Hello,

I need help with a math problem. Essentially three dimensions of a box are provided (i.e., Length, Width, and Height). To manufacture, these boxes a flat board with dimensions 96"x48" is used. The manufacturing process involves calculating how many boxes can be obtained from the "breadth" (i.e., 48") of the board and how many boxes can be manufactured from the "length" of the board (i.e., 96").

Let's take a box with dimensions 12"x12"x5" for example:

Length: 12 inches

Width: 12 inches

Height: 5 inches

First, we calculate the manufacturing length of the box (in its flat form). The calculation is as follows:

Formula for calculating Manufacturing Length (the length that a single flat box takes up):

((2 x (Length + Width) + Flute x 3) + Glue Flap) - Optimization

Constant Values:

Glue Flap (a little area where glue is applied to glue the box together) = 1.5 inches

Flute (thickness of board taken into consideration for the places where the box folds) = 0.16 inches

Optimization (a parameter used to optimize the manufacturing, where more boxes can be made from one sheet in some instances) = 0.2 inches

Example Calculation:

((2 * (12+12) + 0.16 * 3) + 1.5) - 0.2 = 49.78 inches

Therefore, each 12"x12"x5" box will take up 49.78 inches of the length of the board (i.e., 96").

Now that we have the manufacturing length of a single box, we need to calculate the manufacturing breadth (width) of the box (in its flat form). The calculation is as follows:

Formula for calculating Manufacturing Breadth (the width that a single flat box takes up):

(Height +(MIN(Length or Width))+ Flute x 2) - Flute x 2

**I understand that its odd to add the flue x 2 and then subtract it, but this will make sense when you see the excel doc. It is essentially an optimization to get more boxes from a sheet (in some cases)

**MIN means, take the smallest value between Length or Width. Typically, Length should be greater than width, however, it needs to be done this way for convenience. Sofor example, if the width is 6 and the length is 8.

MIN(8 or 6), we use 6.

Constant Value:

Flute (thickness of board taken into consideration for the places where the box folds) = 0.16 inches

Example Calculation:

(5+(MIN(12 or 12))+ 0.16 x 2) - 0.16 x 2= 17 inches

Therefore, each box will take up 17 inches of the breadth (width) of theboard (i.e., 48").

Now that we know that a 12"x12"x5" box (in its flat form) takes up 49.78" per a box from a length of the board (i.e., 96"), and 17" from the breadth (width) of the board (i.e., 48"), we can calculate how many boxes can be made from each dimension of the board, and then how many boxes can be made from the entire sheet.

For the Length of the Board, we do the following calculation:

96"/49.78" = 1.929 boxes

Therefore, we can manufacture 1.929 boxes from the length of the board.

For the Breadth (Width) of the Board, we do the following calculation:

48"/17" = 2.824 boxes

Therefore, we can manufacture 2.824 boxes from the breadth(width) of the board.

Obviously, we cannot obtain 1.929 boxes from the length and 2.824 boxes from the breadth (width) of the board. So, we would have to round down the values to an integer such that a whole number of boxes can be manufactured.

Therefore, in this case we can manufacture 1 box from the length of the board, and 2 boxes from the breadth (width) of the board.

Therefore...

We can obtain a total of 2 boxes from the board:

1 from length x 2 from breadth = 2 boxes total from the board

This is where your help comes in...

As you can see in this particular example if adjustments were to be made to dimensions of the box (i.e., Length, Width, and Height) we can obtain more boxes from a 96"x48" board. Instead of 2.824 boxes, which round down to 2 boxes, we can make minor adjustments to the dimensions of the box to obtain 3 boxes from the breadth, and with minor adjustments to we can also obtain 2 boxes from the length of the board instead of 1, rounded down from 1.929.

Task:

I need you to provide formulas for suggested dimensional changes to optimize the number of boxes that can be obtained from each 96"x48" sheet.

These suggestions should be made only if the decimal value is less than 0.3 or greater than 0.7. So, if 2.10 boxes can be obtained from the length, a suggestion should be made. Or if 2.9 boxes can be obtained from breadth of the board, a suggestion should be made. If 2.5 boxes can be made, no suggestion is required. The suggestions should involve minimal changes to dimensions of the boxes.

2 variety of suggestions are required:

1. To a single dimension (length or width or height)

** Only make one suggestion for one of the dimensions to obtain at least one more box.

1. To two dimensions or more

**Make suggestions for two or all three of the dimensions to obtain at least one more box.

I have prepared an excel sheet that does all the calculations for you and includes when suggestions would be made. All you need to do is create the formulas making these suggestions.