As a Japanese standard university student in the 20th century, I entered Osaka Gas as soon as I graduated. I wanted to do something important for Japan with poor energy resources. At the time of my graduation, Japanese oil companies and upstream companies were not well known among university students and I chose a utility company which had established close and good relationships with end users.
Energy projects in Brazil and their sponsors remain virtually isolated from access to competitive international sources of project finance, as they are denominated in USD, which increase the FX risk.
Our biggest challenge is probably our best asset. We are very thorough when we review a project or accept a mandate for a client.
Dr Nick Cooper was appointed as an executive director and Chief Executive in June 2011. He is a member of the Nomination Committee. Prior to joining Ophir, Nick Cooper was Chief Financial Officer and co-founder of Salamander Energy plc.
KrisEnergy has put on stream two greenfield oil projects in 2015 in the Gulf of Thailand, one of those was our first as the operator, so this has been a big year in our evolution as a relatively young start-up company (2009). Looking ahead, we have two gas developments in Indonesia and one oil development in each of Thailand and Cambodia all planned to go into production in late 2017/early 2018. That should keep us very busy over the next couple of years.
If the oil price stays low there will likely be more company consolidation and sharing of services in the market. We are continually reviewing options in these areas but they need to fit strategically and be financially robust.
Having moved from a construction to operations phase on two major LNG investments, Santos' focus is on delivering every barrel as safely and efficiently as possible.
I firmly believe that true integration and collaboration between operators and service companies is one of the largest opportunities we have and one that is required to really improve productivity at lower costs.
Grow my company's presence in Asia both in terms of trading of energy and securing associated investments in energy infrastructure.
Toby Pierce, Chief Executive Officer, TAG Oil Ltd.
Our biggest priority today is managing our business through low commodity prices and strategically positioning the company for the next upcycle.
AAG is the leading independent CBM producer in China, focus on the development and value optimization of unconventional gas resources to supply clean energy to the Chinese economy.
Our current focus at Mitra is on getting to first production with the gas discoveries that have been made in the Malay Basin in Vietnam.


When you think of spring, do you associate it with flowers, planting your garden, or rain showers and rubber boots splashing in puddles? I wish I could link all of the above to the spring season in the Canadian Rockies but alas, we're still getting weekly (daily) snow storms. For us, spring means that the weather has finally warmed up enough that we can take the kids into the backcountry again after a long winter of staying in resorts, hotels, and front country hostels. And while we LOVE resorts (don't get me wrong,) we also really love wilderness and backcountry trips where we have to travel sans automobile to get to our sleep destination for the night.
A week ago, you hopefully read about our trip into the Elk Lakes Cabin in British Columbia in my first story on Raising Tough Kids. Elk Lakes was the ultimate in difficult winter (spring?) family journeys with a grueling 11km ski/snowshoe required to reach the cabin. Last weekend we did our second wilderness trip of the spring season and it was just a tad easier. We only had to snowshoe about 500 metres to reach our two cabins. And yeah, maybe we should have done this trip first as prep for the harder one.
Last weekend, we travelled the famed Icefields Parkway that connects Banff and Jasper National Parks to reach the Hilda Creek Wilderness Hostel, located a short distance away from the Columbia Icefields Centre. In summer, the Icefields Centre is the place to be if you want to board a giant snow bus for a tour on the Athabasca Glacier and the whole area is crawling with tourists! You wouldn't need to worry about running out of gas, having engine trouble or experiencing an emergency of any kind. (Somebody would be around to find you.) Winter and early spring however is a completely different story. There is nobody hanging out around the Icefields Centre at this time of year. The large building is boarded up and drivers on the highway move along between Jasper and Banff without stopping. Other than the occasional backcountry skiers visiting the area, you are completely on your own!
The first time we wanted to stay at the Hilda Creek Wilderness Hostel, we received quite the "warning" on the phone when I made the booking. We were told that we absolutely HAD to pack out all garbage, that there would be no hostel manager on site, that we might have to melt snow for drinking water if the creek was frozen solid, that there were no services or amenities for at least 100 kilometres, and that we would need solid backcountry experience! We immediately responded with, "sign us up!"
Shoveling out the front door of the main cabin for "light"
The hostel only sleeps 6 so that means we can take one other family with us and reserve the whole hostel for ourselves! I like sharing as much as the next person, but if you don't have to share, why would you? Hilda is the only hostel that can so easily be reserved for a private booking; No other hostel I know of sleeps but 6 people!
The Hilda Creek Hostel may be situated in the middle of nowhere, but the scenery is unparalleled! Last year we took the kids on a hike to see the Athabasca Glacier and this year we took them up the moraines below the Hilda Glacier. Getting to the toe of the Athabasca Glacier is easy - follow the summer road and hiking trail from the Icefields Centre. Reaching the toe of the Hilda Glacier however requires significant backcountry experience in winter . Fortunately, we had no aspirations of touching the Hilda Glacier; We just wanted to do a snowshoe tour out the door of our cabin. Mission accomplished.
The Fastest Way Down - Slide of course!
The main reason we go to Hilda Creek in the winter - to have FUN in the snow. April is still full on winter on the Icefields Parkway but the weather is a bit warmer and so it's our choice time of year to visit.
He does not care that it's April and it's still snowing.
Playing in the snow - kid paradise!
Sledding down the hill below the hostel - FUN every year!
Even the adults had fun with the snow!
Flips off the deck - what else do you do after dinner?
Even the kids had fun jumping off the deck into the waist deep snow!
Honestly, for folks comfortable in the backcountry, Hilda Creek is absolute paradise! We haul our gear to the cabins in sleds and it's close enough to the highway that we can bring everything from coolers of food to six packs of beer, jugs of water (if we were smart,) toys, skis for adults wanting to do turns off Hilda Ridge behind the hostel, and comfort items (you name it - your fav. pillow, the kids' fav. stuffies...) Yes, there is no running water and it's pretty rustic, but it's also an incredible experience to spend the weekend living off the grid completely self sufficient.
We are teaching our kids to appreciate things like clean drinking water that flows from a tap, indoor plumbing, lights that turn on with the flick of a switch and all the modern technology we have so handy at our fingertips from television to refrigerators and microwaves. We had none of that at Hilda Creek and we survived just fine. I'd say we did more than just survive. We had fun.
To read last year's story on Hilda Creek and see our photos hiking to see the Athabasca Glacier, visit this link to my story in Snowshoe Magazine, Spring Adventures on Alberta's Icefields Parkway.
Want to book your own trip to Hilda Creek? Visit the Hostelling International Website for all the information you need. The hostel is a great place to visit in summer as well and you can hike fabulous trails such as Parker Ridge and the Hilda Glacier Trail without ever getting in your car. Within a 5 minute drive, you can reach the Columbia Icefields Centre to see the Athabasca Glacier or you can reach the Wilcox Pass trailhead. These are all STELLAR hikes and absolutely unparalleled in the Rockies. Photos below.
Are you planning your trip yet?
Thanks as always to Hostelling International for assisting with our stay. All opinions and thoughts are my own.
That looks amazing! What a privilege to be able to access real wilderness like that - and it's great that you're taking advantage of it!
Thanks Linda. It is amazing to be able to access wilderness with so little effort. We love our hostels!
Great recap. The Canadian Rockies has so much to offer and I love that you put that out there.
How far in advance do you recommend booking for this hostel?
As far as you are able to. But I'm a planner so I'd say that for anything. If you can do it 6 months in advance, you'll feel safer in knowing you'll get a spot. Long weekends will fill up faster as will weekends in ski season (feb/march).
Can you tell me how good the heating system is and are pots/dishes/cutlery already there? Lastly, is there an oven or only stove top cooking? Thanks!
Thanks for the info. Have booked Hilda Creek hostel for a weekend next February and am looking forward to staying there already!
So happy to hear that you've booked a trip to Hilda. Have a great trip! Let me know how it goes.
I meant to ask how much avalanche risk is at Hilda Creek? I'll check with Parks Canada too but the more info the better!
There is no avalanche risk (to my knowledge) at the hostel itself or in the immediate area. We also hiked up towards the Hilda Glacier from a trail behind the bathrooms and felt it was safe. However, if you want to get on Parker or Hilda Ridge to do skiing or snowshoeing, there is avalanche danger.
Anywhere above the hostel there is avalanche risk. It's lower risk until you get to steeper slopes, but certainly not no risk. Anyone who backcountry skis in that areas carries avy gear and wears beacons from the moment the leave the car or the hostel.
I would not have done that hike towards the Hilda Glacier without avy gear & training. Just not worth the risk.
Also, there is no wood stove, no loft and no oven. Just one three burner propane stove. The wood stove was taken out long ago, and the heat is via propane heaters in the kitchen building and in the bunkroom.
Hi. I have gone through some of my comments and realize I may have responded at one point with a different cabin in mind. Obviously tired and thought I was responding to a different story. As you indicate, there is no oven, only hot plate type stoves, and yes, no loft. Any mention I had of a loft in comments would have referred to the Elk Lakes Cabin. I wrote another story about that cabin as well with a similar title of "raising tough kids." I think you would prefer this cabin in the future as it is a bit more "comfortable."
In the future, it is always recommended to check with the actual website, in this case, Hostelling International, for best information possible. I would prefer to be a source of inspiration only and not a "guide book" if you would. As for avalanche danger, Parks Canada would be able to give you that information as well and they should be consulted for the most up to date conditions so that your family can be safe. I merely provide my "opinions" and again am not a guide book or a trained professional.
Hi Tanya, I made the original 'anonymous' posts back in August and September (but certainly not the last one!). A group of us used my long held reservation and went to Hilda Creek this past weekend and had a WONDERFUL time. The weather was great, the scenery was great and having the hostel all to ourselves made the weekend so much more fun and enjoyable.
I appreciate all the information you post on this website. It has helped my family and friends get out more winter and summer. Lastly, we enjoyed Hilda Creek so much that we've booked the whole hostel for another weekend later this year.
Hi Gerrard. I'm happy that you loved Hilda Creek. We love it too and are back for Easter this year. We find it very rustic and it gives us a backcountry experience that is easy to achieve with young kids. Hope you have fun again later when you go.


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Are you a voracious reader, but not quite that hungry for attention? Join my team and we'll call it a day. When I run out of books to offer you, I am sure I can find some like-minded romance authors from the Love Kissed family to fill in the gaps!


Everyone wants to create a beautiful home. Unfortunately, not everyone has the creativity and knowledge to develop a stylish yet functional space. If you want to feel happier in your property and avoid mistakes, we recommend reading our top tips on how to improve your interior design skills.
Many homeowners make the big mistake of positioning their furniture up against the wall. This is a big arrangement no-no, as you will be unable to emphasize the room's perimeter. Any experienced interior designer will tell you to angle furniture, so move it to the middle of the room or to set it slightly away from the wall.
Less is more when it comes to interior design. Take a good look around every room within your home and ask yourself if all your furniture serves a purpose. If an item serves no purpose and is taking up space, it might be time to remove it from your property altogether. Ask a loved one if they want to take the unnecessary furniture off your hands or donate them to a favorite charity.
Have you been surrounded by the same décor accessories for a long time? If so, it might be time to review each item to ensure they are not outdated. If you don't adore an accessory, remove and replace it with something that will brighten up your living space. Also, never buy an item to simply fill a gap. Each item must add a new element to an interior design and must not detract from the room's focal point.
The lighting you choose will determine the look and feel of your home. After all, artificial lighting can add space, ambience, and color to a living space. You must therefore choose wisely, while thinking about the best fixtures for your design. For example, a beautiful chandelier can add a touch of elegance to a room, while a pendant light can offer a romantic ambience for a cozy, inviting environment. You must therefore browse the many contemporary lighting options online.
Do your guests tend to float from room to room? They might not feel comfortable in one space within your home. Thankfully, there are many ways you can create an inviting space with a few little tweaks to your interior design. For instance, you can transform a cold space by adding area rugs, pillows, and throws, and you can warm up the home with window treatments and an exquisite fireplace. Your guests will never want to leave.
Make a house feel like a home by refining your hallway. For instance, a basket for shoes or a small coat armoire will create a homely atmosphere your guests will love, and it will allow them to take off their shoes and coats to get comfortable and enjoy their time in your home. It's the perfect way to welcome your guests.
Hygge Essentials Giveaway – Enter Now to Win!


Have you ever heard of Chris Kandu?
No, well fingers crossed you will know the name well by the time of the next Olympics in 2016. Chris, aged 18, a pupil at Enfield Grammar, is a high jumper, who has competed for England and Great Britain and is ranked no 2 in the Under 20 class in the world. Last weekend he jumped a new personal best of 2.24m at the England Athletics U20 / U23 Championships winning the competition, just narrowly missing the 2.28m world record.
Chris is an Enfield lad, who lives in Ponders End, along with his sister and their mum, a single parent. He describes himself in self-deprecating terms as "not the brightest of pupils, inclined to misbehave, not focused on my studies"..Whilst sport is his number one priority Chris is studying Business Studies and IT for A Level. However, as most year 13 pupils are counting the days until the end of term, Chris will be embarking on an intensive summer of training and competing. Unfortunately he has had to turn down a place to compete in the Commonwealth Games in Glasgow, as it clashes with World Junior Championships in Portland Oregon, where he a strong prospect of a medal. These junior championship provide Chris with important opportunity to compete at world level, with less than two years to the 2016 Olympic Games.
The education watchdog, Ofsted has reported that sportsmen and women who attended private schools continue to dominate sport at the elite level, particularly in athletics. Despite fewer than one in 10 pupils attending fee-paying schools, more than four in 10 of the British athletes who won medals at London 2012 were schooled privately. Perhaps this is hardly surprising, with access top quality facilities and training opportunities and parents who can afford to take them to compete across the country and more abl to help them to gain scholarships and sponsorships.
Life hasn't been quite so easy for Chris. Whilst he has been fortunate in receiving excellent support from the School and has received some funding from Sports Aid, he narrowly missed out on receiving a grant from British Athletics. As a result there have been events and training opportunities with top athletes that he has had to miss simply because the family couldn't afford the expenses.
As we follow Chris's performance over the summer, what can Enfield do to support this talented young man who is having to jump through too many hoops to further his career, at a time when he should be concentrating solely on the high jump?
A relatively small amount of funding could catapult his career to the next level. No apologies for the sporting puns, Chris Kandu can do, with a little help from Enfield.
Have you ever heard of Chris Kandu? No, well fingers crossed you will know the name well by the time of the next Olympics in 2016.


There are currently 1 Homes for sale in River Oaks. The average price for River Oaks properties is $350,000which is 0% higher than all Homes within a 20 mile radius. The highest priced property in River Oaks is $350,000 while the lowest priced Home is listed at $350,000. Most of the properties available in River Oaks are Homes . Compared to last month, the average price of real estate in the area has gone up 100%.


Welcome to Rich & Davis's newest platform for sharing our devotion in the art of fine picture frame making: The Frame Maker's Blog.
The overwhelming interest and support we have received over the last few years has impassioned us to create a space where we can share our passion in a more intimate and informative way. We are encouraged by the interest in the heritage trade of frame making from around Australia and we feel a responsibility to create a resource of information for those seeking to learn more about picture frame making.
This blog will allow for posts focusing on some of our custom frame making projects, our frame restoration and repair work and posts about individual skills used in our work; such as water and oil-gilding, ornament making and frame design.
This blog will also share information from other reputable sources in the industry, highlights of the picture framing trade and reflections on past and current published or displayed frame-related topics.
We encourage the sharing of information in the frame making industry and we hope that this platform will enable us to share our celebration of the humbled picture frame with the community.


Windsor Board Room at Windsor "Naples' Premier Meetings Venue"
Private Meeting Room for 12 at Windsor "Naples' Premier Meetings Venue"
Our main Boardroom seats 10 comfortably (can and often does hold 12) at a large oval table. This impressive space hosts many Depositions, Mediations, Masterclasses, Mentoring & Training sessions, regional and local sales meetings. , board meetings and national and international video conferences. This space is also preferred by title agents for closings for local developers. Full fast secure internet access, (hard wired or WIFI) 60" TV / Monitor and a full video conferencing suite; 80 inch Matte White Projection Screen. In addition to our Concierge service and lounge, we can cater anything from a breakfast through to a fully cooked meal prepared on site to meet your budget.
55-inch flat screen monitor with SONY Streaming Blue-Ray Player. Includes HDMI and RGB inputs. Cables available for MAC and PC.
Absolutely Amazing. What a hidden gem Windsor is. Everything beautifully decorated and the service is first class. We will be back and highly recommend them to anyone looking for meeting space or an office.


text:
Not Everyone Is Happy And Gay
As predictable as the rising of the sun, there has been a noisy, intemperate, and somewhat loony backlash among the True Believers concerning the Supreme Court's decision to punt on marriage equality. (Just as a light aperitif of sensible comment, here's the invaluable Dahlia on what may still be trapdoorsin what the Nine Wise Souls did.) For example, the National Organization for Marriage went indiscriminately mad, channelling either Thomas Aquinas or Yosemite Sam. It's hard to determine. But the most entertaining reactions came from totally respectable potential presidential candidate Ted Cruz, and his running buddy, Mike Lee, the konztitooshunal skolar from Utah. Both of them leaned hard on the "unelected judges" button, and Lee, who is to jurisprudence what Paul Ryan is to math, was particularly amusing.
"Nothing in the Constitution forbids a state from retaining the traditional definition of marriage as a union between a man and a woman," Lee said in a statement. "Whether to change that definition is a decision best left to the people of each state - not to unelected, politically unaccountable judges. The Supreme Court owes it to the people of those states, whose democratic choices are being invalidated, to review the question soon and reaffirm that states do have that right."
Well, "nothing in the Constitution" forbids a state from retaining the traditional definition of marriage as a union between two white people, either, except that pesky 14th Amendment, or so some unelected and politically unaccountable judges once ruled, anyway. It is helpful here to refer back to Lee's earlier, funnier work regarding another decision rendered by unelected and politically unaccountable judgesin regards to a law openly passed by the national legislature.
"Today's decision in Burwell v Hobby Lobby marks an important victory for religious liberty. Americans do not shed their religious freedoms merely by going into business. The Court's ruling upholds and strengthens the rights of individuals and the rule of law, while protecting the Constitution."
But, apparently, unelected and politically unaccountable judges can require that employees conform in their private lives to the religious whims of their employers, or they can get out. This is based on Lee's strict constructionist viewof the constitutional provisions regarding the right to keep and bear icky ladyparts.
During an appearance on Sirius XM's The Wilkow Majority, host Andrew Wilkow argued that the real question in Burwell v. Hobby Lobby Stores Inc., was about "whether or not a person who runs a business should be forced to provide something that is largely for recreational behavior, if it goes against their religious beliefs." Lee, responded by saying "Yea, that's right, that's right," before claiming that "this administration is using the often coercive power of the federal government to force people into their way of being and their way of existing, their way of believing and thinking and acting."
At least he's not talking about child labor any more. For now. In the vast universe of Not Knowing Dick, Mike Lee is the Padishah emperor.
Everyone Who Turned Out to Be Gay in 2013
Happy Fourth Amendment Day
Happy National Sexytime Day!
North Korea: Everything's Amazing and Everybody's Happy
Happy Birthday, George McGovern

redpajama_set_name: RedPajamaCommonCrawl


4500 Great America Parkway Santa Clara, CA 95054 USA Wireless- G Router WGR614v9 Reference Manual. I lost or cannot use my router configuration CD; what should I do? Netgear wireless g router wgr614v9 setup wizard. May NETGEAR, Inc.
The following article is intended to answer some of the most frequently asked questions regarding NETGEAR routers.
May NETGEAR, Inc. 4500 Great America Parkway Santa Clara, CA 95054 USA Wireless- G Router WGR614v9 Reference Manual. The following article is intended to answer some of the most frequently asked questions regarding NETGEAR routers.
I lost or cannot use my router configuration CD; what should I do?
4 gig sounds like a pipe dream to me! i will say though that there is a proposed benefit in that a user' s current adapters may perform better with the use of the beam forming technology, this is because the new 1900 ac router' s run 2 wireless AC chipsets and the beamforming is used in both 5 gig and 2.


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IR Sports Report
Indians' fall athletes garner state, conference recognition
For the first time in Henlopen Conference girls' volleyball history, a player was recognized with First-Team All-State honors. That first-timer is none other than Indian River High School junior Raychel Ehlers.
Ehlers helped guide the...
Indians pick up first football win of the year
The elusive win finally came last Friday night for the Indian River High School football team, in a 31-12 decision over visiting Seaford, thanks to a defense that was determined not to let a first-half lead go to waste.
Senior linebacker R...
Thursday, October 3, 2019 - 11:29am
Indians drop a close one to Polytech
The Indian River High School football team made the trek north to Dover to take on Polytech last Friday, Sept. 13, only to find themselves on the short end of a 14-13 final score.
With the full moon beaming over the field, the Indians...
Thursday, September 19, 2019 - 9:23am
SMS teams ready to get their seasons going
They get a little later of a start since they aren't permitted to practice in the summer, but the Selbyville Middle School fall sports teams are ready to get things going for their 2019 seasons following two weeks of preparation.
Turnovers spell fate for Indians on the gridiron
Milford capitalizes on several IR miscues for win
The Achilles heel for any football team is generally their ability to take care of the football. Turnovers can give opponents short fields and shift the momentum in games very quickly.
Such was the case in the 2019 season-opener for the...
IR season-opener against Milford moved up by Dorian
All the two-a-days and scrimmages are over. All the days and nights of preparation have led to this weekend. And, just to throw some extra drama into the picture, a looming hurricane named Dorian has pushed up the schedule by a full day.
Thursday, September 5, 2019 - 9:19am
IR football program has high hopes for season
Injuries turned what could have been the picture of a very promising 2018 season for the Indian River High School football team into something of a jigsaw puzzle that was missing several pieces.
Head coach Phillip Townsend and his staff...
Thursday, August 29, 2019 - 11:18am
As new season begins, Indians focused on long grind ahead
The sounds of whistles echoed throughout the air on the ground of the Indian River High School athletic fields this week as coaches welcomed their student-athletes to the fields and gym for the start of the 2019 fall sports season.
Four...
Thursday, August 15, 2019 - 9:43am
More than a football game
IR students take part in 64th annual Blue-Gold game
Like the first one held 64 years ago, this year's annual Blue-Gold football game, played Friday, June 21, at the University of Delaware's Tubby Raymond Field at Delaware Stadium, was more than a football game.
The Delaware Foundation...
Thursday, June 27, 2019 - 9:11am
Blue-Gold game about more than football
The 64th Annual DFRC Blue-Gold All-Star Football Game is an annual showcase for high school seniors that allows them to don their respective schools' football helmets for one last time.
However, the game is not just about the play on the...
Ball 4 All Foundation to host 1st Annual Game Day June 1
All proceeds to benefit Russell White Scholarship and Lower Sussex Indians Pop Warner
The Russell White Scholarship Cornhole Tournament is back.
Thanks to The Ball 4 All Foundation, the tournament that was previously hosted by the Xi Upsilon sorority will continue as part of the 1st Annual Game Day at the Dagsboro fire hall...
Thursday, May 30, 2019 - 10:14am
Thursday, May 23, 2019 - 7:52am
Pop Warner team pushes for big return
The Lower Sussex Indians Pop Warner football team has their game plan in place for the 2019 season. All they need now are some players and cheerleaders to put that plan in motion.
The Indians will be hosting an early registration event at...
Thursday, March 28, 2019 - 8:38am
Schultz selected to participate in DFRC Blue-Gold Game
It is an annual event of note throughout the state, and only select seniors are tapped each year to participate in the all-star showcase known as the DFRC Blue-Gold football game.
This year, Indian River's Zach Schultz will represent the...
'It's about commitment'
After winning their first Division I state championship, the Sussex Central High School football team was honored at the Dec. 17 Indian River School District school board meeting. 'They're pretty special,' head coach John Wells said of his...
SMS football, field hockey finish with winning records
With a chance to finish their 2018 campaign with an undefeated record, the Selbyville Middle School football squad came up just short, with a 6-0 loss to Millsboro in the final game of the season, on Thursday, Nov. 1.
The team battled for...
Thursday, November 15, 2018 - 10:41am
Indians' football season end in rout
By the end of the 2018 season for the Green-and-Gold football team, the IR no longer stood for Indian River. No, it was more fitting, according to senior J.J. Killen, that the IR perfectly described the team as "injured reserve."
Of the 43...
Indians fall behind big early in loss to Raiders
Growing pains come in all forms during the course of a football season.
Unfortunately, for this year's Indian River High School team, those growing pains have been just as tough physically — if not more so — as they have been in other...
Thursday, November 8, 2018 - 9:41am
Indians thump Laurel on Homecoming
When it comes to successful Homecoming football games, last Friday night, Oct. 19, the Indian River football team put on quite a show.
Recording a season-high 569 yards of total offense, the Indians' beat down of the Bulldogs was...
Thursday, October 25, 2018 - 9:49am
SMS Report
Indians girls' volleyball remains perfect
They have yet to be scored upon this season, cruising to six straight shutout wins, including their most recent 2-0 whitewash of Mariner last Wednesday, Oct. 17.
They would be the Selbyville Middle...
Indians football comes back to beat Seaford
It was a long four weeks for the Indian River High School football team. Throughout their four-game losing streak, the Indians continued to hold their heads high and worked extra-hard in practice, knowing that they were just a break or two away...
IR football kicks off season with win
Much like in life, one has to deal with adversity in sports as well, and how people respond to that adversity can go a long way toward determining success or failure.
With their opening game against Glasgow postponed last Friday night,...
Thursday, September 13, 2018 - 10:55am
Optimism surrounds IR football team for '18 campaign
While it was only a year ago, the 2017 season is a million miles in the rearview mirror for the Indian River High School football team and head coach Phillip Townsend.
After struggling through a 1-9 season last year, in which they lost...
Thursday, September 6, 2018 - 3:43pm
Indians hit the practice fields for fall sports
As with the start of any new season, optimism and excitement were in the air all around the campus at Indian River High School on Wednesday morning as the fall sports programs all officially began their 2018 campaigns.
The football, boys'...
Thursday, August 16, 2018 - 12:58pm
Donaway to represent IR in Blue-Gold game
Recent Indian River High School graduate Collin Donaway will be participating in this weekend's Blue-Gold Game at the University of Delaware. The game benefits the Delaware Foundation Reaching Citizens (DFRC).
The Blue-Gold High School...
Friday, June 22, 2018 - 1:43pm


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\section{Introduction}\label{intro}
When high volume production of a standardized commodity is required, assembly lines are appropriate. Assembly lines consist of a set of $ m $ stations ordered along a conveyer belt or a similar material handling system that moves the work-pieces through them. Starting from the first station, the work-piece enters each station and stays there for a fixed time span called cycle time $ c $. During this time, a set of tasks is performed on the work-piece, then the work-piece is moved to the next station to perform the next set of tasks. This process continues until the final product is assembled in the last station. Traditionally the stations are arranged in a straight line and each work-piece moves along this line, such a system is called a straight assembly line.

The best known problem in the literature of assembly line balancing is the Simple Assembly Line Balancing Problem (SALBP) \cite{scholl1999balancing,baybars1986survey}. SALBP is defined as follows:
\begin{itemize}
	\item   A single product is to be manufactured in large quantities in a straight assembly line. The total work required to produce this product, is divided into a set of $ n $ basic tasks $ T=\{1,2,\dots,n\} $. Each task  $ j\in T$ has a positive integral processing time of $ t_j $.
	\item Due to technological constraints, a task cannot be started before all of its predecessors are finished. The precedence constraints can be represented by a graph, vertices of this graph represent the tasks and precedence relations are represented by directed arcs. An arc $ (i,j) $ means that task $ i $ is a predecessors of task $ j $. We denote the set of immediate (all) successors of task $ j $ by  $ F_j (F_j^* ) $ and the set of  immediate (all) predecessors of task $ j $ by $ P_j (P_j^* ) $. Figure \ref{figre1} shows an example of a precedence graph with $ n=11 $ tasks. In this graph, task processing times are also represented above the corresponding vertices.
	\item Assignment of tasks to stations must observe precedence constraints. In other words, task $ j $ cannot be assigned to stations $ k $ unless all of its predecessors are assigned to one of the stations $ 1,2,\dots,k $.
	\item Each task must be assigned to exactly one station. The set of tasks assigned to a station $ k $ is called a work-load of station $ k $ and denoted by $ S_k $. Sum of processing times of tasks assigned to a station $ k $ is called station time and denoted by $ t(S_k ) $. Station times must always observe the cycle time constraint i.e. $ t(S_k )=\sum_{j\in S_k} t_j \le c $.
\end{itemize}
\begin{figure}[H]
	\centering
	\includegraphics{Drawing1}
	\caption{An instance of SALBP-1}
	\label{figre1}
\end{figure}
There are four versions of SALBP:
\begin{itemize}
	\item SALBP-1: minimizing number of stations $ m $ for a fixed cycle time $ c $.
	\item SALBP-2: minimizing cycle time $ c $ for a given number of stations $ m $.
	\item SALBP-E: maximizing line efficiency $ E=\frac{(\sum_{j=1}^n t_j )}{mc} $.
	\item SALBP-F: feasibility problem for given values of $ c $ and $ m $, determining whether or not there is a feasible balance, if so, the problem involves finding a feasible balance.
\end{itemize}
All of four versions of SALBP are known to be NP-Hard \cite{wee1982assembly}. The assumptions of SALBP are very restricting in comparison with the real world assembly line systems. Therefore, researchers recently have focused on identifying and modeling more realistic situations in assembly lines. The resulting problems are called generalized assembly line balancing problems (GALBP). Several generalizations have been studied for the SALBP. Some examples of GALBPs are assembly lines with resource constraints \cite{augpak2005assembly,corominas2011assembly}, assembly lines with setup times between tasks \cite{andres2008balancing,yolmeh2012efficient}, assembly lines with task deterioration and learning effect \cite{toksari2010assembly,shahanaghi2010scheduling}, assembly lines with parallel workstations \cite{buxey1974assembly}, assembly line balancing and supply chain design \cite{paksoy2012supply,yolmeh2015outer} and assembly lines with multi-manned workstations \cite{dimitriadis2006assembly,fattahi2011mathematical}. Some of the latest surveys of assembly line balancing problems are \cite{erel1998survey,rekiek2002state,scholl2006state}.

The surveys suggest that, while many applicable problems have been identified and modeled, developing advanced solution methods for these models still lags behind. Even though there have been significant algorithmic developments to solve the SALBP, the same cannot be said about GALBPs. Therefore, the surveys call for additional research to apply modern solution concepts such as advanced enumeration and bounding techniques to solve the generalized models. In this paper we make the first attempt toward developing a branch, price and remember (BP\&R) algorithm to solve the U shaped line balancing problem (UALBP). We use column generation to obtain tight bounds for this problem. We also use memory to avoid visiting redundant sub-problems. To the best of our knowledge, this is the first attempt to apply a branch, price and remember algorithm to solve a U shaped assembly line balancing problem.

The most closely related work to our study is the branch, bound and remember (BB\&R) algorithm developed by \cite{sewell2012branch} to solve the SALBP. BB\&R combines the idea of branch and bound algorithm with dynamic programming through memorizing already visited nodes in the search tree. Even though our proposed BP\&R also uses memory to avoid redundant nodes, there are some major differences between BP\&R and BB\&R. The first and most important difference is that, unlike BB\&R, BP\&R uses column generation to obtain tight lower bounds for the nodes in the search tree. Column generation algorithm is shown to yield high quality bounds for the SALBP \cite{peeters2006linear}, however its running time was deemed to be too high to be used in a branch and bound algorithm. The main contribution of this paper is devising a scheme to leverage the tight bounds obtained by the column generation algorithm while keeping its run times reasonably low. We do this by relaxing the pricing sub-problem and use of memory. Second difference of BP\&R from BB\&R is in the search strategy. BP\&R uses a best first search strategy, while BB\&R uses cyclic best-first search. Lastly, even though both BB\&R and BP\&R use a modification of Hoffman algorithm to obtain an initial solution, BP\&R uses additional flexibility of U-shaped lines to obtain better solutions.
\begin{figure}[H]
	\centering
	\includegraphics[scale=1.3]{fig2}
	\caption{U-shaped assembly line}
	\label{figre2}
\end{figure}
U shaped lines are a generalization of straight lines where the stations can be arranged in a U shape rather than a straight line. This is done by allowing two work-pieces to be processed simultaneously in a single station. By analogy with SALBP, different problem types can be distinguished: UALBP-1, UALBP-2, UALBP-E and UALBP-F. Figure \ref{figre2} shows an example of U-shaped line. In this figure, the first task of one work-piece, which is starting to be produced and the last task of another work-piece is performed in the same station (S1).
We define the set of forward available tasks as the set of tasks with no unassigned predecessors. In other words, a tasks is called forward available if and only if all of its predecessors have been assigned to a station. Similarly, the set of backward available tasks is the set of tasks with no unassigned successors. The difference between straight and U-shaped lines is that in straight lines, to determine a solution, one should start from the first task and assign tasks into stations while moving forward through the precedence graph i.e. using only forward available tasks. Whereas in a U-shaped line, one can use both forward available and backward available tasks to assign to stations \cite{miltenburg1994u}. Being able to use both forward and backward available tasks to assign to stations results in more possibilities of assigning tasks to stations. Therefore, in comparison with SALBP, UALBP offers more flexibility in assigning tasks to stations, therefore, in many cases, it leads to higher efficiency.

Consider the example of Figure \ref{figre1} with a cycle time $ c=10 $ and objective of minimizing the number of stations i.e. UALBP-1. Figure \ref{figre3} shows an optimal solution with $ m=5 $ stations for this example problem. Consider station 2 in this figure, we have $ S_2=\{3,10\} $. Task 3 is performed on the work-piece whenever it visits station 2 for the first time, which happens when the work-piece is moving from station 1 to station 2, after its predecessor (task 1) has been executed in station 1. When the work-piece returns to station 2 from station 4, all predecessors of tasks 10 have already been performed (in stations 3 to 5) and task 10 can be executed. After performing task 10, the work-piece moves to station 1 to perform the successor of task 10 i.e. task 11. Then the completed product exits station 1. An optimal solution to the corresponding SALBP-1 requires six stations:
$S_1=\{1,2,5\} $, $ S_2=\{4\} $, $ S_3=\{3,7\} $, $ S_4=\{6\} $, $ S_5=\{8,9\} $ and $ S_6=\{10,11\} $. Therefore, knowing that $ \sum_{j\in T} t_j =50 $, for this example line efficiency of optimal U shaped line is 100\% $ (E=\frac{\sum_{j=1}^n t_j}{mc}=\frac{50}{50}) $ , but line efficiency of optimal straight line is 83.33\% $ (E=\frac{\sum_{j=1}^n t_j}{mc}=\frac{50}{60}) $, which highlights the advantage of U shaped lines over straight lines.
\begin{figure}[H]
	\centering
	\includegraphics{fig3}
	\caption{Solution to the example problem with c=10}
	\label{figre3}
\end{figure}
Reviewing the literature of UALBP, \cite{miltenburg1994u} introduced and modeled U-shaped assembly line balancing problem. They developed a dynamic programming approach to solve this problem. \cite{ajenblit1998applying} proposed a genetic algorithm to solve UALBP-1. \cite{urban1998note} developed an integer programming formulation for UALBP-1. \cite{scholl1999ulino} developed a procedure named ULINO (U-Line Optimizer) which is a modification of SALOME \cite{scholl1997salome} to solve all versions of UALBP. ULINO uses a station oriented branching and a depth first search strategy. It also uses several lower bounds and dominance rules. \cite{erel2001balancing} developed a simulated annealing algorithm to solve UALBP-1.
The literature review suggests that, even though many researchers have studied the SALBP, there are few advanced solution methods for the U shaped line balancing problem. The only study that used a branch and bound algorithm to solve this problem is performed by \cite{scholl1999ulino}. They developed an exact algorithm to solve UALBP-1 but also discuss how to use it to solve other versions of the problem. In this paper we develop a branch and price algorithm to solve the UALBP-1. We use a column generation approach to compute tight bounds for this problem. The application of column generation to compute a lower bound is fairly infrequent in the field of assembly line balancing problem. The only study that used column generation to obtain a lower bound for a line balancing problem is \cite{peeters2006linear}. They used Dantzig--Wolfe decomposition to obtain an LP relaxation of SALBP-1 and used column generation algorithm to solve it. To solve the pricing sub-problem, they used a branch and bound algorithm. Their computational results show that, their proposed column generation approach yields very good lower bounds. However, the run times of the column generation was too high to be considered in a branch and bound algorithm as a lower bound. In this paper we tackle the issue of high run times and apply it for the U shaped line balancing problem. Specifically, we relax the sub-problem to obtain a pure knapsack problem, which can be solved in pseudo polynomial time, to obtain a faster column generation. Moreover, we keep a pool of previously generated columns to avoid redundantly generating the same column. Our proposed branch, price and remember also uses memory to avoid visiting redundant nodes in the search tree.

Our proposed BP\&R uses a modification of Hoffman heuristic to obtain a high quality initial solution. Hoffmann algorithm \cite{hoffmann1963assembly} is an efficient heuristic for SALBP-1 that works on a station by station basis. At each iteration, it enumerates all of the possible assignments of tasks to the current station and selects the one with the smallest idle time. This process is repeated until all tasks are assigned. \cite{fleszar2003enumerative} proposed a modification of Hoffman heuristic that works in a bidirectional manner. In other words, their proposed heuristic builds solutions from both sides of the precedence graph and selects the best one.  \cite{sewell2012branch} and \cite{morrison2014application} proposed a modification of Hoffman algorithm that in each iteration, instead of selecting the load with smallest idle time, selects the load with highest value of $ \sum_{j \in U} {(t_j+\alpha w_j+\beta |F_j |-\gamma)} $; where $ F_j (F_j^* ) $ is the set of immediate (all) successors of task $ j $ and $ w_j=t_j+\sum_{k \in F_j^*} t_k $  is the positional weight of task $ j $. $ \alpha,\beta,\gamma $ are coefficients that need to be numerically determined. Our proposed modification of Hoffman heuristic also uses a different criterion to select the load. The difference of our proposed heuristic from the one proposed by  \cite{sewell2012branch} is that our proposed heuristic is designed to be applied on a U shaped line, where forward assignable and backward assignable tasks have different implications on which tasks will be available for the next iteration.

The rest of this paper is organized as follows: in section \ref{sec2} we describe the proposed BP\&R algorithm. Computational results are presented in section \ref{sec3}. In section \ref{sec4}, the main conclusions of the paper and suggestions for future research are given.

\section{The proposed branch, price and remember algorithm}\label{sec2}
BP\&R is a branch, bound and remember (BB\&R) algorithm which uses column generation to obtain lower bounds in each node in the search tree. BB\&R is a branch and bound algorithm that uses memory to avoid revisiting nodes that have already been visited. In BB\&R before branching on a node, it is looked up in the memory to see if it has already been visited. This idea has been used in different fields of combinatorial optimization \cite{jouglet2004branch,morin1976branch}. In the field of assembly line balancing problem, OptPack proposed by \cite{nourie1991finding} to solve SALBP-1 used a variation of BB\&R by storing the sub-problems in a tree structure. \cite{scholl1999balancing} also used a tree structure to store already solved sub-problems.
\cite{sewell2012branch} used BB\&R to great effect to solve the SALBP-1. For the U shaped line balancing problem \cite{scholl1999ulino} also used memory to store the sub-problems in a tree structure. The next subsections present the details of our proposed BP\&R algorithm.

\subsection{Branching}
In this section we illustrate the enumeration procedure for our proposed algorithm. The branch and price algorithm explores nodes of an enumeration tree. Each node represents a partial solution. We denote a partial solution as $ \mathcal{P}=(A,U,S_1,S_2,\dots,S_m ) $, where $ A $ and $ U $ are the sets of assigned and unassigned tasks, respectively; $ m $ is the number of stations used by the partial solution and  $ S_1,S_2,\dots,S_m $ are the sets of tasks assigned to stations 1 to $ m $. Based on these definitions we have $ A=\bigcup \limits_{i=1}^m S_i $  and $ U=T\backslash A $.

Branching on a partial solution (node) means extending it by adding one or many tasks from $ U $ to $ A $ such that the resulting solution is still feasible. There are two main types of branching in the field of assembly line balancing: station oriented branching and task oriented branching. In station oriented branching branches are generated by creating a complete load. However, in task oriented branching, new branches are generated by adding a single task to the current station, if the current station cannot accommodate the task, a new station is created. ULINO, the only branch and bound algorithm available in the literature to solve the U shaped line balancing problem, uses station oriented branching. The proposed BP\&R algorithm also uses station oriented branching for two main reasons: (1) Surveys suggest that station oriented branching tends to yield better results in comparison with task oriented branching. (2) The station oriented branching is a better fit for the BP\&R algorithm. It allows fast computation of column generation lower bound with only minor changes in the restricted master problem.
\subsection{Bounds}
\subsubsection{Upper bound}
Using a heuristic to find a good upper bound can significantly decrease the run time of BP\&R algorithm. We use a modification of the Hoffmann algorithm to obtain a high quality upper bound for the UALBP-1. Hoffmann algorithm \cite{hoffmann1963assembly} is an efficient heuristic for SALBP-1 that generates a solution on a station by station basis. At each iteration, it selects the work-load with smallest idle time. \cite{sewell2012branch} modified the Hoffmann algorithm to search for the load with maximum value of $ \sum_{j \in U} {(t_j+\alpha w_j+\beta |F_j |-\gamma)} $, instead of the load with minimum idle time. In this formula  $ F_j (F_j^* ) $ is the set of immediate (all) successors of task $ j $ and $ w_j=t_j+\sum_{k \in F_j^*} t_k $  is the positional weight of task $ j $. $ \alpha,\beta,\gamma $ are coefficients that need to be numerically determined.

We propose a Modified Hofmann Heuristic for U shaped lines (MHHU) that works as follows: at each iteration find the load that maximizes $ \sum_{j \in F} {(t_j+\alpha w_j+\beta|F_j |-\gamma)} +\sum_{j \in B} {(t_j+\alpha w_j^{'}+\beta |P_j|-\gamma)} $, where $ F $ is the set of  tasks that are assigned in the forward direction in the U shaped line, $ B $ is the set of  tasks that are assigned in the backward direction in the U shaped line,  $ P_j (P_j^* ) $ is the set of immediate (all) predecessors of task $ j $ and $ w_j^{'}=t_j+\sum_{k \in P_j^*} t_k $  is the backward positional weight of task $ j $. MHHU is called in every node in the BP\&R search tree before branching, to check if the current upper bound can be improved.
\subsubsection{Lower bounds}
In this section we present the set of lower bounds used in the proposed BP\&R. BP\&R uses three standard lower bounds available in the literature of assembly line balancing ($ LB_1 $ to $ LB_3 $) and a new column generation based bound (CG). $ LB_1 $ to $ LB_3 $ have been previously used in \cite{sewell2012branch}. For each node $ LB_1 $ to $ LB_3 $ are computed first, if they are not able to prune the node, then CG is computed.
$ LB_1 $ is the total capacity bound which is the simplest available bound in the literature. It follows from the fact that total available time must be greater than or equal to total work content, therefore $LB_1=\ceil{\frac{\sum_{j=1}^{n} t_j}{c}} $ \cite{baybars1986survey}. $ LB_2 $ is obtained by counting the number of tasks $ j $ with $ t_j>\frac{c}{2} $, because these tasks have to be assigned to different stations.
\[ LB_2=\left| \left\lbrace j \in T: t_j>\frac{c}{2} \right\rbrace \right| + \left\lceil \frac{|\{j \in T: t_j=c/2\}|}{2} \right\rceil \]
$ LB_3 $ is obtained by generalizing $ LB_2 $, this lower bound is obtained by adding task weights that are determined as follows: a weight of 1 is given to all tasks $ j $ with $ t_j>\frac{2c}{3} $, a weight of $ \frac{1}{2} $ is given to all tasks with $ t_j \in (\frac{c}{3},\frac{2c}{3}) $. Tasks $ j $ with $ t_j $ equal to $ \frac{c}{3} $ and $ \frac{2c}{3} $ are given weights of $ \frac{1}{3} $ and $ \frac{2}{3} $, respectively.
\[
w_j = \left\{\begin{array}{@{}l@{\quad}l}
1 & \mbox{if $t_j>2c/3$} \\[\jot]
2/3 & \mbox{if $t_j=2c/3$}  \\[\jot]
1/2 & \mbox{if $c/3<t_j<2c/3$} \\[\jot]
1/3 & \mbox{if $t_j=c/3$}
\end{array}\right.
\]
After computing the weights, $ w_j $, $ LB_3 $ is computed as $ LB_3=\ceil{\sum_{j=1}^{n} w_j} $.

If $ LB_1 $ to $ LB_3 $ fail to prune the node, column generation is used to obtain a tighter lower bound for the node. The idea behind this approach is to relax the precedence constraints to obtain a bin packing problem and use a column generation approach to obtain a lower bound for the resulting bin packing problem. There are other algorithms to solve the bin packing problem more efficiently. However, we use column generation because the generated columns for one node in the BP\&R search tree can be used to compute the CG lower bound for other nodes. Therefore, after the first run of column generation i.e. at the root node, the subsequent runs will be faster because of the already generated columns. BP\&R maintains a pool of generated columns and uses this pool whenever CG is called at a node.

The objective of column generation is to compute the lower bound for a partial solution $ \mathcal{P}=(A,U,S_1,S_2,\dots,S_m ) $, which corresponds to a node in the BP\&R search tree. After relaxing the precedence constraints, a bin packing problem with the set of unassigned tasks $ U $, as the set of available items (to be fit in a minimum number of bins i.e. stations) is obtained. We define a packing, or a workload or load, as a set of tasks that can be assigned to one station without violating the cycle time constraint. We denote the set of all possible loads by $ L $. Indexing loads by $ l $, and relaxing the precedence constraints we can formulate the UALBP-1 as following:
\begin{flalign}
\label{eq1}
&min \sum_{l\in L} x_l \\ \label{eq2}
&s.t. \quad  \sum_{l \in L} a_{jl} x_l \ge 1 \quad \forall j \in U \\ \label{eq3}
& \quad \quad x_l \in \{0,1\} \quad   \forall l \in L
\end{flalign}

In this formulation, $ x_l $ is a binary variable which is equal to 1 if load $ l $ is selected and 0 otherwise. $a_{jl} $ is a binary parameter that is equal to 1 if task $ j $ belongs to packing $ l $, zero otherwise. The objective function in this formulation is to minimize the number of selected loads. Number of selected loads is equivalent to the number of stations because each load is assigned to one station. Constraint (2) ensures that each unassigned task $ j\in U  $ is in at least one of the selected loads. Constraint (3) is the integrality constraint for variable $ x_l $. We consider the LP relaxation of (\ref{eq1})-(\ref{eq3}):
\begin{flalign}
\label{eq4}
&min \sum_{l \in L} x_l\\\label{eq5}
&s.t. \quad  \sum_{l \in L} {a_{jl} x_l} \ge 1 \quad \forall j \in U	\\ \label{eq6}
& \quad \quad x_l \ge 0 \quad  \forall l \in L
\end{flalign}

The LP characterized by constraints (\ref{eq4}) to (\ref{eq6}) is called the linear programming master problem (LPM). Note that each column in the LPM corresponds with a work-load. In general the set $ L $, may be exponentially large; however, the number of non-zero variables (the basic variables) in the LPM is equal to the number of constraints, $ |U| $. Therefore, even though the number of possible loads $ L $ is large, only a small number of them is used in the optimal solution. Column generation algorithm uses this idea to start with a subset  $ L^{'} \subseteq L $ of columns and generate columns as needed. The starting subset $ L^{'} $ should be selected such that the following problem is feasible:
\begin{flalign}
&min \sum_{l \in L^{'}} x_l\\
&s.t. \quad   \sum_{l \in L^{'}} a_{jl} x_l \ge 1 \quad \forall j \in U	\\
& \quad \quad x_l \ge 0 \quad \forall l \in L^{'}
\end{flalign}

This problem is called Restricted LPM (RLPM). To initialize the column generation algorithm in the root node, we use the loads generated by the MHHU algorithm. Using variable $ \pi_j $ for constraint $j,$ dual of RLPM is as follows:
\begin{flalign}
&max \sum_{j \in U} \pi_j  \\
&s.t. \quad  \sum_{j \in U} {a_{jl}\pi_{j}} \le 1 \quad \forall l \in L^{'}\\
& \quad \quad  \pi_j \ge 0 \quad \forall j \in U
\end{flalign}

Next step is to find a column (load) in $ L\backslash L^{'} $ that could improve the current optimal solution of RLPM. Given the optimal dual solution $ \boldsymbol{\bar{\pi}}=(\bar{\pi}_1,\bar{\pi}_2,\dots,\bar{\pi}_n) $ of (RLPM), the reduced cost of column $ l \in L\backslash L^{'} $ is $ 1-\sum_{j \in U} a_{jl} \bar{\pi}_j$. Based on the concept of duality in linear programming, optimality of RLPM is equivalent with feasibility of the dual. Therefore, loads that violate constraint $ \sum_{j \in U} {a_{jl}\bar{\pi}_{j}} \le 1 $ can improve the current optimal solution. Therefore we should look for a column (load) $ l $ such that: $ 1-\sum_{j \in U} {a_{jl}\bar{\pi}_{j}} <0 $. Note that $ \bar{\pi}_j $ is fixed, and the problem is to find a load $ l $ with $ a_{jl} $ such that: $ 1-\sum_{j \in U} {a_{jl}\bar{\pi}_{j}} <0 $. This problem is called the pricing sub-problem.
The pricing sub-problem involves finding a set of tasks, a work-load, with a negative reduced cost. Because this work-load does not have to observe the precedence constraints, the problem of finding a work-load with a negative reduced cost is a knapsack problem. To solve this sub-problem, consider the set of unassigned tasks $ U $ and solve the knapsack problem with $ U $ as the set of available items and $ \bar{\pi}_j $ as their values to maximize the overall value $ \sum_{j \in U}{a_{jl}\bar{\pi}_j} $ subject to cycle time constraint. We use the well-known dynamic programing method to solve the generated knapsack problems, which runs in pseudo-polynomial time i.e. $ O(|U|c) $. We also use a dual bound to obtain a lower bound on the number of stations in each iteration

\begin{lemma}\label{lemma1} Let $ v(\text{RLPM}) $ and $ v(\text{LPM}) $ denote the optimum objective function value of the current $ \text{RLPM} $ and $ \text{LPM} $, respectively. Also let $ v(\text{PSP}) $ be the minimum reduced cost obtained by solving the pricing sub-problem to optimality.
	\begin{enumerate}[label=(\roman*)]
		\item \label{parti} We have: $ v(LPM) \ge \frac{v(RLPM)}{1-v(\text{PSP})}$.
		\item \label{partii} If $ \ceil{v(RLPM)}=\ceil{\frac{v(RLPM)}{1-v(\text{PSP})}} $ we can terminate the column generation algorithm with $ \ceil{v(RLPM)} $ as the obtained lower bound on the number of stations.
	\end{enumerate}

\end{lemma}
\begin{proof}
	General form of part \ref{parti} result can be found in \cite{lubbecke2011column}. During each iteration of the column generation algortihm one cannot improve $ v(\text{\textit{RLPM}}) $ by more than $ \sum_{l\in L} x_l $ times the smallest reduced cost $ v(\text{\textit{PSP}}) $. Moreover, we know that $ \sum_{l\in L} x_l =v(\text{LPM}) $ for an optimal solution of the master problem. Therefore we have:
	$  v(\text{\textit{LPM}}) \ge  v(\text{\textit{RLPM}})+(\sum_{l\in L} x_l)v(\text{\textit{PSP}})=v(\text{\textit{RLPM}})+v(\text{\textit{LPM}})v(\text{\textit{PSP}})$. The part \ref{parti} follows from this inequality.\\
	To prove part \ref{partii} we have: $  \frac{v(RLPM)}{1-v(\text{PSP})} \le v(LPM) \le v(RLPM)$. Because the number of stations should be a whole number, we can use $ \ceil{v(LPM)} $ as a lower bound on the number of stations. Moreover, from $ \ceil{v(RLPM)}=\ceil{\frac{v(RLPM)}{1-v(\text{PSP})}} $, we can conclude that $ \ceil{v(LPM)} = \ceil{v(RLPM)}=\ceil{\frac{v(RLPM)}{1-v(\text{PSP})}}$
\end{proof}

\begin{remark}
	Note that in order for the bound in lemma \ref{lemma1} to be valid, the pricing subproblem should be solved to optimality. In general, the dual bound is not monotone over the iterations, this is called the yo-yo effect \cite{lubbecke2011column}.
\end{remark}

Algorithm \ref{alg1} shows the pseudo-code for the overall column generation algorithm.
\begin{algorithm}
	Initialize columns $ \mathcal{L}^\prime $.\\
	Solve RLPM. Let $ \overline{\bm{\pi}}= (\overline{\pi}_1, \overline{\pi}_2, \dots, \overline{\pi}_n) $ be the obtained optimal dual values.\\
	Solve the pricing sub-problem using $ \overline{\bm{\pi}}$ as item values and let $ \textbf{a}^{\textbf{*}}_\textbf{l}=(a_{1l}^*, a_{2l}^*, \dots, a_{nl}^*) $ be the obtained optimal solution.

	\eIf{$ 1-\sum_{j \in U} a_{jl}^* \overline{\pi}_j<0  \text{ and } \ceil{\sum_{j \in U}\overline{\pi}_j} \ne \ceil{\frac{\sum_{j \in U}\overline{\pi}_j}{\sum_{j \in U} a_{jl}^* \overline{\pi}_j}}$}{
		Add the new column $ \textbf{a}^{\textbf{*}}_\textbf{l} $ to RLPM.\\
		Go To Line 2.
	}{
		Return $\sum_{j \in U}\overline{\pi}_j$ as the obtained lower bound.\\
		Terminate the procedure.
	}
	\caption{Pseudo-code for the overall column generation algorithm}
	\label{alg1}
\end{algorithm}
As seen in this pseudo-code, the column generation algorithm starts with a set of initial columns. This set is obtained by the MHHU algorithm in the root node; in other nodes the pool of already generated columns is used as initial set of columns. The RLPM is solved using this set of initial columns and the vector of dual values is obtained. Dual values are then used in the pricing sub-problem to generate a new column. If a new column with a negative reduced cost is obtained and the termination criterion from lemma \ref{lemma1} is not satisfied, it is added to the RLPM and the process is repeated; otherwise the procedure terminates.
\subsection{Use of memory, dominance rules and search strategy }
As its name suggests, BP\&R memorizes already considered nodes. Before considering a node $ \mathcal{N}=(A,U,S_1,S_2,\dots,S_m ) $, BP\&R checks the memory to see if there is a node with the same set of assigned tasks, $ A $, if there is such a node $ \mathcal{M}=(A,U,S_1^{'},S_2^{'},\dots,S_{m^{'}}^{'} ) $ and $ m^{'} \le m $, then $ \mathcal{N} $ is dominated by $ \mathcal{M} $ and $ \mathcal{N} $ can be pruned. OptPack \cite{nourie1991finding}, SALOME \cite{scholl1997salome} and ULINO \cite{scholl1999ulino} applied this dominance rule using a compact tree structure. \cite{sewell2012branch} used a hash table to store the sub-problems. BP\&R Also uses a hash table to store already visited nodes.

Other than the memory based dominance rule, BP\&R uses the maximally loaded dominance rule. This rule states that if a node contains a station that is not maximally loaded it is dominated by a node that contains only maximally loaded stations.

The third dominance rule used in BP\&R is the Jackson rule which was used in SALOME \cite{scholl1997salome} and BB\&R \cite{sewell2012branch}. ULINO \cite{scholl1999ulino} uses a modification of this rule for a U shaped line, which we will directly apply in our proposed BP\&R.

BP\&R uses a best first search strategy where the nodes with higher number of, fixed, stations have the higher priority. If two nodes have the same number of stations, the one with lower CG lower bound is selected to be branched on. A priority queue is used to handle the generated nodes.
Algorithm \ref{alg2} shows the pseudo-code for the overal BP\&R algorithm. The algorithm starts with setting the root node and computing the upper and lower bounds. If these bounds can prove optimality, i.e. $ LB=UB $, then we are done and $ UB $ can be reported as the optimum number of stations. Otherwise if $ LB<UB $, we initialize the branch and bound tree by initializing a Hash table $ H $ and a priority queue $ PQ $ in lines 5 and 6, respectively. In line 7 we push the root node into the priority queue. The while loop in line 8 starts the best first search process. In lines 9 and 10, we remove a node $ \mathcal{M} $ from the top of the priority queue. Line 11 starts the branching process on node $ \mathcal{M} $ by generating all of the possible children of this node. In line 12, for each generated child we check if it is dominated or if it can be pruned by a lower bound argument. If the child node is dominated or can be pruned based on its computed lower bound, it will be discarded. Otherwise, we compute an upper bound based on node $ \mathcal{C} $ using MHHU approach and try to find an improved upper bound. Lines 19 and 20 add the new child node $ \mathcal{C} $ to the priority queue and the Hash table. Note that Algorithm \ref{alg2} does not reflect the subtlety of order of computing $ LB_1 $ to $ LB_3 $ and column generation bound. As mentioned in the lower bounds section, because $ LB_1 $  to $ LB_3 $ are faster to compute, we compute them first in the hope of pruning the node. If they fail to prune the node, we compute the column generation bound.

\begin{algorithm}
	Initialize the root node $ \mathcal{N};$\\
	$ UB :=$ Compute\_upper\_bound $(\mathcal{N});$ \\
	$ LB :=$ Compute\_lower\_bound $(\mathcal{N});$ \\
	\If{$ LB<UB $}{
		Initialize Hash Table $ H ;$\\
		Initialize Priority Queue $ PQ;$\\
		$ PQ.push(\mathcal{N});$\\
		  \While{$ PQ $ is not empty}{
		  	$ \mathcal{M}:=PQ.top(); $\\
		  	$ PQ.pop(); $\\
			\ForEach{child $ \mathcal{C} $ of $ \mathcal{M} $}
			{

				\eIf{C is dominated OR Compute\_lower\_bound $(\mathcal{C})$ $ \ge UB$ }{
					Discard $ \mathcal{C};$\\
					Continue\;}{
						\If{Compute\_upper\_bound $(\mathcal{C})$ $ < UB$ }{
						Update $ UB;$}
					$ PQ.push(\mathcal{C});$\\
					$ H \leftarrow \mathcal{C};$
				}

			}
		}
	}
Output $ UB $ as the obtained optimum number of stations;
	\caption{Pseudo-code for the overall BP\&R}
	\label{alg2}
\end{algorithm}

\section{Computational results} \label{sec3}
In this section we evaluate the BP\&R and CG algorithms in terms of quality of obtained solutions and run times.  The algorithms are coded in C++ and CPLEX 12.6 solver has been used to solve the LPs. The computational experiments are performed on a computer with
Intel Core i7 2.00 GH processor and 8 GB of RAM. We use 269 benchmark problems of SCHOLL data set as well as a new data set generated by \cite{otto2013systematic} to perform the experiments. Both of these data sets are available at the assembly line balancing research homepage (http://alb.mansci.de/). Each instance in the data sets was limited to a total processing time of 500 seconds.
\subsection{Experiments on SCHOLL data set}

In this subsection we run experiments on the 269 benchmark problems of SCHOLL. To compare the lower bounds $ LB_1 $ to $ LB_3 $ and CG with each other, we run these algorithms on all of the 269 benchmark problems. Table \ref{table0} presents the results of this experiment. In this table,  the relative deviation is computed as $ \frac{UB-LB}{UB} \times 100 \%$, where LB is the obtained lower bound and UB is the best known upper bound. Avg. Rel. Dev. and Max. Rel. Dev. rows are the reported average and maximum relative deviations, respectively. Avg. Abs. Dev. and Max. Abs. Dev. rows are the reported average and maximum absolute deviations, respectively. \#Opt Found row reports the number of cases where the obtained lower bound matched the best known upper bound.

As seen in table \ref{table0}, CG outperforms the standard lower bounds $ LB_1 $ to $ LB_3 $ in all of the performance criteria except for the average CPU time. Moreover, the lower bounds obtained by CG match the best known solution in 238 cases and for the remaining cases, it deviates from the best known solution by only 1 station. This shows that CG obtains tight lower bounds for the UALBP-1, however its average CPU time is relatively high. This may raise questions over its applicability in a branch and bound procedure. To address this question, we consider a modification of BP\&R which does not use CG, we refer to this algorithm as BP\&R without CG (BP\&RWOCG).

{\setstretch{1.0}
\begin{table}[H]
	\centering
	\caption{Comparison of lower bounds}

	{
		\begin{tabular}{ccccc}
			\hline
			& $ LB_1 $ & $ LB_2 $ & $ LB_3 $ & CG \\\hline
			Avg. Rel. Dev. (\%)	&3.39  & 63.13& 46.69 & \textbf{0.85}\\
			Max. Rel. Dev. (\%)	& 37.29 &  100& 100 & \textbf{20} \\
			Avg. Abs. Dev. 	&1.23  & 11.39 & 9.25 & \textbf{0.11}\\
			Max. Abs. Dev. 	& 22 & 37 & 36 & \textbf{1}\\
			\#Opt Found 	& 194 & 13 & 14 & \textbf{239}\\
			Avg. CPU (s)	& \textbf{0.00}  &\textbf{ 0.00} & \textbf{0.00} & 4.33\\ \hline
		\end{tabular}
	}
	\label{table0}
\end{table}
}

In the next experiment we investigate the performance of BP\&R and BP\&RWOCG. We run these algorithms on  all of the 269 benchmark problems and report the results in table \ref{table1}. In this table  ``Optimal found" counts the number of cases where the optimal solution is obtained. ``Optimal verified" counts the number of cases where the obtained solution is proven to be optimal. ``Avg. dev." and ``Max. dev." represent the average and maximum relative deviation from lower bound. The relative deviation is computed as $ \frac{UB-LB}{LB} \times 100 \%$, where UB and LB are obtained upper and lower bounds, respectively. ``Avg. CPU verified (s)" shows the average run times in seconds on the problems for which optimality was verified. ``Avg. CPU all (s)" shows the average run time, in seconds, for all instances. For 6 cases, BP\&RWOCG runs out of memory, these cases are not considered when computing the average CPU times for BP\&RWOCG. The results of ULINO algorithm are taken from (Scholl and Klein 1999b). They coded their algorithm using Borlands Pascal 7.0. and performed tests
on an IBM-compatible personal computer with a 80486 DX2-66 central
processing unit and the operating system MS DOS 6.2. They also applied a time limit of 500 seconds per instance (excluding input and output operations).

\begin{table}[H]
	\centering
	\caption{Comparison of BP\&R with BP\&RWOCG and ULINO}
		\begin{tabular}{cccc}
		\hline
		& ULINO & BP\&R & BP\&RWOCG \\\hline
		Optimal found	& 233 & \textbf{255} & 250\\
		Optimal verified	& 233 & \textbf{255} & 233\\
		Avg. Rel. Dev. (\%)	&0.59  & \textbf{0.26}& 0.60 \\
		Max. Rel. Dev. (\%)	& 10 &  \textbf{7.14}& 11.76 \\
		Avg. Abs. Dev. 	&0.13  & \textbf{0.05} & 0.16 \\
		Max. Abs. Dev. 	& \textbf{1} &  \textbf{1} & 4\\
		Avg. CPU verified (s)	& - & 9.11 & 2.34 \\
		Avg. CPU all (s)	& 82.09  & \textbf{34.66} & 59.11 \\ \hline
\end{tabular}
	\label{table1}
\end{table}

As seen in table \ref{table1}, out of 269 benchmark problems, BP\&R is able to solve and verify optimality of 255 problems. Out of these 255 problems, 216 where closed at the root node. Meaning that, for these problems, CG and MHHU where able to prove optimality without branching.

MHHU was able to find the optimal solution in 232 instances out of 269 benchmark problems. In 36 instances, MHHU deviates from optimal solution by only 1 station. For one instance, MHHU deviates from optimal solution by 2 stations.

Comparing BP\&R with ULINO, BP\&R outperforms ULINO in terms of number of optimal solutions found and verified, BP\&R also does better than ULINO in terms of average and maximum deviations from lower bound. ULINO never does better than BP\&R, in terms of upper and lower bounds, for any benchmark problem. However, the run times cannot be compared because of different computer used to run ULINO. It is noteworthy to mention that BP\&R was able to verify the optimal solutions in less than 10 seconds on average.

Comparing BP\&R with BP\&RWOCG, BP\&R outperforms BP\&RWOCG in all performance criteria except for Avg. CPU verified (s), where BP\&RWOCG does better. This highlights the importance of using CG as a lower bound in  BP\&R. Because $ LB_1 $ to $ LB_3 $ are faster than CG, BP\&R uses them first, if they fail to prune a node, CG is called to obtain tighter bounds. For 269 benchmark problems, a total of 4,878,445 nodes where searched. $ LB_1 $ to $ LB_3 $ are called for all of these nodes, out of which 4,335,666 where pruned. Out of the remaining 542,779 nodes CG was able to prune 395,649. On average, CG improved the number of pruned nodes by 9.12 percent.

Table \ref{table2} shows the detailed results for the challenging problems where BP\&R was able to find and verify the optimal solution. Challenging problems are defined to be the ones ULINO was not able to verify optimality. For some of these problems, the optimal solution is verified from SALBP-1 results in the literature. These results are labeled ``Known optimal (or range)" in the table. The instances for which optimality is verified are highlighted in boldface. The open problems solved by BP\&R are determined by a star ``*" besides the results. As seen in table \ref{table2}, BP\&R is able to solve 13 open problems. Moreover, there are 9 problems for which ULINO was not able to verify optimality but the lower bound obtained from solving SALBP-1 verifies optimality of these problems. BP\&R is able to find and verify the optimal solution for these 9 problems without using the SALBP-1 optimal solution as a lower bound.

Table \ref{table3} shows the results obtained for the remaining 14 problems that BP\&R was not able to solve. As seen in this table, 13 of the unsolved problems are ``Arcus 2" with different cycle times, and 1 problem is ``Scholl" with cycle time of 1422. As seen in this table, for all of the 14 problems, both ULINO and BP\&R have the same performance in terms of upper and lower bounds.
\begin{table}[H]
	\centering
	\caption{Solved challenging problems}

		\begin{tabular}{ccccccccccccc}
			\hline
			\multicolumn{3}{c}{Problem} &  & \multicolumn{2}{c}{\begin{tabular}[c]{@{}c@{}}Known optimal\\ (or range)\end{tabular}} &  & \multicolumn{2}{c}{ULINO} &  & \multicolumn{3}{c}{BP\&R}                      \\ \cline{1-3} \cline{5-6} \cline{8-9} \cline{11-13}
			Name        & n     & c     &  & LB                                         & UB                                         &  & LB          & UB          &  & LB           & UB           & CPU              \\ \cline{1-3} \cline{5-6} \cline{8-9} \cline{11-13}
			Arcus 1     & 83    & 3786  &  & \textbf{21}                                & \textbf{21}                                &  & 21          & 22          &  & \textbf{21}  & \textbf{21}  & \textbf{2.04}    \\
			Arcus 2     & 111    & 11570  &  & \textbf{13}                                & \textbf{13}                                &  & 13          & 14          &  & \textbf{13}  & \textbf{13}  & \textbf{65.64}    \\
			Barthol2    & 148   & 85    &  & \textbf{50}                                & \textbf{50}                                &  & 50          & 51          &  & \textbf{50}  & \textbf{50}  & \textbf{1.45}    \\
			Barthol2    & 148   & 89    &  & \textbf{48}                                & \textbf{48}                                &  & 48          & 49          &  & \textbf{48}  & \textbf{48}  & \textbf{0.83}    \\
			Barthol2    & 148   & 93    &  & \textbf{46}                                & \textbf{46}                                &  & 46          & 47          &  & \textbf{46}  & \textbf{46}  & \textbf{0.80}    \\
			Barthol2    & 148   & 97    &  & \textbf{44}                                & \textbf{44}                                &  & 44          & 45          &  & \textbf{44}  & \textbf{44}  & \textbf{0.84}    \\
			Kilbridge   & 45    & 56    &  & \textbf{10}                                & \textbf{10}                                &  & 10          & 11          &  & \textbf{10}  & \textbf{10}  & \textbf{0.11}    \\
			Mukherje    & 94    & 176   &  & 24                                         & 25                                         &  & 24          & 25          &  & \textbf{24*} & \textbf{24*} & \textbf{0.52*}   \\
			Scholl      & 297   & 1394  &  & \textbf{50}                                & \textbf{50}                                &  & 50          & 51          &  & \textbf{50}  & \textbf{50}  & \textbf{20.78}   \\
			Scholl      & 297   & 1515  &  & \textbf{46}                                & \textbf{46}                                &  & 46          & 47          &  & \textbf{46}  & \textbf{46}  & \textbf{22.96}   \\
			Tonge       & 70    & 160   &  & 22                                         & 23                                         &  & 22          & 23          &  & \textbf{22*} & \textbf{22*} & \textbf{7.49*}   \\
			Tonge       & 70    & 176   &  & 20                                         & 21                                         &  & 20          & 21          &  & \textbf{20*} & \textbf{20*} & \textbf{7.19*}   \\
			Warnecke    & 58    & 54    &  & 30                                         & 31                                         &  & 30          & 31          &  & \textbf{31*} & \textbf{31*} & \textbf{0.89*}   \\
			Warnecke    & 58    & 62    &  & 26                                         & 27                                         &  & 26          & 27          &  & \textbf{26*} & \textbf{26*} & \textbf{3.62*}   \\
			Warnecke    & 58    & 65    &  & 24                                         & 25                                         &  & 24          & 25          &  & \textbf{25*} & \textbf{25*} & \textbf{0.49*}   \\
			Warnecke    & 58    & 68    &  & 23                                         & 24                                         &  & 23          & 24          &  & \textbf{23*} & \textbf{23*} & \textbf{8.60*}   \\
			Warnecke    & 58    & 71    &  & 22                                         & 23                                         &  & 22          & 23          &  & \textbf{22*} & \textbf{22*} & \textbf{155.44*} \\
			Warnecke    & 58    & 74    &  & 21                                         & 22                                         &  & 21          & 22          &  & \textbf{22*} & \textbf{22*} & \textbf{3.95*}   \\
			Warnecke    & 58    & 82    &  & 19                                         & 20                                         &  & 19          & 20          &  & \textbf{19*} & \textbf{19*} & \textbf{7.42*}   \\
			Wee-mag     & 75    & 47    &  & 32                                         & 33                                         &  & 32          & 33          &  & \textbf{32*} & \textbf{32*} & \textbf{5.05*}   \\
			Wee-mag     & 75    & 49    &  & 31                                         & 32                                         &  & 31          & 32          &  & \textbf{32*} & \textbf{32*} & \textbf{0.63*}   \\
			Wee-mag     & 75    & 50    &  & 31                                         & 32                                         &  & 31          & 32          &  & \textbf{32*} & \textbf{32*} & \textbf{0.70*}   \\ \hline
		\end{tabular
	}
	\label{table2}
\end{table}

\begin{table}[H]
	\centering
	\caption{Remaining unsolved problems}
	{
		\begin{tabular}{cccccccccccc}
		\hline
		\multicolumn{3}{c}{\multirow{2}{*}{Problem}} & \multirow{2}{*}{} & \multicolumn{2}{c}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Known range\end{tabular}}} & \multirow{2}{*}{} & \multicolumn{2}{c}{\multirow{2}{*}{ULINO}} & \multirow{2}{*}{} & \multicolumn{2}{c}{\multirow{2}{*}{BP\&R}} \\
		\multicolumn{3}{c}{}                         &                   & \multicolumn{2}{c}{}                                                                                &                   & \multicolumn{2}{c}{}                       &                   & \multicolumn{2}{c}{}                       \\ \cline{1-3} \cline{5-6} \cline{8-9} \cline{11-12}
		Name            & n           & c            &                   & LB                                                 & UB                                                 &                   & LB                   & UB                  &                   & LB                   & UB                  \\ \cline{1-3} \cline{5-6} \cline{8-9} \cline{11-12}
		Arcus 2         & 111         & 5785         &                   & 26                                                 & 27                                                 &                   & 26                   & 27                  &                   & 26                   & 27                  \\
		Arcus 2         & 111         & 6016         &                   & 25                                                 & 26                                                 &                   & 25                   & 26                  &                   & 25                   & 26                  \\
		Arcus 2         & 111         & 6267         &                   & 24                                                 & 25                                                 &                   & 24                   & 25                  &                   & 24                   & 25                  \\
		Arcus 2         & 111         & 6540         &                   & 23                                                 & 24                                                 &                   & 23                   & 24                  &                   & 23                   & 24                  \\
		Arcus 2         & 111         & 6837         &                   & 22                                                 & 23                                                 &                   & 22                   & 23                  &                   & 22                   & 23                  \\
		Arcus 2         & 111         & 7162         &                   & 21                                                 & 22                                                 &                   & 21                   & 22                  &                   & 21                   & 22                  \\
		Arcus 2         & 111         & 7520         &                   & 20                                                 & 21                                                 &                   & 20                   & 21                  &                   & 20                   & 21                  \\
		Arcus 2         & 111         & 7916         &                   & 19                                                 & 20                                                 &                   & 19                   & 20                  &                   & 19                   & 20                  \\
		Arcus 2         & 111         & 8356         &                   & 18                                                 & 19                                                 &                   & 18                   & 19                  &                   & 18                   & 19                  \\
		Arcus 2         & 111         & 8847         &                   & 17                                                 & 18                                                 &                   & 17                   & 18                  &                   & 17                   & 18                  \\
		Arcus 2         & 111         & 9400         &                   & 16                                                 & 17                                                 &                   & 16                   & 17                  &                   & 16                   & 17                  \\
		Arcus 2         & 111         & 10027        &                   & 15                                                 & 16                                                 &                   & 15                   & 16                  &                   & 15                   & 16                  \\
		Arcus 2         & 111         & 10743        &                   & 14                                                 & 15                                                 &                   & 14                   & 15                  &                   & 14                   & 15                  \\

		Scholl          & 297         & 1422         &                   & 49                                                 & 50                                                 &                   & 49                   & 50                  &                   & 49                   & 50                  \\ \hline
	\end{tabular}
}
	\label{table3}
\end{table}

\subsection{Experiments on the new data set}
Otto et al \cite{otto2013systematic} published a new problem database of generated instances for the SALBP. In this subsection we use this data set to evaluate the performance of BP\&R algorithm. Eventhough this data set is originally devised for the SALBP we can use BP\&R to solve the U shaped version of the problems inside this data set.

Table \ref{table_5_otto} shows the results of BP\&R on the new data set. As seen in this table, all of the small instances, 99.73 percent of medium instances, 91.05 percent of large instances and 65.9 percent of very large instances are solved by BP\&R. The fourth column, ``\# MHHU found Optimum", shows the number of instances where MHHU found the optimal solution. The fifth column, ``\# Closed in root node", shows the number of instances that where closed in the root node. In other words, for these instances MHHU finds the optimal solution and the lower bounds prove its optimality without need for further branching.

\begin{table}[H]
	\centering
	\caption{Number of solved problem instances}
		{
	\begin{tabular}{ccccc}
		\hline
		Size       & \# BP\&R Solved & \% BP\&R Solved & \# MHHU found Optimum & \# Closed in root node \\ \hline
		Small      & 525             & 100.00          & 486                & 481                 \\
		Medium     & 5236            & 99.73           & 4118               & 4062                \\
		Large      & 478             & 91.05           & 359                & 359                 \\
		Very Large & 346             & 65.90           & 346                & 346                 \\ \hline
	\end{tabular}}
\label{table_5_otto}
\end{table}

Table \ref{Table_6} shows the run times statistics of BP\&R. As seen in this table, all of the run times statistics increase as the problem size increases.

\begin{table}[H]
	\centering
	\caption{Run times of BP\&R (Seconds)}
	{
	\begin{tabular}{ccccc}
		\hline
		Size       & Mean    & Standard deviation & Max     & Min    \\ \hline
		Small      & 0.164   & 0.222              & 2.224   & 0.007  \\
		Medium     & 3.544   & 29.319             & 501.235 & 0.014  \\
		Large      & 60.540  & 146.494            & 509.818 & 0.109  \\
		Very Large & 253.994 & 188.356            & 532.686 & 15.300 \\ \hline
	\end{tabular}}
	\label{Table_6}
\end{table}

An analysis of instances unsolved by BP\&R is presented in Table \ref{Table_7}. This table shows how unsolved instance are divided in terms of problem characteristics. As seen in this table, for problem of medium size, all of the unsolved instances have a mid range order strength (0.6) and a central distribution of task time. Moreover, problems with bottlenecks (BN) in their precedence graph structure tend to be harder to solve for BP\&R. For problems of largers size, problems with central task time distribution still make up the majority of unsolved instances. In terms of order strength, problems with mid range to low order strength tend to be more challenging for BP\&R to solve. Moreover, in terms of graph structure, mixed problems appear to be harder to solve.

\begin{table}[H]
	\centering
	\caption{Unsolved problem statistics}
	{
	\begin{tabular}{ccccccccccccc}
		\hline
		\multirow{2}{*}{Size} &  & \multicolumn{3}{c}{Structure} &  & \multicolumn{3}{c}{Order Strength} &  & \multicolumn{3}{c}{Peak Location} \\ \cline{3-5} \cline{7-9} \cline{11-13}
		&  & BN       & CH      & MIX      &  & 0.2        & 0.6       & 0.9       &  & Bottom    & Central   & Bimodal   \\ \hline
		Medium                &  & 11       & 2       & 1        &  & 0          & 14        & 0         &  & 0         & 14        & 0         \\
		Large                 &  & 9        & 4       & 34       &  & 0          & 24        & 23        &  & 0         & 45        & 2         \\
		Very Large            &  & 51       & 51      & 77       &  & 76         & 76        & 27        &  & 0         & 175       & 4         \\ \hline
	\end{tabular}}
\label{Table_7}
\end{table}

\section{Conclusions and future research}\label{sec4}
In this paper we developed a branch, price and remember algorithm for the U shaped assembly line balancing problem. Computational results show that the proposed algorithm verifies the optimal solution for 255 of the 269 benchmark problems available in the literature. Previous best known exact algorithm is able to solve 233 of the 269 benchmark problems. The proposed approach uses column generation to obtain tight bounds that are computed reasonably fast. It also uses memory to avoid revisiting already visited nodes in the search tree.
The proposed algorithm is designed for the U shaped assembly line balancing problem, but the idea of branch, price and remember can be used to obtain tight bounds for other types generalized assembly line balancing problems such as two sided lines, balancing and scheduling problems with sequence dependent setup times, learning etc. Therefore, as a possible future research in this area, we suggest using the ideas in this paper to develop branch and price algorithms for other types of assembly line balancing problem.\\
\textbf{References}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Evan Longfield, M.D., is a Texas native with roots in San Antonio and Lampasas County. He earned his bachelor's degree with honors in Humanities and Fine Arts at Milligan College. He completed his post baccalaureate premedical requirements at UT: San Antonio while working as a professional photographer. He earned his medical degree at Baylor College of Medicine, where he received a grant to collaborate with the Houston Museum of Fine Arts in offering a medical humanities course. Dr. Longfield completed his residency in Otolaryngology — Head & Neck Surgery (Ear, Nose, and Throat) at the Louisiana State University Health Sciences Center, New Orleans. While in Residency, he presented original research at multiple national meetings and was awarded 1st place at his program's annual resident research day.
He provides comprehensive care to both children and adults. His areas of special interest are the treatment of chronic sinusitis and balloon sinuplasty; pediatric ENT; and treatment of salivary disorders, including sialoendoscopy and thyroid and parathyroid surgeries.
Dr. Longfield welcomes new patients and accepts most insurance plans.


The event was held in the framework of the Embassy Day Asia-Pacific, which is part of activities within the Asia Pacific Week in Berlin (APW) 2018 opened in Berlin on the same day.
Themed "Startups in the context of the 4th Industrial Revolution: Experience from Germany and lessons for Vietnam", the workshop drew many experts, scholars, and representatives from Germany's enterprises, who gave valuable suggestion and advices for Vietnam on how to gain successes in the new field.
Participants to APW 2018 focused their discussion on digitalization, which is an overwhelming trend in the 21st century.
Initiated in 1997 and annually held from 2015, APW is a forum for countries in Asia Pacific and Germany to discuss important topics, from economy, politics, to science research, and socio-culture.
Digitalization, with the focus being the 4th Industrial Revolution, is impacting on and spreading to all areas of life. In 2017, as the chair of the group of 20 developing nations (G20), Germany had made digitization a focus in the agenda of the group.
Digitalization creates great opportunities, but it also comes with many challenges.
Through APW 2018, Germany wants to discuss the issue with its partners in Asia-Pacific, which is considered the most dynamic development area in the world, thus seeking better cooperation opportunities among parties in this new field.


text:
Q: Eclipse generates my classes three times in wtpwebapps i had an issue with PrettyFaces framework that a class is found more than once when using eclipse to run application on server (tomcat 6)
what i found is that eclipse generates the class more than once in wtpwebapps:
\WEB-INF\classes\com\myapp\sample\beans\myclass.class
WEB-INF\classes\target\classes\com\myapp\sample\beans\myclass.class
\WEB-INF\classes\target\app-name\WEB-INF\classes\com\myapp\sample\beans\myclass.class

UPDATE:
i am using m2eclipse plugin, and using maven as my build tool, maybe there's a configuration wrong with them, or something needed to be configured.
UPDATE2:
here's the pom file build, maybe i have something wrong with it:
<build>

        <finalName>appName</finalName>

        <resources>

            <resource>
                <directory>src/main/resources</directory>
                <filtering>true</filtering>
            </resource>

        </resources>

        <plugins>
            <plugin>
                <groupId>org.apache.maven.plugins</groupId>
                <artifactId>maven-compiler-plugin</artifactId>
                <configuration>
                    <source>1.6</source>
                    <target>1.6</target>
                </configuration>
            </plugin>
            <plugin>
                <groupId>org.apache.maven.plugins</groupId>
                <artifactId>maven-dependency-plugin</artifactId>
                <executions>
                    <execution>
                        <id>install</id>
                        <phase>install</phase>
                        <goals>
                            <goal>sources</goal>
                        </goals>
                    </execution>
                </executions>
            </plugin>

            <plugin>
               <groupId>org.apache.maven.plugins</groupId>
               <artifactId>maven-resources-plugin</artifactId>
               <version>2.5</version>
               <configuration>
               <encoding>UTF-8</encoding>
               </configuration>
            </plugin>
        </plugins>
    </build>

any ideas how to fix this issue ?

A: I have you installed the "m2e wtp" connector ?   Maven -> Discovery -> Open Catalog  (in Indigo) search "wtp".
Consider using:
<properties>
    <project.build.sourceEncoding>UTF-8</project.build.sourceEncoding>
</properties>

then there may not be any need for setting UTF-8 in the way you do.  But maybe you have another encoding at this point, but then it would be strange to have *.java one encoding and resources in another.  Best to make the whole project same.
Can you confirm if you are building a WAR and this is what you see in the WAR ?  or if are using the launch Tomcat ability from Eclipse and this is what you see in the project.
Also confirm that you have used the Stop Tomcat, Clean, manually check the directory have cleared out (maybe do a workspace refresh, and another Servers -> Clean, manually delete files in Tomcat area).  Then do a Full Publish (to tomcat) and then Start/Debug tomcat to launch.
I would guess if the configuration you now have is as above (and presuming no parent pom.xml is affecting the build do badly) that the files you see are a result of not cleaning the wtpwebapps area after messing around with the maven config.
There are multiple places to check gets cleaned:
Temporarly Turn off: Project -> Build Automatically
Stop All Servers instances running like: Tomcat
Run As -> Maven -> clean
Servers -> Clean

Open file browser and validate for yourself outside of Eclipse that the following dirs are empty or non-existant:

  workspaceDirectory/projectDirectory/target   (the maven output folder)

  workspaceDirectory/metadata/.plugins/org.eclipse.wst.server.core/tmpX/wtpwebapps/**
  # This one is ok to have the top level contextPath's, provind they are currently active projects you have added to the server.

  $CATALINA_HOME/

Perform workspace refresh (to ensure Eclipse resync file system state to workspace state).  I do this with clicking on any project in Project Explorer, then Ctrl-Clicking it again to deselect it (so nothing is selected but Project Explorer is in focus) then press F5.   Other people use Ctrl-A to select all projects.

  Now do a Project -> Build All
  Now do a Servers -> Publish  (now check the tree you get)

  Now re-enable Project -> Build Automatically (if you run that way)

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