Haulwind

Reblog

Mar 31

QUEUING THEORY [M/M/S and M/M/S/K]

Using material from

Frederick S. Hiller, Gerald J. Lieberman. Introduction to Operations Research. New York: McGraw Hill, 2005. Print.

INPUT

L: Mean entry rate follows POISSON input process

U: Mean service rate follows EXPONENTIAL distribution

S: Number of servers

K: Number of waiting places

FIRST COME FIRST SERVED ASSUMPTION

OUTPUT

A: Average number of customers in the system

B: Average number of customers in the queue

C: Average waiting time in the system

D: Average waiting time in the queue

P: Likelihood that mean entry rate will yield this N customers in the system

Reblog

Feb 16

ANALYTIC HIERARCHY PROCESS

Using material from
Saaty L. Thomas, The analytic hierarchy process: planning, priority setting, resource allocation, McGraw-Hill, 1980.

INPUT

B = [ B1 B2 B3 B4

B5 B6 B7 B8

B9 B10 B11 B12

B13 B14 B15 B16 ]

= [B]

OUTPUT

CONFIDENCE INDEX

WEIGHT OF EACH COLUMN FACTOR

Reblog

Feb 16

ANOVA [Two-Factor Factorial Experiment, Fixed Effects Model]

Using material from
Montgomery C. Douglas and Runger C. George, Applied Statistics and Probability for Engineers, Wiley, 2007.

INPUT

H = [ H1 H2 H3 H4 H5

H6 H7 H8 H9 H10

H11 H12 H13 H14 H15 ]

= [H]

I = [ I1 I2 I3 I4 I5

I6 I7 I8 I9 I10

I11 I12 I13 I14 I15 ]

= [I]

J = [ J1 J2 J3 J4 J5

J6 J7 J8 J9 J10

J11 J12 J13 J14 J15 ]

= [J]

Assuming [H], [I], and [J] are three settings where each setting contains three levels and each level contains 5 replications. Levels and settings could represent two factors.

Null hypothesis: there is no difference between the means of the different levels

Alternative: at least two of the means are different

OUTPUT

sources of variation SS df MS F

Sample A X L

Columns B Y J

Interaction C V K

Within E W

Total D

Level of significance, confidence interval

Hypothesis testing

Reblog

Feb 16

SIMPLEX METHOD [Maximization]

Using material from
Frederick S. Hiller, Gerald J. Lieberman. Introduction to Operations Research. New York: McGraw Hill, 2005. Print.

maximize Z, which is the objective function

where Z = C * X AND A * X ≤ b

where C, X, and b are vectors; and A is a Matrix

e.g. max Z;

C1* X1 + C2* X2 = Z

A1* X1 + A2* X2 ≤ b1

A3* X1 + A4* X2 ≤ b2

A5* X1 + A5* X2 ≤ b3

INPUT

C = { C1, C2 } = L1

b = { b1, b2, b3 } = L2

A = [ A1 A2

A3 A4

A5 A6 ] = [A]

OUTPUT

X = [ X1 X2 ] = LE

Z = Z

Reblog

Feb 16

SIMPLEX METHOD [Maximization, Minimization]

Using material from
Frederick S. Hiller, Gerald J. Lieberman. Introduction to Operations Research. New York: McGraw Hill, 2005. Print.

maximize Z, which is the objective function

where Z = C * X AND A * X ≤ b

where C, X, and b are vectors; and A is a Matrix

e.g. max Z;

C1X1 + C2X2 = Z

A1X1+ A2X2 ≤ b1

A3X1+ A4X2 ≤ b2

A5X1+ A5X2 ≤ b3

MAXIMIZATION INPUT

C = {C1, C2}= L1

b = {b1, b2,b3}= L2

A = [ A1 A2

A3 A4

A5 A6 ]

= [A]

OUTPUT

X = [X1 X2]= LE

Z = Z

Reblog

Feb 16

CHI-SQUARE TEST

Using material from
Montgomery C. Douglas and Runger C. George, Applied Statistics and Probability for Engineers, Wiley, 2007.

INPUT

F = [ F1 F2 F3 F4 F5 F6 F7

F8 F9 F10 F11 F12 F13 F14

F15 F16 F17 F18 F19 F20 F21 ]

= [F]

Assuming these are three separate samples and each sample contains 7 parameters

OUTPUT

Null hypothesis: there is no difference between the means of the different levels

Alternative: at least two of the means are different

Reblog

Feb 16

ANOVA [One Level Analysis of Variance]

Using material from
Montgomery C. Douglas and Runger C. George, Applied Statistics and Probability for Engineers, Wiley, 2007.

INPUT

E = [ E1 E2 E3 E4 E5

E6 E7 E8 E9 E10

E11 E12 E13 E14 E15 ]

= [E]

Assuming these are three separate levels and each level contains 5 replications

[ E1 E2 E3 E4 E5]

[ E6 E7 E8 E9 E10 ]

[ E11 E12 E13 E14 E15 ]

Null hypothesis: there is no difference between the means of the different levels

Alternative: at least two of the means are different

OUTPUT

sources of variation SS df MS F

between groups Z C E

within groups B D

Level of significance, confidence interval

Hypothesis testing

Reblog

Feb 16

CRAMER’S RULE [Solving System of Two Linear Equations]

By Jonathan L. Moscovici

Using material from
Weisstein, Eric W. “Cramer’s Rule.” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CramersRule.html

Here we present a program to solve the system:

Ax + By = C

Dx + Ey = F

Reblog

Feb 16

Quadratic Expression [From Canonical To General Form]

By Jonathan L. Moscovici

Here we present a program to convert quadratic expression in canonical form:

A(x – H)² + K

to general form:

Ax² + Bx + C

Reblog

Feb 16

TRIANGLE AREA

By Jonathan L. Moscovici

Using material from
Kimberling, C. “Triangle Centers and Central Triangles.” Congr. Numer. 129, 1-295, 1998.