System Analysis and Design Tools

Cause-Effect Diagram (Fishbone)

Flow Process Chart

The flow process chart forces a more detailed look at a system.

Modeling Techniques

People/Equipment Requirements

\[M_j = \sum_{i=1}^{n} \frac{P_{ij}T_{ij}}{C_{ij}}\] \[M_j = \textrm{number of machines/people} \\ P = \textrm{production rate} \\ T = \textrm{production time} \\ C = \textrm{production period} \\ n = \textrm{number of products}\]

Material Handling

Euclidian:

\[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]

To optimize material flow:

\[min\quad \sum_{i=1}^mw_i[(x-a_i)^2+(y-b_i)]^{\frac{1}{2}}\] If there are 4 locations with equal weight, the optimal location is the facility within a triangle of the other facilities. If there is no such facility, the optimal location is at the intersection of two lines.

When the weighted costs are proportional to the square of the Euclidean distance, it is called the ‘gravity’ problem.

\[min\quad \sum_{i=1}^mw_i[(x-a_i)^2+(y-b_i)^2]\]

\[x = \frac{\sum_{i=1}^mw_ia_i}{\sum_{i=1}^mw_i}\] \[y = \frac{\sum_{i=1}^mw_ib_i}{\sum_{i=1}^mw_i}\]

Manhattan:

\[|x_2-x_1| + |y_2-y_1|\]

To optimize flow:

\[min\quad \sum_{i=1}^m w_i(|x-a_1|+|y-b_i|)\]

The x value is the median of the location x-coordinates. The y value is the median of the location y-coordinates.

Chebyshev (simultaneous x and y movement)

\[max(|x_2-x_1|,|y_2-y_1|)\]

Relationship Chart

Forecasting Methods

Production Planning Methods

Systems to compute Master Production and Ordering Plan

Material Requirements Planning (MRP)

Manufacturing Resource Planning (MRPII)

Engineering Economics

\[\bigg(\frac{F}{P}\bigg)= (1+i)^N, \quad \bigg(\frac{P}{F}\bigg)= \frac{1}{(1+i)^N} \\ \bigg(\frac{F}{A}\bigg)= \frac{(1+i)^N-1}{i}, \quad \bigg(\frac{P}{A}\bigg)= \frac{(1+i)^N-1}{i(1+i)^N} \\ \bigg(\frac{A}{F}\bigg)= \frac{i}{(1+i)^N-1}, \quad \bigg(\frac{A}{P}\bigg)= \frac{i(1+i)^N}{(1+i)^N-1} \\ \bigg(\frac{P}{G}\bigg)=\frac{1}{i}\bigg[\frac{(1+i)^N-1}{i(1+i)^N}-\frac{N}{(1+i)^N} \bigg], \quad \bigg(\frac{A}{G}\bigg)=\frac{1}{i}-\frac{N}{(1+i)^N-1}\]

*Denominator is current value and Numerator is desired conversion

Depreciation

Modified Accelerated Cost Recovery System (MACRS) – See Tables

\[\text{Straight Line (SL) – } \frac{1}{n}\]

Production Scheduling Methods

Makespan

the time it takes from the start of the first job until the end of the last job

Scheduling Sequence

1. Earliest Due Date – order jobs by due date
2. Shortest Processing Time – order jobs by processing time
3. Critical Ratio – divide time remaining until due date by time left on the machine, order by smallest critical ratio

Johnson’s Optimal Rule for Two Machines

1. Find the shortest processsing times and arbitrarily break ties
2. If the shortest processing time is on Machine A, schedule immediately. If the shortest processing time is on Machine B, schedule it as late as possible.
3. Eliminate the last job scheduled on the list and repeat step 1-2.

Inventory Management and Control

Distribution Methods

Transhipment:

The intermediary storage

Transportation Modes

1. Variable Path
   • truck, vehicle anything that does not have one fixed path
   • versatility
2. Fixed Path
   • conveyor
   • tied to one path

Assignment Problem

Hungarian Procedure:

1. Subtract the minimum of the row from all elements in the row
2. Substract the minimum of the column from all elements in the columns
3. Try to make a valid assignment using the zero elements, if all assigments cannot be made proceed to next step
4. Cover all zeroes with the minimal number of lines
5. From each uncovered element subtract the minimum of the uncovered y, add y to each intersection element. Go to step 3.
6. Transfer the assignment plan to the original cost table.

Controls

An administrative control are training, policies, or procedures.

An engineering control is a physical modification to mitigate hazards.

Noise Dose

Dose

\[D=100*\big(\frac{C_1}{T_1}+\frac{C_2}{T_2}+…+\frac{C_n}{T_n}\big)\le 100\]

Time Weighted Average

\[TWA=16.61log_{10}\big(\frac{D}{100}\big)+90\]

Exposure

Time Weighted Concentration

\[TWA=\frac{\sum_{i=1}^nC_iT_i}{\sum_{i=1}^nT_i}\]

Taylor Tool Life

\[VT^n=C \\ V = \textrm{speed surface feet per minute} \\ T = \textrm{tool life in minutes} \\ C,n = \textrm{constants that depend on material and tool}\]

Work Sampling

\[D = Z_{\alpha/2}\sqrt{\frac{p(-1-p)}{n}}, \quad Z_{\alpha/2}\sqrt{\frac{1-p}{pn}} \\ p = \textrm{proportion of observed time} \\ D = \textrm{absolute error} \\ R = \textrm{relative error} = \frac{D}{p} \\ n = \textrm{sample size}\]

Sample Size

\[E = \frac{z_{\frac{\alpha}{2}}\sigma}{\sqrt{n}}\]

\[n = \bigg( \frac{z_{\frac{\alpha}{2}}\sigma}{E} \bigg)^2\]

Critical Path Method

\[T = \sum_{(i,j)\in CP}d_{ij}\]

Standard Time

\[\textrm{Observed Time * Pace Rating * (1 + personal time allowance) * (1 + fatigue allowance)}\]

Recommended Weight Limit

Units are pounds and inches.

\[RWL = 51\cdot (\frac{10}{H})\cdot (1-.0075|V-30|)\cdot (.82+\frac{1.8}{D})\cdot (1-.0032A)\cdot FM \cdot CM \\ \textrm{H = horizontal location of the load forward of the midpoint of the ankles} \\ \textrm{V = vertical location of the load} \\ \textrm{D = Vertical travel distance between the origin and the destination} \\ \textrm{A=angle of asymmetry between hands and feet} \\ \textrm{FM = frequency multiplier (from table)} \\ \textrm{CM = coupling mulitiplier (from table)}\]

Learning Curve

\[y=kx^n, n=\frac{log_{e}\phi}{log_{e}(2)} \\ \phi=\textrm{learning ratio}=\frac{T(2N)}{T(N)}, \textrm{T(N) = time to produce Nth unit} \\ \textrm{y= time to produce xth unit, k = time to produce first unit, x = cumulative number of units produced}\]

Total Learning Time:

\[T=k\frac{[(x_2+\frac{1}{2})^{n+1}-(x_1+\frac{1}{2})^{n+1}]}{n+1}\]

Remission Line:

\[y=k+\frac{(k-s)(x-1)}{1-x_s}\]

Statistical Process Control

Control vs. Capability

In control if it is within natural variability

Is capable if it is entirely within specification

Process Capability

Actual Capability:

\[C_{pk}=min\bigg(\frac{\mu-LSL}{3\sigma},\frac{USL-\mu}{3\sigma}\bigg)\]

Potential Capability:

\[C_p = \frac{USL-LSL}{6\sigma}\]

Reliability Analysis

Series:

\[R = \prod_{i=1}^n P_i\]

Parallel:

\[R = 1-\prod_{i=1}^n (1-P_i) \]

Hazard Function

\[h(x)=\frac{f(x)}{R(x)} \\ f(x) \text{ = density function, } R(x) \text{ = survival function}\]

Exponential \[h(x)=\lambda\]

Weibull \[h(x)=\frac{\beta}{\alpha}\big(\frac{x}{\alpha}\big)^{\beta-1}\]

Mean Time to Failure

\[\frac{1}{\lambda}, \\ \lambda \text{ = constant failure rate}\]

Six Sigma

Normal Distribution

z-score

\[z=\frac{x-\mu}{\sigma}\]

Confidence Interval

\[\bar{x}\pm\frac{z_{\alpha/2} \sigma}{\sqrt{n}}\]

Two-means comparison:

\[z_0=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}\]

student-t Distribution

t-score:

\[t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\]

Confidence Interval

\[\bar{x}\pm\frac{t_{\alpha/2,n-1}s}{\sqrt{n}}\]

Two-means comparison:

\[t_0=\frac{\bar{x}_1-\bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\]

df for Two Sample t-test:

\[df=\frac{\big(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\big)^2}{\frac{(\frac{s_1^2}{n_1})^2}{n_1-1}+{\frac{(\frac{s_1^2}{n_1})^2}{n_1-1}}}\] Paired t-test:

\[t_0 = \frac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}\]

Hypothesis Testing

Chi-Squared Goodness of Fit

\[\chi^2=\sum_{j=1}^k\frac{(O_j-E_j)^2}{E_j}\]

Linear Regression

\[SSR=\sum_{i=1}^n(\hat{y}_i-\bar{y})^2\]

\[SSE = \sum_{i=1}^n(y_i-\hat{y}_i)^2\]

\[SST = \sum_{i=1}^n(y_i-\bar{y})^2\]

\[R^2=\frac{SSR}{SST} = 1-\frac{SSE}{SST}\]

ANOVA

\[SSA+SSE=SST\]

One-Way

Given Treatment A:

\[SSA+SSE=SST\]

Two-Way

Given treatment factors A & B:

\[SST=SSA+SSB+SSAB+SSE\]

Bayesian Analysis

Bayes’ Theorem

\[P(A|B)=\frac{P(B|A)P(A)}{P(B)}=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^\prime)P(A^\prime)}\]

Distributions

\(\text{Factors for Control Charts}\)

\(\text{Normal Distribution}\)

\(t \text{ Distribution}\)

\(\chi^2 \text{ Distribution}\)

\(F(v_1, v_2) \text{ Distribution}\)

References

[[1]]
Krishnamoorthi K (2006). _A First Course in Quality Engineering:
Integrating Statistical and Management Methods of Quality_.
Pearson Prentice Hall. ISBN 9780131472013, <URL:
https://books.google.com/books?id=lPQJAQAAMAAJ>.

[[2]]
Kutner M, Nachtsheim C, Neter J and Li W (2004). _Applied Linear
Statistical Models with Student CD_. McGraw-Hill
Companies,Incorporated. ISBN 9780073108742, <URL:
https://books.google.com/books?id=0Qq-swEACAAJ>.

[[3]]
Sullivan W, Wicks E and Luxhoj J (2006). _Engineering Economy_,
number v. 1 series Engineering Economy. Pearson/Prentice Hall.
ISBN 9780131486492, <URL:
https://books.google.com/books?id=6pcoAQAAMAAJ>.

[[4]]
Gross D, Shortle J, Thompson J and Harris C (2011). _Fundamentals
of Queueing Theory_, series Wiley Series in Probability and
Statistics. Wiley. ISBN 9781118211649, <URL:
https://books.google.com/books?id=K3lQGeCtAJgC>.

[[5]]
Winston W (2003). _Introduction to Probability Models_. Cengage
Learning. ISBN 9780534405786, <URL:
https://books.google.com/books?id=L9yURQAACAAJ>.

[[6]]
Niebel B and Freivalds A (2003). _Methods, Standards, and Work
Design_, series McGraw-Hill series in industrial engineering and
management science. McGraw-Hill. ISBN 9780072468243, <URL:
https://books.google.com/books?id=ApXSoagqrxcC>.