we are going to continue with the problem that we had been working on about the changes to the model parameter. And in particular, we are going to learn the concept of shadow prices for the two constraints that we're considering in this problem. So now let's look at what happens when we relax the constraint on Machine 2 while keeping the Machine 1's constraint as the original one. So we have the objective function that remains unchanged, 300x1 + 200 x2. The constraint for the first machine is same as the original one, which was 2x1 + x2 is less than equal to 8. But we have relaxed the constraint on Machine 2 this time, so x1 + 2x2 is less than equal to 9. So we have running Machine 1 for one additional hour and we have the nonnegativity constraints. So if you want to plot the feasible region, the boundary line for the first constraint, that is for Machine 1 remains the same as the original one, passing through (0, 8) and (4, 0). And here the dotted line shows the boundary line for the original M2 constraint when x1 + x2 was equal to 8. But now we have the boundary line for the new constraint, for the relaxed constraint, is going to be x1 + 2x2 = 9. So we can substitute some values to identify two points on this line so that we can draw it on this two dimensional plot. So let's say if x1 = 0, we get x2 = 4.5. And when we plot x2 = 0, we get x1 = 9. So these are two points on this line x1 + 2x2 = 9. So this first point is going to be (0, 4.5). And the second one is (9, 0). So we can use these two points to draw this line. So (0, 4.5), that's roughly this point here. This is (0, 4.5) right here. And (9, 0) is this point. So the relaxed constraint of M2, the new constraint would pass through these two points. And it would look something like this. So again, it is parallel to the original constraint but now since it's relaxed, the intercept's changed and this is the new feasible area. So we have got (0, 4.5), (0, 0), (4, 0) and this particular point, we can find by solving these two equations together. x1 + 2x2 = 9 for Machine 2 and 2X1 + X2 = 8, which is this constraint. So if you're solving these two together, we would get the intersection value. And that value turns out to be x1 is going to be 7/3 and x2 is going to be 10/3. So this point has a coordinate of (7/3, 10/3). So as before, we can now use the enumeration method to find which of these four corner points of the feasible region gives the highest value of the objective function. So our objective function remember was 300x1 + 200x2. So now you substitute the values of these four corner points into this objective function and find out which of these corner point gives the highest value. And I've already done that and I can tell you that this point is the new optimal solution. So the value at that point is going to be 300 times 7/3 + 200 times 10/3. And that comes out to be 4,100/3. So, If you had to relax Machine 2 by one unit, the constraint on Machine 2 by one unit, that is let Machine 2 run for one additional hour. Then my profit. At the optimal level would be 41 hundred over three dollars. We can compute the net change in profit over here. In this new case, I'll call this Z dash star. For. Mission Twos relaxation of the machine to constraint minus the star of original. Divided by this change in profit divided by the change in M2's capacity. This metric is going to tell me, if I had to change machine tools capacity by one unit. That is, let it run for one more hour. What is the net change in my profit? So, this is going to be? This value 41 hundred over three, which is by. Optimal value of the objective function for this new set of constraints. Minus the original. Z star, which was 400 over three when none of the constraints were relaxed. Divided by changing empties capacities within hours minus eight and this turns out 100 over three or. $33.33, so by adding one hour to machine two, I would improve my profit by $33.33. So let's compare the net change in the objective function value of the profit by changing the capacity of machine one and machine two. So for machine one we have by adding. One hour. To machine one, while keeping machine,choose running time to be the same as eight hours. We found that. The profit. Increases by. $133.33 which we saw in the last video. And here we saw that by adding one hour to machine two the profit increases by. This amount, which is $33.33. So, this analysis is useful because you can use this analysis to prescribe whether you should run machine one or machine two for one additional hour. If you had a choice to run either of the machine by one additional hour, which machine should you run? So here it clearly says, that if you were to run machine two for one additional hour, then you profit would have increased by $33.33, whereas if you had kept machine one running for one additional hour, your profit would have increased by 133 dollars and 33 cents. After it's reset, so clearly keep running machine one for additional hour results in a higher increase so you can use this calculation to see and compare the cost of running these two machines and comparing the net benefit of running these two machines for one additional hour and see which one is better. So here if you if the two machines cost the exactly the same amount to run then you can see that if I. Keep machine one running for one additional hour. My profit would increase by a larger amount than if I were to keep machine two running for one additional hour. So these type of analysis that we just did by slightly altering the constraint or relaxing the constraint and computing the shadow prices. Can allow us to see which of the constraints relaxation results in a higher improvement in the objective function value. So $33.33 is the shadow price. Of the constraint on machine two. And $133.33 that we computed was the shadow price of the constraint on machine one. So computing this shadow crisis can allow us to see which of these constraints when relaxed by one unit would result in a higher improvement in the optimal value of the objective function. And you can use it to inform decisions like this. For example here, if the two machines cost the same amount, then running machine one for one additional hour would be more beneficial.
