RELIABILITY Explained! Failure Rate, MTTF, MTBF, Bathtub Curve, Exponential and Weibull Distribution

[@ravindrajoshi78]

Great explanation. Easy to understand than any other book. Thank you! God bless you.

[@jonasoliveira9366]

Best content about reliability I've found so far! Great and simple explanation, very helpful! Thanks for the content

[@CQEAcademy]

You're welcome Jonas!

[@brunostonner8549]

Best statistic/quality content on the internet, by far! Great explanation anda phenomenal diction, i speak english as foreign language and can understand ALL of your videos!

[@CQEAcademy]

I really appreciate that, thank you!

[@hudasedaki5529]

Thanks for the nice explanation. It's a bit counter intuitive to say that the average time to failure is 2996 hours even though all the tests lasts to 1000 hour only. so maybe the rest of 14 unit will fail at an hour later after the 1000 hr if we continue testing! A more precise failure rate will be using the units until failure then averaging that time out. good learning everybody !

[@CQEAcademy]

Totally agreed

[@praveenbabu6240]

After looking at different articles, this was the best

[@CQEAcademy]

Awesome, thanks Praveen!!!!

[@warrior_number]

@@CQEAcademy hello Sir. İ need some deep knowledge from the subject "Reliability and safety theory". İ failed at this subject and i need to pass. İ need some private lessons. Could you please help me?

[@CQEAcademy]

@@warrior_number Hey Orxan, unfortunately i don't do any personal coaching, and I don't have a course (yet) for the CRE Exam (Certified Reliability Engineer).

[@billyjames6407]

Important to note for folks is that the graph starting at <a href="#" class="seekto" data-time="1020">17:00</a> is a Probability Density Function (PDF) for the different beta values.

[@CQEAcademy]

That's a great shoutout Billy!! Thanks!

[@AbdulkareemLadoIsmail]

It is pleasure to say thank you for this wonderful contributions to education worldwide.

[@rashikapandit9066]

Thank you so much for uploading this! It was really easy to understand and covered everything I needed 💖

[@CQEAcademy]

Rashika!!!!! I'm so glad you liked it, you're very welcome! -Andy

[@prashkaushal9471]

He is a Master in quality better them books

[@sujeshpachiriyan6653]

very clear presentation from basic to the required level.

[@CQEAcademy]

Glad it was helpful!

[@mulindwamichael3184]

Thanks for the Awesome Video. Helped me to get my coursework Done.

[@andyrobertson566]

You're absolutely welcome, I'm so glad I could help!

[@MS-be9wv]

Awesome, Amazing, Excellent

[@robfielding8566]

thank you! i have been thinking about combining performance measurement (ie: Prometheus counters, health checks) to measure throughput at load, utilization; and somehow combining with measuring the reliability of graphs of services.)

[@chetanmehta3370]

Very informative.. got clear understanding

[@CQEAcademy]

Thanks!

[@mohamedelgamal6333]

One of the best, I have ever watched in reliability. Thank you sir.

[@CQEAcademy]

Wow, thanks!

[@mohammadsaeed3494]

Thank you very much. Great information in a very less time !

[@CQEAcademy]

You're absolutely welcome Mohammad!

[@shih-hsianglin]

Your teaching is sooo clear, and it helps me a lot for this reliability statistics, thx bro

[@CQEAcademy]

Thanks!!!

[@sourabhkumar7787]

A really easy and nice explanation. Thank u so much.

[@CQEAcademy]

Hey Sourabh, I"m glad you like it!!!

[@shaikkareemulla5435]

Thanks a lot. very clear explanation. Great work..........

[@CQEAcademy]

Thanks Shaik!

[@BG-bt5mv]

That is a very clear explanation of the content.

[@CQEAcademy]

Thanks Bhargav!

[@rajasekhar706]

You made it so simple to understand the reliability concept, Thank you

[@CQEAcademy]

You are welcome!

[@dzul5937]

Thanks for the clear explanation. Really appreciate it.

[@CQEAcademy]

You're welcome!

[@mahgoubhamid6642]

Thank you for this vedio and its very simple explanation.

[@CQEAcademy]

Thanks Mahgoub !!

[@tobiashofer3034]

How does this video and Reliability relate to FMEA? I'm doing a ppt on both FMEA and this, but want to know how I can relate the two ideas

[@jaquelinemoreira7385]

Amazing explanation! thank you so much

[@CQEAcademy]

You're very welcome!

[@rahulbhintade6024]

Thank you for explaining basics of reliability... :)

[@CQEAcademy]

You're welcome!

[@foodforthought7062]

Thank you so much for sharing you are the best!

[@CQEAcademy]

Wow, thanks, you are absolutely welcome!!!

[@LeanMeiying]

Thanks, really clear and easy understanding.

[@CQEAcademy]

Glad it was helpful!

[@jajatijena8208]

Wonderful..loved it❤👏👏👏

[@CQEAcademy]

Thanks a lot 😊

[@ChristosDimosthenous-q3h]

Great video!!! Very good analysis and on point! Great work!!!

[@CQEAcademy]

Wow, thanks Christos!!!

[@Phyxa]

you literally save my brainn..thank you sir..have a great day!

[@CQEAcademy]

hahaha, thanks!!!!! You have a great day too!

[@shizotaiko1628]

Best explanation ❤❤

[@CQEAcademy]

Thanks a lot 😊

[@starashioon]

Really good one.thank you so much.

[@CQEAcademy]

You are welcome!

[@jigneshpanchal08]

Nice explanation.. Thanks for an informative video

[@andyrobertson566]

Thanks Jignesh!!!

[@jadoabad]

A great professionally presentation of the concepts as such not easy to digest. Any recommendation for the Certified Reliability Engineer exam? This is the toughest one of ASQs as said but I desparately love to get it done

[@CQEAcademy]

Unfortunately I don't have any direct experience with any of the resources for the CRE exam, so I can't make a good recommendation.

[@rohithchokkapu4405]

thank you so much can we some more vedios brief things about quality and reliability

[@amrmetawie335]

very added value thank you so much

[@greenbeltacademy]

You're absolutely welcome!

[@datalyzer_spcsophusshangha7748]

You are fantastic, explained so well. Thank you

[@CQEAcademy]

Thanks!

[@qualityman1965]

Excellent work. I teach statistics, and reliability engineering. Your videos will be a recommended tutorials for my students.

[@CQEAcademy]

Wow, thanks!!! I appreciate that!

[@elionbio7977]

Just what I needed. Thanks!

[@manalaoahmad]

Hi Andy, great video. I actually got to finaly understand the weibul distribution! Question, say in operating setup (not manufacturing or product testing), failure data for repairable equipment are highly influenced by the maintenance strategy for that specific equipment. Can we use that data to calculate MTBF? Is there MTBF with maintenance and MTBF without maintenance?

[@greenbeltacademy]

Great question! So one of the underlying assumptions here is that when a unit is repaired, the item is repaired to "its original condition" (or near it's original condition). If different maintenance techniques are used, and the unit is not repaired to its original condition, it's hard to define a reliable estimate of MTBF. Does that make sense?

[@ollie599]

Good explanation I appreciate your time on dedicating making it. What are good books to go deep on this topics ??

[@CQEAcademy]

Hey Ollie, one book I've always enjoyed is Practical Engineering, Process, and Reliability Statistics by Mark Allen Durivage

[@gamaliyelsm5256]

Great video, I would like to know 1) how to calculate Beta, 2) I have 50 comp installed, 10 failure removals with mean hours of 700, how can I predict the failure hour of rest components or how can I find where to apply a maintenance so as to preclude the failure? Your reply would be highly helpful. thanks in advance.

[@CQEAcademy]

Hey Gamaliyel!!! Okay, so the beta parameter for your product should be developed through testing. In terms of maintenance that you can do to extend the MTTF, that really depends on your product, and the root causes of the failures that are occurring.

[@cuindless189]

Beta, or Weibull Shape, is calculated using the slope equation. Through MTTF/MTBF testing, you would graph the results on a probability plot and calculate rise over run as normal.

[@yashmilindpotdar8820]

Hello Andy, Thanks for the video. I wanna know one thing if we take multiple samples and run them for specific hours what will be the operation cycle in that case in the MTTF formula or failure rate formula. will it be the (duration for which we run those samples) or (samples multiplied by duration for which we run the samples)

[@MrAcarpioe]

hats off... well explained

[@CQEAcademy]

Thanks Adolfo!

[@sonandpets1215]

Andy, Have you authored any book or recommend one or two that I can buy to prepare for the CQE? I also want to take your course, How long does it take? How much does it cost?

[@rahimnekoeian7930]

Thanks so much for this! I really enjoyed it and learned a lot!

[@CQEAcademy]

Thanks Rahim, I'm glad you liked it, and I appreciate the comment!!

[@smmshow4678]

Very nice presentation

[@andyrobertson566]

Thanks!

[@aldenfarias4873]

Thanks for this

[@greenbeltacademy]

You're welcome!!

[@MohammedKhantheacademician]

How did you get the graph at <a href="#" class="seekto" data-time="840">14:00</a> , Reliability Vs. Time Hours?

[@ahmadta8303]

Thank you very much, it was really helpful.

[@CQEAcademy]

You're welcome!

[@rohithchokkapu4405]

you concpetual explaination is awsome

[@CQEAcademy]

Thanks so much Rohith!!!

[@shridharbendi9087]

Excellent Job!!!❤

[@CQEAcademy]

Thank you!

[@felzebub1762]

Thank you for the upload, very helpful!

[@CQEAcademy]

You're welcome, I love to teach and help people grow!

[@iisasd]

Thank you for these videos, your brilliance is in the way you simplify concepts. Just a question, when calculating the MTBF for repairable machines during the data measurement process in calculating MTBF are only newly 1st time manufactured parts taken into calculation or are repaired parts (2nd repair, 3nd repair, 4th repair, etc is there an upper limit to number of repairs?) also taken in the MTBF calculation ?

[@CQEAcademy]

Great question Harsh! Sooo, that depends on what you assume. If you assume that repaired parts have the same reliability as new parts, then I would include it. Obviously if you think that repaired parts have worse reliability, just know that it'll impact your MTBF. The last comment I'd make is, if your product will routinely be repaired with repaired parts, then i would calculated MTBF for this scenario because this estimate of reliability will reflect your products actual performance. Hopefully that makes sense.

[@iisasd]

​@@CQEAcademy Thank you for your response. This makes great sense. Would you have any suggestions on best industry practices of choosing an optimum product sample size to calculate MTBF and the optimum ratio of repaired product vs new products to use in MTBF & how the selection is done? E.g lets say for low volume manufacturing, 60 products manufactured / year ? Also, on another related topic regarding warranty calculation - does the business strategy team include MTBF when calculating the warranty period of products and are there any situations where the warranty can be also renewed after product has been repaired to new since its MTBF would also be restored? Philosophically, what is considered morally right when providing warranty to customers ?

[@kaiyang8669]

Absolutely amazing. Thanks

[@CQEAcademy]

You're welcome Kai!!!!!

[@kimsenmen7562]

Heyy good video, and ty for expl all this, pl tell me how I can get or calculate betta valuem this example betta=2 howw? ty !

[@CQEAcademy]

Hey There! The beta value usually comes from your failure data - historically, when failure data was plotted on a weibull probability plot, the slope of the line from that data would be the beta value for your product

[@ashkumar8797]

Thanks you so much for this video.

[@yoyochihouse]

Your teaching is much easier to understand compared to my lecturer (from my master's degree, oops).

[@CQEAcademy]

Wow, thanks Christina!!!

[@reydantegarcia8776]

Hello, I am looking for Probability in Quality Control, sadly I can't find this topic on CQE Academy, does anyone know some resources to study that with?

[@karandeep1980]

MTTF formula that i see elsewhere is different . denominator for MTTF you mentioned as No. of failures. online i see Total number of devices. can you please clarify?

[@electricart4446]

Hello, can you, please, explain how you calculated "e"? is a constant? were you take it from? it is not explained

[@CQEAcademy]

Hey There! I probability should have explained this better. e is eulers number, it's a mathematical constant (like pi), and it's equal to 2.71828....

[@KHRAIEF]

thank you for this presentation

[@Alaa-gh93]

I have one thing to add for the MTTF, I believe it should be divided by the total number of units and not the total number of failures. The reason is the units that did not fail should also be included in the mean calculation. The hours of those units were already included when the summation of operational hours was added so it should be counted.

[@brennenmalaga2153]

I am seeing the same thing in other MTTF calculations. Do you know if you did calculate MTTF this way, would you still be able to proceed with the reliability models in the same way as he explained?

[@sheikhshariq1957]

What is the minimum number of product we have to put to trial for determining the MTTF or MTBF? In the above example you have taken 20 units on which basis?

[@CQEAcademy]

Hey Sheikh! There isn't really a set standard for the number of units required for reliability testing. MTTF/MTBF is obviously a sample estimate, and your confidence in that sample estimate grows with more samples, and the number of samples requires can vary from industry/industry and from application to application based on the risk associated with a failure.

[@jannikheidemann3805]

I'm here for the bathtub curve. What is a CQ-Exam?

[@solarpoppop9398]

Very clearly explained.Thanks!

[@CQEAcademy]

Thanks!!

[@ArjunSharma-wi3jp]

Awesome explanation, if i have peak stress and time of peak stress of 10 samples, log normal plotting weibull destribution of its time or stress which will be more helpful ? Which i did using excell and plotting weibull destribution, giving me the modulus intercept and slope , from which i cacluate the beta and eeta. Your insights means a lot to me

[@begguiabdelhadi]

Thanks so much 💚💚💚🇩🇿💚💚, please more videos about reliability and probability and statistics , I must pass the doctor exam (PhD)

[@CQEAcademy]

Will do!

[@jamalabboud9894]

Excellent

[@mmaScholar99]

I have a question about the question on <a href="#" class="seekto" data-time="900">15:00</a> If the probability that it will succeed is 66%, then does that mean the probability that it will fail is 44%? I have a question just like this in my past exam paper and I'm trying to understand it.

[@andyrobertson566]

Hey Leon, Yes, the probability of failure would be 34% (not 44%), but your logic is correct.

[@mmaScholar99]

@@andyrobertson566 Hi, thank you, sorry for late reply, also, last question, are there any feasible ways of increasing the reliability of the system? Like, what can be done?

[@CQEAcademy]

@@mmaScholar99 Hey Leon!!! Great question, and there's really no single answer to that question. The honest truth is, the reliability of your product often depends on the design. The suppliers you choose, the components you choose, the design you choose, and the failure modes associated with that design, all have an impact on the final reliability of your product. If your product is already in production and you cannot easily change the design, then what you can focus on is the process. Improving the process to eliminate failure modes that might reduce the reliability of your product. That sort of thing. Does that make sense?

[@MS-be9wv]

Please, how can I confirm the shape type and B value, how can I say this item should be go through increasing failure or decreasing, or may be exponential?

[@Abir_gaming179]

Hi, mate, great video to see I want to ask a few things: 1. how to calculate MTBF on redundance system? 2. How to define operating hours on a modular system? (eg certain parts are always operating, such as the power supply. But other parts are only active when used) 3. how to calculate MTBF at point 3?

[@CQEAcademy]

Hey There! Okay, so when you have a redundant system, generally we quantify the reliability of each component in that system, then use parallel system calculations (kzbin.info/www/bejne/g5Lcg6ythLmeesk) - or whichever configuration your system is setup as. Great question about operating hours on a modular system, that's somewhat unique to your design so you'll have to come up with a formula/process to calculate operating hours in that scenario. For #3 - you're asking about calculating MTBF in the wear-out period?

[@sachindhiman7116]

Items which are failing on regular basis and repaired again and again, do we calculate reliability of those items also in the same manner...

[@CQEAcademy]

Yes! The mean time between failure (MTBF) would be the metric to use when items get repaired when they break.

[@alexwalton8951]

Great video-can you go through part b of the question below... A non-repairable item exhibits a constant failure rate of 8% per 100 operating hours. A)If the initial stock is 150 units, and all items are operating simultaneously, how many items would be expected to be still operating after 200 hours? B)If the design of the unit is improved so that the failure rate falls by 25%, i.e. to 6% per 100 operating hours, what would be the proportional increase in the number still working after 2000 hours

[@sujanmeshram9345]

When we have to use lambda ×t

[@CQEAcademy]

It depends on what parameter you're given. If you're given lambda (instead of theta) then use lambda*t, if you're given theta, use t/theta.

[@marymulu405]

Very elaborate 😊

[@CQEAcademy]

Thanks!

[@airlanggak5391]

really appreciate your efforts! thanks so much!

[@CQEAcademy]

You're welcome!

[@MegaLitterbox]

Great material. Keep it up.

[@sreekanth4790]

thank you dude

[@CQEAcademy]

You're welcome!

[@javierignacio5250]

Excellent explanatio

[@greenbeltacademy]

Thanks!

[@albertogubernati3531]

Thank you!

[@CQEAcademy]

You're welcome!

[@hanyhegazy9230]

How could I determine beta value

[@CQEAcademy]

Hey Hany, you can only determine your beta value through reliability testing. The beta value used to be derived from a weibull probability plot analyzing reliability data. Now you calculate beta using programs like minitab but it first requires that you perform reliability testing and collect data

[@aravindhanpoopathy6862]

Hi thanks for the wonderful explanation, appreciate it! why is that the exponential dist is applied only for the "useful-Life Period", what is the reason behind it? is it cuz exponential dist doesn't have a shape parameter and Weibull does and therefore Weibull can adapt any part of the bathtub curve for a dataset? Thanks Arav

[@CQEAcademy]

Hey Arav! Great question! So to be fair, the weibull distribution can also be used for the "useful life period", because the weibull distribution can also have a shape parameter (Beta) equal to 1. The reason that the exponential distribution models the useful life period though, is because the useful life period can be characterized as a time period with a constant failure rate (flat line on the bathtub curve). And a constant failure rate (lambda) is modeled by the exponential distribution. I know that can be hard to visualize though and it would take another video to explain this more clearly. I'll add that to future videos!

[@aravindhanpoopathy6862]

@@CQEAcademy Hi I think it makes sense now! Thank you👍👍👍

[@CQEAcademy]

@@aravindhanpoopathy6862 You're welcome!!

[@houdabrahmi7308]

good presentation can I get the pdf file plz

[@CQEAcademy]

Yeah! Just head over to CQEAcademy.com/freecheatsheet

[@aaronkapata7103]

Hi, could you assist with where i can find articles that discuss these concept further? Great explanation!

[@MS-be9wv]

When I say that this device has an MTBF equals 500 yrs, how can I illustrate this to the customer? And you select 20 device in the video, may the reliability value will change if you actually use 50 or 100

[@CQEAcademy]

Great question, and yes, the confidence that you have in your reliability estimate goes up, when you take more samples. In terms of the MTBF of a device, 500 years is a big number, and your customer will obviously want to see the test data that generated this estimate.

[@MS-be9wv]

@@CQEAcademy tell me if the following is right; 500 yrs means reliability is 0.998 and if I produce 10000 device; that means there is a chance to failure of 20 out of the 10000 could fail

[@CQEAcademy]

Hey @@MS-be9wv so normally we talk about reliability in terms of the MTBF (Mean Time Between Failure) or the failure rate (lambda), both of which are typically measured in hours (MTBF) or failures per hour (lambda - failure rate). I think it would be challenging to prove a MTBF of 500 years. How are you getting that number?

[@muhammadzakaria4216]

Is there any software recommendation to solve this topic?

[@CQEAcademy]

Hey Muhammad, I personally don't have a reliability software that I would recommend, but I'd love to have other people chime in with any software that they'd recommend

[@vigneshm827]

Sir how to calculate Slope(B)? Please anyone expert answer!

[@nishantbhosle1268]

Thank you for such great explanations about the topics. Could you please also share the pptx links for your video if possible?

[@CQEAcademy]

Hey Nishant! Those powerpoints are sort of my secret weapon, so I keep those for myself :)

[@songs_gri7107]

Awesome!😀

[@CQEAcademy]

Thanks!

[@mahdipilevar7872]

Thanks

[@CQEAcademy]

You’re welcome!

[@karthiks233]

How do you get the beta value in distribution

[@alfredrosemond8114]

When using a basic calculator, could you walk me through this particular example. I am not getting .6766. Thanks.

[@CQEAcademy]

Hey Alfred, okay to the first order of operation is to take the ratio of 5,000 to 8,000 (0.625) Then, take that result and square it (0.390625). Do not include the negative sign in the square, or you'll lose the negative sign. Lastly, raise that exponent to the -0.390625 or e^(-0.390625) will give to that the probability of 0.6766 or 67.66% I hope that helps! -Andy

[@admorgan967]

@@CQEAcademy Hi Andy, could you please tell me why did you use MTBF not MTTF in this example, since both terms have the same equation? Second, could you simplify solving the exponential equation step by step? Thank you.

[@CQEAcademy]

@@admorgan967 Hey Adam, You're right both MTTF and MTBF have the same equation, the only difference is that MTTF is for non-repairable units, while MTBF is for repairable units, and I chose MTBF randomly. In terms of solving the exponential equation, first convert the fraction into a decimal (-1200/2996) = (-0.400). Then raise "e" to the power of that fraction - e^(-0.400).

[@admorgan967]

@@CQEAcademy hey Andy, in the second example, Weibul distribution, you have mentioned B or beta with a value 2, do you know where can I get this value if I need to perform an actual test? Thank you

[@CQEAcademy]

​@@admorgan967 Hey Adam, to find this shape value you'd have to perform reliability testing to quantify this parameter.

[@Mr.Divorce]

Could one calculate the MTTF/MTBF for the early-failure period?

[@CQEAcademy]

Hey Lee, you could, most definitely. Most textbooks though basically give the same advice that in the early life period, your focus should be more on root-cause analysis and improvements to get that product out of the early life period.

[@Mr.Divorce]

@@CQEAcademy Following up with the MTTF/MTBF calculation, for units that didn't fail, do we need to sensor them from the calculation?

[@CQEAcademy]

@@Mr.Divorce Hey Lee, no you can take credit for the fact that units went through testing and did not fail, that should be captured in the "total test time" when calculating the failure rate.

[@explorer2622]

Hello sir, In the formula of reliability what is e? And how R(1200) = e (-1200/2996) = 0.6699? Can you Please explain this formula?

[@CQEAcademy]

Hey Shailesh!! In that formula, "e" is Eulers number. It's a mathematical constant and it's equal to 2.71828. Think about it like Pi (3.1415926535. . . ) Most calculators have Eulers Number as an available constant to select from. Work that fraction first (-1200/2996 = -0.40053), and then raise "e" to that power (-0.40053) and you should get 0.6699

[@explorer2622]

@@CQEAcademy Thank you for your reply and explanation, I would further like to know why we considered 'e' only? Sorry it might be embarrassing question.

[@jankeshchakravarthy9389]

Hi - In the example with 6 failure, how would we calculate the MTBF for same example assuming all the units ran up to 1000 with 6 failures and 6 repairs?

[@CQEAcademy]

Hey Jankesh, I believe the proper procedure is to capture all of the run time of the units, including before the failure and after the repair, and factor all of that time into the calculation for the MTBF. Don't include any time spent in repair

[@walidhamdy9808]

Question on MTTF/MTBF example : how can one deduce that the MTTF is nearly 3000 hours when none of the device ran more than 1000 hours and 6 of them failed well before 1000 hours? If none had failed, the MTTF using this calculation would be infinite.

[@CQEAcademy]

Walid!!!! that's such an amazing question, and I didn't go into this level of detail in the video, but if this were to happen in real life, what I would recommend would be to continue the testing to failure to confirm the MTTF value, because as you point out it's not a best practice to make predictions for MTTF outside of your testing parameters.

[@etudayeabdulhameed4766]

Please how do I solve this A generating unit contains a large number of n components an a failure of any of these components results in an outage of the unit. Assume there are n component of interest in a generator and each can fail independently of the others, time to failure i can be described by an exponential distribution with parameters i=1,2,3....., n. Determine the system reliability

[@andyrobertson566]

Hey Etudaya, I don't really understand the question, are you looking for a quantitative estimate of the overall system reliability at a given instance in time? Or are you looking for a subjective description of the system and its design?

[@michelevardaro2205]

Question: how would I chose if testing the MTBF for 1000, 5000 or 10000? Amazing videos by the way, thank you!

[@CQEAcademy]

Hey Michele, thanks so much! Okay, so your MTBF value should be calculated from reliability testing. Then, based that reliability data you can make reliability predictions.

[@muhammadyasir1669]

Choosing testing time factor is Reliability testing planning of a product. Reliability testing can be perform in many ways . 1.Time truncated Testing ( Testing till predefined testing time) Both have different ways to calculate the MTBF including censoring data types . 2.Failure Truncated testing ( Testing the product till failure but testing time can increase for more reliable product and less for less reliable product) . To save time and resources we can use the testing strategies according to the product reliability defined by customer requirements and Reliability is part of the design life cycle to save the cost .