OPTIMUM SCHEDULE FOR PREVENTIVE MAINTENANCE: A GENERAL SOLUTION FOR A PARTIALLY SPECIFIED TIME‐TO‐FAILURE DISTRIBUTION

Abstract

The classic problem of determining an optimum schedule for preventive maintenance when the time‐to‐failure (TTF) is random has been previously addressed in the literature under various distributional assumptions. However, no general solution that requires only partial distributional information (in the form of the first few moments) has been suggested. In this paper we develop solution procedures characterized by four features: (1) The solutions are general in the sense that only the first few moments of the TTF distribution need to be specified. (2) The optimal solution is given explicitly in terms of the decision variables, thus allowing simple sensitivity analysis. (3) When the TTF moments are unknown, a new two‐moment (partial and complete) distributional fitting procedure, used in the solution routine, ensures a better representation for the underlying TTF distribution relative to three‐moment or four‐moment distributional fitting. (4) When the TTF observations are truncated, simple routines to calculate maximum likelihood estimates are developed. We demonstrate that the partiál distributional specification, required for the new solution procedures, does not detract meaningfully from the optimality of the solution.

Keywords

DISTRIBUTIONAL FITTING MAXIMUM LIKELIHOOD ESTIMATES PREVENTIVE MAINTENANCE RELIABILITY

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References

Brick

Michael

, and Morganstein

(1989), “Using Statistical Thinking to Solve Maintenance Problems,” Quality Progress, 22, 5, 55–60.

Clements

(1989), “Process Capability Calculations for Non‐normal Distributions,” Quality Progress, 22, 9, 95–100.

Dodson

(1994), “Determining the Optimum Schedule for Preventive Maintenance,” Quality Engineering, 6, 4, 667–679.

Gan

F. F.

and Koehler

K. J.

(1990), “Goodness‐of‐Fit Tests Based on P‐P Probability Plots,” Technometrics, 32, 3, 289–303.

Gan

F. F.

Koehler

K. J.

, and Thompson

J. C.

(1991), “Probability Plots and Distribution Curves for Assessing the Fit of Probability Models,” The American Statistician, 45, 1, 14–21.

Lau

A. H.

and Lau

H. S.

(1993), “A Simple Cost Minimization Procedure for the (Q, R) Inventory Model: Development and Evaluation,” IIE Transactions, 25, 2, 45–53.

Nahmias

(1993), Production and Operations Analysis. 2nd ed., Irwin, Homewood, IL.

Schmeiser

B. W.

and Deutsch

S. J.

(1977), “A Versatile Four Parameter Family of Probability Distributions Suitable for Simulation,” AIIE Transactions, 9, 2, 176–181.

Shore

(1982), “Simple Approximations for the Inverse Cumulative Function, the Density Function and the Loss Integral of the Normal Distribution,” Applied Statistics, 31, 2, 108–114.

10.

Shore

(1986), “Simple General Approximations for a Random Variable and Its Inverse Distribution Function Based on Linear Transformations of a Nonskewed Variate,” Siam Journal on Scientific and Statistical Computing, 7, 1, 1–23.

11.

Shore

(1995a), “Identifying a Two‐Parameter Distribution by the First Two Moments (Partial and Complete),” Journal of Statistical Computation and Simulation, 52, 1, 17–32.

12.

Shore

(1995b), “Fitting a Distribution by the First Two Moments (Partial and Complete),” Journal of Computational Statistics and Data Analysis, 19, 563–577.

13.

Shore

(1996a), “A New Estimate of Skewness with MSE Smaller Than That of the Sample Skewness,” Communications in Statistics (Simulation and Computation), 25, 2, 403–414.

14.

Shore

(1996b), “Approximating an Unknown Distribution When Distributional Information is Extremely Limited,” Under refereeing.