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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsMensuration : Length, Area, and VolumeMeasurement Methods

### Measurement by superimposition

To measure a length, area, or volume one of the following methods is used.

•  Measurement by Superimposition

•  Measurement by Calculation

•  Measurement by Equivalence

The First two methods, "superimposition" and "calculation" are simple. Let us review them quickly. These two do not need much explanation and quite easy to understand.

This topic covers the measurement by superimposition.

click on the content to continue..

Consider the shape given in the figure. It is a rectangle of length 7cm and width 5cm. The figure also provides the reference unit-square on the right. What is the area of the shape?

• 35cm^2
• 35cm^2
• area cannot be found using the given information

The answer is "35cm^2". The area is computed by creating a grid of unit-squares. The number of unit squares contained within the shape is the area of the rectangle.

Area of the given shape is calculated by superimposing the reference unit-squares. Based on this, the area of a square or rectangle is simplified to the formulas

text(Area of a square ) = text( side)^2

text(Area of a rectangle ) = text( length ) xx text( width)

Consider the shape given in the figure. The measurements are not provided. The reference unit-square is shown on the top-right corner. What is the area of the given shape?

• area cannot be calculated
• approximate area can be calculated by superimposing a grid of unit-squares
• approximate area can be calculated by superimposing a grid of unit-squares

The answer is "approximate area can be calculated by superimposing a grid of unit-squares".

The area is approximately 15cm^2.
.
The area is approximated to the count of large squares within the figure.

Measurement by superimposition provides accurate results for

•  area of squares and rectangles : The unit-square has 90^@ angles at the vertices. And so, the unit-square fits within Squares and Rectangles.

•  volume of cubes and cuboids : The unit-cube has 90^@ angles at the vertices. And so, the unit-cube fits within cubes and cuboids.

For shapes like triangles or prisms, measurement requires some geometrical calculation or the measurement will be approximate.

Measurement by Superimposition : Length, Area, or Volume can be measured by superimposition of corresponding unit-measures. This method suits best for

•  area of squares and rectangles

•  volume of cubes and cuboids

slide-show version coming soon

format: plain


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# Beginner level : Bit error rate trend

Please correct me where wrong and I would really appreciate intuitive explanation for the following fundamental questions. These may appear trivial but as I am not from Electrical background, these concepts are quite baffling for me.

1. It is preferred to have a low BER. Does the theoretical BER serves as the lower or the upper bound? If the simulated / emperical bit error rate for an uncoded AWGN channel, no pulse-shaping and no spreading techniques, after blind equalization is lower than the theoretical BER - what can be inferred? Basically, I am comparing equalization algorithm in literature and have encountered several variants of Constant Modulus Algorithm. Each paper claims to perform better than the other eventhough the improvement is negligible when compared to the complex modification done to the algorithm. So, this Question stems from the doubt that is when can we say that the equalizer performance is good? When the BER reaches the analytical BER or if it is even lower than the analytical BER?

1. What should be the ideal trend of BER at low and high signal-to-noise ratio (SNR)? I have found that the performance of equalizers is good at low SNR and worsens for higher SNR. In general, we seek to develop techniques that can do channel estimation or system identification at low SNR. But, in doing so that technique must also perform good at high SNR because the effect of measurement noise is negligible. What is the tradeoff?

Thank you, and please let me know if any modifications/ improvements need to be made to make my Question clear, if otherwise.

A lower BER would seem to be better, who wants corrupt data after all?

However, it's like the bankers' (apparently) mantra that 'if you are not cheating, you're not trying hard enough'. If in a working comms system you have a BER rate much lower than your correction can handle, you are operating the system with too much power, or too few users per cell, in other words with an inefficient use of resources. The power control algorithm of most modern comms systems reduces the output power (in the mobile to save battery life, in the base station to reduce interference to other users) when the receiver reports back an unnecessarily good BER.

Error correction/detection systems allow you correct small numbers of errors and detect and resend blocks with large numbers of errors. This enables the system to achieve effectively zero error when appropriate (file transfer) and tolerable error when it doesn't need to (speech, video).


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## FANDOM

506 Pages

ISO 639 is the set of international standards that lists short codes for languages. It was also the name of the original standard, approved in 1967 and withdrawn in 2002.

ISO 639 consists of different parts, of which four parts have been approved (parts 1, 2, 3 and 5). The other parts are works in progress.

## The six parts of the standardEdit

Standard Name (Codes for the representation of names of languages -- ...) First Edition Current No. In List
ISO 639-1 Part 1: Alpha-2 code 1967 (as ISO 639) 2002 185
ISO 639-2 Part 2: Alpha-3 code 1998 1998 >450
ISO 639-3 Part 3: Alpha-3 code for comprehensive coverage of languages 2007 2007 7704 + local range
ISO/DIS 639-4 Part 4: Implementation guidelines and general principles for language coding (in DIS stage) - -
ISO 639-5 Part 5: Alpha-3 code for language families and groups 2008-05-15 2008-05-15 114
ISO/FDIS 639-6 Part 6: Alpha-4 representation for comprehensive coverage of language variation (in FDIS stage) -  ?

Each part of the standard is maintained by a maintenance agency, which adds codes and changes the status of codes when needed.

## Characteristics of individual codesEdit

Scopes:

• Individual languages
• Macrolanguages (part 3)
• Collections of languages (part 1, 2, 5) (part 1 contains only 1 collection: bh; most collections are in part 2, and a few were added in part 5)
• Group
• Rest group
• Dialects
• Reserved for local use (part 2, 3)
• Special situations (part 2, 3)

Types (for individual languages):

• Living languages (part 2, 3) (all macrolanguages are living languages[1])
• Extinct languages (part 2, 3) (437[2], four in part 2 chb, chg, cop, sam; none in part 1)
• Ancient languages (part 1, 2, 3) (112[3], 19 are in part 2; and 5 of them, namely ave, chu, lat, pli and san, also have a code in part 1: ae, cu, la, pi, sa)
• Historic languages (part 2, 3) (63[4], 16 of them are in part 2, none has part 1 code)
• Constructed languages (part 2, 3) (19[5], 9 in part 2: epo, ina, ile, ido, vol, afh, jbo, tlh, zbl; five in part 1: eo, ia, ie, io, vo)

Bibliographic and terminology codes

• Bibliographic (part 2)
• Terminology (part 2)

## Use of ISO 639 codesEdit

The language codes defined in the several sections of ISO 639 are used for bibliographic purposes and, in computing and internet environments, as a key element of locale data. The codes also find use in various applications, such as Wikipedia URLs for its different language editions.

## Code spaceEdit

### Alpha-2 code spaceEdit

"Alpha-2" codes (for codes composed of 2 letters of the basic Latin alphabet) are used in ISO 639-1. When codes for a wider range of languages were desired, more than 2 letter combinations could cover (a maximum of $26^2=676$), ISO 639-2 was developed using Alpha-3 codes (though the latter was formally published first).Template:Fact

### Alpha-3 code spaceEdit

"Alpha-3" codes (for codes composed of 3 letters of the basic Latin alphabet) are used in ISO 639-2, ISO 639-3, and ISO 639-5. Mathematically, the upper limit for the number of languages and language collections that can be so represented is $26^3=17,576$.

The common use of Alpha-3 codes by three parts of ISO 639 requires some coordination within a larger system.

Part 2 defines four special codes mul, und, mis, zxx, a reserved range qaa-qtz (20 × 26 = 520 codes) and has 23 double entries (the B/T codes). This sums up to 520 + 23 + 4 = 547 codes that cannot be used in part 3 to represent languages or in part 5 to represent language families or groups. The remainder is 17,576 – 547 = 17,029.

There are somewhere around six or seven thousand languages on Earth today[6]. So those 17,029 codes are adequate to assign a unique code to each language, although some languages may end up with arbitrary codes that sound nothing like traditional name(s) of that language.

### Alpha-4 code spaceEdit

"Alpha-4" codes (for codes composed of 4 letters of the basic Latin alphabet) is proposed to be used in ISO 639-6. The upper limit for the number of languages and dialects that can be represented is $26^4=456,976$.

Cite error: <ref> tags exist, but no <references/> tag was found


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# Cars and their torque

1. Nov 21, 2008

### thecombover

Some buddies and I were discussing cars and their torque. We were wondering why the amount of toruqe that is exerted by the flywheel is not the same amount of torque at the end of the drive shaft. Does it have something to do with the longer the drive shaft the less torque? or does it deal with the extra mass added by the extra length? If anybody knows it would be great to find out thanks!!!

2. Nov 21, 2008

### daveyman

Re: Toruqe

I don't know much about cars, but I do know something about torque.

The equation for torque is $$\tau = rF\sin{\theta}$$.

The variable r represents the distance from the center of rotation to the point that is rotating. The variable F is the force vector of the point (mass times acceleration in a particular direction). The sine term represents the angle between the axis and the direction that the point is rotating. In this case, this angle is 90 degrees. Since the sine of 90 degrees is 1, all we have to worry about is r and F.

Now if you imagine a point on the outer edge of the flywheel, you could determine r (the distance from the center) and F (the mass of the "point" multiplied by its acceleration).

If you imagine a point on the outer edge of the drive shaft, you would get different values for r and F. So, according to the definition of torque, you would get a different value.

Although the torque is not the same, power must be (nearly) the same or we would have a violation of conservation of energy. So now we turn to the definition of power (in terms of torque).

$$Power=Torque\times 2\pi \times Rotational Speed$$

So whichever object (flywheel or drive shaft) has less torque, it must also have more rotational speed. This means both objects are capable of delivering the same amount of power and no conservation laws are violated (whew!).

Last edited: Nov 21, 2008
3. Nov 21, 2008

### Phrak

Re: Toruqe

The flywheel is mounted to the engine crank. Between the crank and the drive shaft is the transmission. The transmission changes the torque developed on the crank and applies it to the drive shaft.

4. Nov 21, 2008

### Naty1

Re: Toruqe

As Phrak post implied, transmissions change gear ratios and those transform torque relationships. Its effectively the same idea as using a long breaking bar on a socket when trying to break loose a tight nut: a longer bar permits more torque but you have to apply more motion to the end to get it. An engine has little torque at low RPM and so high gear ratios enable the engine to rev up more easily to an RPM range where more torque is produced.

HP = (torque x RPM)/5252

Derivation and good discussion at:
http://www.vettenet.org/torquehp.html


text:
December 05, 2020, 09:59:44 AM
Forum Rules: Read This Before Posting

### Topic: I'm new, where can I find this book?  (Read 21057 times)

0 Members and 1 Guest are viewing this topic.

#### BrokeThePope

• Guest
##### I'm new, where can I find this book?
« on: July 26, 2005, 09:57:22 PM »
I am looking for this book, but I'm having problems finding one and dont have timeon this public computer.

The golden book of chemistry experiments;: How to set up a home laboratory -- over 200 simple experiments

thanks

#### billnotgatez

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##### Re:I'm new, where can I find this book?
« Reply #1 on: July 27, 2005, 04:39:13 AM »
From the following web site  -

The Golden Book of Chemistry

Experiments|Ebook|PDF|114 pages|size 26,5 Mb|

More than 200 color pictures and clear, easy-to-follow directions show junior chemists how to prepare a laboratory at home, how to buy and make apparatus, and how to set up fascinating, informative experiments. In addition, there are facts about famous chemists, their contributions, and chemistry in nature and industry.

This book was pulled from all public libraries and store shelves by the government. It was said that the experiments and information contained herin were too dangerous for the general public. How'd I luck up on a copy? Well, a friend of mine who could remember checking out and reading the book when he was a kid, thought he would try to find a copy for himself just for sentimental reasons. To his surprise this book is extremely rare and expensive if you do find a copy for sale. With books going from $375 for a very damaged copy to upwards of$2,000 for a decent copy he knew it would be a task.

Edit: Russian laissez fair copyright laws not withstanding, it would be best if you send this information over our private message service.
« Last Edit: July 27, 2005, 05:50:33 PM by Mitch »

#### billnotgatez

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##### Re:I'm new, where can I find this book?
« Reply #2 on: July 27, 2005, 09:36:18 PM »
Greetings Mitch   et al,

I had assumed that the site was allowed to do its thing by the author since there is obvious de facto censorship. Obviously he is not benefiting from the high prices for this book caused by the banning (unless he is selling them for 2k). It never occurred to me that there might be some copyright issues. I did not notice the Russian extension on the link.

I agree that keeping the link off your web site is prudent, since the patriot act has been enacted. I doubt that you could take up the gauntlet against censorship and win.

If this book is so good then it is a shame that it has such a high price due to the censorship. It makes one wonder if high praise for science literacy is just lip service.

In any case if I can find it using Google, then anyone can if they know it is there.

I thoroughly enjoy this forum and hope it thrives. Some of the censorship here I do not agree with but I like it here too much to quibble.

Regards,
Bill

#### Mitch

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##### Re:I'm new, where can I find this book?
« Reply #3 on: July 28, 2005, 12:25:25 AM »
There was a discussion on the matter between the admins on the Staff forum. In my opinion the link had two strikes against it. One, it gave unsound chemical advice targeted to an audience that doesn't know better. Second, no matter the current state of sales for a book, we're not going to encourage the practice of free downloading versus legal purchasing.

Censorship, as it pertains to the probable possibility that an uneducated youth is going to get himself injured or killed by trying something posted or linked to, will be a permanent fixture of the site, if not for legal reasons, then because the staff has sound judgement.

Being an admin, I of course think the amount of censorship at the website is very minimal, so my perspective is a bit skewd on the subject. But if others have more general complaints they can direct them to the staff forum.
Most Common Suggestions I Make on the Forums.
1. Start by writing a balanced chemical equation.
2. Don't confuse thermodynamic stability with chemical reactivity.
3. Forum Supports LaTex

#### limpet chicken

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##### Re:I'm new, where can I find this book?
« Reply #4 on: July 28, 2005, 01:06:28 AM »
The light blinds
So behold darkness as our new light
In our darkness we can see
So with others blindness
We take flight.

#### Donaldson Tan

• Editor, New Asia Republic
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##### Re:I'm new, where can I find this book?
« Reply #5 on: August 05, 2005, 08:03:21 PM »
we foster safe learning in an open-minded environment. that's why there's minimal and sensible cencorship in this forum. meanwhile, feel free to try out the instant messaging feature (available to all registered users).
"Say you're in a [chemical] plant and there's a snake on the floor. What are you going to do? Call a consultant? Get a meeting together to talk about which color is the snake? Employees should do one thing: walk over there and you step on the friggin� snake." - Jean-Pierre Garnier, CEO of Glaxosmithkline, June 2006

#### bunch1962

• Guest
##### Re:I'm new, where can I find this book?
« Reply #6 on: November 23, 2005, 12:28:59 PM »
I have an original copy of "The Golden Book of Chemistry Experiments" that is in near-mint condition. I checked it out of my elementary school library when I was 8 years old and my friends and I thought it was the greatest book we'd ever seen. I loved the book so much that my parents bought it for me for Christmas. We performed all of the experiments in it. I don't think it was a dangerous book at all because it promoted mature, scientific behavior. The chemicals it recommended using were dangerous in many cases if used improperly, but the book was geared toward their safe usage for scientific study. Some of the chemicals included in the book's experiments were:

Carbon Tetrachloride
Hydrochloric acid
Lye
Sodium Bisulfite (Sani Flush)
Wood alcohol

Some of the toxic substances produced by experiments in the book included:

Amonia
Chlorine gas
Hydrogen gas
Hydrogen sulfide
Sulfuric acid

All of these substances were produced in small quantities and the aparati that were used usually included some sort of neutralizing component. I did all of the experiments in the book and performed some for my 4th, 5th, 6th, 7th and 8th grade science classes.

The point is that there used to be a lot of chemistry sets and books like this one and kids could only mess with these things after they'd demonstrated they were responsible enough to do so on their own, or if they were too young, they did experiments under their parents' supervision. When I was 9 I got a Gilbert Chemistry Set (which was a fantasy gift) and I recall it having Cyanide in it.

Anyhow, I've always loved that old book and am surprised it was banned.  :twitchsmile:

#### buckminsterfullerene

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##### Re:I'm new, where can I find this book?
« Reply #7 on: November 28, 2005, 04:35:15 PM »
you can just get almost any chemistry book and find dangerous chemicals in it..
but how is hydrogen gas dangerous..
as far as to my knowledge, H2 is soo light that in air it would disperse very fast, making it less a risk of explosion or fire hazard then propane and methane, and all that nice heavier compounds that kill many people.  I mean it has been proven that the most likely cause of the Hindenburg accident was not the hydrogen but the spark caused on the highly flammable coating outside of the vessel.
it would only be dangerous under high concentrations in a relatively small area....
and i really do not see anything wrong with the golden book of chemistry experiements, i have the .pdf copy of it, and really its a collecteble, is not like i could print out several copies to sell, i mean i probably could, but it would not be original now would it.. there are some factors that would apply to the copyright laws, and generally when a book is more than a certain number of years (forgot how many) the book becomes something that can be copied and resold by virtually anyone, this is the case with Romeo and Juliet, or any of the shakespeare's works, i think even any of the arthur conan doyle's works can be found online with out any risk from copyright laws.  you cannot say that you wrote it still, but who would do that , and i am pretty sure that a banned book may be somewhere.. but really i like it, it tells you where you can get some materials from grocery stores, and hardware stores etc. etc.
currently a student attending high school in South Florida, capital of all the hurricanes that come through the US, and the sunshine state.  My interests falls into electrochemistry going to renewable resources of energy, i like hydrogen fuel cells and solar energy

#### mike

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##### Re:I'm new, where can I find this book?
« Reply #8 on: November 28, 2005, 06:58:32 PM »
Quote
I don't think it was a dangerous book at all because it promoted mature, scientific behavior. The chemicals it recommended using were dangerous in many cases if used improperly, but the book was geared toward their safe usage for scientific study.

So you say the book isn't dangerous, however the chemicals it recommends are?? Sure the book may promote safe use of the chemicals but it is naive to think that children (or adults) won't misuse or misunderstand the book.

Quote
Carbon Tetrachloride
Hydrochloric acid
Lye
Sodium Bisulfite (Sani Flush)
Wood alcohol

Some of the toxic substances produced by experiments in the book included:

Amonia
Chlorine gas
Hydrogen gas
Hydrogen sulfide
Sulfuric acid

These are not chemicals that "kids" should be playing with. Although not all necessarily "toxic".

Quote
I recall it having Cyanide in it.

Thankfully times have changed! Why does a chemistry set need cyanide? I assume this is sodium cyanide or similar and not cyanide gas?

Quote
you can just get almost any chemistry book and find dangerous chemicals in it..

True, but why should we promote it?

Quote
but how is hydrogen gas dangerous..

Read the MSDS!! It is extremely flammable!

Quote
and i really do not see anything wrong with the golden book of chemistry experiements,

Ok so I haven't read the book, but my first guess is that it doesn't reference any MSDS or other up-to-date safety literature (this is one thing I can think that is probably wrong with it)

Quote
there are some factors that would apply to the copyright laws

Correct, maybe go and find out the specifics then, different laws for different countries and different circumstances....

Please practice chemistry safely and promote the safe use of chemicals.
There is no science without fancy, and no art without facts.

#### buckminsterfullerene

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##### Re:I'm new, where can I find this book?
« Reply #9 on: November 28, 2005, 09:32:56 PM »
Quote
Quote
but how is hydrogen gas dangerous..
.

Read the MSDS!! It is extremely flammable!

but i said that is dangerous under high concentrations, the lightness of hydrogen means that the hydrogen molecules would disperse very fast in air, and would thus not pose such a great threat.  though i agree that it is extremely flammable, even more per gram than other fuels, but other fuels are more dangerous in the fact that they are heavier and would not disperse as fast in air.

hmmm.... about the cyanide, i remmember a when i was like 9 or 10 got a chemstry set, contained cyanide inside, that was only about 8 or 7 years ago, but do not know what kind of cyanide it was, but it was solid, maybe blue colored.

P.S. the book was banned in 1960
« Last Edit: November 28, 2005, 09:40:50 PM by 3.3.141592653 »
currently a student attending high school in South Florida, capital of all the hurricanes that come through the US, and the sunshine state.  My interests falls into electrochemistry going to renewable resources of energy, i like hydrogen fuel cells and solar energy

#### mike

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##### Re:I'm new, where can I find this book?
« Reply #10 on: November 28, 2005, 09:52:49 PM »
Quote
but i said that is dangerous under high concentrations, the lightness of hydrogen means that the hydrogen molecules would disperse very fast in air, and would thus not pose such a great threat.

Remember that high and low concentrations are relative terms. Have you checked the flammability limits for hydrogen? This will tell you at what concentrations hydrogen is safe and dangerous. Although you are probably correct in saying that not enough hydrogen would be produced to  pose a serious danger. The problem comes when the student, on successfully completing the reaction decides it would be great to see what happens when the reaction is scaled up etc etc....

I am intrigued as to what use the cyanide would have had in the chemistry set, anyone know?

And by the way, who banned the book in 1960? and where is it banned from?

Cheers
There is no science without fancy, and no art without facts.

#### buckminsterfullerene

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##### Re:I'm new, where can I find this book?
« Reply #11 on: November 28, 2005, 11:54:26 PM »
the book was banned in the US in 1960, it was banned by the government, it is said that the reason that they stated was because the information contained in the book was to dangerous for the general public..

i asked my self the same question about the cyanide, i was afraid to take out hearing how dangerous it was, but they were atleast sensible to put outside the container DO NOT INJEST   (like that would do much), bought it in a toy store (out of all places)

currently a student attending high school in South Florida, capital of all the hurricanes that come through the US, and the sunshine state.  My interests falls into electrochemistry going to renewable resources of energy, i like hydrogen fuel cells and solar energy

#### jdurg

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##### Re:I'm new, where can I find this book?
« Reply #12 on: November 29, 2005, 08:35:35 AM »
I do not know of any cyanide compounds which are blue colored.  What you may have had was a cyan colored dye which probably had the cyanide name in there.  (As cyan blue like color).  I know that the chemical dye "cyanine" is still used extensively today.  So there's a high chance that you had cyanine dye in your set and NOT cyanide salt.

For the hydrogen, if memory serves me correctly, it forms an explosive mixture with air at concentrations of as little as 5%.  So if you have a 10 liter container and there was 500 mL of hydrogen gas, there would be an explosive mixture.  In addition, in experiments that generate hydrogen gas, the concentration of H2 at the site of generation is ENORMOUSLY higher than that of the surrounding room.  Even if the H2 does disperse quickly, any source of ignition where the H2 is being generated can cause a vicious fire or explosion.  Finally, while I hate to say it, the general public is incredibly ignorant in terms of safety so it has to be assumed that any possible accident can and will occur if a misinformed individual is doing it.  I know that in every experiment I've done involving hydrogen, the gas is collected into some type of container.  We aren't just letting H2 generate and escape out into the atmosphere.
"A real fart is beefy, has a density greater than or equal to the air surrounding it, consists

#### Borek

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##### Re:I'm new, where can I find this book?
« Reply #13 on: November 29, 2005, 09:00:23 AM »
I do not know of any cyanide compounds which are blue colored.

Prussian blue:

3FeCl2 + 4FeCl3 + 18KCN -> 18KCl + Fe4[Fe(CN)6 ]3

or

3Fe2+ + 4Fe3+ + 18CN- -> Fe4[Fe(CN)6 ]3

EBAS rulez

I decided to implement exporting balanced equations in UBBC so you will see more reaction equations from me now
« Last Edit: November 29, 2005, 09:00:48 AM by Borek »
ChemBuddy chemical calculators - stoichiometry, pH, concentration, buffer preparation, titrations.info, pH-meter.info

#### mike

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##### Re:I'm new, where can I find this book?
« Reply #14 on: November 29, 2005, 07:15:23 PM »
There are a few Gilbert chemistry set photos on the net and from what I can tell the "cyanide" is sodium ferrocyanide (although I don't think this is blue?? maybe yellow??)
There is no science without fancy, and no art without facts.


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# A crate measures 4 feet by 8 feet by 12 feet on the inside...

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A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A. 2
B. 4
C. 6
D. 8
E. 12

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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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20 Dec 2015, 22:06
1
A crate that is 4 by 8 by 12 has 3 faces.

Face 1: L=4, B=8, H=12. In trying to fit a circle into a 4 by 8 rectangle, diameter can only be maximized to be 4. As such, the volume of the cylinder is
pi X (4/2)^2 X 12 = 48pi

Face 2: L=4, B=12, H=8. Same as above, the circle has to fit into a 4 by 12 rectangle, diameter is 4. Volume of cylinder is
pi X (4/2)^2 X 8 = 32pi

Face 3: L=8, B=12, H=4. Lastly, the circle has to fit into a 8 by 12 rectangle, diameter is 8. Volume of cylinder is
pi X (8/2)^2 X 4 = 64pi

The volume in the 3rd scenario is the largest. As such, to maximize the volume of the cylinder, the radius has to be 8/2=4.
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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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20 Dec 2015, 22:33
1
Sash143 wrote:
A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A. 2
B. 4
C. 6
D. 8
E. 12

We can find the radius of all the three cases of cylinders.

The only crux to find the answer faster is that:
Voulme is pi*r^2*h. The volume is a function of r^2. so r has to be the highest to find the largest volume.

So r=4 for the surface 8*12 face.

Volume = 64pi
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A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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17 Jul 2016, 03:15
1
Sash143 wrote:
A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A. 2
B. 4
C. 6
D. 8
E. 12

Just glancing at the dimensions of the crate makes us realise that it is a cuboid.

To fit the cylinder with largest radius inside this cuboig , we should make the base of the crate as wide as possible
so we will take the base as 12 feet by 8 feet
Now since the limiting number in the base is 8 feet; therefore a cylinder {we can visualise that a cylinder's width is its diameter } can only fit inside the crate if it is 8 feet or less.
Therefore the radius of the cylinder will become$$\frac{Diameter}{2}===> \frac{8}{2} = 4$$
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A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone  [#permalink]

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03 Nov 2016, 10:19
A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A) 2
B) 4
C) 6
D) 8
E) 12
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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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03 Nov 2016, 10:57
SW4 wrote:
A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A) 2
B) 4
C) 6
D) 8
E) 12

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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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12 Nov 2016, 00:46
3
Sash143 wrote:
A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

A. 2
B. 4
C. 6
D. 8
E. 12

Check the solution as attached
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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside...  [#permalink]

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15 Sep 2017, 11:17
If the diameter is 4 then vol is 4x8pie
if diameter is 8 => vol = 16x4
diameter 12 not possible => diameter = 8 = radius 4 OPTION B
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Re: A crate measures 4 feet by 8 feet by 12 feet on the inside... &nbs [#permalink] 15 Sep 2017, 11:17
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Demand Forecast of Weapon Equipment Spare Parts Based on Improved Gray-Markov Model

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VOLUME 5 , ISSUE 3 (Oct 2020) > List of articles

### Demand Forecast of Weapon Equipment Spare Parts Based on Improved Gray-Markov Model

Keywords : Grey Theory, Markov Model, Spare Parts Forecast

Citation Information : International Journal of Advanced Network, Monitoring and Controls. Volume 5, Issue 3, Pages 47-56, DOI: https://doi.org/10.21307/ijanmc-2020-027

Published Online: 14-October-2020

### ARTICLE

#### ABSTRACT

The demand for spare parts of weapons and equipment is time-varying and random. It is difficult to predict the demand for spare parts. Therefore, on the basis of gray GM(1,1), a state transition probability matrix based on improved state division is used to establish a demand forecast model for weapon equipment and spare parts. The model not only considers the characteristics of the GM(1,1) model’s strong handling of monotonic sequences, but also extracts the characteristics of random fluctuation response of data through the transformation of the state transition probability matrix, avoiding the phenomenon of the worst prediction results when the maximum probability state is not the actual state. It is proved through experiments that the prediction result based on the improved gray-Markov model is superior to the traditional model and the classic gray-Markov prediction model, and the accuracy of the improved model is about 1.46 times higher than that of the gray model.

## I. INTRODUCTION

Weaponry is composed of many parts, and these parts need to be repaired and replaced during use. In order to shorten the short interval between the repairs of weaponry and equipment, increase normal working hours, reserve a reasonable number of spare parts for replacement at any time. Weaponry spare parts are effective measures to improve the availability of weaponry and equipment, reduce life cycle costs, and ensure the effectiveness of combat effectiveness on time [1]. In the issue of spare parts support, the prediction of spare parts demand is an important means to promote “precision support”, and is also one of the key and difficult points in the research of equipment comprehensive support.

There are many reasons for the occurrence of spare parts demand. In addition to the reliability rules of weapons and equipment itself, the use of equipment, repair strategies and maintenance methods will affect the number and time of spare parts demand. Spare parts demand shows randomness and fluctuation Therefore, equipment managers urgently need to find a method to predict the random demand for repair spare parts.

At present, there are many technologies for demand planning, forecasting and decision-making, mainly including time series forecasting models, regression analysis methods, support vector machines, neural networks, gray forecasting and decision making, Markov forecasting, Combined optimization decision-making, and their Between each other. Among them, the method based on the time series prediction model requires a large amount of historical data, and the data must not have periodic changes or mutations. Neural network-based spare parts demand prediction method requires a large number of statistical samples, and the prediction results are highly subjective and random. These factors greatly limit the application of artificial intelligence methods [2]. Therefore, how to improve the prediction accuracy of the demand for spare parts for weapons and equipment is an important link for effectively guaranteeing the supply of spare parts for our military equipment.

The gray prediction model is suitable for solving the problems of small samples, poor information and uncertainty. It has less calculation and is convenient and practical. It has a good prediction effect for short-term prediction, but it has a poor fit for long-term prediction and volatile data series. The Markov theory describes the influence of random factors and the internal laws of transition between states through the state transition probability, which can effectively make up for the deficiencies of the gray model [3]. To this end, this paper proposes to predict the demand for spare parts for weapons and equipment based on an improved gray Markov model, in order to effectively improve the prediction accuracy of random volatility data and broaden the application range of gray theory.

## II. RESEARCH STATUS OF SPARE PARTS DEMAND MANAGEMENT

With the development and progress of economic globalization and science and technology, people have higher requirements for product quality and production efficiency, and there are more and more complex equipment in industrial production. In the process of using complex equipment, failures will inevitably occur due to factors such as maintenance damage, wear, corrosion, and expiration of life. In order to restore the normal operation of the equipment in time, minimize the economic loss caused by equipment failure or shutdown. [4] Companies generally purchase and store a certain number of accessories in advance, which are called spare parts. Spare parts, as support materials for the daily maintenance and emergency handling of equipment, are an important factor in ensuring the normal operation of complex equipment. Accurate and timely spare parts supply can ensure the continuity of production operations of the enterprise. Spare parts are an important material basis for equipment support work. Spare parts management Work has become an important part of equipment support work. Spare parts planning is affected by spare parts demand, so an effective spare parts demand forecasting model will provide an important basis for spare parts management decision making, and it is also the basis for quickly responding to changes in customer needs and improving corporate service levels.

The demand for spare parts is very special. In most cases, the demand for spare parts occurs in uncertain and irregular time intervals, and the quantity is also unstable and changeable. Strictly speaking, demand is usually divided into: slow-moving spare parts, intermittent demand spare parts. The consumption of spare parts is extremely special. Some spare parts consume a large amount, while some spare parts consume a small amount, and have not even been consumed in a few years. This greatly increases the difficulty of accurately predicting the consumption of spare parts. In fact, in addition to conventional methods for forecasting spare parts, it is more important to study forecasting methods for uncertain or intermittent demand. Spare parts demand forecasting is a very important part of equipment management, and it is the basis of inventory management. Accurately planning the supply of spare parts can reduce the huge budget spent on spare parts, supply the required spare parts in time, improve the availability of equipment and the completeness of weapons and equipment, ensure that the equipment can complete production tasks and normal operations on time, and guarantee weapons and equipment In military exercises and battles, fighters will not be missed. On the other hand, accurate demand forecasting will have a very important impact on the formulation of spare parts inventory strategies and the construction of inventory models [5].

In many complex equipment companies, spare parts inventory management has not attracted enough attention. In general: On the one hand, there are generally backward spare parts inventory management methods, improper inventory management methods, difficult spare parts search, and excessive backlog of a large number of unimportant spare parts, resulting in high inventory costs of enterprises; on the other hand, some spare parts Inventory will not be able to meet equipment maintenance or customer demand changes. The shortage of key spare parts may cause equipment delays in maintenance or even shutdown accidents occasionally. Inventory issues increasingly become a bottleneck restricting the survival and development of complex equipment companies. For companies, the most important issue is how to use inventory management strategy and inventory management system optimization to greatly reduce the amount of inventory funds occupied by spare parts, and then increase Capital turnover and corporate economic benefits.

## III. THEORETICAL OVERVIEW OF GREY SYSTEM AND MARKOV CHAIN

### A. Basic concepts of gray system

The naming of the gray system is different from the naming methods of other systems, it is named according to the degree of mastery of the information. The completely unknown information is represented by “black”, this system is called the black system, and the information that is clearly grasped is represented by “white”, this system is called the white system. According to this law, it is clear that “gray” is in the middle ground, and our grasp of this part of information is in a state of “ambiguity”. This system is called the gray system.

The most widely used grey system in the field of forecasting is the GM(1,1) model. Due to its small sample data, simple calculation and other advantages, it has been widely used in various fields such as society, economy, ecology, agriculture, etc., especially in the case of small samples, poor information and uncertain systems and lack of data, it has also been successful. Application, which determines that the GM(1,1) model in the gray system occupies an important position in the fields of prediction and decision-making. In order to expand the scope of application and prediction accuracy of the GM(1,1) model, many scholars have done a lot of theoretical research. These studies mainly focus on: processing the original data, constructing the background value in the GM(1,1) model, discussing the method of determining the initial value in the GM(1,1) model, combining the gray system theory and other theoretical models Combine. Among them, there are two types of concepts in optimized combination forecasting. One is a forecasting method that selects appropriate weights for the weighted average of the forecast results obtained from several forecasting methods. The key is to determine the weighting coefficient of each individual forecasting method. Compare in several prediction methods, choose the prediction model with the best fit or the smallest standard deviation as the best model for prediction. Combination forecasting is to play its role when a single forecasting model cannot completely describe the changing law of the forecast.

### B. Basic principles of the gray system

Using the information currently available to explore and predict unknown information is the most important theory of the gray system, which is the process from “grey” to “white” until the research purpose is achieved. The basic principles of the gray system are:

#### 1) Principle of difference information

“Difference” is information, and all information must be different. Saying “thing A is different from thing B” means that A has special information that B does not, which is the so-called difference. The basic way people understand the world is to observe the differences of different things in the objective world.

#### 2) The principle of non-uniqueness of solution

When the information is not fully grasped, the solution obtained for it is uncertain and non-unique. This is caused by the incompleteness of the gray system information and cannot be avoided.

#### 3) Principle of least information

How to use the least information that has been mastered to maximize the effect is an important feature of the gray system to study unknown information.

#### 4) Cognitive basis principle

Information is the basis of cognition. All cognition must be based on the information you have.

#### 5) New information priority principle

The new information has priority in cognition, and its use value is greater than the old information.

#### 6) The principle of immortality

Incomplete information and uncertainty are universal. Information is completely relative and temporary. The completeness of information is specific to a specific environment, and the objective environment is constantly changing, which will also be accompanied by the emergence of new uncertainties.

### C. Grey prediction model

The gray prediction model uses GM (1,1) modeling on the known information to find the simulated value, and then predicts the unknown information. This determines the change range of the gray forecast object, and its range must be bounded and related to time. As the core model of grey prediction, GM(1,1) model is widely used in practical research. The establishment of the model is based on the data in the time series. Through this model, the law of data change is analyzed, and the internal relationship is analyzed from the external relationship of various factors to find the hidden law, thereby generating the corresponding data sequence, and performing differential equation modeling. Forecast the development trend of things.

Predicting the gray system is to mine and discover the development laws of the system by processing the original data and establishing the gray model, so as to predict the future state of the system. Since the gray system contains both known information and unknown information, its prediction is the prediction of the gray process that is related to time and changes in a certain direction. Although the laws shown by some gray processes are random and disorderly, their essence is bounded, orderly, and potentially regular. There are four main types of gray system forecasts:

• a) Sequence prediction: refers to the prediction of the behavior of the system variables in the future. After the qualitative analysis of the data series, the appropriate sequence operator is selected, and the gray model is established based on the number sequence after the operation of the operator. The accuracy of the model It can be used to predict after inspection.

• b) Catastrophe prediction: refers to the prediction of the abnormal value of the gray system, that is, the prediction of the time when a given gray number occurs, so as to provide guidance for the work of relevant departments.

• c) System prediction: refers to the prediction of multiple variables in the system together, which is mainly used to predict the relationship between system variables and reveal the development and changes of the system.

• d) Topological prediction: also called waveform prediction. When the original data fluctuates greatly, the gray model is used to predict the waveform of future behavior data.

### D. Markov process

Markov process has great significance in stochastic process and is widely used in biology, physics and other sciences. The definition of Markov chain was first proposed in the early nineteenth century, and then people began to study and explore a random process with no aftereffect. Ineffectiveness means that the state of the future is only related to the present and not related to the past. Markov has made great contributions to probability theory, number theory, and differential equations[6]. During 1906-1912, he proposed and studied Markov chains. His research results are of great help to probability theory. The random process he studied is called Markov process.

Markov processes are continuous or discrete according to their state and time parameters, and are divided into three categories

• 1) Time and state are both discrete Markov processes, called Markov chains.

• 2) The Markov process with continuous time and discrete state is called continuous-time Markov chain.

• 3) Time and state are continuous, which is called Markov process.

The research object of Markov model is mainly the state of the system and the transition between states. The main purpose of establishing Markov model for calculation analysis is to predict the possible state of variables in the future by analyzing the current state and change trend of system variables, and then provide corresponding theoretical basis for decision-making.

The steps of using Markov process to predict include:

Step1 Reasonably divide the state. When the sample has less data, the number of states should be as small as possible, so that each state contains more data, which can more objectively reflect the transfer law between each state;

Step2 Calculate the transition probability between each state in the Markov process of the system, and determine the corresponding state transition matrix;

Step3 According to the initial state of the system and the transition probability matrix to predict the state in the future.

## IV. IMPROVED GRAY MARKOV MODEL

### A. Gray model

Grey system theory was first proposed by our famous scholar Professor Deng Julong in 1982. It is a theory of gray system modeling, prediction, analysis, control and decision-making. Grey prediction method is a new type of nonlinear prediction technology in the late 1990s, which is a method system based on sequence operators and gray sequence production. The gray prediction model uses accumulation, accumulation or step ratio generation techniques, and uses differential fitting to directly convert the time series into a first-order univariate constant coefficient differential equation. According to the principle of continuity of prediction, the gray theory can be used to get more Accurate prediction results [7]. For data with relatively short data series, less information, and less regularity, data that meets the characteristics of poor information systems, the gray model can show superiority and advanced. The gray model is usually divided into a first-order single-variable model and a first-order Combined model. The model is used in this paper.

• 1) The original data sequence required for the design of spare parts for weapons and equipment is:

##### (1)
$X(0)={X(0)(1),X(0)(2),⋯,X(0)(n)}$

among them:x(0)(k)>0,k=1,2,…,nº

• 2) Accumulate the original sequence once:

##### (2)
$X(1)={x(1)(1),x(1)(2),⋯,x(1)(n)}$

among them:

$x(1)(k)=∑i=1kx(0)(i)(k=1,2⋯,n)$

• 3) Generate a sequence of immediate means:

##### (3)
$Z(1)={z(1)(2),z(1)(3),⋯,z(1)(n)}$

among them:

$z(1)(k)=12[x(1)(k)−x(1)(k−1)]$

• 4) Establish model whitening equation:

##### (4)
$dz(1)dx+az(1)=b$

Among them:a and b are undetermined parameters

• 5) The gray equation of the model is:

##### (5)
$x(1)(k)+az(1)(k)=b$

• 6) The corresponding time response equation is:

##### (6)
$x(1)(k+1)=[x(0)(1)−ba]e−at+ba$

• 7) The least square estimates of parameters a and b are:

##### (7)
$A=[ab]=(BTB)−1BTYB=[−z(1)(2)1−z(1)(3)1⋮1−z(1)(n)]Y=[x(0)(2)x(0)(3)1x(0)(n)]$

• 8) Reducing the prediction results to the prediction sequence:

##### (8)
$x^(0)(k+1)=x^(1)(k+1)−x^(1)(k)k=1,2,⋯,n−1$

among them: k = 1,2, …, n–1

It can be seen from the above that the model predicts the original sequence after one accumulation. Because the one-time accumulation sequence is monotonic, the model is suitable for predicting the data of the exponential change law, and the prediction error is large for random volatility data.

The GM(1,1) forecasting model occupies an important position in the grey system theory. The starting point of the model research is to explore valuable information in its own time series and explore the law of the research content without considering the impact of the research content. The gray system prediction model takes a small amount of data information as the research object. Its research characteristics are simple calculation, simple principle and high prediction accuracy. The model has good prediction accuracy for modeling small amounts of information, but the prediction accuracy for irregularly changed and fluctuating data will be greatly reduced, so the gray prediction model does not have high prediction results for any data. The Markov chain prediction model is different from the gray system model. It makes up for the shortcomings of the gray prediction model and can study data with large volatility. Its model requires the prediction object to have Markov properties. In this chapter, the two models are reasonably combined to form a gray Markov prediction model. The GM(1,1) model is used to characterize the trend of the original data sequence, and the Markov prediction model is applied to the simulated data obtained by the GM(1,1) model. The length of the ruler is selected and the shortness of the inch is compensated to improve the accuracy.

### B. Grey Markov model

According to the gray system theory, the gray mean model and the Markov model are fused, and the gray Markov model is used to predict the demand for weapon equipment and spare parts. Specifically, the gray mean model is used to predict the future demand for weapon equipment and spare parts [8]. The state transition matrix determines the possible state of future spare parts, and finally corrects the prediction result based on the ratio between the predicted value and the actual value, so as to realize accurate prediction of weapon equipment spare parts. After determining the state division and state transition matrix, the general gray Markov model uses the maximum value of the current transition probability as the next transition value [9]~[11]. This method ignores the possibility of other transition probabilities and based on the last state. For prediction, it is easy to be affected by randomness, which makes the prediction accuracy low, so the gray Markov model is improved.

The modeling idea of the gray-Markov combination model is to first establish a gray prediction model to obtain a prediction sequence, and then use the relative difference sequence of the prediction sequence and the actual sequence to divide the state space, find the interval of the predicted value, and follow the prediction [12]. The interval modifies the results of the model prediction, increases the credibility of the prediction, calculates the transition probability matrix from the point where the original data sequence falls into each state, and estimates the future change trend based on the state transition probability matrix.

Calculate the state transition probability matrix and obtain the predicted value according to the gray model $x^(0)(k)(k=1,2,⋯n)$. Using the curve as a reference, it is divided into several bar regions parallel to the trend curve, and each region constitutes a state. In this way, a non-stationary random sequence that matches the characteristic points of the Markov chain is divided into n states. Any state is

##### (9)
$⊗i=[⊗1i,⊗2i]$

among them:

$⊗1i=x^(0)(k)+Ai,⊗2i=x^(0)(k)+Bi$

The state transition probability is:

##### (10)
$pij(k)=Mij(k)Mi,i=1,2⋯,n$

In the formula:Mij(k) is the number of original data that state ⊗i transfers to state 1 after ⊗j steps; Mi is the number of original data in state ⊗i.

The specific steps based on the improved gray Markov prediction model are:

Step1: According to the original data x(k), find the coefficient a, b in the gray model, and get the fitting value $x^(k)$ corresponding to the actual data.

Step2: The range of the relative residual value sequence is obtained from the relative residual value formula.

Step3: According to the actual situation, use the residual formula to divide the relative residual value range into states: ⊗1, ⊗2, …, ⊗L

Step4: Based on the improved Markov chain state transition probability determination method, the one-step state transition probability matrix is calculated:

##### (11)
$P=(p11p12⋯p1Lp21p22⋯p2L⋮⋮⋮pL1pL2⋯pLL)$

among them:

$pij≥0,∑j=1Lpij=1$

Suppose a prediction system has ⊗1, ⊗2, ⊗3, ⊗4 four states, the last data in actual data x(k) is in state ⊗4, then the initial distribution is I(0) = (0 0 0 1), and the predicted value of the next data is I(1) = I(0). P, based on which the state interval in which the predicted data is located can be obtained.

The gray Markov model is a model that combines the gray model and the Markov model. Its prediction process is: firstly generate gray according to the original data sequence, determine the parameters of the gray model according to the generated sequence; then use the established gray model Carry out simulation prediction, test the accuracy of the established model according to the predicted value and actual value; then appropriately divide the gray prediction result into several states, establish a Markov model, and predict and modify the state of the gray prediction result. According to the combination of gray model and Markov model, traditional gray Markov model prediction methods mainly include the following two:

Use the Markov process to predict the sign of the gray prediction result residual model. After processing the given time series, a gray model is established, and the residual data series can be obtained by comparing the predicted result of the gray model with the actual value. The absolute value of the residual data sequence is used as the original data, and the gray prediction is performed to obtain the gray residual model. In order to improve the prediction accuracy, the Markov process is used to predict the sign of the residual model at the future time.

Use Markov process to predict the relative error of gray prediction results. Establish a gray model based on the original data series, divide the state of the relative error of the gray prediction result, and predict the state of the relative error of the gray prediction result at the future time according to the current state transition, thereby improving the accuracy of the gray prediction result Sex. The essence of the first combination forecasting method is to correct the gray residual model to improve the accuracy of the forecast; the essence of the second combination forecasting method is to directly correct the gray forecast results, so it is better than the first combination forecasting method. The calculation process is simple.

When the Markov process is used to predict the relative error of the gray results, there are two ways: one is to directly predict the state of the system in the future based on the initial state of the system and the n-step state transition matrix; the other is It is based on the initial state of the system and the one-step state transition matrix to predict the state of the next moment, and then based on the state of the next moment and the one-step state transition matrix to predict the state of the system at a later time. The advantage of the first method is that when predicting the state of the system at any time, the initial state used is known and accurate, and the various states of the system at a certain time are considered, and The probability of overestimation and underestimation is small; the disadvantage is: when predicting the state of the system at a distant time, a multi-step state transition matrix is needed. The calculation process of the multi-step state transition matrix is more complicated, and when the number of steps is large The accuracy of the corresponding matrix cannot be guaranteed. The advantage of the second method is that only one step of the state transition matrix is required for Markov prediction, the calculation process is simple, and when the predicted state is accurate, the prediction result is very ideal; the disadvantage is: when the state of the system at a certain moment When the forecast is inaccurate, the forecast result will have a big deviation, and it will affect the forecast accuracy at subsequent moments. In the following articles, the model corresponding to the first calculation method is called the first gray Markov model, and the model corresponding to the second calculation method is called the second gray Markov model.

The improved gray Markov model is a reasonable combination of the first and second gray Markov models, that is, two calculation methods are used to predict the corresponding values of the system parameters at the future time, and then the two results are calculated. Calculate the average value. The improved gray Markov model is optimized for Markov process prediction, so the gray modeling part and The GM(1,1) model is consistent, but the difference is the correction effect of the Markov process on the gray prediction results. The specific steps for the establishment of the improved gray Markov model mainly include: firstly generate gray for the prediction sequence, obtain the parameters of the gray model according to the generated data sequence, determine the gray model and make predictions; then divide the relative error of the gray prediction results State, determine the state transition of the Markov process; then according to the calculation method of the first gray Markov model and the second gray Markov model to obtain the corresponding correction value of the gray prediction result, so as to obtain the improved gray The predicted value of the Markov model.

## V. CASE ANALYSIS

Finally, complete content and organizational editing before formatting. Please take note of the following items when proofreading spelling and grammar:

### A. Experiment Verification

Taking the consumption of a spare part of a certain type of self-propelled artillery of the Military Machinery Maintenance Institute as an example, Table I. is the historical data of the consumption of a certain spare part during the continuous service of a certain type of self-propelled artillery of 2016~2019 for 3 years, with the serial number of the spare part as the horizontal axis, and the demand for spare parts For the vertical axis, the improved gray Markov model is used to predict the future demand for spare parts.

##### TABLE I

HISTORICAL DATA OF SPARE PARTS CONSUMPTION OF A CERTAIN EQUIPMENT

The following data is used as a sample, and the improved gray-Markov model and other models are used to predict the number of spare parts for the eighth time, and a comparative analysis is made.

Using the improved gray-Markov model to predict the demand for spare parts:

From the above data can be obtained

X(0) ={20,14,18,9,10,11,6,5}

It can be obtained by smoothing the historical data of the demand for spare parts of weapons and equipment

λX = {1.4268,0.7778,2.0000,0.9000,0.9091, 1.8333,1.2000}

Therefore, X(0) needs to be exchanged for translation. Takec=32.5, after calculation, we can get

$Y(0)={52.5,46.5,50.5,41.5,42.5,43.5,38.5,37.9}λY={1.1290,0.9208,1.2169,0.9765,0.9770,1.1299,1.0267}$

So, the Y(0) sequence can be operated with the gray model, and finally subtract c to obtain $X^(0)$

Using Equation 2 to do an accumulation generator for we can get

Y(1) = {52.5,99.149.5,191,233.5,277.5,315.5,353}

The sequence is generated by using the immediate mean for, which can be obtained according to Equation 3.

Z(1) = {75.75.124.25,170.25,212.25,255.25,296.25,234.25}

Establish prediction equations according to formulas 4, 5, and 6

$y^(1)(k)=−1216.14×e−0.0405413(k−1)+1268.64$

Find the reduction value according to equations 7 and 8

$Y^(0)={52.5,48.3176,46.398,44.5545,42.7844,41.0845,39.4522,37.8848}X(0)={20.0000,15.8176,13.8980,12.0545,10.2844,6.9522,5.3848}$

Through the prediction of the gray model, you can get the fitted figure of the simulated value and the actual value after data processing as shown in Figure 1.

##### Figure 1.

Fitting diagram of simulated value and actual value

Obtain the relative residual sequence according to Equation 9

$ε(0)={−12.9831,22.7892,−33.9394,−2.8437,21.9588,−15.8705,−7.6955}$

To divide the state, the upper and lower limits of the state are

${Θmin=min(ε(0))−0.05×|max(ε(0))−min(ε(0))|Θmax=max(ε(0))−0.05×|max(ε(0))−min(ε(0))|$

The range of the residual error is expanded by 5% above and below, calculated

$Θ=(−36.7758,25.6256)$

The interval between the cells in the state is taken to be equal, and the number of states is sequentially selected from 2 to be calculated. The maximum value is the sequence length. From this, 7 sets of trial calculation results can be obtained.

Calculate the one-step state transition probability matrix according to Equation 12

$P=[00100.080.4200.5000.32490.67510.50.500]$

(1) Use the state transition probability matrix to obtain the prediction results, as shown in Table II

##### TABLE II

FORECAST RESULTS

The state transition probability matrix is used to predict the consumption of spare parts for a certain equipment spare part in the future time period, but its accuracy and accuracy need to be calculated using error analysis formulas, and compared with other gray models or classic gray-Markov models.

### B. Model comparison

Average error calculation：

##### (12)
$E¯=∑k=2n|e(k)|n−1$

First, according to the average error calculation formula, calculate the average error of the model when different states are divided. As shown in Table III, it can be seen that when the number of states is 4, the accuracy of the model is the highest.

##### TABLE III

MODEL AVERAGE ERROR UNDER DIFFERENT STATE DIVISION

Second, the classic gray-Markov model is used for prediction, and the prediction accuracy is compared, as listed in Table IV.

##### TABLE IV

FORECAST ACCURACY OBTAINED BY USING DIFFERENT MODELS

When the number of state divisions is 2, 3, the accuracy of the improved model is about 1.06 times and 1.23 times higher than that of the model, which is the same as the accuracy of the classic gray Markov prediction model. When the number of state divisions is 4, the accuracy of the improved model is about 1.46 times higher than that of the model, and the classical Markov prediction model cannot achieve the prediction effect because it cannot build a successful Markov chain state transition probability matrix. As the number of state divisions increases, the average accuracy of the improved gray Markov model is on the rise. The improved model supports more number of state divisions, so it is better than the classic gray Markov prediction model. The application environment of the gray prediction method is relatively loose. It does not need to determine whether the data changes follow the same type of distribution. It does not require large sample statistics to make predictions. It has certain application prospects.

## VI. CONCLUSION

In the field of spare parts support for weapons and equipment, forecasting has always been a relatively difficult problem. Due to the time-varying and random characteristics of weapon equipment spare parts demand, this paper proposes to predict the demand of weapon equipment spare parts based on the improved gray-Markov prediction method. By comparing the prediction results, the modified model is superior to the traditional gray model and classic the gray-Markov model has high prediction accuracy and provides a reliable scientific basis for the evaluation of weapon equipment reliability and the maintenance of its equipment.

## ACKNOWLEDGMENT

This work is supported by the Shanxi Natural Science Basic Research Project (2019JM-603) for its fund support.

## References

1. Shao Yanjun. Research on Preventive Maintenance Strategy of Weapon Equipment Based on Fault Prediction [D]. North University of China, 2013.
2. Yan Xiaoyao, Yan Li. Application of neural network in the forecast of aviation weapon equipment spare parts demand [J]. Today Keyuan, 2008 (20): 56-57.
3. Zhou Hao, Huang Shanzhong. Equipment consumption prediction based on GM(1,1) and grey Markov model[J]. Journal of Wuhan University of Technology (Transportation Science and Engineering Edition). 2015, 39(06): 1166-1168+1174.
4. Qiu Chundong, Yang Yuzhi, Wang Chunyun. Prediction of CT tube failure interval based on gray Markov chain model [J]. Modern Instruments and Medical, 2019, 25(06): 86-88+66.
5. Xu Ning, Dang Yaoguo, Ding Song. GM(1,1) model background value optimization method based on error minimization[J]. Control and Decision. 2015,30(02): 283-288.
6. Lv Z, Hou H, Yi Z, Singh HP, editors. Grey prediction model application in early warning of ice-coating lines[C]. Recent Developments in Control, Automation and Power Engineering (RDCAPE), 2015 International Conference on; 2015 12-13 March 2015.
7. Ma Liangli, Li Gang, Tao Daoqiang. Fault prediction method based on gray GM(1,1) model[J]. Computer Applications and Software. 2013, 30(04): 198-200.
8. Shi Pei, Gao Shan, Su Yanqin. An improved GM(1,1) model used in equipment fault prediction [J]. Computer Measurement and Control. 2012, (05): 1176-1178.
9. Zhang Wenjie, Yuan Hongping. Research on the failure prediction of energy-saving equipment based on grey Markov model[J]. System Science and Mathematics, 2019, 39(01): 65-75.
10. Li Li, Xiong Wei, He Jie, Yuan Xufeng, Zou Xiaosong. Equipment failure rate prediction based on improved grey Markov model [J]. Electric Power Science and Engineering, 2015, 31(08): 20-24.
11. Hong Zhanpeng, Chen Zhenghua. Gray Markov Model for Metro Car Door Failure Prediction [J]. Equipment Manufacturing Technology, 2017 (03): 223-226.
12. Ma Chunmao, Shao Yanjun, Pan Hongxia, Liu Yongjiang. Research on prediction of equipment failure interval based on gray Markov model [J]. Journal of Ordnance Engineering, 2013, 34(09): 1193-1196.

### FIGURES & TABLES

Figure 1.

Fitting diagram of simulated value and actual value

### REFERENCES

1. Shao Yanjun. Research on Preventive Maintenance Strategy of Weapon Equipment Based on Fault Prediction [D]. North University of China, 2013.
2. Yan Xiaoyao, Yan Li. Application of neural network in the forecast of aviation weapon equipment spare parts demand [J]. Today Keyuan, 2008 (20): 56-57.
3. Zhou Hao, Huang Shanzhong. Equipment consumption prediction based on GM(1,1) and grey Markov model[J]. Journal of Wuhan University of Technology (Transportation Science and Engineering Edition). 2015, 39(06): 1166-1168+1174.
4. Qiu Chundong, Yang Yuzhi, Wang Chunyun. Prediction of CT tube failure interval based on gray Markov chain model [J]. Modern Instruments and Medical, 2019, 25(06): 86-88+66.
5. Xu Ning, Dang Yaoguo, Ding Song. GM(1,1) model background value optimization method based on error minimization[J]. Control and Decision. 2015,30(02): 283-288.
6. Lv Z, Hou H, Yi Z, Singh HP, editors. Grey prediction model application in early warning of ice-coating lines[C]. Recent Developments in Control, Automation and Power Engineering (RDCAPE), 2015 International Conference on; 2015 12-13 March 2015.
7. Ma Liangli, Li Gang, Tao Daoqiang. Fault prediction method based on gray GM(1,1) model[J]. Computer Applications and Software. 2013, 30(04): 198-200.
8. Shi Pei, Gao Shan, Su Yanqin. An improved GM(1,1) model used in equipment fault prediction [J]. Computer Measurement and Control. 2012, (05): 1176-1178.
9. Zhang Wenjie, Yuan Hongping. Research on the failure prediction of energy-saving equipment based on grey Markov model[J]. System Science and Mathematics, 2019, 39(01): 65-75.
10. Li Li, Xiong Wei, He Jie, Yuan Xufeng, Zou Xiaosong. Equipment failure rate prediction based on improved grey Markov model [J]. Electric Power Science and Engineering, 2015, 31(08): 20-24.
11. Hong Zhanpeng, Chen Zhenghua. Gray Markov Model for Metro Car Door Failure Prediction [J]. Equipment Manufacturing Technology, 2017 (03): 223-226.
12. Ma Chunmao, Shao Yanjun, Pan Hongxia, Liu Yongjiang. Research on prediction of equipment failure interval based on gray Markov model [J]. Journal of Ordnance Engineering, 2013, 34(09): 1193-1196.


text:
# Cross-Price Elasticity of Demand

## The cross-price elasticity of demand measures the change in demand for one good in response to a change in price of another good.

#### Key Points

• Complementary goods have a negative cross-price elasticity: as the price of one good increases, the demand for the second good decreases.

• Substitute goods have a positive cross-price elasticity: as the price of one good increases, the demand for the other good increases.

• Independent goods have a cross-price elasticity of zero: as the price of one good increases, the demand for the second good is unchanged.

#### Terms

• A good with a negative cross elasticity of demand, meaning the good's demand is increased when the price of another good is decreased.

• A good with a positive cross elasticity of demand, meaning the good's demand is increased when the price of another is increased.

#### Figures

1. ##### Complements

Two goods that complement each other have a negative cross elasticity of demand: as the price of good Y rises, the demand for good X falls.

2. ##### Substitutes

Two goods that are substitutes have a positive cross elasticity of demand: as the price of good Y rises, the demand for good X rises.

3. ##### Independent

Two goods that are independent have a zero cross elasticity of demand: as the price of good Y rises, the demand for good X stays constant.

The cross-price elasticity of demand shows the relationship between two goods or services. More specifically, it captures the responsiveness of the quantity demanded of one good to a change in price of another good. Cross-Price Elasticity of Demand (EA,B) is calculated with the following formula:

$E_{A,B} = \frac { \% Change\; in\; Quantity\; Demanded\; for\; Good\; A }{ \%\; Change\; in\; Price\; of\; Good\; B }$

The cross-price elasticity may be a positive or negative value, depending on whether the goods are complements or substitutes. If two products are complements, an increase in demand for one is accompanied by an increase in the quantity demanded of the other. For example, an increase in demand for cars will lead to an increase in demand for fuel. If the price of the complement falls, the quantity demanded of the other good will increase. The value of the cross-price elasticity for complementary goods will thus be negative (Figure 1).

A positive cross-price elasticity value indicates that the two goods are substitutes. For substitute goods, as the price of one good rises, the demand for the substitute good increases. For example, if the price of coffee increases, consumers may purchase less coffee and more tea. Conversely, the demand for a substitute good falls when the price of another good is decreased. In the case of perfect substitutes, the cross elasticity of demand will be equal to positive infinity (Figure 2).

Two goods may also be independent of each other. In this instance, if the price of one good changes, demand for the other good will stay constant. For independent goods, the cross-price elasticity of demand is zero (Figure 3): the change in the price of one good with not be reflected in the quantity demanded of the other.

#### Key Term Glossary

Complement
A good with a negative cross elasticity of demand, meaning the good's demand is increased when the price of another good is decreased.
##### Appears in these related concepts:
consumer
An individual who trades money for goods.
##### Appears in these related concepts:
demand
The desire to purchase goods and services.
##### Appears in these related concepts:
elasticities
Elasticity is the ratio of the percentage change in one variable to the percentage change in another variable. It is a tool for measuring the responsiveness of a function to changes in parameters in a unitless way.
##### Appears in these related concepts:
elasticity
The sensitivity of changes in a quantity with respect to changes in another quantity.
##### Appears in these related concepts:
goods
That which is produced, then traded, bought, or sold, and finally consumed.
##### Appears in these related concepts:
perfect substitute
Goods that have a constant marginal rate of substitution are perfect substitutes.
##### Appears in this related concept:
price
The cost required to gain possession of something.
##### Appears in these related concepts:
Price
The quantity of payment or compensation given by one party to another in return for goods or services.
##### Appears in these related concepts:
price elasticity
A measure of how a product's demand and supply will change if you adjust its price.
##### Appears in these related concepts:
Substitute
A good with a positive cross elasticity of demand, meaning the good's demand is increased when the price of another is increased.
##### Appears in these related concepts:
Substitute Good
A good that fulfills a consumer need in a way that is similar to another good.
##### Appears in this related concept:
value
The amount (of money or goods or services) that is considered to be a fair equivalent for something else.