Preventive Maintenance, Concepts, Modeling, and Analysis

Abstract

The purpose of this chapter is to present elements of planned and preventive maintenance concepts and models. Preventive maintenance concepts are presented, followed by a host of diagnostic techniques which are useful for implementing condition-based maintenance programs. Then, this chapter presents a host of mathematical models for optimal preventive maintenance and replacement, followed by mathematical models for inspections. The inspections models aid in determining frequency of inspections that are useful for planning effective maintenance actions. In addition, the chapter presents the concepts of imperfect maintenance and delay time modeling. Several examples are provided to illustrate the concepts and the models.

Exercises

• 1. Why is preventive maintenance in general is more effective than other types of maintenance strategies?

• 2. Visit a plant near your area and identify the need for CBM. List the diagnostic techniques the plant needs. Identify the type of monitoring and equipment needed to make the CBM operational.

• 3. What are the cost components involved in the equipment life cycle? How do you estimate each component?

• 4. Visit a nearby plant and select a critical equipment and design a planned maintenance program for it.

• 5. An equipment time to failure follows an exponential distribution with mean equal 500 days. The cost of preventive and failure maintenance are $50 and $300, respectively. Determine t _p the optimal time for preventive replacement using the following policies:

  • (a) Type I policy.

  • (b) Type II policy .

• 6. Solve the same problem in 7 if the failure distribution is lognormal with scale parameter μ = 2 and shape parameter σ ² = 0.01.

• 7. Solve the optimal t _p in example 2 using the golden section method.

• 8. Determine the optimal t _p for the problems in example 1 and 2 using Newton method .

• 9. If an equipment has a failure time distribution F(t) and density function f(t), show the following:

  • (a) $\mu = \int_{0}^{\infty } {\overline{F} (t){\text{d}}t}$

  • (b) $r(t) = {\raise0.7ex\hbox{${f(t)}$} \!\mathord{\left/ {\vphantom {{f(t)} {1 - F(t)}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${1 - F(t)}$}}$.

  • (c) The expected number H(t _p) of failures in the interval (0, t _p) is $H(t_{\text{p}} ) = \int_{0}^{{t_{\text{p}} }} {r(t){\text{d}}t}$

• 10. Develop an optimization method to find the optimal k, T ₁, …, T _k for the generalized type I and type II policies in Sect. 3.6.4.

• 11. Consider the inspection model in Sect. 3.8.2 and show that a breakeven inspection interval exists for any failure distribution. Show that the breakeven point is unique in the cases of the exponential and Weibull distributions (results are in [9]).

• 12. Consider the model in Sect. 3.8.2 and derive the optimal inspection schedule for the exponential distribution with a parameter λ in terms of p, C _r, C _I, and λ. Find the schedule if λ = 0.01, p = 500, C _r = 4000, and C _i  = 50.

• 13. Consider the model in Sect. 3.8.3 and show that if the failure rate distribution is exponential, then the inspections are equally spaced, i.e., T _i+1 −T _i  = T _i  − T _i−1.

• 14. Use the model given in Sect. 2.8.4 for coordinating the inspections of five machines. The data for the machines are given below.

  Machine i	$C i	λ i	$r i	$s i
  1	5	0.01	100	10
  2	10	0.05	200	100
  3	5	0.08	100	10
  4	5	0.02	300	100
  5	10	0.1	200	50

  A = 25, N = 5 (Data source is reference [3])

• 15. Suppose the failure distribution for an equipment is Weibull with shape parameter α = 2 and scale parameter β = 50. Then, show the following:

  • (a) The failure rate function is given by, $r(t) = \beta \alpha \, t^{\alpha - 1}$

  • (b) Show that r(t) is monotonically increasing.

  • (c) Compute T* for the models given for approaches 1 and 3 for modeling imperfect maintenance.