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Modeling and Analyzing a Joint Optimization Policy of Block-Replacement and Spare Inventory With Random-Leadtime Runqing Huang, Lingling Meng, Lifeng Xi, and C. Richard Liu
Abstract—We consider a generalized joint optimization policy of block replacement & periodic review spare inventory with random lead time. According to the relationship between geometric area in the graph of inventory level over time, and holding or shortage costs, a model analyzing four mutually exclusive & exhaustive possibilities is developed for the expected average cost per unit time, and is based on the stochastic behavior of the assumed system. The model reflects the cost of inventory holding, spare shortage, replacement, and ordering. And for the first time known to the authors, we deliver the sufficient and necessary conditions of the existence and uniqueness of the minimum in the joint models of this type. Because the model and its analysis are general, one existing result is shown to be subsumed by this model with some modifications. Some numerical cases verify the deduction, and give a general searching solution procedure. Finally, we introduce some discussions related to the models. The models mentioned in the paper can be readily applied in many fields such as economical fields, financial engineering, armament administration, and even medical fields, with some modifications. And the mathematical deduction in the paper will be a guideline for analyzing related stochastic models.
NOTATION pre-arranged PR interval actual cycle length, the time between successive PR maximal inventory level -expected inventory level just after realization of planned PR. the duration where the original inventory level decreases linearly from to 0 without considering the inventory of new delivery spares in the cycle number of -identical, -independent components in the system pdf for component failure time ,
Index Terms—Block replacement, corrective replacement, existence and uniqueness, joint optimization, renewal process, spare provisioning.
CR DV PR
ACRONYM1 corrective replacement decision variable preventive replacement
Manuscript received June 6, 2006; revised August 1, 2007; accepted September 9, 2007. This work was supported in part by the National Natural Science Foundation of China under Grant 50128504. Associate Editor: R. H. Yeh. R. Huang was with the Department of Industrial Engineering & Management, Shanghai Jiao Tong University, Shanghai 200240, China. He is now with the city government of Shanghai, China (e-mail: [email protected]). L. Meng is with the Computer Science & Technology Department and the Department of Educational Information Technology, East China Normal University, Shanghai 200062, China (e-mail: [email protected]). L. Xi is with the Department of Industrial Engineering & Management, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). C. R. Liu is with the School of Industrial Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TR.2008.916887 1The
singular and plural of an acronym are always spelled the same.
,
mean, and standard deviation of component renewal function component renewal density number of component CR during , when the system has only 1 component of a given type number of component CR during , when the system has components of a given type pdf for pre-arranged procurement lead time, delivery lag time of the ordered component, and it may be a positive, negative or zero pdf of ordering time, delivery time, -expected total cost of system maintenance per unit time -expected replacement cost in a -expected inventory cost in a -expected ordering cost in a -expected holding cost in a -expected shortage cost in a cost of component PR, CR set-up cost for placing an order cost of a spare purchased holding cost per spare per unit time downtime cost due to shortage of spares per component per unit time -expected value of any variable
0018-9529/$25.00 © 2008 IEEE
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Most notations in this paper are the same as those in [1]. Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.
I. INTRODUCTION HIS paper is mainly concerned with a joint optimization of block replacement & periodic review spare inventory policy with random lead time. Joint considerations of maintenance replacement policy & inventory have been treated extensively in the literature, e.g., [1]–[10]. However, most of the treatments focus on the age replacement policy [2]–[9], and suppose a one-unit system [2], [5], [6], [8]–[10]. In addition, related analytic deductions are few, while some literatures use numerical analysis or simulation to realize optimization, e.g. [1], [3], [4], [7]. To the best of the authors’ knowledge, related theoretic deductions were done primarily in [5], [6], [8]–[11]. But, those are either concerned with joint optimization of age replacement policy & inventory [5], [6], [8]–[10], or only concerned with the optimization of the maintenance policy [11]. The literature covering theoretic deduction of joint consideration of block replacement & spare inventory are very scarce. Reference [1] developed a stochastic mathematical analytic model to determine the jointly optimal “block replacement” & “periodic review spare-provisioning policy” with constant lead time. However, the condition of the existence & uniqueness of a minimum wasn’t deduced. In addition, the lead time was considered fixed. From a more practical view-point, the actual lead time is usually variable, either later or earlier, so it’s necessary to further research the policy with random lead time. In this paper, through introducing artfully a r.v. which is a delivery lag of the ordered component, we extend the policy and model in [1] deeply, and set up a more general model using the -expected total cost of system maintenance per unit time over an infinite time span as the objective function. More importantly, when we suppose that the maximal inventory level is the only decision variable (DV) of the objective function, then for the first time known to the authors, we deduce successfully the condition of existence & uniqueness of a minimum in the model of joint optimization of block replacement & periodic review spare inventory policy. Our results are concise, and generalize the results in [1] to some extent. Section II describes the policy, and develops the model, including necessary assumptions, and lemma. Section III deduces the condition of existence & uniqueness of a minimum. Section IV verifies the model with some numerical tests. And the final section introduces a special case, the general applicability, and some discussions related the model.
T
II. POLICY AND MODEL A. Assumptions 1) The ordering time is constant, and shorter than the prearranged PR interval T.
2) The procurement lead time is -independent of the quantity of spares being ordered. 3) There is no quantity discount, so the spare cost does not depend on the ordered quantity. 4) The inventory of spares for a given component type is replenished entirely -independently of inventories of spares for other types of system components. 5) The holding cost in a given cycle is proportional to the geometric area with positive inventory in the graph of inventory level over time in a given cycle. 6) The shortage cost in a given cycle is proportional to the geometric area with negative inventory in the inventory graph over time in a given cycle (viz. proportional to the cumulative component downtime due to a shortage of spares during this cycle). 7) During the cycle, the inventory level decreases linearly. Note: Reference [1] has tested the assumption of linear decreasing of the inventory level. The test results shown in its Table 2 show that the presumption of linearity does not appreciably affect the optimal solution. So, based on their test results, we also adopt the same assumption. is the pdf of with . Gen8) is a r.v., and erally, the ordered spares can’t be delivered before the orduring a cycle. So, here we recommend a dering time -uniform distribution
or -triangle distribution
9) 10) 11) 12) 13)
Note: In fact, more general pdf will also have no effect on the final deduction results in our model. So in the following deduction, we often relax the integrating range such to . as The component failure times are i.i.d. The replacement time is negligible. Without loss of generality, only research the cycle that begins at time 0 in the infinite time span. After a PR, the procedure is repeated. All failures are instantly detected & replaced. Note: Some assumptions are the same as those in [1].
B. Lemma 1 According to Leibnitz’s Rule [12], for
where, is a continuous function which depends on the param1) eter , and has a continuous derivative for ; 2) , , , is a constant respectively; , and are also continuous in both , and 3) in some region of the plane including , and ;
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and, for an improper integral,
let
also satisfy the conditions stated above, and converges uniformly in , if , then we have
for . Because the Proof is simple, it is omitted.
Fig. 1. Variation of the inventory level over time (adapted from [1]).
C. Description of Policy and Model
(the amounts of spares that are needed to replenish failed components, but not yet received). Fig. 1 can be viewed as a series of cycles; a “cycle” is defined as the time interval between successive PR.
We consider a system with i.i.d. components, and the component hazard function increases with time. Maintenance of the system is performed mainly according to the block replacement policy. If spares are available, the failed component is replaced immediately upon failure (viz. CR), while all operating components are replaced at prearranged time intervals of length (viz. PR). However, if the ordered spares arrive after the prearranged interval of , PR of the system will have to be postponed, then performed once the ordered spares arrive. , the actual So in our model, when the delivery time equals ; when , the equals cycle length . Before the actual cycle , the CR policy is performed as long as an inventory of spares are available. The downtime of any components due to shortage of spares (time between the failure moment of the component and the arrival moment of the new component ordered) represents a loss of the system operational time. The inventory of spares is replenished at a single location according to the periodic review inventory policy. Under the above policy, although the actual delivery time may not be punctual, it , (see is reasonable to choose the reorder points at assumption 1), and 2)). At each reorder point, the spares of expected CR, and PR numbers in one expected cycle are ordered to replenish the failed components. Considering the random lead time, we define a variable that denotes the delivery lag time of components ordered. It is cento . tered at the prearranged time , and varies from When the delivery time is punctual, it equals 0. So
D. Model Development Based on the above description, we choose the expected cost rate for an infinite time span as criteria of optimality where the . DV is the maximal inventory level According to the famous renewal reward theorem (see [13]), the expected cost rate for an infinite time span is the expected cost per cycle divided by the expected cycle length. Because the time between successive PR is a cycle, (2) Hence, the expected cycle length is
(3)
(1)
The expected cost per cycle is the sum of the replacement, ordering, holding, and shortage costs.
Demand for spares during a time cycle interval, and consequently the variation of the level of spares in the inventory, depends on the number of component CR & PR in this interval [1]. Fig. 1 illustrates the variation of the inventory level over time. The heavy line represents the inventory level; the dashed curve under the axis represents the “negative inventory” level
(4) Next, we deduce the four terms in the r.h.s. of (4) one by one.
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1) Expected Replacement Cost in One Cycle: When the system under consideration is relatively large2, the could be approximated by the -normal pdf with mean , and standard deviation
of
The of component CR as a function of can be described therefore equals the value by the ordinary renewal process. of in time . In addition [1], (5) Because the spare numbers being ordered in every cycle equal the numbers of the spares of expected CR, and PR numbers in one expected cycle, the number of ordered spares is
where equals the number of spares for PR, and equals the number of spares for CR in one expected cycle. We can see, is also defined. once is defined, the Hence, the expected replacement cost in one cycle is
Fig. 2. Sub-case 1: the ordered spares arrive before the pre-arranged PR time, and the original inventory alone would not be enough to supply spares for CR during the cycle.
(6) 2) The Expected Ordering Cost in One Cycle:
(7) 3) The Expected Inventory Holding Cost in One Cycle: Four mutually exclusive & exhaustive possibilities may exist in every cycle (see Figs. 2–6): i. When the new spares ordered arrive before the pre-arranged PR interval, the original inventory is still enough in one cycle. Here, the ordered spares arrive before . ii. When the new spares ordered arrive before the pre-arranged PR interval, the original inventory is not enough to supply the components for CR in one cycle. Here, the ordered spares arrive before , but after . iii. The new spares ordered arrive after the pre-arranged PR interval, but the original inventory is enough to supply the components for CR in one cycle. Here, the ordered spares arrive after , but before . iv. The new spares ordered arrive after the pre-arranged PR interval, but the original inventory is not enough to supply the components for CR in one cycle. Here, the ordered spares arrive after , and . According to assumption 5, the holding cost in a given cycle is proportional to the geometric area with positive inventory in 2For
complex industrial systems, this statement is usually true.
Fig. 3. Sub-case 2: the ordered spares arrive before the pre-arranged PR time, and the original inventory alone would be enough to supply spares for CR during the cycle.
the inventory graph over time in a given cycle. The geometric area with positive inventory practically equals the product of the average positive inventory level (the average number of spares physically located in the inventory during the cycle), and the entire given cycle time. Then, when we research the expected inventory holding cost in one cycle, we only need to focus on the geometric area with positive inventory in the inventory level-time figure. Based on the above possibilities, the detailed deduction is as follows. , viz. i. In fact, this case includes two possibilities (see Figs. 2 and 3), where the original inventory exists simultaneity with new arriving spares being ordered at some time in one cycle. Sub-case 1 If the new spares being ordered did not arrive, the original inventory alone would not be enough to supply spares for CR during one cycle.
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Fig. 5. The ordered spares arrive after the prearranged PR time, and the original inventory always is enough for CR during the cycle.
Fig. 4. When the ordered spares arrive before the prearranged PR time, the original inventory already has been consumed out.
Sub-case 2 Even if the new spares being ordered did not arrive, the original inventory alone would be enough to supply spares for CR during one cycle is the duration in which the original spare In sub-case 1, to 0 supposing that the inventories decrease linearly from inventory of new delivery spares in the cycle was not considered. . Then, using simple trigonometry, we have Here, the system has no spare shortage, but has extra holding inventory. Hence, when Fig. 6. The ordered spares arrive after the prearranged PR time and the original inventory is not enough for CR during the cycle.
according to the assumption 7, DCEF is a parallelogram, and the geometric area with positive inventory in Fig. 2 equals , in which
The geometric area with positive inventory in the cycle is
(9) Hence, the two sub-cases can be incorporated into one case. When Then, the geometric area with positive inventory during the cycle is
(8) In sub-case 2
(10) holds. , viz. (see Fig. 4). ii. are both before the prearHere, the delivery time , and ranged PR time, and the delivery time is after . At the B point , the new delivery spares replenish the failed components at once. According to assumption 10, the replacement time is negligible. Then, the remaining inventory spares will still be consumed linearly until the PR time point. Using Fig. 4, at the B point, the spares in shortage are equal , . After these spares to the length of are decreased, the rest have been kept in the warehouse, and supplied for CR.
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BCHG is a trapezoid. And the geometric area with positive , in which inventory in Fig. 4 is equal to
4) The Expected Shortage Cost in One Cycle: According to assumption 6, similarly as the analysis of holding cost, the shortage cost in a given cycle is proportional to the geometric area with negative inventory in the inventory graph over time in a given cycle. Based on the above four possibilities, we can easily find that shortage only occurs in case (ii), and case (iv): A) For the case (ii) (see Fig. 4): Because spares for CR exceed the original inventory greatly, although the ordered spares arrive in advance, shortage still occurs before the delivery time. Then, when
Hence, when
we obtain the geometric area with positive inventory during the cycle is (11) iii. (see Fig. 5) When ventory in the cycle is
(15) the geometric area with positive inB) For the case (iv) (see Fig. 6): When (12)
iv. (see Fig. 6). is before In fact, this case also includes two possibilities: is after the prearranged PR the prearranged PR time ; and time , and before the delivery time . However, the two subcases also can be incorporated into one case. the geometric area with positive inventory When in the cycle is
(16) So, allowing for both possible variants of a cycle, the a cycle of length can be expressed as
in
(13) Considering the above four mutually exclusive & exhaustive possibilities in a cycle, in order to acquire the , we suppose are independent. So, the “expected value of the rv of , and a function of two jointly distributed random variables” [14], can be described as
(17) According to (4), the is constituent of the costs calculated by (6), (7), (14), and (17), respectively. Finally, the objective function of the model can be expressed as (18) shown at the bottom of the next page. And because the maximal inventory level could appear only at the ordering time point when the ordered spares arrive without , so we have any delay,
(14) (19)
HUANG et al.: JOINT OPTIMIZATION POLICY OF BLOCK-REPLACEMENT AND SPARE INVENTORY
Note: Because
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. , we have
In addition, from a practical view, the pre-arranged PR interval should not be less than the pre-arranged procurement lead time. And, because a component usually fails at or near time , beyond has no practical meaning. So should be further restricted to
III. MODEL ANALYSIS Define
(20) The monotonicity, and extremum of
depend on the
.
(18)
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Let the integrands, and upper and lower limit functions of satisfy the conditions stated as Lemma 1 respecthe r.h.s. in tively. Observe the first two terms of the r.h.s. in (20). According to Lemma 1, we have
Substitute
into
and respectively, then holds. Hence, the derivative of the first two terms in the r.h.s. of with respect to is
(22) Every term in the r.h.s. of (22) is grater than or equal to 0. Therefore, (23)
The derivatives of the remaining terms in the r.h.s. of can be similarly obtained. respect to
with
Finally, we have Theorem 1 as follows. Theorem 1: For the objective function & only one making minimized iif
, there exists one
Proof of theorem 1: Once other parameters are given, the , and are a simple kind of linear relation (19). So
. Hence, replacing tion is reasonable. First, proof of sufficiency: Define
with
in related deduc-
(24)
(21)
is a strict monotone increasing funcAccording to (23), . tion for Then, we can prove that the sufficient & necessary condition is of
Next, (25)
Please refer to the Appendix proof. Next, we can prove (26)
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Because
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strictly increasing function, hence Appendix proof, we have
. According to the
IV. MODEL TESTING can be given, and submitted into (19) one by one. Then the corresponding from (19) is submitted into (18). And the can be identified. Generally, only a numerical minimum of solution can be obtained. After obtaining the optimal value, the is also identified. optimal maximal inventory level A. Model Parameters In the model testing, we use the model parameters in [1]. They are reviewed as follows: ; is -normal, and the value of parameters of are weeks, weeks; weeks; units; units; units; units; units/week; units/week; In addition, we suppose that is a -uniform distribution as in assumption 8, and weeks. Note: In the computation, our model is tested with “Mathematica” software. Mathematica works well with the improper integral (eg. It uses a well-defined to represent the “infinity” in numerical calculation). The detailed procedure refers to related references. This paper doesn’t give unnecessary details. B. Model Output, and Analysis
so
holds. Then, when
,
hold According to the Intermediate Value Theorem, and , the solution of the monotonicity of exists, and is unique. Considering (23), there exists one (also, there exists one unique unique ), minimized. making Next, proof of necessity: , making When there exists one unique minimized, holds. And because is a
First, test the above data with Theorem 1. We can easily find holds. In addition, considering that , the closest, is 128. Then, for the objective function , there smallest , making minimized. exists one unique Fig. 7 is the variation of the derivative of with respect to . The derivative always increases, and passes the axis of the abscissa. When the derivative is smaller than 0, the variation is great; while is beyond 146, the derivative apover proaches a constant of about 0.59. When the maximal inventory with respect to is closest level is 138, the derivative of , should have a minimal to zero. Then, when value. From Fig. 8, the minimum position ( , and ) can be easily found. , varies from 9428.83 units/week For to 777.148 units/week. Because the downtime cost due to a shortage of spares per component per unit time is much higher than the cost of holding one spare in the inventory per unit time, we can see that a moderate overstocking would not appreciably
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Fig. 9. Variation of C
Fig. 7. Variation of the derivative of C
with respect to S
over S
T
(
(
T
= 12).
.
Fig. 10. Variation of C
Fig. 8. Variation of C
over S
= 23).
affect the total cost of system maintenance form Fig. 8. This reasoning is confirmed by the results of our model testing, and is also in good agreement with results in [1]. is 138, the So, when the DV of maximal inventory level -expected total system maintenance cost per unit time can be minimized. The numerical analysis is accordant well with the theoretical analysis, and validates the theoretical deduction about the conditions of existence & uniqueness of the minimum. In addition, it also gives a general procedure on finding solutions to the model.
over S
(
T
= 44).
about the conditions of existence & uniqueness of the minimum once more. , there Through a repeating procedure, for every . Then, the global is a corresponding unique & local minimal minimum of is 717.51 units/week at , and . See Fig. 11.
V. MODEL APPLICABILITY, AND DISCUSSION A. Special Model Case When is fixed at 0, we have is transformed to the model in [1]:
, so our model
C. Model Extension Analysis Further, if both and are regarded as DV as in [1], the global minimum can also be easily found. The minimum of with respect to is the every “local unique minimum” of global minimum. , and Two extreme cases are researched: one is . See Figs. 9 and 10. Both of them another is have a unique minimum. Their minimums are 876.11 units/ , and 1597.29 units/week at week at respectively. We have validated that other have the similar behavior. The numerical results are accordant well with the theoretical analysis, and validate well the theoretical deduction
(27)
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pany (as a technician “warehouse”), and/or preventive replacement cycle can be jointly, and readily solved. C. Discussion
Fig. 11. Variation of local minimal C
over T .
Here, according to Theorem 1,
Hence, for any
In the paper, convenient for analytic deductions, we suppose is the only DV of the objecthat maximal inventory level tive function, and let the pre-arranged PR interval be fixed. and are both DV Now, if we extend it, and suppose that as [1], it is an open question on how to decide the conditions about optimum through analytic deductions. We think an analytical approach will be a difficult, even an unfeasible one. In fact, even though we only research the special case (27), as both DV of as [1] did, we and regard the and encounter some difficulties; although the conditions of the existence & uniqueness of the joint optimization minimum can be preliminarily deduced (where [15] is partly referred to) [16]. Generally, joint optimization of two or more activities in an analytical way means a very complex behavior. We only regard our paper as a preliminary research on joint optimization of several activities. And this paper intends to start further discussion on this question.
, it has only one unique , making the
in [1] minimized. Further, the is relaxed to a DV as [1], and the global minimum can be easily found just to find the minimum of every with respect to . “local minimum” of
APPENDIX I Proof of the Conditions of
B. Model’s General Applicability Field The models mentioned can be readily applied to optimize maintenance procedures for a variety of industrial systems, and also to update the maintenance policy in situations where block replacement preventive maintenance is already in use. This application has been verified by [1] using field data on electric locomotives in Slovenian Railways. The theoretical deduction in this paper can help engineers decide whether the minimum of the model exists, and what defines the minimum of the model. It is a guideline to help engineers in practical applications of models. In addition, similar model extensions are numerous, as long as the system has similar behavior characters of both “block replacement,” and “inventory”. So, the models mentioned can be readily extended to such areas as financial engineering (e.g. block renewal of currency in circulation), armament administration (e.g. ammunition block renewal), and even medical fields after some appropriate changes. For example, consider the joint optimization of the preventive replacement cycle length, and the number of maintenance technicians in a certain maintenance service company. Here, the original maintenance policy depicted in [1] stays unchanged; the spare inventory are substituted by the workload of technicians; and the relevant original costs in [1] are substituted by corrective replacement costs, preventive replacement costs, wages, call-back pay fee of technicians, and the downtime costs caused by a shortage of technicians, respectively. Then, through models similar to (18) or (27) in paper, the optimal number of technicians in the maintenance service com-
According to assumption 8,
then,
:
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where
ACKNOWLEDGMENT The authors thank the reviewers, and the Editor, Dr. Ruey Huei Yeh, Dr. Jason Rupe, and Dr. Edward A. Pohl for their comments and advice which have greatly improved the content, and presentation of this paper. REFERENCES [1] A. Brezavscek and A. Hudoklin, “Joint optimization of block-replacement and periodic-review spare-provisioning policy,” IEEE Trans. Reliability, vol. 52, pp. 112–117, Mar. 2003. [2] W. P. Liao and J. Yuan, “Optimal replacement for a one-unit system subject to delivery and test,” Journal of Quality in Maintenance Engineering, vol. 4, no. 1, pp. 51–65, 1998. [3] A. B. M. Zohrul Kabir and S. H. A. Farrash, “Simulation of an integrated age replacement and spare provisioning policy using SLAM,” Reliability Engineering & System Safety, vol. 52, no. 2, pp. 129–138, 1996. [4] A. B. M. Zohrul Kabir and A. S. Al-Olayan, “Stocking policy for spare part provisioning under age based preventive replacement,” European Journal of Operational Research, vol. 90, no. 1, pp. 171–181, 1996. [5] M. J. Armstrong and D. R. Atkins, “Joint optimization of maintenance and inventory policies for a simple system,” IIE Trans. (Institute of Industrial Engineers), vol. 28, no. 5, pp. 415–424, 1996. [6] M. J. Armstrong and D. A. Atkins, “Note on joint optimization of maintenance and inventory,” IIE Trans. (Institute of Industrial Engineers), vol. 30, no. 2, pp. 143–149, 1998. [7] A. B. M. Kabir and S. H. A. Farrash, “Fixed interval ordering policy for joint optimization of age replacement and spare part provisioning,” International Journal of Systems Science, vol. 28, no. 12, pp. 1299–1309, 1997. [8] S. H. Sheu and Y. H. Chien, “Optimal age-replacement policy of a system subject to shocks with random lead-time,” European Journal of Operational Research, vol. 159, no. 1, pp. 132–144, 2004. [9] S. H. Sheu and W. S. Griffith, “Optimal age-replacement policy with age-dependent minimal-repair and random-leadtime,” IEEE Trans. Reliability, vol. 50, no. 3, pp. 302–309, 2001. [10] Y. T. Park and K. S. Park, “Generalized spare ordering policies with random lead time,” European Journal of Operational Research, vol. 23, no. 3, pp. 320–330, Mar. 1986.
[11] M. Berg, “A marginal cost analysis for preventive replacement policies,” European Journal of Operational Research, vol. 4, no. 2, pp. 136–142, Feb. 1980. [12] M. R. Spiegel, Schaum’s Outline of Theory and Problems of Advanced Calculus. New York: McGraw-Hill, 1963. [13] S. M. Ross, Applied Probability Models With Optimization Applications: Holden-Day. : , 1970. [14] I. Olkin, L. J. Gleser, and C. Derman, Probability Models and Applications. New York: Macmillan Publishing Co. Inc., 1980. [15] E. Smeitink and R. Dekker, “A simple approximation to the renewal function,” IEEE Trans. Reliability, vol. 39, pp. 71–75, Apr. 1990. [16] R. Huang, “Research on Key Challenges for the Joint Optimization of Maintenance and Spare Parts Inventory,” (in Chinese) Doctoral dissertation, Shanghai Jiao Tong University, , 2006.4. Runqing Huang has a PhD from the Department of Industrial Engineering & Management, Shanghai Jiao Tong University, Shanghai 200240, P. R. China. Now, he works as a civil servant of Shanghai City. Dr. Huang got his Master degree, and Bachelor degree from Shanghai Jiao Tong University, and Xi’an Jiao Tong University respectively. He has worked as an Assistant Director of the Management Committee at Shanghai Nanhui Industrial Zone, and an Assistant Researcher of Shanghai Quality Management Academy, P. R. China. His present research interests include modeling strategic decisions, inventory theory and modeling, mass customization, and Logistics.
Lingling Meng is a PhD Candidate at the Computer Science & Technology Department, and a teacher with the Department of Educational Information Technology, East China Normal University, Shanghai 200062, P. R. China. Ms. Meng’s present research interests include metadata theory and application, data mining, information retrieval, and computer assisted instruction,
Lifeng Xi is a Professor at the Department of Industrial Engineering & Management, and vice dean of the School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China. Prof. Xi’s research interests include quality theory and management, maintenance systems and applications, and modern manufacturing theory and technology.
C. Richard Liu is a Professor of Industrial Engineering at Purdue University, West Lafayette, USA. He was elected Fellow of ASME in 1992, and is an internationally recognized expert in the area of design and manufacturing engineering. Prof. Liu’s research interests include process mechanics, and its relationship to the reliability of the component produced; precision and dynamics of machine tools; CAD/CAM/CIM, and Internet-based Manufacturing; and computable representation, classification, analysis, and integration of design concept, geometry and manufacturability. He has also contributed to industrial product and process research resulting in significant commercial impacts.
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Modeling and Analyzing a Joint Optimization Policy of Block-Replacement and Spare Inventory With Random-Leadtime Runqing Huang, Lingling Meng, Lifeng Xi, and C. Richard Liu
Abstract—We consider a generalized joint optimization policy of block replacement & periodic review spare inventory with random lead time. According to the relationship between geometric area in the graph of inventory level over time, and holding or shortage costs, a model analyzing four mutually exclusive & exhaustive possibilities is developed for the expected average cost per unit time, and is based on the stochastic behavior of the assumed system. The model reflects the cost of inventory holding, spare shortage, replacement, and ordering. And for the first time known to the authors, we deliver the sufficient and necessary conditions of the existence and uniqueness of the minimum in the joint models of this type. Because the model and its analysis are general, one existing result is shown to be subsumed by this model with some modifications. Some numerical cases verify the deduction, and give a general searching solution procedure. Finally, we introduce some discussions related to the models. The models mentioned in the paper can be readily applied in many fields such as economical fields, financial engineering, armament administration, and even medical fields, with some modifications. And the mathematical deduction in the paper will be a guideline for analyzing related stochastic models.
NOTATION pre-arranged PR interval actual cycle length, the time between successive PR maximal inventory level -expected inventory level just after realization of planned PR. the duration where the original inventory level decreases linearly from to 0 without considering the inventory of new delivery spares in the cycle number of -identical, -independent components in the system pdf for component failure time ,
Index Terms—Block replacement, corrective replacement, existence and uniqueness, joint optimization, renewal process, spare provisioning.
CR DV PR
ACRONYM1 corrective replacement decision variable preventive replacement
Manuscript received June 6, 2006; revised August 1, 2007; accepted September 9, 2007. This work was supported in part by the National Natural Science Foundation of China under Grant 50128504. Associate Editor: R. H. Yeh. R. Huang was with the Department of Industrial Engineering & Management, Shanghai Jiao Tong University, Shanghai 200240, China. He is now with the city government of Shanghai, China (e-mail: [email protected]). L. Meng is with the Computer Science & Technology Department and the Department of Educational Information Technology, East China Normal University, Shanghai 200062, China (e-mail: [email protected]). L. Xi is with the Department of Industrial Engineering & Management, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). C. R. Liu is with the School of Industrial Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TR.2008.916887 1The
singular and plural of an acronym are always spelled the same.
,
mean, and standard deviation of component renewal function component renewal density number of component CR during , when the system has only 1 component of a given type number of component CR during , when the system has components of a given type pdf for pre-arranged procurement lead time, delivery lag time of the ordered component, and it may be a positive, negative or zero pdf of ordering time, delivery time, -expected total cost of system maintenance per unit time -expected replacement cost in a -expected inventory cost in a -expected ordering cost in a -expected holding cost in a -expected shortage cost in a cost of component PR, CR set-up cost for placing an order cost of a spare purchased holding cost per spare per unit time downtime cost due to shortage of spares per component per unit time -expected value of any variable
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Most notations in this paper are the same as those in [1]. Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.
I. INTRODUCTION HIS paper is mainly concerned with a joint optimization of block replacement & periodic review spare inventory policy with random lead time. Joint considerations of maintenance replacement policy & inventory have been treated extensively in the literature, e.g., [1]–[10]. However, most of the treatments focus on the age replacement policy [2]–[9], and suppose a one-unit system [2], [5], [6], [8]–[10]. In addition, related analytic deductions are few, while some literatures use numerical analysis or simulation to realize optimization, e.g. [1], [3], [4], [7]. To the best of the authors’ knowledge, related theoretic deductions were done primarily in [5], [6], [8]–[11]. But, those are either concerned with joint optimization of age replacement policy & inventory [5], [6], [8]–[10], or only concerned with the optimization of the maintenance policy [11]. The literature covering theoretic deduction of joint consideration of block replacement & spare inventory are very scarce. Reference [1] developed a stochastic mathematical analytic model to determine the jointly optimal “block replacement” & “periodic review spare-provisioning policy” with constant lead time. However, the condition of the existence & uniqueness of a minimum wasn’t deduced. In addition, the lead time was considered fixed. From a more practical view-point, the actual lead time is usually variable, either later or earlier, so it’s necessary to further research the policy with random lead time. In this paper, through introducing artfully a r.v. which is a delivery lag of the ordered component, we extend the policy and model in [1] deeply, and set up a more general model using the -expected total cost of system maintenance per unit time over an infinite time span as the objective function. More importantly, when we suppose that the maximal inventory level is the only decision variable (DV) of the objective function, then for the first time known to the authors, we deduce successfully the condition of existence & uniqueness of a minimum in the model of joint optimization of block replacement & periodic review spare inventory policy. Our results are concise, and generalize the results in [1] to some extent. Section II describes the policy, and develops the model, including necessary assumptions, and lemma. Section III deduces the condition of existence & uniqueness of a minimum. Section IV verifies the model with some numerical tests. And the final section introduces a special case, the general applicability, and some discussions related the model.
T
II. POLICY AND MODEL A. Assumptions 1) The ordering time is constant, and shorter than the prearranged PR interval T.
2) The procurement lead time is -independent of the quantity of spares being ordered. 3) There is no quantity discount, so the spare cost does not depend on the ordered quantity. 4) The inventory of spares for a given component type is replenished entirely -independently of inventories of spares for other types of system components. 5) The holding cost in a given cycle is proportional to the geometric area with positive inventory in the graph of inventory level over time in a given cycle. 6) The shortage cost in a given cycle is proportional to the geometric area with negative inventory in the inventory graph over time in a given cycle (viz. proportional to the cumulative component downtime due to a shortage of spares during this cycle). 7) During the cycle, the inventory level decreases linearly. Note: Reference [1] has tested the assumption of linear decreasing of the inventory level. The test results shown in its Table 2 show that the presumption of linearity does not appreciably affect the optimal solution. So, based on their test results, we also adopt the same assumption. is the pdf of with . Gen8) is a r.v., and erally, the ordered spares can’t be delivered before the orduring a cycle. So, here we recommend a dering time -uniform distribution
or -triangle distribution
9) 10) 11) 12) 13)
Note: In fact, more general pdf will also have no effect on the final deduction results in our model. So in the following deduction, we often relax the integrating range such to . as The component failure times are i.i.d. The replacement time is negligible. Without loss of generality, only research the cycle that begins at time 0 in the infinite time span. After a PR, the procedure is repeated. All failures are instantly detected & replaced. Note: Some assumptions are the same as those in [1].
B. Lemma 1 According to Leibnitz’s Rule [12], for
where, is a continuous function which depends on the param1) eter , and has a continuous derivative for ; 2) , , , is a constant respectively; , and are also continuous in both , and 3) in some region of the plane including , and ;
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and, for an improper integral,
let
also satisfy the conditions stated above, and converges uniformly in , if , then we have
for . Because the Proof is simple, it is omitted.
Fig. 1. Variation of the inventory level over time (adapted from [1]).
C. Description of Policy and Model
(the amounts of spares that are needed to replenish failed components, but not yet received). Fig. 1 can be viewed as a series of cycles; a “cycle” is defined as the time interval between successive PR.
We consider a system with i.i.d. components, and the component hazard function increases with time. Maintenance of the system is performed mainly according to the block replacement policy. If spares are available, the failed component is replaced immediately upon failure (viz. CR), while all operating components are replaced at prearranged time intervals of length (viz. PR). However, if the ordered spares arrive after the prearranged interval of , PR of the system will have to be postponed, then performed once the ordered spares arrive. , the actual So in our model, when the delivery time equals ; when , the equals cycle length . Before the actual cycle , the CR policy is performed as long as an inventory of spares are available. The downtime of any components due to shortage of spares (time between the failure moment of the component and the arrival moment of the new component ordered) represents a loss of the system operational time. The inventory of spares is replenished at a single location according to the periodic review inventory policy. Under the above policy, although the actual delivery time may not be punctual, it , (see is reasonable to choose the reorder points at assumption 1), and 2)). At each reorder point, the spares of expected CR, and PR numbers in one expected cycle are ordered to replenish the failed components. Considering the random lead time, we define a variable that denotes the delivery lag time of components ordered. It is cento . tered at the prearranged time , and varies from When the delivery time is punctual, it equals 0. So
D. Model Development Based on the above description, we choose the expected cost rate for an infinite time span as criteria of optimality where the . DV is the maximal inventory level According to the famous renewal reward theorem (see [13]), the expected cost rate for an infinite time span is the expected cost per cycle divided by the expected cycle length. Because the time between successive PR is a cycle, (2) Hence, the expected cycle length is
(3)
(1)
The expected cost per cycle is the sum of the replacement, ordering, holding, and shortage costs.
Demand for spares during a time cycle interval, and consequently the variation of the level of spares in the inventory, depends on the number of component CR & PR in this interval [1]. Fig. 1 illustrates the variation of the inventory level over time. The heavy line represents the inventory level; the dashed curve under the axis represents the “negative inventory” level
(4) Next, we deduce the four terms in the r.h.s. of (4) one by one.
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1) Expected Replacement Cost in One Cycle: When the system under consideration is relatively large2, the could be approximated by the -normal pdf with mean , and standard deviation
of
The of component CR as a function of can be described therefore equals the value by the ordinary renewal process. of in time . In addition [1], (5) Because the spare numbers being ordered in every cycle equal the numbers of the spares of expected CR, and PR numbers in one expected cycle, the number of ordered spares is
where equals the number of spares for PR, and equals the number of spares for CR in one expected cycle. We can see, is also defined. once is defined, the Hence, the expected replacement cost in one cycle is
Fig. 2. Sub-case 1: the ordered spares arrive before the pre-arranged PR time, and the original inventory alone would not be enough to supply spares for CR during the cycle.
(6) 2) The Expected Ordering Cost in One Cycle:
(7) 3) The Expected Inventory Holding Cost in One Cycle: Four mutually exclusive & exhaustive possibilities may exist in every cycle (see Figs. 2–6): i. When the new spares ordered arrive before the pre-arranged PR interval, the original inventory is still enough in one cycle. Here, the ordered spares arrive before . ii. When the new spares ordered arrive before the pre-arranged PR interval, the original inventory is not enough to supply the components for CR in one cycle. Here, the ordered spares arrive before , but after . iii. The new spares ordered arrive after the pre-arranged PR interval, but the original inventory is enough to supply the components for CR in one cycle. Here, the ordered spares arrive after , but before . iv. The new spares ordered arrive after the pre-arranged PR interval, but the original inventory is not enough to supply the components for CR in one cycle. Here, the ordered spares arrive after , and . According to assumption 5, the holding cost in a given cycle is proportional to the geometric area with positive inventory in 2For
complex industrial systems, this statement is usually true.
Fig. 3. Sub-case 2: the ordered spares arrive before the pre-arranged PR time, and the original inventory alone would be enough to supply spares for CR during the cycle.
the inventory graph over time in a given cycle. The geometric area with positive inventory practically equals the product of the average positive inventory level (the average number of spares physically located in the inventory during the cycle), and the entire given cycle time. Then, when we research the expected inventory holding cost in one cycle, we only need to focus on the geometric area with positive inventory in the inventory level-time figure. Based on the above possibilities, the detailed deduction is as follows. , viz. i. In fact, this case includes two possibilities (see Figs. 2 and 3), where the original inventory exists simultaneity with new arriving spares being ordered at some time in one cycle. Sub-case 1 If the new spares being ordered did not arrive, the original inventory alone would not be enough to supply spares for CR during one cycle.
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Fig. 5. The ordered spares arrive after the prearranged PR time, and the original inventory always is enough for CR during the cycle.
Fig. 4. When the ordered spares arrive before the prearranged PR time, the original inventory already has been consumed out.
Sub-case 2 Even if the new spares being ordered did not arrive, the original inventory alone would be enough to supply spares for CR during one cycle is the duration in which the original spare In sub-case 1, to 0 supposing that the inventories decrease linearly from inventory of new delivery spares in the cycle was not considered. . Then, using simple trigonometry, we have Here, the system has no spare shortage, but has extra holding inventory. Hence, when Fig. 6. The ordered spares arrive after the prearranged PR time and the original inventory is not enough for CR during the cycle.
according to the assumption 7, DCEF is a parallelogram, and the geometric area with positive inventory in Fig. 2 equals , in which
The geometric area with positive inventory in the cycle is
(9) Hence, the two sub-cases can be incorporated into one case. When Then, the geometric area with positive inventory during the cycle is
(8) In sub-case 2
(10) holds. , viz. (see Fig. 4). ii. are both before the prearHere, the delivery time , and ranged PR time, and the delivery time is after . At the B point , the new delivery spares replenish the failed components at once. According to assumption 10, the replacement time is negligible. Then, the remaining inventory spares will still be consumed linearly until the PR time point. Using Fig. 4, at the B point, the spares in shortage are equal , . After these spares to the length of are decreased, the rest have been kept in the warehouse, and supplied for CR.
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BCHG is a trapezoid. And the geometric area with positive , in which inventory in Fig. 4 is equal to
4) The Expected Shortage Cost in One Cycle: According to assumption 6, similarly as the analysis of holding cost, the shortage cost in a given cycle is proportional to the geometric area with negative inventory in the inventory graph over time in a given cycle. Based on the above four possibilities, we can easily find that shortage only occurs in case (ii), and case (iv): A) For the case (ii) (see Fig. 4): Because spares for CR exceed the original inventory greatly, although the ordered spares arrive in advance, shortage still occurs before the delivery time. Then, when
Hence, when
we obtain the geometric area with positive inventory during the cycle is (11) iii. (see Fig. 5) When ventory in the cycle is
(15) the geometric area with positive inB) For the case (iv) (see Fig. 6): When (12)
iv. (see Fig. 6). is before In fact, this case also includes two possibilities: is after the prearranged PR the prearranged PR time ; and time , and before the delivery time . However, the two subcases also can be incorporated into one case. the geometric area with positive inventory When in the cycle is
(16) So, allowing for both possible variants of a cycle, the a cycle of length can be expressed as
in
(13) Considering the above four mutually exclusive & exhaustive possibilities in a cycle, in order to acquire the , we suppose are independent. So, the “expected value of the rv of , and a function of two jointly distributed random variables” [14], can be described as
(17) According to (4), the is constituent of the costs calculated by (6), (7), (14), and (17), respectively. Finally, the objective function of the model can be expressed as (18) shown at the bottom of the next page. And because the maximal inventory level could appear only at the ordering time point when the ordered spares arrive without , so we have any delay,
(14) (19)
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Note: Because
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. , we have
In addition, from a practical view, the pre-arranged PR interval should not be less than the pre-arranged procurement lead time. And, because a component usually fails at or near time , beyond has no practical meaning. So should be further restricted to
III. MODEL ANALYSIS Define
(20) The monotonicity, and extremum of
depend on the
.
(18)
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Let the integrands, and upper and lower limit functions of satisfy the conditions stated as Lemma 1 respecthe r.h.s. in tively. Observe the first two terms of the r.h.s. in (20). According to Lemma 1, we have
Substitute
into
and respectively, then holds. Hence, the derivative of the first two terms in the r.h.s. of with respect to is
(22) Every term in the r.h.s. of (22) is grater than or equal to 0. Therefore, (23)
The derivatives of the remaining terms in the r.h.s. of can be similarly obtained. respect to
with
Finally, we have Theorem 1 as follows. Theorem 1: For the objective function & only one making minimized iif
, there exists one
Proof of theorem 1: Once other parameters are given, the , and are a simple kind of linear relation (19). So
. Hence, replacing tion is reasonable. First, proof of sufficiency: Define
with
in related deduc-
(24)
(21)
is a strict monotone increasing funcAccording to (23), . tion for Then, we can prove that the sufficient & necessary condition is of
Next, (25)
Please refer to the Appendix proof. Next, we can prove (26)
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Because
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strictly increasing function, hence Appendix proof, we have
. According to the
IV. MODEL TESTING can be given, and submitted into (19) one by one. Then the corresponding from (19) is submitted into (18). And the can be identified. Generally, only a numerical minimum of solution can be obtained. After obtaining the optimal value, the is also identified. optimal maximal inventory level A. Model Parameters In the model testing, we use the model parameters in [1]. They are reviewed as follows: ; is -normal, and the value of parameters of are weeks, weeks; weeks; units; units; units; units; units/week; units/week; In addition, we suppose that is a -uniform distribution as in assumption 8, and weeks. Note: In the computation, our model is tested with “Mathematica” software. Mathematica works well with the improper integral (eg. It uses a well-defined to represent the “infinity” in numerical calculation). The detailed procedure refers to related references. This paper doesn’t give unnecessary details. B. Model Output, and Analysis
so
holds. Then, when
,
hold According to the Intermediate Value Theorem, and , the solution of the monotonicity of exists, and is unique. Considering (23), there exists one (also, there exists one unique unique ), minimized. making Next, proof of necessity: , making When there exists one unique minimized, holds. And because is a
First, test the above data with Theorem 1. We can easily find holds. In addition, considering that , the closest, is 128. Then, for the objective function , there smallest , making minimized. exists one unique Fig. 7 is the variation of the derivative of with respect to . The derivative always increases, and passes the axis of the abscissa. When the derivative is smaller than 0, the variation is great; while is beyond 146, the derivative apover proaches a constant of about 0.59. When the maximal inventory with respect to is closest level is 138, the derivative of , should have a minimal to zero. Then, when value. From Fig. 8, the minimum position ( , and ) can be easily found. , varies from 9428.83 units/week For to 777.148 units/week. Because the downtime cost due to a shortage of spares per component per unit time is much higher than the cost of holding one spare in the inventory per unit time, we can see that a moderate overstocking would not appreciably
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Fig. 9. Variation of C
Fig. 7. Variation of the derivative of C
with respect to S
over S
T
(
(
T
= 12).
.
Fig. 10. Variation of C
Fig. 8. Variation of C
over S
= 23).
affect the total cost of system maintenance form Fig. 8. This reasoning is confirmed by the results of our model testing, and is also in good agreement with results in [1]. is 138, the So, when the DV of maximal inventory level -expected total system maintenance cost per unit time can be minimized. The numerical analysis is accordant well with the theoretical analysis, and validates the theoretical deduction about the conditions of existence & uniqueness of the minimum. In addition, it also gives a general procedure on finding solutions to the model.
over S
(
T
= 44).
about the conditions of existence & uniqueness of the minimum once more. , there Through a repeating procedure, for every . Then, the global is a corresponding unique & local minimal minimum of is 717.51 units/week at , and . See Fig. 11.
V. MODEL APPLICABILITY, AND DISCUSSION A. Special Model Case When is fixed at 0, we have is transformed to the model in [1]:
, so our model
C. Model Extension Analysis Further, if both and are regarded as DV as in [1], the global minimum can also be easily found. The minimum of with respect to is the every “local unique minimum” of global minimum. , and Two extreme cases are researched: one is . See Figs. 9 and 10. Both of them another is have a unique minimum. Their minimums are 876.11 units/ , and 1597.29 units/week at week at respectively. We have validated that other have the similar behavior. The numerical results are accordant well with the theoretical analysis, and validate well the theoretical deduction
(27)
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pany (as a technician “warehouse”), and/or preventive replacement cycle can be jointly, and readily solved. C. Discussion
Fig. 11. Variation of local minimal C
over T .
Here, according to Theorem 1,
Hence, for any
In the paper, convenient for analytic deductions, we suppose is the only DV of the objecthat maximal inventory level tive function, and let the pre-arranged PR interval be fixed. and are both DV Now, if we extend it, and suppose that as [1], it is an open question on how to decide the conditions about optimum through analytic deductions. We think an analytical approach will be a difficult, even an unfeasible one. In fact, even though we only research the special case (27), as both DV of as [1] did, we and regard the and encounter some difficulties; although the conditions of the existence & uniqueness of the joint optimization minimum can be preliminarily deduced (where [15] is partly referred to) [16]. Generally, joint optimization of two or more activities in an analytical way means a very complex behavior. We only regard our paper as a preliminary research on joint optimization of several activities. And this paper intends to start further discussion on this question.
, it has only one unique , making the
in [1] minimized. Further, the is relaxed to a DV as [1], and the global minimum can be easily found just to find the minimum of every with respect to . “local minimum” of
APPENDIX I Proof of the Conditions of
B. Model’s General Applicability Field The models mentioned can be readily applied to optimize maintenance procedures for a variety of industrial systems, and also to update the maintenance policy in situations where block replacement preventive maintenance is already in use. This application has been verified by [1] using field data on electric locomotives in Slovenian Railways. The theoretical deduction in this paper can help engineers decide whether the minimum of the model exists, and what defines the minimum of the model. It is a guideline to help engineers in practical applications of models. In addition, similar model extensions are numerous, as long as the system has similar behavior characters of both “block replacement,” and “inventory”. So, the models mentioned can be readily extended to such areas as financial engineering (e.g. block renewal of currency in circulation), armament administration (e.g. ammunition block renewal), and even medical fields after some appropriate changes. For example, consider the joint optimization of the preventive replacement cycle length, and the number of maintenance technicians in a certain maintenance service company. Here, the original maintenance policy depicted in [1] stays unchanged; the spare inventory are substituted by the workload of technicians; and the relevant original costs in [1] are substituted by corrective replacement costs, preventive replacement costs, wages, call-back pay fee of technicians, and the downtime costs caused by a shortage of technicians, respectively. Then, through models similar to (18) or (27) in paper, the optimal number of technicians in the maintenance service com-
According to assumption 8,
then,
:
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where
ACKNOWLEDGMENT The authors thank the reviewers, and the Editor, Dr. Ruey Huei Yeh, Dr. Jason Rupe, and Dr. Edward A. Pohl for their comments and advice which have greatly improved the content, and presentation of this paper. REFERENCES [1] A. Brezavscek and A. Hudoklin, “Joint optimization of block-replacement and periodic-review spare-provisioning policy,” IEEE Trans. Reliability, vol. 52, pp. 112–117, Mar. 2003. [2] W. P. Liao and J. Yuan, “Optimal replacement for a one-unit system subject to delivery and test,” Journal of Quality in Maintenance Engineering, vol. 4, no. 1, pp. 51–65, 1998. [3] A. B. M. Zohrul Kabir and S. H. A. Farrash, “Simulation of an integrated age replacement and spare provisioning policy using SLAM,” Reliability Engineering & System Safety, vol. 52, no. 2, pp. 129–138, 1996. [4] A. B. M. Zohrul Kabir and A. S. Al-Olayan, “Stocking policy for spare part provisioning under age based preventive replacement,” European Journal of Operational Research, vol. 90, no. 1, pp. 171–181, 1996. [5] M. J. Armstrong and D. R. Atkins, “Joint optimization of maintenance and inventory policies for a simple system,” IIE Trans. (Institute of Industrial Engineers), vol. 28, no. 5, pp. 415–424, 1996. [6] M. J. Armstrong and D. A. Atkins, “Note on joint optimization of maintenance and inventory,” IIE Trans. (Institute of Industrial Engineers), vol. 30, no. 2, pp. 143–149, 1998. [7] A. B. M. Kabir and S. H. A. Farrash, “Fixed interval ordering policy for joint optimization of age replacement and spare part provisioning,” International Journal of Systems Science, vol. 28, no. 12, pp. 1299–1309, 1997. [8] S. H. Sheu and Y. H. Chien, “Optimal age-replacement policy of a system subject to shocks with random lead-time,” European Journal of Operational Research, vol. 159, no. 1, pp. 132–144, 2004. [9] S. H. Sheu and W. S. Griffith, “Optimal age-replacement policy with age-dependent minimal-repair and random-leadtime,” IEEE Trans. Reliability, vol. 50, no. 3, pp. 302–309, 2001. [10] Y. T. Park and K. S. Park, “Generalized spare ordering policies with random lead time,” European Journal of Operational Research, vol. 23, no. 3, pp. 320–330, Mar. 1986.
[11] M. Berg, “A marginal cost analysis for preventive replacement policies,” European Journal of Operational Research, vol. 4, no. 2, pp. 136–142, Feb. 1980. [12] M. R. Spiegel, Schaum’s Outline of Theory and Problems of Advanced Calculus. New York: McGraw-Hill, 1963. [13] S. M. Ross, Applied Probability Models With Optimization Applications: Holden-Day. : , 1970. [14] I. Olkin, L. J. Gleser, and C. Derman, Probability Models and Applications. New York: Macmillan Publishing Co. Inc., 1980. [15] E. Smeitink and R. Dekker, “A simple approximation to the renewal function,” IEEE Trans. Reliability, vol. 39, pp. 71–75, Apr. 1990. [16] R. Huang, “Research on Key Challenges for the Joint Optimization of Maintenance and Spare Parts Inventory,” (in Chinese) Doctoral dissertation, Shanghai Jiao Tong University, , 2006.4. Runqing Huang has a PhD from the Department of Industrial Engineering & Management, Shanghai Jiao Tong University, Shanghai 200240, P. R. China. Now, he works as a civil servant of Shanghai City. Dr. Huang got his Master degree, and Bachelor degree from Shanghai Jiao Tong University, and Xi’an Jiao Tong University respectively. He has worked as an Assistant Director of the Management Committee at Shanghai Nanhui Industrial Zone, and an Assistant Researcher of Shanghai Quality Management Academy, P. R. China. His present research interests include modeling strategic decisions, inventory theory and modeling, mass customization, and Logistics.
Lingling Meng is a PhD Candidate at the Computer Science & Technology Department, and a teacher with the Department of Educational Information Technology, East China Normal University, Shanghai 200062, P. R. China. Ms. Meng’s present research interests include metadata theory and application, data mining, information retrieval, and computer assisted instruction,
Lifeng Xi is a Professor at the Department of Industrial Engineering & Management, and vice dean of the School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, P. R. China. Prof. Xi’s research interests include quality theory and management, maintenance systems and applications, and modern manufacturing theory and technology.
C. Richard Liu is a Professor of Industrial Engineering at Purdue University, West Lafayette, USA. He was elected Fellow of ASME in 1992, and is an internationally recognized expert in the area of design and manufacturing engineering. Prof. Liu’s research interests include process mechanics, and its relationship to the reliability of the component produced; precision and dynamics of machine tools; CAD/CAM/CIM, and Internet-based Manufacturing; and computable representation, classification, analysis, and integration of design concept, geometry and manufacturability. He has also contributed to industrial product and process research resulting in significant commercial impacts.
