TITLE: Power law correlation between with three parameters
QUESTION [0 upvotes]: I made a CFD model of the Centrifugal Atomization Process. Studies show that the mean particle size $d_{32}$ can be expressed as a function of the three operating dimensionless numbers (Reynolds, Ekman and Ohnesorge).
Previous works made up correlations in power law as
$$d_{32}^* = \frac{d_{32}}{D} = \mathrm{const.} \times \mathrm{Ek}^{\alpha_1} \times \mathrm{Re}^{\alpha_2} \times \mathrm{Oh}^{\alpha_3} $$
from parametric studies where $d_{32}^*$ is calculated with two dimensionless numbers fixed and varying one.
I have the data from these parametric studies and I would like to create one of this correlation but I have no idea how to do it. The following tab is an example of (fake) data where the first row in the initial state ; row 2 and 3 are data for the Reynolds number parametric study ; row 4 and 5 are data for the Ekman number parametric study ; and row 6 and 7 are data for the Ohnesorge number parametric study.
Edit: I wrote the real data.

I guess the answer is easy to find on internet but I guess I don't have the right vocabulary I don't find anything.
Thank you for your help.
Best.

REPLY [0 votes]: Solver is an absolute BEAST when it comes to solving least squares problems.  As an example, if you want to do a least squares curve fit on a linear equation, you would take your desired equation of $y=mx+b$, and modify it such that it reads $y-mx-b=0$.  This modified equation can be slightly changed such that it becomes a least squares problem by squaring it thusly: $(y-mx-b)^2 = 0$.  The left hand side of this equation represents the error squared term of the equation if starting guesses for m and b are incorrect.  This equation ONLY converges to a minimum value when the best values of m and b are selected.  This means that for this particular problem, the objective function for the linear curve fit is that last equation, and Solver should minimize this equation by manipulating the values of m and b.
As practice, I have given the following two problems to high school students in order to teach them the technique.  After those two problems, I posted how I set up your problem.  The two practice problems are:
Sample Solver Problem 1 (Inseparable variables)

Open a new Excel spreadsheet.
At spreadsheet cell A1, enter “x”.
Go down one cell to A2, and enter an arbitrary starting value (e.g., 2).
Assume that you are solving the equation: $x^3 + x^{1/2} =100$            (1)
Rewrite equation (1) to make this a least squares problem, which Solver is an absolute expert at solving: $(x^3 + x^{1/2} - 100)^2 = 0$                            (2)
Enter “err^2” as a title in cell A5.
Enter equation (2) in cell A6, using the Excel format “=(a2^3+a2^0.5-100)^2”.    When finished with this equation hit the “enter” key to enter it.  Note that “a2” refers to cell a2, which is where your starting guess for the correct value of “x” is located.
Select the Data tab on the menu bar, and start Solver.
For Solver Parameters (in the pop-up screen), choose “Set Objective”, clear out what is initially in the box, and click on cell A6 in the spreadsheet, which is the cell that you are trying to minimize.
Since this is a minimization problem, select “Min” in the Solver Parameters “Equal To” area
Click in the “By Changing Cells” box of the Solver Parameters pop-up
Click on cell A2 of the spreadsheet to select the value that you are manipulating in order to minimize equation (2).
Select “Solve” in the Solver Parameters pop-up screen.
Your answer should be 4.608135.

Sample Solver Problem 3 (Curve Fitting)

Open an Excel spreadsheet
Assume that you are trying to perform a least squares regression on a linear equation.  Further assume that this problem has 5 data points.  Thus, y = mx + b for this example, but note that the actual equation that you can fit with this method is limited only by your imagination.
At cell A1, enter label “m”, and at cell B1, enter label “b”.
At cell A2, enter a starting guess for m (e.g., 2).
At cell B2, enter a starting guess for b (e.g., 2).
At cell A5, enter label “x”, to designate the column that will be used for independent variable values.
At cell B5, enter label “y”, to designate the column that will be used for dependent variable values.
Starting at cell A6, enter 5 x values into the spreadsheet, going down in column A.
Starting at cell B6, enter 5 y values into the spreadsheet, going down in column B.
At cell C5, enter label “y-pred”, to designate the column that will be used for the predicted value of y values
At cell C6, enter the equation that predicts the y value for the given x value.  This equation is “=$A$2*A6+$b$2”.
Copy cell C6 into cells C7 through C10.
At cell D5, enter label “err^2”.
At cell D6, enter the equation for the error squared term, based on the chosen values of m and b.  This equation is “=(B6-C6)^2”.
Copy cell D6 into cells D7 through D10.
Go to cell D11, and sum all of the error squared terms in column D by entering the equation “=SUM(D6:D10)”.
Select the Data tab from the top menu bar and start Solver.
In the Solver Parameters pop-up screen, select “Set Objective” and clear the contents of the box.
Go to the spreadsheet and click on cell D11, which is the sum of all of the error squared terms.
In the Solver Parameters pop-up screen, select “Min” on the “Equal To:” line.
In the Solver Parameters pop-up screen, go to the box “By Changing Cells:” and click in the box.
Go to the spreadsheet, and highlight cells A2 and B2 by selecting cell A2, holding down the left mouse button, and dragging the mouse to cell B2.
Once cells A2 and B2 have been selected, choose “Solve” in the Solver Parameters pop-up screen.
When Solver converges, your answers for m and b will be in cells A2 and B2.
Verify “m” and “b” by using Excel’s plotting and trend line functions

Your problem:
1) Set up your problem as a curve fitting problem
2) Your objective function should be $(d_{32} - const * E_k^{alpha1} * R_e^{alpha2} * O_h^{alpha3})^2$
3) Your 4 manipulated variables should be const, alpha1, alpha2, and alpha3.  Choose a value other than 0 or 1 as starting guesses.  Note that I chose a value of 2 for all of my starting guesses for these four unknown values.
4) Tell solver to minimize the sum of the error squared terms that the objective function is calculating, by manipulating the stated manipulated variables.  When you run Solver, it should give you the values that I previously posted.


TITLE: If $X \sim \text{Exp}(1)$, show that $\sqrt X$ follows a Rayleigh distribution.
QUESTION [2 upvotes]: I have a task to show that if $$X  \sim \text{Exp}(1)$$ then $$\sqrt X \sim \mathrm{Rayleigh}\left(\frac{1}{\sqrt 2}\right)$$
How do I start from here?
For $X\sim$ exponential, I know that the pdf is defined as
$$f_X(x) = \lambda e^{-\lambda x}$$
and for Rayleigh
$$f_X(x) = \frac{x}{\lambda ^2} e^{\frac{-(\lambda x)^2}{2}}$$

REPLY [3 votes]: How do I start?

Notice that $Y = \sqrt X$ is one to one over the possible values of $X$. Hence you can you use the one to one transformation; $X = Y^2$, and so
$$f_Y(y) = \frac{f_X(y^2)}{\left|\frac{dy}{dx}\right|_{x = y^2}}.$$
Otherwise, you can use the cdf method,
$$P(Y\leq y) = P(\sqrt X\leq y) = P(X\leq y^2).$$


TITLE: Understanding Bra Ket notation with two items inside.
QUESTION [0 upvotes]: When working in quantum mechanics, an operator can be written as a "ket".
$\left| n \right> $
However, what does it mean when we have two items inside the ket? Such as $\left| l ,m \right>$. My notes say that "Here we adopt the Dirac notation in a way such that $\left|l,m\right> \equiv \psi^l_m$". I am confused as to what $\left| l ,m \right>$ is. Does it have something to do with there now being two dimensions? How would lowering and raising operators work on it?

REPLY [0 votes]: Does it have something to do with there now being two dimensions?

Hamiltonians with symmetrical 2D potentials that we can solve explicitly often have eigenfunctions that can be naturally labeled by two parameters. If we have an irregular potential ("random" bumps and valleys), there will not necessarily be a natural two parameter labeling. In either case, they are just labels - we could label them $\left| a \right>, \left| b \right>, \left| c \right>, $ ... .


TITLE: Sequence convergence and limits
QUESTION [1 upvotes]: if I have a sequence $(x_n)_{n\geq0}$ such that $\lim_{n \to\infty } x_n = c$.
Then why can I write:
 $\left | c-x_n \right | = \lim_{m \to\infty } \left | x_m - x_n \right |$
I tried to work it out but I get an inequality: $\left | c-x_n \right | \leqslant \left | c-x_m \right | + \left | x_m-x_n \right |$ Then as m tends to infinity the first term of the right side tends to zero. Is this the right way? What am I missing that makes both sides equal?
Thanks in advance.

REPLY [1 votes]: Since $x_m \to c$, we have for any $\epsilon > 0$, $|x_m -c| < \epsilon$ when $m$ is sufficiently large.
Using the reverse triangle inequality,
$$||x_m-x_n|-|c-x_n|| = ||x_m-x_n|-|x_n-c|| \\\leqslant |(x_m-x_n)+(x_n-c)| \\= |x_m-c| < \epsilon.$$
Hence,
$$\lim_{m \to \infty}|x_m - x_n| = |c - x_n|$$


TITLE: Size of complement of context-free language
QUESTION [6 upvotes]: Let $L$ be a context-free language, $\bar L$ be its complement and $\bar L_n$ be the length $n$ words in $\bar L_n$.

What is known about $|\bar L_n|$?

Note that it is known that $|L_n|$ is either polynomial (if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|\bar L_n|$.
Warning, $L$ is allowed to be ambiguous!

REPLY [11 votes]: From the proof that determining if a CFL ${L}$ = $\Sigma^*$ is undecidable, the set of strings $ID_0\#ID_1^R\#ID_2\#ID_3^R\#\ldots\#ID_t$ where $ID_0,ID_1,\ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|\overline{L}_n|$ can basically be any computable function less than exponential.


TITLE: Can we always find a primitive element that is a square?
QUESTION [13 upvotes]: Let $L/\mathbb Q$ be a finite field extension. The Primitive Element Theorem says that there is an element $\alpha \in L$ so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$ so that $L=\mathbb Q(\beta^2)$ ?

REPLY [13 votes]: $\newcommand{\Q}{\mathbb Q}$
I think this works. Let $K/\Q$ be a finite field extension. Note that $K$ has finitely many subfields, call them $F_1,\dots,F_t$ and let $W$ be their union. Then any element in $K \setminus W$ is necessarily a primitive element. So it suffices to show their exists some $\alpha \in K \setminus W$ that has a square root. Pick $\alpha \in K \setminus W$, note that for $k \in \Q$ we still have $\alpha_k=\alpha + k \in K \setminus W$ otherwise $\alpha \in W$. Consider $\alpha_k^2=(\alpha+k)^2=\alpha^2+2k\alpha+k^2$ suppose that for every $k \in \Q \setminus {0}$ we have $\alpha_k^2 \in W$. Then by the pigeonhole principle we must have some $j$ and $k_1 \neq k_2$ such that $\alpha_{k_1}^2,\alpha_{k_2}^2 \in F_j$. So we have
$$\frac{\alpha_{k_1}^2-\alpha_{k_2}^2-(k_1^2+k_2^2)}{2(k_1-k_2)} \in F_j$$
but
$$ \frac{\alpha_{k_1}^2-\alpha_{k_2}^2-(k_1^2+k_2^2)}{2(k_1-k_2)} =\frac{2\alpha(k_1-k_2)}{2(k_1-k_2)}=\alpha.  $$
Which would imply that $\alpha \in F_j \subset W$ so we have some $k \in \Q\setminus \{0\}$ such that $(\alpha+k)^2 \in K \setminus W$ and thereby $\Q((\alpha+k)^2)=K$.
I think this could be turned into a constructive argument fairly easy. Take one of the primitive elements given to you by the primitive element theorem then add $t+1$ non-zero rational numbers to it and the square of one of those will be the desired element.
Ammendum: This could also extend to a proof that there always exists an element $\alpha \in K$ such that $\Q(\alpha^n)=K$. Essentially use the same argument except find $t$ such distinct $c_i$ so that $\alpha_{c_i}^n \in F_t$ then notice that the matrix given by $A_{ij}=\binom{m}{j}c_i^j$ has determinant a constant multiple of the Vandermonde matrix $A_{ij}=c_i^j$. Thereby the determinant is nonzero and so we can find a linear combination of the $\alpha_{c_i}^n$ that gives us $\alpha$.


TITLE: Finding a limit of a rational function
QUESTION [2 upvotes]: $\lim_{x \rightarrow 27} \frac{x^{1/3} - 3}{x - 27}$.
Hello,
I am stuck on this one. I am sure there is a simple step but I am not seeing it. Thanks in advance for the help.

REPLY [0 votes]: Hint. For any differentiable function near $a$, one has
$$
\lim_{x \to a}\frac{f(x)-f(a)}{x-a}=f'(a).
$$ Then just apply it to $f(x)=x^{1/3}-3,\, a=27$.


TITLE: Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?
QUESTION [2 upvotes]: We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, \dots, \|\cdot \|_\infty$ etc. ? And if not how can you possibly prove that any norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$ if you don't know what 'form' the norm takes?

REPLY [1 votes]: For the sake of completeness, here's an answer to your questions:
1) Consider, on $\mathbb{R}^2$, the norm
$$
\|(x,y)\|=2|x|+3|y|
$$
(of course, any other choice of positive coefficients will to, too). You can also do the same kind of variation with the $p$-norms. Notice that if you take any subadditive positive function on $\mathbb{R}$ ($f(t)>0$ for all $t\ne0$, $f(0)=0$, $f(s+t)\leq f(s)+f(t)$), then
$$
\|(x,y)\|_f=f(x)+f(y)
$$
defines a norm, and even (for $p\geq1$)
$$
\|(x,y)\|_{f,p}=\left(f(x)^p+f(y)^p\right)^{1/p}.
$$
2) The answer to your second question can be seen from the proof of Fabian quoted above: in a finite-dimensional space any norm is bounded above by a multiple of the infinity norm; this together with the compactness of any closed and bounded set (again a consequence of the finite-dimensionality) guarantees the equivalence of your norm with the infinity norm.


TITLE: Is there relation that is symmetrical, transitive and non-reflexive?
QUESTION [2 upvotes]: We must show that there exists some kind of $\alpha$ relation
 $\alpha ⊆ X \times X$  which has these conditions :
if this relation is I and II type.
I) symmetrical:
if $∀x,x' ∈ X : (x, x') ∈ \alpha ⇒  (x', x) ∈ \alpha$
II) transitive:
if $∀x, x', x'' ∈ \alpha : (x, x') ∈ \alpha (x',x'') ∈ \alpha ⇒ (x, x'') ∈ \alpha$

then there must be result of non-reflexivity for this $\alpha$ relation III) nonreflexive:
if $∀x ∈ X : (x, x) $ $\notin \alpha$
Is there any kind of relation like this?

REPLY [6 votes]: If $X=\{x\}$ or $X=\emptyset$, then it's easy to show that $\alpha=\emptyset$.
Suppose $X$ has more than one element. Take $x\ne y$ - elements of $X$. Suppose also that $(x,y)\in\alpha$, then $(y,x)\in\alpha$, hence $(x,x)\in\alpha$, which leads to contradiction.
Thus, $\alpha=\emptyset$.


TITLE: Trying to establish the following identity involving sums using residues
QUESTION [1 upvotes]: If $2z - 1 $ is not an integer, then
$$ \frac{1}{\cos( \pi z) } = 1 + \frac{4}{\pi} \sum_{n=1}^{\infty} \left[ \frac{ 2z -1}{(2z-1)^2-4n^2} + \frac{4}{1-4n^2} \right]$$

################Attempt################.

I consider $f(z) = \frac{1}{\cos \pi z} -1 = \frac{ 1 - \cos \pi z}{\cos \pi z}$. We know $\cos \pi z = 0 \iff \pi z = (2n +1) \frac{ \pi}{2} \iff z = n + \frac{1}{2}$. So $f(z) $ has simple poles at $a_n = n + \frac{1}{2} $. The residues at $z = a_n$ are given by $b_n = (-1)^n $ and consider square $C_k$ with corners $ \pm k \pm ki $. Then by residue theorem
$$ \int_{C_k} f(z) dz = 2 \pi i \sum_{n= - \infty}^{\infty} \{ b_n : b_n \in C_n \} $$
But, then I feel like this wont lead me anywhere. Am I on the right track?

REPLY [1 votes]: I think what you want to prove is that
$$\sum_{n=-\infty}^{\infty} f(n) = -\sum_k \operatorname*{Res}_{\zeta=\zeta_k} \left [\pi \, \cot{(\pi \zeta)} \, f(\zeta)\right ] $$
where $\zeta_k$ are the non-integer poles of $f$.  You do this by showing that
$$\lim_{N \to \infty} \oint_{C_N} d\zeta \, \pi \, \cot{(\pi \zeta)} \, f(\zeta) = 0$$
on the square $C_N$ with vertices $(\pm N \pm 1/2 )(1+i)$.
Once you show that, then you can evaluate
$$\sum_{n=-\infty}^{\infty} \left [\frac{2 z-1}{(2 z-1)^2-4 n^2} + \frac{4}{1-4 n^2} \right ] = \frac{\pi}{4} \sum_k \operatorname*{Res}_{\zeta=\zeta_k} \left ( \cot{(\pi \zeta)} \,  \left [\frac{2 z-1}{\zeta^2-(z-1/2)^2} + \frac{4}{\zeta^2-1/4} \right ]\right ) $$
where $\zeta_{1,2} = \pm (z-1/2)$.  (Note that singularities at $\zeta=\pm 1/2$ are removable.)  Thus, the double infinite sum is equal to
$$\frac{\pi}{4} [-\tan{(\pi z)}] \left (1+\frac{4}{z(z-1)} \right ) + \frac{\pi}{4} \tan{(\pi z)} \left (-1+\frac{4}{z(z-1)} \right )= -\frac{\pi}{2} \tan{(\pi z)}$$
The sum, however, is also equal to $2 S + 4 + \frac1{2 z-1} $.  Thus,

$$\sum_{n=1}^{\infty} \left [\frac{2 z-1}{(2 z-1)^2-4 n^2} + \frac{4}{1-4 n^2} \right ] = -\frac12 \left [\frac{\pi}{2} \tan{(\pi z)} + 4 + \frac1{2 z-1} \right ]$$

This agrees with numerical results I derived in Mathematica.  I am not sure where your result comes from, but it is not correct.


TITLE: For what values of $n$ is it possible to divide into three piles of equal weight?
QUESTION [1 upvotes]: For $n \in \mathbb{N},n \geq 3$ we consider $n$ weights where the $k^{th}$ weight has a weight of precisely $k$ grams. For what values of $n$ is it possible to divide into three piles of equal weight?
My work
It is obvious that $\sum_{i=1}^n i= n(n+1)/2$ should divide $3$. This means $n\equiv 0,2 \pmod 3$.
But how do I show this for all such numbers?
Eg. $n=5 \Rightarrow \{1,4\},\{2,3\},\{5\}$

REPLY [1 votes]: If $n=9 \Rightarrow \{9,6\},\{8,7\},\{1,2,3,4,5\}$
I did this by folowing a simple algorithm :
Let $n \in \mathbb{N}$ (verifying the condition) we consider the set $A =\{1,2,...,n\}$
Let $S_1 ,S_2, S_3$ be the three subsets of A we're looking for.
We use the notation :  $sum(S)= \sum_{i\in S}i$
Then we proceed like this for the two first sets $S_1,S_2$ :

if $sum(S_i)<{n\over3} $ :

if $sum(S_i)+max(A)<{n\over3}$ :

We move $max(A)$ from $A$ into $S_i$

else :

We move ${n\over 3} - sum(S_i)$ from $A$ into $S_i$

Then we will have $S_3 = A$ ( remember each time we added an element to a previous set we deleted it from A )
I hope this helps =)


text:
\begin{document}

\title{A Discrete Event Simulation Model for Coordinating Inventory Management and Material Handling in Hospitals}

\author[1a]{Amogh Bhosekar}
\affil[1]{\small Department of Industrial Engineering, Clemson University,
Freeman Hall, Clemson, SC 29634 \authorcr
  Email: abhosek@clemson.edu \authorcr
    Email: tisik@clemson.edu}

\author[2]{Sandra Ek\c{s}io\u{g}lu}
\affil[2]{Department of Industrial Engineering, University of Arkansas,
Bell Engineering Center, Fayetteville, AR 72701 \authorcr
  Email: sandra@uark.edu}
\author[1b]{Tu\u{g}\c{c}e I\c{s}{\i}k}
\author[3]{Robert Allen}
\affil[3]{Perioperative Services, Prisma Health,
701 Grove Rd, Greeville, SC 29605 \authorcr Email: robert.allen@prismahealth.org}

\date{}

\maketitle

\vspace{11in}

\title{\large \bf A Discrete Event Simulation Model for Coordinating Inventory Management and Material Handling in Hospitals}

\begin{abstract}
{
For operating rooms (ORs) and hospitals, inventory management of surgical instruments and material handling decisions of perioperative services are critical to hospitals’ service levels and costs. However, efficiently integrating these decisions is challenging due to hospitals’ interdependence and the uncertainties they face. These challenges motivated the development of this study to answer the following research questions: (R1) \emph{How does the inventory level of surgical instruments, including owned, borrowed and consigned, impact the efficiency of ORs?} (R2): \emph{How do material handling activities impact the efficiency of ORs?} (R3): \emph{How do integrating decisions about inventory and material handling impact the efficiency of ORs?} Three discrete event simulation models are developed here to address these questions. Model 1, \emph{Current}, assumes no coordination of material handling and inventory decisions. Model 2, \emph{Two Batch}, assumes partial coordination, and Model 3, \emph{Just-In-Time} (JIT), assumes full coordination. These models are verified and validated using real life-data from a partnering hospital. A thorough numerical analysis indicates that, in general, coordination of inventory management of surgical instruments and material handling decisions has the potential to improve the efficiency and reduce OR costs. More specifically, a \emph{JIT} delivery of instruments used in short-duration surgeries leads to lower inventory levels without jeopardizing the service level provided.

}
\end{abstract}

\textbf{Keywords: }{OR in Health Services, Simulation,  Inventory Management, Automated Guided Vehicles, Data Analytics}

\section{Introduction}\label{intro}

\textbf{Motivation:} The cost of supply chain activities that support ORs contributes to a hospital’s total expenses, accounting for as much as 40\% of the operating budget in hospitals [\citenum{dobson2015configuring}]. Holding inventory of supplies and surgical instruments makes up about 10\% to 18\% of these expenses [\citenum{volland2017material}]. For example, a 2014 study conducted in acute care hospitals in California suggests that an OR costs, on average, between \$36 and \$37 each minute. The use of surgical instruments costs between \$2.50 and \$3.50 each minute [\citenum{childers2018understanding}]. Hospitals maintain large inventories to ensure that the required instruments are available for a scheduled surgery, since the lack of an instrument leads to delays. A study conducted by \Mycite{wubben2010equipment} suggests that 45.9\% of the delays in an OR happened because an instrument was unavailable [\citenum{wubben2010equipment}]. These delays resulted in longer working hours for doctors and staff, and thus, additional costs for the hospital. A surgery delayed due to a lack of instruments also negatively impacts the quality of care and adverse effects can occur [\citenum{wubben2010equipment}].

Increasing inventory levels may not necessarily eliminate these delays since some delays occur because of inefficiencies in the material handling process. For example, congestion, due to the movement of Automated Guided Vehicles (AGVs) along the narrow corridors of a hospital, can lead to delays. Based on our review of the literature, very little research evaluates the effects of inventory and material handling decisions relative to the efficiency of and service levels provided by ORs in hospitals. This gap in the literature is the main motivation for this research.

{\bf Background:} Surgeries performed in most hospitals are categorized as elective or emergency [\citenum{gupta2008appointment}]. This research focuses on elective surgeries. An elective surgery is scheduled within 12 weeks, and the exact timing of the surgery and the room assignment are finalized between 24 and 48 hours before the day of the surgery. The OR scheduler makes these assignments after considering the availability and preferences of the surgeon and surgical staff, as well as the availability of the required equipment and instruments. These assignments impact the availability of instruments for the rest of the day. Emergency surgeries are incorporated into the daily schedule since the timing of intervention is critical for patients’ safety [\citenum{gupta2008appointment}]. To maintain high service levels, some hospitals develop plans to ensure that sufficient ORs, instruments, and equipment are available. The limited availability of instruments restricts hospitals from utilizing ORs and surgeons’ time efficiently. In most hospitals, an instrument is not used more than once the same day because it should be decontaminated and sterilized before reuse. This process can take up to 3 or 4 hours. Most of the time, an OR operates 12 hours each day, and most of the surgeries last no more than 5 hours.

\textbf{Research Questions:} Surgical instruments are categorized as: (1) owned by the hospital, (2) borrowed from other hospitals or doctors, or (3) consigned by a vendor who owns the instrument [\citenum{chobin2015best}]. The process of adopting a new surgical instrument is initiated upon a surgeon’s request. Next, the hospital evaluates whether buying, renting, or consigning this instrument is the best option. The main factors impacting this decision are the selling price of the instrument and the frequency of its use. Typically, a hospital would not purchase an instrument if only used in rare or specialty surgeries [\citenum{chobin2015best}]. In such a case, the hospital would consign or borrow the instrument and pay the owner upon its use. A hospital has several other reasons to borrow instruments, such as to match demand and supply, accommodate doctors’ requests for scheduling consecutive surgeries during a given day, continue operations on a limited budget, or mitigate a lack of storage space [\citenum{seavey2010reducing}]. According to \Mycite{seavey2010reducing}, such practices lead to inefficiencies because borrowed instruments create an additional workload for the sterile processing department (SPD) since the hospital is required to maintain documentation and pack and sterilize instruments [\citenum{seavey2010reducing}]. Additionally, some instruments have special cleaning procedures, which may differ from other procedures for instruments owned by the hospital. Following these procedures adds to employees’ workloads. Also, consigned instruments stored at the hospital occupy additional storage space. A recent study conducted in a major US academic hospital suggests that half of the instruments are consigned, and their cost is, on average, 12\% more than instruments owned by the hospital [\citenum{mandava2017how}]. These challenges motivate the first research question: \textbf{(R1)} \emph{How does the inventory level of surgical instruments, including owned, borrowed, and consigned, impact the efficiency of ORs?} Typically, the inventory level is determined by the total number of surgeries scheduled in a day, the daily schedule of surgeries that use the same instrument, the processing capacity of the central sterile storage division (CSSD), and the schedule of material handling activities. This study presents data-driven, discrete event simulation (DES) models and a numerical study that evaluates how inventory levels impact the utilization of instruments and the delay of surgeries. The models are validated with data from a partnering US-based hospital.

Instruments are delivered to ORs via containers called case carts that are transported by carriers, such as staff or AGVs. After a surgery, soiled instruments are delivered to the CSSD using the same carriers. Inefficiencies in material handling activities lead to delays, which impact the availability of instruments. Additionally, the duration of a surgery is uncertain; thus, a surgery may take longer than planned, keeping instruments unavailable. For these reasons, some hospitals deliver instruments to a storage area beside the OR the night before the surgery. Such a practice ensures that the instruments required are available during the surgery. As a result, the same instrument cannot be reused in other surgeries scheduled on the same day. Alternatively, an instrument could be delivered to an OR directly from the CSSD a short time before the start of a surgery, the JIT delivery approach. Such a practice increases the utilization of instruments used in short-duration surgeries performed earlier in the day. This approach can also lead to lower inventory levels and lower inventory holding costs. However, such an approach requires coordination of material handling, instrument decontamination and sterilization, and OR scheduling. These challenges motivate the second research question: \textbf{(R2)} \emph{How do material handling activities impact the efficiency of ORs?} A numerical analysis via the proposed DES models is conducted to answer this question. These models take different approaches to the handling of instruments. Each material handling approach follows a different schedule of delivering case carts to ORs. For each approach, a numerical study is conducted to evaluate how the number of carriers impacts travel time, congestion, the utilization of carriers, delivery time, and the utilization of instruments.

Coordinating decisions about material handling schedules and instrument inventory is challenging. For example, the decision to reduce the inventory of an instrument limits the time that instrument is available. This, in turn, negatively impacts the flexibility of scheduling a surgery needing the instrument and, therefore, the service level provided. The problem becomes even more challenging when other important considerations are added, such as the inconsistent material handling system, stochastic demand, and uncertain surgery duration. These challenges motivate the following research question: \textbf{(R3)} \emph{How do integrating decisions about inventory and material handling impact the efficiency of ORs?} To answer this question, another numerical study using the DES models is conducted.

\textbf{Contributions:} The proposed research offers several important contributions: (\textit{i}) This study highlights the role of coordinating decisions between material handling and inventory management in improving OR efficiency and reducing costs while maintaining high service levels. In particular, this work demonstrates that JIT delivery of surgical cases for short-duration surgeries can potentially improve the efficiency of ORs and reduce the cost of healthcare. (\textit{ii}) This study develops a real-life case study using data from a US-based hospital. The proposed material handling approaches, which are intuitive and easy to implement, are verified and validated using historical data. The results of the proposed analysis have inspired the partner hospital featured in this study to make improvements in material handling and inventory management practices. While other healthcare facilities may not choose to implement the models presented here, they can learn from these practices.

\textbf{Outline:} The rest of this article is organized as follows: Section 2 reviews the literature relevant to this work, and Section 3 provides a detailed description of the problem. Section 4 describes the proposed simulation models, and Section 5 introduces the case study. Section 6 discusses the results of the experiments, and finally, Section 7 summarizes the results and presents concluding remarks.

\section{Literature Review}
The main stream of existing literature relevant to this work is inventory management of reusable surgical instruments. Since AGVs are used as carriers by this study's partner hospital, as well as in many others, the literature that discusses the use of AGV systems for material handling in hospitals is also reviewed.

{\bf Inventory Management of Reusable Surgical Instruments:}

The cost of ORs is impacted by the availability of surgical supplies and implants [\citenum{chasseigne2018assessing}]. Surgical supplies include soft goods and instruments required for surgery. These supplies can be either reusable or disposable. Numerous studies show that the cost of reusable supplies is significantly lower than the cost of disposable supplies [\citenum{demoulin1996cost}, \citenum{eddie1996comparison}, \citenum{schaer1995single}, \citenum{manatakis2014reducing}, \citenum{adler2005comparison}], {and} disposable supplies negatively impact the environment [\citenum{adler2005comparison}]. \Mycite{chasseigne2018assessing} conduct a study to evaluate the cost of opened, unused soft goods and instruments in a French hospital. They reported that wasted supplies have a median cost of \euro 4.1 per procedure, which accounts for about 20.1\% of the cost of surgical supplies. However, most hospitals do not have standardized procedures to manage the inventory of surgical supplies [\citenum{ahmadi2018inventory}]. In their review paper, \Mycite{ahmadi2018inventory} indicate that inventory management of sterile instruments requires three important considerations: instrument and quantity assignment for each tray-type, the tray-type’s assignment to a surgeon or procedure, and the number of trays carried by the hospital. Decisions related to the first two considerations are impacted by the surgeon’s preferences, indicated in the doctor preference card (DPC).

The cost of surgical supplies can be reduced in several ways, including by 1) improving the accuracy of the DPCs, 2) increasing surgeon awareness, and 3) standardizing surgical techniques. Accuracy of the DPC can be improved by reviewing it periodically [\citenum{harvey2017physician}, \citenum{farrelly2017surgical}] or by recording which instruments are used on a tray and removing the instruments that are not used [\citenum{nast2019decreasing}, \citenum{dyas2018reducing}]. For example, \Mycite{harvey2017physician} show that engaging physicians in the review of the corresponding DPC led to the removal of 109 disposable supplies and the elimination of 3 reusable instrument trays. Consequently, the cost of a case cart was reduced by \$16 on average. According to a survey conducted by \Mycite{jackson2016surgeon}, surgeons often underestimate the cost of expensive items and overestimate the cost of less expensive items due to internal bias and cost ignorance [\citenum{jackson2016surgeon}]. Thus, the cost of a surgical procedure can be reduced by increasing surgeons’ awareness of standardized operating equipment and the cost of instruments [\citenum{gitelis2015educating}, \citenum{avansino2013standardization}]. Finally, work by \Mycite{skarda2015one} shows that standardization of surgical techniques can significantly reduce operating costs without impacting the quality of a procedure [\citenum{skarda2015one}].

\Mycite{stockert2014assessing} indicate that tailored and streamlined tray compositions lead to significant cost savings [\citenum{stockert2014assessing}]. Additionally, surgeons prefer trays that have fewer unsolicited instruments [\citenum{dobson2015configuring}, \citenum{stockert2014assessing}]. Several optimization models have been developed to solve the tray optimization problem and address tray composition and inventory management for reusable surgical instruments. The objective of this problem is to minimize an OR's cost by optimizing the number of trays utilized and the amount of inventory supplied. The problem also addresses surgeon preferences for instruments. \Mycite{dobson2015configuring} develop a linear integer programming formulation and propose a heuristic algorithm to obtain a solution to this problem [\citenum{dobson2015configuring}]. \Mycite{reymondon2008optimization} propose a resource sharing method for reusable devices [\citenum{reymondon2008optimization}]. The objective is to minimize storage, processing, and wastage costs for supplies that have not been used. \Mycite{van2008optimizing} propose a deterministic model that minimizes the storage and delivery cost of instruments by optimizing tray composition [\citenum{van2008optimizing}]. \Mycite{ahmadi2019bi} present a bi-objective optimization model for configuration of surgical trays with ergonomic considerations [\citenum{ahmadi2019bi}]. The first objective function minimizes the total number of assembled tray types, and the second objective function minimizes the total number of instruments that were not requested. They use the $\epsilon$- constraint method to obtain the Pareto-optimal front. \Mycite{dollevoet2018solution} develop an exact integer linear programming formulation, a row and column generation approach, a greedy heuristic, and some metaheuristics. These approaches are evaluated based on the average computation time, the average value of the objective function, and the number of solutions for which optimality is proven.

{\bf AGV Systems and Operations in Hospitals:}

The existing literature on AGV systems focuses on fleet size selection. Simulation and optimization models are proposed to identify AGV fleet size (\Mycite{choobineh2012fleet}, \Mycite{maxwell1982design}, \Mycite{arifin2000determination}, \Mycite{rajotia1998determination}, \Mycite{sinriech1992economic}, \Mycite{egbelu1987use}, \Mycite{tanchoco1987determination}, and \Mycite{Bhosekar:2018}).

\Mycite{katevas2001mobile} discusses several factors that must be considered to design a mobile robotic system for healthcare applications, and his work provides several guidelines for researchers to improve these designs. \Mycite{rossetti2000simulation} compare the manual delivery of clinical and pharmaceutical items with the performance of robotic courier delivery [\citenum{rossetti2000simulation}]. They use cost, turnaround time, variability of turnaround time, cycle time, and utilization as performance measures. The proposed simulation model shows that using robotic delivery is economically viable and improves the performance measures listed above. \Mycite{rossetti2001multi} use Analytic Hierarchy Process to build a decision problem that evaluates the performance of a robotic healthcare delivery system based on technical, economical, and several other factors. Their proposed simulation model assesses the technical {factors} that include speed of robot and human couriers based on the arrival rates of visitors who request the elevator, elevator availability, the arrival rates of the delivery items that request robots, and robot availability. In their case study, \Mycite{chikul2017technology} compare three supply chain models that use: a) manual inventory check and delivery, b) RFID inventory check and manual delivery, and c) manual inventory check and AGV-based material handling. This study shows that combined RFID tracking and AGV-based delivery maximizes cost savings in the supply chain model and yields ergonomic benefits due to reduced manpower requirements. Via a simulation-based case study, \Mycite{pedan2017implementation} identify potential benefits of utilizing an AGV system in a hospital. Finally, \Mycite{fragapane2018material} evaluate the impact that material and information flow have on costs at a Norway hospital.

{Different from this literature which focuses either on improving the management of inventory of surgical instruments, or improving material handling activities in hospitals, our proposed work focuses on evaluating the impacts of integrating material handling and inventory management decisions. Via our  numerical analysis we show that coordinating these decisions leads to reduced inventory levels, reduced  number of AGVs used and increased utilization of AGVs, while the quality of service provided remains intact. }

\section{Problem Description}\label{mhp}
\textbf{Material Handling and Inventory Management Processes:} Figure \ref{fig: MHProcess} describes a typical material handling process for the delivery of surgical case carts in a hospital. The process begins by creating a detailed schedule of surgeries. This schedule is prepared by the OR manager. Based on the schedule and the doctors’ preferences, a list of instruments and soft goods is prepared and submitted to the materials division (MD). For each surgery, a clean case cart is loaded with the requested instruments, soft goods, and implants. These case carts are moved to pick-up/drop-off stations for carriers to pick up. Then, the clean case carts are moved from the MD to the case cart storage area (CCSA). At the CCSA, each case cart is inspected to ensure that it contains the required materials. The case carts are held at the CCSA until they are moved to the corresponding OR at the time of the surgery. The case carts are delivered to ORs prior to the surgeries. ORs are divided into separate cores based on the specialties they serve. Specialty instruments and implants required for the surgical cases, which are stored in the OR cores, are added to the case carts before the surgery. After the surgery, the instruments and case carts are considered soiled and should be decontaminated. The soiled carts and instruments are transported to the CSSD by the carrier. The instruments and case carts are washed and sterilized at the CSSD. The specialty surgical instruments are returned to the corresponding OR cores. This process ensures the availability of instruments before the scheduled surgery. Figure \ref{fig: Model1} presents the locations of the departments and paths traversed by the carriers (i.e., AGVs) at our partnering hospital.

\begin{figure}[ht]\centering{
{\includegraphics[width=0.75\textwidth]{Flow_Chart.pdf}}
\caption{Material Handling Process \label{fig: MHProcess}}
}
\end{figure}

\begin{figure}[ht]
{\includegraphics[width=\textwidth]{Model1.pdf}}
\caption{Map - GMH Floor Map}
{\label{fig: Model1}}
\end{figure}

\begin{table}[ht]
\centering
\caption{AGV Movements by the Time of the Day \label{Table: ByTime}}
{\begin{tabular}{llllll}
\hline
Route & Time & No. of & \multicolumn{2}{c}{Travel Time [Min]} & Coefficient of \\\cline{4-5}
 & Interval & Trips & Average & Std. Dev.  & Variation \\ \hline
 & 12 am-3 am & 131 & 4.66 & 10.26 & 2.2\\
 & 3 am-6 am & 227 & 5.96 & 9.55 & 1.6 \\
 & 6 am-9 am & 112 & 5.27 & 6.17 & 1.17 \\
Materials Department - & 9 am-12 pm & 80 & 5.8 & 2.4 & 0.41 \\
Case Cart Storage Area & 12 pm-3 pm & 101 & 5.33 & 3.69 & 0.69 \\
 & 3 pm-7 pm & \textbf{1,416} & \textbf{8.94} & \textbf{6.49} & 0.73 \\
 & 7 pm-9 pm & 254 & 5.67 & 4.45 & 0.78\\
 & 9 pm-12 am & 196 & 4.88 & 5.72 & 1.17\\ \hline
  & 12 am-3 am & 44 & 7.28 & 6.16 & 0.85\\
 & 3 am-6 am & 53 & 6.87 & 6.16 & 0.9\\
 & 6 am-9 am & 146 & 5.55 & 3.27 & 0.59\\
2nd Floor Soiled & 9 am-12 pm & 981 & 4.56 & 4.04 & 0.88 \\
Cart Storage - CSSD & 12 pm-3 pm & 882 & 5.17 & 2.56 & 0.49\\
 & 3 pm-7 pm & 753 & \textbf{9.71} & \textbf{8.51} & 0.88 \\
 & 7 pm-9 pm & 126 & 6.65 & 3.04 & 0.46\\
 & 9 pm-12 am & 77 & 5.45 & 1.01 & 0.19\\ \hline
\end{tabular}}\\
{\footnotesize{This table was obtained from the prior research. [\citenum{Bhosekar:2018}]}}
\end{table}

To reduce the cost of inventory, hospitals need to coordinate inventory management and material handling decisions. This coordination becomes ever more important in face of uncertainty. For example, if surgery duration and travel time of carriers is fixed, hospitals can calculate the necessary inventory levels with certainty and decide how many instruments to loan or consign. However, in order to ensure high service level under uncertainty, many hospitals keep large inventories, loan instruments, and prepare/deliver case carts one day before the surgery. We observed at our partner hospital that, if the delivery of instruments from the CSSD to ORs was completed within a short time before the start of the surgery, the instruments could be reused within the same day. Such an approach has the potential to lead to a reduction in the cost of using loaned or consigned instruments. This observation led the development of the material handling process proposed next.

\begin{table}[ht]
\centering
\caption{A List of Instruments Used \label{Table: IAGMH}}
{\begin{tabular}{r|rr}
\hline
\multicolumn{1}{c|}{\bf Type} & {\bf Total Number} & {\bf Total in \%}
\\ \hline
Loaner & 266 & 5\% \\
Consigned & 1,095 & 19\% \\
Owned & 3,507 & 61\% \\
Other Services & 927 & 16\% \\ \hline
{\bf Total} & 5,795 & 100\% \\ \hline
\end{tabular}}
{}
\end{table}

\textbf{Experimental Setup:} The movement of clean instruments to ORs and the movement of soiled instruments to the CSSD affect inventory {availability} and the starting times of surgeries. This is the reason why some hospitals, like our partner hospital, prepare and deliver the surgical case carts to the CCSA one day in advance. Such a practice ensures the availability of surgical instruments, but a number of inefficiencies results regarding material handling and inventory management. For example, Table \ref{Table: ByTime} summarizes the data obtained on the travel times of AGVs during different times of the day at the partner hospital. The data shows that the average travel time  and the corresponding standard deviation are highest during 3 pm to 7 pm. This is because the clean surgical carts are being delivered to the CSSA for elective surgeries. The consequent increase in the number of AGV movements leads to congestion and thus, longer travel times for every AGV that uses the same path. These delays lead to an increased inventory of instruments since an instrument cannot be reused in different surgeries scheduled on the same day. Table \ref{Table: IAGMH} lists the types of the total number of instruments used by the partner hospital and the percentage of each type. Notice that about 24\% of the instruments used are either loaned or consigned.

This study evaluates three approaches to delivering surgical supplies to ORs and compares their performances. The first approach, Model 1, is referred to as the \emph{Current} approach and assumes that materials required by surgeries are delivered to the CCSA the night before the surgery. The \emph{Current} approach is the ongoing practice of the partner hospital in the data presented here.

Next, the \emph{Two Batch} approach, Model 2, assumes that materials required by surgeries scheduled in the morning are delivered to the CCSA the previous evening, and the materials required by surgeries scheduled in the afternoon are delivered in the morning on the day of the surgery. This approach provides the opportunity to reuse the instruments from the surgeries scheduled later in the day. Since the CSSD works 24 hours each day, the instruments can be washed overnight and delivered in the morning. Since instruments are delivered a few hours in advance of the surgery, the staff has an abundant amount of time to intervene if an instrument becomes unavailable. Thus, the risk of instruments not being delivered on time is only minimal and does not affect the quality of care in the hospital.

Finally, the \emph{Just-in-Time} approach, Model 3, assumes that materials required are delivered shortly before the start of the surgery. The time between the delivery of surgical supplies and the surgery, referred to as the delivery interval, needs to be determined and impacts the inventory levels. The inventory level required increases with the delivery interval. For example, consider two surgeries that require the same instrument and are scheduled on the same day. In the current system, a hospital must have two sets of the identical instruments since they are delivered to the CCSA the day before the surgery. If the delivery interval is 1 hour, the instrument can be sterilized and delivered before the subsequent surgery, provided that the two surgeries are scheduled several hours apart. In this case, the hospital needs only one instrument. However, if the delivery interval is chosen to be less than 3 hours, then, to avoid any delays of the second surgery, two instruments are needed. Note that the implementation of JIT and other lean methods in healthcare, unlike with manufacturing, should be considered with caution because such practices could delay surgeries and jeopardize the well being of patients.

{Three performance measures are used to compare the proposed approaches: (\textit{i}) the average delay of a surgery's start time, which is a measure of the service level provided, (\textit{ii}) the number of instruments inventoried, which  measures the efficiency of the inventory system, and (\textit{iii}) the number of carriers required for each proposed material handling approach, which measure the efficiency of the material handling systems.}

The delays of a surgery start time are separated into two categories: delays due to carriers, e.g., long travel time because of congestion or unavailability of carriers, and delays due to unavailable instruments. Delays due to carriers can create challenges for the JIT approach, but these delays can be reduced by optimizing fleet size. Delays due to unavailable instruments are caused either by delays in the delivery of soiled case carts or by an increase in the number of emergency surgeries. A delay in the delivery of soiled case carts subsequently delays the cleaning process of instruments and carts, which leads to the delay of the start of the next surgery that uses the same instrument. Delays due to unavailable instruments can be reduced by optimizing the inventory level. In this research, simulation experiments are conducted to determine the optimal fleet size and inventory level under each proposed material handling approach. Based on the results of these experiments, the delivery interval that optimizes the performance measures identified is also determined.

Each of the proposed material handling approaches requires a different number of carriers to deliver materials on time. This number is impacted by the surgery schedule and the material handling process. For instance, the number of carriers needed for the JIT delivery approach is lowest since the delivery of case carts is spread throughout the day. The number of carriers needed by the \emph{Current} delivery approach is larger because the delivery of case carts is completed within a short time period. The number of carriers needed also depends on the total number of cases scheduled and the spread of the schedule. A tight schedule would require more carriers to complete material handling on time.

{\bf Limitations of this Research:}

{\bf Model:} The research proposed here is conducted in collaboration with Greenville Memorial Hospital (GMH), a US-based hospital located in Greenville, South Carolina. The models presented here are  motivated by the material handling and inventory management practices at GMH. The research team worked closely with the perioperative services department, which consists of the MD, the CSSD, and the OR Division. GMH uses AGVs to transport surgical case carts to and from ORs. The problem setting proposed here and the assumptions made are influenced by the practices at GMH. The models presented here are a valuable contribution to the literature because, based on a careful review of the literature, similar practices are followed by other hospitals for material handling and inventory management of surgical instruments.

\noindent {\bf Data:} Nine months of real-life data are used to develop the case study. This data includes information about the number of surgical cases each day and ranges over a time period long enough to observe how seasonality impacts the number of surgical cases. Ideally, larger amounts of data would be available, but that does not apply here.

\section{Simulation Model}

DES models are developed to evaluate and compare the three approaches proposed for the delivery of surgical supplies. These models are created in ARENA simulation software by Rockwell Automation. An entity type represents a surgery type, and each entity represents a surgical case of a particular type. An entity has three attributes: \emph{duration, starting time, and type}. \emph{Duration} is randomly generated using the distributions listed in Table 3. These distributions are derived from the data collected at GMH. The \emph{starting time} and \emph{type} are fed to the model from the actual data. Other entities are used to control the movement of AGVs and elevators, as well as to handle other specific requirements, such as calculating the value of certain variables (e.g., the number of AGVs to activate each day). ORs, case carts, cart washers, and elevators were modeled as resources. Variables are used to track the number of busy resources. A guided path transporter network is developed with intersections and links to replicate the movement of AGVs along the corridors of the hospital. This network was constructed using actual distances obtained from a GMH floor map. The links of the network are unidirectional, bidirectional, or spurs (dead ends). The intersections represent the areas where two or more links intersect. The intersections allow AGVs to make turns and move from one link to the next, following their routes. Intersections are also used to represent pick-up/drop-off stations. A spur link marks the end of a route. Departments can only handle a certain number of AGVs, and their processing capacity is limited by variables.

\begin{table}[ht]
\caption{Input Parameters: Surgery Duration \label{Table: LOS}}
{\footnotesize \begin{tabular}{l|cc|l|l|c}
\hline
Service & From & To & Distribution & Expression (Length of Surgery) & Squared Error \\ \hline
ENT Surgery& 00:00 & 08:00 & Lognormal & LOGN(2.02, 2.12) & 0.008 \\
 & 08:00 & 14:00 & Lognormal & LOGN(1.62, 1.2) & 0.007 \\
 & 14:00 & 00:00 & Lognormal & LOGN(1.23, 0.672) & 0.003 \\ \hline
Gynecology Service & 07:00 & 08:00 & Beta & 0.01 + 4.81 * BETA(2.85, 4.03) & 0.009 \\
 & 15:00 & 16:00 & Lognormal & 0.27 + LOGN(0.965, 0.511) & 0.005 \\
 & 16:00 & 07:00 & Lognormal & LOGN(1.65, 0.859) & 0.011 \\ \hline
Neurological Surgery  & 00:00 & 09:00 & Gamma & GAMM(0.494, 5.44) & 0.007 \\
 & 09:00 & 13:00 & Erlang & ERLA(0.454, 5) & 0.005 \\
 & 13:00 & 00:00 & Beta & 12 * BETA(4.95, 25.8) & 0.028 \\ \hline
Ortho Trauma Surgery  & 0:00 & 8:00 & Erlang & ERLA(0.587, 5) & 0.002 \\
 & 08:00 & 14:00 & Lognormal & LOGN(2.57, 1.29) & 0.004 \\
 & 14:00 & 00:00 & Lognormal & LOGN(2.09, 0.983) & 0.004 \\ \hline
Pediatric Surgery & 00:00 & 00:00 & Lognormal & LOGN(1.35, 0.693) & 0.011 \\ \hline
Urology Surgery & 00:00 & 07:00 & Lognormal & LOGN(1.63, 0.975) & 0.001 \\
 & 07:00 & 08:00 & Lognormal & LOGN(1.2, 0.821) & 0.007 \\
 & 08:00 & 00:00 & Erlang & ERLA(0.244, 4) & 0.011 \\ \hline
Vascular Surgery  & 00:00 & 07:00 & Beta & 0.03 + 8.97 * BETA(0.97, 1.78) & 0.004 \\
 & 07:00 & 09:00 & Gamma & GAMM(0.608, 3.58) & 0.017 \\
 & 09:00 & 14:00 & Gamma & GAMM(0.42, 4.39) & 0.025 \\
 & 14:00 & 00:00 & Triangular & TRIA(0.13, 0.83, 3.54) & 0.011 \\ \hline
\end{tabular}}
{}
\end{table}

The first DES model, Model 1, depicts the \emph{Current} material handling approach used at GMH. Figure \ref{fig: SimulationFlowchart} describes this model and Table \ref{tab:param} summarizes the values of its input parameters. In this model, the release of entities begins at 3 pm. The start time of these entities takes place after 6 am the next day. Next, the availability of instruments is checked using the decide module. If an instrument is not available, the case cart is held at MD until the instrument becomes available. An available instrument seizes a case cart and is delivered to the CCSA. There, the entity is held until the scheduled start time of the surgery. At this point in time, the entity seizes an available OR for the duration of the surgery. At the end of a surgery, the OR is released, and the corresponding case cart and instrument are moved to the CSSD to wash and sterilize. The resources at the CSSD are seized for the duration of service. The variables which record the number of busy units are updated when resources are released.

\begin{figure}[ht]
{\includegraphics[width=\textwidth]{flowchartmodel.pdf}}
\caption{Flowchart of the Simulation Model \label{fig: SimulationFlowchart}}
{}
\end{figure}

The second DES model, Model 2, depicts the \emph{Two-Batch} material handling approach. In this model, entities are released twice a day, at 6 am and 3 pm. The entities released at 6 am have a start time between after 12 pm the same day. Next, these entities follow a similar procedure as described above in Model 1. Entities released at 3 pm have a start time between 6 am and noon the next day. These entities are held until the next morning using the hold module, and then, they follow the procedure outlined above.

The third DES model, Model 3, depicts the \emph{JIT} approach. In this model, entities are released one hour prior to their \emph{start time}. This delivery interval was chosen based on the results obtained in Table \ref{Table:CCS}. Next, these entities follow the same procedure outlined above. In every model, the delivery of soiled case carts begins as soon as a surgery is completed.

\begin{table}[h]
\caption{Summary of Input Parameters }
\label{tab:param}
\resizebox{\textwidth}{!}{
\begin{tabular}{lll}
\hline
\textbf{Parameter} & \textbf{Source} & \textbf{Description} \\ \hline
Entity Creation Time & Surgery schedule data & Read from the data \\
Attribute Duration & Surgery schedule data & Random variable from \\
 &  & the corresponding distribution \\
Network link distances & GMH floor maps & Read from the data \\
No. of Case carts & AGV system data & 110 \\
No. of ORs & GMH Survey & 32 \\
No. of loading personnel & GMH Survey & 4 \\
No. of AGVs & AGV system data & {[}6,8,10{]} \\
Capacity of elevators & GMH Survey & {[}2,2{]} \\
Capacity of cart washers & GMH Survey & 3 \\
Cart loading delay & GMH Survey & Triangular(2,3,5) minutes \\
Cart washing delay & GMH Survey &  20 Minutes \\
Elevator movement delay to carry AGV & GMH Survey &  40 seconds \\
Cart loading unloading delay & GMH Survey &  15 seconds \\
Instrument washing delay & GMH Survey &  3 hours \\ \hline
\end{tabular}
}
\end{table}

\section{A Case Study}
{\bf Input Data Analysis:}
The main objective of the data collection and analysis is to evaluate the impacts that the \emph{Current} material handling approach has on the inventory of surgical instruments. The data collected is used to develop the DES models.

The data is presented in the following sets: The first data set provides information about the surgeries scheduled at GMH from Jan. 1, 2018, until Sept. 11, 2018. This data includes the surgery identification number (ID), OR’s ID , the date of the surgery, the scheduled start and finish times of a surgery, the type of surgery (i.e., vascular, orthopedic, neurological etc.), information about the surgeon, the primary procedure, and the instruments requested. The second data set provides information about the surgical instruments used at GMH. This data presents the instrument ID, the type of surgery the instrument is used for, the inventory level, and information regarding its ownership.

The hospital offers 46 different surgical services. Our experimental analysis focuses on the following 7 types of surgeries: ENT, pediatric, ortho trauma, neurology, gynecology, urology, and vascular. We focus on these surgical services because they are scheduled multiple times each day. Therefore, there is an opportunity to reduce the size and cost of inventory by reusing some of the instruments. The duration of a surgery is calculated using the actual start and finish times.

Surgeries are grouped based on service type, duration, and scheduled start times. For each service type, an hypothesis test is conducted to evaluate whether the duration of surgeries within each service type differ based on the starting time of the given surgery. When differences were observed, the distribution of surgery duration was estimated separately. Otherwise, the data was used to derive a single distribution for surgeries of the same type that were started at different times of the day. The results of the hypothesis test generated the input parameters used in the simulation model. For example, the surgery duration differs based on the time of the day the surgery is scheduled, by day of the week, and also by service type. A continuous distribution was fitted using the Input Analyzer of Rockwell Automation to represent the surgery duration. Table \ref{Table: LOS} shows the service types, distribution of the length of surgeries, and the squared error. The real-life scheduled start times of the surgeries are used in the simulation model obtained from the data set and presented here.

Table \ref{Table: NOS} summarizes the total number of surgical cases scheduled between Jan. 1, 2018, and Sept. 11, 2018. Here, only the surgery types that were scheduled more frequently are listed. Each of these surgery type is scheduled more than once a day and requires multiple instruments of the same kind. For each surgery type, only one set of instruments, common to all the surgical cases of that type, is used. Table \ref{Table: IISI} lists the instruments selected for this study and their corresponding inventory levels.

GMH carries multiple instruments for each surgery type for three main reasons: First, the same surgery could be scheduled more than once in the same day if the hospital follows a block schedule approach. This approach assigns the same block of time to a surgeon or a group of surgeons who perform similar procedures every week because surgeons perform back-to-back specialty surgeries in the assigned blocks and use similar instruments. The Current material handling system requires that every instrument is available one day before the surgery. Second, surgeons of different specialties may request the same instrument for the same procedure. Third, the hospital carries safety stock to respond to instrument-related incidents, such as dropping or breaking an instrument during a surgical procedures.

\begin{table}[ht]
\caption{Input Parameters: Number of Surgeries \label{Table: NOS}}
\resizebox{\textwidth}{!}{
{\begin{tabular}{l|ccccccc|c}
\hline
\textbf{Service} & \textbf{Sunday} & \textbf{Monday} & \textbf{Tuesday} & \textbf{Wednesday} & \textbf{Thursday} & \textbf{Friday} & \textbf{Saturday} & \textbf{Total} \\ \hline
\begin{tabular}[c]{@{}c@{}} ENT  Surgery\end{tabular} & 25 & 295 & 148 & 264 & 231 & 302 & 20 & 1,285 \\
\begin{tabular}[c]{@{}c@{}}Gynecology  Service\end{tabular} & 17 & 227 & 133 & 181 & 176 & 198 & 13 & 945 \\
\begin{tabular}[c]{@{}c@{}}Neurological  Surgery\end{tabular} & 22 & 174 & 168 & 293 & 163 & 225 & 17 & 1,062 \\
\begin{tabular}[c]{@{}c@{}}Ortho Trauma Surgery\end{tabular} & 2 & 205 & 171 & 174 & 206 & 207 & 56 & 1,021 \\
\begin{tabular}[c]{@{}c@{}}Pediatric  Surgery\end{tabular} & 62 & 145 & 248 & 153 & 276 & 158 & 79 & 1,121 \\
\begin{tabular}[c]{@{}c@{}}Urology   Surgery\end{tabular} & 39 & 293 & 333 & 298 & 382 & 466 & 72 & 1,883 \\
\begin{tabular}[c]{@{}c@{}}Vascular   Surgery\end{tabular} & 61 & 141 & 242 & 224 & 241 & 235 & 81 & 1,225 \\ \hline
{\bf Total} & 228 & 1,480 & 1,443 & 1,587 & 1,675 & 1,791 & 338 & 8,542 \\ \hline
\end{tabular}}
{}
}
\end{table}

\begin{table}[ht]
\centering
\caption{Number of Instruments in the Inventory \label{Table: IISI}}
{\begin{tabular}{llc}
\hline
\textbf{Service Type} & \textbf{Instrument} & \textbf{Inventory} \\ \hline
{ENT Surgery} & Set T \& A  GMMC  1047 & 10 \\
{Gynecology Service} & Set D \& C mini  GMMC  15896 & 10 \\
{Neurological Surgery} & Set Back Neuro  GMMC  1341 & 12 \\
{Ortho Trauma Surgery} & Set Minor Ortho  GMMC  100031 & 17 \\
{Pediatric Surgery} & Set Pediatric Minor  GMMC  1247 & 8 \\
{Urology Surgery} & Ureteroscope 7.5 Comp  GMMC  12656 & 18 \\
{Vascular Surgery} & Probe Doppler Pencil 8.1 GMMC  1824 & 25 \\ \hline
\end{tabular}}
{}
\end{table}

{\bf Verification and Validation:} Verification and validation procedures are used to compare the conceptual model with the proposed DES models. The development of the DES models is guided by the process flowchart and uses input data provided by GMH staff, who examined and approved these models. Additionally, the approach proposed by \Mycite{sargent2010verification} is adopted to verify and validate the DES models. \textbf{Data Validity:} The input data analysis section describes our data collection and analysis. This analysis indicates that our data is correct and adequately used. \textbf{Conceptual Model Validation:} The proposed conceptual model is validated via \emph{face validation} by GMH staff and via \emph{traces} following specific entities through the model. Flowcharts of the conceptual model are verified by GMH staff. Computerized Model Verification: The DES models are verified via techniques listed in \Mycite{sargent2010verification}. These techniques include animation, comparison with other models, and running several replications of the model. \textbf{Operational Validation:} A thorough sensitivity analysis was conducted to check the accuracy of the DES models. In the sensitivity analysis, the number of resources used (i.e., the number of AGVs, the number of instruments, etc.) changed, so the impact of these changes on the behavior of the model outputs was monitored. For example, the model outputs, after changing the number of AGVs, equal the average and standard deviation of travel time of the AGVs. Next, hypothesis tests were conducted to evaluate whether the difference between the outputs of DES models and the real-world data are statistically different. At a $p$-value of 0.05, the test indicates that the difference is not statistically significant.

\section{Discussion of Results}

The results from the DES models are used to address the research questions outlined in Section \ref{intro}.

\textbf{R1: How does the inventory level of surgical instruments, including owned, borrowed and cosigned, impact the efficiency of ORs?} A simulation-optimization experiment is conducted using ARENA Opt-Quest to answer this question. The objective of the simulation-optimization is to minimize the total delays at the start of a surgery by changing the inventory level. The delay of a surgery is calculated as the difference between the \textit{Actual Start Time} and the \textit{Scheduled Start Time}. The decision variables of type \textit{integer} are the number of instruments in the inventory for each of the seven service types (see Table \ref{Table: IISI}). In order to reduce the computational time of the simulation-optimization experiments, a lower bound, based on the data collected at GMH, is developed on the number of instruments inventoried. Let $n$ be the maximum number of surgeries scheduled in a day for each service type. The lower bound equals $\ceil{n/2}$. A lower bound is added for each surgery type via these constraints: (\textit{i}) the number of instruments used $\leq$ number of instruments in the inventory, and (\textit{ii}) the number of instruments used $\geq$ lower bound.

Experiments are conducted for three different scenarios. Scenario 1 assumes that the available inventory of instruments equals the current inventory level of GMH. Consider this inventory level to be an upper bound. Scenario 2 assumes that the available inventory of instruments equals the lower bound. Scenario 3 assumes that the available inventory of instruments equals the average value of the upper and lower bounds. Table \ref{Table:R1} summarizes the results of these experiments, and the following observations result:

\emph{Observation 1:} The \emph{Current} material handling approach, Model 1, is the most sensitive to changes in the inventory level, compared to \emph{Two Batch}, Model 2, and \emph{JIT}, Model 3. A decrease of inventory level, from Scenario 1 to Scenario 2, leads to an increase of the average delay from 0.42 to 31 minutes per surgery in the \emph{Current} approach. In the \emph{Two Batch} approach, the corresponding average delay increases from 0.01 to 5.12 minutes, and in the \emph{JIT} approach from 0.00 to 1.47 minutes per surgery (see Table \ref{Table:R1}).

\emph{Observation 2:} The \emph{Current} material handling approach requires additional levels of inventory to maintain the same service level, as measured by the average delay per surgery, compared to the proposed \emph{Two Batch} and \emph{JIT approaches} (see Table \ref{Table:R2}).

\emph{Observation 3:} The \emph{JIT} approach leads to reduced inventory levels of instruments used in short-duration surgeries without reducing the service level.

Table \ref{Table: NOS} presents the total number of neurological surgeries conducted at GMH during the 9-month period reviewed here. This number averages about 4.2 surgeries per day. Table \ref{Table: NOS} also presents the number of pediatric surgeries during the same time period, which corresponds to about 4.4 surgeries per day. The duration of neurological surgeries is about 1 hour longer than for pediatric surgeries. An hypothesis testing ($p$-value = 0.05) was conducted to evaluate the difference between the duration of neurological and pediatric surgeries. This test indicates that the difference is statistically significant (see Table \ref{tab:LengthComparison}). The results of Table \ref{Table:R2} show that the number of instruments required by neurological surgeries is higher than pediatric surgeries in Models 2 and 3 versus Model 1. This is because instruments used in Pediatric surgeries can be reused in the same day due to the shorter duration of these surgeries.

\begin{table}[H]
\caption{The Average Delay per Surgery}
\label{Table:R1}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccccccccccc}
\hline
\multicolumn{7}{c}{\textbf{Number of Instruments per Service Type}} & \textbf{} & \multicolumn{3}{c}{\textbf{Average Delay/Surgery (Minutes)}} \\ \hline
\multicolumn{1}{c|}{\textbf{Scenario}} & \textbf{ENT} & \textbf{Gynecology} & \textbf{Neurological} & \textbf{Ortho Trauma} & \textbf{Pediatric} & \textbf{Urology} & \multicolumn{1}{c|}{\textbf{Vascular}} & \textbf{Model 1} & \textbf{Model 2} & \textbf{Model 3} \\ \hline
\multicolumn{1}{c|}{\textbf{1}} & 10 & 10 & 13 & 17 & 8 & 18 & \multicolumn{1}{c|}{16} & 0.42 & 0.01 & 0 \\
\multicolumn{1}{c|}{\textbf{2}} & 6 & 6 & 5 & 9 & 4 & 10 & \multicolumn{1}{c|}{6} & 31.27 & 5.12 & 1.47 \\
\multicolumn{1}{c|}{\textbf{3}} & 8 & 8 & 9 & 13 & 6 & 14 & \multicolumn{1}{c|}{10} & 3.58 & 0.25 & 0.01 \\
\hline
\end{tabular}
}
\end{table}

\begin{table}[H]
\caption{Inventory Level of Instruments}
\label{Table:R2}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccccccccc}
\hline
 &  & \multicolumn{7}{c}{\textbf{Number of Instruments per Service}} \\ \hline
\multicolumn{1}{c|}{\textbf{Model}} & \multicolumn{1}{c|}{\textbf{Delay/ Surgery (Minutes)}} & \textbf{ENT} & \textbf{Gynecology} & \textbf{Neurological} & \textbf{Ortho Trauma} & \textbf{Pediatric} & \textbf{Urology} & \textbf{Vascular} \\ \hline
\multicolumn{1}{c|}{\textbf{1}} & \multicolumn{1}{c|}{0.42} & 10 & 10 & 13 & 17 & 8 & 18 & 16 \\
\multicolumn{1}{c|}{\textbf{2}} & \multicolumn{1}{c|}{0.41} & 6 & 10 & 13 & 13 & 6 & 18 & 12 \\
\multicolumn{1}{c|}{\textbf{3}} & \multicolumn{1}{c|}{0.41} & 6 & 6 & 9 & 9 & 4 & 16 & 12 \\ \hline
\end{tabular}
}
\end{table}

\begin{table}[H]
\caption{Comparison of Two Service Types}
\label{tab:LengthComparison}
\centering
\resizebox{0.55\textwidth}{!}{
\begin{tabular}{l|cc}
\hline
\textbf{Statistics} & \textbf{Neurology Surgery} & \textbf{Pediatric Surgery} \\ \hline
\textbf{Sample Size} & 1062 & 1121 \\
\textbf{Average Length} & 2.31 & 1.36 \\
\textbf{95\% CI} & (2.24,2.38) & (1.31,1.40) \\
\textbf{Standard Deviation} & 1.16 & 0.77 \\ \hline
\end{tabular}
}

\end{table}

\textbf{R2: How do material handling activities impact the efficiency of ORs?} Two sets of experiments are conducted. The first set focuses on the impact that changing the number of AGVs has on the performance of the material handling system. This performance is measured via the average travel time per trip, the total travel time, and the corresponding standard deviations. The delivery time of clean and soiled case carts is analyzed as the number of AGVs increases from 6 to 8 to 10. Experiments with fewer than 6 AGVs led to extensive delays in delivering all the case carts in \emph{Current} system, which requires employees to work overtime, so these experiments are not considered in this analysis.

The second set of experiments focuses on the impact that changing the timing of delivery has on the performance of the material handling system. For this purpose, the performances of Models 1, 2, and 3 are compared. The results of these experiments are summarized in Tables \ref{Table:CCS} and \ref{Table:SCS} and Figures \ref{fig:EffectofMaterialHandlingCCS} and \ref{fig:EffectofMaterialHandlingSCS}. The following observations result:

\emph{Observation 1:} The average daily travel time of clean case carts is longest in the \emph{Current} material handling approach and shortest in the \emph{JIT} approach (see Figures \ref{fig:cm1}, \ref{fig:cm2}, and \ref{fig:cm3}). The \emph{Current} approach has the longest travel time due to congestion since the delivery of clean case carts for elective surgeries takes place during 3-7pm.

\emph{Observation 2:} The average daily travel time of clean case carts increases with the number of AGVs (see Figures \ref{fig:cm1}, \ref{fig:cm2}, and \ref{fig:cm3}). This increase is highest in the \emph{Current} material handling approach.

\emph{Observation 3:} The average daily travel time of clean case carts in the \emph{JIT} approach is not impacted by the increase in the number of AGVs since the delivery of case carts is spread over the day. These deliveries do not cause congestion (see Figure \ref{fig:cm3}).

\emph{Observation 4:} The average daily travel time of soiled case carts for every material handling approach is slightly impacted by the increase in the number of AGVs (see Figures \ref{fig:dm1}, \ref{fig:dm2}, and \ref{fig:dm3}). Note that the difference in the average travel time per trip is small but still statistically significant. The change in travel time due to the increase in the number of AGVs for every Model is small because soiled case carts are delivered to CSSD right after the surgery; thus, they are delivered throughout the day, and these deliveries have a minimal impact on congestion.

\begin{table}[ht]
\caption{Sensitivity Analysis of Clean Case Carts Delivery}
\label{Table:CCS}
\centering
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c|cccc}
\hline
\textbf{} & \textbf{} & \multicolumn{4}{c}{\textbf{Travel Time (Minutes)}} \\ \hline
\textbf{No. of AGVs} & \textbf{Model} & \textbf{Average} & \textbf{StDev} & \textbf{CI for Average} & \textbf{CI for StDev} \\ \hline
\textbf{6} & \textbf{1} & 4.83 & 0.31 & (4.83, 4.84) & (0.30, 0.31) \\
\textbf{} & \textbf{2} & 4.96 & 0.17 & (4.95, 4.96) & (0.17, 0.18) \\
\textbf{} & \textbf{3} & 3.31 & 0.27 & (3.31, 3.32) & (0.26, 0.27) \\ \hline
\textbf{8} & \textbf{1} & 6.53 & 0.60 & (6.51, 6.54) & (0.59, 0.61) \\
\textbf{} & \textbf{2} & 6.47 & 0.64 & (6.46, 6.48) & (0.62, 0.65) \\
\textbf{} & \textbf{3} & 3.43 & 0.36 & (3.42, 3.44) & (0.35, 0.36) \\ \hline
\textbf{10} & \textbf{1} & 8.02 & 1.16 & (7.99, 8.05) & (1.14, 1.17) \\
\textbf{} & \textbf{2} & 7.66 & 1.17 & (7.63, 7.69) & (1.14, 1.19) \\
\textbf{} & \textbf{3} & 3.46 & 0.39 & (3.46, 3.47) & (0.38, 0.39)
\\ \hline
\end{tabular}
}

\end{table}

\begin{figure}[H]
    \centering
    \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{cm1.pdf}
        \caption{Total Travel Time: Model 1}
        \label{fig:cm1}
    \end{subfigure}
    ~

    \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{cm2.pdf}
        \caption{Total Travel Time: Model 2}
        \label{fig:cm2}
    \end{subfigure}
    ~

     \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{cm3.pdf}
        \caption{Total Travel Time: Model 3}
        \label{fig:cm3}
    \end{subfigure}

    \caption{Sensitivity Analysis of Clean Case Carts Delivery}
    \label{fig:EffectofMaterialHandlingCCS}
\end{figure}

\begin{table}[ht]
\centering
\caption{Sensitivity Analysis of Soiled Case Carts Delivery}
\label{Table:SCS}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{c|c|cccc}
\hline
\textbf{} & \textbf{} & \multicolumn{4}{c}{\textbf{Travel Time (Minutes)}} \\ \hline
\textbf{No. of AGVs} & \textbf{Model} & \textbf{Average} & \textbf{StDev} & \textbf{CI for Average} & \textbf{CI for StDev} \\ \hline
\textbf{6} & \textbf{1} & 5.46 & 0.10 & (5.46, 5.46) & (0.094, 0.097) \\
\textbf{} & \textbf{2} & 5.44 & 0.09 & (5.44, 5.44) & (0.087, 0.089) \\
\textbf{} & \textbf{3} & 5.39 & 0.07 & (5.39, 5.39) & (0.069, 0.071) \\ \hline
\textbf{8} & \textbf{1} & 5.60 & 0.18 & (5.59, 5.60) & (0.179, 0.186) \\
\textbf{} & \textbf{2} & 5.53 & 0.15 & (5.53, 5.54) & (0.149, 0.154) \\
\textbf{} & \textbf{3} & 5.40 & 0.07 & (5.39, 5.40) & (0.072, 0.074) \\ \hline
\textbf{10} & \textbf{1} & 5.75 & 0.29 & (5.75, 5.76) & (0.290, 0.299) \\
\textbf{} & \textbf{2} & 5.64 & 0.23 & (5.63, 5.64) & (0.232, 0.238) \\
\textbf{} & \textbf{3} & 5.40 & 0.07 & (5.39, 5.40) & (0.072, 0.074) \\ \hline
\end{tabular}
}
\end{table}

\begin{figure}[ht]
    \centering
    \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{dm1.pdf}
        \caption{Total Travel Time: Model 1}
        \label{fig:dm1}
    \end{subfigure}
    ~

    \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{dm2.pdf}
        \caption{Total Travel Time: Model 2}
        \label{fig:dm2}
    \end{subfigure}
    ~

     \begin{subfigure}[b]{0.3\textwidth}
        \includegraphics[width=\textwidth]{dm3.pdf}
        \caption{Total Travel Time: Model 3}
        \label{fig:dm3}
    \end{subfigure}

    \caption{Sensitivity Analysis of Soiled Case Carts Delivery}
    \label{fig:EffectofMaterialHandlingSCS}
\end{figure}

\textbf{R3: How does integrating decisions about inventory and material handling impact the efficiency of ORs?}

\emph{Observation 1:} The average delay per surgery and the total number of delays are lowest in the \emph{JIT} material handling approach. These statistics are highest in the \emph{Current} approach. A successful implementation of \emph{JIT} requires coordination of material handling and inventory decisions, and numerical results show that this coordination leads to improved OR efficiency.

This observation is true at different inventory levels, represented by Scenarios 1, 2, and 3 in Table \ref{Table:R1}; for different material handling approaches, represented by Models 1, 2, and 3 in Tables \ref{tab:AvgDelayByType} and \ref{tab:delayfrequency}; and for different material handling capacities, represented by the number of AGVs in Tables \ref{tab:AvgDelayByType} and \ref{tab:delayfrequency}.

\emph{Observation 2:} The average inventory level is lowest in the \emph{JIT} material handling approach. This observation is supported by the results of Tables \ref{Table:R1} and \ref{Table:R2}.

\textbf{Recommendations:} The following recommendations are made based on the observations presented above:

\emph{Recommendation 1:} Coordinating material handling and inventory management decisions has the potential to improve the efficiency and reduce the cost of ORs without jeopardizing the service level provided. To facilitate this coordination, the requirements for each surgery, the number of available instruments, and the location of instruments must be known at all times. Transparent information technology systems will facilitate the coordination of decisions.

\emph{Recommendation 2:}  Hospitals should consider implementing a \emph{JIT} material handling approach for instruments used in short-duration surgeries because such an approach leads to lower inventory levels without jeopardizing the service level provided.

\emph{Recommendation 3:} Hospitals should frequently reevaluate their material handling system to identify improvements. For example, GMH currently uses 10 AGVs. Using only 6 or 8 AGVs leads to reduced congestion along the corridors of the hospital and leads to shorter delivery times. The remaining AGVs can be used for transportation of trash, linen, and pharmaceuticals, among other items.

\begin{table}[H]
\caption{Average Delay by Service Type (Hours)}
\label{tab:AvgDelayByType}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccc|ccc|ccc}
\hline
\textbf{} & \multicolumn{3}{c|}{\textbf{Model 1}} & \multicolumn{3}{c|}{\textbf{Model 2}} & \multicolumn{3}{c}{\textbf{Model 3}} \\ \hline
\textbf{Service Type} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} \\ \hline
ENT  & 0.060 & 0.006 & 0.001 & 0.009 & 0.000 & 0.000 & 0.001 & 0.000 & 0.000 \\
Gynecology  & 0.132 & 0.011 & 0.001 & 0.014 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Neurological  & 1.672 & 0.124 & 0.005 & 0.330 & 0.011 & 0.000 & 0.085 & 0.000 & 0.000 \\
Ortho trauma  & 0.018 & 0.000 & 0.000 & 0.001 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Pediatric  & 1.186 & 0.244 & 0.042 & 0.266 & 0.020 & 0.002 & 0.030 & 0.001 & 0.000 \\
Urology  & 0.245 & 0.024 & 0.001 & 0.014 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
Vascular  & 0.324 & 0.003 & 0.000 & 0.028 & 0.000 & 0.000 & 0.003 & 0.000 & 0.000 \\ \hline
\multicolumn{3}{c}{\footnotesize{*The total number of replications is 30.}}
\end{tabular}
}
\end{table}

\begin{table}[H]
\caption{Frequency of Delayed Surgeries}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|ccc|ccc|ccc}
\hline
 & \multicolumn{3}{c|}{\textbf{Model 1}} & \multicolumn{3}{c|}{\textbf{Model 2}} & \multicolumn{3}{c|}{\textbf{Model 3}} \\ \hline
\textbf{Service Type} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} & \textbf{6 AGV} & \textbf{8 AGV} & \textbf{10 AGV} \\ \hline
{ENT} & 378 & 41 & 5 & 175 & 14 & 2 & 17 & 0 & 0 \\
{Gynecology} & 580 & 60 & 5 & 138 & 10 & 0 & 10 & 0 & 0 \\
{Neurological} & 6,254 & 595 & 22 & 3,352 & 314 & 0 & 1,820 & 0 & 0 \\
{Ortho Trauma} & 120 & 0 & 0 & 63 & 0 & 0 & 0 & 0 & 0 \\
{Pediatric} & 4,859 & 1,293 & 255 & 2,532 & 377 & 52 & 743 & 20 & 0 \\
{Urology} & 2,261 & 268 & 8 & 652 & 39 & 0 & 8 & 0 & 0 \\
{Vascular} & 1,722 & 24 & 0 & 615 & 5 & 0 & 85 & 0 & 0 \\ \hline
\multicolumn{3}{c}{\footnotesize{*The total number of replications is 30.}}
\end{tabular}
}
\label{tab:delayfrequency}
\end{table}

\section{Summary and Concluding Remarks}

The proposed research and the models presented here are motivated by the opportunities for improvement observed in GMH’s inventory management and material handling. Based on their \emph{Current} material handling system, case carts loaded with instruments required by surgeries scheduled during a day are delivered, via AGVs, to ORs within a few hours or the evening before the surgery. This delivery schedule leads to (\textit{i}) increased inventory of instruments, whether owned, loaned, or cosigned; (\textit{ii}) increased traffic and congestion, which delay the deliveries of case carts and the delivery other materials that use AGVs; and (\textit{iii}) delayed surgery start times. The inefficiencies identified motivated the following research questions: \textbf{(R1)}: \emph{How does the inventory level of surgical instruments, including owned, borrowed and cosigned, impact the efficiency of ORs?} \textbf{(R2)}: \emph{How do material handling activities impact the efficiency of ORs?} \textbf{(R3)}: \emph{How does integrating decisions about inventory and material handling impact the efficiency of ORs?}

In order to address these research questions, two new material handling approaches are proposed and compared to the approach already in place. Along with the already-existing \emph{Current} approach, the \emph{Two Batch} approach delivers surgical carts to ORs twice a day, in the morning and in the evening, and the \emph{JIT} approach delivers a surgical carts to an OR before the surgery. Three DES models are developed for each of the three approaches, and they are verified and validated using real-life data collected at a partnering hospital.

A thorough sensitivity analysis of the DES models is conducted and leads to a number of observations and recommendations. The \emph{Current} material handling approach is most sensitive to changes in the inventory level, requires the highest levels of inventory to maintain a high service level, and leads to congestion and delays of the delivery of surgical case carts. Both the \emph{Two Batch} and \emph{JIT} approaches outperform the Current material handling approach. The implementation of the \emph{JIT} approach leads to the greatest improvements in OR efficiency and service levels. Based on these observations, hospitals should identify opportunities to coordinate material handling and inventory management decisions since it leads to improved efficiency for ORs. New, data-based approaches to material handling and inventory management, like the \emph{JIT} delivery of surgical cases, have the potential to improve the efficiency of short-duration surgeries in hospital ORs.

\subsection*{ACKNOWLEDGMENTS} This research is partially funded via a Spark Research Grant awarded by Material Handling Institute (MHI) and College-Industry Council on Material Handling Education (CICMHE). The authors are grateful to HMI and CICMHE for their support. The authors are thankful to the staff of Greenville Memorial Hospital for their continuous support in this research by providing the data, and the expertise necessary to verify and validate the models developed.

\bibliographystyle{plainnat}
\bibliography{FinalPaper}
\end{document}

config: arxiv


TITLE: How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$?
QUESTION [3 upvotes]: How many integer pairs (x, y) satisfy $x^2 + 4y^2 − 2xy − 2x − 4y − 8 = 0$?
My Attempt
Let $f(x,y)=x^2 + 4y^2 − 2xy − 2x − 4y − 8$ . So $f(x,0)=x^2 − 2x − 8$ . $f(x,0)$ has two roots $x=4 , -2$ . So (4,0),  (-2,0) are solution of the given equation. Same way solving for $f(0,y)=0$ we get $ y=2 , -1$ are roots and hence (0,2),  (0,-1) are solutions. I have tried to factorize $f(x,y)$ or writing it as sum of squares but could not succeed. Is there any other solutions? How do I find them?

REPLY [0 votes]: As the equation given is that of a conic curve (an ellipse), one can enlarge the issue, and look for all points with rational (instead of integer) coordinates. It is maybe worth to recall a classical method for doing so.
(a) Choose a point with rational coordinates; I take here $P(0,2)$.
(b) Consider all lines through $P$, with general equation $y-2=t(x-0)$. The second point of intersection with the conic curve has coordinates:
$$x=\dfrac{6\,\left( 1 - 2\,t \right) }{1 - 2\,t + 4\,t^2} \ , y = \dfrac{2\,\left( 1 + t - 2\,t^2 \right) }{1 - 2\,t + 4\,t^2}$$
(c) Impose a rational slope $t=p/q$, $p$ and $q$ integers.
One obtains in this way the general form of all rational points on the cubic:
$$x=\dfrac{6 q \,\left( q - 2\,p \right)}{4\,p^2 - 2\,p\,q + q^2} \ , \ y=\frac{2\,\left( -2\,p^2 + p\,q + q^2 \right) }{4\,p^2 - 2\,p\,q + q^2}$$


TITLE: Volume and Surface Area of Banana using Calculus
QUESTION [4 upvotes]: I'm trying to find the volume and surface area of a curved surface which is similar to that of a banana using calculus, however I am having difficulty because normally, one would due this using a volume of revolution, but there is no straight axis through which the curves modelling the banana could be revolved. Could someone please give some suggestions as to how I could approach this?

REPLY [1 votes]: You might be interested in Cavalieri's principle:

If two bananas are such that every slice has the same area, then they have the same volume.

This way, if you can find the area of every slice of your banana, you can compute the volume of another banana, one which is a solid of revolution, that has the same area for each slice (so you would just integrate the area over the length).

I don't have an easy way to find the surface area, you might have to read up on surface integrals to do that.


TITLE: List all cosets of An in Sn
QUESTION [0 upvotes]: Looking for all cosets of $A_n$ in $S_n$.
I know there are only 2 distinct cosets according to Lagrange's Theorem and was told they were sets of all even and odd permutations. It makes sense intuitively but how should I prove it?
Thanks in advance.

REPLY [2 votes]: You probably know that $A_n$ is normal. Then, since cosets partition the group, the other coset is $S_n - A_n$. You can also write that as $xA_n$ for $x \notin A_n$.


TITLE: Derivative of $\sin (2x)\cos(3x)$
QUESTION [0 upvotes]: What is the derivative of $\sin (2x)\cos(3x)?$

Is it
 $\cos (2x) \cos(3x)-\sin (2x) \sin(3x)$ or $2\cos (2x) \cos(3x)-3\sin (2x) \sin(3x)?$
The answer is given the latter but I only get the former.
Please offer your assistance for the same.
Thankyou

REPLY [2 votes]: The product rule states:
$$\frac d {dx}(f(x)\times g(x))=f'(x)\times g(x) + g'(x)\times f(x)$$
So let f(x) = sin(2x) and g(x) = cos (3x).  You should be familiar with the derivative of sin(ax) and cos(ax), so just plug those in.
Useful:
$$\frac d {dx} \sin(ax) = a\cos(ax)$$
$$\frac d {dx} \cos(ax) = -a\sin(ax)$$

REPLY [1 votes]: HINT:
Why don't we use $$2\sin B\cos A=\sin(A+B)-\sin(A-B)$$ to find $$2\sin2A\cos3A=\sin5A-\sin A$$


TITLE: Why capacitance is given as constant value in Farads, and not as max charge in Coulombs?
QUESTION [2 upvotes]: The equation for capacitance is $Q=CV$ or $V={1\over C}Q$.
I don't understand what is the physical meaning of this "$C$":

Does the charge in a system changes linearly with voltage under all circumstances?
Wouldn't it be more comfortable to define electrostatic energy storage capability of a capacitor (or any other system) in terms of max possible charge, $Q_{max}$?

(related question: Why does the area of the plates affect the capacitance?)

REPLY [1 votes]: The equation for capacitance is Q=CV or V=1CQ. I don't understand what is the physical meaning of this "C":
Does the charge in a system changes linearly with voltage under all circumstances?

This first part is the statement of the behavior of an "ideal" capacitor. Because it is an idealization, it is easy to characterize as a linear component whose behavior can be easily calculated. As a concept this gives it general applicability in many real-world situations, where important and valid results can be obtained by approximating the messy, imprecise, non-linear behaviors of things with idealized components given some constraints (such as limiting the system characterization to a small range of voltages and/or currents).
Non-ideal capacitors will not behave in this way, but this linear model of capacitance is so useful, and so generally applicable in modern, everyday electronic circuit theory, that the messy, general, nonlinear forms are not even considered.
As for the area of the plates... it seems intuitively obvious that the storage of charge in the "capacitor" must be directly proportional to some measure relating to "capacity". In a freely conducting medium, the electric field is uniform, so the only other parameter which can provide more capacity is the area of the plate: more charges require more area to spread out over, if the electric field is to remain uniformly at one value.
For the nature of the question, this seems like the best answer that can be afforded "intuitively" without going into great detail about Gauss's law and other aspects of electrostatics.


TITLE: How to perform vertical line test algebraically?
QUESTION [2 upvotes]: I understand how a function should have one value mapped to a single value in the range. Thus, when graphing the function I understand how a vertical line can determine if a point or multiple points intersect the vertical line. This is visual. However, how can we perform the same test algebraically across a domain of the function?
I am learning calculus from the start again since college, so in case my question is naive or obvious I apologize.

REPLY [1 votes]: To say that a "relation" (or oftentimes "graph") is a "function" from A to B (A is the domain and B is the range) is to say that for any number a (in A) (so any number we take a "y value" at), there is exactly one number that this a gets mapped to. In practice, this typically amounts to checking how the "relation" is defined, and comparing it with this "exactly one" condition. Let's say our "relation" is f(x) = x (so the "identity function", sends numbers to themselves), we start with a number a, and f sends this a to f(a), which is just a in this case, so we start with a, and this a gets sent just to a, so where the function sends any number a (to itself) is definitely unique. $f(x) = x^2$ is another function, to check this is a function amounts to just checking that when you take any number x and square it, the output is unique (there is a single output, for example, -3 gets sent to the unique output $(-3)^2 = 9$) In this way, $x^2$ is a function. This process doesn't have to be "algebraic" it is just "clear" (or "rigorous") (more so than looking at a picture at least).


TITLE: Does every function have an integral?
QUESTION [1 upvotes]: I tried to find the integral $\int f(x)dx$ of $f(x)=\sqrt{1+\dfrac{1}{x^4}}$. I couldn't solve this manually and used https://www.integral-calculator.com/ for finding the integral. Surprisingly, a message was displayed.

Antiderivative or integral could not be found. Note that many functions don't have an elementary antiderivative.

Now, I was confused as integral $\int f(x)dx$ represented the area under the curve $y=f(x)$. Now, the curve $y=\sqrt{1+\dfrac{1}{x^4}}$ has some area under it. So, why doesn't it have an integral?

REPLY [1 votes]: To answer the question in the title (completely separate from the question of elementary-function antiderivatives):

Does every function have an integral?

No, there are functions that do not have any defined integral at all. But the exact category of such functions depends on which definition of integral you're using: Riemann, Lebesgue, or less commonly, something else.
For example, the characteristic function of the rational numbers (function with value 1 at a rational number, 0 elsewhere) is not Riemann integrable, but it is Lebesgue integrable. On the other hand, the sinc function taken over the entire real number line is not Lebesgue integrable.
If you're interested in pursuing that more, consider reading the Wikipedia article on the Lebesgue integral, particularly sections on limitations of the various integral definitions.


TITLE: The negation of $(\forall \varepsilon >0) (\exists N\in \mathbb{N})(\forall n\in\mathbb{N})$ if $ n\geq N,$ then $ |a_{n}-a|<\varepsilon $
QUESTION [1 upvotes]: I am trying to negate the following expression:
$(\forall \varepsilon >0) (\exists N\in \mathbb{N})(\forall n\in\mathbb{N})$: if $n\geq N, $ then $ |a_{n}-a|<\varepsilon $.
My negation is the following:
$(\forall \varepsilon >0) (\forall N\in \mathbb{N})(\forall n\in\mathbb{N})$: if $n\geq N,$ then $ |a_{n}-a|\geq  \varepsilon $
Is it correct?
Thanks

REPLY [0 votes]: Following Bram's recipe:
$$\exists \varepsilon>0: \forall N \in \mathbb{N}: \exists n \in \mathbb{N}: (n > N) \land |a_n -a | \ge \varepsilon$$
of in words: there is some distance $\varepsilon$ such that infinitely many terms in the sequence have distance at least $\varepsilon$ from $a$.


TITLE: Angle between two planes: $ x+2y-6 $ and $ y=0 $
QUESTION [0 upvotes]: Plane 1: $x + 2y - 6 = 0$
Plane 2: $y = 0$
--
I can see that the normal vector of the first plane is {1, 2, 0} but what would be the the normal vector of the second one?

REPLY [1 votes]: For the plane $ax+by+cz=d$ the normal is $(a, b, c)$. For the plane $y = 0$ we may write $ 0x+1y+0z=0$. Hence the normal is $\boxed{(0, 1, 0)}$.

REPLY [1 votes]: Your first normal vector is not correct.
The normal vector should be $(1, 2, 0)$
You might ask: Why we do not coins $6$? No matter what constant we take, those planes are parallel, and thus have the same normal vector.
In terms of the second equation, write it as
$$0\cdot x + 1 \cdot y + 0 \cdot z = 0$$
Could you find its normal vector?


TITLE: Show $U = \frac{(n-1)S^2}{\theta^2} $ is a pivotal quantity
QUESTION [0 upvotes]: Let Y1,...Yn be a random sample from $N(Mean = \theta,Var = \theta)$
Show $U = \frac{(n-1)S^2}{\theta^2} $ is a pivotal quantity.
$ S^2 = \sum_{i=1}^{n} (Y_i - \overline{Y})/(n-1) $
so $U = \frac{(n-1)S^2}{\theta} $ is a chi-square but what about over $\theta^2$?
What distribution is U?

REPLY [0 votes]: Remember that if $X_1,\ldots X_n$ are a sample from the normal($\mu,\sigma^2$) distribution, then $\sum(X_i-\bar X)^2/\sigma^2$ has chi-square distribution with $n-1$ degrees of freedom.
To apply this to your problem, define $$Z_i:=\frac{Y_i-\theta}\theta.$$ Argue that each $Z_i$ has standard normal distribution. Next, use algebra to show that
$$\sum(Y_i-\bar Y)^2=\theta^2\sum(Z_i-\bar Z)^2.
$$
From this you should be able to conclude that $$U:=\frac{(n-1)S^2}{\theta^2}=\frac{\sum(Y_i-\bar Y)^2}{\theta^2}$$ has chi-square distribution with $n-1$ df, which is free of $\theta$, and is therefore pivotal.


TITLE: May Champernowne constants $C_m$ be related to other numbers than $m$?
QUESTION [0 upvotes]: [This question is related to another question concerning normal numbers I asked at Math SE.]

Has it ever been found worth to ask the question if the Champernowne constants $C_m$, especially $C_2$ might be related to other (known or nameable, algebraic or transcendental, probably normal) numbers like $\sqrt{2}$, $\pi$ or $e$? Is there any chance, or are there good arguments that $C_2$ probably won't be related to e.g. $\pi$, i.e. there's no closed formula $\phi(\cdot)$ saying $C_2 = \phi(\pi)$.
An argument might go like this:

The value of $C_2$ depends on the arbitrary base 2 while the value of $\pi$ doesn't.

But maybe 2 isn't so arbitrary? (To say the very least: 2 is a prime number. And there's a deep connection between prime numbers and $\pi$.)

As infinite series they are defined by

(http://mathworld.wolfram.com/ChampernowneConstant.html)
and take values like
$C_2 = 0.862240125868054571557790283249394578565764742768299094516\dots$
$C_3 = 0.598958167538433992500172217929436590978208768676105936754\dots$

We obviously don't see at a glance that $C_2 = \frac{\pi}{3} = 1.047\dots$ or  $C_2 = \frac{\pi}{4} = 0.785\dots$ or $C_2 = \frac{\pi^3}{2^5} = 0.968\dots$. (
Endless combinations abound.)
But who assures us - just for example - that no enumerable sequence $(\alpha_k)$ with $\alpha_k \in \mathbb{Q}$ exists with $C_2 = \sum \alpha_k \pi^k$?

Note that many infinite series resulting in a closed expression over $\pi$ – first of all Leibniz' formula for $\pi$ –  came somehow as a surprise. Can this be turned around?

REPLY [3 votes]: One argument against $\pi$ being related to $C_2$: while $C_2$ is normal, its normality is really terrible in the sense that the convergence of its digit-sequence frequencies to uniform is extremely slow. Almost all real numbers, and (presumably) all algebraic expressions in $\pi$, should be not only normal but should approach the limiting frequencies about as fast as sequences of coin tosses would.
You can find out more about this concept of "strong normality" in this paper of Belshaw and Borwein.


TITLE: $f$ is a entire function, $f(\mathbb{R}) \subseteq \mathbb{R} , f(\mathbb{iR}) \subseteq \mathbb{R}$
QUESTION [1 upvotes]: $f$ is a entire function, $f(\mathbb{R}) \subseteq \mathbb{R} , f(\mathbb{iR}) \subseteq \mathbb{R}$, Please prove that $f$ is a even function.
I wanted to consider $f(z)-f(-z)$ and show that it is zero.So I considered the category of singularity $\infty$, but I got no clue.
Thank you!

REPLY [1 votes]: Robert Israel's answer is probably the easiest approach, but here's another one.  Prove that if $f(\mathbb{R})\subseteq\mathbb{R}$, then $f'(\mathbb{R})\subseteq \mathbb{R}$ as well.  Thus by induction on $n$, $f^{(n)}(\mathbb{R})\subseteq \mathbb{R}$.  In particular, $f^{(n)}(0)\in\mathbb{R}$, so the coefficients of the Taylor series of $f$ at $0$ are real.  Now apply the same argument to $g(z)=f(iz)$, and use this to show that the odd coefficients of the Taylor series of $f$ must vanish.  Conclude that $f$ is even.