1. The previous cubic is entered in Excel in away that may be read easily Note the ‘ name box’ is used here to give the numerical content of the cell a name, not just cell coordinates, that may then be used in equations. Subscripted variables may be represented with underscores eg x_1, a_2 etc. Excel has vector and matrix functions but it is not represented as done in Matlab name box Cell C12 is named ‘a_2’ (see the name box) and its value is -3.5, it is calculated as the sum of the 3 coefficients x_1, x_2 & x_3 This is just text reminding the user what this s.s. is about This is just text reminding the user the name of the variable in the cell to its right

2. A column of independent variables xs are entered, associated dependent ys are calculated. The box is highlighted and a scatter diagram is chosen from the Chart Wizard I copied the content of C18 to E18, delete the ‘=‘ sign leaving a text version of the otherwise unseen equation behind cell C18

3. Right clicking on the chart allows the selection ‘Chart Options’ Providing choices of how the chart is presented, ie titles, axes labels, grid density, max & min limits of the graph. MS decides the major & minor divisions and if they are initially to be visible

4. Right clicking on data point allows selection of data points altering their default point presentation in size, type, colour and fill And selection of line of ‘best’ fit ie the Tredline

5. Here selecting a cubic trendline must provide us with a near perfect fit. The full equation of the Trendline line may also be shown. This is a useful tool to arrive at an equation representing noisy data.

6. Title and purpose of this spread sheet Name of s.s. file Description of variable Symbol used in equations Value of variable units Equation to calculate variable An example of a layout for a s.s. The intentions is to make the file easily readable, used by others expanded by yourself and assessed by your supervisor. If your marker or supervisor cannot understand they will not give you the benefit of the doubt they will mark you down.

7. We copied a single pair of independent and dependent variables to B26 & C26. Select Tools/Solver, this permits the selection of the object function, ie the ‘ Target Cell ’ and the independent variable(s), ie ‘ Changing Cells ’. Note that the starting values of the search for the maximum ‘y’ are the values in the cells B26 & C26. The solver is a numeric not an a symbolic tool, it will gives only approximations. Cells B26 and C26

8. Here the correctness of the Solver ’s solution can only be verified within the resolution of the graph, but the range and density of the gridlines on the graph can be adjusted to give better resolution. The equation of the Trendline is available - good for random variables Local maximum, approximately (-6,15)

9. We now seek to find the min value of the cubic, using x = -4 as the starting point. The Solver finds the nearest or local condition required of the object function. The solver does not execute a global or a guided search. This initial condition for x & y tend to negative infinity, to limit the search we can add constraints.

10. In the search for some ‘optimal’ solution in design, where there are multiple variables, it is often necessary to use constraining relationships to eliminate nonsense answers. We have a rectangular prism representing a container for which we want to minimise the cost of the material or limit the heat transfer through the surfaces. The question is what are the proportions of the sides so that the surface area is a minimum. If we search for the smallest surface area, L, W & H will all tend to 0. Giving a box of no volume. An appropriate constraint may be taken to be that the volume be =1.

11. The search is then for the dimensions of a rectangular prism with the smallest surface area but fixed volume. Here we used a Volume =1 as the constraint. Fortunately essentially all practical engineering problems have fairly easily recognisable solutions, obvious constraints and limited pitfalls, unlike the sort of functions that mathematicians are fond of examining. What do you expect the lengths of the sides to be?

12. The search for the lengths of the length sides of a 13 dimension prism, of min surface area, but which meets 10 boundary conditions, ie all volumes from the 3 rd dimension to the 13 th are all required to be 1. Limits to: time, cycles, tolerance, method of approximations etc. The previous example is deceptively easy and quick. In an attempt to search for the limitations in the capacity of PCs and this solver, the 3 dimensional prism can be extended to higher dimensions

13. Not so long ago a PC would bog down long before the 13 th dimension, not at all now. A few years ago a 1 mega-flop computer would cost $1 M. To day new PCs can do much better. It would take a human with a calculator more than 2 years to do a mega- flop. This indicates to me that the ultimate power of PCs has barely been explored.

14. The function for the 13 th dimension surface area has 78 non linear terms !!

15. A real and very useful application is to find the parameters of a pair of gears that have an equal margin of safety against failure in 4 areas, two in each gear: Compressible failure A, on each. Tensile failure at B, at the base of opposite teeth. A chain is no stronger that its weaker link. The lowest factor of safety limits the capacity of the gear pair

16. Each variable shown here is a function of other variables. There are 2 sets of such equations one for each gear In a gear box there are many such sets, and in an industrial plant there may be many such boxes