Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach

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1 Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 Published online January 7, 6 ( doi:.648/j.ajtas.s.65.4 IN: (Print); IN: (Online) Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah Niholas A. Nehval, *, Gundars Berzins, Vadims Danovis 3 Department of Mathematis, Balti International Aademy, Riga, Latvia Department of Management, University of Latvia, Riga, Latvia 3 Department of Marketing, University of Latvia, Riga, Latvia address: nehval@junik.lv (N. A. Nehval), gundars.berzins@lu.lv (G. Berzins), vadims.danovis@lu.lv (V. Danovis) o ite this artile: Niholas A. Nehval, Gundars Berzins, Vadims Danovis. Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah. Amerian Journal of heoretial and Applied tatistis. peial Issue: Novel Ideas for Effiient Optimization of tatistial Deisions and Preditive Inferenes under Parametri Unertainty of Underlying Models with Appliations. Vol. 5, No. -, 6, pp. -8. doi:.648/j.ajtas.s.65.4 Abstrat: Age replaement strategies, a unit is replaed upon failure or on reahing a predetermined age, whihever ours first, provide simple and intuitively attrative replaement guidelines for tehnial units. Within theory of stohasti proesses, the optimal preventive replaement age, in the sense of leading to minimal expeted osts per unit of time when the strategy is used for a sequene of similar units over a long period of time, is derived by appliation of the renewal reward theorem. he mathematial solution to the problem of what is the optimal age for replaement is well known for the ase when the parameter values of the underlying lifetime distributions are known with ertainty. In atual pratie, suh is simply not the ase. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. he risk assoiated with using estimates rather than the true parameters is alled estimation risk and is often ignored. When data are limited and (or) unreliable, estimation risk may be signifiant, and failure to inorporate it into the model design may lead to serious errors. Its expliit onsideration is important sine deision rules that are optimal in the absene of unertainty need not even be approximately optimal in the presene of suh unertainty. In the present paper, for effiient optimization of statistial deisions under parametri unertainty, the pivotal quantity averaging () approah is suggested. his approah represents a new simple and omputationally attrative statistial tehnique based on the onstrutive use of the invariane priniple in mathematial statistis. It allows one to arry out the transition from the original problem to the equivalent transformed problem (in terms of pivotal quantities and anillary fators) via invariant embedding a sample statisti in the original problem. In this ase, the statistial optimization of the equivalent transformed problem is arried out via anillary fators. Unlike the Bayesian approah, the proposed approah is independent of the hoie of priors. his approah allows one to eliminate unknown parameters from the problem and to find the better deision rules, whih have smaller risk than any of the well-known deision rules. o illustrate the proposed approah, the numerial examples are given. Keywords: Age Replaement, Parametri Unertainty, Equivalent ransformed Problem, Optimization. Introdution Attention to age replaement has predominantly been based on a lassial operational researh (OR) perspetive, the probability distribution for the lifetime of the unit is assumed to be known. Age replaement strategies for tehnial units desribe that a unit is replaed preventively upon reahing age, or orretively upon failure before, preventive replaements are typially less expensive than orretive replaements. he lassial mathematial approah for determining the optimal replaement age is based on the renewal riterion, whih impliitly assumes that the same replaement strategy is used over a very long period of time, onsisting of many yles, one yle is the period of random length between two onseutive replaements; see, for example, Barlow and Hunter [], Barlow and Proshan [, 3]. hey disussed the problem of determining an optimal preventive replaement age to minimize the long-run average expeted ost per unit time over the infinite horizon (the average ost in short). ine then, this basi model has been generalized and modified by many

2 Niholas A. Nehval et al.: Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah authors to handle more pratial situations, as summarized in Asher and Feingold [4], Nakagawa [5], and Valdez-Flores and Feldman [6]. In pratie, this proedure is also used even though one realizes that the resulting optimal strategy may only be used for a few suh yles, for example, beause the unit would normally undergo some tehnial updates within reasonable period of time, or one wishes to keep the option open to hange the poliy in light of new information that may our during the proess. It should be noted that Barlow and Hunter [] onsidered only minimal repair and preventive replaement as maintenane ativities. Minimal-repair and replaement are often used as pratial maintenane ativities of real reliability systems. A minimal repair is the maintenane ativity to repair the failed system so that its funtion is reovered, without hanging its age, while a replaement restores the entire system into the new ondition so that it behaves as a new system. Further, replaement is lassified into preventive replaement or failure (or orretive) replaement aording as whether the system is in operation or in failure. Phelps [7] introdued failure replaement as a maintenane ativity, and disussed an optimal maintenane problem with minimal repair and failure replaement under the average ost riterion (sine it was assumed in this model that the required osts for preventive and failure replaements are equal, the system should be replaed only when it is failed). ahara and Nishida [8] disussed the maintenane problem with both preventive replaement and failure replaement whih have different osts. he above studies onern, in general, the lassial age replaement model and its some modifiations.. Classial Age Replaement Model In the well-known lassial model for age replaement (Barlow and Proshan [, 3]), the failure time of the unit (omponent) is assumed to be an absolutely ontinuously distributed random variable X with known probability distribution, with umulative distribution funtion (df) F(x) Pr(X x), probability density funtion (pdf) f(x), hazard rate h(x) f(x)/( F(x)), and expeted value E(X). It is assumed that h(x) is monotonously stritly inreasing, whih is often onsidered to be a natural assumption for situations age replaement may be ost effetive (Barlow and Proshan [, 3]) o avoid mathematial ompliations, it is assumed that F(), F(x) > for all x >, and E(X) <. he simple age replaement poliy of replaing a omponent when it fails or at time, whihever omes first, requires fixed osts and for replaing a failed and non-failed item, respetively, with > >. Let N (t) and N (t) denote the number of failed and non-failed units, respetively, whih are replaed in (, t]. he optimum age replaement interval, *, is hosen so as to minimize the limiting expeted ost per unit time, C() (Barlow and Proshan [], Glasser [9], heaffer []), C( ) lim E{ N ( t) + N( t)}/ t t ( ) F( ), F () F F. () It an be shown (Berg []) that if h() inreases in and if, lim h ( ) >, ( ) E{ X} then there exists a unique minimum of () and * is the unique solution of the optimality equation (3) h( ) F x + F. (4) Conditions for the existene of a unique minimum are established and an equation haraterizing the optimal poliy and minimal average osts is derived. In this paper, for effiient optimization of statistial deisions (based on a past random sample of lifetimes) under parametri unertainty, the pivotal quantity averaging () approah is suggested. his approah represents a simple and omputationally attrative statistial tehnique based on the onstrutive use of the invariane priniple in mathematial statistis. It allows one to arry out the transition from the original problem to the equivalent transformed problem (in terms of pivotal quantities and anillary fators) via invariant embedding a sample statisti in the original problem (Nehval and Vasermanis []). In this ase, the statistial optimization of the equivalent transformed problem is arried out via anillary fators. Unlike the Bayesian approah, the proposed approah is independent of the hoie of priors. It allows one to eliminate unknown parameters from the problem and to find the better deision rules, whih have smaller risk than any of the well-known deision rules. 3. Innovative Age Replaement Model In this paper, to handle more pratial situations, an innovative model for age replaement is proposed. he osts inluded in the age replaement model are not assumed to be onstants. Consider a unit (omponent) with a stohasti lifetime X with df F θ (x) and pdf f θ (x), θ is a parameter (in general, vetor). It is assumed that f θ (x) is ontinuous in x. If one adopts a poliy of replaing the unit either at failure ('orretive replaement' at ost ( X)), or at some speified replaement age ('preventive replaement' at ost (X ) > ), whihever ours first, the ost pet unit is

3 Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 3 ( X ) if X <, ( ) ( X ) if X, and are assumed to be known onstants with > >, a logial requirement to make preventive replaement possibly worthwhile; while the usage per unit is he expeted ost per unit is (5) t( ) min( X, ). (6) Cθ ( ) Eθ { ( )} ( x) fθ + ( x ) fθ + Eθ { X} ( + ) Fθ, (7) while the expeted usage per unit is θ ( ) xfθ + fθ Fθ. (8) he performane index (objetive funtion) of the innovative age replaement model is given by C θ Λ θ θ ( ) + Eθ { X} ( + ) Fθ Fθ + E { X} θ ( + ). Fθ It measures the effetiveness of a replaement poliy over an unlimited time span, beause it represents the long-run average ost per unit of time of utilization. Remark. If if X <, ( ) if X, (9) () then the performane index of the lassial age replaement model takes plae: ( ) F θ Λ θ. Fθ () 4. Optimization of Replaement Poliy Under Complete Information 4.. Optimality Equation o obtain the optimality equation for, the performane index (9) will be differentiated with respet to and set equal to zero to provide the optimal value of as follows: Fθ ( ) Fθ Fθ ( ) Eθ { X} d θ Λ. d () Fθ By requiring this equation to be zero, the desired optimality equation for is obtained as Fθ x Fθ Eθ { X}. (3) his equation represents a neessary ondition for the existene of an optimal solution, If arg F F θ θ Eθ { X}. (4) Eθ { X} Fθ <, (5) it follows from (3) that the optimal solution (4) exists and is unique. o make sure that the optimal solution (4) represents a minimum of (9), not a maximum, the seond derivative of the performane index (9) is alulated and evaluated at * : d Λ θ ( ) [ + Eθ { X}] fθ ( ) > d Fθ (6) whih is positive (i. e., (4) represents a minimum point of (9)), as desired. hus, the following theorem has been proven. heorem. Consider units (omponents) with a stohasti lifetime X with df F θ (x) and pdf f θ (x), θ is a parameter (in general, vetor). It is assumed that f θ (x) is ontinuous in x. hen the optimal preventive replaement age, * C arg min θ Λ θ θ ( )

4 4 Niholas A. Nehval et al.: Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah + Eθ { X} ( + ) Fθ arg min Fθ arg min ( + Eθ { X} ) Fθ ( + ) E { } θ X θ + Λ ( + ) Fθ + ( + ) 46.. exp( / ) 4.3. Numerial Example (Continued) (5) arg F F θ θ Eθ { X}, (7) representing a minimum point of the performane index Λ θ () of the innovative age replaement problem, exists and is unique if and only if E θ {X}<. 4.. Numerial Example Consider units with a stohasti lifetime X with X ~ EXP( θ ( µ, )), (8) x µ Fθ exp, x µ fθ exp, f h θ θ, x > µ, Fθ (9) () () Eθ { X} µ +, var{ X}, () > is a sale parameter and µ is both a loation and a threshold parameter. When µ one gets the well-known one-parameter exponential distribution. he aim is to determine an optimal preventive replaement age *. Let us assume that µ,,,. (3) It follows from (4) that the optimal preventive replaement age is given by arg F F θ θ Eθ { X} arg exp (4) It follows from (9) that the minimal value of the performane index is If µ,,,, (6) then it follows from (4) that the optimal replaement age is given by arg F F θ θ Eθ { X} µ arg exp µ µ (7) It follows from (9) that the minimal value of the performane index is equal to E { } θ X θ + Λ ( + ) Fθ µ µ + + ( + ) µ exp 5. Optimization of Replaement Poliy Under Parametri Unertainty 5.. ale Family of Lifetime Distributions (8) Consider a situation desribed by a sale family of probability distribution funtions, indexed by a parameter, x F Pr( X x) F, < x <, (9) > is a sale parameter, the distribution of Z X / does not depend on any unknown parameter. Let X, X,, X n denote a random sample of omponent lifetimes, eah possessing a distribution funtion (9) suh that F() and

5 Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 5 E {X} is finite. It is assumed that the sale parameter is unknown, but there exists some statisti (X, X,, X n ) suh that, V (3) represents a pivotal quantity whose distribution does not depend on any unknown parameter. In this ase, the expeted ost per unit (7) and the expeted usage per unit (8) are transformed, respetively, as follows: and x x C ( ) + F ( + ) F x x x x + F d ( + ) F d ρv ρv + ( + ) C ( ρ, V ) (3) x x x ( ) F F d ρv ( ρ, V ), (3) ρ (33) is an anillary fator. Using the pivotal quantity averaging approah, one an obtain and C ( ρ) E{ C ( ρ, V )} ρv E ρv + ( + ), (34) ρv ( ρ) E{ ( ρ, V )} E, (35) C ρ Λ ρ ( ρ) ρv E ρv + ( + ) ρv E ρe{ V} + ( + ). ρv E (36) hus, the unknown parameter is eliminated from the problem. Now, an optimal statistial deision (based on pivotal quantity averaging) for a preventive replaement age is determined as ρ, (37) ( ρ ) ρ arg min Λ ρ ρe{ V} + arg min ( + ). ρ ρv E (38) Comparison of statistial deisions. For omparison, a maximum likelihood estimate of is determined as C ˆ arg min ˆ Λ ˆ ( ) + E ˆ { X} ( + ) Fˆ arg min F ˆ + E ˆ { X} arg min ( + ) F ˆ

6 6 Niholas A. Nehval et al.: Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah arg F ˆ ( ) Fˆ E ˆ { X}, (39) ˆ is a maximum likelihood estimate of. In this ase, the index of improvement perentage in effetiveness of a replaement poliy of the pivotal quantity averaging approah as ompared with the maximum likelihood approah is given by Λ ( ρ ) Λ ( ρ ) Iimp.per (, ) %, (4) Λ ( ρ ) ρ. (4) 5.. Loation-ale Family of Lifetime Distributions Consider a situation desribed by a loation-sale family of probability distribution funtions, indexed by a parametri vetor θ ( µ, ), x µ Fθ Pr( X x) F, < x <, (4) < µ < is a loation parameter and > is a sale parameter, the distribution of Z (X µ) / does not depend on any unknown parameters. Let X, X,, X n denote a random sample of omponent lifetimes, eah possessing a distribution funtion (4) suh that E θ {X} is finite. It is assumed that the sale parameter is unknown, but there exists some statisti (X, X,, X n ) (, ) suh that µ V, V (43) represent pivotal quantities with probability distributions that do not depend on any unknown parameter. In this ase, the expeted ost per unit (7) and the expeted usage per unit (8) are transformed, respetively, as follows: x µ x µ Cθ ( ) + F ( + ) F x µ x µ + F d x µ x µ ( + ) F d and ( ρ + / ) V + ρv + V ( + ) C ( ρ, V, V ) (44) x x µ x µ θ ( ) F F d ρv + V ( ρ, V, V ), (45) ρ (46) is an anillary fator. Using the pivotal quantity averaging approah, one an obtain and C ( ρ) E{ C ( ρ, V, V )} E ( ρ + / ) V + ρ V + V ( + ), (47) ρv + V ( ρ) E{ ( ρ, V, V )} E, (48) C ρ Λ ρ ( ρ) ρv + V E ( ρ + / ) V + ( + ) ρv + V E ( ρ + / ) E{ V} + ( + ). ρv + V E (49)

7 Amerian Journal of heoretial and Applied tatistis 6; 5(-): -8 7 hus, the unknown parametri vetor θ is eliminated from the problem. Now, an optimal statistial deision (based on pivotal quantity averaging) for a preventive replaement age is determined as + ρ, (5) ( ρ ) ρ arg min Λ ρ ( ρ + / ) E{ V} + arg min ( + ). ρ ρv + V E (5) Comparison of statistial deisions. For omparison, a maximum likelihood estimate of is determined as C ˆ( ) arg min Λ ˆ( ) θ θ ˆ( ) θ + E ˆ{ X} ( + ) Fˆ θ θ arg min Fˆ θ + E ˆ{ X} arg min θ ( + ) Fˆ θ arg F ˆ( ) Fˆ E ˆ{ X}, θ θ θ (5) ˆ θ is a maximum likelihood estimate of θ. In this ase, the index of improvement perentage in effetiveness of a replaement poliy of the pivotal quantity averaging approah as ompared with the maximum likelihood approah is given by Λ imp.per (, ) ρ Λ I ρ %, (53) Λ ( ρ ) 5.3. Numerial Example ρ. (54) One-parameter exponential distribution of lifetime. Let us assume that a unit's lifetime is represented by a random quantity x X ~ F exp, (55) the parameter is unknown. he aim is to determine an optimal statistial deision for preventive replaement age. Let X, X,, X n be a sample of independent and identially distributed random quantities from (55). It is known that the statisti n Xi (56) i is suffiient for with the probability density funtion n s f s s exp, s, ( n) n > Γ (57) the pivotal quantity V/ has the probability density funtion n f ( v) v exp( v), v >. Γ( n) (58) If n,,,, it follows from (38) that ( ρ ) ρ arg min Λ ρ ρe{ V} + arg min ( + ) ρ ρv E ρn + arg min ( ).368, / ( ) n + ρ + ρ (59) ρ n + Λ ( ρ ) ( ) , / ( ) n + (6) + ρ ρ ; (6) [ ] arg min Λ ˆ

8 8 Niholas A. Nehval et al.: Optimization of tatistial Deisions for Age Replaement Problems via a New Pivotal Quantity Averaging Approah + E ˆ { X} arg min ( + ) , F ˆ (6) ρ n + Λ ( ρ ) ( ) , / ( ) n + (63) + ρ ρ.46. (64) Comparison of statistial deisions. hus, the index of improvement perentage in effetiveness of a replaement poliy of the pivotal quantity averaging approah as ompared with the maximum likelihood approah is given by I imp.per (, ) Λ ( ρ ) Λ ( ρ ) % 3.66%. Λ ( ρ ) 6. Conlusion (65) In this paper, an innovative age replaement model is proposed. It is suitable for situations the osts inluded in the model are not assumed to be onstants. For effetive optimization of statistial deisions for age replaement problems under parametri unertainty, the pivotal quantity averaging () approah is suggested. It represents a new simple and omputationally attrative statistial tehnique based on the onstrutive use of the invariane priniple in mathematial statistis. his tehnique allows one to eliminate unknown parameters from the problem and to find the better deision rules, whih are more effetive than any of the well-known deision rules. he methodology, whih is suggested in this paper, may be found to be useful for the use in many industries with parametri unertainty of underlying models to better manage the information that we already have. While the details of problems onsidered in the paper an hange signifiantly from one industry to the next, the fous is always on making better statistial deisions and preditive inferenes and not manually with guess work and intuition but rather sientifially with models and tehnology, all implemented with disiplined proesses and systems. hus, the paper fouses on the development and implementation of new methodologies and tehnologies that should help one to effetively solve many real pratial problems. Referenes [] R. E. Barlow and L. C. Hunter, Optimal preventive maintenane poliies, Operations Researh, vol. 8, pp. 9, 96. [] R. E. Barlow and F. Proshan. Mathematial heory of Reliability. New York: Wiley, 965. [3] R. E. Barlow and F. Proshan, tatistial heory of Reliability and Life esting: Probability Models. New York: Holt, Rinehart, and Winston, 975. [4] H. Asher and H. Feingold, Repairable ystems Reliability: Modeling, Inferene, Misoneptions and their Causes. New York: Marel Dekker, In., 984. [5]. Nakagawa, ummary of periodi replaement with minimal repair at failure, Journal of the Operations Researh oiety of Japan, vol. 4, pp. 3 8, 98. [6] C. Valdez-Flares and R. M. Feldman, A survey of preventive maintenane models for stohastially deteriorating single-unit systems, Naval Researh Logistis, vol. 36, pp , 989. [7] R. I. Phelps, Optimal poliy for minimal repair, Journal of the Operational Researh oiety, vol. 34, pp , 983. [8] A. ahara and. Nishida, Optimal replaement poliy for minimal repair model, Journal of the Operations Researh oiety of Japan, vol. 8, pp. 3 4, 975. [9] G. J. Glasser, he age replaement problem, ehnometris, vol. 9, pp. 83 9, 967. [] R. L. heaffer, Optimum age replaement poliies with an inreasing ost fator, ehnometris, vol. 3, pp , 97. [] M. B. Berg, A marginal ost analysis for preventive maintenane poliies, European Journal of Operational Researh, vol. 4, pp. 36 4, 98. [] N. A. Nehval and E. K. Vasermanis, Improved Deisions in tatistis. Riga: Izglitibas soli, 4.