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% Document Info
\author{musikdr}
\pdfinfo{
  /Title (ise-463-mid-term.pdf)
  /Creator (Cheatography)
  /Author (musikdr)
  /Subject (ISE 463 MID-TERM Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{ISE 463 MID-TERM Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{musikdr} via \textcolor{DarkBackground}{\uline{cheatography.com/93594/cs/20651/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}musikdr \\
  \uline{cheatography.com/musikdr} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 3rd October, 2019.\\
   Updated 3rd October, 2019.\\
   Page {\thepage} of \pageref{LastPage}.
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\end{multicols}}




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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.55618 cm} x{0.9154 cm} x{2.10542 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Little's Law}}  \tn
% Row 0
\SetRowColor{LightBackground}
WORK IN PROGRESS & WIP = TH*CT &  \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Cycle Time & CT & WIP/TH \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Throughput & TH & WIP/CT \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
Bottleneck Rate & rb & =1/Max Avg Processing Time \tn 
% Row Count 6 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Raw Processing Time & To & Sum of Avg Processing Time \tn 
% Row Count 8 (+ 2)
% Row 5
\SetRowColor{white}
Critical WIP & Wo & rb*To \tn 
% Row Count 9 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.60195 cm} x{2.05965 cm} p{0.9154 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{BEST CASE PERFORMANCE}}  \tn
% Row 0
\SetRowColor{LightBackground}
CT BEST & if w\textless{}=Wo & To \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
 & otherwise & W/rb \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
TH BEST & if w\textless{}=Wo & w/To \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
 & otherwise & rb \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.19696 cm} p{1.92234 cm} p{0.4577 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{WORST CASE PERFORMANCE}}  \tn
% Row 0
\SetRowColor{LightBackground}
CTworst & =w*To &  \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
THworst & 1/To &  \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{1.29402 cm} x{3.68298 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{PRACTICAL WORST CASE}}  \tn
% Row 0
\SetRowColor{LightBackground}
CTpwc & To+((w-1)/rb) \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
THpwc & (w/(Wo+w-1))rb \tn 
% Row Count 2 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sample Midterm Qsle Midterm Qs (Cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{b) A company supplying seats to an auto assembly plant sends trucks to its customer at an average rate of 6 trucks per day. Given the travel time to the customer is an average of three days, what is the average number of trucks in transit at any given time?}} \newline % Row Count 6 (+ 6)
TH = 6 trucks/day \newline % Row Count 7 (+ 1)
CT = 3 days \newline % Row Count 8 (+ 1)
WIP = TH x CT = 18 trucks% Row Count 9 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33427 cm} p{0.54924 cm} x{1.69349 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{PREEMPTIVE ONLY}}  \tn
% Row 0
\SetRowColor{LightBackground}
Natural Proc. Time & to &  \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
STD of Nat. Proc. Time & σo &  \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
SCV of Nat. Proc. Time & co\textasciicircum{}2\textasciicircum{} & σo\textasciicircum{}2\textasciicircum{}/to\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
STD of Nat. Proc. Time & mf &  \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Mean Time to Repair & mr &  \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
STD of Time to Repair & σr &  \tn 
% Row Count 10 (+ 2)
% Row 6
\SetRowColor{LightBackground}
Mean Availability & A & mf/(mf+mr) \tn 
% Row Count 11 (+ 1)
% Row 7
\SetRowColor{white}
SCV of Time to Repair & cr\textasciicircum{}2\textasciicircum{} & σr\textasciicircum{}2\textasciicircum{}/mr \tn 
% Row Count 13 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Mean Eff. Time with Preemptive Outages & \seqsplit{te(Po)} & to/A \tn 
% Row Count 15 (+ 2)
% Row 9
\SetRowColor{white}
SCV of Eff. Time with Preemptive Outages & ce(PO)\textasciicircum{}2\textasciicircum{} & co\textasciicircum{}2\textasciicircum{}+(1+cr\textasciicircum{}2\textasciicircum{}){\emph{A}}(1-A)*mr/to \tn 
% Row Count 18 (+ 3)
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\begin{tabularx}{5.377cm}{x{1.96811 cm} p{0.4577 cm} x{2.15119 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{PREEMPTIVE PLUS NON PREEMPTIVE}}  \tn
% Row 0
\SetRowColor{LightBackground}
Mean batch size & Ns &  \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Mean batch size & ts &  \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
STD of Setup Time & σs &  \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
Mean Eff. Time with Preemptive Outages & te & te(PO)+ts/Ns \tn 
% Row Count 6 (+ 3)
% Row 4
\SetRowColor{LightBackground}
std. dev. Squared of eff. Time & σe\textasciicircum{}2\textasciicircum{} & te(po)\textasciicircum{}2\textasciicircum{} x cd(po)\textasciicircum{}2\textasciicircum{}+(rs\textasciicircum{}2\textasciicircum{}/Ns) + (Ns-1/Ns\textasciicircum{}2\textasciicircum{}) x ts\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 10 (+ 4)
% Row 5
\SetRowColor{white}
SCV of Eff. Time with Preemptive Outages & ce\textasciicircum{}2\textasciicircum{} & σe\textasciicircum{}2\textasciicircum{}/te\textasciicircum{}2\textasciicircum{} \tn 
% Row Count 13 (+ 3)
% Row 6
\SetRowColor{LightBackground}
Mean Utilization & u & te/ta \tn 
% Row Count 14 (+ 1)
% Row 7
\SetRowColor{white}
SCV of interarrival times & ca\textasciicircum{}2\textasciicircum{} &  \tn 
% Row Count 16 (+ 2)
% Row 8
\SetRowColor{LightBackground}
SCV of interdeparture times & cd\textasciicircum{}2\textasciicircum{} & 1 + (1-u\textasciicircum{}2\textasciicircum{}) x (ca\textasciicircum{}2\textasciicircum{}-1) + u\textasciicircum{}2\textasciicircum{}(ce\textasciicircum{}2\textasciicircum{}-1) \tn 
% Row Count 19 (+ 3)
% Row 9
\SetRowColor{white}
CTQ &  & (ca\textasciicircum{}2\textasciicircum{} + ce\textasciicircum{}2\textasciicircum{})/2 x u/(1-u) x te \tn 
% Row Count 21 (+ 2)
% Row 10
\SetRowColor{LightBackground}
CT &  & CTQ + te \tn 
% Row Count 22 (+ 1)
% Row 11
\SetRowColor{white}
WIP &  & CT/ta \tn 
% Row Count 23 (+ 1)
% Row 12
\SetRowColor{LightBackground}
WIP &  & CT/ta \tn 
% Row Count 24 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sample Midterm Qsle Midterm Qs (Cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{b) A company supplying seats to an auto assembly plant sends trucks to its customer at an average rate of 6 trucks per day. Given the travel time to the customer is an average of three days, what is the average number of trucks in transit at any given time?}} \newline % Row Count 6 (+ 6)
TH = 6 trucks/day \newline % Row Count 7 (+ 1)
CT = 3 days \newline % Row Count 8 (+ 1)
WIP = TH x CT = 18 trucks% Row Count 9 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.18988 cm} x{2.78712 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Sample Midterm Qs}}  \tn
% Row 0
\SetRowColor{LightBackground}
SCV of Effective processing time & ce\textasciicircum{}2\textasciicircum{} = ((σo\textasciicircum{}2\textasciicircum{} + (σs\textasciicircum{}2\textasciicircum{}/N) + (Ns-1)/Ns\textasciicircum{}2\textasciicircum{} x ts\textasciicircum{}2\textasciicircum{})/te \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Utilization & u = (te/ta) = (te(np)/ta) \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Utilization (coffee sho) & u = ra/re=te/ta \tn 
% Row Count 7 (+ 2)
% Row 3
\SetRowColor{white}
WIP   (M/M/1) & u/(1-u) \tn 
% Row Count 8 (+ 1)
% Row 4
\SetRowColor{LightBackground}
WIP   (M/M/1) & u/(1-u) \tn 
% Row Count 9 (+ 1)
% Row 5
\SetRowColor{white}
CT  (M/M/1) & te/(1-u) \tn 
% Row Count 10 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sample Midterm Qs (Cont)}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{\bf{3. a) Compute the average cycle time at machine 1.}} \newline % Row Count 2 (+ 2)
CTq1 =( Ca\textasciicircum{}2\textasciicircum{} + Ce\textasciicircum{}2\textasciicircum{})/2 x (u/(1-u)) x te \newline % Row Count 3 (+ 1)
u = (te/ta) \newline % Row Count 4 (+ 1)
{\bf{b) Compute the mean and coefficient of variation of the time between departures from Machine 1.}} \newline % Row Count 6 (+ 2)
ta(2) = td(1) =ta(1) \newline % Row Count 7 (+ 1)
cd\textasciicircum{}2\textasciicircum{} = 1 + (1-u\textasciicircum{}2\textasciicircum{}) x (ca\textasciicircum{}2\textasciicircum{} -1) + u\textasciicircum{}2\textasciicircum{}(ce\textasciicircum{}2\textasciicircum{} - 1) \newline % Row Count 9 (+ 2)
u2 = te2/ta2 = 18/22 \newline % Row Count 10 (+ 1)
{\bf{c) Compute the average cycle time at machine 2}} \newline % Row Count 12 (+ 2)
CT(2) = CTq(2) + te(2)  (note use u2 to calculate CTq(2) \newline % Row Count 14 (+ 2)
{\bf{d) Calculate the total CT and WIP of the system (combining machine 1 and 2).}} \newline % Row Count 16 (+ 2)
Total CT = CT(1) + CT(2) \newline % Row Count 17 (+ 1)
From Little's Law \newline % Row Count 18 (+ 1)
Total WIP = CT x TH = CT/ta \newline % Row Count 19 (+ 1)
{\bf{e) Now suppose the line must produce both products in equal proportion, i.e., one unit of Product 1 for each unit of Product 2. Estimate the bottleneck rate and raw process time of the line under this product mix.}}  \newline % Row Count 24 (+ 5)
Hint: Think about the what the average processing time will be at each machine. \newline % Row Count 26 (+ 2)
Processing time at (M1 + M2)/\#of machines (2 calculations ..1 for each product) \newline % Row Count 28 (+ 2)
rb = 1/largest processing \newline % Row Count 29 (+ 1)
to = to1+to2 (answer from processing time)% Row Count 30 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{p{0.54747 cm} x{4.42953 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Kendall Notation}}  \tn
% Row 0
\SetRowColor{LightBackground}
M & Memoryless or exponential \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
D & Deterministic \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
G & General \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
G/G/1 & not exponential, gives approx CT and CTq \tn 
% Row Count 5 (+ 2)
% Row 4
\SetRowColor{LightBackground}
M/M/1 & exponential, infinite source population unlimited queue length \tn 
% Row Count 7 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{M/M/1 Queing}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{WIP = u/(1-u) \newline % Row Count 1 (+ 1)
CT = WIP/ra = te/(1-u) \newline % Row Count 2 (+ 1)
CTq = CT-te =(u x te)/(1-u) \newline % Row Count 3 (+ 1)
WIPq =raCTq=u\textasciicircum{}2\textasciicircum{}/(1-u)% Row Count 4 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{G/G/1 QUE}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{CT = ((ca\textasciicircum{}2\textasciicircum{} + ce\textasciicircum{}2\textasciicircum{})/2)(u/(1-u) x te% Row Count 1 (+ 1)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{M/M/1/b}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{WIP = u/(1-u) - ((b+1)u\textasciicircum{}b+1\textasciicircum{}) / 1-u\textasciicircum{}b+1\textasciicircum{} \newline % Row Count 1 (+ 1)
TH = ((1-u\textasciicircum{}b\textasciicircum{}) / (1-u\textasciicircum{}b+1\textasciicircum{}) x ra \newline % Row Count 2 (+ 1)
• Smaller buffer sizes bring greater losses relative to uncapacitated system% Row Count 4 (+ 2)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}