So we are now going to solve this problem using Excel solver to prescribe the right product mix decision for this company. So here we have the Excel sheet with the different numbers laid out. We can now use solver. We go to the data tab and then click solver. We need to set the objective function in the cell containing that, which is going to be L5 over here. Then this is a maximization problem, so we're going to select Max and the decision variables will be in the cells B5 through K5. And we're going to add the constraints. So one constraint comes from the fact that the D has to be less than equal to, The value of 20 here in E6. So we'll add that, another one will come from the fact that. E, the decision variable value of E has to be less than equal to, 20 from G6. And in addition, we'll have these constraints from on the values of M1 on to M3. Being less than equal to 128 each. And finally, we have the constraint that the three quality constraints should hold, so the left hand side of the expressions should be equal to the right hand side value, which is 04, these equality's. So we have set these up. We can now use simplex to solve this problem. And look at what values we get. So we're getting the optimal solution as to produce 942.5 units of product C. 20 units of product E. And you're not producing any of the other products. The optimal decision would be to just take 2 product C and product E, and only produce 20 units of product. You're not producing anything more than that, and you are not producing any of the other products. And in doing so, you would run into the time constraints on machine one, that you are satisfying this boundary. This is becoming a binding constraint, so the upper bound was 128 hours available. So you're already hitting that just by producing these two products, and your total profit will be $1,777.625. And this is going to be your solution, and that it would prescribe to this plant like they have to manufacture 942.5 units of product C and 20 units of product E in order to get the maximum profit. If they wanted to relax the number of hours that and one can run. If you were to increase this to let's say, 135 then you probably can get to a higher profit value. So let's make it 135. And let's resolve this, so we have 1,777 as the solution. Now we might be able to get a little bit more because we are adding number of hours to machine one, and now we can go up to 1875, 1874.75. Again, this turns out to be the binding constraint. So machine one is still the constraint you can keep on, if the company where to be able to relax it, they can see that maybe there are ways to add more resources so that machine one can be kept running for a longer time. Or maybe you buy more machine ones, in order to increase the total number of machine hours available for machine one. And let's see if you solve it, whether some of the other machines become binding or not. So if you're still even with 200 hours if you keep machine one running for 200 hours, you're still seeing that machine one is the binding constraint. So machine one is a critical equipment here. If you are able to improve the total number of available hours for machine one, your profit will keep on increasing till it hits another one. So here, it has already hit machine two's upper bound. So just by increasing machine one alone, you probably might not be able to do much. More than this going forward, you might have to relax some of the other machine hours as well in order to be able to get even more profit. But as you can see here, as the constraint values change, the right hand side changes your strategy. The optimal strategy is different. You are getting a different outcome in terms of the total profit achievable and you're being able to produce different units of the product. So here you are producing, if you have 200 hours available on machine one, it would be producing 887.39 units of product C. You reproducing 20 units of product D and plus 456.087 units of product D, so a total of 476.087 units of product D. If you have 200 hours of machine one available. And you can see like, this way you can figure out what is the optimal decision for these type of product mix problem where you have nonlinear functions for the profit margins. And you just introduced different decision variables. Split it up as D and D2, E and E2, in order to tackle these kind of problems. So these are very common problems that you would encounter in the industry that where you can apply optimization to find the best course of action.