The document describes the process for producing salt water taffy on an industrial production line. It begins with mixing the raw ingredients together in a 10:3:1 ratio by weight. The mixture is then baked and cooled before being stretched, folded, and cut into individual pieces on an extruding machine. To produce 2500kg of taffy per day, the optimal machine configuration was calculated to be 21 mixers, 42 cooling trays, 14 pullers, and 28 extruders. This would require 2520kg of the raw ingredients mixed in the proper ratio. Mathematics is used to determine rates, ratios, and optimize the system to run continuously at full capacity.

1. Taffy Production Line

2. History of Salt Water Taffy
As far as the name, legend has it that salt water taffy
moniker came from an Atlantic City store owner named
David Bradley. In 1883, a major storm caused the tide to
rise and flood his store, thoroughly soaking his candy
supply with Atlantic Ocean water. When a young girl
came into the store asking for taffy, Bradley offered her
some of what he jokingly called his new salt water taffy.
The girl loved the treat so much that she shared some
with her friends. It wasn’t long before just about
everyone was referring to taffy as salt water taffy.

4. Problem
Modern production lines have to be designed to be as fast as possible with almost
no error. When producing millions of products a day how does a company keep
track of it all? This project helps students to visualize how a product is produced.
The project describes how Salt Water Taffy is made. The process starts with mixing
raw materials. The mixture is then baked in ovens and subsequently cooled. The
taffy ingredients are pulled, folded and twisted (color can also be added at this step).
Finally, the mixture is put into an extruding and cutting machine which produces 7-
gram pieces of wrapped taffy.

5. Problem
In this project students become design engineers. Students will be
able to produce—and explain the system capable of producing—
2500 kg of taffy in a 24 hour period once the system is running at
100% capacity,I

6. Background of mathematics involved
The production of taffy needs to be done as efficiently as possible while adhering to the set
of specifications below.
The taffy is produced by the following process:
● Three different ingredients, X, Y and Z, are first combined in a large container in a ratio
by weight of 10:3:1.
● Four separate types of machines are required: Mixers, Cooling Trays, Pullers, and
Extruders.
● The 20-kilogram batches of the raw materials are mixed and heated for a period of 4
hours.
● This 20-kilogram syrup is cooled for a period of 8 hours.
● The Puller/Stretcher takes 30 kilogram batches and kneads them for 4 hours.
● The Extruder (which also cuts and wraps) takes 8 hours to process 30 kg of taffy.

7. The chart below summarizes the four machine types, capacities, and required times.
Machine Capacity (kg) Time Required (hours)
Mixer 20 4
Cooling Tray 20 8
Puller 30 4
Extruder 30 8

8. Assumptions
In order to devise a solution to this problem, it was necessary to make assumptions about various
components of the process.
● System of machines runs continuously; transition time not significant to our
calculations
● First product will be available at hour 24
● No mechanical-related setbacks; system operates flawlessly and without interruption
● All machines are operating at full capacity

9. Assumptions cont.
In order to devise a solution to this problem, it was necessary to make assumptions about various components of
the process.
● Process is entirely mechanical (in other words, manual labor rates not considered)
● Moderate expectations for quality
● No apparent equipment deterioration due to heating, cooling and/or heavy use; all
current rates are maintained
● Goal is to minimize equipment, maximize production
● At no point in this process can materials be recycled

10. The use of mathematics in the taffy production
process is related to the following topic
● Problem Solving
● Basic arithmetic skills
● Rates and ratios

11. Determining optimal machine configuration:
The machines have different capacities, but no machine should ever be
idle or running with less than maximum capacity. Use the ratios of one
machine’s capacity to that of the others’ capacities to determine how
many machines of each type are needed for continuous, full-capacity
operation.
Number of Mixers/Ovens:
M
Number of Cooling Trays:
C
Number of Pullers:
Mixer/Oven capacity:
20 kg
Cooling Tray capacity: 20 kg
Puller Capacity:
30 kg

12. For continuous operation to be possible, the ratios of the numbers of machines must
equal the ratios of their capacities.
Determining optimal machine configuration:
Coefficients should be whole numbers, since the
factory can’t have partial machines. Multiply the
equation by 3.
So an appropriate ratio M:C:P:E is 3:3:2:2

13. The Cooling Trays and Extruders require eight hours to perform their task,
whereas the Mixers/Ovens and Pullers only require four hours.
With the ratio of machines above, the three Mixers would have to be idle for four
hours after finishing their task, waiting for the Cooling Trays to finish. Since
Cooling Trays and Extruders need twice as much time as the other machines,
double the number of these two types of machines to make continuous
operation possible.
Determining optimal machine configuration:

14. For example, with two “sets” of three Cooling Trays, these machines can
handle the output of one “set” of three Mixers that works twice as fast.
Thus a “unit” of machines that can operate continuously at full capacity
consists of:
3 Mixers/Ovens
6 Cooling Trays
2 Pullers
4 Extruders
Determining optimal machine configuration:

15. First we need to determine how much a single “unit” of efficiently configured
machines (from the previous section) produces in one day. The unit of 3
Mixers/Ovens, 6 Cooling Trays, 2 Pullers, and 4 Extruders operates continuously at
full capacity. When the set of 3 Mixers is loaded with ingredients, the amount
loaded is 3 times the capacity of each mixer.
This 60 kg of ingredients leaves the mixer after four hours. At that time, another 60
kg of ingredients is placed in the mixer. Since the unit of machines operates
continually, if 60 kg of ingredients are inserted every four hours, then 60 kg of taffy
are produced every four hours at the other end of the production line. Thus one unit
of machines produces (60 kg / 4 hours)(24 hours / day) = 360 kg/day

16. Number of machines needed to produce 2500 kilograms
of taffy per day:
Then we determine how many units of machines are needed to produce a total of at
least 2500 kg of taffy per day.
The number of units must be rounded up to an integer. Thus, 7 units of machines
are required for this level of production.

17. Number of machines needed to produce 2500 kilograms
of taffy per day:
Finally we can now determine the total number machines in 7 complete units. Since
each unit contains 3 Mixers/Ovens, 6 Cooling Trays, 2 Pullers, and 4 Extruders, the
total numbers are:
21 Mixers/Ovens
42 Cooling Trays
14 Pullers
28 Extruders

18. Amount of materials needed to produce 2500 kilograms
of taffy per day:
7 units of machines are each producing 360 kg of taffy per day. The total
amount of taffy produced per day is therefore
Three ingredients, X, Y, and Z are combined in a 10:3:1 ratio by weight to
make the taffy. The total weight must be 2520 kg.

19. Amount of materials needed to produce 2500 kilograms
of taffy per day:
Calculating the 10:3:1 ratio for an entire day

20. Calculating the 10:3:1 ratio for a single Mixer/Oven

22. References
Cims. (2010). Worcester Polytechnic Institute (WPI).
https://web.wpi.edu/academics/math/CIMS/IMPHSS/Descriptions/taffy.html
Warrell. (2016, August 9). Salt water taffy is made with what?!? Warrell Creations.
https://www.warrellcorp.com/blog/salt-water-taffy/