Exercises. 2. A batch of items contains 4% of defectives. If 15 items are taken from the batch, what is the probability that 4 will be defective?

Transcription

1 Exercises 1. A manufacturing unit has 40 workstations, each with two independent machines, Mi, M2. A statistical study has shown that the probabilities of failing during a working day are: for Mi, 0.08; for M2, (a) Find the probability that a station will be out of service (i.e. both machines failed) some time during a day. (b) Find the probability that at least one machine fails. (c) Find the probability that only M2 fails. (d) Find the probability that one, and only one, machine fails at each station. (e) Let x, a random variable, be the number of stations at which one, and only one, machine has failed. Assuming the x has a Poisson distribution, find the probability that there are six stations with one, and only one, failed machine. (f) Find the probability that there are (strictly) more than four stations with one, and only one, failed machine. (g) Justify the assumption of the Poisson law. 2. A batch of items contains 4% of defectives. If 15 items are taken from the batch, what is the probability that 4 will be defective? 3. In a group of machines one relay fails per day, on average. What is the minimum replacement stock that should be carried so as to cover a week's failures (five working days) with a maximum 5% risk of running out of replacements? 4. In a metal-cutting workshop the percentage of scrap is used as an indicator of the need to check the machines. The width of the pie ces cut has a normal distribution with mean 150mm and the allowable limits are 150 ± If the acceptable percentage of sc rap is 10%, what is the maximum allowable standard deviation? 5. Destructive tests are used to assess the quality of certain essential parts of a machine. A certain batch of ten contains two defective items. For

2 Exercises 177 the test three are chosen at random and if a single one fails the batch is rejected. Wh at is the probability of rejecting this batch? 6. Two conveyors are opera ted in active rendundancy. Their reliability function is exponential, R(t) = exp( - At) and the MTBF of each is 54 h. What is the reliability of the system at the end of 16 h? 7. The compression springs in a car's shock absorbers follow a log-normal distribution with parameters m = 7: 1, (J = 2. (a) After how long should they be changed if a 90% reliability is to be guaranteed? (b) What is the MTBF? 8. An investigation of the MTBF of solar cells gave the following results for a batch of 150: Time interval No. of cells failing (a) Wh at values does this give for the failure rates over the different intervals? (b) Find these rates also by fitting a Weibull distribution 9.1. The maintenance service for a factory wishes to optimize the preventive maintenance schedule for the machines it is responsible for; it has the following histories: Machine 1 Machine 2 Times between No.of Times between No. of breakdowns breakdowns breakdowns breakdowns 400h 1 400h

3 178 Exercises Assuming a Weibulllaw (a) Compute the parameters. (b) Compute the MTBF. (c) Derive a preventive maintenace policy that will guarantee fault-free operation with probability 95% The history of a certain machine is as foliows: No. of breakdowns 1 Time in operation 24h Can one say that these figures support the hypothesis that the time of fault-free operation is exponentially distributed with F(t)= l-exp(tj42), with a 5% risk of rejecting the hypothesis if it is true? (a) Suggest a goodness-of-fit test for the hypo thesis. Tables of the X 2 and Kolmogorov distributions are provided; choose the one you consider the more suitable. (b) Apply the test and give the conclusions reached The following figures are the times between successive break downs of a motor: Analysing these by the Weibull method: (a) What are the values of the parameters y, 1], ß? (b) What type oflaw is found? (c) On which part of the 'bathtub' curve do these figures lie? (d) Compute the MTBF in two different ways. (e) Deduce, graphically, the reliability at t = 500 h The times between failures for the electronic system of a certain machine are: 24.5, 35.5, 38.5, 39.5, 42.5, 57.5, 62.5 h

4 Exercises 179 Can the exponential model F(t) = 1 - exp(0.0235t) be accepted with 5% risk? Suggest a goodness-of-fit test: (a) Select the more suitable, X 2 or Kolmogorov. (b) Apply the test and give the conclusions reached The following lifetimes are recorded for a certain system: 1860,2500,2900, 3600, 3950, 5100, 6300 (a) Are these fitted by a Weibulllaw? Find the parameters. (b) Compute E(t) A system is represented by the diagram Initially, Unit 1 is operating; if it fails, the switch S switches it out and Unit 2 in. A,/1 The following assumptions are made: S is perfectly reliable (reliability = 1); the repair time is distributed exponentially with parameter (= l/mttr); the time offault-free operation of each unit is distributed exponential with parameter ( = I/MTBF); at t=o P(O) = 1, P(1)=P(2)=0 (a) We wish to investigate the availability of the system. Give the Markov diagram. (b) Give the change-of-state equations. (c) Apply the Laplace transform and express the resulting equations in matrix form. (d) Compute the a vailability function. (e) What is the asymptotic availability as t = oo? The problem concerns an aircraft blind-ianding system (BLS). The level of safety required is that the probability of failure of any one of the aircraft's safety-critical systems during a single flight must not exceed 10-6 ; since there are 10 such systems, including BLS, the

5 180 Exercises probability for each must not exceed 10-7 BLS consists of 1000 components, each having an exponentiallifetime with parameter = 105 failures per hour; and, in any case, operates only for the 6-minute approach to landing: (a) Compute the prob ability of BLS failing during a flight. (b) Does this satisfy the safety criterion? (c) If not, wh at active redundancy provision would make it acceptable? 12. Compute the reliability of the following redundant systems for an operating time of 500 h. The parameters are mean faults per hour. (a) AA = , AS = Construct the PERT chart for the following set of tasks Give the critical path and the minimum time to complete the project Draw the Gantt diagram. Task A B C D E F G Time (h) Antecedents A A B W W F Description of task

6 Exercises 181 H 2 W,F I 4 W,E J 2 W K 4 W,J L 2 G,E,C M 2 J, F, H, K N 1.5 G,L,F 0 1 D,N,L P 1 D,R,M,Q Q 2 R 3 W,F S 0.5 O,R,M T 2.5 S,O U 2 P, T, S, 0 V 1 U W The consumption of certain replacement items is as follows: Feb. Mar. Apr. May Jun. July Aug. Sep. Oct. Nov Given that the ownership cost is 25%; the cost of handling an order is 800 fr.; the average cost of an item is 150 fr.; the annual consumption is 318; the delay between ordering and receipt is 2 months; the permissible risk of running out of stock is 10%; find (a) the economic size of order; (b) the optimum safety level of stock. 15. The manager of a car-hire firm wishes to find if there is any relation between the number of serious breakdowns experienced by a car and the number of drivers who have used it. He has these records: No. of breakdowns No. of drivers

7 182 Exercises Is this evidence for such a relation? The maintenance engineer wishes to optimize the safety level stock he should hold. He knows that: break downs during the delay between ordering and receipt follow a Poisson distribution, with a me an of 2; a failure costs 1500 fr.; the ownership cost of 100 fr. per item. What is the value of the optimum stock? 17. A road haulage company has this information concerning its vehicles: Cost to buy is fr. Resale value is 8(t) = exp(-0.15t) tin 1000h in use Operating cost is ljj(t) = {exp(0.41) -1} What is the optimum time at which to replace a vehicle? 18. A machine produces items for which two dimensions A, Bare required to be within the limits: A: 25 ± 0.1, B: 30 ± 0.1. After machining, the items are measured and it is found that the measurements have independent normal distribution with parameters: A: mean , standard deviation 0.05 B: mean , standard deviation 0.04 What percentage of scrap will be produced? 19. A group of machines suffers from random fracturing of certain drive shafts, with records showing that there have been five such breakages over aperiod of 45 months. How many spares should be held, if: the cost of a breakage, in lost production, is 6000 fr. per hour a new shaft costs 1800 fr. the time to repair the damage is 1 ~ hours labour costs for repair are 150 fr. per hour the delay between ordering and receipt is two months the ownership cost of stock is 20~~ 20. An electronics manufacturer win a contract with a large company. To

8 Exercises 183 assess its ability to fulfi1 the terms of the contract it examines its output over 100 working days, finding the following distribution of numbers of items produced: Daily production No. of days with this production (a) Find the mean and standard deviation. (b) The contract calls for 50 items per day over aperiod of 100 days. Wh at is the probability of fulfilling it, assuming a normal distribution for the daily production? (c) 3100 items are produced in the first 60 days, when a breakdown stops production for 3 days. What is then the probability offulfilling the contract? 21. Production losses in a sheet metal workshop lead to risks of running out of stock of a certain item X. An investigation of the stock holdings shows that an unnecessarily large stock of another item Y is held. The mean monthly demands are: Jan. X 3 Y 31 Feh. Mar. Apr. May Jun. Jly Aug. Sep. Oct. Nov. Dec and the mean holding of Y is 60 items. If the ownership costs of X and of Y are 20%; the cost of handling an order is 500 fr.; X and Y cost 300 fr. and 100 fr. each, respectively; the orderjreceipt delays for X and Y are 1 month and 2 months respectively; (a) Give the most appropriate provisioning policy for X and for Y.

9 184 Exercises (b) Compute - the economic order size; - the value of the order; - the safety-ievel stocks, allowing a 2.5% risk of running short. (c) Give the graphical representation of the stock management. 22. The manager of a maintenance service is seeking to establish the best policy for provisioning certain replacement items. He has three suppliers, firms F1, F2 and F3, and has made the standard A-B-C dassification of the items he buys. He has already decided that the dass C items shall be bought from F3. Class A items are to be bought from whichever of F1 and F2 offers the better terms, and dass B from the other - so as to spread the orders; the A items are to be ordered according to a strict program, the B by an order-on-warning policy. The following information is available: The monthly demand for the A items is: Jan. Feb. Mar. Apr. May Jun. July Aug. Sep. Oct. Nov. Dec The terms offered for supplying this item are: F1: unit price 430fr. (exc1. taxes): delivery charges 5% of order, carriage free for orders over 4000 fr. F2: unit price 460 fr. (excl. taxes), carriage free The orderjreceipt delay is 15 days for each supplier. (a) Decide the supplier for the A items. (b) Give the annual number of orders, the ordering interval, the quantities ordered and the total cost of the provisioning. The cost of handling an order is 100 fr. Interest charges on stocks held are 15% (c) Compute the safety-ievel stock; this must cover 15 days' demand. (d) Determine who should supply the various products: Reference Annual demand Price R (Class A)

10 Exercises A factory makes a certain item in large numbers, the manufacturing process involving two independent phases. The item can suffer from two faults A and B which can arise in the first and second phases respectively. Experience has shown that A arises in 2% of cases, B in 8% Compute the probability that an item chosen at random: (a) will have both faults; (b) will have at least one of the two faults; (c) will have one and only one of the two faults; (d) will have neither fault A sam pie of 200 items is drawn from stock; we are interested in the random variable X equal to the number ofthese items having fault A: (a) We assume that X has a Poisson distribution: what grounds have we for this? Wh at is the value of the parameter? (b) What is the probability that among the 200 items, 10 have fault A? A sampie of 300 is drawn; we are interested in the random variable Y equal to the number of items having fault B. (a) We assume that Y has a Gaussian distribution: what grounds have we for this? What are the parameters? (b) Compute: prob(y < 24) prob(20< Y < 25) prob(y< 30 Y> 24) ANSWERSTOPROBLEMS 1. (a) P(MI and M2 fail) = P(MI fails)' P(M2 fails), as they are independent = ; (b) P(at least one machine fails) = P(Ml) + P(M2) - P(Ml)' P(M2) = ; (c) P(only M2 fails) = ; (d) P(one, and only one, machine fails at each station) = P(Ml). 1 - P(M2) + P(M2) 1 - P(Ml) = ; (e) m = np = 40 x = 3.87; (f) P( x 4) = 36%; (g) n is large and the parameter m is known.

11 186 Exercises 2. P = k= b = P = R = (a) t = 83 h; (b) MTBF=8109h. 8. Values of ), (failure rates) over successive intervals: (a) y = 0, ß = 1.6, IJ = 680 h; (b) MTBF = 612h; (c) t=106h (a) Kolmogorov test; (b) Accept the hypo thesis (a) y=0,ß=1,1j=1500; (b) Exponentiallaw; (c) Maturity phase, since A is constant; (d) MTBF j = [(1 + I/ß) MTBF 2 = 1/),; (e) R(t) = 71% (a) Kolmogorov test; (b) Accept the exponential model (a) Weibulllaw with y > 0; (b) y = 1200, ß = 1.4, IJ = (a) J1 J1 (b) P~(t)= -APO(t) + f1p j (t) pi! (t) = ÄP ou) - (fl + },)P 1 (t) + flp 2(t) P~ (t) = ),P 1 (t) - flp 2 (t)

12 Exercises 187 z + ).,u IF (d) A(t)=-_,u _ + _ z {r1exp(+rzt)rzexp(r1t)} A z +,uz + All A ),11 (e) as t-4 CD A(t)-4( ),,u)/().z ).11) (a) Probability of not failing during a flight = ; (b) This does not satisfy the safety criterion; (c) Provide active redundancy by doubling the number of components and arranging them in series-parallel. 12. (a) 0.997; (b) PER T chart on p. 111 Minimum time for completion 15 min. The Gantt diagram is shown here. A C B 0 R Q S T U V W E F G H L M N 0 p --.J

13 o

14 Exercises (a) Qa= 116; (b) Sa = Yes, p = x 10 3 h % (a) m = 51.23, ()' = (b) 53%. (c) 48.4%. 21. (a) X: order on stock warning. Y: order at fixed intervals. (b) for X: Qe = 34, Ss = 9, Pe = 15. for Y: T = 4 months 15 days, Ss = (1) Choose F1. (2) Number of orders = 12; P = 1 month 4 days; Qe = 42 items; Approx. cost = 138, 016 fr. (3) R 103 from F 1; R 104, 105, 109, 110 from F2; R 102, 106, 107, 108 from F Faults A, B occur independently with probabilities PA = 2%, PB = 8%. l(a) Prob ability that an item has both faults = P(A (l B) = PA' PB = 0.2% x 0.8% = 0.16%. l(b) Probability that an item has at least one of the two faults: = P(AuB) = PA + P B - PA'PB = 9.84%. l(c) Probability that an item has one, and only one, fault: = PA'PB + PB'PA = P A(l- PB) + PB(1- PA) = 9.68% = 2 x x 92 = 9.68% Alternatively, probability = P(A u B) - P(A (l B) = = 9.68%.

15 190 Exercises l(d) Probability that an item has neither fault = P>P B = 90.16%. Alternatively = 1 - P(A ub) = = 90.16%. 2(a) Sampie size n = 200 is large, prob ability 2% (= 0.02) is small. Poisson parameter is the me an m = np = 200 x 0.02 = 4. 2(b) P(x = 10) = exp( - 4) 4 10 /1O! = = 0.53%. 3(a) For n large and p not small the binomial distribution B(n, p) tends to the Gaussian N(m, 0"). For the Poisson law m = np = 300 x 8% = 24 and Cf = -Jm = (b) Probability of Y> 24. A better approximation is to take 0.5 from 24, for then we go from a discrete to a continuous law. P(Y < 23.5) = F(v) = F[( )/4.48] = F( ) = 1 - F(0.102) = = 0.460, etc.

16 Annexe

17 Binomial law k Cumulative function Pk = Ic;p'(l - p)n-, o SampIe size k p = 1% p = 2% p = 3% p = 4% p = 5% p = 6% p = 7% p = 8% p = 9% p = 10% p = 20% p = 30% p = 40% p = 50% N= N=lO

18 N=

19 Binomiallaw (cont.) SampIe slze k p = 1 % p = 2% p = 3% p = 4% p = 5% p = 6% p = 7% p = 8% p = 9% p = 10% p = 20% p = 30% p = 40% p = 50%._----~-._ N=

20

21 BinomiaI Iaw (cont.) Sampie size k p = 1% p = 2% p = 3% p = 4% p = 5% p = 6% p = 7% p = 8% p = 9% p = 10% p = 20% p = 30% p = 40% p = 50% N=

22 SampIe size k N= p= 1% p=2% p= 3% p=4% p= 5% p=6% p= 7% p= 8% p=9% p= 10% p=20% p= 30% p=40% p= 50%

23 Binomiallaw (cant.) Sampie size k -_.._ N= p= 1% p=2% p=3% p=4% p=5% p=6% p=7% p=8% p=9% p= 10% p = 20% p = 30% p = 40% p = 50%

24 Poisson law Cumulative function P K = I 'k _AA e -- O~k~K k! K A= 0.1 A=0.2 A = 0.3 A = 0.4 }, = 0.5 A=0.6 A=0.7 A=0.8 A= K A = 1.0 A = 1.5 A=2.0 A= 2.5 A = 3.0 A = 3.5 A=4.0 A=4.5 A=

25 Poisson law (folli.) K A. = 5.5 A = 6.0 ), = 6.5 ).= 7.0 A. = 7.5 A. = 8.0 A= 8.5 A. = 9.0 A. =

26 Poisson law (Cont.) K A= 10 A= 11 A= 12 A= 13 A= 14 A= 15 A= 16 A= 17 A=

27 Poisson law (cont.) K ). = 19.1,=20.1, = 21.1,=22.1,=23.1,=24.1,= ü

28 Normal (Gaussian) distribution The table gives the cumulative function far the reduced normal distribution (mean = 0, variance = 1), that is, the prob ability F(x) of observing a value less than x. ~ -00 o x +00 X _._

29 Normal (Gaussian) distribution (cant.) x

30 For X? 3, F(x) can be füund from the follüwing table üf 1 ~ F(x). x The abüve tables are für x > 0; für negative values F( ~ x) = 1 ~ F(x) e.g. F( ~ 0.94) = 1 ~ F(0.94) = 1 ~ =

31 Student's distribution The table gives the values of t which will be exceeded with probability!x ~ A o t x

32 l

33 Chi-square (X 2 ) The table gives the values of X 2 which will be exceeded with probability Cl P(X2)~ o x 2 Cl v For v> 30 the values of J2X2 - J2v - 1 can be taken as following the reduced normal law.

34 Fisher-Snedecor distribution The table gives the values of F which will be exceeded with probability!x, where F=(XUVI)/(X~/V2)Xi, X~ have VI' V1 degrees offreedom respectively. ~ o I F cx)' (1= (1= V

35 Fisher-Snedecor distribution (cont.) ()(= ()(=

36 Kolmogorov-Smirnov test The table gives the critical values for the Kolmogorov- Smirnov test Significance level N >35 - jn jn jn jn jn

37 Weibulllaw: determination of MTBF The table gives the values of A, B which enable the mean and standard deviation to be calculated from the Weiball parameters ß, 1'/, y. Mean = AI'/ + Y Standard deviation = BI'/ ß A B ß A B ß A B Gamma distribution r(x) x r(x) x r(x)

38 Method of median ranks (Johnson) SampIe size Rank order

39 Method of median ranks (Johnson) (cant.) Sampie size Rank order

40 Method of median ranks (Johnson) (cant.) Rank order Sampie size