text:
TechData Sheet
Data Sheet—Friday, April 28, 2017
Adam Lashinsky
Heather Clancy
Welcome to let's-sort-through-all-the-tech-financials Friday, Data Sheet readers. This is Heather Clancy, understudying Adam Lashinsky for one final performance.
I take a special interest in matters Microsoft. That's partly because it was one of the first software companies I ever covered in depth as a junior tech reporter two decades ago. Mostly, it's because I'm as curious as the next person as to whether the 42-year-old company can steal the spotlight from 20-something Amazon when it comes to replacing corporate data centers (at least most of them) with cloud computing services.
Microsoft's latest financial results—delivered last night alongside acts from tech powerhouses Amazon, Intel, and Google parent Alphabet—talk up all the right things. It recorded a very respectable 11% revenue growth for the unit that represents its cloud offerings—the Azure business alone grew a scene-grabbing 93%. And just this week, the company scored another huge Fortune 500 account: shipping company Maersk. I'm sure that follows CEO Satya Nadella's turnaround script just fine.
But the acknowledged market leader, Amazon Web Services, managed to upstage those results. It pulled off an overall revenue expansion of almost 43% during the same period. Plus, AWS added several marquee names to its cast of customers, including the parent company of Dunkin' Donuts and Baskin-Robbins, and the newly public Snap Inc., which will buy $1 billion in AWS services over the next five years.
You can also expect Google to add dramatic tension. While it doesn't break out the revenue from its cloud services (yet), the "other" category into which it lumps those results grew close to 50% in the first three months of 2017. Just last weekend, Google cloud guru Diane Greene said she believes her company can steal the spotlight from AWS by 2022.
Clearly, the last scene in the cloud drama is still being written—Fortune's Barb Darrow offers her own review of what's up next. Meanwhile, this will make for good theater. Some creative improvisation required.
Speaking of bowing out, this is my last day curating this newsletter, which I helped launch in August 2014. It's been simultaneously the most exhausting and enriching experience of my career, and I'm grateful to Fortune for including me as part of the cast for so long. And now, exit stage left.
@greentechlady
heather@heatherclancy.com
A cheat sheet for last night's earnings bonanza. Intel's data center business didn't grow as fast as expected. Alphabet recorded a surge in mobile and video advertising, despite boycotts from some huge advertisers over its YouTube content policing policies. Amazon's Prime subscription business grew almost 50% amid a monster quarter. And GoPro is managing to get more efficient, without compromising the release schedule for its next digital camera. (Reuters, Fortune, Reuters, Reuters, Fortune)
The dispute between Apple and Qualcomm has escalated. Apple has stopped paying Qualcomm-related royalties to its iPhone contract manufacturers, according to a statement made by the chipmaker Friday morning. That forced Qualcomm to slash its revenue projections for the second quarter by about $500 million to between $4.8 billion and $5.6 billion. Qualcomm's licensing agreements are with Apple's manufacturers directly, who in turn charge Apple. (Fortune)
Uber moves controversial self-driving car exec into another role. Anthony Levandowski—the former Google engineer in the middle of a bitter intellectual property fight with Alphabet's Waymo division—is no longer in charge of autonomous vehicle research at the ride-sharing company. Waymo has accused Levandowski of taking information about its LiDAR sensing technology when he left to start Otto, the self-driving truck company Uber bought last year. (Fortune, New York Times)
It looks like Apple has a case of Venmo envy. The technology giant is working on an alternative to PayPal's peer-to-peer mobile payments platform, reports Recode. Given how well the app did in PayPal's latest quarter—transaction volumes doubled to $6.8 billion—the revelation isn't so surprising and this rumor has surfaced before. (Fortune, Recode)
The smartphone market grew last quarter, no thanks to Apple and Samsung. Unit shipments for the category grew 4.3%, reports International Data Corp. Chinese handset makers like Huawei, Oppo, and Vivo—which target lower price points—were responsible for most of the growth. (Fortune)
Chinese ride-sharing giant Didi Chuxing closes record $5.5 billion round. The new funding reportedly raises the valuation for Uber's rival to more than $50 billion. Didi Chuxing hasn't disclosed who participated, but the buzz is that both SoftBank and Silver Lake Kraftwerk were involved. (Bloomberg, TechCrunch)
It looks like analytics software firm Domo isn't going public soon. Instead, the Utah-based startup has raised at least half of a new $200 million funding round that will help it grow as a private company for a bit longer, according to paperwork it filed this week and flagged Thursday by Bloomberg. BlackRock led that infusion, which values the company around $2.3 billion. (Bloomberg)
Cloudera will undershoot its last valuation when it starts trading today. The data management software firm priced its initial public offering at $15 per share, raising about $225 million and giving it a market capitalization of about $1.9 billion. That's higher than expected but still much lower than the $4.1 billion it was once worth as a private company. (Reuters)
THE DOWNLOAD
Aurora is the secretive new startup from the best minds in self-driving cars. Before Google launched its moonshot factory, Uber disrupted the taxi industry, and Tesla delivered its first electric vehicle—Sterling Anderson, Drew Bagnell, and Chris Urmson were working on self-driving cars.
Their positions at Google, Tesla, and Uber, perhaps the most watched autonomous vehicle programs in the world, are an indication of their esteem. Now, they're striking out on their own—and likely competing with at least one of their former employers—with a self-driving car company called Aurora Innovation. Fortune's Kirsten Korosec has more details about the venture's, which settled a lawsuit with Tesla just last week.
How McAfee's CEO is securing the future. Chris Young has a long to-do list. As CEO of McAfee, which popularized antivirus software in the 1980s and 1990s, he has to prove to investors that his company can operate on its own following a $4.2 billion spinout from Intel in April.
He needs to show peers that McAfee can succeed in a new era. And he must teach customers that the company can once more be on the cutting edge. Fortune's Michal LevRam spoke with Young ahead of his company's spinoff about how he plans to execute.
Facebook Admits It's Being Used by Political Actors to Shape Opinion, by Mathew Ingram
A Hacker Group Hijacked Some Medium Blogs (Including Ours), by Robert Hackett & Jeff John Roberts
Google Snags Former Salesforce Cloud Exec, by Barb Darrow
Why Comcast Already Sounds So Bored With Wireless, by Aaron Pressman
You Shouldn't Get Too Excited About Flying Cars, by Aric Jenkins
Apple's iPhone 7 Performance Is a Mixed Bag, by Don Reisinger
How Keyboard Lovers Can Get the New BlackBerry KeyOne Phone, by Aaron Pressman
MIT figured out a way to "print" a building in less than a day. The company is experimenting with 3D printer-slash-robotics that could be used to create structures in difficult-to-reach locations. (Fortune)
JiveWorld: Strategies and technologies for workplace collaboration. (May 1-3; Las Vegas)
Data Citizens: Strategies for data governance. (May 2-3; New York)
Apttus Accelerate: Perspectives on automating the "quote-to-cash" process. (May 2-4; San Francisco)
Collision: A tech conference created by the organizers of Europe's Web Summit. (May 2-4; New Orleans)
Red Hat Summit: The premier open source technology event. (May 2-4; Boston)
Knowledge17: ServiceNow's annual customer gathering. (May 7-11; Orlando, Fla.)
Gartner IT Operations Strategies & Solutions Summit: How to accommodate cloud services and other "digital" technologies. (May 8-10; Orlando, Fla.)
Gartner Digital Marketing Conference: Actionable advice about martech. (May 10-12: San Diego)
Outperform: The PROS annual conference about omnichannel commerce technology. (May 10-12; Chicago)
Build: Microsoft's annual conference for software developers. (May 10-12; Seattle)
MarkLogic World: Enterprise database strategies and insights. (May 16-17; Chicago)
Google I/O: Alphabet's annual developer conference. (May 17-19; Mountain View, California)
Epicor Insights: Strategies for retail and resource planning. (May 22-25; Nashville)
Couchbase Connect: How the database powers digital experiences. (May 22-23; New York)
Signal: Twilio's annual developer confab. (May 24-25; San Francisco)
Apple Worldwide Developer Conference: An annual gathering of iOS, macOS, and watchOS coders. (June 5-9; San Francisco)
Zuora Subscribed: A conference dedicated to the subscription economy. (June 5-7; San Francisco).
Pure//Accelerate: The future of data storage. (June 12-14; San Francisco)
HR Tech World: An event about people, technology, and organizations. (June 14-15; San Francisco)
MongoDB World: A gathering of the world's fastest-growing database community. (June 20-21; Chicago)
Cisco Live: Education for technology innovators. (June 25-29; Las Vegas)
.NEXT: Nutanix's conference on the future of enterprise cloud services. (June 28-30; Washington, D.C.)
Fortune Brainstorm Tech: An invitation-only summer retreat for business leaders. (July 17-19, Aspen, Colo.)
Microsoft Ignite: Hands-on learning and industry insights for business leaders. (Sept. 25-29; Orlando, Florida)
Tableau Conference: Tableau's annual customer conference (Oct. 9 -12; Las Vegas)
This edition of Data Sheet was curated by Heather Clancy.
Find past issues. Sign up for other Fortune newsletters.


Vancouver, WA, August 14, 2014: ICD High Performance Coatings is pleased to invite you to our booth (#2246) at the upcoming GlassBuild America show, Las Vegas, September 9-11.
We will be showcasing our well known industry standard OPACI-COAT-300® Water-Based Silicone Spandrel, as well as a new and exciting metallic spray line of the same tried and true product.
For automotive obscuration band and appliance glass printing applications, we are very excited to feature our newest product CeramiGlass™. This product provides solutions to common problems in surface two (S2) automotive windshield applications using ceramic enamels. From outstanding green strength to no optical burn, this product helps your windshields fulfill the demands of rapidly evolving automaker designs.
CeramiGlass™ is also available as a dot/line screen print alternative to ceramic enamel, with no VOC's, less energy use in manufacturing, and broad color range that includes custom color matching.


NWBioTrust team from Fred Hutch, UW Medicine, ITHS Biomedical Informatics, Seattle Children's and Seattle Cancer Care Alliance.
The University of Washington departments of Anatomic Pathology and Laboratory Medicine, the Institute of Translational Health Sciences, Fred Hutchinson Cancer Research Center and Seattle Children's have come together to launch NWBioTrust, a resource that is increasing the availability of high-quality, human blood and tissue samples and clinical data for innovative biomarker discovery.
Dr. Gail Deutsch, associate professor of pediatric pathology at Seattle Children's.
Dr. Steve Schmechel, who joined the UW Department of Pathology in 2012, brought considerable experience in biospecimen resources from the University of Minnesota and now co-directs NWBioTrust and oversees the collection of surgical tissue specimens at UW.
Together the team has built a robust and efficient biospecimen resource to connect annotated human blood and tissue samples from consenting individuals to potentially high-impact research studies. Such samples and the associated clinical data are crucial to the development of novel diagnostic and therapeutic technologies that can improve disease detection and treatment and benefit the health of the state's citizens.
Through the NWBioTrust portal, researchers can access a central Web-based system to search existing banked cancer specimens or request the collection of samples from surgeries at UW and clinics at Seattle Cancer Care Alliance that match the detailed requirements of their research protocols. The online system allows investigators to register their studies, submit sample requests and track the delivery process.
While the NWBioTrust will fill a critical gap in the development of new treatments and diagnostic and prognostic tests, it also provides a novel opportunity for patients to take an active role in the discovery of a cure for their disease. Patients can participate in the program by providing their consent to donate tissue and/or blood samples, many of which are left over from clinical procedures and would otherwise go to waste. As part of this process, NWBioTrust is dedicated to ensuring that all patient privacy and consent regulations are strictly followed.
To learn more about NWBioTrust, please visit www.nwbiotrust.org.


Cornell's "Celebrate Urban Birds" is having a fun contest -- Funky Nests in Funky Places. Get out your pictures of nests in strange places or try your hand at some art project - making a nest and enter to win from a neat variety of things from bug repellent wrist bands to several books - even a NEW Leica Camera!!
"The dove thought it was an ideal location, apparently (right next to a lantern, you'll notice)! In my opinion, this bird's feathers are so smooth and perfect you almost can't tell the bird is "real" (not stuffed, or fake...)....but she is! She was also indifferent to my presence....caring less that I was inches away from her, taking pictures. If only all my avian subjects were so helpful! (Barn swallow mothers dive bomb me, when I even approach their nests of babies...!) I think the location of the Baker's Rack nest is especially funny, since Baker's racks are often found in kitchens, with cookbooks on them (for instance). Ms. Mourning Dove decided to cook up a little recipe of her own (for babies)!"


text:
Alabaster to launch Government Academy in September
The city of Alabaster approved the purchase of property for a new multi-use facility at a public hearing on Monday, Sept. 12. (File)
By EMILY REED | Special to the Reporter
ALABASTER – Beginning in September, the city of Alabaster will launch the first Alabaster Government Academy.
The purpose of the Academy is to give residents and business owners a "hands-on" look at their city.
"Our goal is to foster a greater connection between the city's government and Alabaster residents by giving them a chance to get an in-depth and hands-on look at what each city department does," said city Public Information Officer Neal Wagner.
Wagner is instrumental in creating the program, which will be held twice a month until the end of February 2023.
"There are not many programs like this in Alabama, so we are excited to launch our inaugural academy in September," Wagner said. "On our side, it will also help us to identify ways to more effectively communicate with the community and work with neighborhood leaders."
The Academy is an adult education program, which is open to residents 18 and older.
Wagner said applicants who are in leadership positions with their HOAs, in the faith community, in their neighborhoods, etc. are encouraged to apply.
During the Academy, each city department will have its own night to be featured.
"Participants will have a chance to tour the police and fire departments, City Hall, Environmental Services, Public Works and much more," Wagner said. "The classes will be interactive to allow participants to get a hands-on feel for what each department does."
Each class night will be led by the department's respective department head, and the city Administration Department is organizing the academy.
"It will really be a unique experience since there aren't many programs like this in the state," Wagner said. "Participants will get to access places that aren't typically open to the public, and they will have a much greater understanding of how the city operates once they complete the academy. It also may lead to opportunities to serve on various city boards."
The program is free for those interested in participating.
More information about the Academy, including an application, can be found on the city's website by visiting Cityofalabaster.com/556/Alabaster-Government-Academy.
More News Main
Champy's in Alabaster celebrates 10 years of business

redpajama_set_name: RedPajamaCommonCrawl


One of the most significant recently has been the growth of companies creating dedicated centers of excellence (CoE) to drive improved results, consistency, and economies of scale.
Since tactical social media initiatives are still quite common, organizations are finding that company-related social media activity is taking place across a larger and larger swath of functions in burgeoning social businesses. As this happens, the need is felt to make social media more coherent, less duplicative, more answerable to the business, and better organized and managed.
Not only is it extremely common to find at least two major social media efforts in most enterprises today (usually a set of social marketing and CRM initiatives that is externally facing and a social collaboration effort that is inwardly focused), but there is often multiple instances in of each, especially in larger and geographically distributed firms and conglomerates.
Other duplication typically abounds as well, both in the primary social platforms being used and the secondary capabilities like marketing automation, social analytics, security, compliance, and identity, as well as points of integration with regular line of business applications. So too with staff and roles such as social marketing leads, community management, social publishing, and customer care.
As a result, as senior executives these days have generally moved beyond asking why they should invest in social media (most organizations are clearly doing so today), but rather how they can establish better organization, discipline, and cost effectiveness, while still broadly unleashing the social capabilities of their organizations. Some of this is to help answer the long-standing question of who is ultimately in charge of social media? Is it a central function? Or does it make more sense as a decentralized set of loosely coordinated activities on the edge, closer to where work gets done.
For most companies, instead of creating a new social media department, they are creating centers of excellence that support the rest of the organization, yet actually has dedicated staff and resources that can establish consistent guidelines, unify social tools and platforms where it makes sense, collect techniques and best practices, provide education and bootstrapping, and establish a central monitoring and dispatch capability to respond and engage in social media at scale.
The early days of establishing a center of excellence tends to be politically fraught, with concerns about over-centralization and prized social media functions at a local level getting moved to a larger bureaucracy in another area within the company. But the first step is clearly understanding who is engaging in important social media activity within the company today. The pattern tends to be the same: There are one or two major initiatives, 5-10 departmental initiatives, and a lot of smaller informal or shadow activities.
Going forward, all of these social media activities will receive support from the CoE, which will — over time — introduce consistency across tools, policy, and processes. This takes time to establish but as a service bureau and support unit, the CoE has the means and resources to help all of the business units succeed, from ramping up a social media effort, to managing it, and then optimizing it.
One of the essential social business functions, community management, is frequently centralized under the CoE, though subject matter-specific communty managers typically remain distributed across the organization in their particular line of business. Executives are also key stakeholders of the CoE. They can use its existence to better exert oversight, participate in governance, set goals, and monitor the progress of social media across the organization using reports, dashboards, and other types of data provided by the CoE team.
While a large part of the CoE's function is to respond to requests for assistance from across the organization, a lot takes place on a regular basis that was formerly managed locally, informally, and in a fragmented fashion. Proactive activities include centralized listening, dispatch of key events to internal stakeholder, reporting, measurement, community management, education and training, risk management, governance, and the coordination of crises in social media. Plus evaluation of tools, new social media channels, and enterprise-wide social media strategy development and planning.
As a result of all of this, I'm often asked how many full time staff a CoE must have in order to be useful and adequately support the demands that will be put upon it. Typically, most enterprises, even large ones, are reluctant to have more than a handful of FTEs supporting social media across the company, at least at first. Today, we typically see CoE staff sizes based on level of social media maturity. Thus, immature organizations usually have 1-2 FTEs initially. This makes it difficult to have an acceptable service level, even with volunteer efforts from other employees. Organizations that have moved beyond the early stages typically have a CoE leader (often called the social media manager), a social media manager, a content manager, and a couple of community managers. The most mature organizations frequently have 5-10 FTEs in their CoE, including the community management team.
Social media is greatly enabled by the right social technology, maintained in an effective architecture. The specific details vary by industry, organization, and even by the part of the organization. The CoE has the unenviable job of sorting out the confluence of social silos and activities and bringing some semblance of order and consistency. Most CoEs quickly realize there is no one tool or silver bullet when it comes to social technology. All of CoEs manage large and growing portfolio of social tools, platforms, and standards, with a focus on careful introduction of new tools and rationalization of existing ones so that fragmentation, risk, excess effort, and duplication are managed to an acceptable level.
Together, the stakeholders of the CoE — particularly the internal customers and the executive team (which are typically collectively the ones contributing funding to the new unit) — set the objectives for the CoE. This includes everything from a vision and charter to an expected level of service and detailed business goals. At a high level, the expectation is that the CoE will help drive higher level of business performance, consistency, efficiency, risk, and ROI tracking. Most CoEs I've seen review operating goals carefully with stakeholders at least once a year, usually right before the yearly budget cycle starts.
One of the most vital outputs of the work of the CoE is a body of knowledge about how best to apply social media in the organization. Most of the processes and activities in the CoE generate valuable information that other stakeholders can learn from.
CoEs that have the greatest impact actively capture and share this knowledge, often in a social tool/repository, and ensure everyone can access it and learn from it. The social media knowledge base contains best practices, lessons learned, playbooks for common social activities, a trouble ticket system, FAQ, policy framework, and most recently, a content directory which contains the social media assets of the organization.
A few years ago, I explored the early days of social media organization and we've seen it come quite a long way in the interim. Companies are readily funding CoEs as social becomes infused with so much of what we do. As digital business becomes an ever more dominant part of our organizations, the CoE is becoming the de facto social media department, with the caveat that is must have full contact with and actively support the rest of the organization for it to hope to scale and add sufficient value to warrant the investment.
For now, most organizations are still experimenting with how they should best organize for social media, and the CoE seems to be the leading model at the moment. There are certainly others however, and I'd love to hear your stories on how you company is structuring its social media efforts.


PlayaEvents : 2014 Event — Queeravansary!
BloAsis welcomes all Queer Qamps in Black Rock to join us for a Fun Themed opening party. Come get to know your Queer and Queer Friendly neighbors, make new friends, and prepare for the coming week.


I continue my deep dive into DotNetNuke 7's new DAL 2 with a look at the features that make up the repository component.
The pattern at its simplest provides simple CRUD (Create, Retrieve, Update and Delete) methods. In the DotNetNuke implementation there are a few extra overloads to provide flexibility. Listing 1 shows the complete Interface.
There are six methods that get (or retrieve) items from the database. Four of these methods return collections and 2 return single items.
These six methods are in pairs – 2 Get methods which return a collection, 2 GetPage methods that return a "page" of items and 2 GetById methods that return a single item. The 2 methods differ in each case by the addition of a TScopeType type parameter and parameter.
Let's first take a look at the two Get methods. Listing 2 shows the use of the parameter-less overload to retrieve all of the tasks from the database.
The pattern should be familiar by now. The repository is created by the data context and then the Get method is executed. This returns an IEnumerable which is then converted to a List and returned.
Under the covers the repository together with PetaPoco generates the SQL necessary to fetch all the tasks. PetaPoco generates a SQL statement to return every column in the table.
The other overload is used in conjunction with the Scope Attribute. In fact if the type of the repository does not have the Scope attribute then an exception is thrown. We can see how this works if we add a ModuleID property to the TaskInfo class (Listing 4) and a ModuleID column to the Tasks table. We can then use the Get overload as shown in Listing 5 resulting in the generated SQL shown in Listing 6.
As can be seen the generated query is parameterized reducing the risk of SQL injection attacks.
The other two methods (GetPage) that return a collection of items from the database are similar to the Get methods. Both methods return a "page" of data, and have two additional parameters - compared with their Get counterparts - the page index and the page size. Listing 7 shows an example of how to use these methods and Listing 8 shows the generated SQL.
PetaPoco handles the generation of the appropriate "paging" SQL and the repository returns an IPagedList of T. This collection interface and its associated concrete PagedList of T  implementation are new (in DotNetNuke 7) custom collections to support this API.
Essentially the IPagedList of T interface just extends the core .NET IList of T collection to add properties that provide metadata about the page of data, as well as the collection as a whole. For example, there are properties which indicate whether there is a previous page or a next page, and there are properties which indicate the number of pages in the complete data set, as well as the total no of items.
In fact the RepositoryBase of T class uses this logic internally to return a page of data from the cached collection – see Listing 10.
As the IPagedList inherits from IList and PagedList inherits from List - both can essentially be used wherever their base class is used and both implement IEnumerable so they can be used with LINQ.
In addition to the methods that return collections there are two methods that return single objects. Again these methods only differ as to whether they are scoped or not. By default the convention is that the column name for the property used as an "id" is "ID", but this can be controlled by the PrimaryKey attribute.
Listing 11 shows an example of using the GetById method and Listing 12 shows the generated SQL.
The sql code that is following each method here is very helpful for the concept understanding. I wish you put some more of it for the rest of the methods.


text:
Case Study: ECM is Key Ingredient for Poultry and Pork Feed Producer
In 72 years in business, comprehensive agricultural company The Wenger Group has grown to where it operates seven mills meeting the needs of much of the mid-Atlantic's poultry and pork producing region. Last year as the family-owned company decided to diversify its three areas of operation into distinct subsidiaries, it focused on feed production, nutrient solutions, and poultry services but included improved record keeping as a must-have for success.
The company assessed its business processes and found many were too time-intensive and costly due to their dependence on paper documents. According to Michelle Lombardo Smith, The Wenger Group's document management coordinator, continuing with these legacy paper-based processes—that required printing, mailing, filing and digging through boxes and filing cabinets—would be highly inefficient and limit planned growth opportunities. This led The Wenger Group IT Manager Nick Szmyd to consider implementing electronic records and enterprise content management (ECM) software.
From an administrative perspective, electronic documentation is much easier to manage than paper on many levels, says Smith. First, electronic records enable a company to dramatically improve access to company records, producing immediate and extensive time savings.
The ECM software allows for more efficient handling of those electronic records by providing managers with a comprehensive overview of operational workflows. ECM also makes it much easier for managers to identify costly inefficiencies while enhancing oversight so business leaders understand what improvements need to be made to what tasks in what business process.
Despite the presence of many electronic records and ECM products on the market, Szmyd and the selection team felt that the Laserfiche ECM system was the best purchase option. The system's Windows-based platform made it easy for employees to learn, and, equally important, according to Szmyd, is the system's comprehensive suite of ECM products that allow the company to diagram and automate workflows that enhance administrative efficiency while providing ever-expanding opportunity for business process optimization.
"Laserfiche's business process management, workflow and electronic forms components stood out as solid tools for helping us manage the complexity of our work processes," says Smith. "As an ISO-certified organization, we must carefully maintain the documentation on our processes and use many forms to capture daily work outputs that support this documentation. With such a large amount of paper and emails being routed to multiple people for information, we immediately recognized how the ECM software, and electronic workflow and forms specifically, could help us streamline so many operations at The Wenger Group."
Though still early in the implementation of ECM, The Wenger Group has great expectations. Within the next two years, Smith expects to report a six-figure savings from reduced printing and paper handling costs. The organization will also free up space in an offsite storage facility that was previously occupied by file cabinets and other forms of paper storage.
Beyond these savings, The Wenger Group's ECM deployment streamlined critical business operations, such as the company's safety data sheets management process. Federally mandated safety data sheets "provide comprehensive information about a substance or mixture for use in workplace chemical management." Safety data sheets provide a readily available list of what chemicals are in the mills for local public safety officials in the event of an emergency. The high availability of safety data sheets helps The Wenger Group mitigate the risk of expensive fines for failing to meet local, state and federal regulatory obligations.
In early 2016, The Wenger Group implemented Laserfiche's workflow and electronic forms functionality to facilitate inspection, authorizing and filing of the safety data sheets in a remotely accessible central electronic repository. The software enables The Wenger Group to digitally capture safety data sheets, including key product identifiers automatically route it to the mills and electronically file it.
The digitization and automation of the process not only streamlines the tasks involved, but it also makes the information instantly searchable and standardizes the way it is filed, making retrieval of information even easier. This meant less headaches when managing the regulatory demands placed on the various mills The Wenger Group operates. The savings have been immediate and significant.
"With our safety data sheet project, we went from a total time, in working days, from 27 to nine (projected over the course of a year) to compile and maintain these documents," Smith says. "We look at these time savings as key metrics in estimating our overall success. We have seen this project as giving us a new way to work within our larger community—particularly the local fire departments, for which we will be providing access to our safety data sheets for emergency response at any of our mills or offices."
The safety data sheets are just the first step in what Wenger management expect to be a very lucrative, long-term embrace of ECM technology. Just in the training session to use workflow solution, Smith saw that the exercise sparked discussion and theorization about how similar uses of electronic forms and automation could be deployed throughout The Wenger Group's three new divisions to eliminate paper processes and enhance operational efficiency.
"We see this as a way to completely redesign our documentation library and better support our production environment with more timely flow of information," Smith says.
Although The Wenger Group is only six months into its deployment of ECM, Smith says the company has only unlocked less than 10 percent of the potential. The IT department continues to delve deeper into what the technology can do. As team members are being brought up to speed on the basics of the new system, they are excited about the potential of eliminating the hours each week spent burrowing through filing cabinets.
"We are still familiarizing our team members with the new processes and all of the capabilities and efficiencies," Smith says. "We have streamlined the process of updating local, offline copies for disaster recovery using Laserfiche Workflow. This enables us to make safety data sheet records instantly available at mills and remote sites to address any emergency situation that arises anywhere."
The same benefits apply for the many non-emergency situations that arise. When audits are conducted, there can be so many forms and documents required that it becomes a labor-intensive process to ensure all forms are filled out properly, are legible and are readily available—hence Laserfiche ensures regulatory compliance. Securing and maintaining that information in electronic forms is a much more efficient process that ensures legibility, accuracy and ease of access. This has enabled The Wenger Group to greatly reduce the complexity of information handling among departments and different geographic locations.
Smith shared that "My list of projects for Laserfiche is quite long!" In the next year, The Wenger Group expects to apply these technologies to the company's invoicing, accounts payable and human resources departments, and Smith predicts extensive use of Laserfiche throughout the enterprise.
Reference: https://www.msdsonline.com/resources/ghs-answer-center/ghs-101-safety-data-sheets-sds
Latest in Industry 4.0
Sony Built a Car
Warehouse Workers Adapting to Robots
Matt O'Brien
Manufacturing 2020: 3 Major Predictions
Antony Bourne
Can 'Robotic Blacksmithing' Revive Mfg?
Glenn S Daehn
More in Industry 4.0
Warehouse Solutions Increase Worker Productivity
Zebra Technologies has announced a series of new warehouse offerings designed to optimize workflows, maximize productivity in the warehouse and improve employee onboarding.
How AI Improves Industrial Procurement
Artificial Intelligence can help dissect patterns - good and bad - in revealing the nuances of buying behavior, and how they constantly evolve.
Google's Co-Founders Step Down
Larry Page and Sergey Brin will relinquish their roles as CEO and president, respectively, of Google parent Alphabet.
Rachel Lerman
Make Bottleneck Management a Priority
It's time for digital transformation to address the biggest problem in manufacturing: bottlenecks.
Ali Hasan Raza
ERP into the Future
ID chats with DDI Systems' Jennifer Luizzi about ongoing trends in ERP systems, and what's to come.
How to Use Blockchain for Transparency Across Your Supply Chain
Typically, managing contractual obligations for every supplier and sub-supplier is risky and uncertain because it relies on trusting the supplier as an intermediary.
What Does the Future of Manufacturing Technologies Look Like in 2019?
Technology is evolving and is putting its mark on the manufacturing industry. Each year is bringing innovation, and new technologies are used to create smart products which meet customers' expectations.
Are robot baristas the future of South Korea's vibrant coffee culture?
GM Adds Trickier Roads to its Semi-Autonomous Driving System
General Motors is adding 70,000 miles of roads across the U.S. and Canada to the area where its Cadillac Super Cruise semi-autonomous driving system can run, including some with cross traffic similar to those that have confused Tesla's Autopilot system.
4 TIPS TO STAY COMPETITIVE IN MANUFACTURING
In today's rapidly evolving manufacturing landscape, many companies are turning to technology to gain a competitive advantage. Smart manufacturers are leveraging technology to increase efficiency, reduce/ mitigate human error, and optimize their supply chain in order to increase their profitability. Manufacturing in the future will require investing in technology in order to be more connected and provide more visibility for all stakeholders, both internally and across the supply chain. This e-book
World's Top Steelmaker to Cut Output Further in Europe
The world's largest steel maker, ArcelorMittal, on Wednesday announced further cuts to production at several plants in Europe due to weak demand and high imports, reflecting troubles in the wider industry.
IoT Conference Traces Changing Industry
At IoT World, held in Santa Clara, California, from May 13-16, 2019, manufacturers joined companies from a wide number of industries to network and explore how the Internet of Things is changing the way people do business.
Megan Crouse
Servitization for Sensors
Is S2aaS a viable business model?
Competitive Collaboration: Are You Seizing the True Value of Digital Ecosystems?
Even in the fast-paced blur of today's manufacturing landscape, one thing is clear: no one can innovate alone.


text:
\section{Introduction}
Order statistics have been widely studied from the past few decades. It is the arrangement of random variables according to their strength. Let $X_1, \ldots, X_n$ be a set of random variables then the increasing arrangement of these random variables is $X_{1:n} \leq X_{2:n} \ldots \leq X_{n:n}$, here $X_{1:n}$ is the smallest ordered statistic and $X_{n:n}$ is the largest ordered statistic. The terms in the form of $X_{k:n}$ are the k-th ordered statistic. An interpretation of these ordered statistic can be observed in terms of system whose components are these n random variables. The smallest ordered statistic is the minimum of all the n random variables, i.e., $X_{1:n} = \min(X_1, X_2, \ldots, X_n)$, this represents the structure of a series system. Similarly the largest ordered statistic $X_{n:n} = \max(X_1, X_2, \ldots, X_n)$ represents the parallel system and $X_{k:n}$ represents the $n-k+1$ out of $n$ system. The parallel and series systems are the primary units of complex coherent systems. Moreover, they are often used in statistics, applied probability, auction theory, actuarial science etc. \\
In this paper we shall focus on the study of stochastic comparison of smallest and largest ordered statistic whose components follow Weibull- G distribution and Gompertz Makeham distribution. Weibull and Gompertz Makeham distributions are widely used to study the nature of human death distribution as they contain minimum number of parameters and the distribution functions are in closed form. Both of them are limit distributions of extreme value theory and a comparative analysis of both the distributions have been studied by \cite{15}. Firstly let us understand how the Weibull-G distribution evolved. The distribution originated from T-X (Transformed transformer) family of distributions, proposed by Alzaatreh et al.\cite{6} as a new method of generating continuous distributions. The T-X family of distributions can fit highly left tailed, right tailed, unimodal, bimodal distributions. This family of distributions was greatly inspired from the Beta- Normal distributions proposed by Eugene et al.\cite{5}. Let $X$ and $T$ be two continuous random variables such that the density function, distribution function(cdf) and survival function(sf) of $X$ are $f(x), F(x), \overline{F}(x)$ respectively, and $T$ has support on $[a, \infty)$ with density function $u(x)$. Let $X_{T}$ be a random variable following T-X family of distributions, the cdf of $X_{T}$ is
\begin{equation}\label{TX}
H_{TX}(x)= \int_a^{w(F(x))}u(t) dt, x \in \mathbb{R},
\end{equation}
where $w(F(x))$ is continuously differentiable and monotonically non-decreasing in the interval $[a, \infty)$. Let $X$ follow a scale family of distribution say $X \sim F(\gamma x)$, $\gamma >0$ is the scale parameter. The Weibull-G distribution is generated when the random variable $T$ follows the Weibull distribution ($T \sim W(\alpha, \beta)$, $\alpha >0, \beta >0$) with parameters $\alpha$ and $\beta$ and the function $w(F(x)) = \dfrac{F(x)}{1-F(x)}$ in $\eqref{TX}$. The resulting distribution function of $X_{WG}$ is
$$H_{WG}(x) = 1- e^{-\alpha\left(\dfrac{F(\gamma x)}{1- F(\gamma x)}\right)^{\beta}}, ~x >0, \alpha >0, \beta >0, \gamma>0,$$
here $\alpha, \gamma$ are the scale parameters and $\beta$ is the shape parameter. We shall denote the distribution as $W-G(\alpha, \beta, \gamma)$.\\
The generalised distributions are extremely useful for practical purposes as they have more number of parameters. The stochastic comparison of such distributions are also necessary. Recently Chowdhury et al.\cite{n} have studied stochastic comparison of the smallest ordered statistic when the parameters are vector majorized. In this paper we shall use the same notations as used in Chowdhury et al.\cite{n} for convenience. The Weibull-G distribution is an advanced distribution that can obtain the odds that an individual will die prior to time $X$, with cdf $F$, where these odds follow another life distribution $T$ as observed by Cooray \cite{4}. Bourguignon et al.\cite{11} have studied the mathematical properties of Weibull-G distribution and observed that this distribution nicely fits real data sets. \\

The Gompertz Makeham distribution evolved from the well known Gompertz law and Makeham law respectively, provided by two researchers \cite{12} and \cite{13} in the 19th century. It was observed that the death rate at any age is the sum of two terms in which one term contributes as age-independent (Makeham term) and the other term is the age dependent term (Gompertz term). \cite{14}, \cite{15}, \cite{16}, \cite{17}, \cite{18} have studied various interesting properties of Gompertz Makeham (GM) distribution. Let $X$ be a continuous random variable following GM distribution with positive parameters $\alpha, \beta, \lambda$, we shall denote this as $GM(\alpha, \beta, \lambda)$. The distribution function of $X$ is
\begin{equation}
F_{GM}(x) = 1- e^{-\lambda x - \dfrac{\alpha}{\beta}(e^{\beta x}-1)}, x >0,  \alpha >0, \beta >0, \lambda >0.
\end{equation}
Here the parameter $\alpha$ denotes the initial mortality and $\beta$ denotes the increase in mortality with increasing age, both of them are Gompertz term whereas the parameter $\lambda$ is a Makeham term and it denotes the risk of death due to unexpected causes such as accidents, infections, weather conditions etc. In recent times, few extensions of Gompertz Makeham distribution are available such as the bivariate Gompertz Makeham life distribution \cite{20} and the transmuted Gompertz Makeham distribution \cite{21}, \cite{22}. In this paper we shall study the stochastic comparison results for series and parallel systems comprising of n components (say) where each component follow GM distribution.
 Many researchers have studied the stochastic comparison of Weibull distributions, Li and Li \cite{10}, Torrado and Kochar \cite{9} are among them. Fang and Zhang\cite{7} have compared two parallel systems comprising of Exponentiated Weibull components in terms of usual stochastic, dispersive and likelihood ratio ordering. Later Kundu and Chowdhury\cite{1} also studied parallel systems whose components follow Exponentiated Weibull distribution with respect to reversed hazard rate and likelihood ratio ordering. The above mentioned studies were carried out for vector- majorized parameters. The univariate majorization arises during income allocation or in comparison of only one attribute. In real life situations when allocation of more than one attribute are compared, one can use multivariate majorization. Studies have been conducted for multivariate chain majorization by various researchers. Fang and Balakrishnan\cite{3} studied Exponential-Weibull distribution and obtained usual stochastic and hazard rate ordering for smallest ordered statistic with vector majorized components and usual stochastic ordering of largest ordered statistic when the components were chain majorized. Biswas and Gupta\cite{8} observed usual stochastic ordering for parallel system with Exponentiated Gumbel Type-II distributed components for chain majorized parameters. \cite{23} observed various stochastic ordering for series and parallel systems whose components follow Kumaraswamy's and Frechet distribution. Moreover \cite{KS} obtained several stochastic ordering results for Kumaraswamy-G distributed components. Recently \cite{24} observed stochastic orderings for Extended Inverse Lindley distribution with heterogeneous components. The stochastic orders mentioned here are available in details in the book by Shaked and Shantikumar \cite{2b}. Some of the orders that have been used in the paper are mentioned in the next section. \\
In this paper we have discussed the stochastic comparison of the smallest and largest ordered statistic from independent heterogeneous Weibull-G random variables. Let $X_1, X_2, \ldots, X_n$ be independent random variables with $X_i \sim W-G(\alpha_i, \beta_i, \gamma_i), i=1,2,\ldots, n$. Furthermore, let $Y_1, Y_2, \ldots, Y_n$ be another set of independent random variables with $Y_i \sim W-G(\alpha_i^*, \beta_i^*, \gamma_i^*), i=1,2,\ldots, n$. When $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and the matrix containing the parameters $\alpha_i, \gamma_i$ changes to another matrix containing the parameters $\alpha_i^*, \gamma_i^*, i=1,2, \ldots, n $, in the sense of multivariate chain majorization, we study the hazard rate ordering of the smallest ordered statistic. Next, when $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $\gamma_1 = \gamma_2 = \ldots = \gamma_n= \gamma_1^* = \gamma_2^* = \ldots = \gamma_n^*$ and $(\alpha_1, \alpha_2,\ldots, \alpha_n) \prec_{w} (\alpha_1^*, \alpha_2^*,\ldots, \alpha_n^*)$, we establish reversed hazard rate ordering of the largest ordered statistic. Also, when $\alpha_1 = \alpha_2 = \ldots = \alpha_n= \alpha_1^* = \alpha_2^* = \ldots = \alpha_n^*$, $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $(\gamma_1, \gamma_2, \ldots, \gamma_n) \prec^w (\gamma_1^*, \gamma_2^*, \ldots, \gamma_n^*)$, we observe the usual stochastic ordering of the largest ordered statistic when the baseline distribution of $X$ is Exponential.\\
The paper is organized as follows: The definitions of various terms used in the paper are mentioned in Section 2. The results for smallest ordered statistic with chain majorized parameters and for largest ordered statistic with vector majorized parameters are discussed in Section 3 and 4. The conclusion is given in Section 5.

\section{Definitions}

We consider $X$ and $Y$ to be two absolutely continuous random variables with distribution functions $F(x)$ and $G(x)$; survival functions as $\overline{F}(x)$ and $\overline{G}(x)$; probability density functions as $f(x)$ and $g(x)$.
Then the two random variables can be compared with each other with the help of various stochastic orders such as:
\begin{enumerate}
\item $X$ is smaller than $Y$ in the usual stochastic order ($X \leq_{st} Y$) if and only if $$\overline{F}(x) \leq \overline{G}(x) ~\forall x \in (-\infty, \infty).$$

\item $X$ is smaller than $Y$ in hazard rate order ($X \leq_{hr} Y$) if and only if
 $$\dfrac{\overline{G}(x)}{\overline{F}(x)}  \text{ increases in }x \in (-\infty,max(u_X,u_Y))$$ where $u_X$ and $u_Y$ are the right end-points of the supports of $X$ and $Y$ respectively.

\item $X$ is smaller than $Y$ in the reversed hazard rate order $X \leq_{rh} Y$ if and only if
$$\dfrac{G(x)}{F(x)} \text{ increases in}~ x \in (\min(l_X,l_Y),\infty)$$ where $l_X$ and $l_Y$ are the left end-points of the supports of $X$ and $Y$ respectively.
\item $X$ is smaller than $Y$ in the likelihood ratio order ($X \leq_{lr}Y$) if and only if $$\dfrac{g(x)}{f(x)} \text{ increases in x over the union of the supports of X and Y.}$$

\end{enumerate}

 The following concept of majorization is available in Marshall et al.(\cite{2}).
 \subsection{Vector Majorization}
Consider two n-dimensional real valued vectors $\underline{a} = (a_1, \ldots, a_n)$ and $\underline{b} = (b_1, \ldots, b_n)$, such that they are arranged as $a_{1:n} \leq a_{2:n} \leq \ldots \leq a_{n:n}$ and $b_{1:n} \leq b_{2:n} \leq \ldots \leq b_{n:n}$. Then the vector
\begin{enumerate}
\item $\underline{a}$ is \emph{majorized} by $\underline{b}$   ( $\underline{a} \prec \underline{b}$ ) if
\begin{equation}
 \sum_{i=1}^{n}a_{i:n} = \sum_{i=1}^{n}b_{i:n} ~\text{and}~ \sum_{i=1}^{k}a_{i:n} \geq \sum_{i=1}^{k}b_{i:n} ~\forall~k = 1,\ldots, n-1;
 \end{equation}
\item $\underline{a}$ is \emph{weakly submajorized} by $\underline{b}$ ( $\underline{a} \prec_{w} \underline{b}$ ) if
\begin{equation}
  \sum_{i=1}^k a_{n-i+1:n} \leq \sum_{i=1}^k b_{n-i+1:n} ~\forall~ k = 1, \ldots, n;
\end{equation}
\item $\underline{a}$ is \emph{weakly supermajorized} by $\underline{b}$ ( $\underline{a} \prec^{w} \underline{b}$ ) if
\begin{equation}
\sum_{i=1}^k a_{i:n} \geq \sum_{i=1}^k b_{i:n} ~\forall~ k = 1, \ldots, n;
\end{equation}
\end{enumerate}
Clearly, $\underline{a} \prec^{w} \underline{b} \Leftarrow \underline{a} \prec \underline{b} \Rightarrow \underline{a} \prec_{w} \underline{b}$.

\subsection{Multivariate chain Majorization}
A permutation matrix $\Pi$ is a square matrix where each row and column has exactly one entry $``1"$ and all the other entries are zero. The identity matrix of order $n$ is a permutation matrix and thus interchanging its rows and columns one can obtain $n!$ permutation matrices. A T-transform matrix is of the form
$$T^{\lambda}_{i,j} = \lambda I_n + (1-\lambda)\Pi_{i,j},$$
where $0 \leq \lambda \leq 1$, $\Pi_{i,j}$ is a $n \times n$ permutation matrix that interchanges the i-th row with the j-th row and $I_n$ is the nth order identity matrix. Let $T^{\lambda_1}_{i,j} = \lambda_1 I_n + (1-\lambda_1)\Pi_{i,j}
^{(1)}$ and $T^{\lambda_2}_{i,j} = \lambda_2 I_n + (1-\lambda_2)\Pi_{i,j}
^{(2)}$ be two T-transform matrices, where $0 \leq \lambda_1, \lambda_2\leq 1$. If
$\Pi_{i,j}^{(1)} = \Pi_{i,j}^{(2)}$, then $T^{\lambda_1}_{i,j}$ and $T^{\lambda_2}_{i,j}$ have the same structure otherwise they are different. A square matrix of order n, $Q= \{q_{uv}\}$ is doubly stochastic if $\displaystyle\sum_{u=1}^n q_{uv} = 1$, $\forall v=1,2,\ldots,n$ and $\displaystyle\sum_{v=1}^n q_{uv} = 1$, $\forall u=1,2,\ldots,n$.

Consider two $m \times n$ matrices $A=\{a_{ij}\}$ and $B= \{b_{ij}\}$, the respective rows are $a_1^{R}, \ldots, a_m^{R}$ and $b_1^{R}, \ldots, b_m^{R}$. Then
\begin{enumerate}
\item $A$ is said to chain majorize $B$ ($ A \gg B$), if there exists a finite set of $n \times n$ T-transform matrices $T^{\lambda_1}, \ldots, T^{\lambda_k}$ such that $B = A T^{\lambda_1}, \ldots, T^{\lambda_k} $;
\item $A$ is said to majorize $B$ ($A \succ B$), if there exists a $n\times n$ doubly stochastic matrix $Q$ such that $B= AQ$.
\end{enumerate}
 Since product of T-transforms is a doubly stochastic matrix thus
 $$ A \gg B \Rightarrow A \succ B.$$
 When $m \geq 2$ and $n \geq 3$, majorization does not imply chain majorization.

\subsection{Schur-convexity (Schur-concavity)}
A real valued function $\psi$ defined on a subset of $\mathbb{R}^n$ is \emph{Schur-convex (Schur-concave)} if
\begin{equation}
\underline{a} \prec \underline{b} \Rightarrow \psi(\underline{a}) ~\leq (\geq)~ \psi(\underline{b}),
\end{equation}
where $\underline{a} = (a_1, \ldots, a_n)$ and $\underline{b} = (b_1, \ldots, b_n)$ are two real valued vectors.

\begin{lem}[Theorem 3.A.4, see {Marshall et al.(1979)\cite{2}}]
Let $A \subset \mathbb{R}$ be an open interval , a function $\psi: A^n \rightarrow \mathbb{R}$ be continuously differentiable then $\psi$ is Schur-convex (Schur-concave) on $A^n$ if and only if $\psi$ is symmetric on $A^n$, and
\begin{equation*}
(a_i - a_j)\left(\dfrac{\partial \psi(\underline{a})}{\partial a_i} - \dfrac{\partial \psi(\underline{a})}{\partial a_j}\right) \geq (\leq) ~ 0 ~~~~\forall \underline{a} \in A^n,
\end{equation*}
where $\underline{a} = (a_1,\ldots, a_n)$.
\end{lem}

\begin{lem}[Proposition 3.C.1, see {Marshall et al.(1979)\cite{2}}]\label{lemma2}
	\normalfont If $\mathbb{A} \subset \mathbb{R}$ is an interval and $h: \mathbb{A} \rightarrow \mathbb{R}$ is convex (concave), then $\psi(\underline{a}) = \displaystyle\sum_{i=1}^n h(a_i)$ is Schur-convex (Schur-concave) on $\mathbb{A}^n$, where $\underline{a} = (a_1, \ldots, a_n)$.
\end{lem}
\begin{lem}[Theorem 3.A.8, see {Marshall et al.(1979)\cite{2}}] \label{lemma3}
\normalfont	Let $S \subset \mathbb{R}^n$, a function $f: S \rightarrow \mathbb{R}$ satisfying $$\underline{a} \prec_w \underline{b} ~(\underline{a} \prec^w \underline{b}) \mbox{ on S } \Rightarrow f(\underline{a}) \leq f(\underline{b})$$ if and only if $f$ is increasing (decreasing) and Schur-convex on $S$.
\end{lem}

\begin{lem}[Proposition 15.B.1, see {Marshall et al.(1979)\cite{2}}]
\normalfont A function $\psi: \mathbb{R}^4 \rightarrow \mathbb{R} $ is differentiable and satisfies
$$\psi(A) \leq (\geq) \psi(B) ~~~~ \forall A \prec \prec B $$
if and only if the following conditions are satisfied:
\begin{enumerate}
\item $\psi(A) = \psi(A \Pi)~~~\forall$ permutation matrices $\Pi$,
\item $\displaystyle\sum_{i=1}^2 (a_{ij} - a_{ik})\left(\dfrac{\partial \psi(A)}{\partial a_{ij}} - \dfrac{\partial \psi(A)}{\partial a_{ik}}\right) \geq (\leq) 0$ $~~~~~\forall j, k =1,2$.
\end{enumerate}
\end{lem}
\begin{lem}\label{lh1}
\normalfont Let $h_1:[0,\infty) \rightarrow \mathbb{R}$ be defined as
\begin{equation} \label{h1x}
h_1(x) = e^{x} - xe^{x}-1,
\end{equation}
then $h_1(x) \leq 0$ $\forall x >0$.
\end{lem}
\bf{Proof}
\normalfont The derivative of $h_1(x)$ with respect to $x$,
\begin{align*}
h_1^{\prime}(x) &= -xe^{x}\\
& <0.
\end{align*}
Therefore $h_1(x)$ is a decreasing function and $h_1(0)=0$, then $x>0 \Rightarrow h_1(x) \leq 0$.

\begin{lem}\label{lh2}
\normalfont Let $h_2:[0,\infty) \rightarrow \mathbb{R}$ be defined as
\begin{equation}\label{h2x}
h_2(x) = xe^{x}-2e^{x} +x+2,
\end{equation}
then $h_2(x) \geq 0$ $\forall x>0$.
\end{lem}
\bf{Proof}
\normalfont The derivative of $h_2(x)$ with respect to $x$,
\begin{align*}
h_2^{\prime}(x)&= xe^{x} -e^{x} +1\\
&=-h_1(x)\\
&>0.
\end{align*}
Therefore $h_2(x)$ is an increasing function and $h_2(0)=0$ then $x >0 \Rightarrow h_2(x) \geq 0$.\\

 Let us define
 $$P_n =
\begin{cases}
 (a,b) = \begin{bmatrix}
a_{1}&\cdots &a_{n} \\
b_{1}&\cdots &b_{n}
\end{bmatrix}: a_i, b_j >0, (a_i-a_j)(b_i-b_j)\geq 0, i,j = 1,2,\cdots,n.
\end{cases}
 $$
 This shall be used in the result section of the paper.
\section{Results}
Let $X_1, X_2, \ldots, X_n$ be independent random variables with $X_i \sim W-G(\alpha_i, \beta_i, \gamma_i), i=1,2,\ldots, n$ and let $Y_1, Y_2, \ldots, Y_n$ be another set of independent random variables with $Y_i \sim W-G(\alpha_i^*, \beta_i^*, \gamma_i^*), i=1,2,\ldots, n$. When $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and the matrix containing the parameters $\alpha_i, \gamma_i$ changes to another matrix containing the parameters $\alpha_i^*, \gamma_i^*, i=1,2, \ldots, n $,  in the sense of multivariate chain majorization, we study the hazard rate ordering of the smallest ordered statistic. The first result is studied when the value of $n=2$.
\subsection{Results for smallest ordered statistic with multivariate chain majorized components}
\begin{theorem}\label{thm1}
\normalfont Let $X_1,X_2$ and $Y_1,Y_2$ be 2 pairs of independent random variable such that $X_i \sim W-G(\alpha_i, \beta, \gamma_i)$ and $Y_i \sim W-G(\alpha_i^*, \beta, \gamma_i^*)$ for $i=1,2$.
Whenever $\beta \geq 2$, $w(x)$ is an increasing convex function, also $w^{\prime \prime}(x)$ is increasing and $\begin{bmatrix}
\alpha_1 & \alpha_2\\
\gamma_1 & \gamma_2
\end{bmatrix} \in P_2$, then
\begin{equation*}
\begin{bmatrix}
\alpha_1 & \alpha_2\\
\gamma_1 & \gamma_2
\end{bmatrix} \prec \prec \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\gamma_1^* & \gamma_2^*
\end{bmatrix} \Rightarrow X_{1:2} \geq_{hr} Y_{1:2}.
\end{equation*}

\end{theorem}
\begin{proof}
\normalfont
The reliability function of $X_{1:2}$ is
\begin{equation}
\overline{G}_{X_{1:2}}(x)=e^{\displaystyle-\sum_{i=1}^2 \alpha_i(w(\gamma_i x))^{\beta}}, ~\alpha_i >0, \beta >0, \gamma_i >0 ~~\forall i=1, 2, \ldots, n.
\end{equation}

Taking logarithm on both sides,
\begin{equation} \label{rWG}
-\ln\overline{G}_{X_{1:2}}(x) =  \displaystyle\sum_{i=1}^2 \alpha_i(w(\gamma_i x))^{\beta},
\end{equation}
differentiating \eqref{rWG} with respect to $x$, we obtain the hazard rate function as
$$r_{X_{1:2}}(x)= \beta \displaystyle\sum_{i=1}^2 \alpha_i \gamma_i (w(\gamma_i x))^{\beta -1}w'(\gamma_i x).$$
Consider $\psi(\underline{\alpha}, \underline{\gamma}) = (\alpha_1 - \alpha_2)\left(\dfrac{\partial r_{X_{1:2}}(x)}{\partial \alpha_1} - \dfrac{\partial r_{X_{1:2}}(x)}{\partial \alpha_2}\right)+(\gamma_1 - \gamma_2)\left(\dfrac{\partial r_{X_{1:2}}(x)}{\partial \gamma_1} - \dfrac{\partial r_{X_{1:2}}(x)}{\partial \gamma_2}\right)$.\\
Hence we have,
$$\psi(\underline{\alpha}, \underline{\gamma}) = \beta [(\alpha_1 - \alpha_2)(f(\gamma_1)-f(\gamma_2))+(\gamma_1 - \gamma_2)(\alpha_1 g(\gamma_1 x)-\alpha_2 g(\gamma_2 x))] ,$$
such that $f(\gamma) = \gamma(w(\gamma x))^{\beta - 1}w'(\gamma x)$ and $g(t) = (w(t))^{\beta -2}(w(t)w'(t)+ (\beta -1)t(w'(t))^2 + t w(t) w''(t))$. \\
We observe that
$$\dfrac{d}{d \gamma}f(\gamma) = (w(\gamma x))^{\beta -2} (w(\gamma x)w'(\gamma x)+ \gamma x (\beta -1)(w'(\gamma x))^2 + \gamma x w(\gamma x)w''(\gamma x))$$
 is positive for $\beta \geq 2$ and $w(.)$ is a convex and increasing function in its domain. Moreover,
 \begin{align*}
 g'(t) = &2(\beta -1)(w(t))^{\beta -2}(w'(t))^2 + t(\beta -1)(\beta -2)(w(t))^{\beta -3}(w'(t))^3 + 3(\beta -1)t (w(t))^{\beta -2}w'(t)w''(t) \\
 & + 2(w(t))^{\beta -1}w''(t)+ t (w(t))^{\beta -1} w'''(t)
 \end{align*}
 is positive as $\beta  \geq 2$ and $w'''(t) \geq 0$. \\
 Hence, $ (\underline{\alpha}, \underline{\gamma}) \in P_2 \Rightarrow \psi(\underline{\alpha}, \underline{\gamma}) \geq 0$. Using Lemma 2.4 we conclude here that $r_{X_{1:2}}(x) \leq r_{Y_{1:2}}(x) $, i.e., $X_{1:2} \geq_{hr} Y_{1:2}$.
 \end{proof}

 \begin{example}
\normalfont Let $X_1, X_2$ be independent random variables such that $X_i \sim W-Exp(\alpha_i, \beta, \gamma_i)$, $i=1,2$. Also let $Y_1, Y_2$ be another pair of independent random variable such that $Y_i \sim W-Exp(\alpha_i^*, \beta, \gamma_i^*), i=1,2$ (the baseline distribution is Exponential with cdf, $F(x) = 1- e^{-\gamma x})$. The parameters are given in the form of matrices as
 \begin{equation*}
 \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\gamma_1^* & \gamma_2^*
\end{bmatrix} = \begin{bmatrix}
4.8 & 3.4\\
2.5 & 1.6
\end{bmatrix} \in P_2 \text{ and } \begin{bmatrix}
\alpha_1 & \alpha_2\\
\gamma_1 & \gamma_2
\end{bmatrix} = \begin{bmatrix}
4.03 & 4.17\\
2.005 & 2.095
\end{bmatrix}.
 \end{equation*}
 It can be observed that $\begin{bmatrix}
\alpha_1 & \alpha_2\\
\gamma_1 & \gamma_2
\end{bmatrix} = \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\gamma_1^* & \gamma_2^*
\end{bmatrix} T^{0.45}$, and it satisfies the condition

 \begin{equation*}
\begin{bmatrix}
\alpha_1 & \alpha_2\\
\gamma_1 & \gamma_2
\end{bmatrix} \prec \prec \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\gamma_1^* & \gamma_2^*
\end{bmatrix} \Rightarrow X_{1:2} \geq_{hr} Y_{1:2}.
\end{equation*}
The plot of the difference $r_{Y_{1:2}} - r_{X_{1:2}}$ is
\begin{figure}[h]
		\centering
			\includegraphics[scale=0.3]{eg1}

			 \caption{The figure shows the graph of $r_{Y_{1:2}} - r_{X_{1:2}}$ for $\beta = 3$.}
	\end{figure}

 \end{example}

 The next result is observed for $n>2$.
 \begin{theorem}\label{thm2}
 \normalfont Let $X_1, \ldots, X_n$ be a set of independent random variables such that $X_i \sim W-G(\alpha_i, \beta, \gamma_i), i=1,\ldots,n$. Also let $Y_1,\ldots,Y_n$ be another set of random variable such that $Y_i \sim W-G(\alpha_i^*, \beta, \gamma_i^*), i=1,\ldots,n$. If
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} \in P_n
 \end{equation*}
 and
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1^* & \ldots & \alpha_n^*\\
 \gamma_1^* & \ldots & \gamma_n^*
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} T_{i,j}^{\lambda},
 \end{equation*}
 then $\beta \geq 2, ~w^{\prime}(x) \geq 0, ~w^{\prime\prime}(x) \geq 0,  ~w^{\prime\prime\prime}(x) \geq 0$ $\Rightarrow X_{1:n} \geq_{hr} Y_{1:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont We observe that $X_k$ and $Y_k$ have the same distribution (the parameters $\alpha_k = \alpha_k^*$ and $\gamma_k = \gamma_k^*$) $\forall k \neq i,j$ as the T-transform matrix $T_{i,j}^{\lambda} = \lambda I_{n} + (1-\lambda)\Pi_{i,j}$, where $\Pi_{i,j}$ interchanges the $i^{th}$ row with the $j^{th}$ row. Therefore applying Theorem \ref{thm1} the result follows.
 \end{proof}
 A finite product of T-transform matrices with the same structure is a T-transform matrix with the same structure. The above theorem can be applied in this case.  The finite product of T-transform matrix with different structures may or may not be a T-transform matrix in these circumstance the next theorem is useful.

 \begin{theorem}
 \normalfont Let $X_1, \ldots, X_n$ be a set of independent random variables such that $X_i \sim W-G(\alpha_i, \beta, \gamma_i), i=1,\ldots,n$. Also $Y_1,\ldots,Y_n$ be another set of random variable such that $Y_i \sim W-G(\alpha_i^*, \beta, \gamma_i^*), i=1,\ldots,n$. Presume $k \geq 2$, if
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} \in P_n,
 \end{equation*}
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} T^{\lambda_1}\ldots T^{\lambda_i} \in P_n, \text{for } i=1,\ldots, k-1,
 \end{equation*}
 and
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1^* & \ldots & \alpha_n^*\\
 \gamma_1^* & \ldots & \gamma_n^*
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} T^{\lambda_1}\ldots T^{\lambda_k},
 \end{equation*}
 then $\beta \geq 2, ~w^{\prime}(x) \geq 0, ~w^{\prime\prime}(x) \geq 0,  ~w^{\prime\prime\prime}(x) \geq 0$ $\Rightarrow X_{1:n} \geq_{hr} Y_{1:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont Let us fix
 \begin{equation}
 \begin{bmatrix}
 \alpha_1^{(j)} & \ldots & \alpha_n^{(j)}\\
 \gamma_1^{(j)} & \ldots & \gamma_n^{(j)}
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \gamma_1 & \ldots & \gamma_n
 \end{bmatrix} T^{\lambda_1}\ldots T^{\lambda_j}, j=1, \ldots, k-1.
 \end{equation}
 Consider $Y_1^{(j)},\ldots, Y_n^{(j)}, j=1, \ldots, k-1$, be sets of independent random variables with $Y_i^{(j)} \sim W-G(\alpha_i^{(j)}, \beta, \gamma_i^{(j)}), i=1, \ldots, n$ and $j=1, \ldots, k-1, k \geq 2$. It has been assumed that
 \begin{equation}
 \begin{bmatrix}
 \alpha_1^{(j)} & \ldots & \alpha_n^{(j)}\\
 \gamma_1^{(j)} & \ldots & \gamma_n^{(j)}
 \end{bmatrix} \in P_n, \text{ for } j=1,\ldots, k-1.
 \end{equation}
 Using theorem $\ref{thm2}$ repeatedly we observe that $X_{1:n} \geq_{hr} Y_{1:n}^{(1)} \geq_{hr} \ldots Y_{1:n}^{(k-1)} \geq_{hr} Y_{1:n}$.
 \end{proof}
 \subsection{Results for largest ordered statistic with vector majorized components}
 When $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $\gamma_1 = \gamma_2 = \ldots = \gamma_n= \gamma_1^* = \gamma_2^* = \ldots = \gamma_n^*$ and $(\alpha_1, \alpha_2,\ldots, \alpha_n) \prec_{w} (\alpha_1^*, \alpha_2^*,\ldots, \alpha_n^*)$, we observe the following result for  the largest ordered statistic.
 \begin{theorem}
\normalfont Let $X_1, X_2, \ldots, X_n$ be a set of independent random variables from Weibull- G distributed family of distribution such that $X_i \sim W-G(\alpha_i, \beta, \gamma)$ for $i=1,2, \ldots, n$. Let $Y_1, Y_2, \ldots, Y_n$ be another set of independent random variables such that $Y_i \sim W-G(\alpha_i^*, \beta, \gamma)$ for $i = 1, 2,\ldots , n$. Then $\underline{\alpha} \prec_{w} \underline{\alpha}^* \Rightarrow X_{n:n} \leq_{rh} Y_{n:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont The distribution function of $X_{n:n}$ is
 \begin{equation}
 H_{X_{n:n}}(x)= \prod_{i=1}^n \left(1- e^{-\alpha_i(w(\gamma x))^\beta}\right), ~~x>0, ~\alpha_i >0, ~\beta>0, ~\gamma>0 ~\forall ~i=1,2,\ldots,n.
 \end{equation}
 Taking logarithm on both sides and differentiating with respect to $x$ we obtain the reversed hazard rate function as
 \begin{equation}\label{rhnn}
\tilde{r}_{X_{n:n}}(x) = \beta \gamma\left(\dfrac{F(\gamma x)}{1-F(\gamma x)}\right)^ {\beta -1} \dfrac{f(\gamma x)}{(1-F(\gamma x))^2} \sum_{i=1}^n \dfrac{\alpha_i e^{-\alpha_i (w(\gamma x))^{\beta}}}{1- e^{-\alpha_i (w(\gamma x))^{\beta}}}.
 \end{equation}
 For an easier understanding we rewrite equation \eqref{rhnn} as
 \begin{equation}
 \tilde{r}_{X_{n:n}}(x) =  \beta \gamma\left(\dfrac{F(\gamma x)}{1-F(\gamma x)}\right)^ {\beta -1} \dfrac{f(\gamma x)}{(1-F(\gamma x))^2} \sum_{i=1}^n g(\alpha_i),
 \end{equation}
 such that $g(\alpha) = \dfrac{\alpha}{e^{\alpha z} -1}$, where $z(x, \beta, \gamma) \equiv z = (w(\gamma x))^{\beta}$.
 It is required to show that the function $g(\alpha)$ is decreasing and convex.
 Computing $g^{\prime}(\alpha)$ and $g^{\prime\prime}(\alpha)$ we observe that,
 \begin{align}
 g^{\prime}(\alpha) &= \dfrac{e^{\alpha z}- \alpha ze^{\alpha z} - 1}{(e^{\alpha z}-1)^2} \label{g1},\\
 g^{\prime\prime}(\alpha) &= \dfrac{ze^{\alpha z}(\alpha ze^{\alpha z}-2e^{\alpha z}+ \alpha z +2)}{(e^{\alpha z}-1)^3} \label{g2}.
 \end{align}
 From \eqref{g1}, $g^{\prime}(\alpha) \overset{sign}{=} h_1(\alpha z) = e^{\alpha z}- \alpha ze^{\alpha z} - 1$ and $h_1(t)$ is a decreasing function of $t$. We recall lemma 2.5, then $\alpha > 0 \Rightarrow g^{\prime}(\alpha) \leq 0$.
 Similarly from \eqref{g2}, $g^{\prime\prime}(\alpha) \overset{sign}{=} h_2(\alpha z) = \alpha ze^{\alpha z}-2e^{\alpha z}+ \alpha z +2$. Using lemma $\ref{lh2}$ we observe that $g^{\prime\prime}(\alpha) \geq 0$. Thus $g(\alpha)$ is a decreasing convex function. Hence using lemma 2.2, we conclude that $\tilde{r}_{X_{n:n}}(x)$ is a Schur-convex function, therefore, we obtain $\underline{\alpha} \prec_{w} \underline{\alpha}^* \Rightarrow \tilde{r}_{X_{n:n}}(x) \leq_{rh} \tilde{r}_{Y_{n:n}}(x)$.
 \end{proof}

 When $\alpha_1 = \alpha_2 = \ldots = \alpha_n= \alpha_1^* = \alpha_2^* = \ldots = \alpha_n^*$, $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $(\gamma_1, \gamma_2, \ldots, \gamma_n) \prec^w (\gamma_1^*, \gamma_2^*, \ldots, \gamma_n^*)$, we observe the usual stochastic ordering of the largest ordered statistic where the baseline distribution of $X$ is Exponential.

 \begin{theorem}
 \normalfont Let $X_1,X_2, \ldots,X_n$ be a set of independent random variables such that
$X_i \sim W-Exp(\alpha,\beta,\gamma_i)$ for $i = 1, 2, \ldots, n$. Let $Y_1, Y_2, \ldots , Y_n$ be another set of independent random
variables such that $Y_i \sim W-Exp(\alpha,\beta, \gamma_i^* )$ for $i = 1, 2, \ldots, n$. Then $\underline{\gamma} \prec^w \underline{\gamma}^* \Rightarrow X_{n:n} \leq_{st} Y_{n:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont The baseline distribution is Exponential of the form $F(\gamma x) = 1- e^{-\gamma x}, x>0$. Therefore the function $w(\gamma x) = e^{\gamma x} -1$.
 The distribution function of $X_{n:n}$ is given by
 \begin{equation}\label{baseexp}
 F_{X_{n:n}}(x) = \prod_{i=1}^n \left(1-e^{-\alpha(e^{\gamma_i x}-1)^{\beta}}\right), x>0, ~\alpha>0, ~\beta>0, ~\gamma_i >0 ~\forall ~i=1,2,\ldots, n.
 \end{equation}
 Differentiating $\eqref{baseexp}$ with respect to $\gamma_i$,
 \begin{align*}
 \dfrac{\partial F_{X_{n:n}}(x)}{\partial \gamma_i} &= \alpha \beta x F_{X_{n:n}}(x) \dfrac{(e^{\gamma_i x}-1)^{\beta -1}e^{\gamma_i x}}{e^{\alpha(e^{\gamma_i x}-1)^{\beta}}-1}\\
 & > 0.
 \end{align*}

 Let $\psi_1(\gamma) = \dfrac{(e^{\gamma x}-1)^{\beta -1}e^{\gamma x}}{e^{\alpha(e^{\gamma x}-1)^{\beta}}-1}$, differentiating $\psi_1(\gamma)$ with respect to $\gamma$, we obtain
 \begin{align*}
 \psi_1^{\prime}(\gamma) &= -\dfrac{x(e^{\gamma x}-1)^{\beta-2}e^{\gamma x}}{(e^{\alpha(e^{\gamma x}-1)^{\beta}}-1)^2}(((\alpha \beta (e^{\gamma x}-1)^{\beta}-\beta)e^{\gamma x}+1)e^{\alpha(e^{\gamma x}-1)^{\beta}}+\beta e^{\gamma x} -1)\\
 & \overset{sign}{=} - (((\alpha \beta (e^{\gamma x}-1)^{\beta}-\beta)e^{\gamma x}+1)e^{\alpha(e^{\gamma x}-1)^{\beta}}+\beta e^{\gamma x} -1), \gamma >0, x>0.
 \end{align*}

Let us assume $e^{\gamma x} =t$, then the above equation (ignoring the -ve sign )can be rewritten as,
$$\psi_2(t)= ((\alpha \beta (t-1)^{\beta}-\beta)t+1)e^{\alpha(t-1)^{\beta}}+\beta t -1, ~t>1,$$
also $\psi_2(1) = 0$. Differentiating $\psi_2(t)$ with respect to $t$,
$$\psi_2^{\prime}(t)= \dfrac{\beta}{t-1}((({\alpha}^2\beta(t-1)^{2\beta}+ \alpha(t-1)^{\beta}-1)t+1)e^{\alpha(t-1)^{\beta}}+t-1).$$
Again consider,
$$\psi_3(t) = (({\alpha}^2\beta(t-1)^{2\beta}+ \alpha(t-1)^{\beta}-1)t+1)e^{\alpha(t-1)^{\beta}}+t-1,$$
differentiating $\psi_3(t)$ we obtain
\begin{align*}
\psi_3^{\prime}(t)&= 1-e^{\alpha(t-1)^{\beta}}+ e^{\alpha(t-1)^{\beta}}(\alpha(t-1)^{\beta}+{\alpha}^2\beta(t-1)^{2\beta}+2{\alpha}^2{\beta}^2t(t-1)^{2\beta-1}+{\alpha}^2{\beta}^2t(t-1)^{3\beta -1}\\
& ~~+{\alpha}^2\beta t(t-1)^{2\beta -1}+\alpha\beta{(t-1)}^{\beta -1}).
\end{align*}
Let us consider $\phi(z) = 1-e^{z}+ z e^{z}$, where $z= \alpha(t-1)^{\beta}$. Using lemma 2.5, we observe that $\phi(z)>0 ~~\forall z >0$, i.e., $\psi_3(t) >0$. Consequently $\psi_2^{\prime}(t) >0$ and for $t>1$, $\psi_2(t) >0$. Hence $\psi_1^{\prime}(\gamma) <0$ for $\gamma >0$.
 Consider
 \begin{align*}
 \Delta &= (\gamma_i -\gamma_j)\left(\dfrac{\partial F_{X_{n:n}}(x)}{\partial \gamma_i} - \dfrac{\partial F_{X_{n:n}}(x)}{\partial \gamma_j}\right)\\
 &= \alpha \beta x F_{X_{n:n}}(x) (\gamma_i -\gamma_j)(\psi_1(\gamma_i)-\psi_1(\gamma_j))\\
 & \leq 0,
 \end{align*}
 i.e., $F_{X_{n:n}}(x)$ is increasing and Schur-concave function with respect to the parameter $\gamma_i$ $\forall i =1,2, \ldots, n$. Hence $\underline{\gamma} \prec^w \underline{\gamma}^* \Rightarrow F_{X_{n:n}}(x) \geq F_{Y_{n:n}}(x)$.
 \end{proof}

\section{Gompertz-Makeham distribution}
In this section we shall discuss the stochastic ordering results for Gompertz Makeham distribution. When the parameter $\lambda$ is kept constant and the remaining two parameters are varied in the sense of multivariate chain majorization we observe  a hazard rate ordering between $X_{1:2}$ and $Y_{1:2}$.
\subsection{Results for smallest ordered statistic with multivariate chain majorized components}
\begin{theorem}
\normalfont Let $X_1,X_2$ and $Y_1,Y_2$ be 2 pairs of independent random variable such that $X_i \sim GM(\alpha_i, \beta_i, \lambda)$ and $Y_i \sim GM(\alpha_i^*, \beta_i^*, \lambda)$ for $i=1,2$. Let $\begin{bmatrix}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2
\end{bmatrix} \in P_2$, then
$$\begin{bmatrix}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2
\end{bmatrix} \prec \prec \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\beta_1^* & \beta_2^*
\end{bmatrix} \Rightarrow X_{1:2} \geq_{hr} Y _{1:2}.$$
\end{theorem}
\begin{proof}
\normalfont The reliability function of the smallest ordered statistic $X_{1:2}$ is
\begin{equation}
\overline{F}_{X_{1:2}}(x) = e^{-2\lambda x}\prod_{i=1}^2 e^{-\dfrac{\alpha_i}{\beta_i}(e^{\beta_i x}-1)}, x> 0, \alpha_i>0, \beta_i >0, \lambda>0.
\end{equation}
The corresponding hazard rate function is given by
\begin{equation}
r_{X_{1:2}}(x) = 2\lambda + \sum_{i=1}^2 \alpha_ie^{\beta_i x} , x>0.
\end{equation}
Consider
\begin{align*}
\phi(\underline{\alpha},\underline{\beta}) &= (\alpha_1 - \alpha_2)\left(\dfrac{\partial r_{X_{1:2}}(x) }{\partial \alpha_1} - \dfrac{\partial r_{X_{1:2}}(x) }{\partial \alpha_2}\right) + (\beta_1 - \beta_2)\left(\dfrac{\partial r_{X_{1:2}}(x) }{\partial \beta_1}- \dfrac{\partial r_{X_{1:2}}(x) }{\partial \beta_2}\right)\\
&= (\alpha_1 - \alpha_2)(e^{\beta_1 x} - e^{\beta_2 x})+ x(\beta_1 -\beta_2)(\alpha_1e^{\beta_1 x} - \alpha_2e^{\beta_2 x})\\
& \geq 0.
\end{align*}
Since we have considered the parameter matrix $\begin{bmatrix}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2
\end{bmatrix} \in P_2$, then either $\alpha_1 > \alpha_2, \beta_1 > \beta_2$ or $\alpha_1 < \alpha_2, \beta_1 < \beta_2$. In both the cases the sign of $\phi(\underline{\alpha},\underline{\beta})$ is always positive. Hence we can conclude using Lemma 2.4 that $\begin{bmatrix}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2
\end{bmatrix} \prec \prec \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\beta_1^* & \beta_2^*
\end{bmatrix} \Rightarrow r_{X_{1:2}} \leq r_{Y _{1:2}}$, i.e., $X_{1:2} \geq_{hr} Y _{1:2}$.

\end{proof}
As in Example 1, with the same set of parameters, we can observe the following example.
 \begin{example}
\normalfont Let $X_1, X_2$ be independent random variables such that $X_i \sim GM(\alpha_i, \beta_i, \lambda)$, $i=1,2$. Also let $Y_1, Y_2$ be another pair of independent random variable such that $Y_i \sim GM(\alpha_i^*, \beta_i^*, \lambda), i=1,2$. The parameters satisfy all the given conditions and are given in the form of matrices as
 \begin{equation*}
 \begin{bmatrix}
\alpha_1^* & \alpha_2^*\\
\beta_1^* & \beta_2^*
\end{bmatrix} = \begin{bmatrix}
4.8 & 3.4\\
2.5 & 1.6
\end{bmatrix} \in P_2 \text{ and } \begin{bmatrix}
\alpha_1 & \alpha_2\\
\beta_1 & \beta_2
\end{bmatrix} = \begin{bmatrix}
4.03 & 4.17\\
2.005 & 2.095
\end{bmatrix}.
 \end{equation*}
 The plot of the difference $r_{Y_{1:2}} - r_{X_{1:2}}$ is
\begin{figure}[h]
		\centering
			\includegraphics[scale=0.3]{eg2}

			 \caption{The figure shows the graph of $r_{Y_{1:2}} - r_{X_{1:2}}$ for any value of $\lambda$.}
	\end{figure}

 \end{example}
The above theorem can be extended to n random variables using the next theorem.
\begin{theorem}
 \normalfont Let $X_1, \ldots, X_n$ be as set of independent random variables such that $X_i \sim GM(\alpha_i, \beta_i, \lambda), i=1,\ldots,n$. Also $Y_1,\ldots,Y_n$ be another set of random variable such that $Y_i \sim GM(\alpha_i^*, \beta_i^*, \lambda), i=1,\ldots,n$. If
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \lambda_1 & \ldots & \lambda_n
 \end{bmatrix} \in P_n
 \end{equation*}
 and
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1^* & \ldots & \alpha_n^*\\
 \lambda_1^* & \ldots & \lambda_n^*
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \lambda_1 & \ldots & \lambda_n
 \end{bmatrix} T_{i,j}^{\delta},
 \end{equation*}
 then $X_{1:n} \geq_{hr} Y_{1:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont Proceeding in a similar way as in Theorem 3.2, the parameters $\alpha_k = \alpha_k^*$ and $\lambda_k = \lambda_k^*$ corresponding to the r.v. $X_k$ and $Y_k$ respectively $\forall k \neq i,j$. The T-transform matrix $T_{i,j}^{\delta} = \delta I_{n} + (1-\delta)\Pi_{i,j}$ where $\Pi_{i,j}$ interchanges the $i^{th}$ row with the $j^{th}$ row. Therefore the result follows from Theorem 4.1.
 \end{proof}

 \begin{theorem}
 \normalfont Let $X_1,\ldots, X_n$ be a set of independent random variables such that $X_i \sim GM(\alpha_i, \beta_i, \lambda), i=1,\ldots,n$. Also let $Y_1, \ldots, Y_n$ be another set of random variables such that $Y_i \sim GM(\alpha_i^*, \beta_i^*,\lambda), i=1,\ldots,n$. For $k\geq 2$, if
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \beta_1 & \ldots & \beta_n
 \end{bmatrix} \in P_n,
 \end{equation*}
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \beta_1 & \ldots & \beta_n
 \end{bmatrix} T^{\delta_1}\ldots T^{\delta_i} \in P_n, \text{for } i=1,\ldots, k-1,
 \end{equation*}
 and
 \begin{equation*}
 \begin{bmatrix}
 \alpha_1^* & \ldots & \alpha_n^*\\
 \beta_1^* & \ldots & \beta_n^*
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \beta_1 & \ldots & \beta_n
 \end{bmatrix} T^{\delta_1}\ldots T^{\delta_k},
 \end{equation*}
 then $X_{1:n} \geq_{hr} Y_{1:n}$.
 \end{theorem}
 \begin{proof}
 \normalfont Consider
 \begin{equation}
 \begin{bmatrix}
 \alpha_1^{(j)} & \ldots & \alpha_n^{(j)}\\
 \beta_1^{(j)} & \ldots & \beta_n^{(j)}
 \end{bmatrix} = \begin{bmatrix}
 \alpha_1 & \ldots & \alpha_n\\
 \beta_1 & \ldots & \beta_n
 \end{bmatrix} T^{\delta_1}\ldots T^{\delta_j}, j=1,\ldots, k-1.
 \end{equation}
 Let us assume $Y_1^{(j)},\ldots, Y_n^{(j)}, j=1,\ldots, k-1$, be sets of independent random variables with $Y_i^{(j)} \sim GM(\alpha_i^{(j)}, \beta_i^{(j)}, \lambda)$, $i=1,\ldots,n$ and $j=1,\ldots, k-1, k \geq 2$. Since
$$\begin{bmatrix}
\alpha_1^{(j)} & \ldots & \alpha_n^{(j)}\\
 \beta_1^{(j)} & \ldots & \beta_n^{(j)}
 \end{bmatrix} \in P_n, \text{ for } j=1,\ldots, k-1.$$
 Thus using Theorem 4.2 repeatedly we obtain $X_{1:n} \geq_{hr} Y_{1:n}^{(1)} \geq_{hr} \ldots, Y_{1:n}^{(k-1)} \geq_{hr} Y_{1:n}$.
 \end{proof}
\subsection{When the components are chain majorized}
In this section we shall observe the stochastic behaviour of GM distributed components when only one parameter is varied and all the other parameters are kept constant. Firstly we shall observe the behaviour of the parameter $\lambda$ in the sense of usual stochastic ordering.
\begin{theorem}
\normalfont Let $X_1,X_2, \ldots,X_n$ be a set of independent random variables such that
$X_i \sim GM(\alpha,\beta,\lambda_i)$ for $i = 1, 2, \ldots, n$. Let $Y_1, Y_2, \ldots , Y_n$ be another set of independent random
variables such that $Y_i \sim GM(\alpha,\beta, \lambda_i^* )$ for $i = 1, 2, \ldots, n$. Then $\underline{\lambda} \prec^{w} \underline{\lambda}^* \Rightarrow X_{1:n} =_{st} Y_{1:n}$.
\end{theorem}
\begin{proof}
\normalfont The survival function of $X_{1:n}$ is
\begin{equation}
\overline{F}_{X_{1:n}}(x) = e^{-\left(\displaystyle\sum_{i=1}^n \lambda_i\right) x} e^{-n \dfrac{\alpha}{\beta}(e^{\beta x}-1)}, x> 0, \alpha>0, \beta >0, \lambda_i>0.
\end{equation}
Consider $\phi_1(\underline{\lambda}) = (\lambda_i -\lambda_j)\left(\dfrac{\partial \overline{F}_{X_{1:n}}(x)}{\partial \lambda_i} - \dfrac{\partial \overline{F}_{X_{1:n}}(x)}{\partial \lambda_j}\right)$, where $i \neq j$. Now $\dfrac{\partial \overline{F}_{X_{1:2}}(x)}{\partial \lambda_i} = -x \overline{F}_{X_{1:2}}(x)$. We can thus observe that $\phi_1(\underline{\lambda}) =0$ for every $\alpha >0, \beta>0$ and $\overline{F}_{X_{1:2}}(x)$ is decreasing with respect to each $\lambda_i$, $i=1,2,\ldots,n$. Using lemma 2.1 the result follows.
\end{proof}

In a similar manner, we have obtained the stochastic comparison results for the maximum ordered statistic, $X_{n:n}$ also. The following theorem  presents a usual stochastic ordering between $X_{n:n}$ and $Y_{n:n}$.
\begin{theorem}
\normalfont Let $X_1,X_2, \ldots,X_n$ be a set of independent random variables such that
$X_i \sim GM(\alpha_i,\beta,\lambda)$ for $i = 1, 2, \ldots, n$. Let $Y_1, Y_2, \ldots , Y_n$ be another set of independent random
variables such that $Y_i \sim GM(\alpha_i^*,\beta, \lambda)$ for $i = 1, 2, \ldots, n$. Then $\underline{\alpha} \prec^{w} \underline{\alpha}^* \Rightarrow X_{n:n} \leq_{st} Y_{n:n}$.
\end{theorem}
\begin{proof}
\normalfont The distribution function of $X_{n:n}$ is
\begin{equation}
F_{X_{n:n}}(x)= \prod_{i=1}^n \left( 1- e^{-\lambda x - \dfrac{\alpha_i}{\beta}(e^{\beta x}-1)} \right), x >0, \alpha_i >0, \beta>0, \lambda >0.
\end{equation}
Consider $\phi_2(\underline{\alpha}) = (\alpha_i -\alpha_j)\left(\dfrac{\partial F_{X_{n:n}}(x)}{\partial \alpha_i} - \dfrac{\partial F_{X_{n:n}}(x)}{\partial \alpha_j}\right)$, for $i \neq j$. Now,
$$\dfrac{\partial F_{X_{n:n}}(x)}{\partial \alpha_i} = \dfrac{1}{\beta} \dfrac{F_{X_{n:n}}(x) (e^{\beta x}-1)}{e^{\lambda x + \dfrac{\alpha_i}{\beta}(e^{\beta x}-1)}-1}.$$
Thus, $$\phi_2(\underline{\alpha}) = \dfrac{1}{\beta}(e^{\beta x}-1)F_{X_{n:n}}(x)(\alpha_i -\alpha_j)\left(p(\alpha_i) - p(\alpha_j)\right), \alpha_i \neq \alpha_j$$ where $p(\alpha) = \dfrac{1}{e^{\lambda x+\dfrac{\alpha}{\beta}(e^{\beta x}-1)}-1}$.\\
 We observe that the quantity $p(\alpha)$ is decreasing with respect to $\alpha$, since\\
  $p^{\prime}(\alpha) = -\left(e^{\lambda x+\dfrac{\alpha}{\beta}(e^{\beta x}-1)}-1\right)^{-2}\dfrac{(e^{\beta x}-1)}{\beta}e^{\lambda x + \dfrac{\alpha}{\beta}(e^{\beta x}-1)}$. Consequently, $\phi_2(\underline{\alpha}) \leq 0$. Hence using lemma 2.1, $F_{X_{n:n}}(x)$ is Schur-concave with respect to $\underline{\alpha} = (\alpha_1,\alpha_2,\ldots,\alpha_n)$. Also, $F_{X_{n:n}}(x)$ is increasing w.r.t each $\alpha_i$, $i=1,2,\ldots,n$. In other words $-F_{X_{n:n}}(x)$ is Schur-convex and decreasing with respect to $\underline{\alpha}$. Using Lemma 2.3, we observe that $\underline{\alpha} \prec^{w} \underline{\alpha}^* \Rightarrow F_{X_{n:n}}(x) \geq_{st} F_{Y_{n:n}}(x)$ and the result follows.
\end{proof}
 \section{Conclusion}
 We have observed the following results:
  Let $X_1, X_2, \ldots, X_n$ be independent random variables with $X_i \sim W-G(\alpha_i, \beta_i, \gamma_i), i=1,2,\ldots, n$. Furthermore, let $Y_1, Y_2, \ldots, Y_n$ be another set of independent random variables with $Y_i \sim W-G(\alpha_i^*, \beta_i^*, \gamma_i^*), i=1,2,\ldots, n$. When $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and the matrix containing the parameters $\alpha_i, \gamma_i$ changes to another matrix containing the parameters $\alpha_i^*, \gamma_i^*, i=1,2, \ldots, n $ in the sense of multivariate chain majorization, we study the hazard rate ordering of the smallest ordered statistic.\\
  Next, when $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $\gamma_1 = \gamma_2 = \ldots = \gamma_n= \gamma_1^* = \gamma_2^* = \ldots = \gamma_n^*$ and $(\alpha_1, \alpha_2,\ldots, \alpha_n) \prec_{w} (\alpha_1^*, \alpha_2^*,\ldots, \alpha_n^*)$, we establish reversed hazard rate ordering of the largest ordered statistic. Also when $\alpha_1 = \alpha_2 = \ldots = \alpha_n= \alpha_1^* = \alpha_2^* = \ldots = \alpha_n^*$, $\beta_1 = \beta_2 = \ldots = \beta_n= \beta_1^* = \beta_2^* = \ldots = \beta_n^*$ and $(\gamma_1, \gamma_2, \ldots, \gamma_n) \prec^w (\gamma_1^*, \gamma_2^*, \ldots, \gamma_n^*)$, we observe the usual stochastic ordering of the largest ordered statistic when the baseline distribution of $X$ is Exponential.\\
  A similar set of random variables following Gompertz Makeham distribution $(X_i \sim GM(\alpha_i, \beta_i, \lambda_i) \text{ and } Y_i \sim GM(\alpha_i^*, \beta_i^*, \lambda_i^*))$ are considered. We observed hazard rate ordering for the sample minimum when the parameters $\lambda_1 =\lambda_2=\ldots, \lambda_n = \lambda_1^* =\lambda_2^*=\ldots, \lambda_n^* = \lambda$ and the other remaining parameters are related in the sense of multivariate chain majorization. Whereas usual stochastic ordering has been observed for the sample maximums when the parameters $\beta_1=\beta_2=\ldots, \beta_n = \beta_1^*=\beta_2^*=\ldots, \beta_n^*$ and $\lambda_1=\lambda_2=\ldots, \lambda_n = \lambda_1^*=\lambda_2^*=\ldots, \lambda_n^*$ while the parameter $\alpha$ is only varied using vector majorization. Also we observe that the age independent parameter $\lambda$ has absolutely no effect on the ordered statistics.

 \section*{Acknowledgments}
  The first author would like to thank IIT Kharagpur for research assistantship.

{\small

redpajama_set_name: RedPajamaArXiv