 Hi, this is Dr. Don. I have for you a linear optimization, a linear programming problem, and we're going to solve it using Excel with the solver add-in. Now this particular problem uses binary constraints, binary decision variables. So it's a little more interesting than the standard. We have a company that's budgeting and they're given the R&D vision, a challenge that they've got seven research projects that they would like to fund, but there's not enough people and not enough money to go around for all seven of the projects. So they've got to figure out which of the projects to fund in order to maximize return, maximize profit. Now they've developed this table with the seven projects, the number of engineers, the staff that's required from within the company, and then they've got some outside consulting costs for each of the seven projects. And then if they complete the projects, they will earn a profit shown in this row. So let's go over and set it up to use Excel solver. I've brought over that table at the top and I've added over here the first part the decision variables that we're going to use. We've got seven projects, so you would have seven variables and these are going to be our decision variables, but they're a special kind, they're binary. Think about it, you can either fund a project or not fund a project. It's either on or off and we model that using binary variables, a zero for off, a one for on. So I've set up a row here with my decision variables and I like to have ones in there when I'm building the model just so I can see them more clearly what's going on. As far as engineers use, I'm just going to click in this cell and we are going to take equal and we're going to take our decision variable and then multiply that times the number of engineers and it enter and then I can just drag that over to get that equation, that formula copied across for all of those in that row. Do something similar for budget, equal decision variable times the, consulting cost, if we use that hit enter, I got 230,000 and of course we just copy that across. Now finally we look at the profit, we do the same thing, equal decision variable, yes or no, times that profit, enter, that's 590 and I'm just going to copy that across. So that's a big part of our decision model. Our objective function, that's the big equation we're trying to solve, would just be to maximize profit and so we've got the profit earned if we did all seven of those. So we're just going to sum that up, I'm going to hit equal sum, double click that to select it and I'm going to sum up those seven projects, the profit and those seven, enter. So theoretically if I could do all seven projects I would get 3.855 million dollars in profit but I can't do all seven and down here for our constraints we set them up traditionally in a left-hand side and a right-hand side with the math operator we need in between. So our left-hand side would be the engineers used, again it's sum equal and I'm going to take my engineers use row and sum that up. So I would use 57 and of course I've got a maximum of 35 from the problem statement. Do the same sort of thing for the consulting budget equal sum, double click and my consulting budget right there, enter. So now I've got my left-hand side of my constraints and my right-hand side they're both left standard equal. So now we need to set up Solver. I'm going to go over here and click on Solver and we get our dialog box open up there. We want to set the objective cell, click in there and my objective function is there. I want to maximize, so I'm going to select that and then we've got our decision variables, select then that box and then just drag across to select that range. Now I've got to add my constraints. So I'm going to click on add and the first one is the number of engineers. So my left-hand side cell reference, 57, math operator less than or equal, there's our options and our right-hand side is 35. So I'm going to add another one and now we've got our consulting budget of 1.655, again less than or equal and then our right-hand side and then finally I've got to add my binary constraints. My cell references are these decision variables and we want them to be binary, yes or no. So that's it, I can click okay and down at the bottom we check to make sure that we've got unconstrained variables as non-negative, we're going to use the basic non-linear engine and just click solve. Okay, we've got our solution. I'm going to go ahead and select the answer report real quick, click okay and we get the answer report. We look down at the bottom here, we can see our constraints. Engineers is not constrained, we've got a slack of one engineer's not binding and then of course our consulting budget, we've got some slack there again, $130,000 is not binding. So we go back to Solver tab to get our solution. You can see we've got a zero for project one, a one for project two, so we're going to do project two, three and four, we've got a zero for project five, so we don't do five, we do six and we don't do seven. So we've got one, two, three, four projects we do and of course that gives us our 2.575 million dollars of profit that the most we can get, 34 out of our 35 engineers and 1.07 million out of our 1.2 million budget for consulting. So I hope this helps.