# Fashion Designer

**Coming soon: Launch worksheet**

**Coming soon: Launch solution **

## The Scenario

You are a world class fashion designer, sought after by fashion shows and celebrities from all around the world. It’s not easy staying at the top of your game however – one mistake and you could fall from the top of the fashion world. It’s critical that you get all the key elements of being a successful fashion designer right.

## Math Scenario 1 – Metres of Fabric

Being a hands on designer, you still insist on doing the initial prototype designs yourself – once you feel you’re on a winner you can then get one of your many talented staff to finalize the design.

You’re currently working on a beautiful formal gown for a famous celebrity’s Hollywood event. You’ve travelled to Italy to seek out the finest materials and are considering how much you’ll need. You can’t decide between two different fabrics – one that comes in a wide, 150 cm wide roll, the other in a 110 cm wide roll. You estimate you’ll need about 12 square metres of fabric for each prototype.

How many metres of each fabric should you buy?

**150 cm roll**

*Length required = area required / width*

*Length required = 12 / 1.5*

*Length required = 8 m*

**110 cm roll**

*Length required = area required / width*

*Length required = 12 / 1.1*

*Length required = 10.91 m*

## Math Scenario 2 – Runway Traffic

At your upcoming fashion show, you’ve been given a 35 minute slot to show off your latest fashion line. That time includes two music performances of 4 minutes where you won’t be able to use the catwalk. Your models will take an average of 40 seconds to do a complete catwalk run. How many different outfits will you be able to showcase?

First convert the catwalk time to minutes to be consistent:

*40 seconds = 40 seconds * 1 / 60 minutes / second*

40 seconds = 0.6667 minutes

*Number of outfits = time available / time per catwalk*

*Number of outfits = (35 – 2 * 4) / 0.6667*

*Number of outfits = 40.5*

Looks like you’ll be able to show off 40 outfits during the show – enough to cover your entire new fashion line hopefully!

## Math Scenario 3 – Economies of Scale

The fashion show is a runaway (haha pun) success and you receive large orders from the major retailers all around the world. It’s time to scale up production.

With no chance of being able to meet demand, you need to be selective. Only the largest orders will get serviced first, and then as demand lessens you can go to the smaller retailers.

To work out the minimum viable order size, you examine your production costs. Any order that has a production cost per dress of less than $60 is viable:

*Cost per dress = 50 / exp(0.0002 * n) + 50*

*where n is the number of dresses in the order and ‘exp’ is the exponential function*

We can plot a graph of production cost versus the order size:

From inspecting the graph, it looks like the minimum viable order size is around the 8000 dresses mark. We can work this out more accurately using logarithms:

*Cost per dress = 50 / exp(0.0002 * n) + 50*

*Cost per dress – 50 = 50 / exp(0.0002 * n)*

*exp(0.0002 * n) * (Cost per dress – 50) = 50*

*exp(0.0002 * n) = 50 / (Cost per dress – 50)*

Take the natural log *ln* of both sides:

*ln[exp(0.0002 * n)] = ln[50 / (Cost per dress – 50)]*

*0.0002n = ln[50 / (Cost per dress – 50)]*

*n = ln[50 / (Cost per dress – 50)] / 0.0002*

Substitute in the maximum allowable cost per dress:

*n = ln[50 / (60 – 50)] / 0.0002*

*n = 8047 dresses*

Pretty close to the answer we got by plotting the graph.

## Math Scenario 4 – Expanding your Designer Team

You have become a victim of your own success! Rapidly growing orders means you can no longer afford to do initial prototype designs yourself, although you’ll still look over them.

That means you need to hire some top creative fashion talent, train them up to meet your standards and ideals and let them loose on your upcoming fashion launches.

To attract the top talent, you’ll need to offer a competitive package – a **salary**, a yearly **bonus** plus **equity** in the company. You figure you need a team of 5 full time designers to meet the growing demand.

Your finance department estimates the extra cash flow from your increased sales will give you an increased net profit per year of $12.5 million dollars. They will allow you to use up to 20% of that extra profit as salary and bonus. They suggest that you offer a 20% end of year bonus on top of salary.

What’s the maximum size salary and bonus you can offer that meets all these conditions?

*Cash available per designer = increased profit * profit percentage usable / number of new designers*

*Cash available per designer = 12500000 * 0.2 / 5*

*Cash available per designer = $500000*

Half a million dollars – that should be attractive. But we’ve still got to divide it up into salary and bonus:

*Total cash package = salary + bonus*

The bonus has to be 20% of the base salary, so *bonus = 0.2 * salary*:

*Total cash package = salary + bonus*

*Total cash package = salary + 0.2 * salary*

*Total cash package = 1.2 * salary*

*salary = total cash package / 1.2*

*salary = 500000 / 1.2*

*salary = $416667*

And:

*bonus = 0.2 * 416667*

*bonus = $83333*

So your offer to designers will be a base salary of $416667 per year, with an end of year bonus on top of that of $83333.

## Math Scenario 5 – Optimising Production Processes

Although you only needed to buy an approximate amount of fabric for the prototype gowns, for mass production your designers will need to work out exactly how much fabric they need per clothing item. One factory (Factory A) charges more per fabric area but only charges for the fabric actually used (it recycles the rest) while the second (Factory B) charges less per fabric but charges you for the entire rectangle of fabric stock including the off cuts.

For a new skirt design, the fabric pattern is shown above (shaded area is the fabric).

The distance *a* is the waist measurement divided by 2 pi. The length *b* is the length of the skirt plus some allowance for the hem*.*

Calculate the amount of fabric used for both factories, for a 30 inch (76.2 cm) waist and a 50 cm length skirt + hem. Then work out how much cheaper Factory B must be per fabric used than Factory A for it to be the better choice.

First we can work out what *a* is given the waist measurement:

*2 * PI * a = 76.2*

*a = 12.13 cm*

**Factory A**

*Fabric used = large quarter circle area – small quarter circle area*

*Fabric used = 0.25 * PI * R^2 – 0.25 * PI * r^2*

*Fabric used = 0.25 * PI * (a + b)^2 – 0.25 * PI * a^2*

*Fabric used = 0.25 * PI * (12.13 + 50)^2 – 0.25 * PI * 12.13^2*

*Fabric used = 2916 cm^2*

**Factory B**

*Fabric used = size of smallest rectangle containing the pattern*

*Fabric used = (a + b) * (a + b)*

*Fabric used = (12.13 + 50) * (12.13 + 50)*

*Fabric used = 3860 cm^2*

Now calculate how much cheaper Factory B has to be:

*Required price ratio = Factory A area / Factory B area*

*Required price ratio = 2916 / 3860*

*Required price ratio = 0.7554*

To be competitive, Factory B will need to charge no more than 75.54% of the rate per fabric used that Factory A charges.

## Real Life Example – Complex Fabric Patterns

Real life fabric patterns can be very complex. Optimizing the production process can involve working out the best way to arrange various parts of the design so that there is minimal wasted fabric – a problem that is sometimes quite complex to solve.

Other considerations are the order in which complex, multi-piece fashion items are made, especially given the rise in automated, robotic production systems that do much of the work a human used to.