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Question 14S.2: Comparing Preventive Maintenance and Breakdown Maintenance w...

Comparing Preventive Maintenance and Breakdown Maintenance when Breakdown Time is Normally Distributed

Another approach that might be used relates to the time before a breakdown occurs. Suppose that the average time before breakdown is normally distributed and has a mean of 3 weeks and a standard deviation of .60 week. If breakdown cost averages $1,000 and preventive maintenance costs $250, what is the optimal maintenance interval?

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Begin by computing the ratio of preventive cost to the breakdown cost:

\frac{Preventive  cost}{Breakdown  cost} = \frac{\$250}{\$1,000} = .25

Find the number of standard deviations from the mean represented by an area under the normal curve equal to .25 using Appendix B, Table B. It is –.67. Use this value of z to compute the maintenance interval:

Table B.1  Areas under the standardized normal curve, from −∞ to − z

.09 .08 .07 .06 .05 .04 .03 .02 .01 .00 z
.0002 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 −3.4
.0003 .0004 .0004 .0004 .0004 .0004 .0004 .0005 .0005 .0005 −3.3
.0005 .0005 .0005 .0006 .0006 .0006 .0006 .0006 .0007 .0007 −3.2
.0007 .0007 .0008 .0008 .0008 .0008 .0009 .0009 .0009 .0010 −3.1
.0010 .0010 .0011 .0011 .0011 .0012 .0012 .0013 .0013 .0013 −3.0
.0014 .0014 .0015 .0015 .0016 .0016 .0017 .0018 .0018 .0019 −2.9
.0019 .0020 .0021 .0021 .0022 .0023 .0023 .0024 .0025 .0026 −2.8
.0026 .0027 .0028 .0029 .0030 .0031 .0032 .0033 .0034 .0035 −2.7
.0036 .0037 .0038 .0039 .0040 .0041 .0043 .0044 .0045 .0047 −2.6
.0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060 .0062 −2.5
.0064 .0066 .0068 .0069 .0071 .0073 .0075 .0078 .0080 .0082 −2.4
.0084 .0087 .0089 .0091 .0094 .0096 .0099 .0102 .0104 .0107 −2.3
.0110 .0113 .0116 .0119 .0122 .0125 .0129 .0132 .0136 .0139 −2.2
.0143 .0146 .0150 .0154 .0158 .0162 .0166 .0170 .0174 .0179 −2.1
.0183 .0188 .0192 .0197 .0202 .0207 .0212 .0217 .0222 .0228 −2.0
.0233 .0239 .0244 .0250 .0256 .0262 .0268 .0274 .0281 .0287 −1.9
.0294 .0301 .0307 .0314 .0322 .0329 .0336 .0344 .0351 .0359 −1.8
.0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 .0436 .0446 −1.7
.0455 .0465 .0475 .0485 .0495 .0505 .0516 .0526 .0537 .0548 −1.6
.0559 .0571 .0582 .0594 .0606 .0618 .0630 .0643 .0655 .0668 −1.5
.0681 .0694 .0708 .0721 .0735 .0749 .0764 .0778 .0793 .0808 −1.4
.0823 .0838 .0853 .0869 .0885 .0901 .0918 .0934 .0951 .0968 −1.3
.0985 .1003 .1020 .1038 .1056 .1075 .1093 .1112 .1131 .1151 −1.2
.1170 .1190 .1210 .1230 .1251 .1271 .1292 .1314 .1335 .1357 −1.1
.1379 .1401 .1423 .1446 .1469 .1492 .1515 .1539 .1562 .1587 −1.0
.1611 .1635 .1660 .1685 .1711 .1736 .1762 .1788 .1814 .1841 −0.9
.1867 .1894 .1922 .1949 .1977 .2005 .2033 .2061 .2090 .2119 −0.8
.2148 .2177 .2206 .2236 .2266 .2296 .2327 .2358 .2389 .2420 −0.7
.2451 .2483 .2514 .2546 .2578 .2611 .2643 .2676 .2709 .2743 −0.6
.2776 .2810 .2843 .2877 .2912 .2946 .2981 .3015 .3050 .3085 −0.5
.3121 .3156 .3192 .3228 .3264 .3300 .3336 .3372 .3409 .3446 −0.4
.3483 .3520 .3557 .3594 .3632 .3669 .3707 .3745 .3783 .3821 −0.3
.3859 .3897 .3936 .3974 .4013 .4052 .4090 .4129 .4168 .4207 −0.2
.4247 .4286 .4325 .4364 .4404 .4443 .4483 .4522 .4562 .4602 −0.1
.4641 .4681 .4721 .4761 .4801 .4840 .4880 .4920 .4960 .5000 −0.0

Table B.2  Areas under the standardized normal curve, from −∞ to + z

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
.7 .7580 .7611 .7642 .7673 .7703 .7734 .7764 .7794 .7823 .7852
.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993
3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995
3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997
3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

Mean + z standard deviations = 3 – .67(.60) = 2.598 (round to 2.6 weeks)

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