It will be a happy holidays as Jets/Titans didn't go down as the last NFL game ever played.
We only had five games to recap this week with a ton of blowouts on Separation Sunday, but at least we got the game of the year. San Francisco at New England was deserving of a monster recap, as we look at all the teams to come back from 28+ points, finding that only two of the seven have actually gone on to win the game. Also look at the teams who have come back from 21+ down in the fourth quarter and lost, which includes the Patriots three times now. Dallas and Pittsburgh also played an overtime thriller, and the Cowboys are on their biggest run of clutch wins in team history.
The NFL is a week-to-week league, and after Sunday's results, everything changed. Again. The biggest winners were San Francisco and Denver, as they now have the inside track to the No. 2 seed in each conference. Denver should be in Super Bowl or bust mode this season, as they may never have a better chance at a ring.
It is hard to define a marquee game during the regular season, but I found it even harder to find ones that were really competitive this season. It's been a lot of blowouts, so what does that mean for the playoffs? A look at the previous five seasons.
Note: I still contend the title of the article should be 2012 and not 2013, but not everyone agrees.
J.J. Watt led a dominant attack against Andrew Luck and the Colts' offense in a 29-17 win. Luck was pressured on a season-high 51.5 percent of his drop backs. In his last four games Luck has been pressured on 40.8 percent of his drop backs compared to 27.1 percent in the first 10 games. That explains the drop in offensive efficiency.
I thought this came out as the best edition of the year. A look at why the Bengals need to stop being the Bungles and take care of Pittsburgh. The stat of the year candidate on Adrian Peterson. He has seven runs of 50+ yards in his last seven games alone. That's as many as the Houston Texans have ever had (174 games), and as many as the Colts and Patriots have combined for in their last 412 games.
The Bears are who we thought they were, or are they? Once 7-1, the Bears are hoping to avoid becoming the fourth team since 1978 to start 7-1 and miss the playoffs. Included is a look at why the Bears have faltered, and which company they could be joining should they complete one of the NFL's greatest second-half collapses. Anyone remember the 1993 Miami Dolphins or 1996 Washington Redskins?
The week is about to soon start with the Falcons/Lions, so let me make sure I get this in on time.
Well that's obvious, but what did he want with me?
Look at my stats; they're one hell of a sight!
With the stash of automatic handguns he just bought.
St. Rex continued to vent his frustrations.
So why did you take him fifth in the draft?
It's a little too late for a return, Jets.
Is it Sunday yet? Bring on the best Sunday schedule of the regular season.
For the first time ever, two rookie quarterbacks (Robert Griffin III and Kirk Cousins) combined to lead a fourth-quarter comeback win. Could it have been avoided if the Ravens went for a two-point conversion to take a late 9-point lead? We have stats on that. Also, the Colts won a game when trailing by 13 points at halftime, something teams only do 11.5 percent of the time. Tony Romo had his 17th comeback win, which is the most in Dallas history. Even the Rams are getting heavily back into comeback wins after having two from 2007-11.
Mike Wallace wanted Larry Fitzgerald money, then Vincent Jackson money, now he might be lucky to get Antonio Brown money. A look into why he has struggled in 2012, and the warning signs were there before the season even started.
Andrew Luck can play 18 years and never have a game with a more misleading TD:INT ratio (1:2). That's what happens when you have a pick six that was a sack blown by the refs, a Hail Mary INT before halftime, and a dropped 35-yard TD that could have been the game-winner. Instead Adam Vinatieri made his 30th game-winning field goal and Luck set a rookie regular season record with his sixth game-winning drive.
Peyton Manning will look to keep his 8-game winning streaks alive for both the Broncos and against the Baltmore Ravens. The Giants will take on Matt Ryan in the Georgia Dome, where he is 32-5 but has struggled this season. Jay Cutler always struggles with the Packers, home or road. Finally we will see if the 49ers can slow down the Patriots in a "No. 1 offense vs. No. 1 defense" matchup. Hopefully we'll just have a great game somewhere, as there have been too many marquee duds this year.
"You want to be playing your best football in December." It's a cliche, but is it true? I examined the final four regular season games for all 120 playoff teams from 2002-11 to find out if playing well late has any impact on playoff success. The truth is it does not, and more than your record, it pays off to be playing well (scoring differential). Teams who finish 2-2 are just as likely to win in the playoffs and a Super Bowl as teams who finish 4-0 in the regular season. Not to mention the whole playoff rest strategy some teams use plays directly against building momentum for the playoffs.
In summary, just another myth. The Giants are known for getting hot twice, but what about a horrible loss at home to the Redskins in December in both 2007 and 2011? The only thing that matters is how you play that day in the postseason where anything can happen in the one and done.
For a change of pace this week, I did a Q&A previewing Colts/Texans with Stephanie Stradley for the Houston Chronicle. I call the Colts the worst 9-4 team in NFL history, but if Houston lets them stay around late, the Colts will definitely have a shot to pull the upset.
Part of me wanted to take Philadelphia on Thursday night, but fortunately I didn't drop the ball on that one and took the Bengals. That'll conclude the Thursday night games this season.
With the news out of Dallas on Josh Brent killing teammate Jerry Brown in a drunk-driving accident, this is another NFL weekend with tragedy coming before the games. Hopefully the NFL and really everyone out there can learn lessons from these incidents. How hard is it not to get into a car when you're drunk? If we are still too stupid to make these decisions, then maybe we need cars that will make them for us.
Before I get confused for Bob Costas during one of his halftime speeches, onto the usual Saturday recap.
The Colts were able to rally from a 12-point deficit in the final 4:02 for another stunning comeback victory over the Lions. Pittsburgh surprised everyone with Charlie Batch leading the team to a road win in Baltimore. The two wins shape up yet another AFC playoff picture with the usual suspects. Also: Tony Romo tied Troy Aikman for most 4QC wins in Dallas history (16). He did it by attacking "The Todd Bowles Movement." This is the most jam-packed edition of Captain Comeback yet with 11 game recaps. It was a good week of games.
True story: this article originally had a general headline (see URL), but after finishing it, the results were pretty clear: Andrew Luck owns clutch moments in a way no rookie really has. We know all about his five game-winning drives, but his performance on third down, especially third and long, is something we just never see from a young QB. The surprise is just how historically bad RGIII has been at converting on third and long.
The war is clearly raging between these fanbases, so arm yourself with some real data that matters, unless you think throwing a short pass on 3rd and long and punting is good.
Find out how Andrew Luck threw 30 incompletions, three interceptions, had a season-high in three-and-out drives, but still managed to grow his legend with another comeback win and career-high four touchdown passes. The young Jedi just had to use the Schwartz.
If the numbers stay on their current pace, then the 2012 NFL season could be the most efficient ever in terms of passing, rushing, kicking and punting, otherwise known as the primary acts of a football game. Details on each category and the record-breaking numbers the league is averaging. It is also the most pass-happy season ever with teams throwing the ball on 57.95 percent of their plays.
In-depth previews for Raiders/Broncos, Chargers/Steelers, Saints/Giants and Texans/Patriots plus predictions for every game.
Last week we looked at turnover rates using all fumbles. This week I took the lost fumble data that goes back to 1992 and looked at which quarterbacks have the most and least fumble luck, because recoveries are in fact random, and the data bears that out. Tom Brady is completely average (42.5%) in this regard, while Matthew Stafford has only lost four of his 14 fumbles, and Kyle Orton has the worst luck, losing 60% of his fumbles. Finally an updated turnover rate list using fumbles lost is included for the 59 quarterbacks from the last two decades.
Was there even any doubt on the Thursday game? Peyton Manning has now won at least eight straight games in a season for the seventh time in his career. If only I had his consistency at making these game picks.
Mark Sanchez played poorly and was pulled on Sunday in place of the backup quarterback, who promptly led the offense on a game-winning drive for the New York Jets over the Arizona Cardinals.
This scenario has been expected for many months now, but no one imagined it would have been Greg McElroy at quarterback rather than Tim Tebow.
These are the moments Tebow has built his NFL career on, but he was nowhere to be found Sunday, missing the game with a rib injury.
McElroy, on the very first drive of his career, led a game-winning touchdown march, hitting Jeff Cumberland for the easiest 1-yard touchdown pass you may ever see. He even contributed to the last drive which consumed the final 7:55 off the clock in a 7-6 win.
Rarely does a quarterback lead a fourth-quarter comeback and game-winning drive in his first NFL game, but I have started to compile a list of occurrences.
Not sure anyone on the list but McElroy was able to accomplish it on their very first drive. If you broke it down that way, he'd be batting 1.000 in clutch drive success rate.
Is it the most impressive list? Not at all, but the win should create another quarterback controversy in New York.
Just not the one we all expected.
Getting right down to business.
Joe Flacco hits the first "Dump Mary" in NFL history, with Ray Rice taking it the distance on 4th and 29. NFL teams are 6/35 (17.1%) on 4th and 20+ since 2000 in clutch situations. Full table included of the successes. Also this week: Houston wins in OT for the second straight week with the help of Walt Coleman explaining another terrible, obscure NFL rule. Matt Ryan sets more records and the Steelers make asses out of themselves in Cleveland.
It's a slideshow, but don't worry, it's still over 4,100 words. A look at the 10 teams who could least afford to lose their starting QB, with some surprises in the top 5.
The "Fire Wanny!" movement is understandable. Dave Wannstedt's defense was the most vanilla up front all season out of the 11 games the Colts have played, rushing four or less on 40 of 43 drop backs for Andrew Luck. However, they did get an impressive amount of pressure, so it was not all bad. Though, Reggie Wayne was the most wide open he has been all season, and the Colts receivers had their best YAC performance of the season.
Since the NFL moved to eight divisions in 2002, the league averages 5.2 new division winners and 6.3 new playoff teams. This season, they may be lucky to only get two new division winners and three new playoff teams. The numbers could be even smaller if the Packers win the NFC North again. This comes after a season with a record seven new division winners in 2011.
Even if you don't want to read this one, it's a great piece to bookmark as I compiled the division winners and all 12 playoff seeds for 2002-11 into tables. Might be the only place on the internet you can get all of that info on one page.
This week we look at: why the Falcons need to get over (and did) the Saints hump. Are the Colts the worst 7-4 team ever? Todd Bowles is having the worst career start ever for a defensive coordinator in Philadelphia.
Updating (but not rehashing) an article from last year, here is a look at the career QB Turnover Rates, which includes interceptions and fumbles (all fumbles). Aaron Rodgers (2.65) has the best mark ever, while Rex Grossman (5.30) is the worst active QB, and exactly double the rate of Rodgers. How fitting. A full table for 171 quarterbacks in NFL history (min. 1,500 attempts).
Did Drew Brees' birthmark expand after he threw five interceptions on Thursday night? I picked the Falcons to win, but that was still a shocking way for the game to play out.


The most surprising thing about the Republican presidential debate was the questions, and how tough many of them were. Most especially the questions aimed at Donald Trump. Those were so fierce - not least the calling-out on an independent candidacy at the beginning, which ostensibly wasn't aimed at a single candidate (though it really was) - that a clear goal on the part of Fox of seriously damaging Trump was evident. If Trump's constituency were of a different kind, it might have worked, too. The questions hit home on such matters as party loyalty, violation of core party stands and more, matters that would kill off most candidates. But while the questions highlighted, they did not unearth. Trump's threat to run a third-party candidacy has been in the news, as had nearly everything else the Fox questioners brought up. Was Trump damaged by the debate? We'll find out more soon in the after-party polling, but I'd guess not. I think it's more likely Fox drove a wedge between itself and some of Trump's constituency, which may be led by the candidate to now view Fox as just another part of the establishment. And did other candidates gain? Maybe Marco Rubio, a little, since he came cross as polished at least and got some easy questions. In the kids' table debate, Carly Fiorina was described as projecting a strong presence, but she's way back in the pack to start. Jeb Bush did himself little good - he didn't crash, but he came across like a dull corporate attorney. Did the debate change the contours of the race? Probably not much. And on to round two.
In our household when we turn to "the news" on television, that has for many years meant Jon Stewart and Stephen Colbert - no other regular "news" TV programming need apply, so poor has most of the quality gotten. (CSPAN is welcome, and scattered individual programming, but nothing else on a nightly basis.) So this has been a significant year: First Colbert and now Stewart, as of last night, have departed. The news won't be the same. But the future beckons. Larry Wilmore, while not yet the equal of either of those two, has been gaining some strength. And while we as yet have no idea of what job Stewart's successor will do, we do know that others can do the job well: A year ago, John Oliver did a terrific job filling in for three months, and he was promptly grabbed away to do his own program elsewhere. So good luck to the new order.


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Home / News / Tax Tips and News Let property: improvement v repair
Let property: improvement v repair
Newsletter issue - December 2021.
The tax rules for property businesses have changed considerably since 2010, for example the restriction of relief for financing costs to the equivalent of a basic rate tax reducer (at best). Almost all these changes have been made with the aim of making property letting less attractive. This makes it all the more important to ensure legitimate deductions are correctly identified and claimed.
One area where there is often confusion is the issue of whether work done on a property constitutes a repair, in which case the cost is allowable as a deduction against income (i.e. a revenue cost), or an improvement which is capital in nature, and can only be deducted against the eventual sales proceeds.
The problem is that it's not always a clear distinction. As an example, it should be fairly clear that building a new extension is a capital cost. However, would this still be the case if the extended part became damaged and had to be replaced?
HMRC's guidance in the property income manual is a helpful starting point. Their approach is that where there is significant capital improvement the cost is likely to be capital in nature. However, where the capital improvement is incidental, the whole cost can be included as a revenue expense.
Of particular interest is the point that where a repair or replacement of part of a building leads to an element of capital improvement, but this is due to use of modern materials, the cost remains revenue in nature. The examples given are:
wooden beams with steel girders, and
lead pipes with copper or plastic pipes.
As a general rule, the landlord should therefore be able to offset anything that involves a replacement with broadly equivalent materials.
Where there is both capital expenditure and revenue expenditure, a reasonable apportionment may be made if, for example, a single invoice has been issued.
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redpajama_set_name: RedPajamaCommonCrawl


This Metropolis poster is hailed as the most valuable one sheet in the world.
A "Metropolis" movie poster hailed as the most valuable one sheet in the world was auctioned by the U.S. Bankruptcy Court in Los Angeles on Thursday for $1.2 million.
Ralph DeLuca's bid beat out three other collectors to claim the prize he compares to Leonardo da Vinci's Mona Lisa. He says he got a deal, too.
"I was expecting to pay $1.5 to 1.6 million at least," DeLuca told TheWrap hours after making the purchase. "Had this been a proper auction, the Metropolis alone would have been bought for $1.6 million."
Fortunately for his wallet, that was not the case.
Previous owner Kenneth Schacter bought the German artwork for Fritz Lang's 1927 science-fiction film in 2005 for a record $690,000, but had to give it up after filling for bankruptcy.
Thursday's exchange was expected to produce a new record poster price tag, however, the rare movie memorabilia came bundled with several other classic posters for films such as "King Kong" and "The Invisible Man," so the sale did not qualify.
There's a good chance a future sale will, though. DeLuca, who also lays claim to owning the largest movie photo collection, believes "the greatest movie poster in the world" would sell for two million dollars.
If he decides to give it up, that is.
Since there are only three other surviving copies of the particular piece designed by German artist Heinz Schulz-Neudamm -- one rumored to belong to Leonardo DiCaprio, one to the Museum of Modern Art and the other to the Austrian National Library museum -- it would be hard to give up the poster. But DeLuca's not ruling out the possibility.
"I'm not saying I would never sell it," he said when asked about his plans for the vintage design. "Someone could make me an offer I couldn't refuse."


text:
\section{Introduction}
Agile manufacturing is essential to industries where demand varies.
Dynamic Lot Sizing (DLS) models deliver production schedules that accommodate dynamic time dependent demand.
The aim of this paper is to present a general DLS MIP formulation.

This section develops a taxonomy of Dynamic Lot-Sizing (DLS) models
and establishes the context to Economic Lot Sizing Problems (ELSP).
Used solution methods will be briefly mentioned with a focus on MIP approaches.
Essential insights and problem formulations are extracted using a consistent notation throughout this paper.
Occasionally, new formulation DLS MIP formulations will be given.

\subsection{DLS models}
In 1958 \citet{wagner2004dynamic} produced a ``landmark'' paper introducing a dynamic version of the
Economic Lot Size (or Economic Order Quantity) model.
Varying deterministic demand $d_j$ has to be fulfilled to create the production schedule $q=(q_1,\ldots,q_n)$ for one product over $n$ periods.
The inventory entering period $t$ is $I=I_{0}+\sum_{j=1}^{t-1} q_{j}-\sum_{j=1}^{t-1} d_{j} $,
where $I_0$ is the initial stock, $q_t$ the manufactured goods and $d_t$ the demand during period $t$.
They formulated a minimal cost policy with holding costs $c^h_t$ and setup costs $c^o_t$ for periods $t \in \set{1,\dots,n}$:
\lequ{WagnerBell}{f_{t}(I)=\min _{q_{t} \geq 0 \atop I+q_t \geq d_t} \left[c^h_{t-1} I+\omega_t c^o_t+f_{t+1}\left(I+q_t-d_t\right)\right],}
where $\omega_t=1$ represents setup occurring when $q_t>0$ and zero otherwise.
As can be seen the model does not allow shortages and is recursive.
They postulated three basic theorems about characteristics of an optimal program.
These allowed them to rewrite the policy as an more efficient recursive minimal cost program.
I would like to give an alternative MIP formulation and call it
\textit{ Dynamic Lot Size }(DLS) model \textit{with one product} (\textit{DLS-1p}):
\LP{ \label{LP:DLS}\objd[min]{q,\omega}{\tp{I} c^h  +\tp{\omega} c^o}}
   { \sto{I_t=I_{t-1}+q_t-d_t,~ t \in T = \sett{n},}
   	 \sto{\omega\leq Mq, ~\omega \geq mq, }
   	 \sto{q\in \real^n_+, \omega\in\binary^n,I_0\in\real_+,}
   }
where $e=\ones$, $M=\tp{e}d$ and $m=\frac{1}{M}$ ($M$ and $m$ can be tightened).
We used $\omega\leq Mq, ~\omega \geq mq$ to represent the logical constraints:
\qbox{\tcb{If} $q_t>0$ \tcb{then} $\omega_t = 1$ \tcb{else} $\omega_t=0$.}
Alternatively, the constraints
\lequ{bigM}{q_t\leq M\omega_t,~t\in T}
could have been used.
These are valid because of the following arguments.
If $q_t=0$ then $\omega_t=0$ or 1,
but due to the minimisation objective $\omega_t=0$.
If $q_t>0$ then $\omega_t \meq 1$, because of (\ref{eq:bigM}).
If we assume that $\omega_t=0$ then (\ref{eq:bigM}) ensures that $q_t=0$.
On the other hand, if $\omega_t=1$ then $q_t<M$.
This proves the validity of (\ref{eq:bigM}) resembling the logical constraint.

\begin{example}{Wagner and Within.}
Let $d=\tp{\smat{69&29&36&61&61&26&34&67&45&67&79&56}}$,
$c^h=\tp{\smat{1&1&\ldots&1}}$
and $c^o=\tp{\smat{85&102&102&101&98&114&105&86&119&110&98&114}}$.
This example was used in 1958 to illustrate the functionality of the recursive minimal cost program.
It gave the optimal cost 864. The here introduced MIP formulation gives the same result.

The special steady state data $d = 52.5e$, $c^h=e$ and $c^o=102.8e$,
i.e. $\bar{d}=\frac{1}{n}\tp{e}d = 52.5$, $\bar{c}^h = 1$ $\bar{c}^o=102.8$
returns the optimal production quantity 105 every other week.
This is comparable to the economic order quantity (EOQ)
$Q=\sqrt{2\bar{d}\bar{c}^o/\bar{c}^h}\approx 103.9$,
which is valid in the continuous case (i.e. the period $t\rightarrow0$).
\end{example}

The DLS-1p can be extended to a \textit{DLS with multiple products (DLS-mp)}.
\citet{boctor2004models} review models and algorithms for the \textit{dynamic-demand joint replenishment problem (DJRP)}.
This is a DLS-mp with an additional ``joint'' ordering cost (or common setup cost) regardless of the product(s) that are ordered (produced).
The common ordering cost are known as general/common setup or changeover cost in a manufacturing context.
This model will be abbreviated with \textit{DLS-mp-cs} instead of DJRP to be consistent with previous notations.
The MIP after converting it to this paper's notation and using logical IP constraints is:
\LP{ \objd[min]{q,\Omega, \omega}{
		\sum_{t \in T}\left[C^o_{t} \Omega_{t}+\sum_{p \in P} \left[c^o_{tp} \omega_{tp}+c^h_{tp} I_{tp}\right]\right]}}
{   \sto{I_t=I_{t-1,p}+q_{tp}-d_{tp},~ t \in T,p\in P,}
	\sto{q_{tp}\leq M\omega_{tp}, ~\sum_{p\in P} \omega_{tp} \leq m \Omega_t,}
	\sto{Q=(q_{tp})\in \real^{n\times m}_+, (I_{tp}) \in \real^{n\times m}, I_0\in \real^p_+,}
    \sto{(\omega_{tp}) \in \binary^{n\times m}, \Omega\in \binary^p_+.}}
Here, a common setup (ordering) cost $C^o_t$ is used
in addition to the individual setup (ordering) cost $c^o_{tp}$
for product $p\in P=\sett{m}$
with corresponding decision variables $\Omega_{t}$ and $\omega_{tp}$.
This formulation has two interesting features: (1) the common ordering cost,
and (2) the logical constraints.
When the common ordering cost is zero the program (\ref{LP:DLS}) defines the DLS-mp.
However, the number of decision variables must be reduced by dropping $\Omega$
and the following constraints have to be used:
\lequ{limiting}{\omega_{tp} \leq M q_{tp}, ~\omega_{tp} \geq m q_{tp}.}
By the way, the objective function can be rewritten using matrices/vectors
(for readability transpositions are implicit):
\lequ{obj-DLS-mp-cs}{C^o\Omega+\sum_{p \in P}(c^o_p\omega_p+c^h_p I_p).}

Next, I will mention a few works that illustrate other DLS-mp flavours.
\cite{gilbert2000coordination} looked at creating production schedules for
multiple products with a single production line.
A production capacity requirement $\sum_{p\in P}q_{tp} \meq \hat{q}_t$ was introduced.
Moreover, the model introduced constant priced products $(r_p)$,
which allowed the formulation as a profit maximisation program.
Their revenue component defines a revenue/demand parameters $\beta_{tp} := d_{tp}/d_p$,
where $d_p$ was used for a demand intensity (related to price).
This makes the price demand dependent.
Price changes lead to demand changes due to price elasticity.
However, it was assumed that this will not happen.
Moreover, the program gets the costs by solving the DLS with common setup costs.
Overall, some interesting first steps into considering price adaptations
and more importantly (in regard to DLS) a capacity requirement.
The capacity requirement will be picked up in Section \ref{sec:MIP} as production limit.

\cite{bruggemann2000discrete} introduced a DLS with batch production requirements.
This addresses an essential short-coming of the previous models.
Their model formulation assumes that a batch is produced
over several consecutive periods.
This requires the introduction of decision variables and constraints
indicating the start and end of the production period.
The increased complexity motivated them to introduce
a ``two-phase'' (1. feasible solution search, 2. cost optimisation) heuristic based on simulated annealing.
Test-instances (products,periods)$ = \set{(3,30),(6,60),(8,80),(10,100)}$
were used to gain performance insights.
The (3,30) instance was compared to the optimal solution having a 12.53\% difference.
I will propose a different approach to consider batch-sizes.
Rather than using a small period (e.g. hours)
that spans overs several periods,
a large period (e.g. week) is used that allows the production of several batches.
This approach will be defined later in more detail
and overcomes the issue of the large number of decision variables.
Another idea to enhance their work are ``batch-interruption'' penalties in the objective function.
Overall, such DLS models are classified as \textit{DLS with batch production} (\textit{DLS-bp}) systems.

\cite{absi2009multi} proposed a multi-item capacitated lot-sizing problem with safety stocks and demand shortage costs.
They used a Lagrangian relaxation algorithm and a dynamic programming algorithm to solve this problem.
Generally, \textit{DLS models with capacity limit} will be classified as \textit{DLS-cl} systems.
Safety stocks are essential and their immediate consideration
when creating the production schedule seems promising.
This addresses shortages indirectly.
Interestingly, most DLS models found in this review do not allow shortages.
An alternative is to introduce safety buffers after having created the production schedule.
Here, the uncertainty (e.g. $N(\bar{d}_p,\sigma_{dp})$) in demand can be taken into account for each product
and the uncertainty in lead time (e.g. $N(\bar{L}_p,\sigma_{Lp})$).
Hence, the safety buffer is:
\lequ{safetyBuffer}{b_p = Z\sqrt{\bar{d}_p^2 \sigma_{Lp}^2 +  \bar{L}_p^2 \sigma_{dp}^2}.}
Please see \cite[p223]{Garn2018Dec}, \cite[p121ff]{ghiani2004introduction} or \cite{silver1998inventory} for more details.

\citet{lu2011dynamic} provided a dynamic lot sizing for multiple products introducing a joint replenishment model,
where production quantities are of same size for all products per periods $q_{ti} = q_{tj}, i,j\in P$,
i.e. $q_t$ can be used.
Lost sales are considered by rejecting production requirements, i.e. no back-ordering.
This implies that no shortage decision variables are required for this formulation.
However, the rejected quantity $y_{tp}$ has to be recorded.
In addition to a common setup cost a common production cost $\pi_t$ is used.
The resulting MIP \footnote{The transpose operator has been used implicitly to improve the readability, i.e. $C^o \Omega := \tp{(C^o)} \Omega = \sum_{t\in T} C^o_t \Omega_t $, similarly variables non-negativity was assumed} is:
\LP{ \obj{C^o\Omega + \pi q_t + \sum_{p \in P} (c^h_p I_p + c^s_p y_p) }}
   { \sto{ I_{tp} = I_{t-1,p}+y_{tp} + q_t - d_{tp},}
   	 \sto{q\leq M \Omega,~Y\leq D.} }
NP-hardness of the above problem was shown using the subset-sum approach.
The NP-hardness motivated the development of a heuristic.
The \textit{Dominant Product Approach} was used,
which is actuated from the real-world observation
that a few products generate the majority of economic benefits.
This is commonly known as the Pareto effect.
The proposed algorithm chooses the dominant product to generate the initial schedule.
Roughly speaking, the production of the remaining products is mainly guided by rejecting quantities.
The second heuristic aggregates demand of all products during a time period; and runs a LP.

\citet{lee2005heuristic} gave a heuristic algorithm for a multi-product dynamic lot-sizing and shipping problem.
Basically, the meaning of the common ordering cost $C^o$ was substituted by freight cost of a container.
The difference is that instead of a setup decision $\Omega_t \in \binary$
an integer decision describing the number of containers was used.
Additionally, there is a capacity constraint $\hat{Q}$ for the containers.
The resulting model is:
\LP{ \objd[min]{q,\Omega, \omega}{
		\sum_{t \in T}\left[C^o_{t} \Omega_{t}+\sum_{p \in P} \left[c^o_{tp} \omega_{tp}+c^h_{tp} I_{tp}\right]\right]}}
{   \sto{I_t=I_{t-1,p}+q_{tp}-d_{tp},~ t\in T,p\in P,}
	\sto{\sum_{p\in P} q_{tp}\leq \hat{Q} \Omega_{t},~t\in T}}
A local search heuristic was proposed to solve this problem.
The performance was tested using randomly (uniform, normal) generated demands.
Products varied between 3 and 10 and periods varied between 4 and 18.

\citet{minner2009comparison} introduced three local search heuristics for the multi-product dynamic lot-sizing with limited warehouse capacity and provided a MIP implementation.
The heuristics are only a few percentage points away from the best solution found.

Figure \ref{fig:DLS-classification} shows the proposed classification scheme.
\begin{figure}[htbp]\centering
	\includegraphics[width=\columnwidth, height=4cm, keepaspectratio]{DLS-classification}
	\caption{DLS classification}\label{fig:DLS-classification}
\end{figure}
For instance, DLS-mp-cs-cl represents a Dynamic Lot Sizing model with multiple products, common setup and capacity limit.

\subsection{ELSP}
The Economic Lot Sizing Problem (ELSP) is closely related to DLS models.
In general, ELSP attempts to find cycles given constant demand rates.
However, there are several ELSP formulations that allow varying or stochastic demand.
Another main difference is that shortage costs are usually considered within the ELSP.

I have reviewed and classified the ELSP in \cite{Garn2015754}
and the interested reader is referred to this source.
However, I will add a few reference, which are relevant in regard to MIP formulations.
One of the first and most well-known ELSP formulations is by \citet{bomberger1966dynamic}.
He presented a dynamic programming approach
using the Bellman equation to solve a lot scheduling problem.
This ELSP does not allow shortages and is similar to the DLS,
but instead of having dynamic demand a constant demand rate per product is given.
In Section \ref{sec:Case-Studies} I will introduce a MIP that offers a solution
for their classic test-instance.
Another influential paper that took the ELSP into the MIP domain was by
\citet{elmaghraby1978economic}, who discussed the ELSP and gave analytical insights.
The paper used the Bomberger data to demonstrate the concepts.
There are a few more papers, which look at the ELSP using MIP.
\citet{goyal1975scheduling} gave another algorithm to find a solution to the ELSP.
Again, the Bomberger data was used, but credited to \cite{madigan1968scheduling}.
\citet{cooke2004finding} find effective schedules for the economic
lot scheduling problem using a simple mixed integer programming approach.
\cite{salvietti2008profit} look at a profit-maximising economic lot scheduling problem with price optimisation.
A column-generation method was used in combination with a LP to solve the problem.
\cite{sun2010economic} consider the economic lot scheduling problem under extended basic period and power-of-two policy.
Power-of-two means that multiples of twos extend the basic period.

In the above literature review the main focus was on model/problem formulations
as Mixed Integer Programs.
Now, the focus will shift to how these MIP can be solved in general.

\subsection{Solution methods}
Exact algorithms for the DLS models and ELSPs can be placed into four main categories:
dynamic programming, branch-and-bound, branch-and-cut and Dantzig-Wolfe decomposition.
\cite{williams1974structured} gave a structured linear programming model in the food industry, where the
Dantzig-Wolfe decomposition was regarded to be ``worthwhile'' for a multiperiod model.

A procedure to tighten bounds was explained in \cite{balas1975facets}.
The main idea is to find minimum covers for constraints.
$S \subseteq \setn$ is called a \textit{cover} for $\tp{a}\omega \leq \hat{a}$ with $a\in \real^n, \omega \in \binary^n$
if $\sum_{i \in S} a_i > \hat{a}$.
$Q$ is a \textit{minimum cover} if $\sum_{i \in Q} a_i > \hat{a}$ for all $Q \subset S$.
The interesting consequence \cite[Theorem 3]{balas1969structure} is that
$\sum_{i\in S} \omega_i \leq |S|-1 \equi \tp{a}\omega \leq \hat{a} $.
That means the number of decision variables can be reduced.
This is desirable, because the number of decision variables increases the solution space.
For instance, assume there are $n$ binary variables then there are $2^n$ solutions.
This emphasises the importance of reducing the number of variables.
However, branch-and-bound algorithms cut off many of the $2^n$ solutions,
which are infeasible or too large.

An interesting question is whether and how significant it is to use heuristics
prior to starting exact methods.
Most IP engines use heuristics such as rounding and diving prior to running the branch-and-bound algorithm.
Usually those heuristics  cover the whole solution space.
\cite{linderoth1999computational} review several of the recent IP solution methods and their computational performance.

Recently, an interesting heuristic approach was introduced that focuses on essential
decision variables and then extends the solution space systematically.
For more details see the kernel optimisation methods introduced in
\cite{guastaroba2014heuristic}, \cite{guastaroba2017adaptive} and \cite{angelelli2010kernel},
although they were used for other problem classes.
This should not be confused with the kernel squeeze feature where
decision variables with zero value and a zero dual solution cause pivots to reduce the solution space.
This helped \cite{mcbride1988integer} to find solutions for
the optimal production line selection problem.

Runtime issues with exact algorithms encouraged researchers to propose heuristical approaches
not connected at all to exact approaches.
Greedy-add \citep{federgruen1994joint}, greedy-drop,
extended-Silver-Meal \citep{silver1973heuristic},
general-part-period-balancing \citep{iyogun1991heuristic} and many more are such heuristics.

The above exact solution methods are typically implemented/used within optimisation engines.
Next, I will mention popular Solvers (optimisation engines)
and sketch out how they work.

\subsection{Optimisation engines \label{sec:OptimisationEngines}}
The models and case studies in this paper were solved using the following three optimisation engines (solvers):
\begin{itemize}
	\item Matlab's optimisation engine,
	\item math.smartana.org optimisation engine: GLPK,
	\item OpenSolver's optimisation engine: COIN-OR CBC linear solver.

\end{itemize}

Matlab's optimisation engine uses the following approach to solve MIPs:
\begin{enumerate}
	\item Preprocess LP by removing redundant variables and constraints;
	\item Solve MIP as relaxed LP;
	\item Preprocess MIP ty tightening bounds, removing redundant constraints, strengthening constraints and fixing integer variables;
	\item Generate cuts such as MI rounding cuts, Gomory cuts, clique cuts, cover cuts, flow cover cuts, strong Chvatal-Gomory cuts, etc.;
	\item Run heuristics to find integer feasible solutions
	such as rounding, ``diving'' - partial depth-first search \citep{witzig2019conflictdriven}, relaxation induced neighbourhoods \citep{danna2005exploring};
	\item Run branch-and-bound algorithm.
\end{enumerate}

The GLPK (Gnu Linear Programming Kit) performance was analysed in \cite{pryor2011faster}.

The CBC (Coin-or branch and cut) uses LP relaxations, heuristics and cut generations.
Cut generation include rounding cuts, Gomory cuts, clique cuts, Knapsack cover, flow cover and many more.
Hint: For practitioners the OpenSolver Add-In for Excel may be of interest.
This solver uses the CBC library by default but offers access to other commercial solvers such as Gurobi.

The most popular commercial solvers are Cplex, Xpress and Gurobi.
\cite{meindl2012analysis} analysed the performance of several commercial, free and open-source solvers.
Here, CBC was shown to perform better than GLPK and LP\_Solve.
There are several challenges with measuring performance such as
Solvers being trained for certain test-instances and their variability.
The performance variability has been discussed in \cite{lodi2013performance}.

This section provided several DLS MIP formulations.
It mentioned methods that can solve these MIP,
and provided a brief insight into optimisation engines used to solve DLS models and ELSP.
The next section introduce a novel general DLS-mp MIP.
This will be followed by case studies demonstrating the performance and limitations.
The Bomberger case study shows how the DLS approach can be used to solve the ELSP.
The conclusion summarises essential findings.

\section{MIP for the general DLS-mp \label{sec:MIP}}
In the previous section several DLS models were introduced.
Each had a certain specialisation.
The here introduced DLS-mp unifies several of them.
This model introduces several new constraints
that take into account production limitations during a time period.

There are $m$ products $P=\sett{m}$.
The number of time-slots (weeks) is $T=\sett{n}$.
I will use the index $t$ for time and $p$ to denote a product.
Generally, $t$ describes a period (basic-period, time-window and time-slot are used synonymously).
Without restricting generality, I will use ``week'' to describe $t$, e.g. hours, days and months are equally valid.
There are three types of cost considered: holding, shortage and setup cost.
The holding costs for products are $c^h \in  \real_{+}^{m}$.
The shortage costs are $c^s \in \real_{+}^{m}$, and
changeover costs are $c^o \in  \real_{o}^{m}$.

Let $I_0 \in  \real^m_+$ be the vector of initial stock, i.e. $I_{0p}$ is the initial stock for product $p$.
The stock for all products during all periods is $I \in \real_{+}^{n \times m}$,
i.e. $I_{tp}$ represents the stock in period $t$ for product $p$.
The shortage quantity is $S \in \real_{+}^{n \times m}$ and the initial shortage is $S_0 \in \real^m$.
Note that $I_{tp}>0 \ie S_{tp}=0$ and $S_{tp}>0 \ie I_{tp}=0$.
The decisions whether to produce $p$ in $t$ is given by $X=\left(x_{tp}\right) \in \binary^{n \times m }$.
The decisions about the corresponding quantity of $p$ to produce in $t$ are captured in $Q = \left(q_{tp}\right) \in \mathbb{R}_{+}^{n \times m}$.
Note that $x_{tp}=1 \ie q_{tp}>0$.
The quantities are produced in batches of size $\hat{b}_{p} \in \real_+$ dependent on the product $p$.
Furthermore, the total amount that can be produced during a time-slot is limited to $\hat{q}$.
It should be mentioned here, that these limitations resulted from existing practical requirements.
More flexibility can be introduced by using $\hat{q}_{tp}$ to set a production quantity target for each period and each product.
However, I will refrain from this here.

The lists below summarise the indices, parameters and decision variables explained previously.

\noindent Dimensions and indices:\\
\begin{tabular}{ll}
	$m$  & Number of products \\
	$P$  & Product set \\
	$p$  & Product index \\
	$n$  & Number of time-slots/production windows \\
	$T$  & Time-slot set \\
	$t$  & Time-slot index \\
\end{tabular

\noindent Parameters:\\
\begin{tabular}{ll}
	$c^h_p$ & Holding cost for product $p$ (per unit per $t$) \\
	$c^s_p$ & Shortage cost for product $p$ (per unit per $t$) \\
	$c^o_p$ & Changeover cost from any product to product $p$ \\
	$\hat{b}_p$ & Batch size of product $p$ for $t$ \\
	$\hat{q}$ & Production limit during $t$ \\
	$\hat{x}$ & Max. number of products produced during $t$ \\
	$\hat{\omega}$ & Max. number of changeovers during $t$ \\
\end{tabular

\noindent Quantities:\\
\begin{tabular}{ll}
	$I_{0p}$ & Initial stock for product $p$ \\
	$I_{tp}$ & Stock/inventory for product $p$ per $t$ \\
	$S_{0p}$ & Initial shortage for product $p$ \\
	$S_{tp}$ & Shortage for product $p$ per $t$ \\
\end{tabular

\noindent Decision variables:\\
\begin{tabular}{ll}
	$X_{tp}$ & Decision to produce product $p$ in $t$ \\
	$Q_{tp}$ & Quantity of product $p$ produced in $t$  \\
	$\omega_{tp}$ & Changeover to product $p$ at $t$ \\
\end{tabular

\noindent The objective is to minimise the total costs:
\lequ{objwb}{
	\begin{split}
		&\min \text{Cost} =\min \set{ \text{holding} + \text{shortage} + \text{setup} }=\\
		=&\min_{I,S,\omega} \set{\sum_{p\in P} \left[ c^h_p \sum_{t \in T} I_{tp} +  c^s_p \sum_{t\in T} S_{tp} + c^o_p \sum_{t\in T} \omega_{tp} \right] }
	\end{split}
}

The inventory needs to fulfil the balance equation:
\lequ{inventoryBalance}{\left(I_{t-1,p}-S_{t-1,p}\right)+q_{t p}-d_{t p} \stackrel{!}{=} I_{t p}-S_{t p},}
where $t \in T$ and $p \in P$. Note that $I_{0p}$ and $S_{0p}$ are known.
The previous period's stock/shortage is topped up by the production quantity in the current period
and reduced by the current demand. This sets the period's stock/shortage.
For instance, assume last week's stock was 10 units.
This week's production quantity is 40 units and there is a demand of 20.
Hence, the current week's stock is $10+40-20 = 30$.

In the ELSP only one product is permitted to be produced per period (e.g. week):
\lequ{nbProductsProducedOne}{\sum_{p \in P} x_{tp} \meq 1, ~t \in T.}
However, there are other interpretations:
(1) although there is only one production line several products can be produced during a sufficiently large period;
(2) there are several production lines available to produce a single batch during a period;
(3) several production lines and several products are produced simultaneously during a period.
To take care of the above interpretations constraint (\ref{eq:nbProductsProducedOne}) is
changed to
\lequ{nbProductsProducedMany}{\sum_{p \in P} x_{tp} \leq \hat{x}, ~t \in T,}
where $\hat{x}$ is the maximum number of different products produced simultaneously.
Furthermore, the production limit constraint (\ref{eq:productionLimit}) ``kicks in''.

The amount produced is exactly the same as the batch size:
\lequ{batchLimit}{q_{tp} \meq  x_{tp} \hat{b}_{p}, ~t\in T, p \in P.}
Here, it can make sense to allow the production of multiple batch sizes,
which would require the decision variable $x_{tp}$ to be in $\setN_0$ rather than $\binary$.

The total amount produced cannot exceed the time-slot's (e.g. weekly) production limit:
\lequ{productionLimit}{\sum_{p \in P} q_{tp} \leqslant \hat{q},~t \in T.}

Now, the ``logical'' IP constraints that identify changeovers are given by:
\lequ{changeoverConstraints}{
	\begin{aligned}
		&x_{t+1,p}-x_{tp} \geq 2\omega_{t+1, p}-1,\\
		&x_{t+1,p}-x_{tp} \leq 2\omega_{t+1, p},\\
		& t \in \sett{n-1},~ p \in P.
	\end{aligned}
}
The initial setups (changeovers) are identical to whether the product is produced:
\lequ{initialSetupConstraints}{
	\omega_{1, p} \meq x_{1,p} ,~ p \in P.
}

The logical constraints require validity checks.
The problem is simplified by considering one transition.
Let $x_1$ and $x_2$ be the current and next production state.
If $x_1=1$ then something is produced otherwise not.
A setup-changeover happens when $(x_1,x_2)=(0,1)$. This is reflected with $\omega = 1$.
The discontinuation of producing $(x_1,x_2)=(1,0)$ is also a changeover.
However, in this formulation it is assumed that the discontinuation-changeover activities and associated costs
are integrated within the setup-changeover, i.e. $\omega = 0$.
If $(x_1,x_2) \in \set{(0,0),(1,1)}$, then no changeover takes place, i.e. $\omega = 0$.
This represents the logical operation $\neg x_1 \land x_2$ or $x_1<x_2$ shown in Table \ref{tab:lop}.
\qtable{Logical operation $\neg x_1 \land x_2 \ie \omega$.}{lop}{r|r|r|r}
{$x_1$ & $x_2$ & $\neg x_1$ & $\omega$}
{0     & 0   & 1  & 0 \\
	0     & 1   & 1  & 1 \\
	1     & 0   & 0  & 0 \\
	1     & 1   & 0  & 0 \\}
The corresponding logical constraint is:
\qbox{\tcb{If} $x_1<x_2$ \tcb{then} $\omega = 1$ \tcb{else} $\omega=0$.}
This can be written as IP constraints (motivated by Figure \ref{fig:logicalconstraints}):
\lequ{logicalConstraintsSimple}{
\begin{aligned}
&x_2-x_1 \geq 2\omega-1,\\
&x_2-x_1 \leq 2\omega.
\end{aligned}}
In order to show that this is a valid formulation
the logical constraints must be equivalent to the IP-constraints.

Suppose that $(x_1, x_2,\omega)$ is a feasible solution for the IP formulation.
If $\omega=1$ then $x_2-x_1 \geq 1$ and $x_2-x_1 \leq 2$.
The left of Figure \ref{fig:logicalconstraints} shows the constraints and  feasible region.
\begin{figure}[htbp]\centering
	\includegraphics[width=\linewidth]{LogicalConstraints}
	\caption{Logical constraints}\label{fig:logicalconstraints}
\end{figure}
This implies that $(x_1,x_2) \meq (0,1)$, because this is the only feasible solution fulfilling both constraints.
The other points $\set{(0,0),(1,0),(1,1)}$ are infeasible solutions.
If $\omega=0$ then $x_2-x_1 \geq -1$ and $x_2-x_1 \leq 0$.
The right of Figure \ref{fig:logicalconstraints} shows the constraints and feasible region.
This implies that $(x_1,x_2) \in \set{(0,0),(1,0),(1,1)}$, because a feasible solution was assumed.
The other point $(x_1,x_2)=(0,1)$ is an infeasible solution.
This proves the validity of (\ref{eq:logicalConstraintsSimple})
and thus the validity of (\ref{eq:changeoverConstraints}).

$\omega_{tp}$ represents a changeover.
If only one product can be produced during a period, then
constraint (\ref{eq:nbProductsProducedOne}) will suffice,
and ensure only one changeover.
However, if multiple products can be produced simultaneously,
then constraint (\ref{eq:nbProductsProducedMany}) is used.
However, it may be necessary to limit the number of changeovers
to $\hat{\omega} = (\hat{\omega}_t) \in \real_+^n$:
\lequ{changeOverLimit}{\sum_{p\in P} \omega_{tp} \leq \hat{\omega}_t,~t\in T.}
This can happen if the number of resources available to
execute the changeovers is constrained.

\section{Case studies \label{sec:Case-Studies}}

Two cases studies are presented.
The first demonstrates the functionality of the general DLS model introduced in the previous section as DLS-mp-bp-cl-lc-sc system (see Figure \ref{fig:DLS-classification}).
The second one illustrates how a classic ELSP
can be solved using a DLS-mp-qc-st system approach.

\subsection{Food manufacturer}
The case study, which was introduced in \cite{Garn2015754} will be used here.
For the convenience of the reader a brief summary is given.
A company produces several SKUs (stock keeping units) and requires a production schedule.
The case study consists out of $m=10$ SKUs produced over a period of two years.
The demand data is given on a weekly basis ($n=104$), i.e. $D\in \real^{104\times 10}$.
The demand was 404k packages in the first year and 608k in the second packages.
Figure \ref{fig:total_demand_per_week} shows the total weekly demand.
\begin{figure}[htbp]\centering
	\includegraphics[width=\linewidth]{total_demand_per_week}
	\caption{Total demand per week.}\label{fig:total_demand_per_week}
\end{figure}
This demand is assembled via 10 individual SKUs shown in Figure \ref{fig:total_demand_per_week}.
\begin{figure*}[htbp]\centering
	\includegraphics[width=\linewidth]{TS-ten-products-comments-v3}
	\caption{Demand and characteristics of each SKU.}\label{fig:total_demand_per_week}
\end{figure*}

Holding costs varied between \$1.10 and \$3.00 per package per week.
The shortage costs are in the range \$4.60 and \$8.60 per package per week.
All packages can be back-ordered.
The  unit-costs for all products are fixed over the entire two years.
Changeover (setup) costs are between \$5,200 and \$8,300.
They only depend on the product that is produced, i.e. independent of which product was produced before.
The detailed costs, batch-sizes and aggregated demands are shown in Table \ref{tab:cost-factors}.
\input{tab-cost-factors}
The table illustrates that three products (G02SN, G01SL and O08SN)
have a demand of 77.2\%.
Hence, in case of complexity issues the focus should be on these three products.
Accordingly, the products were ordered by their total demand.
The weekly production capacity of 5,000 units is the same as the maximum batch-size.
Furthermore, all other batch sizes suggest that producing more than one product a week makes sense.

The case study's complexity was increased by 10 weeks and one product.
Hence, 100 test-instances based on this one case study were created.
The Matlab optimisation engine (see Section \ref{sec:OptimisationEngines}) was used
to solve the MIP explained in Section \ref{sec:MIP}.
The time to solve the MIP problem was limited to two hours.
46 out of 100 test-instances obtained an optimal solution within the set time limit.
Figure \ref{fig:cs-runtime} shows the total runtime, i.e. time to formulate problem plus time for solving it.
\begin{figure}[htbp]\centering
	\includegraphics[width=\linewidth]{cs-runtime}
	\caption{Runtime for test-instances of case study.}\label{fig:cs-runtime}
\end{figure}
For instance, the runtime to find the production schedule for the first 50 weeks
and the top three products is 53.5 minutes.
This involves the usage of a virtual machine with 4 cores with 2GHz allocated Intel CPUs
and 1GB RAM (out of 8GB) available.
The total number of decision variables used was $|T|\times|P|\times 4 = 50\times 3\times 4$,
where the last 4 represents the number of decision variable matrices ($\set{Q,X,\omega,I,S}$).
As expected the MIP finds optimal solutions with moderate complexity
within the two-hours time window.
It is interesting to observe that the optimisation engine fails faster to find optimal solutions,
when the number of products increases rather than when time increases.
For instance, for up to 4 products and 70 weeks ($280\times 4$ variables) an optimal solution is still found.
However, at 6 products and 30 weeks ($180\times 4$ variables) only a sub-optimal solution is found.

Figure \ref{fig:cs-cost} shows the cost increase over the weeks and products produced.
\begin{figure}[htbp]\centering
	\includegraphics[width=\linewidth]{cs-cost}
	\caption{Cost for test-instances of case study.}\label{fig:cs-cost}
\end{figure}
If the demand is roughly uniform, one would expect a linear growth of costs.
However, the demand increases as shown in Figure \ref{fig:total_demand_per_week}.
Moreover, there are outliers, which put additional strain/costs on the system
- see \cite{Garn2015} for details.
The above explains why cost are increasing apparently exponential.

Next the effect on runtime and solution quality will be considered if an initial solution is provided.
Here, the initial solution for $m=1$ is not to produce anything,
and for $m>1$ to use the solution found for $m-1$ and not to produce product $m$.
This is a similar idea as introduced in \cite{Garn2015754} in the iterative schedule generation algorithm.
The following results do not support the results expected intuitively.
Runtime savings would have been expected.
Especially, since the time to find an initial solution was not taken into account.
Surprisingly,  optimal solutions for only 28\% of the test-instances were found,
when an initial solution was provided. Previously 46\% optimal solutions were determined.
62\% of the instances had longer runtimes.
Solution quality is 51\% of the time worse than previously.
Since the experiments were run on shared VM,
the unexpected results may be due to shared memory and CPUs being affected,
or because of an implementation issue within Matlab's optimisation engine.

Overall, the case study demonstrated that optimal solutions
can be found for most test instances.
The number of products influences the runtime significantly.
Since the top three products dominate the economic benefits, this issue can be mitigated.

\subsection{Bomberger}
The classic test data from \cite{bomberger1966dynamic} was used.
This ELSP focuses on production cycles.
Daily demand for each product per day $d_p$ is given and assumed to be constant (over the entire time-horizon).
The test-instance investigated periods were 20, 35 and 40 days.
Practically, it does not make sense to plan the schedule for more than two years.
A year has $y=240$ days in this problem instance.
That means, if $\tau=20$ then there are $n=24$ periods to consider.
The period $\tau$ are long in comparison to the setup time $\tau^o$,
which would allow us to neglect the setup times.
However, it is equally convenient to consider the setup time
within the maximum quantity producible during $\tau$.
It appears reasonable to require setup costs in each period,
even when there is a continuation of the production into the next period.
This is due to the extensive length of the production period.
The production rate is given in units per day.
Hence, the maximum is $\hat{q}_p = (\tau-\tau^o) b_p$,
where $b_p$ represents the production rate for product $p$ (Note: ensure identical units such as days for $\tau$ and $\tau^o$).
Assume that there is a minimum amount that has to be produced, e.g. $\check{q}_p = \frac{b_p}{8}$ (i.e. 1 hour runtime).
The amount produced is captured in the decision variable $q_{tp} \in \real_{+}$.
The number of days a product is produced is abbreviated by $x_{tp}\in \real_{+}$, i.e. fractional days are allowed.
The fact whether a product is produced is equivalent to a changeover/setup and represented by $\omega_{tp}\in \binary$.
The daily holding cost is defined as $f=0.10$  multiplied by the cost of
labour and material (i.e. piece cost $c^m$).
Hence, the holding cost per unit during $\tau$ is $c^h = 0.1 c^m \tau$.
This interpretation introduces imprecision,
e.g. if something is produced on the last day of $\tau$ then the entire period needs to be paid.
Yet, in real-world scenarios it is common to arrange a minimum payable period (e.g. one week)
and a minimum quantity (e.g.one pallet) as payment to third-party warehouses.
Usually initial inventories $I_{0p} \in \real_{+}$ are present.
However, in this test-instance $I_{0p}=0$.
Furthermore, the Bomberger test-instance does not permit shortages.
The above allows us to give a simplified formulation
of the general DLS introduced in Section \ref{sec:MIP}:
\LP{\label{MIP:Bomberger}
	\objd[min]{I,\omega}{\sum_{p \in P}\sum_{t \in T} (c^h_p I_{tp} + c^o_p \omega_{tp})}}
{
	\sto{I_{tp} = I_{t-1,p} + q_{tp} - d_{p} \mcomment{inv. balance}}
	\sto{\sum_{p\in P} (x_{tp} + \omega_{tp}\tau^o_p) \leq \tau \mcomment{max time}}
	\sto{ x_{tp} b_{p} = q_{tp} \mcomment{production rate}}
	\sto{q_{tp}\leq \hat{q}_p \mcomment{quantity limit}}
	\sto{q_{tp} \leq \hat{q}_p \omega_{tp} \mcomment{if $q_{tp}$ then $\omega_{tp}$}}
	\sto{q_{tp} \geq \check{q}_p \omega_{tp} \mcomment{if $\omega_{tp}$ then $q_{tp}$}}
	\sto{I_{tp}, q_{tp}, x_{tp}\in \real^+,~\omega_{tp} \in \binary \mcomment{variables}.}
}
It is assumed that $t \in T$ and $p\in P$,
where $T=\sett{n}$ is the number of periods during the schedule horizon ($n=\frac{y}{\tau}$);
and $P=\sett{m}$ represents $m$ products.
All factors have to be scaled to have consistent units.

Upper bounds for the optimisation variables are $I_{tp} \leq n \times \max_p \set{d_p}$,
$q_{tp} \leq \max_p \set{\hat{q}_p}$ and $x_{tp}\leq \tau$.

The parameters: setup cost $c^o$, piece cost $c^m$, production rate $b$, demand rate $d$ and setup time $\tau^o$
are shown in Table \ref{tab:BombergerData}.
\begin{table}[htbp]
	\centering
	\caption{Bomberger data.}
	\adjustbox{width=\columnwidth}{ \begin{tabular}{c|cc|cc|c}
		\toprule
		Part Nb & Setup Cost & Piece Cost & Prod. Rate & Demand & Setup Time \\
		& [\$] & [\$/unit] & [units/day] & [per day] & [hours] \\
		\midrule
		1    & 15   &        0.0065  &          30,000  &          400  & 1 \\
		2    & 20   &        0.1775  &           8,000  &          400  & 1 \\
		3    & 30   &        0.1275  &           9,500  &          800  & 2 \\
		4    & 10   &        0.1000  &           7,500  &       1,600  & 1 \\
		5    & 110  &        2.7850  &           2,000  &            80  & 4 \\
		6    & 50   &        0.2675  &           6,000  &            80  & 2 \\
		7    & 310  &        1.5000  &           2,400  &            24  & 8 \\
		8    & 130  &        5.9000  &           1,300  &          340  & 4 \\
		9    & 200  &        0.9000  &           2,000  &          340  & 6 \\
		10   & 5    &        0.0400  &        15,000  &          400  & 1 \\
		\bottomrule
	\end{tabular}
	\label{tab:BombergerData
\end{table

In general, the assertion:$\frac{d_p}{b_p} \leq 1$ must hold for a feasible solution.
Product (part nb, item) 8 has a $\frac{d_p}{b_p} = \frac{340}{1300}\approx 0.262$ fraction, which is the maximum.
This value can be interpreted that it takes a quarter of a day to produce the required demand for that day.
If there exists a feasible solution, then
\lequ{BB-necessary-condition}{\sum_{p\in P} \left(\frac{d_p\tau-I_0}{b_p} \right) \leq \tau,}
because the demand must be fulfilled already in the first cycle.
In the Bomberger example the sum is 21.4 days,
which is well below the $\tau$ (35 and 40 days) used in Bomberger's paper.
For $\tau=20$ we will assume an initial inventory of $I_0=\frac{d\tau}{2}$.
Bomberger stated a similar necessary condition for repetitive schedules
but allowing different periods for each product.

Solving this MIP (\ref{MIP:Bomberger}) gives a daily cost of \$444 for $\tau=20$.
This is a higher solution value than Bomberger's.
This may be due to the used period length.
For instance, a period of $\tau=80$ and $\tau=120$  finds a solution value which is \$130 and \$7.33 (with no initial inventory).
Here, the holding and setup cost drop significantly.
Only as much is produced as there is consumed.
Bomberger's cycle lengths are up to 400 days,
which appear to be unreasonably high for a manufacturing environment.
Furthermore, demand was assumed as uniform, which will not happen in real world scenarios.

A period of five and ten working-days is examined next.
An initial inventory for five days is provided for all products.
The minimum quantity produced is set to a two hours runtime, i.e. $\check{q} =\frac{1}{4}b$.
The average daily costs are \$241.78 (relative gap: 36.8\%, after 90min runtime) for $\tau=5$
and \$109.88 (optimal) for $\tau=10$.
$\omega_p$ reveals almost cyclic solutions.
The average number of days between setups is:
\smat{38.8& 5.0& 5.0& 5.0& 9.8& 21.7& 31.8& 5.0& 5.0& 9.8} for $\tau=5$
and \smat{20.0& 10.0& 10.0& 10.0& 10.0& 20.4& 30.0& 10.0& 10.0& 10.2} for $\tau=10$.

This case study demonstrated that the ELSP can be solved using a typical DLS modelling approach.

\section{Conclusions}
DLS has received a fair amount of attention in the past,
but several essential constraints were omitted
such as production limits and shortages.
Sometimes, unnecessary (from an industry point of view) constraint specifications  were introduced.
For instance, holding costs varying at every time period.
Interesting first steps into conjoint sales price considerations have appeared,
but more has to be done for it to be useable in practice.
This work has unified several of the previous studies and suggested alternative formulations and approaches when appropriate.

The main contributions of this work are the classification of DLS models
and a novel DLS MIP formulation,
which is suitable for agile manufacturing in an industrial environment
allowing shortages and introducing production \& changeover limits
(i.e. a DLS-mp-bp-cl-lc-sc system).
In the context of the MIP the validity of changeover constraints was proven.
The capacitated DLS models for multiple products and ELSPs have been implemented as MIPs using Matlab's optimisation engine.
A case-study consisting out of 200 test-instances was executed.
Interesting run-time and cost insights were gained, such as product variety causes more performance issues than longer time periods.
The classic ELSP Bomberger test instance was solved using a MIP.

To conclude, the here introduced MIPs can be used for a variety of industry solutions in the area of agile manufacturing.
A unified approach in formulating and classifying DLS models and ELSP was proposed.

\section{Appendix}
Table \ref{tab:cs-runtime} and \ref{tab:cs-cost} show the runtimes and costs for test-instances resulting from the food manufacturer case study respectively.
\input{tab-cs-runtime}
\input{tab-cs-cost}

\iffalse
Manufacturing $m$ products during a certain timespan to maximise the profit subject to time and material constraints
is a classic yet essential topic:
\lequ{classicProduction}{\max_q \set{\tp{p}q: \tp{\tau}q\leq \hat{\tau }, \tp{\kappa}q \leq \hat{\kappa}},}
where $q$ is the produced quantity, $p$ the profit per unit, $\tau$ the time required to produce one unit,
$\hat{\tau}$ the time limit, $\kappa$ the required raw material and $\hat{\kappa}$ the available raw material.
\fi
\bibliographystyle{elsarticle-harv}

redpajama_set_name: RedPajamaArXiv


ALA notable author Iain Lawrence pens a moving tale with a Dickensian atmosphere about a young man's life aboard a cruel 19th-century prison ship. Fourteen-year-old Tom Tin is forced to scavenge on the grimy streets of London after his father is sent to debtor's prison. Tom's survival is threatened by the sinister lowlifes that inhabit his hopeless world. But when he is unjustly arrested for murder, he finds a place that is even worse than the dark city streets.


Botanical coordinator Igone Amoztegui decided that it was time to rebuild the brand's image from the ground up. Once Igone Natural's logo had been restyled, it was time to rethink the brand's entire image, making changes to packaging, labels and overall digital presence.
Reducing the size of brand labels offered multiple benefits. Not only did this help the marvellous textures of Igone Natural's creams and ointments to truly shine through, but it also minimised the amount of ink and paper used to produce them, making the brand more environmentally friendly. The new design is decidedly professional and minimalist, with typography featuring as the main element of focus.


In the video below you will see a series of classic Hatha Yoga postures, which is also the opening sequence in the Ashtanga practice. Any serious ashtanga and/or vinyasa practitioner should have this down. So many other postures and movements come from this...along with sun salutations.
Please note...in this video I go thru the sequence quickly...it is normally five breaths in each posture.
Here you'll find demonstrations and tutorials going over postures, movements, and details of Ashtanga and Vinyasa practice.


This is an introduction to the simple meditation technique by the name of Sahaja Yoga, when the Kundalini passes throughout the central nervous system by way of three energy channels, these are often refereed to in the Sanskrit language as 'nadis'. The three channels or nadis. The channels on the nervous system are the left channel, the channels of our emotions, desires and our past, the right channel we can say acts as the 'doer' this is the channel that puts our desires into action and the centre channel is represented by the autonomic nervous system the part of the nervous system responsible for control of the bodily functions not consciously directed, such as breathing, the heartbeat, and digestive processes.
All three work perfectly together to integrate and balance the flow of your Kundalini energy, once it has been awakened by this simple meditation technique. Each channel has a number of chakras placed on them, each individual chakra has a very specific role in maintaining our well being, the full array of moods and our physical body related to our health.
The chakras are located on all three channels. When the Kundalini energy rises through the central channel, the autonomic nervous system, it nourishes each chakra, the centre, left and right side of each chakra. At the 7th or chakra named as the Sahasrara chakra in the Sanskrit language, we achieve a connection with the all-pervading divine energy. The opening of the Sahasrara chakra facilitates the flow of the divine energy through this opening in the Sahasrara into the three channels/nadis nourishing the chakras and bringing about a change in our emotional and physical well being.
All this happens in a split second, spontaneously and instantaneously. When we first begin to meditate a few strands of the thread-like Kundalini energy rise, but the more we meditate the stronger the connection becomes with the divine energy and we are able to touch on the inner joy in a much deeper way.


All are made of ripstop canvas and are available in olive green or sand.
A must have for any outdoor adventurer; you won't go thirsty if you invest in our water bag. It's made from a natural flax material and comes standard with a cork nozzle.


So….my very first shoot in the UK was this absolutely beautiful christening in Wiltshire for the sweetest one year old you have ever seen! The backdrop for my shoot was a stunning village, a 12th century church and a charming pub called 'The Horseshoe Inn'….oh, and really lovely people that made me feel so welcome. Sigh……what a great way to start off my photography business in the UK.
My approach to shooting a Christening is to shoot a selection of posed photographs, especially with close family members and friends mixed with a more documentary style shooting of the christening events as they unfold. The mix of both formal posed shots peppered with documentary style shooting, lends itself very well to documenting the day and giving you fantastic memories to look back on.


text:
Q: Does IndexedDB persist with a new service-worker? Does the indexedDB persist when I unregister a service worker and install a new one?
If I would manually unregister the service-worker like this
        navigator.serviceWorker
          .getRegistrations()
          .then(function (registrations) {
            for (let registration of registrations) {
              registration.unregister();
            }
          });
        window.location.reload();

what happens with the indexedDB that is created in my service worker? Is it dependent on the specific sw it was created in?

A: Unregistering a service worker does not affect site storage, and that includes IndexedDB and the Cache Storage API. All previously saved data will persist independent of a service worker registration.

redpajama_set_name: RedPajamaStackExchange


text:
The Miracle of Ruth and Naomi's Journey with Grief
Series: Sermons
Sarah Stewart
A friend recently noted "how a person reacts to your sadness says a lot about how long they're going to be in your life."Glenn Ringtved, Danish author of the children's book, Cry, Heart, But Never Break captures how it feels to face losing such a dear one. Ringtved's masterful story features four children who grapple with their grandmother's imminent death. In the tale, Death is depicted as a hooded stranger whom they attempt to stall by serving cup after cup of coffee at the kitchen table. When the hour has come for him to head upstairs, where their elderly grandmother lies in bed, the youngest pleads, "But, why must she die?" Death indulges her with an allegory to illustrate the inextricable connection between life and death. Death tells of two brothers, Sorrow and Grief, whose mundane existence is transformed the day they encounter, fall in love with, and commit to sharing their lives with their soulmates, two sisters named Joy and Delight.
I cannot help but think of this tender story, in this season of our life in the communion of saints, a time when we slow down to hold space for those who have died. At Wednesday's requiem service we will toll bells and light candles, praying the names of the ones lovingly buried from this community of faith. Sadly, no rituals of remembrance can erase the ache of their absence. But we can companion those who mourn, honoring their sorrow. Together, we turn toward God who promises to redeem our deepest losses, even when that future hope feels impossibly far away. The Book of Ruth illuminates how such a sacred journey can unfold, as communal grace reveals God with us. The narrative bears out like the Psalmist's ancient witness: "those who go out weeping, bearing seed for sowing, come home with shouts of joy, shouldering their sheaves."
January 22, 2023 | The Third Sunday after the Epiphany | The Rev. Dr. Dwight Zscheile
January 15, 2023 | Martin Luther King, Jr. Sunday | Dr. AnneMarie Mingo
Sermon by Dr. AnneMarie Mingo on Martin Luther King, Jr., Sunday, January 15, 2023
"Rise and Shine" | The Rev. Dr. Andrew K. Barnett | January 8, 2023
"Arise, shine; for your light has come, and the glory of God has shone upon...
The Rev. Dr. Simon Mainwaring | January 1, 2023
Sermon by The Rev. Simon Mainwaring on Sunday, January 1, 2023.
The Rev. Dr. Jennifer McBride | December 25, 2022
Sermon by The Rev. Dr. Jennifer McBride on Sunday, December 25, 2022.
"Lo, I am with you to the end of the age" | The Rev. Dr. Andrew Barnett | December 18, 2022
"You are not alone." Sermon by The Rev. Dr. Andy Barnett on Sunday, December 18, 2022.
The Rev. Nontombi Naomi Tutu | December 11, 2022
Sermon by The Rev. Nontombi Naomi Tutu on Sunday, December 11, 2022.
Rev. Natosha Reid Rice | December 4, 2022
Sermon by Rev. Natosha Reid Rice, Minister for Public Life, on Sunday, December 4, 2022.
Dr. Jennifer McBride | November 27, 2022
Sermon by Dr. Jennifer McBride on Sunday, November 27, 2022.
The Rev. Naomi Tutu | November 20, 2022
Sermon by The Rev. Naomi Tutu on Sunday, November 20, 2022.
Being Christian after an Election | The Rev. Dr. Andrew Barnett | November 13, 2022 | 11:15 a.m.
"Being Christian after an Election" by The Rev. Dr. Andy Barnett on Sunday, November...
The Fruits of the Spirit | The Rev. Dr. Andrew Barnett | November 13, 2022 | 9:00 a.m.
"The Fruits of the Spirit" : The Rev. Dr. Andy Barnett on Sunday, November 13, 2022...
The Rev. Dr. Simon J. Mainwaring | November 6, 2022
The Rev. Dr. Simon J. Mainwaring, All Saints' Rector, preaches on All Saints' Day - a...
The Rev. Dr. Simon J. Mainwaring | October 30, 2022
The Rev. Dr. Simon J. Mainwaring, All Saints' Rector, preaches on Luke 19:1-10 on this...
The Rev. Natosha Reid Rice | October 23, 2022
The Rev. Natosha Reid Rice, All Saints' Minister for Public Life, preaches on Luke...
The Rt. Rev. Robert C. Wright | October 16, 2022
The Rt. Rev. Robert C. Wright, Bishop of the Diocese of Atlanta, visited All Saints...
Dr. Jennifer McBride | October 9, 2022
On this Pride Sunday at All Saints', Dr. Jenny McBride, our new Associate for...
The Rev. Naomi Tutu | October 2, 2022
The Rev. Naomi Tutu preaches on the Parable of the Mustard Seed from Luke 17:5-10...
The Rev. Dr. Simon J. Mainwaring | September 25, 2022 | 11:15 a.m.
Sermon by The Rev. Dr. Simon Mainwaring on Sunday, September 25, 2022.
The Rev. Dr. Andrew K Barnett | September 25, 2022 | 9:00 a.m.
Sermon by The Rev. Dr. Andy Barnett on Sunday, September 25, 2022.
The Rev. Naomi Tutu | September 18, 2022
The Rev. Dr. Simon Mainwaring | September 11, 2022
Watch Simon's sermon from Sunday, September 11, 2022.
"The Cost of Discipleship" | The Rev. Dr. Andrew K. Barnett | September 4, 2022
Watch Andy's sermon from Sunday, September 4. "One of the hallmarks of discipleship is...
The Rev. Natosha Reid Rice | August 28, 2022
The Rev. Sarah C. Stewart | August 21, 2022
The Rev. Dr. Simon Mainwaring | August 14, 2022
The Rev. Dr. Gary Mason | August 7, 2022
In his sermon on Genesis 15, The Rev. Dr. Gary Mason focuses in on the life of...
"God bless us, everyone" | The Rev. Dr. Andy Barnett | July 31, 2022
We are constantly having our lives transformed by the spiritual practice of...
"From Ritual to Relationship" | The Rev. Natosha Reid Rice | July 24, 2022
The Rev. Dr. Simon J. Mainwaring | July 17, 2022
"This is not looking at all, but sight, vision, seeing at the depth of the divine...
The Rev. Naomi Tutu | July 10, 2022
"Pledging Ourselves to God's Dreams: Unity, Healing, and Flourishing" | July 3, 2022
Sunday Sermon | June 26, 2022
Sunday Sermon, June 19, 2022
Sermon by the Rev. Natosha Reid Rice, Sunday, June 19, 2022
Sermon by the Rev. Karen Evans, Sunday, June 12, 2022
You Can't Stop the Spirit: She is like a mountain, old and strong
Sermon by the Rev. Dr. Andrew Barnett, Sunday, June 5, 2022
Sunday Sermon, May 29, 2022
Sermon by the Rev. Dr. Simon Mainwaring, Sunday, May 29, 2022
The Church Emerging
Sermon by The Rev. Dr. Simon Mainwaring, Sunday, May 22, 2022.
Sermon by the Rev. Karen Evans for Sunday, May 15, 2022
Senior Sermons from Margaret, Caroline, Madeleine, and Jack - Sunday, May 8, 2022
Sermons from Margaret Pope, Caroline Stanford, Madeleine Coleman, and Jack Srinivasan...
Following Jesus into a New Normal
Sermon by The Rev. Dr. Dwight Zscheile for Sunday, May 1, 2022
Easter Sunday Sermon
Sermon by the Rev. Dr. Simon J. Mainwaring for Easter Sunday, April 17, 2022.
The Wondrous Cross: Living as We See
Sermon by the Rev. Sarah C. Stewart for Good Friday, April 15, 2022.
A New Commandment
Sermon by the Rev. Dr. Andrew K. Barnett for Maundy Thursday, April 14, 2022.
The Passion of Our Lord for Palm Sunday
A reading of The Passion of Our Lord for Palm Sunday, April 14, 2022.
The Rev. Dr. Lauren Winner
Sermon by the Rev. Dr. Lauren Winner for Sunday, April 3, 2022.
Welcome Home | March 27
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, March 27, 2022.
Can You Handle the Truth
Sermon by the Rev. Natosha Reid Rice for Sunday, March 20, 2022.
God is my refuge and strength: whom shall I fear?
Sermon by the Rev. Dr. Andrew K. Barnett at the 9:00 a.m. service on March 13, 2022.
Jeopardy, Possibility, Hope: Being In the City, For the City
Sermon by the Rev. Sarah C. Stewart for the 11:15 a.m. service on Sunday, March 13.
The Rt. Rev. Mary Gray-Reeves
Sermon by the Rt. Rev. Mary Gray-Reeves for Sunday, March 6, 2022.
Holy Loitering
Sermon by the Rev. Dr. Simon J. Mainwaring for Ash Wednesday, March 2, 2022.
A Revolution of Power
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, February 27, 2022.
The Honorable Rev. Dr. Raphael Warnock | CEEP/EPN Closing Worship Service
Sermon by the Honorable Rev. Dr. Raphael Warnock for the CEEP/EPN Closing Worship...
A Vision for All the Saints | The Multicultural Church with the Rev. Nancy Frausto | February 20, 2022
The Rev. Nancy Frausto is Director of Latinx Studies at the Seminary of the Southwest...
The Rev. Nancy Frausto
Sermon by the Rev. Nancy Frausto for February 20, 2022.
A Deeper Happiness
Sermon by the Rev. Dr. Simon J. Mainwaring for February 13, 2022.
Tap Into Your Super Power
Sermon by the Rev. Natosha Reid Rice for Sunday, February 6, 2022.
Sermon by the Rev. Dr. Andrew K. Barnett for the 11:15 a.m. service on Sunday, January...
Surprising Medicine, Amazing Grace
Sermon by the Rev. Sarah C. Stewart for the 9:00 a.m. service on Sunday, January 30, 2022.
January 23, 2022 - Geoffrey Hoare
Sermon by the Rev. Geoffrey Hoare for January 23, 2022.
January 16, 2022 - Luther Smith
Sermon by the Rev. Dr. Luther Smith for Sunday, January 16, 2022.
I Have Called You By Name
Sermon by the Rev. Dr. Simon J. Mainwaring for January 9, 2022.
By Another Road
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, January 2, 2022.
A Bigger Story
Sermon by the Rev. Dr. Simon J. Mainwaring for Christmas Eve, December 24, 2021.
The In-Between Times
Sermon by the Rev. Dr. Andrew K. Barnett for Sunday, December 19, 2021.
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, December 12, 2021.
Generation to Generation
Sermon by the Rev. Zachary C. Nyein on his last Sunday at All Saints', December 5, 2021.
The Kingdom of God is Near
Sermon by the Rev. Dr. Andrew K. Barnett for Sunday, November 28, 2021.
We Are On Trial
Sermon by the Rev. Natosha Reid Rice for Sunday, November 21, 2021.
The Pilgrim House
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, November 14, 2021.
Called to Be Saints
Sermon by the Rev. Dr. Simon J. Mainwaring for All Saints' Sunday, November 7, 2021.
Sermon by the Rev. Dr. Andrew K. Barnett for the All Souls Requiem service on...
God is Calling
Sermon by the Rev. Natosha Reid Rice for Sunday, October 24, 2021.
We Are All Priests
Sermon by the Rev. Zachary C. Nyein for Sunday, October 17, 2021.
People of the Question
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, October 10, 2021.
The Advance Team
Sermon by the Rev. Natosha Reid Rice for the Festival of Song on Sunday, October 3, 2021.
Sermon by the Rev. Karen Evans for Sunday, October 3, 2021.
Courage Cannot Be Quarantined
Sermon by the Rev. Natosha Reid Rice for Sunday, September 26, 2021.
Let's not miss the One who is always drawing near
Sermon by the Rev. Sarah C. Stewart for Sunday, September 19, 2021.
A Future Bigger Than Our Past
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, September 12, 2021.
Who From Our Labors, Rest
Sermon by the Rev. Simon J. Mainwaring for Sunday, September 5, 2021.
Risky Questions
Sermon by the Rev. Zachary C. Nyein for Sunday, August 29, 2021.
The Assurance of Things Hoped For
Sermon by the Rev. Andrew K. Barnett for Sunday, August 22, 2021.
Making The Most Of The Time
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, August 15, 2021.
Sermon by the Rev. Karen Evans for Sunday, August 8, 2021.
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, August 1, 2021.
Speaking of (Our) Sin
Sermon by the Rev. Sarah C. Stewart for Sunday, July 25, 2021.
Rebel Wilson Doesn't Care About You
Sermon by the Rev. Zachary C. Nyein for Sunday, July 18, 2021.
What is your why?
Sermon by the Rev. Natosha Reid Rice for Sunday, July 11, 2021.
Sermon by the Rev. Zachary C. Nyein for Sunday, July 4, 2021.
Divine Touch
Sermon text by the Rev. Dr. Simon J. Mainwaring for Sunday, June 27, 2021.
How to Face Goliath: Living Like the Battle Belongs to the Lord
Audio-only recording of the Rev. Sarah C. Stewart's sermon in Ellis Hall on Sunday...
A Call to Grow Up
Sermon by the Rev. Natosha Reid Rice for Sunday, June 13, 2021.
God is in the struggle
Audio-only recording of the Rev. Dr. Andrew K. Barnett's sermon in Ellis Hall on...
Safety in Numbers Will Come When We Are Possessed by God's Spirit of Love
Sermon by the Rev. Sarah C. Stewart for Sunday, June 6, 2021.
The Holy Trinity
The School of Love
Sermon by the Rev. Dr. Simon J. Mainwaring for Sunday, May 30, 2021.
Sermon by the Rev. Dr. Simon J. Mainwaring.


These Nike Accelerate Shorts are crafted with an elasticated waistband and 2 open pockets. They feature flat lock seams to prevent chafing and DriFit technology to keep the skin cool, dry and comfortable. Tyto šortky jsou lehká konstrukce v barvě bloku. They are designed with a printed logo and are complete with Nike Swoosh branding.


P.S. This is my 800th post!!!
well,i don't celebrate it,BUT,i do go to nazerath cause it beutifull in there,and there is good candies over there.
Well Thanks Dog. Happy Hanuka to ya.
In a surprise move today The Federal Aviation Administration and Homeland Security joined forces to unilaterally deny domestic air space clearance to the traditional Christmas Eve sleigh ride by Santa Claus.
Santa's ride has been bringing joy and happiness to millions of children for generations. Irate parents worldwide are said to be absolutely distraught. "Who's going to eat the milk and cookies for a start?" asked one parent.
Homeland Security has said that any penetration into American air space will be considered an act of aggression as part of their duty in keeping American airspace and borders safe. A spokesman from Homeland Security told us, "We just can't have every jolly, rosy cheeked, white haired fruitcake in the world flying around American air space unsupervised."
FAA are scratching their heads about where to begin certifying Santa's sleigh as safe for flight. There are no standards regulating the stability and performance of reindeer, also the FAA simply don't know how to inspect a machine which, in their minds, "totally defies the laws of physics".
* As a powered aircraft, Santa's sleigh requires a proper licensed propulsion system (Reindeer fed on magic dust are not mentioned in the regs and can't be used).
* The Sleigh must have a valid maintenance certificate and must meet aerodynamic requirements and be tested in a wind-tunnel.
Santa Claus is said to be very upset by the reaction from these US agencies and feels the actions are a direct violation of the Children's basic human rights.
Homeland Security laid down the following ultimatum to Santa Claus, "If we see any flying sleighs in the skies over America you can bet you bottom dollar there'll be a F-16 with sidewinders blasting it outta the sky."
Sorry i've been away from the forum SSSSSSSSSOOOOOOOOOOOOoooooooooooooo long, very busy over here with my new job and PPL test study (check ride is booked for the 24th January YAY!!) anyway enough excuses.
We are off on Christmas holidays to Vanuatu, leave tomorrow, I have even booked a dual flight in Air Club Vila's C172 to go for a sceanic...WOOO HOOO!!! so MERRY CHRISTMAS and A HAPPY NEW YEAR to everyone.
We are off on Christmas holidays to Vanuatu, leave tomorrow, I have even book a flight in Air Club Vila's C172 to go for a sceanic...WOOO HOOO!!! so MERRY CHRISTMAS and A HAPPY NEW YEAR to everyone.
Have fun in Vanuatu mate!!! Be sure to get back, so we can have chat again!
Have a wonderful time!!!!!!!! Happy holly-daze to you!!
"What the *#@! is the gun for!!??!!!
"I'm really NOT SUPPOSE to tell you this but,,,,,,,, "
"You're gonna' loose an engine on take off..."
Has this been posted somewhere on the forum yet?
"St. Nicholas One, turnin left onto final.
But a Rutan-built sleigh, with eight Rotax Reindeer!!
As he passed all fixes, he called them by name.
On Comet! On Cupid! What pills was he takin?
They phoned to my office, and I heard it with dread.
Then I heard "Left at Charlie" and "Taxi to parking."
His boots were as black as a crop dusters belly.
Then he put on his headset, and I heard him yell "Clear!!"
Turn right three-two-zero at pilots discretion".
"Your traffics a Grumman, inbound from the west."
Merry Christmas to all!! I have traffic in sight."


Situated adjacent to the beach and just minutes from LEGOLAND California, Carlsbad Village shopping and more, this villa-style resort offers comfortable accommodations, complete with 40-inch flat-screen TVs, full kitchens, workspaces, and more. Guests can also enjoy amenities like heated pools, a 24-hour fitness center, complimentary shuttle access to the resort's local beauty spa and restaurant.
Beach is across the street.
With a stay at Carlsbad Seapointe Resort in Carlsbad, you'll be near the beach and convenient to Flower Fields of Carlsbad and Legoland California. This family-friendly condo is within close proximity of Sealife Aquarium and Carlsbad Premium Outlets.
Make yourself at home in one of the air-conditioned rooms featuring fireplaces. Rooms have private balconies. Kitchens are outfitted with refrigerators and microwaves. Cable programming along with DVD players are provided for your entertainment.
Be sure to enjoy recreational amenities, including an outdoor pool, a sauna, and a fitness center. Additional amenities include concierge services, wedding services, and a picnic area.
Have you been to Carlsbad Seapointe Resort? Share your knowledge and be the first to review this.
The staff is so friendly and accommodating. They made every effort to see that we were comfortable and enjoying our stay. They had lots of activities for all ages. The rooms were neat and clean and very comfortable.
We love Seapoint! We come every year and always have a great time. Convenient location without the crowds of San Diego or Orange County. Right across PCH from the beach. And the resort has two beautiful pools and hot tubs one for kids and one for adults. The rooms are recently remodeled, modern, and clean - I'm a freak about unclean hotel rooms. Our family of four is comfortable for a week here. All units have balconies that face the ocean, no rooms without a view! Each unit has a full kitchen so we don't have to eat out for all meals. Every unit has a sofa sleeper too. There is a laundry room, fitness center, tennis court, basketball court, pool table, ping pong table - and we have used it all. Everything is new, clean, and well maintained. The Coaster train station is literally located behind the resort so we always take the train to Old Town or downtown, it's so easy and cheap, don't have to drive in traffic or pay for parking in town, and the train is fun! The resort also has a schedule of fun activities every day of the week. Dexter and his associate Rick make the whole trip extra fun for the kids! I can't say enough good things about Dexter! All the staff are super friendly and helpful - reception, concierge, maintenance and cleaning staff. They really do make you feel at home. The only not great things about Seapoint are 1) the bed mattresses are not good, but the sheets and pillows are nice, 2) the resort is on the train tracks and the trains that don't stop blow their whistle scaring the hell out of us every time (but no trains at night), 3) the beach is right across the street which is great but you have to cross the highway and the beach is rocky and very narrow, at high tide there is no beach, 4) the maid service is scheduled only one day a week so if you need more towels or anything just give the front desk a call - no biggie. Overall, I highly recommend Seapoint Carlsbad for a family vacation, clean and friendly, no crowds, convenient location to see all the sights in San Diego. LEGOland is only a few blocks away. And you can't beat waking up to a view of the ocean every day with ocean sunsets every night.
"Never disappointed staying at Carlsbad Seapointe."
Beautiful room, great weather, every member of the staff was welcoming and friendly. Let me give a special thanks to Ms. N Flores from Housekeeping who kept our room neat and clean. Five star rating to all the staff! FYI - April is a great month to travel to Carlsbad (no May grey or June gloom!).
"2019 Visit to Carlsbad Seapointe Resort"
Well run, well maintained, condo on the ocean with lots of excellent restaurants accessible without wrestling with freeway traffic. Units are very well furnished and decorated. Especially loved the set in round backlit magnifying mirrors.
We just returned from our stay at Carlsbad Seapointe Resort. My family had a great time and were impressed by the great customer service and amenities the resort has to offer. The location was convenient to the beach as well as the little downtown village of Carlsbad. We were especially pleased with our housekeeper, Hortencia and the maintenance guy Donald. Thank you for your wonderful service and making our stay a relaxing one.


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December 2011 Prying Publisher
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Hot Dogs and Horticulture! This March marks Harper's 71st Anniversary! We'll be celebrating by having John H. Harper Jr. ( the second generation of the Harpers, Jay, Jeff, and Jerry Harper here to answer all your garden questions at 10am. That is over 200 years of combined garden and landscape experience in one place. If you ever wanted to ask a question, are an experienced gardener and want to compare notes, or are just getting started an more rare opportunity would be hard to come by!
We'll also have the grill ready and have a great lunch for you FREE afterwards-come celebrate, learn, or just say hey!!


Until such measures are implemented, however, the question of employment status is still subject to significant uncertainty, resulting in numerous cases over the last couple of years, most notably in relation to "gig economy" type business arrangements such as Deliveroo and Uber.
There is no fixed "test" which can give a definitive answer on employee status, and the facts of the individual case will always be important. One factor often mentioned is the obligations to provide service personally; in other words, is the individual required to carry out the work themselves or are they free to appoint a substitute to do this for them?
In a recent Employment Appeal Tribunal ("EAT") decision, the question of substitution was as a key consideration in a rather old-fashioned kind of working arrangement. The nephew of an elderly gentleman in Chatfield-Roberts v Phillips and another UKEAT/0049/18/LA appealed a decision of the Employment Tribunal hearing that the live-in carer he had engaged to look after his uncle, Ms Phillips had been his employee. Ms Phillips had been on the books of 'Universal Aunts', which provided carers on a rotational basis to clients with carers normally moving on every three to four weeks. Mr Chatfield-Roberts was however keen to secure more long-term, consistent care for his uncle and requested that Ms Phillips commit to a minimum of 6 months, with the engagement in the end running for over three years.
Ms Phillips lived in the property and worked six days a week, with her day off covered by another 'Aunt' initially, and later by the gentleman's cleaner. On the occasions where she was on annual leave and on jury service, Ms Phillips contacted Universal Aunts for a replacement, to whom Mr Chatfield-Phillips continued to pay a monthly fee. Ms Phillips initially submitted invoices for her fee but later Mr Chatfield-Roberts paid her directly by standing order, with overtime payments being dealt with by way of supplementary payments. Ms Phillips did her own tax returns but was not required to supply any of her own equipment or make any expenditure to carry out her duties. She received instructions from and gave updates to Mr Chatfield-Roberts by telephone and during his visits to his uncle.
When her engagement was terminated, the question of whether she had been employed by Mr Chatfield-Roberts arose as a preliminary point in relation to whether or not this dismissal had been fair. The EAT upheld the decision of the first instance employment tribunal in finding that, on the facts, Ms Phillips had been employed.
In reaching the decision, the EAT considered factors such as the level of control over the work carried out by Ms Phillips, and the fact that her tasks had been set by Mr Chatfield-Roberts, with an expectation that she would carry out the tasks required. One of the main factors considered was the question of whether the fact that Ms Phillips organised a substitute to cover when she was not at work was a sufficient ground to determine that she was not employed.
The EAT referred to earlier the Court of Appeal decision in the Pimlico Plumbers, in which distinctions were drawn between different types of rights to substitution. For example, where the individual carrying out the work has an unconstrained right to provide a substitute at any point, this is inconsistent with employment. Where, however, there is a need to obtain the consent of the person to whom services are provided and that person has discretion to refuse that consent, this is more consistent with employment. If, as in this case, the right to provide a substitute is only engaged where the individual is unable to carry out the work, this is consistent with the requirement to personally perform the obligation, pointing towards an employment relationship. The EAT also took into account that Ms Phillips was not arranging her own choice of substitute; rather she was using the service which Mr Chatfield-Roberts paid for to cover only her days off, holidays and jury service.
This case illustrates that simply including a right for the individual to provide a substitute is not sufficient to avoid employment status. Much will depend on the facts and the extent to which the right to provide a substitute is actually capable of being utilised – for example, by requiring Ms Phillips to commit to a minimum 6 month term, it was evident that it was she personally who was required. In many cases, individuals providing services will have been selected on the basis that it is their particular skills that the employer wishes to utilise, and the fact that cover is arranged only when that individual is, in limited circumstances, unable to attend does not negate an employment relationship. The issue becomes more complex where there is a right to provide a substitute and the client or "employer" has limited ability to veto the selection, in which case other factors such as level of control over the tasks carried out will require to be looked at in conjunction.