Operations Research: Linear Programming 1 [Matrices 1]

	OPERATIONS RESEARCH
	6/5/2015 10:42:43
#	Linear Programming [Matrix Variables and Constraints]
1	Study of Operations Research Information from Matrices.
	Given the matrix below:
	Cost Table	Dist Center D	Dist Center E		# of Rows	# of Columns
	Plant A	80	215		3	2
	Plant B	100	108
	Plant C	102	68
	The following information can be determined:
	VARIABLES
	Variables = # of data in matrix, or counting data cells.
	So for above matrix we can use the following excel formula
	Variables (=COUNTA(C8:D10))	6
	CONSTRAINTS
	Constraints =the number of rows + the number of columns.
	Constraints (=SUM(F8+G8))	5

Normal Distribution

Below is my Google Docs Spreadsheet for Normal Distributions:

This sheet you cannot change because I have it locked, but if you would like access so you can use my sheet, and just put in your data and get the answer - email me: messenger1964@yahoo.com

Link to Normal Distribution Page - CLICK HERE!
(Use Math Stat Dist IV tab at bottom)

Normal Distribution
	a	μ	X	σ	Answer Φ(a)
		4.4	2	1.6	-1.5
	b	μ	X	σ	Answer Φ(b)
		10	10.3	0.25	1.2
Formula (p.d.f)
f(x) = 1/(σ√2π)*e^(-(x-μ)^(2)/2σ^(2))	a	⇒	⇒	⇒	0.000000085818477
∀ x ∈ R (-∞, +∞)
Google Sheets Formula =					(=(1/(2*PI())*EXP(-D3-C3)^(2))/2*E3^(2))
Converting P(X) to P(Z) (raw Z score)					Answer Z
P((X-μ)/σ)				a	-1.5
Example: P(X<10.1)= P((X-μ)/σ)<P(10.1-10/.25)				b	1.2
if u = 10, X = 10.1 and σ = .25
Using Standard Norm Table Formulas:
P{Z<a} = Φ(a)			⇒	⇒	-1.5
P{Z>a} = 1-Φ(a)			⇒	⇒	2.5
P{Z<-a} = 1-Φ(a)			⇒	⇒	2.5
P{Z>-a} = Φ(a)			⇒	⇒	-1.5
P{a<Z<b} = Φ(b)-Φ(a)		a<b	⇒	⇒	2.7
P{-a<Z<b} = Φ(b)+Φ(a)-1		a<b	⇒	⇒	-1.3
P{-b<Z<a} = Φ(b)+Φ(a)-1		a<b	⇒	⇒	-1.3
P{-b<Z<a} = Φ(b)-Φ(a)		a<b	⇒	⇒	2.7
Finding Percentiles of Normal Distributions:		Z	μ	σ	Answer (X)
Z = (X-μ)/σ so X = μ+σ*Z		-0.45	4.4	1.2
X = μ+σ*Z =
X=					3.86
Google Sheets Formula =					(=E29+F29*D29)
E[X] =
E[X]=μ		⇒	⇒	⇒	4.4
E[X^(2)] =
σ^(2)+μ^(2)		⇒	⇒	⇒	21.92
Google Sheets Formula =					(=E3^(2)+C3^(2))
Variance
Var = σ^(2) = σ^(2)		⇒	⇒	⇒	2.56
Standard Deviation
St.Dev = √σ		⇒	⇒	⇒	1.6
E[X^(3)] =
6*θ^(3)		⇒	⇒	⇒	24.576
Third Moment of X about the Mean =
E[(X-E[X])^(3)]=
2*θ^(3)		⇒	⇒	⇒	8.192
Moment Generating Function
M(t) = E[e^(tX)] =
e^[μt+((σ^(2)*(t^(2))/2]