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United Airlines Complaints>Making Keystone Cops look like the Manhattan Project
jodyrrr 8:05 PM Jun 22, 2016
On June 21, 2016, our United Express flight 5547 out of YVR to LAX had an engine issue just before takeoff. We went back to the terminal and were told it would take a few minutes. We were then told they needed a new part. We were then told they couldn't get the part, but we could get on a SkyWest flight. We were handed what we thought were boarding passes. We were not informed by the United ticketing personnel that we were required to check in again, so we sat and waited. Then, a fellow passenger realized that a check in was required just before the flight was about to board. Suddenly, there was 60 people waiting in line to check in to a flight that was leaving in 40 minutes. Because of computer issues, the line was not moving. I waited 30 minutes before being told that we could not get on the flight. 60 of us cried out in unison. We were told by SkyWest to go back to United. The clamor was impressive. The SkyWest person then threatens to call security to get us to leave the SkyWest gate. At this point, it's 3 hours after the original take off time.
The United ticketing folks then told us that a United flight from SFO terminating at YVR would be turned around to LAX for us. They said it was arriving at 5. Then I got a text saying 6 something, then 7 something, then 8 something. The plane finally arrives and we're on by 9, a full 8 hours after our original take off time.
For all this trouble, we were offered a measly $10 meal ticket to use at the airport. When we were finally aboard, I suggested that perhaps a free drink and free wifi coupon were in order. I was told by the flight attendant that neither was possible. To their credit, throughout all of this both the United ticketing folks and flight attendants handled the situation quite well outside the alternative flight ticketing fiasco we endured, which made the Keystone Cops look like the Manhattan Project.
While all this is happening, I'm complaining on Twitter. I get a response from United telling me to go to a website to be compensated for the trouble. When I get there and take more of my time to fill out the forms, I'm offered a paltry $50 credit on my next United flight, which isn't ever likely to happen at this point. It was like having salt poured into an open would and was frankly, very insulting.
In the end, we are just meat with money to United, and offering a ridiculously low $50 credit to be used on their woefully inadequate airline is clear proof of this.
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Lightweight Styling ©Dartho


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Kirk Douglas for "It Runs in the Family"
Tuesday, April 15th 2003 12:14 am
At 87, Kirk Douglas is frail, suffering the clear aftermath of the stroke that impeded his speech. Yet there remains a power to the man. Age has not dampened but rather strengthened his sense of humour.
Promoting his first major film since the stroke, It Runs in the Family, produced by and co-starring his famous son Michael, the legendary actor arrives to be interviewed with his grandson Cameron, who makes his acting debut in the same film. "You better be nice to him because he's going to become a famous actor. So you have to be nice to him. I'm nice to him because he'l give me a job in his next picture", he quips.
Asked where his sense of humour comes from, he suggests reading from his latest book in which he recounts, with both humour and pathos, life after his stroke. "In the book, I had my operator's manual. One, when things go bad, always remember it could be worse. Two, never, never give up and keep working on your speech and on your life. Three, never lose your sense of humour: Laugh at yourself, laugh at others. Those things are the most important. Everybody has depression. I found out that the way you deal with depression is you try to think of other people, other things, because if a person is so self-centred, oh woe is me. Of course, my wife was very helpful. I would be in bed and she've say come on, get your ass out of bed and work with your speech therapist."
Douglas did just that. He may not quite be the Hollywood idol of The Vikings, Lust for Life or Spartacus, but Kirk Douglas remains every bit the movie star. These days, the classical Hollywood star thrives on his work, and had a ball returning front and centre to It Runs in the Family, a sometimes wacky comedy about fathers and sons, grief and family bonding. It gave the actor, who rediscovered his Jewishness following a near fatal helicopter accident some years back, the chance to work with much of his family, including of all people his ex-wife Diana, who plays his wife in the film.
"It was wonderful because we have always had a good relationship and she's a very good actress, so that was very easy working with her. I think we both like working together and Diana is a very good friend of my wife. They have lunch together. It's part of our life. You can't cut out part of your life. I had two sons with her. AS a matter of fact, after I got married to my wife, Anne, my wife was pregnant and I was shooting a picture in Oregon called Indian Fighter and there were good parts for my ex-wife Diana, so I gave her the part. So, we had Diana and I in Oregon making the movie and my wife Anne was taking care of little Michael and Joel. So, we go back a long way", Douglas recalls, forcing a half smile.
As for working with Michael, with whom he has a had a sometimes testy fraternal relationship over the years, Douglas jokes that his son was a tough boss. "He doesn't pay well but he's fair. He'l give you a shot and helps you as best as he can. It was a wonderful experience." Kirk says that he regrets not having worked with Michael earlier in their lives. "I always wanted to work with Michael. When I had my stroke, I thought well, I will never work as an actor again because what can an actor do who can't talk? He waits for silent pictures to come back. It was a big thrill to finally do a picture with Michael and to top it off, I never expected to be working with Cameron. So, it was a big thrill. When he found It Runs in the Family it was before my stroke and after I had my stroke, Michael tried to encourage me. He said well, Dad, listen. Just work with your speech therapist. Then we'l make the movie. That made me mad. I said Michael, why don't YOU work with my speech therapist and then when you talk the way I talk, we'l do the movie. We all laughed at that. But, first of all, it's hard to find scripts of any kind that's good. I mean, nobody sets out to make a bad movie and how many bad movies are there? But we were lucky with this picture, I think, because it was something that people can relate to. Every family is dysfunctional and every family has problems. The sons are always doing this or that and fathers like Michael have a lot of problems with Cameron, but I'm the grandfather. I have no problems." Kirk Douglas remains a Hollywood legend who knows how to make an exit.
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redpajama_set_name: RedPajamaCommonCrawl


Thank you soooooo very much. I am thrilled that after all these years we have managed to secure something together. THRILLED. I really do hope we can continue to land orders this size together. So professional.
Simba are the #1 Towel Supplier & Manufacturer in Australia.
When a client needed 1200 x Custom Made Towels for an event, Simba was the only supplier I considered for the project.
Aside from meeting the deadline, the final decoration result was fabulous. The CEO even noted that 'people were taking them on holidays & photographing them on the beach…and one even made it to Germany!'.
Really appreciate your help and efficiency !
I would just like to say thank you for our recent purchase to all we came into contact with.
During a tricky time for you all - transition to a great new bigger space we placed an order and were treated with the usual professionalism we have always had.
We then came back for more and even though we changed our order late in the day, it was not a problem to remove extra box and re-invoice.
Our towels arrived today and they look and feel fantastic!!
Thanks so much for doing a great job.. I'm sure they will fly out the door.
With a heavy heart, I would like to inform you about my decision to resign from my current job. It was a great experience having an opportunity to work with such a professional vendor like you and having received finest services from you.On my farewell, I wish to express my deepest gratitude and thankfulness to you for the timely delivery of your good and services. The level of professionalism that we shared would be missed. Throughout my employment with my company you always proved to be a great support. I greatly appreciate your services and generosity. Goodbye and wish you all the best for your future endeavours. Thanks a lot for everything.
Thanks very much Sarah: Our client is SO happy to have these so fast we do appreciate it.
Zooming in on the images Mary sent us, the embroidery looks absolutely spectacular! Maybe we need to have you do all our embroidery from this day forth!!!
Thought I'd share this with you as you did these towels for us.
They look awesome and the client is super happy with them!
So thank you for the great job that you did.
Thanks so much for all your assistance. We are looking forward to receiving the towels.
Both myself and Adrian are extremely impressed with this job. Your service and speed has been phenomenal.
I cannot thank you enough. We will definitely be working with you again in the future.
Just wanted to pass on my thanks for my tea towels; I picked them up today and I am very, very happy with them.
Thanks for all your help, you made the process very easy and personal. I will definitely be purchasing again; hopefully larger quantities!
"I just wanted to pass on the wonderful feedback that I have been receiving about our delegates gifts this year.
It is always nice to get positive feedback."
In this project, I had the pleasure to work with you & your very Professional & Competent team.
Thankyou so much for all your help & please relay my thanks to everyone.
Hopefully, I will have an opportunity to work with you again in the near future.
Dear Mehul, Mary and The Team!
You guys have been amazing with our recent orders and look forward to a prosperous growth in towel jobs next year with a JPS & Simba Partnership.
Your help, guidance, late night calls and emails are always appreciated.
We have been using Simba Towels as our main supplier now since 2003. Since the start, the service, the price and the quality we have recieved has been second to none. I would be very happy and extremely confident, to recomend Simba Towels to any prospective customer.
Although we completed delivery of the 'Size Matters' beach promotion kits almost two years ago, we felt it important to let you know that the sample we kept for testing has proven to be an astonishing performer. Despite daily use, the towel stands out as the best value for money product in the group, hard wearing and still looking like new, it has really surpassed allour expectations.
Thank you all very much for your brilliant work !
Bless you Sarah; you're a gem and we truly appreciate it.
Not only Express Post but efficiently, you've even provided the Con Note Number; wish all suppliers were like you.
Good morning Mary :)...here's the address for the towels.
Thanks so much for your help...amazing turnaround here..big brownine points with client (deserved too! )...job really well done to you and your team!


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La recette du retour de Simona Halep: nourrir avec confiance «Jeu par jeu»
Warrior-like Simona Halep shared how she found the way to fight back in a herculean way to overcome strong-hitting Ekaterina Alexandrova after being 6/3, 2/0 down. Simona showed us many times that she is capable of making comebacks in the harshest moments of a match and doing so involves her well-known fighting spirit.
"I got the confidence game by game, even if she was leading me. I still thought that there is a chance to win, and I have just to fight", said Simona after the match. The match ended 6/3, 5/7, 4/6 after Simona managed to prevail in the second set after breaking at 5-5.
After an intense third set, Halep managed to get in a 0-40 situation on Alexandrova's latest service game before serving well and concealing the victory. However, the Romanian felt the toughness of her opponent, so she was even more proud of her victory.
"She kept the rhythm and the level of her game very high today the whole match. So I needed something extra to finish it to be well. I knew that she's gonna play really, really well, and she did. But I didn't give up, and I'm very pleased with that", said Simona.
Even if Halep injured" her left foot> in Toronto, she did not show signs of any pain in the affected area. In the 1/8 finals, Simona will face Madison Keys, another hard-hitting player, so she will have to rethink her game-plan. "Keys is hitting really strong, the ball, and is playing flatter", concluded Halep about her next opponent.
Soutenue par Virginia Ruzici, Simona Halep cherche…
Je fais confiance à Rafael Nadal, je ne fais pas…
Simona Halep: «Serena Williams doit jouer plus…
WTA Madrid: Simona Halep en tête Johanna Konta.…
WTA Madrid: Naomi Osaka et Simona Halep entrent dans quarts
Simona Halep: «Le titre de Wimbledon battant Serena…
← Novak Djokovic: «Le succès du tennis canadien est très important impressionnant' Lucas Pouille sur Novak Djokovic face à: "Tout le monde a leurs chances .. ' →


Boldo is a small tree that grows on the sunny sides of Chilean mountains. It can reach 5 to 6 meters and is covered with small, aromatic leaves. Andean Indians have apparently known of this plant's medicinal value for quite some time. Eucalyptol, flavonoids and several alcaloids, including boldine which is apparently responsible for most of the plant's therapeutic effect. A toxic volatile oil, which contains ascaridole, is also present in the leaves.
Dried leaves: 60 to 200 grams 3 times a day.
Tea: 1 g of dried leaves in 150 ml of boiling water for 5 to 10 minutes.
There is insufficient reliable information to conclude that boldo is effective in any other indication.
When used in moderation, no toxicity has been associated with the use of boldo. It may cause heartburn and excessive doses may irritate the urinary tract.
It should be avoided in the presence of renal disorders. Boldo is contraindicated in people with active liver disease and bile duct blockage and should be used with great caution in people with gallstones (biliary lithiasis).
Theoretically, boldo may increase the effect of anticoagulants (Coumadin) and antiplatelet agents. Before taking boldo, check with your pharmacist to make sure that there are no interactions with your regular medication.
Because of the possible presence of a toxic volatile oil (ascaridole), boldo use is not advisable during pregnancy and breast-feeding.
Boldo's medicinal virtues have been known to man for thousands of years. Over thirteen thousand years old fossilized leafs, with the imprints of human teeth on them, have been found.


The MEP Youth Education Advisor provides educational services and support to eligible children (ages 3-21) enrolled in the Migrant Education Program in Tennessee. The MEP Youth Education Advisor will assess the academic needs of the students and provide supplemental academic support services such as homework help and tutoring, and other services. Additional activities include the identification and recruitment of migrant students eligible for participation in the Migrant Education Program. The successful candidate will demonstrate strong experience working in an educational setting, particularly with young students and families, excellent organizational skills, and the ability to successfully collaborate with school personnel, students, and other support agencies. The ideal candidate must also possess a thorough understanding of the region where they will work. Fluency in Spanish and experience working with immigrant populations is required.
Only candidates who can demonstrate total commitment to the mission and values of Conexión Américas and the Migrant Education Program should apply. Please read about our mission and values on our website at www.conexionamericas.org. Applications sent on job boards will not be considered and all application materials must be emailed.
Please submit cover letter, resume and salary history to Sara Del Castillo, Director of People & Culture at sara@conexionamericas.org. No phone calls or in-person inquiries, please.
Position Description: The MEP Youth Education Advisor reports to the Associate Director of the Migrant Education Program. This is a full-time positions salary depends on experience and/or qualifications. The Migrant Education Program (MEP) is a program established by the United States Department of Education. Conexión Américas is responsible for the implementation of the Migrant Education Program in the state of Tennessee. The fundamental purpose of the program is to remove barriers to academic success for migrant children (ages 3-21), and to promote academic achievement.
Connects with local school administrators, businesses, and other organizations in the community to identify and recruit eligible students.
Provides referrals to educational, social, and other services as needed.
Coordinates wrap – around services for students and families.
Serves as a liaison between school districts, school personnel, families and the Migrant Education Program.
Plans and conduct outreach in assigned counties.
Assists with recruitment and enrollment of migrant students.


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The San Fernando Valley
A century-old water tunnel in the Valley will be restored to capture Sierra snowmelt
It dates to the year the Valley became part of LA
By Bianca Barragan Jul 21, 2017, 11:50am PDT
LADWP restores water tunnel
By fall, the restored Maclay Highline will help transport 130 acre-feet of water per day. Read more here: http://www.dailynews.com/20170720/la-restores-historic-water-tunnel-to-turn-sierra-snow-runoff-into-drinkable-water
Posted by Los Angeles Daily News on Friday, July 21, 2017
The Los Angeles Department of Water and Power is restoring an underground water tunnel in the San Fernando Valley that dates back to 1915—the same year that the Valley officially became part of the city of Los Angeles.
The Los Angeles Daily News reports the tunnel is being refurbished to capture water runoff from the Sierras, which was inundated with snow this winter.
The tunnel is part of a larger system, called the Maclay Highline, that runs from "the L.A. Aqueduct Cascades in Sylmar to a group of meadows in Pacoima."
Once restored, the tunnel will carry a significant amount of water—130 acre-feet a day—to the Pacoima Spreading Grounds, where it will filter down into the city aquifer and become drinking water. (One acre-foot can supply two households with water for a year.)
"We do have, on the books, a huge groundwater treatment system being constructed," says Michael Grahek, manager of southern aqueduct and Owens Lake Operations and maintenance for LADWP. "This ties into that. We will have groundwater for years to come."
Read the full story over at the Daily News.
LA restores historic water tunnel to turn Sierra snow runoff into drinkable water [Daily News]
California's mountains are officially buried in snow [Curbed LA]
Does LA Have Enough Water for the Next 25 Years? [Curbed LA]
Next Up In History
How the Third Street Promenade became too successful for its own good
LA's 'most recognizable and beloved' building
LACMA is beloved. Its design never was.
When the plague came to Los Angeles
LA's most beautiful storage building was also a speakeasy
Temples to white supremacy


Is Apple Partnering With Valve To Release A Gaming Console?
When Apple's CEO is seen in public and especially when he visits another tech company, people tend to notice and analysts start to fret. A couple of days ago, Tim Cook paid a visit to Valve's officials in their Bellevue headquarters, and tech voices already foresee a partnership.
Valve is the gaming company behind hugely popular titles such as Half-Life, Team Fortress and Portal. Valve also owns an online gaming platform, Steam, which has similarities with Apple's App Store. Steam is a game network distribution with more than 30 million users worldwide.
The company is also rumoured to be working on its own console called Steam Box. With this track record, Valve might raise Apple's interest for additional features to be integrated into Apple HDTV.
Analysts suggest that Apple's assault on our living rooms will not be limited to a high-end TV set, but it would include a revolutionary home console.
A recent CultOfMac report points out that Apple might plan to release its own game console based on complex sensors, much like Microsoft's Kinect.
It's here, where Valve comes into the picture. Valve's job listings page suggests that the company is building a new console based on biometric technology.
We previously reported, Apple was working on a physical controller for the iOS.


text:
import is from 'util/is';

var registry = {};

function getRegister (name) {
    return registry[name];
}

function getOrCreateRegister (name) {
    return registry[name] || (registry[name] = {
        loaded: false,
        objects: [],
    });
}

export function one (name, callback, context = null) {
    var register = getOrCreateRegister(name);
    var objects = register.objects;
    var i = 0;

    if (register.loaded) {
        setTimeout(callback.bind(context));
        return;
    }

    for (i = 0; i < objects.length; ++i) {
        if (objects[i].type === 'one' && objects[i].callback === callback) {
            return;
        }
    }

    objects.push({
        type: 'one',
        callback: callback,
        wrapper: callback.bind(context),
    });
}

function fireCallbacks (register) {
    register = is(register, 'Object') ? register : getOrCreateRegister(register);
    register.loaded = true;

    var i = 0;
    var objects = register.objects;
    var cbObj;

    while (i < objects.length) {
        cbObj = objects[i];

        cbObj.wrapper();

        // removing callbacks changes the register's length.
        // increment i only if the length hasn't decreased by 1 due to splice
        if (cbObj.type === 'one') {
            objects.splice(i, 1);
        } else {
            i++;
        }
    }
}

export function fileWasLoaded (filename) {
    var register = getRegister(filename);

    if (register) {
        fireCallbacks(register);
    }
}

redpajama_set_name: RedPajamaGithub


At Woodyatt Curtains we specialise in window dressings and have been trading online since 2003. We were originally trading as Nets 2 Go, but due to us selling a wider range of products we rebranded to Woodyatt Curtains in early 2012.
We are a British company and our curtains are sourced from all over the world. The ranges and designs that we offer are regularly being updated ensuring that our website provides customers with the best products available. Due to our massive buying power we are able to negotiate big discounts with all of our suppliers and we pass these savings on to our customers. So they can then purchase the best value net curtains online. All of our products have a low price promise and have different delivery times. These are quoted on each individual product page.
Net Curtains are generally delivered within 5 working days. Ready made curtains are 5 - 7 working days and Made to Measure curtains are 10 - 14 days.


Aaron Gustafson is the Founder and Principal Consultant of Easy! Designs, a web development consultancy. He is Group Manager of the Web Standards Project (WaSP) where he has spearheaded both Web Standards Sherpa and a small business outreach effort.
If you've been working on the web for any amount of time, you've likely heard (or even used) the term "progressive enhancement" before. As you probably know, it is the gold standard of how to approach web design. But what is progressive enhancement really? What does it mean? How does it work? And how does it fit into our workflow in a time of rapidly evolving languages and browsers? In this session, Aaron Gustafson will answer all of these questions and provide concrete takeaways that will help you improve your web design skills.


Located east of Montreal in the affluent Repentigny neighbourhood is Galeries de Repentigny. Strategically situated on the highly trafficked Blvd Brien and in close proximity to major Hwy 40, this centre offers easy accessibility and high traffic counts. With major tenants including Super C, Pharmaprix and Tim Hortons, Galeries de Repentigny is a convenient shopping destination serving the local and surrounding residential communities. Adjacent to Galeries Brien Ouest and Galeries Brien Est the synergy of these three centres provide a complete shopping experience.


text:
\section{Introduction}\label{se:introduction}
In experimental designs, the completely randomization is usually  recommended to avoid the bias caused by the factors that are not controlled. However, in many industrial or agricultural experiments, there often exist factors whose levels are difficult to change. These factors are called the whole-plot factors. The other factors whose levels are easy to change are called the subplot factors. If whole-plot factors exist in an experiment and the completely randomization is conducted, then the experimental cost will increase due to frequently changing levels of the whole-plot factors. To reduce the cost, the two-stage randomization strategy is usually suggested.
First the randomization is conducted only for the treatments of whole-plot factors, call the whole plots.
Then in each whole plot, the second randomization is conducted for the treatments of subplot factors, called the subplots.
This kind of experiments was first introduced by Fisher (1925) and the design used for the experiment is referred to as the split-plot design (SPD).

The two-stage randomization results in the multistratum structure of the split-plot design and leads to two error terms, the whole-plot errors and the subplot errors.
Therefore, traditional analysis methods for completely randomized designs are no longer appropriate and the optimal completely randomized designs may not be optimal for the split-plot experiment.
In literature, many approaches for constructing optimal split-plot designs have been proposed.
A common construction method is based on the minimum aberration criterion, which can be found in Huang et al. (1998), Binham and Sitter (1999a, 1999b, 2001, 2003), Mukerjee and Fang (2002), and Tichon et al. (2012).
The minimum aberration criterion aims to find the optimal design which minimizes the alias of the important effects.
An idea of this method is that the powers to detect significant effects are not the same for whole-plot factors and subplot factors. It is usually assumed that the subplot variability is smaller than the whole-plot variability, which implies that the power to detect significant subplot effects is greater than the power to detect significant whole-plot effects. Hence, subplot factors are considered more important and should be given shorter world length than whole-plot factors. The design that sequentially minimizes the wordlength patterns is selected as the minimum aberration split-plot design.
Since the minimum aberration criterion minimizes the alias, the optimal split-plot design constructed by this criterion should have less bias of the estimation for the important effects.

Another widely used criterion for constructing optimal split-plot designs is the $D$-optimal criterion, which can be found in Lesinger et al. (1996), Goos and Vandebroek (2001, 2003, 2004) and Jones and Goos (2007).
The basic idea of this construction method is that a good split-plot design should have higher estimation ability for the fitted model.
This estimation ability can be evaluated by measuring the determinant of the information matrix of a design, called the $D$-efficiency.
The $D$-optimal split-plot design is the one which has the highest $D$-efficiency among all of designs.
Since maximizing the $D$-efficiency of a design is equivalent to minimizing the determinant of the variance-covariant matrix of the estimation, the $D$-optimal split-plot design should have smaller variances and covariances for the estimation of the effects. A good review of split-plot designs can be found in Jones and Nachtsheim (2009).

Although the minimum aberration criterion and the $D$-optimal criterion are commonly used for constructing optimal split-plot designs, the optimal designs selected by the two criteria may have higher mean square error (MSE), which consists of the square of the bias matrix and the variance-covariance matrix of the estimation.
The minimum aberration criterion focuses on minimizing the bias of the estimation but less considers minimizing the variances and covariances. Therefore, the optimal split-plot design constructed by this criterion may have higher mean square error due to higher variances or covariances of the estimation.
On the contrary, the $D$-optimal criterion aims to minimize the variances and covariances but ignores the bias of the estimation. When the fitted model is
misspecified, there exists a bias for the estimation. Therefore, the optimal split-plot design constructed by the $D$-optimal criterion may have higher mean square error due to higher bias of the estimation.

In this paper, we take the model misspecification into account. When the fitted model differs from the underlying true model, a good split-plot design should be able to simultaneously control the variance-covariance matrix and the bias matrix of the estimation.
An appropriate criterion to deal with the model misspecification is the $D$-optimal minimax criterion proposed by Zhou (2001, 2008), Wilmut and Zhou (2011), Lin and Zhou (2013) and Yin and Zhou (2014). This criterion is usually applied on the construction of the robust completely randomized design for model misspecification. In this paper, we extend the application of the $D$-optimal minimax criterion to the split-plot designs and provide a general form of the loss function used by this criterion. This general form of the loss function allows us apply the $D$-optimal minimax criterion to selecting optimal design for split-plot experiments or completely randomized experiments with or without model misspecification.
To more efficiently construct and search for the robust split-plot design, we combine the point-exchange algorithm proposed by Goos and Vandebroek (2001) and the anneal algorithm proposed by Zhou (2001). The update formulas suggested by Arnouts and Goos (2010) are applied to increasing the computing speed for calculating the loss function for the $D$-optimal minimax criterion.

The rest of this paper is organized as follows. Section~\ref{se:cri} introduces the split-plot design and the $D$-optimal minimax criterion with the general form of the loss function, which can be used for selecting robust split-plot designs or completely randomized designs. Section~\ref{se:au} provides an algorithm which combines the point-exchange algorithm and the anneal algorithm for constructing and selecting the robust split-plot design for model misspecification. The update formulas are introduced and modified for the $D$-optimal minimax criterion. Section~\ref{se:ex} provides two examples to demonstrate how to apply our proposed method to obtaining robust split-plot designs. Section~\ref{se:co} is the conclusions and remarks.

\section{Background and criterion}
\label{se:cri}
Let $\cal H$ denote an $N$-run full factorial design for factors $F_1,\cdots,F_m$ with levels $s_1,\cdots,s_m$, respectively, where $N=\prod_{i=1}^ms_i$ and the levels of factors are coded as orthogonal contrasts.
Let ${\bf H}$ be the $N\times N$ matrix whose first column is all ones for the grand mean and the other $N-1$ columns are the contrasts of all the main effects and interactions of the full factorial design.
The $i$th row of ${\bf H}$ is corresponding to the $i$th design point (run) in $\cal H$.

\subsection{Estimation of the split-plot design}
A split-plot design with $n$ runs can be selected from the $N$ rows of $\cal H$ without replacement by arranging the design points that have the same level combinations of the whole-plot factors into a whole plot.
Assume that $\cal D$ is an $n$-run split-plot design with $m_w$ whole-plot factors and $m_s=m-m_w$ subplot factors, where the total number of whole plots is $b$ and the number of subplots in the $i$th whole plot is $n_i$, $i=1,\cdots,b$.
Let $R$ be a requirement set containing $p$ effects which usually includes all the main effects of whole-plot factors and subplot factors and some interactions.
Then the linear model for $R$ is
\begin{equation}\label{eq:m1}
{\bf Y}={\bf X}_1\boldsymbol\beta_1+{\bf Z}\boldsymbol\gamma+\boldsymbol\epsilon,
\end{equation}
where ${\bf Y}$ is the $n\times 1$ vector of response, $\boldsymbol\beta_1$ is the $(1+p)\times 1$ vector of the grand mean and the effects in $R$, ${\bf X}_1$ is the $n\times (1+p)$ matrix of the orthogonal contrasts for $\boldsymbol\beta_1$, ${\bf Z}$ is an $n\times b$ indicator matrix with entries $z_{li}=1$ if the $l$th run of $\cal D$ belongs to the $i$th whole-plot and $z_{li}=0$ otherwise, $\boldsymbol\gamma$ is the $b\times 1$ vector of random whole-plot errors, and $\boldsymbol\epsilon$ is the $n\times 1$ vector of random subplot errors. It is assumed that $\boldsymbol\gamma$ and $\boldsymbol\epsilon$ are independent and have mean zero and variance-covariance matrix $\sigma_\gamma^2{\bf I}_b$ and $\sigma_\epsilon^2{\bf I}_n$, respectively.

Since there exists a multistratum structure for the split-plot design, the variance-covariance matrix of ${\bf Y}$ is
\begin{equation}\label{eq:sigma.spd}
{\boldsymbol\Sigma}=\sigma_\epsilon^2({\bf I}_n+d\bf{ZZ'}),
\end{equation}
where $d=\sigma_\gamma^2/\sigma_\epsilon^2$.
If the entries of ${\bf Y}$ are grouped per whole plots, then equation (\ref{eq:sigma.spd}) can be written as the $n\times n$ block diagonal matrix
\begin{equation*}\label{eq:v}
{\boldsymbol \Sigma}=\left[\begin{array}{cccc}\boldsymbol \Sigma_1& {\bf 0}& \cdots & {\bf 0}\\
{\bf 0}&\boldsymbol \Sigma_2 & \cdots & {\bf 0} \\
 \vdots & \vdots & \ddots & \vdots \\
 {\bf 0} & {\bf 0} & \cdots & \boldsymbol \Sigma_b
 \end{array}\right],
\end{equation*}
where
$\boldsymbol \Sigma_i=\sigma_\epsilon^2({\bf I}_{n_i}+d{\bf 1}_{n_i}{\bf 1}_{n_i}')$ for $i=1,\cdots, b$.
The generalized least square estimate (GLSE) of $\boldsymbol\beta_1$ is
\begin{equation*}\label{eq:beta.spd}
\hat{\boldsymbol \beta}_1=({{\bf X}_1'\boldsymbol\Sigma}^{-1}{{\bf X}_1})^{-1}{{\bf X}_1}'{\bf \Sigma}^{-1}{\bf Y},
\end{equation*}
and the variance-covariance matrix of $\hat{\boldsymbol\beta}_1$ is
\begin{equation*}\label{eq:cov.spd}
cov(\hat{\boldsymbol\beta}_1)=({{\bf X}_1'\boldsymbol\Sigma}^{-1}{{\bf X}_1})^{-1}.
\end{equation*}
The estimates of $\sigma_\gamma^2$ and $\sigma_\epsilon^2$ can be obtained by the restricted maximum likelihood (REML) method (see Letsinger et al., 1996).

\subsection{General form of the $D$-optimal minimax criterion}\label{se:ld}
Let ${\bf H}=({\bf H}_1,{\bf H}_2)$, where ${\bf H}_1$ is the $N\times(p+1)$ submatrix of ${\bf H}$ with the column ${\bf1}_N$ for the grand mean and the columns for the $p$ effects in the requirement set $R$, and ${\bf H}_2$ is the $N\times(N-p-1)$ submatrix of ${\bf H}$ with the columns for the effects not in $R$. Since the columns of ${\bf H}$ are orthogonal, it is obvious that
\[
{\bf H}'{\bf H}=\left(\begin{array}{cc}{\bf V}_1&\bf 0\\\bf 0&{\bf V}_2\end{array}\right),
\]
where both ${\bf V}_1={\bf H}_1'{\bf H}_1$ and ${\bf V}_2={\bf H}_2'{\bf H}_2$ are diagonal matrices.
If there exist significant effects that are not included in $R$, then model (\ref{eq:m1}) is misspecified. The underlying true model with small departures from (\ref{eq:m1}) can be written as
\begin{equation*}
{\bf Y=X}_1\boldsymbol\beta_1+{\bf X}_2{\boldsymbol\beta_2}+{\bf Z}\boldsymbol\gamma+\boldsymbol\epsilon,
\end{equation*}
where $\boldsymbol\beta_2$ is the $(N-p-1)\times 1$ vector of all the effects not in $R$ and ${\bf X}_2$ is the $n\times(N-p-1)$ matrix of the orthogonal contrasts for $\boldsymbol\beta_2$. The unknown parameter vector is assumed satisfying $\frac{1}{N}\boldsymbol\beta_2'{\bf V}_2\boldsymbol\beta_2\leq\alpha^2$, where $\alpha\geq 0$ controls the seriousness of departures. Note that model (\ref{eq:m1}) is correct if $\alpha=0$.

When model (\ref{eq:m1}) is misspecified $(\alpha>0)$, the generalized least square estimate of $\boldsymbol\beta_1$ is biased with
\[
\begin{array}{rl}
bias(\hat{\boldsymbol\beta_1})&=E(\hat{\boldsymbol\beta_1})-\boldsymbol\beta_1\\
&=({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1}{\bf X}_1'{\boldsymbol \Sigma}^{-1}{\bf X}_2{\boldsymbol\beta}_2.\\
\end{array}
\]
Then the mean square error of $\hat{\boldsymbol\beta}_1$ is
\begin{equation}\label{eq:mse}
\begin{array}{rl}
MSE(\hat{\boldsymbol\beta}_1,{\bf X}_1,\boldsymbol\beta_2)&=cov(\hat{\boldsymbol\beta_1})+bias(\hat{\boldsymbol\beta_1})bias(\hat{\boldsymbol\beta_1})'\\
&=({\bf X_1'\Sigma}^{-1}{\bf X_1})^{-1}+({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1}{\bf X}_1'{\boldsymbol \Sigma}^{-1}{\bf X}_2{\boldsymbol\beta}_2{\boldsymbol\beta}_2'{\bf X}_2'{\boldsymbol \Sigma}^{-1}{\bf X}_1({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1}.\\
\end{array}
\end{equation}
A robust split-plot design should be able to simultaneously control the variances/covariances and the bias of the estimation. To construct and obtain the robust split-plot design for model misspecification, we adopt the $D$-optimal minimax criterion and provide a general form of the loss function which can be applied to searching for the optimal design with or without multistratum structure.
Define the loss function of design $\cal D$ with respect to the requirement set $R$ as
\begin{equation}\label{eq:ld0}
L_R({\cal D})=\max_{\boldsymbol\beta_2\in\Theta}|MSE(\hat{\boldsymbol\beta}_1,{\bf X}_1,\boldsymbol\beta_2)|,
\end{equation}
where $\Theta=\{\boldsymbol\beta_2|\frac{1}{N}\boldsymbol\beta_2'{\bf V}_2\boldsymbol\beta_2\leq\alpha^2\}$ and $|\cdot|$ is the determinant of a matrix.
Let the singular value decomposition (SVD) of $\boldsymbol\Sigma$ be ${\bf U}\Lambda{\bf U}'$
where ${\bf U}$ is an $n\times n$ unitary matrix and $\Lambda$ is an $n\times n$ diagonal matrix with non-negative eigenvalues of $\boldsymbol\Sigma$. Define $\boldsymbol\Sigma^{-2}={\bf U}\Lambda^{-2}{\bf U}'$. Then
Equation (\ref{eq:ld0}) can be written as
\begin{equation}\label{eq:ld}
L_R({\cal D})=\frac{1+N\alpha^2\lambda_{max}\left({\bf V}_1^\frac{1}{2}({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1}{\bf X}_1'{\bf \Sigma}^{-2}{\bf X}_1{\bf V}_1^{-\frac{1}{2}}-{\bf V}_1^{-\frac{1}{2}}{\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1{\bf V}_1^{-\frac{1}{2}}\right)}{|{\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1|},
\end{equation}
where $\lambda_{max}(\cdot)$ is the greatest eigenvalue of a matrix.
The split-plot design that minimizes the loss function among all of the possible designs is call the $D$-optimal minimax split-plot designs, which is robust for model misspecification.

Equation (\ref{eq:ld}) is a general form for the $D$-optimal minimax criterion which can be applied on both split-plot designs and completely randomized designs with or without model misspecification as follows.
\begin{itemize}
\item[I.] When $d=0$, $\boldsymbol\Sigma$ reduces to $\sigma_\epsilon^2{\bf I}_n$ and equation (\ref{eq:ld}) reduces to
\[
\begin{array}{rl}
L_R({\cal D})&=\frac{1+N\alpha^2\lambda_{max}(\sigma_\epsilon^{-2}{\bf I}_n-\sigma_\epsilon^{-2}{\bf V}_1^{-\frac{1}{2}}{\bf X}_1'{\bf X}_1{\bf V}_1^{-\frac{1}{2}})}{\sigma_{\epsilon}^{-2(p+1)}|{\bf X}_1'{\bf X}_1|}\\
&=\sigma_{\epsilon}^{2(p+1)}\frac{1+\frac{N\alpha^2}{\sigma_\epsilon^{2}}\left\{1-\lambda_{min}({\bf V}_1^{-\frac{1}{2}}{\bf X}_1'{\bf X}_1{\bf V}_1^{-\frac{1}{2}})\right\}}{|{\bf X}_1'{\bf X}_1|},\\
\end{array}
\]
where $\lambda_{min}(\cdot)$ is the smallest eigenvalue of a matrix. Therefore, equation (\ref{eq:ld}) reduces to the form of the $D$-optimal minimax criterion given in Lin and Zhou (2013) for selecting the robust completely randomized design with model misspecification.

\item[II.] When $\alpha=0$, equation (\ref{eq:ld}) reduces to $L_R({\cal D})=1/|{\bf X}_1'\boldsymbol\Sigma^{-1}{\bf X}_1|$. Minimizing $L_R({\cal D})$ is equivalent to maximizing $|{\bf X}_1'\boldsymbol\Sigma^{-1}{\bf X}_1|^{1/(p+1)}$ and hence the $D$-optimal minimax criterion is equivalent to the $D$-optimal criterion for selecting the optimal split-plot design without model misspecification.

\item[III.] When both $d=0$ and $\alpha=0$, equation (\ref{eq:ld}) reduces to $L_R({\cal D})=1/|{\bf X}_1'{\bf X}_1|$. Minimizing $L_R({\cal D})$ is equivalent to maximizing $|{\bf X}_1'{\bf X}_1|^{1/(p+1)}$ and hence the $D$-optimal minimax criterion is equivalent to the $D$-optimal criterion for selecting the optimal completely randomized design without model misspecification.
\end{itemize}

\subsection{Scale invariance}
The $D$-optimal minimax criterion with the loss function as equation (\ref{eq:ld}) is scale invariant. Suppose that contrasts of the effects are rescaled by
\[
\tilde{\bf H}=\{\tilde{\bf H}_1,\tilde{\bf H}_2\}=\{{\bf H}_1{\bf C}_1,{\bf H}_2{\bf C}_2\},
\]
where ${\bf C}_1=diag(1,c_1,\cdots,c_p)$, ${\bf C}_2=diag(c_{p+1},\cdots,c_{N-1})$, and $c_1,\cdots,c_{N-1}$ are positive constants. Then
$\tilde{\bf V}_1={\bf C}_1{\bf V}_1{\bf C}_1$ and the fitted model with the rescaled effects can be written as
\begin{equation}\label{eq:m2}
{\bf Y}=\tilde{\bf X}_1\boldsymbol\beta_1+{\bf Z}\gamma+\epsilon,
\end{equation}
where $\tilde{\bf X}_1={\bf X}_1{\bf C}_1$. Let $L_{R({\bf X}_1)}({\cal D})$ and $L_{R(\tilde{\bf X}_1)}({\cal D})$ represent the loss functions of $\cal D$ corresponding to model (\ref{eq:m1}) and model (\ref{eq:m2}), respectively. Then we obtain
\begin{equation}\label{eq:sc}
L_{R(\tilde{\bf X}_1)}({\cal D})=L_{R({\bf X}_1)}({\cal D})/\prod_{i=1}^pc_i^2.
\end{equation}
If $\cal D$ is a $D$-optimal minimax design for model (\ref{eq:m1}), it minimizes $L_{R({{\bf X}}_1)}({\cal D})$. Thus $\tilde{{\bf X}}_1={\bf X}_1{\bf C}_1$ minimizes $L_{R(\tilde{{\bf X}}_1)}({\cal D})$ and is a $D$-optimal minimax design for model (\ref{eq:m2}).
Therefore, the $D$-optimal minimax criterion is scale invariant.

\iffalse
\[
\begin{array}{rl}
l_A&=\max_{\frac{1}{N}\boldsymbol\beta_2'V_2\boldsymbol\beta_2\leq\alpha^2}tr(MSE)\\
&=tr(({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1})(1+N\alpha^2\lambda_{max}(({\bf X}_1'{\bf \Sigma}^{-1}{\bf X}_1)^{-1}{\bf X}_1'{\bf \Sigma}^{-1}{\bf \Sigma}^{-1}{\bf X}_1-{\bf V}_1^{-1})\\
\end{array}
\]
\fi

\section{Algorithm and update formulas}\label{se:au}
To construct and search for the robust split-plot design, we develop an efficient algorithm by combining the anneal algorithm and the point-exchange algorithm.
The annealing algorithm has been shown effective for constructing the $D$-optimal designs or $D$-optimal minimax completely randomized designs (see Fang and Wines, 2002; Haines, 1987; Meyer and Nachtsheim, 1988; Zhou, 2001, 2008, 2011) and
the point-exchange algorithm is efficient for constructing optimal split-plot designs (see Goos and Vandebroek, 2001, 2003, 2004).
Another important issue for constructing the robust split-plot design is that the computing is intensive to calculate the inverse and determinant of the updated designs.
To save the computational cost, we apply and modify the update formulas suggested by Arnouts and Goos (2010).

\subsection{Design construction algorithm}\label{se:al}

Let $A$ be the candidate set of the whole plots, $E$ be the candidate set of the subplots, $T_0$ be the initial temperature, $a_b$ ($\leq b$) and $e_{n_i}$ ($\leq n_i$), $i=1,\cdots, b$, be the maximum numbers of whole plots and subplots that are allowed to change in a design to generate a new design, $N_T$ be the number of designs searched at each temperature, and $M_0$ be the total number of temperature changes.

\begin{itemize}
\item[] Step 1. Randomly generate an initial split-plot design ${\cal D}_0$ by selecting $b$ whole plots ${\bf w}_i$, $i=1,\cdots,b$, from the candidate set of whole plots $A$ and selecting $n_i$ subplot ${\bf t}_{ij}$, $j=1,\cdots,n_i$, for the $i$th whole plot from the candidate set of subplots $E$. Make sure that the design point $({\bf w}_i,{\bf t}_{ij})$ for $i=1,\cdots,b$, $j=1,\cdots n_i$ are selected without replacement. Let $J$ be the number of temperature changes in the algorithm and set $J=1$ at beginning.

\item[] Step 2. Compute the loss function $L_R({\cal D}_0)$.
For each $i$, define a subset $E_i$ including all the points in $E$ that are not ${\bf t}_{ij}$, $j=1,\cdots,n_i$.

\item[] Step 3. Implement point exchange for the whole plots.
\begin{itemize}
\item[(a)] Randomly choose a number $a$ from set $\{1,\cdots,a_b\}$. Select $a$  whole plots ${\bf w}_{i_l}$, $l=1,\cdots,a$, randomly and replace them by $a$ design points selected randomly from $A$ to obtain an updated design ${\cal D}^*$.
\item[(b)] Compute the loss function $L_R({\cal D}^*)$. If $L_R({\cal D}^*)<L_R({\cal D}_0)$, then accept the updated design. Otherwise, accept the updated design ${\cal D}^*$ with probability $p_0=exp\{-[L_R({\cal D}^*)-L_R({\cal D}_0)]/T_0\}$. If the updated design is accepted, then let ${\cal D}_0={\cal D}^*$.
\end{itemize}

\item[] Step 4. Conduct the point interchange for the whole plots.
\begin{itemize}
\item[(a)] Swap design points between the $i$th whole plot and $l$th whole plot, where  $i\neq l$, to obtain an updated design ${\cal D}^*$.
\item[(b)] Same as  Step 3 (b).
\end{itemize}

\item[] Step 5. Perform the point exchange for the subplots.
\begin{itemize}
\item[(a)] For the $i$th whole plot, $i=1,\cdots,b$, randomly choose a number $e$ from set $\{1,\cdots,e_{n_i}\}$. Select $e$ subplot ${\bf t}_{ij_l}$, $l=1,\cdots,e$, randomly and replace them by $e$ design points selected randomly from $E_i$ to obtain an updated design ${\cal D}^*$.
\item[(b)] Same as Step 3 (b).
\end{itemize}

\item[] Step 6. Repeat Step 2 to Step 5 $N_T$ times and then go to Step 7.

\item[] Step 7. Reduce the temperature by a positive factor $f$ ($<1$), i.e., $T_0=fT_0$, and set $J=J+1$. If $J\leq M_0$, then go to Step 2. Otherwise, finish the process.

\end{itemize}

The final design ${\cal D}_0$ obtained from the algorithm can be consider as the $D$-optimal minimax split-plot design.

\iffalse
In the point-exchange algorithm is that the candidate set need to be generated. If the number of factors increases, the size of the candidate set becomes very large and hence cannot be generated. To overcome this problem, Goos (2007) proposed another algorithm, called the coordinate exchange algorithm, which is a candidate-free algorithm with the same spirit of the point-exchange algorithm.
The main difference between the two algorithm is that the design points for exchange or interchange in the coordinate exchange algorithm are not selected from the candidate set but randomly generated. Since the coordinate exchange algorithm is more efficiency for large number of factors, we adopt it for searching the $D$-optimal minimax split-plot design. The detail algorithm is given in the Appendix.

Two algorithms are commonly used for constructing optimal minimum $D$-optimal minimax designs. The second is the coordinate-exchange program introduced by Goos 2008 for constructing optimal split-plot designs.

Two algorithms are commonly used for searching optimal split-plot designs. The first algorithm is the point-exchange algorithm suggested by Goos (2001, 2003).
This algorithm has four main steps as follows.
\begin{itemize}
\item[] Step 1. Randomly generate an initial split-plot design from a candidate set. The initial design is record as the best design and the value of it's $L_D$ is calculated.
\item[] Step 2. Perform the point exchange for the subplot. For each whole plot ${\bf w}_i$, $i=1,\cdots,b$, the design point of the subplot ${\bf t}_{ij}$, $j=1,\cdots,n_i$, is substituted with a point in the candidate set of the subplot $G_i$.
\item[] Step 3. Conduct the point interchange for the whole plots. This step swaps design points between the $i$th whole plot and $l$th whole plot where $i\neq l$.
\item[] Step 4. Implements the point exchange for the whole plot. This step substitutes the design point of the $i$th whole plot, $i=1,\cdots,b$ by the point in the candidate set of the whole plot $B$.
\end{itemize}
In Step 2 to Step 4, the value of $L_D$ for the updated design is evaluated. If $L_D$ of the updated design is less than the value of $L_D$ of the best design, then the best design is replaced by the updated design and repeat Steps 2 to 4. If not, go back to Step 1.
Repeat this process for $t$ times, the final best design which has the greatest value of $L_D$ in the search is reported as the optimal robust split-plot design against model misspecification.

In the point-exchange algorithm is that the candidate set need to be generated. If the number of factors increases, the size of the candidate set becomes very large and hence cannot be generated. To overcome this problem, Goos (2007) proposed another algorithm, called the coordinate exchange algorithm, which is a candidate-free algorithm with the same spirit of the point-exchange algorithm.
The main difference between the two algorithm is that the design points for exchange or interchange in the coordinate exchange algorithm are not selected from the candidate set but randomly generated. Since the coordinate exchange algorithm is more efficiency for large number of factors, we adopt it for searching the $D$-optimal minimax split-plot design. The detail algorithm is given in the Appendix.
 \fi

\subsection{Update formulas}
If a matrix can be expressed in the form
\begin{equation*}\label{eq:M}
{\bf M}+{\bf Q}{\bf D}{\bf P},
\end{equation*}
then we can use the following formulas to calculate the determinant and the inverse of the matrix:
\begin{equation}\label{eq:up1}
|{\bf M}+{\bf Q}{\bf D}{\bf P}|=|{\bf M}||{\bf D}||{\bf D}^{-1}+{\bf P}{\bf M}^{-1}{\bf Q}|,
\end{equation}
and
\begin{equation}\label{eq:up2}
({\bf M}+{\bf Q}{\bf D}{\bf P})^{-1}={\bf M}^{-1}-{\bf M}^{-1}{\bf Q}({\bf D}^{-1}+{\bf P}{\bf M}^{-1}{\bf Q})^{-1}{\bf P}{\bf M}^{-1}.
\end{equation}
The second formula is called the Sherman-Morrison-Woodbury formula and the proofs of them were given by Harville (1997).

To apply the two formulas, let ${\bf M}_1={\bf X}_1'\boldsymbol\Sigma^{-1}{\bf X}_1$, ${\bf M}_2={\bf G}_1'\boldsymbol\Sigma^{-1}{\bf G}_1$, and ${\bf M}_3= {\bf G}_1'\boldsymbol\Sigma^{-2}{\bf G}_1$, where ${\bf G}_1={\bf X}_1{\bf V}_1^{-\frac{1}{2}}$. Then equation (\ref{eq:ld}) can be written as
\begin{equation}\label{eq:ld1}
L_R({\cal D})=\frac{1+N\alpha^2\phi_R({\cal D})}{\pi_R({\cal D})},
\end{equation}
where $\phi_R({\cal D})=\lambda_{max}({\bf M}_2^{-1}{\bf M}_3-{\bf M}_2)$ and $\pi_R({\cal D})=|{\bf M}_1|$.
Let ${\bf X}_{1i}$ and ${\bf G}_{1i}$ be the submatrices of ${\bf X}_1$ and ${\bf X}_1{\bf V}_1^{-\frac{1}{2}}$, respectively, corresponding to the $i$th whole plot. Let ${\bf f}({\bf w}_i,{\bf t}_{ij})$ and ${\bf g}({\bf w}_i,{\bf t}_{ij})$ be the rows of ${\bf X}_1$ and ${\bf X}_1{\bf V}_1^{-\frac{1}{2}}$, respectively, corresponding to the design point of the $j$th subplot in the $i$th whole plot.
If the point exchange or interchange process of the algorithm updates ${\bf M}_1$, ${\bf M}_2$, ${\bf M}_3$ to ${\bf M}_1^*$, ${\bf M}_2^*$, ${\bf M}_3^*$, and
$a$ ane $e$ are small, then formulas (\ref{eq:up1}) and (\ref{eq:up2}) are efficient for calculating the invariances and determinants of ${\bf M}_1^*$, ${\bf M}_2^*$, ${\bf M}_3^*$ for the updated design.
We summarize the results of the update formulas for Step 3 to Step 5 in Table~\ref{tb:up}.

\begin{table}[h]
\begin{center}
\caption{Update formulas for Step 3 to Step 5 of the algorithm}\label{tb:up}
\begin{tabular}{rl}
\hline
(a)&Update formulas\\
&$\begin{array}{l@{=}l}
|{\bf M}_1^{*}|&|{\bf M}_1||{\bf D}_1||{\bf D}_1^{-1}+{\bf P}_1{\bf M}_1^{-1}{\bf P}_1'|\\
{\bf M}_2^{*}&{\bf M}_2+{\bf P}_2{\bf D}_1{\bf P}_2'\\
{\bf M}_2^{*-1}&{\bf M}_2^{-1}-{\bf M}_2^{-1}{\bf P}_2'({\bf D}_1^{-1}+{\bf P}_2{\bf M}_2^{-1}{\bf P}_2')^{-1}{\bf P}_2{\bf M}_2^{-1}\\
{\bf M}_3^{*}&{\bf M}_3+{\bf P}_2{\bf D}_2{\bf P}_2'\\
\end{array}$\\
(b)&Step 3. Exchange for whole plots\\
&$\begin{array}{l@{=}l}
{\bf P}_1&({\bf X}_{1i_1}',\cdots,{\bf X}_{1i_a}',{\bf X}_{1i_1}^{*'},\cdots,{\bf X}_{1i_a}^{*'},{\bf X}_{1i_1}'{\bf 1}_{n_{i_1}},\cdots,{\bf X}_{1i_a}'{\bf 1}_{n_{i_a}},{\bf X}_{1i_1}^{*'}{\bf 1}_{n_{i_1}},\cdots,{\bf X}_{1i_a}^{*'}{\bf 1}_{n_{i_a}})'\\
{\bf D}_1&\sigma_\epsilon^{-2}diag(-{\bf I}_{n_{i_1}},\cdots,-{\bf I}_{n_{i_a}},{\bf I}_{n_{i_1}},\cdots,{\bf I}_{n_{i_a}},\frac{d}{1+dn_{i_1}},\cdots,\frac{d}{1+dn_{i_a}},-\frac{d}{1+dn_{i_1}},\cdots,-\frac{d}{1+dn_{i_a}})\\
{\bf P}_2&({\bf G}_{1i_1}',\cdots,{\bf G}_{1i_a}',{\bf G}_{1i_1}^{*'},\cdots,{\bf G}_{1i_a}^{*'},{\bf G}_{1i_1}'{\bf 1}_{n_{i_1}},\cdots,{\bf G}_{1i_a}'{\bf 1}_{n_{i_a}},{\bf G}_{1i_1}^{*'}{\bf 1}_{n_{i_1}},\cdots,{\bf G}_{1i_a}^{*'}{\bf 1}_{n_{i_a}})'\\
{\bf D}_2&\sigma_\epsilon^{-4}diag(-{\bf I}_{n_{i_1}},\cdots,-{\bf I}_{n_{i_a}},{\bf I}_{n_{i_1}},\cdots,{\bf I}_{n_{i_a}},\frac{2d+d^2n_{i_1}}{(1+dn_{i_1})^2},\cdots,\frac{2d+d^2n_{i_a}}{(1+dn_{i_a})^2},-\frac{2d+d^2n_{i_1}}{(1+dn_{i_1})^2},\cdots,-\frac{2d+d^2n_{i_a}}{(1+dn_{i_a})^2})\\
\end{array}$\\
(c)&Step 4. Interchange for whole plots\\
&$\begin{array}{l@{=}l}
{\bf P}_1&({\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1l}'{\bf 1}_{n_l},{\bf X}_{1i}^{*'}{\bf 1}_{n_i},{\bf X}_{1l}^{*'}{\bf 1}_{n_l})'\\
{\bf D}_1&\sigma_\epsilon^{-2}diag(\frac{d}{1+dn_i},\frac{d}{1+dn_l},-\frac{d}{1+dn_i},-\frac{d}{1+dn_l})\\
{\bf P}_2&({\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1l}'{\bf 1}_{n_l},{\bf G}_{1i}^{*'}{\bf 1}_{n_i},{\bf G}_{1l}^{*'}{\bf 1}_{n_l})'\\
{\bf D}_2&\sigma_\epsilon^{-4}diag(\frac{2d+d^2n_i}{(1+dn_i)^2},\frac{2d+d^2n_l}{(1+dn_l)^2},-\frac{2d+d^2n_i}{(1+dn_i)^2},-\frac{2d+d^2n_l}{(1+dn_l)^2})\\
\end{array}$\\
(d)&Step 5. Exchange for subplots\\
&$\begin{array}{l@{=}l}
{\bf P}_1&({\bf f}({\bf w}_i,{\bf t}_{ij_1}),\cdots,{\bf f}({\bf w}_i,{\bf t}_{ij_e}),{\bf f}({\bf w}_i,{\bf t}_{ij_1}^*),\cdots,{\bf f}({\bf w}_i,{\bf t}_{ij_1}^*),{\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1i}^{*'}{\bf 1}_{n_i})'\\
{\bf D}_1&\sigma_\epsilon^{-2}diag(-1,\cdots,-1,1,\cdots,1,\frac{d}{1+dn_i},-\frac{d}{1+dn_i})\\
{\bf P}_2&({\bf g}({\bf w}_i,{\bf t}_{ij_1}),\cdots,{\bf g}({\bf w}_i,{\bf t}_{ij_e}),{\bf g}({\bf w}_i,{\bf t}_{ij_1}^*),\cdots,{\bf g}({\bf w}_i,{\bf t}_{ij_e}^*),{\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1i}'{\bf 1}_{n_i})'\\
{\bf D}_2&\sigma_\epsilon^{-4}diag(-1,\cdots,-1,1,\cdots,1,\frac{2d+d^2n_{i}}{(1+dn_{i})^2},-\frac{2d+d^2n_{i}}{(1+dn_{i})^2})\\
\end{array}$\\
\hline
\end{tabular}
\end{center}
\end{table}

In Step 3, assume that the design point of whole-plot factors ${\bf w}_{i_l}$ is substituted by ${\bf w}_{i_l}^*$, $l=1,\cdots,a$. This point exchange for whole-plot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*}$, ${\bf G}_1^{*}$, ${\bf M}_1^{*}$, ${\bf M}_2^{*}$, and ${\bf M}_3^{*}$. Let ${\bf X}_{1i_l}^{*}$ and ${\bf G}_{1i_l}^{*}$ be the submatrices of ${\bf X}_1^{*}$ and ${\bf G}_1^{*}$, respectively, corresponding to the $i_l$th whole plot, $l=1,\cdots,a$.
Then equation (\ref{eq:ld1}) for the updated design in Step 3 can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*}|$, ${\bf M}_2^*$, ${\bf M}_2^{*-1}$, and ${\bf M}_3^{*}$, respectively, listed in Table~\ref{tb:up}~(a) where ${\bf P}_1$, ${\bf D}_1$, ${\bf P}_2$, and ${\bf D}_2$ are given in Table~\ref{tb:up}~(b).

In Step 4, assume that the design points of whole-plot factors ${\bf w}_{i}$ and ${\bf w}_{l}$ are interchanged. This point interchange for whole-plot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*}$, ${\bf G}_1^{*}$, ${\bf M}_1^{*}$, ${\bf M}_2^{*}$, and ${\bf M}_3^{*}$.
Let ${\bf X}_{1i}^{}$ and ${\bf X}_{1l}^{}$ (${\bf G}_{1i}^{}$ and ${\bf G}_{1l}^{}$) be the submatrices of ${\bf X}_1$ (${\bf G}_1$) corresponding to the $i$th and $l$th whole plots, respectively, and ${\bf X}_{1i}^{*}$ and ${\bf X}_{1l}^{*}$ (${\bf G}_{1i}^{*}$ and ${\bf G}_{1l}^{*}$) be the submatrices of ${\bf X}_1^{*}$ (${\bf G}_1^{*}$) corresponding to the $i$th and $l$th whole plots, respectively.
Then equation (\ref{eq:ld1}) for the updated design in Step 4 can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*}|$, ${\bf M}_2^*$, ${\bf M}_2^{*-1}$, and ${\bf M}_3^{*}$, respectively, listed in Table~\ref{tb:up}~(a) where ${\bf P}_1$, ${\bf D}_1$, ${\bf P}_2$, and ${\bf D}_2$ are given in Table~\ref{tb:up}~(c).

In Step 5, assume that the design point of subplot factors ${\bf t}_{ij_l}$ is substituted by ${\bf t}_{ij_l}^*$, $l=1,\cdots,e$. This point exchange for subplot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*}$, ${\bf G}_1^{*}$, ${\bf M}_1^{*}$, ${\bf M}_2^{*}$, and ${\bf M}_3^{*}$. Let ${\bf X}_{1i_l}^{*}$ and ${\bf G}_{1i_l}^{*}$ be the submatrices of ${\bf X}_1^{*}$ and ${\bf G}_1^{*}$, respectively, corresponding to the $i$th whole plot and ${\bf f}({\bf w}_i,{\bf t}_{ij_l}^*)$ and ${\bf g}({\bf w}_i,{\bf t}_{ij_l}^*)$ be the rows of ${\bf X}_1^{*}$ and ${\bf G}_1^{*}$, respectively, corresponding to the $j_l$th design point in the $i$th whole plot, $l=1,\cdots,e$. Then equation (\ref{eq:ld1}) for the updated design in Step 5 can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*}|$, ${\bf M}_2^*$, ${\bf M}_2^{*-1}$, and ${\bf M}_3^{*}$, respectively, listed in Table~\ref{tb:up}~(a) where ${\bf P}_1$, ${\bf D}_1$, ${\bf P}_2$, and ${\bf D}_2$ are given in Table~\ref{tb:up}~(d).

\iffalse
In Step 2, assume that the design point of whole-plot factors ${\bf w}_{i}$ is substituted by ${\bf w}_{i}^*$. This point exchange for whole-plot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*(wp)}$, ${\bf G}_1^{*(wp)}$, ${\bf M}_1^{*(wp)}$, ${\bf M}_2^{*(wp)}$, and ${\bf M}_3^{*(wp)}$. Let ${\bf X}_{1i}^{*(wp)}$ and ${\bf G}_{1i}^{*(wp)}$ be the submatrices of ${\bf X}_1^{*(wp)}$ and ${\bf G}_1^{*(wp)}$, respectively, corresponding to the $i$th whole plot. Then equation (\ref{eq:ld1}) for the updated design can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*(wp)}|$, ${\bf M}_2^{*-1(wp)}$, and ${\bf M}_3^{*(wp)}$, respectively, as follows
\begin{center}
\begin{tabular}{l@{=}l}
$|{\bf M}_1^{*(wp)}|$&$|{\bf M}_1||{\bf D}_1||{\bf D}_1^{-1}+{\bf P}_1^{-1}{\bf M}_1^{-1}{\bf P}_1'|$,\\
${\bf M}_2^{*-1(wp)}$&${\bf M}_2^{-1}-{\bf M}_2^{-1}{\bf P}_2'({\bf D}_1^{-1}+{\bf P}_2{\bf M}_2^{-1}{\bf P}_2')^{-1}{\bf P}_2{\bf M}_2^{-1}$,\\
${\bf M}_3^{*(wp)}$&${\bf M}_3+{\bf P}_2{\bf D}_2{\bf P}_2'$,\\
\end{tabular}
\end{center}
where
${\bf P}_1=({\bf X}_{1i}',{\bf X}_{1i}^{*(wp)'},{\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1i}^{*(wp)'}{\bf 1}_{n_i})'$,
${\bf D}_1=\sigma_\epsilon^{-2}diag(-{\bf I}_{n_i},{\bf I}_{n_i},\frac{d}{1+dn_i},-\frac{d}{1+dn_i})$,
${\bf P}_2=({\bf G}_{1i}',$ ${\bf G}_{1i}^{*(wp)'},{\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1i}^{*(wp)'}{\bf 1}_{n_i})'$, and
${\bf D}_2=\sigma_\epsilon^{-4}diag(-{\bf I}_{n_i},{\bf I}_{n_i},-\frac{2d+d^2n_i}{(1+dn_i)^2},\frac{2d+d^2n_i}{(1+dn_i)^2})$.

In Step 3, assume that the design points of whole-plot factors ${\bf w}_{i}$ and ${\bf w}_{l}$ are interchanged. This point interchange for whole-plot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*(ww)}$, ${\bf G}_1^{*(ww)}$, ${\bf M}_1^{*(ww)}$, ${\bf M}_2^{*(ww)}$, and ${\bf M}_3^{*(ww)}$.
Let ${\bf X}_{1i}^{(ww)}$ and ${\bf X}_{1l}^{(ww)}$ (${\bf G}_{1i}^{(ww)}$ and ${\bf G}_{1l}^{(ww)}$) be the submatrices of ${\bf X}_1^{(wp)}$ (${\bf G}_1^{(wp)}$) corresponding to the $i$th and $l$th whole plots, respectively, and ${\bf X}_{1i}^{*(ww)}$ and ${\bf X}_{1l}^{*(ww)}$ (${\bf G}_{1i}^{*(ww)}$ and ${\bf G}_{1l}^{*(ww)}$) be the submatrices of ${\bf X}_1^{*(ww)}$ (${\bf G}_1^{*(ww)}$) corresponding to the $i$th and $l$th whole plots, respectively.
Equation (\ref{eq:ld1}) for the updated design can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*(ww)}|$, ${\bf M}_2^{*-1(ww)}$, and ${\bf M}_3^{*(ww)}$, respectively, as follows
\begin{center}
\begin{tabular}{l@{=}l}
$|{\bf M}_1^{*(ww)}|$&$|{\bf M}_1||{\bf D}_3||{\bf D}_3^{-1}+{\bf P}_3^{-1}{\bf M}_1^{-1}{\bf P}_3'|$,\\
${\bf M}_2^{*-1(ww)}$&${\bf M}_2^{-1}-{\bf M}_2^{-1}{\bf P}_4'({\bf D}_3^{-1}+{\bf P}_4{\bf M}_2^{-1}{\bf P}_4')^{-1}{\bf P}_4{\bf M}_2^{-1}$,\\
${\bf M}_3^{*(ww)}$&${\bf M}_3+{\bf P}_4{\bf D}_4{\bf P}_4'$,\\
\end{tabular}
\end{center}
where
${\bf P}_3=({\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1l}'{\bf 1}_{n_l},{\bf X}_{1i}^{*(ww)'}{\bf 1}_{n_i},{\bf X}_{1l}^{*(ww)'}{\bf 1}_{n_l})'$,
${\bf D}_3=\sigma_\epsilon^{-2}diag(\frac{d}{1+dn_i},\frac{d}{1+dn_i},-\frac{d}{1+dn_i},-\frac{d}{1+dn_i})$,
${\bf P}_6=({\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1l}'{\bf 1}_{n_l},{\bf G}_{1i}^{*(ww)'}{\bf 1}_{n_i},{\bf G}_{1l}^{*(ww)'}{\bf 1}_{n_l})'$, and
${\bf D}_4=\sigma_\epsilon^{-4}diag(-\frac{2d+d^2n_i}{(1+dn_i)^2},-\frac{2d+d^2n_i}{(1+dn_i)^2},\frac{2d+d^2n_i}{(1+dn_i)^2},\frac{2d+d^2n_i}{(1+dn_i)^2})$.

In Step 4, assume that the design points of subplot factors ${\bf t}_{ij}$ is substituted by ${\bf t}_{ij}^*$. This point exchange for subplot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*(sp)}$, ${\bf G}_1^{*(sp)}$, ${\bf M}_1^{*(sp)}$, ${\bf M}_2^{*(sp)}$, and ${\bf M}_3^{*(sp)}$. Let ${\bf X}_{1i}^{*(sp)}$ and ${\bf G}_{1i}^{*(sp)}$ be the submatrices of ${\bf X}_1^{*(sp)}$ and ${\bf G}_1^{*(sp)}$, respectively, corresponding to the $i$th whole plot and ${\bf f}({\bf w}_i,{\bf t}_{ij}^*)$ and ${\bf g}({\bf w}_i,{\bf t}_{ij}^*)$ be the rows of ${\bf X}_1^{*(sp)}$ and ${\bf G}_1^{*(sp)}$, respectively, corresponding to the $j$th design point in the $i$th whole plot. Applying the update formulas (\ref{eq:up1}) and (\ref{eq:up2}), equation (\ref{eq:ld1}) for the updated design can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*(sp)}|$, ${\bf M}_2^{*-1(sp)}$, and ${\bf M}_3^{*(sp)}$, respectively, as follows
\begin{center}
\begin{tabular}{l@{=}l}
$|{\bf M}_1^{*(sp)}|$&$|{\bf M}_1||{\bf D}_5||{\bf D}_5^{-1}+{\bf P}_5^{-1}{\bf M}_1^{-1}{\bf P}_5'|$,\\
${\bf M}_2^{*-1(sp)}$&${\bf M}_2^{-1}-{\bf M}_2^{-1}{\bf P}_6'({\bf D}_5^{-1}+{\bf P}_6{\bf M}_2^{-1}{\bf P}_6')^{-1}{\bf P}_6{\bf M}_2^{-1}$,\\
${\bf M}_3^{*(sp)}$&${\bf M}_3+{\bf P}_6{\bf D}_6{\bf P}_6'$,\\
\end{tabular}
\end{center}
where
${\bf P}_5=({\bf f}({\bf w}_i,{\bf t}_{ij}),{\bf f}({\bf w}_i,{\bf t}_{ij}^*),{\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1i}^{*(sp)'}{\bf 1}_{n_i})'$,
${\bf D}_5=\sigma_\epsilon^{-2}diag(-1,1,\frac{d}{1+dn_i},-\frac{d}{1+dn_i})$,
${\bf P}_6=({\bf g}({\bf w}_i,{\bf t}_{ij}),$ ${\bf g}({\bf w}_i,{\bf t}_{ij}^*),{\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1i}^{*(sp)'}{\bf 1}_{n_i})'$, and
${\bf D}_6=\sigma_\epsilon^{-4}diag(-1,1,-\frac{2d+d^2n_i}{(1+dn_i)^2},\frac{2d+d^2n_i}{(1+dn_i)^2})$.
 \fi

\iffalse
\begin{table}
\begin{center}
\begin{tabular}{l}
\hline
update formula for point exchange of the whole plots\\
$\begin{array}{ll}
{\bf P}_3=&({\bf X}_{1i_1}',\cdots,{\bf X}_{1i_w}',{\bf X}_{1i_1}^{*(wp)'},\cdots,{\bf X}_{1i_w}^{*(wp)'},\\&{\bf X}_{1i_1}'{\bf 1}_{n_{i_1}},\cdots,{\bf X}_{1i_w}'{\bf 1}_{n_{i_w}},{\bf X}_{1i_1}^{*(wp)'}{\bf 1}_{n_{i_1}},\cdots,{\bf X}_{1{i_w}}^{*(wp)'}{\bf 1}_{n_{i_w}})'\\
{\bf D}_3=&\sigma_\epsilon^{-2}diag(-{\bf I}_{n_i},{\bf I}_{n_i},\frac{d}{1+dn_i},-\frac{d}{1+dn_i})\\
{\bf P}_4=&({\bf G}_{1i}',$ ${\bf G}_{1i}^{*(wp)'},{\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1i}^{*(wp)'}{\bf 1}_{n_i})'\\
{\bf D}_4=&\sigma_\epsilon^{-4}diag(-{\bf I}_{n_i},{\bf I}_{n_i},\frac{d}{1+dn_i}(\frac{dn_i}{1+dn_i}-2),-\frac{d}{1+dn_i}(\frac{dn_i}{1+dn_i}-2))
\end{array}$
\end{tabular}
\end{center}
\end{table}

In Step 2, assume that the design point of whole-plot factors ${\bf w}_{i_l}$ is substituted by ${\bf w}_{i_l}^*$, $l=1,\cdots,w$. This point exchange for whole-plot factors updates ${\bf X}_1$, ${\bf G}_1$, ${\bf M}_1$, ${\bf M}_2$, and ${\bf M}_3$ to ${\bf X}_1^{*(wp)}$, ${\bf G}_1^{*(wp)}$, ${\bf M}_1^{*(wp)}$, ${\bf M}_2^{*(wp)}$, and ${\bf M}_3^{*(wp)}$. Let ${\bf X}_{1i}^{*(wp)}$ and ${\bf G}_{1i}^{*(wp)}$ be the submatrices of ${\bf X}_1^{*(wp)}$ and ${\bf G}_1^{*(wp)}$, respectively, corresponding to the $i$th whole plot. Applying the update formula ()(), equation (\ref{eq:ld1}) for the updated design can be calculated by replacing $|{\bf M}_1|$, ${\bf M}_2^{-1}$, and ${\bf M}_3$ by $|{\bf M}_1^{*(wp)}|$, ${\bf M}_2^{*-1(wp)}$, and ${\bf M}_3^{*(wp)}$, respectively, as follows.
\begin{center}
\begin{tabular}{ll@{=}l}
&$|{\bf M}_1^{*(wp)}|$&$|{\bf M}_1||{\bf D}_3||{\bf D}_3^{-1}+{\bf P}_3^{-1}{\bf M}_1^{-1}{\bf P}_3'|$,\\
&${\bf M}_2^{*-1(wp)}$&${\bf M}_2^{-1}-{\bf M}_2^{-1}{\bf P}_4'({\bf D}_3^{-1}+{\bf P}_4{\bf M}_2^{-1}{\bf P}_4')^{-1}{\bf P}_4{\bf M}_2^{-1}$,\\
&${\bf M}_3^{*(wp)}$&${\bf M}_3+{\bf P}_4{\bf D}_3{\bf P}_4'$,\\
\end{tabular}
\end{center}
where
${\bf P}_3=({\bf X}_{1i}',{\bf X}_{1i}^{*(wp)'},{\bf X}_{1i}'{\bf 1}_{n_i},{\bf X}_{1i}^{*(wp)'}{\bf 1}_{n_i})'$, ${\bf D}_3=\sigma_\epsilon^{-2}diag(-{\bf I}_{n_i},{\bf I}_{n_i},\frac{d}{1+dn_i},-\frac{d}{1+dn_i})$, ${\bf P}_4=({\bf G}_{1i}',$ ${\bf G}_{1i}^{*(wp)'},{\bf G}_{1i}'{\bf 1}_{n_i},{\bf G}_{1i}^{*(wp)'}{\bf 1}_{n_i})'$, and ${\bf D}_4=\sigma_\epsilon^{-4}diag(-{\bf I}_{n_i},{\bf I}_{n_i},\frac{d}{1+dn_i}(\frac{dn_i}{1+dn_i}-2),-\frac{d}{1+dn_i}(\frac{dn_i}{1+dn_i}-2))$.
\fi

\section{Examples}\label{se:ex}
We apply the construction algorithm developed in Section~\ref{se:al} and use the general form of the loss function for $D$-optimal minimax criterion introduced in Section~\ref{se:ld} to obtain the robust split-plot designs for model misspecification. Two examples are given to demonstrate our methods. The first example is for two-level split-plot designs and the second example is for mixed-level split-plot designs.

\begin{table}[h]
\caption{$D$-optimal split-plot design (${\cal D}_1$ with $\alpha=0$) and $D$-optimal minimax split-plot design (${\cal D}_2$ with $\alpha=1$) for $\sigma_\epsilon^2=\sigma_\gamma^2=1$.}\label{tb:ex1}
\begin{center}
\begin{tabular}{cc}
$\begin{array}{c|rrrrr}
\multicolumn{6}{c}{\mbox{Design }{\cal D}_1}\\
\hline
WP	&	F_1	&	F_2	&	F_3	&	F_4	&	F_5	\\
\hline
1	&	1	&	-1	&	1	&	1	&	1	\\
	&	1	&	-1	&	-1	&	-1	&	-1	\\
	&	1	&	-1	&	-1	&	-1	&	1	\\
	&	1	&	-1	&	1	&	1	&	-1	\\
\hline
2	&	-1	&	1	&	1	&	1	&	-1	\\
	&	-1	&	1	&	1	&	-1	&	-1	\\
	&	-1	&	1	&	-1	&	1	&	1	\\
	&	-1	&	1	&	-1	&	-1	&	1	\\
\hline
3	&	-1	&	-1	&	-1	&	-1	&	-1	\\
	&	-1	&	-1	&	1	&	-1	&	1	\\
	&	-1	&	-1	&	-1	&	1	&	-1	\\
	&	-1	&	-1	&	1	&	1	&	1	\\
\hline
4	&	1	&	1	&	1	&	-1	&	-1	\\
	&	1	&	1	&	-1	&	1	&	1	\\
	&	1	&	1	&	-1	&	1	&	-1	\\
\hline
\multicolumn{6}{l}{\phi_R({\cal D}_1)=.6733}\\
\multicolumn{6}{l}{\pi_R({\cal D}_1)^{1/(1+p)}=6.7468}\\
\multicolumn{6}{l}{L_R({\cal D}_1)^{1/(1+p)}=.2188}\\
\end{array}$
&
$\begin{array}{c|rrrrr}
\multicolumn{6}{c}{\mbox{Design }{\cal D}_2}\\
\hline
WP	&	F_1	&	F_2	&	F_3	&	F_4	&	F_5	\\
\hline
1	&	1	&	1	&	-1	&	1	&	1	\\
	&	1	&	1	&	1	&	-1	&	-1	\\
	&	1	&	1	&	1	&	1	&	-1	\\
	&	1	&	1	&	-1	&	-1	&	1	\\
\hline
2	&	-1	&	-1	&	-1	&	-1	&	1	\\
	&	-1	&	-1	&	1	&	-1	&	-1	\\
	&	-1	&	-1	&	1	&	1	&	1	\\
	&	-1	&	-1	&	-1	&	1	&	-1	\\
\hline
3	&	1	&	-1	&	-1	&	-1	&	-1	\\
	&	1	&	-1	&	1	&	-1	&	1	\\
	&	1	&	-1	&	-1	&	1	&	-1	\\
	&	1	&	-1	&	1	&	1	&	1	\\
\hline
4	&	-1	&	1	&	-1	&	-1	&	1	\\
	&	-1	&	1	&	1	&	1	&	-1	\\
	&	-1	&	1	&	-1	&	1	&	-1	\\
\hline
\multicolumn{6}{l}{\phi_R({\cal D}_2)=.6323}\\
\multicolumn{6}{l}{\pi_R({\cal D}_2)^{1/(1+p)}=6.7339}\\
\multicolumn{6}{l}{L_R({\cal D}_2)^{1/(1+p)}=.2176}\\
\end{array}$
\end{tabular}
\end{center}
\end{table}

\iffalse
\begin{figure
\centering
\begin{minipage}{.4\textwidth}
\includegraphics[width=\textwidth]{a=0,d=1-1.pdf}
\caption{Design ${\cal D}_1$}
\end{minipage}
\begin{minipage}{.4\textwidth}
\includegraphics[width=\textwidth]{a=1,d=1-1.pdf}
\caption{Design ${\cal D}_2$}
\end{minipage}
\end{figure}
\fi

\textbf{Example 1.}
Consider to construct a split-plot design with fifteen runs and five two-level factors $F_1,\cdots,F_5$, where the two levels of factors are coded as $(-1,+1)$.
Suppose that the first two factors $F_1$ and $F_2$ are hard-to-change factors arranged into four whole plots ($b=4$) and the last three factors $F_3,$ $F_4$, $F_5$ are easy-to-change factors where the numbers of subplots in the $i$th whole plot are $n_i=4$ for $i=1,2,3$ and $n_i=3$ for $i=4$.
If we want to investigate all the main effects, the interaction of $F_1$ and $F_2$, and the interaction of $F_1$ and $F_3$, then the requirement set is $R=\{x_1,x_2,x_3,x_4,x_5,x_1x_2,x_1x_3\}$, where $x_i$ is the main effect of $F_i$, $i=1,\cdots,5$.
The number of effects in the requirement set $R$ is $p=7$ and $N$ in this case is $2^5=32$.
Let the candidate set of whole plots $A$ be the $2^2$ full factorial design and the candidate set of subplots $E$ be the $2^3$ full factorial design.
We apply the algorithm by setting $T_0=.001$, $M_0=50$, $N_T=100$, $a_b=3$, $e_{n_i}=3$, and $f=.8$ to search for the $D$-optimal minimax split-plot design ($\alpha=1$) and the $D$-optimal split-plot design ($\alpha=0$) with $\sigma_\epsilon=1$ and $\sigma_\gamma=1$.
The update formulas given in Table~\ref{tb:up} are used for increasing the computing speed for obtaining the loss function of the updated designs, $L_R({\cal D}^*)$.
Table~\ref{tb:ex1} lists two optimal split-plot designs.
The $(1+p)$th root of the determinant of the information matrix of ${\cal D}_1$ with respect to the requirement set $R$ is $\pi_R({\cal D}_1)^{1/(1+p)}=6.7468$. Since $\pi_R({\cal D}_1)^{1/(1+p)}$ is minimum among all of the designs we construct, ${\cal D}_1$ is the $D$-optimal split-plot design.
However, if the fitted model is misspecified and there exist small departures from the underlying true model with $\alpha=1$,
then the $(1+p)$th root of the loss function of ${\cal D}_1$ is $L_R({\cal D}_1)^{1/(1+p)}=.2188$, which is greater than $L_R({\cal D}_2)^{1/(1+p)}=.2176$.
It implies that design ${\cal D}_1$ has higher bias of the estimation than design ${\cal D}_2$.
Since $L_R({\cal D}_2)^{1/(1+p)}$ is minimum among all of the designs we construct, ${\cal D}_2$ is the $D$-optimal minimax split-plot design. Therefore, when  $\alpha=1$, design ${\cal D}_2$ is the optimally robust split-plot design for model misspecification.
We look into the allocations of the design points in ${\cal D}_1$ and ${\cal D}_2$ and show their structures in Figure~\ref{fi:ex1}.
We find that, in each whole plot with $n_i=4$ in ${\cal D}_1$, the connection of two points of the subplots can be parallel to the connection of the other two points. However, for the whole plot $(-1,-1)$ in ${\cal D}_2$, it is impossible to connect two points which is parallel to the connection of the other two points. The different structures between the two designs result in less bias of the estimation for design ${\cal D}_2$ when the model is misspecified with $\alpha=1$.

\iffalse
while those of design $D_2$ are $\phi_R(D_2)=.6323$ and $\pi_R(D_2)=4,227,858$.
Design $D_1$ has $\phi_R(D_1)=.6323$ and $\pi_R(D_1)=4,227,858$. Design $D_2$ has $\phi_R(D_2)=.6733$ and $\pi_R(D_2)=4,294,967$,
The $\phi_2$ of $D_2$ is maximum and hence is $D$-optimal split-plot design. However, the $\phi_1$ of $D_1$ is less than $\phi_1$ of $D_2$. In there exist a differ with $\alpha\leq .1266$, $D_2$ is also the $D$-optimal minimax split-plot design. When $\alpha> .1266$, $D_1$ is also the $D$-optimal minimax split-plot design.
\fi

\begin{figure}
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{a=0,d=1-1.pdf}
  \caption{Design ${\cal D}_1$}
  \label{fig:sfig1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{a=1,d=1-1.pdf}
  \caption{Design ${\cal D}_2$}
  \label{fig:sfig2}
\end{subfigure}
\caption{Structures of ${\cal D}_1$ and ${\cal D}_2$}
\label{fi:ex1}
\end{figure}

\begin{table}
\caption{Designs ${\cal D}_3$ and ${\cal D}_4$ in Example 2.}
\begin{center}
\begin{tabular}{cc}
$\begin{array}{c|rrr}
\multicolumn{4}{c}{\mbox{Design }{\cal D}_3}\\
\hline
WP	&	F_1	&	F_2	&	F_3	\\
\hline
1	&	-1	&	0	&	0	\\
	&	-1	&	1	&	1	\\
\hline
2	&	1	&	0	&	2	\\
	&	1	&	0	&	1	\\
\hline
3	&	-1	&	2	&	2	\\
	&	-1	&	2	&	0	\\
	&	-1	&	0	&	1	\\
\hline
4	&	1	&	2	&	1	\\
	&	1	&	1	&	2	\\
	&	1	&	1	&	0	\\
\hline
\multicolumn{4}{l}{\phi_R({\cal D}_3)=.9074}\\
\multicolumn{4}{l}{\pi_R({\cal D}_3)^{1/(1+p)}=4.5472}\\
\multicolumn{4}{l}{L_R({\cal D}_3)^{1/(1+p)}=.2925}\\
\end{array}$
&
$\begin{array}{c|rrr}
\multicolumn{4}{c}{\mbox{Design }{\cal D}_4}\\
\hline
WP	&	F_1	&	F_2	&	F_3	\\
\hline
1	&	-1	&	2	&	2	\\
	&	-1	&	1	&	1	\\
\hline
2	&	1	&	2	&	0	\\
	&	1	&	0	&	2	\\
\hline
3	&	-1	&	1	&	0	\\
	&	-1	&	0	&	2	\\
	&	-1	&	2	&	1	\\
\hline
4	&	1	&	2	&	1	\\
	&	1	&	0	&	0	\\
	&	1	&	1	&	2	\\
\hline
\multicolumn{4}{l}{\phi_R({\cal D}_4)=.6667}\\
\multicolumn{4}{l}{\pi_R({\cal D}_4)^{1/(1+p)}=4.5472}\\
\multicolumn{4}{l}{L_R({\cal D}_4)^{1/(1+p)}=.2842}\\
\end{array}$
\end{tabular}
\end{center}
\end{table}

\textbf{Example 2.}
Consider an experiment with factors $F_1$, $F_2$, and $F_3$, where $F_1$ has two levels and $F_2$ and $F_3$ have three levels. Assume that $F_1$ is a hard-to-change factor arranged into four whole plots. The other two factors $F_2$ and $F_3$ are easy to change and the numbers of subplots in the $i$th whole plot are $n_i=2$ for $i=1,2$ and $n_i=3$ for $i=3,4$.
If we are interested in estimating all the main effects, the interaction between $F_1$ and $F_2$, and the interaction between $F_1$ and $F_3$, then the requirement set is $R=\{x_1,x_{2L},x_{2Q},x_{3L},x_{3Q},x_1x_{2L},x_1x_{2Q},x_1x_{3L},x_1x_{3Q}\}$, where $x_1$ is the main effect of $F_1$ and $x_{iL}$ and $x_{iQ}$ are the linear and quadratic components of the main effect of factor $F_i$, $i=2,3$.
The two levels of $F_1$ are coded as $(-1,+1)$ and the three levels $(0,1,2)$ of $F_i$ are coded as $\sqrt \frac{3}{2}(-1,0,+1)$ for $x_{iL}$ and $\sqrt\frac{1}{2}(+1,-2,+1)$ for $x_{iQ}$, $i=2,3$.
The candidate set of the whole plots $A$ is the $2^1$ full factorial design and the candidate set of the subplots $E$ is the $3^2$ full factorial design.
We apply the algorithm by setting $T_0=.001$, $M_0=50$, $N_T=100$, $a_b=3$, $e_{n_i}=2$, and $f=.8$ to construct the $D$-optimal minimax split-plot design with $\sigma_\epsilon^2=1$, $\sigma_\gamma^2=1$, and $\alpha=1$.
Table 2 lists two designs ${\cal D}_3$ and ${\cal D}_4$.
Both ${\cal D}_3$ and ${\cal D}_4$ have the same $(1+p)$th root of the determinants of the information matrices, $\pi_R({\cal D}_3)^{1/(1+p)}=\pi_R({\cal D}_4)^{1/(1+p)}=4.5472$. Since $\pi_R({\cal D}_3)^{1/(1+p)}$ and $\pi_R({\cal D}_4)^{1/(1+p)}$ are maximum among all of the designs, both ${\cal D}_3$ and ${\cal D}_4$ are $D$-optimal split-plot designs.
However, if model is misspecified with $\alpha=1$, then design ${\cal D}_3$ has $\phi_R({\cal D}_3)=0.9074$ and ${\cal D}_4$ has $\phi_R({\cal D}_4)=0.6667$. The value of $\phi_R({\cal D}_4)$ is minimum among all of the designs. Therefore, ${\cal D}_4$ is the $D$-optimal minimax split-plot design.
Since $\pi_R({\cal D}_4)^{1/(1+p)}$ is maximum and $\phi_R({\cal D}_4)$ is minimum, the value of $L_R({\cal D}_4)$ does not depend on $\alpha$ and is minimum among all of the designs.
Therefor, ${\cal D}_4$ is the optimally robust split-plot design with or without model misspecification.
This example shows that the $D$-optimal minimax split-plot design could be also the $D$-optimal split-plot design.

\iffalse
to construct a mixed-level split-plot design with one two-level whole-plot factor coded by $(-1,+1)$ for levels (0,1) and two three-level subplot factors where the linear effects and the quadratic effects are coded as $(-1,0,+1)$ and $(+1,2,+1)$, respectively, for levels $(0,1,2)$. This design has for whole plots and
Assume that the requirement set is $R=\{x_1,x_{2L},x_{2Q},x_{3L},x_{3Q},x_1x_{2L},x_1x_{2Q}\}$, where $x_1$ is the main effect of $F_3$ and $x_{iL}$ and $x_{iQ}$ are the linear and quadratic components of the main effect of factor $F_i$, $i=2,3$.

The $D$-optimal minimax designs with $n=10$ and $(n_1,n_2,n_3,n_4)=(2,3,2,3)$ are constructed. Table 2 represents design $D_1$ and $D_2$, where $D_1$ is a $D$-optimal design but
\fi

\section{Conclusions and remarks}\label{se:co}
Many approaches for constructing optimal split-plot designs could be found in literature. However, the optimal designs obtained by these methods might be unrobust for model misspecification.
If there exist significant effects that are not included in the model, then the estimation of effects could be highly biased.
In this paper, we take the model misspecification into account.
We extend the application of the $D$-optimal minimax criterion to the split-plot design and provide a general form of the loss function for the criterion.
This general form of the loss function can be used for finding the optimal design for split-plot experiments and complete randomized experiments with or without model misspecification.
By combining the anneal algorithm and the point-exchange algorithm, we develop a new construction algorithm to efficiently obtain the robust split-plot design for model misspecification.

There exist two articles that are related to our work.
The first article by Smucker et al. (2012) provides a method to obtain the model-robust designs for split-plot experiments. The authors argued that many methods in literature rely on the a priori assumption that the form of the regression function is known. They relaxed this assumption by allowing a set of model forms to be specified. This method uses a scaled product of $D$-optimal criterion to produce designs that account for all models in the set.
This method is innovative and the optimal split-plot design constructed by it is robust if the specified set of model forms includes the true model.
However, in practice the underlying true model is usually complicated and unknown. It is not easy to specify a set that including the unknown true model.
In this paper, we relax this constraint for the specified set.
Our method dose not rely on the knowledge of the true model and allows the fitted model differing from the unknown true model.
By minimizing the loss function, the optimal split-plot design obtained by our method can simultaneously control the variances/covariances and the bias of the estimation and hence is robust for model misspecification.

Another article by Mann et al. (2014) is close to our work. Both of their method and our method use the $D$-optimal minimax criterion but with different model setting. The method proposed by Mann et al. (2014) assumed that the block effects are fixed while our method assumes that the block effects are random.
The former is usually used for finding the robust block design with model misspecification and the later is used for obtaining the robust split-plot design.

In summary, a good split-plot design should be able to control both the variances/covariances and the bias of the estimation. The method we propose can achieve this goal and construct the robust split-plot design for model misspecification.

\iffalse
\section*{Acknowledgments}
The authors gratefully acknowledge the helpful comments by the Editor and referees. The research of Chang-Yun Lin was supported by the National Science Council of Taiwan (Grant no. NSC 103-2118-M-005-003-MY2). For all correspondence, please contact Dr. Chang-Yun Lin.
\fi

\iffalse

redpajama_set_name: RedPajamaArXiv


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Ashley Edwards Walker
Genius New Postpartum Underwear Makes Recovery Way Easier
Mesh panties are about to be a thing of the past.
When preparing for birth, moms-to-be are told to stock up postpartum essentials like maxi pads, ice packs and mesh underwear from the hospital—but trying to layer them all together to soothe your tender parts is a ridiculously difficult process recovering moms have been forced to contend with for far too long. So three Chicago moms decided to do something about it.
To ensure women get the postpartum care they need, Eden Laurin, Aubrey Howard and Mia Clarke founded a new company called Nyssa. The first product they're launching is a patent-pending postpartum underwear, designed to provide comfort following vaginal and c-section births.
Friends for a decade, the four women all gave birth within the same year and were shocked that, after their bodies had gone through labor and delivery, they were sent home with nothing more than a few pads and an ice pack to DIY their own postpartum underwear.
"You're physically wrecked, and yet you're responsible for taking care of a tiny human, and you're trying to deal with all your own emotions, and the last thing you should have to do is try to DYI or hack a solution together for your own physical recovery," Howard told CBS Chicago.
After spending a year chatting with other moms and designers, they came up with Fourthwear Underwear, which can be worn high up over the stomach area or folded down lower depending on where the wearer is in her recovery process. It also features—get this!—a slot to place an ice or heat pack in the vaginal/perineal area or at the site of a c-section incision, a design addition that any mom will tell you is pretty darn genius.
"The product can be worn in multiple ways, depending on your comfort level," Howard said. "If you're a little bit bigger, it's really comfortable to wear it just over your belly. If you've lost a lot of weight, you can really easily fold it down."
In addition to the underwear, the co-founders have also launched a podcast called " The Unmentionables." With Clarke as host, the series aims to spark conversations around what moms and babies experience during the fourth trimester, a term that refers to the first three months after giving birth. Recent episodes featured Sara Reardon, a pelvic floor therapist who runs the Instagram account The Vagina Whisperer; Dr. Rpua Mahadevan, an integrative pediatrician, Trystan Angel Reese; a trans parent and activist; and Lauren Smith Brody, author and founder of The Fifth Trimester.
"We really want to change the narrative around the postpartum experience, and we really want to build a community of people who are supportive of each other in this incredibly transformational time," Howard said.
The postpartum recovery process is way harder than it has to be, and brands are finally starting to figure out what they can do to change that. Earlier this year, Frida Mom launched another postpartum recovery product line, featuring an upside-down peri-bottle, disposable cotton underwear, medicated pad liners in the shape and size of your pad, perineal healing foam instead of spray, and several other items to help women recover post-birth.
PHOTO: Courtesy Nyssa
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Greetings my dear friends and sisters in Christ!
These past weeks have been very difficult for me. Not only have I found my walk with Jesus struggling (on my own account), but I have faced loneliness, bitterness, pain, and anger seemingly all at once. Who knew (at least I didn't) that I could have such strong feelings? I am blessed because I have had relatively few hurts in my life, but hurt still hurts. And when I am not in close contact with my Heavenly Father, these trials are even more difficult to face and conquer. I only write this to deeply encourage you to always walk in close contact with Jesus. He alone can bear our most burdensome and sometimes silly fears and struggles. I'm not sure of all I am supposed to learn through these past experiences except that God is good, He is in control, and He offers forgiveness. . .as should I. Please pray for me as I go through this time, that I may come out closer to my Jesus and a shining example of what His child ought to be!
I think I will apply Romans 12 to my heart and life. It is such an encouraging and challenging chapter!
On a different, but exciting note, I have joined up with Emily Elizabeth from My Life Belongs to Him , purposing to have a productive summer. This is such a blessing to me as I try to stay accountable and use my summer for the glory of God. I'm a little behind for week two, but I tried to be productive. First, I washed the windows! It is such a blessing to have clean windows and I am so thankful my Momma taught me. :o) A second thing is that I started a new jumper (please disregard the fact that it has remained untouched for several days. . .). So with that said, I had better get started on this week's tasks. I hope to make a few phone calls today and finish a few letters. Also, we are leaving for Wyoming tomorrow in order to visit/pick up my dear sister Allie. She has been away working at a dude ranch and so I would like to clean up her bedroom, tidy my own, and pack my bag. Very exciting!
I think this is the first comment I've left your on lovely blog! Amazing, yes? I've loved reading your posts ever since I found Authentic Virtue. 🙂 They've been such a blessing to me, and I wanted to say thank you.
I'll be praying that Yahweh will be with you as you go through this time in your life. 🙂 Have a blessed and safe trip!
I really enjoy reading about your accomplishments! You are certainly productive! I'm anxious for your post later this week!
I'm sure you can't wait to see your sister! I'll be praying that she has a safe flight or that you have a safe drive, whichever the case may be!
She is hosting a giveaway for a $20 gift certificate. And she is going to give it to the person that can round up the most followers! So, please go over and follower her, saying that 'Johanna and An Old-Fashioned Girl' sent you?
Just wanted to tell you that I like your new blog look! And I never noticed that your Mom's name was the same as mine! I've never had anybody with the same name as me, not that I've known!
Shana, Oh thank you! I am so glad and blessed to know that you are being blessed. And thank you for your prayers, I trust that they have been a great help to me!
Addie, Thank you too for your prayers and thoughts. I am always blessed by your encouraging posts and am thankful for your comments!
Kendra, Thank you! My Mom loves to find others who have her name as well! 🙂 What is your middle name (if you don't mind).


Bangladesh axed six players from national contracts after a year of poor performances with players who escaped the chop being denied a pay rise, an official said on Thursday.
Opening batsmen Soumya Sarkar and Imrul Kayes were among the six to miss out on a contract as Bangladesh cricket warned players to shape up, selector Habibul Bashar said.
"For contracts, we mostly preferred players who would perform regularly over the next one year," Bashar said.
"Some players have lost their regular spot in the team. They needed to be sent a message. Getting dropped does not mean the door is closed for them. Everyone will get a fair chance in future."
The top remunerated players - Shakib, Mushfiqur and Mortaza - are paid $5,121 per month from the board. The board also said it would review contracts for around 80 first-class players.


Construction and renovation projects are governed by four committees and design guidance is provided by the Architectural Review Council.
The Architectural Review Council is comprised of faculty from the UF College of Design, Construction and Planning (DCP), staff from the UF Department of Planning, Design and Construction (PDC), as well as local architectural alumni and practicing architects. The Council serves in an advisory capacity to the Faculty Senate Land Use Committee and provides the professional expertise needed by Land Use to fully evaluate the work submitted by our professional consultants. Rather than dictate design, the ARC will guide the design process by asking the right questions, and will lead our consultants, as well as the greater University community, to a deeper understanding of the role of architecture on campus.
Responsibilities of the committee include participating in the development and updating of the Master Plan and monitoring the execution of the Master Plan. In addition the committee recommends policies for land use and facility development.
Responsibilities include overseeing projects that affect the use of University lakes, including guidelines for use of such lakes in order to preserve their ecological integrity and research capabilities. The committee is also charged with the anagement and well being of natural areas containing non-domesticated plants and animals. It guides policies regarding the removal of trees and other vegetation and provides input to the University Land Use and Facility Planning Board regarding planning of major landscape elements such as green space, open space, and significant architectural features to ensure their compatibility with existing and planned landscaping and master planning.
Responsibilities include assisting the University in implementing its Policy on Historic Preservation. It also assists the University in fulfilling federal and state obligations with respect to its historic and archeological resources.
Meets on the 2nd Tuesday of each month at 2:00 p.m.
Responsibilities include reviewing and recommending rules and regulations governing the traffic, parking and registration of vehicles, bicycles, scooters and motorcycles on campus. In addition the committee reviews and recommends site locations for new parking facilities on campus and existing parking lot enhancements. Special parking requests from University departments, students, staff and faculty are reviewed. Any significant findings regarding all parking and transportation matters are brought before the committee.


When my order for Rapier Miniatures Parthians was delivered, Rapier kindly sent me some samples. One was a pack of early Imperial Romans and the other was some American Civil War figures. The American Civil War figures were good as they kind of allowed a like for like comparison between the Rapier figures and Baccus/Adler Miniatures.
The first comparison is between the Heroics and Ros Prussian musketeer and some Rapier ACW infantry. A you can see there is a considerable size difference between the Rapier and the Heroics. I certainly would not consider mixing these figures together in the same unit.
Even thought the difference in height is about 1mm, it is very apparent which is the larger figure and which is the smaller.
When comparing against the Adler Miniatures the size difference is a little harder to see. Unfortunately the only figure I had easily to hand was a plumed figure. Certainly the Adler head is much larger than the Rapier head and this makes the Adler figure look larger than the Rapier overall. However, allowing for the height difference bought on by the large plume, there is not too much of a difference between the figures. I think that it would be possible to mix Rapier figures and Adler figures within the same unit.
Given that the Baccus figures look slightly less bulky than the Adler, I also think it would be possible to mix Baccus and Rapier within the same unit. I will be in a better position soon to make that comparison however as I have a batch of Baccus Sassanians on order and due for delivery in the next two or three days.


text:
Health Alert HEALTH ALERT: Severe Heat Warning Read On...
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The county clerk is an elected official who serves as the clerk of the state supreme and county courts and is responsible for providing files, records, and indexes of papers in legal actions and proceedings. The clerk also maintains judgment dockets and records of monies paid or ordered to be paid to court.
In addition to court duties, various statutes require that the clerk's office record deeds, mortgages, and liens; file maps, financing statements, and military discharges; process pistol permits and passports; qualify notary public appointments in the county; file oaths of office of county officials and employees; and act as the agent of the commissioner of motor vehicles to issue motor vehicle registrations and licenses.
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Important Pistol Permit Notice
If you missed the January 31, 2018 Deadline for the Pistol Permit Re-Certification County Clerk Mary Strickland encourages you to do it as soon as possible!
"If you missed the January 31, 2018 deadline to re-certify don't let that deter you from completing it as soon as possible" Strickland stated. "I still encourage you to re-certify on the State Troopers website if you are able to, it is the quickest way to get it processed."
Press Release from January 3, 2018
GENESEO, NY – Livingston County Clerk Mary Strickland along with Sheriff Thomas Dougherty are encouraging pistol permit holders in Livingston County to recertify their permits. As part of the NY SAFE Act, if your permit was issued before January 15, 2013, the deadline to submit your recertification is January 31, 2018 and every five years thereafter. If your permit was issued on or after January 15, 2013, the deadline to recertify is five years after the date the permit was issued and every five years thereafter. As a permit holder, it is your responsibility to recertify your permit whether you receive a notification letter or not.
"Our office is recommending that if you are able to recertify on the New York State Police website that you choose that method. If you choose to electronically submit the form you will be able to print off a copy of your confirmation immediately," said Strickland. "If you are unable to access the website, we have the paper form available in our office that will need to be mailed into Albany, and can be picked up anytime during regular office hours."
"Although I continue to be opposed to the SAFE act and how the state has mishandled this very important topic, I do urge you to recertify your permits," stated Sheriff Dougherty. "I do this for your own best interest as I do not want a good, law abiding citizen to be stripped of their 2nd amendment rights by the state of NY and subsequently put yourself in jeopardy of state law enforcement knocking on your door to confiscate your permit and your firearms. If you do not recertify your permit, the New York State Police can, and most likely will, petition the court to revoke your pistol permit. If your pistol permit is revoked by the court, you can not possess any firearms at all, including long guns. Bottom line, take the time to recertify and keep your rights intact."
With four weeks left, approximately 70% of the pistol permit holders in Livingston County that need to recertify by January 31, 2018 have completed the recertification process.
According to the NY SAFE Act, failure to re-certify within the prescribed period "shall act as a revocation of such license" which may result in the seizure of all firearms.
Forms need to be mailed to:
New York State Police
Pistol Permit Bureau
Albany, New York 12226-2252
Link to NYS Pistol Permit Recertification & Frequently Asked Questions
https://troopers.ny.gov/Firearms/
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