Explains two basic types of optimization problems for blends, namely the simpler where total contents have requirements, and the slightly more complicated where final concentrations have requirements. Both are linear programs that can reliably be solved in a spreadsheet, we walk through this process. The point is that you can save money on making mixtures or blends, it is very important and not too difficult.

1. Elementary optimization
Special Topics in Industrial Chemistry
Seppo Karrila
November 2014

2. Executive summary
• What is optimization
– Important terms
• Tools you can easily use
– How to use within a spreadsheet
• Some examples of Linear Programs
– A raw material mix with requirements on
“concentrations”, so there are dilution effects
– Diet problem: satisfy minimum daily requirements

3. Industry view
• The purpose of research is to enable decisions
• We want to make “the best most profitable
decisions”
– Research must provide a prediction: what
happens when we decide this or that
– Alternative decisions must be compared, based on
numbers (calculated profit)
• Finding “the best” is called “optimization”

4. Optimization
• You are in control of some decision variables
• Your decisions affect an outcome that you can
calculate from a numerical model
• How do you get the best outcome?
– Maximize profit
– Minimize cost
– In general, optimize an objective function
• You get the optimum (maximum or minimum)
with an optimal decision
• Rules that your variables have to satisfy are
called constraints

5. Example: making a blend
• Raw materials have each their characteristics
• Blending them affects the characteristics as if
these were concentrations
mtot c = m1 c1 + m2 c2
mtot = m1 + m2
 c = (m1 c1 + m2 c2)/(m1 + m2)

6. A coffee blend
• Aroma and strength follow similar rules as
concentrations. Must have blend aroma >=78,
and strength >= 16 .
• Availability: Brazilian < = 1,500 lb
Colombian <= 1,200 lb, Peruvian <= 2,000 lb
• Make 4,000 lb of blend at lowest cost !

7. Why this is really important
• In industrial production, if you save 0.1 % in raw
material costs, you have earned your wages.
• You can apply this to making any raw material
mixtures
– Available raw materials are often already mixtures
• Purification adds costs. Why purify if you will mix again?
– Your blend must satisfy some “quality” requirements
– You want to minimize the cost

8. What is the aroma rating of the blend?
• Apply the “concentration rule”:
• What is required of this characteristic:

9. Manipulate this to get a simpler
inequality
• This is the quality criterion for aroma
• Similarly, you get the criterion for strength of the
blend:

10. Putting the model together
Minimize
• Quality criteria
• Required
production
• Availabilities

11. So it looks terrible?
• This type of systems are solved routinely
• Any spreadsheet program you use can solve
them
– The solution technique is reliable:
if there is a solution it is found,
if there is none then you will be told so
• This is an example of a “linear program”

12. You need to know how to handle this
• So we will go through the solution with a
program that you certainly can access:
– Google drive spreadsheet
• Otherwise, you can do exactly the same in
Excel
– You may have to activate Solver add-in, if you have
not used it before

13. In Google Drive, install an add-on
• Search for “solver”, add it to your spreadsheet

14. Open solver
• Press “Insert
Example” to see a
similar problem

15. The solution to our problem
• Note: ALL calculations done with “sumproduct” !
• Method: Simplex. (LP = Linear Programming.)

16. You can check aroma and strength of
the solution
• I left these out, they are not needed to solve
the problem
• Note that LP is reliable. You can try other
solution methods, they may give worse
“solutions.”
– Important to have “Assume Non-negative”
selected in “Options”, otherwise you get wrong
results also

17. A Reference
• K.A. Baker: Optimization Modeling with
Spreadsheets
– Explains several other problem types that can be
solved with Linear Programming
– Covers some further cases:
• Integer variables (numbers without decimals)
• Some non-linear programming

18. Another typical LP problem: the diet problem
• Make the lowest cost food mix that satisfies nutrition
requirements, namely “daily dose” of constituents
– You could have lots of details, about various vitamins and
minerals. Some are bad in overdose, you can have max limits!
– If you are feeding an army, a small change in cost will be a lot of
money. How about feeding 50 cows or 100 pigs?

19. This problem has “minimum
requirements” for total content in
blend
• Dilution by other components does not matter, we are not
concerned with concentrations
– The concentration problem in the coffee mix example is slightly more
difficult than this one. That is why we went it through in detail, so you
can do those problems in the future…
• You could also have maximum limits, for example for some
contaminant in a reaction mixture
• In real world research (experiments) are used to determine limits
– How much contamination can you tolerate in recycled plastic, by
another type of plastic? There is no ready-to-use model, you will have
to make it.

20. My personal view
• You must be able to “write in equations” and
solve small LP problems
– They come up all the time, at least if you know to look
for them
• Leave big or difficult problems to specialists
– Small: Excel handles up to 200 decision variables
– But to solve in Excel as LP, you have to write
everything in “sumproduct” formulas
– When this does not work, go see a specialist who has
experience with some actual optimization software

21. Work in classroom
• Write down the diet problem as equations
– What are the decision variables?
• Can they be negative?
– What is the objective, is it minimized or
maximized?
– What are the constraints, or quality
requirements?
• Solve this in Excel, or in Google spreadsheet!

22. Key points to know and remember
• To optimize something, you need
– ONE SINGLE numeric objective to minimize or maximize,
often cost or income, or profit
– The values you can choose (decide) are your decision
variables
– Requirements on quality, limits on availability are
constraints
• When all your computations are
sumproduct(decision_vars, constants)
– Then you have a Linear Program
– These are easy to solve reliably, even in spreadsheets
• Making blends often gives linear programs
– Chemical or food industries are all about making blends