Decision Modeling Using Boolean Algebra in

SpreadsheetsBryan Barney Reagan, Ph.D.

Saint Leo University

Law of Excluded MiddleAristotle (384 – 322 BC)

“For every assertion must, as is admitted, be either true or false, ….”

- Aristotle, Section 1, Part 4, “Categories”, The Organon, Translated by E. M. Edghill (2011) Page 15.

Bust of Aristotle. Marble, Roman copy after a Greek bronze original By Lysippos from 330 BC; Photo http://commons.wikimedia.org/wiki/Lysippos

Roots of X*X = XGeorge Boole (1815 -1864 A.D.)

Founded the field of Boolean Algebra in his two works: Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854)

In order to use simple mathematical operations to model logic. In order for X*X to equal X, he used the values zero (0) and one (1) for false and true. (Boole, George, Investigation of the Laws of Thought, Chapter 3. pp. 39-51)

http://www.maa.org

Boole’s Notation

Operation English BooleNegation NOT (1-A)

Conjunction AND A*B

Disjunction OR A+B

Aristotle BooleFalse 0

True 1

Negation

A NOT AFalse True

True False

A NOT A = (1-A)0 1

1 0

Conjunction

A B A AND BFalse False FalseFalse True FalseTrue False FalseTrue True True

A B A*B0 0 00 1 01 0 01 1 1

Disjunction

Disjunction is denoted with +, but it is not simple addition since it must not produce results greater than 1.We can derive an expression for “A + B” using DeMorgan’s Thereom, with which Boole was familiar.

A + B = 1 – ((1-A)*(1-B))= 1 – (1 – A – B + AB) = 1 – 1 + A + B – AB = A + B - AB

Disjunction

A B A OR BFalse False FalseFalse True TrueTrue False TrueTrue True True

A B A + B – (A*B)0 0 00 1 11 0 11 1 1

Distributive Properties in Boolean Algebras

Logical Values in Spreadsheets

Modern spreadsheets such as Excel support the Logical Values TRUE and FALSE , which may be entered as text or the result of comparisons using relational operators.

Comparison Type SampleEquality =(B12 = 42)

Non-Equality =(B12 <> 42)Less Than =(B12 < 42)

Less Than or Equal to =(B12 <= 42)Greater Than =(B12 > 42)

Greater Than or Equal To =(B12 >= 42)

Logical /Boolean Operators in Excel

Operator Excel NotationNegation =NOT(B12)Conjunction =AND(B12, C12)

=AND(B12:C12)Disjunction =OR(B12, C12)

=OR(B12:C12)

Conditional Formulas in Excel

=IF( Test Expression, Value if True, Value if False)

=IF(B12>500, 400, 0)

=IF(AND(B12>500,C12>0), 400, 0)

=IF(B12=MAX($B$2:$B$26),”*”,””)

Conversion Between Logical and Binary Values

• To convert from Logical to Binary, Excel allows multiplication of the values {False, True} by 1 to produce {0, 1}.

= B12 * 1

• To convert from Binary values to Logical, you can use the OR function with a single parameter to convert an value from {0, 1} to {False, True}.

= OR(C12)

Solver Constraints

Thank you Bryan.Reagan@SaintLeo.edu