 Hi, this is Dr. Don. I have a problem that we want to solve. And in it, we need to use linear optimization, and we're going to use Excel with Solver. In this problem, we have a partnership, Fast Homes FHP. They have a 45-acre parcel outside Atlanta, and it's restricted that they can build no more than nine units per acre. They want to build condos there, one, two, and three bedroom units, and they bestimated the construction costs will be 300,000, 500,000, and 750,000, respectively for those. From their experience, they think 10% of the construction cost will be the profit. They have funding of $175 million, and from their experience, they think that the best mix would be a minimum of 25% of the units to be one bedroom, 25% of the total number of units to be two, and the 20% minimum to be three bedroom. So we need to determine how many of each type to maximize. The basic procedure for an optimization model, we identify an objective function. In this case, that's going to be we want to maximize profit. We need to find the decision variables, the things we can change. We need help deciding how much to change them. We've got to know all the constraints, and then we've got to write these objective function of constraints as a mathematical expression, and we're going to do that in Excel. So let's boogie on over to Excel. I've already built the decision model, and I'll walk you through how it's set up, and it's designed to make it easy to plug into Solver. Okay, so here's what I've done. I've built the basic model up top, and I've got here the one, two, and three bedrooms in the totals, and I've set this up. These will be the number of units that we're building, and I like to start with just putting one for those. This is the total of those three, and you can see that's the equation there. These are the cost per units. This total cost, again, has an equation there of just the number built times the cost per unit. We've got profit of 10%, and so the profit is just equal to this 10% times the total cost. And we add that up to get the total profit, and here we've got our total cost. So that is our decision model and the objective function, which is this total profit that we want to maximize built into a set of mathematical equations, just a simple decision model. Down here I have the constraint, and I like to set them up, and what is known as the left-hand side, that's what you're constraining, and then the right-hand side has the value against what you're constraining it. And then you have an operator, and then the solver is either greater than or equal, equal or less than or equal. So I've put those in there. Total cost has got to be less than or equal to the budget of $175 million. And again this cell here is just linked to that cell there to bring that down. And the same thing is true for the number of units. This is linked up to that cell, and total units is linked to that cell. So I've got this decision model linked to my constraints down here, and there's again the operators, and here's the right-hand side. I've got the budget. The percentage that we want to target is 25% of this total in B6, and again 25% in the total in B6, 20% for three bedrooms minimum, and then we've got our zoning requirement. We have 45 acres with a max of 9 per acre, so 405 units is the max that we can have, and the total number of units needs to be less than or equal to that. So we've got our model and our constraint set up. Let's go to, okay, we click on solver, and we get our dialog box up here. We're going to set our objective, which is to maximize the profit. Our decision variables, we're going to change those. That's these three cells there. And then we've got to add our constraints. So I'm going to click on add. And the first constraint I'm going to add is the limitation on our budget. So here's our total cost has got to be less than or equal to the budget. I'm going to add another. Now XL is pretty smart, so I can take this limitation on having a minimum number of each one of these units. I'm going to highlight those cells that I've got to have. The operator is greater than. We want to have at least as much as these minimums over there. And then finally, I've got to add a constraint on the total number of units. Has to be less than or equal to the zoning restriction max of 405. And there is one more I almost forgot. Since we're building things, we can't build a partial unit. So I'm going to constrain those decision variables to be integers and hit OK. Now we've got the model setup. We've got solver setup. I'm going to just leave everything here default and click on solve. And we get a message that we've got a solution. Click OK to keep the solution. And you can see that we've got 78 one bedrooms, 96-2, 138. We've got a total cost of 174.9, which is just slightly less than our budget. And we've got a profit, of course, which is 10% of that of almost $17.5 million. So our constraint on maximum are units. We've got some room there if we could figure out how to do it. And we've got all of the other constraints met there as well. So solver did us a good job here. I hope this helps.