Study on Operational Availability Relational Model of RMS

Abstract. This paper is about a research of the relationship between equipment’s reliability, maintainability, supportability and operational availability. It investigates the influencing factors of operational availability, analyzes the maintenance strategies of equipment and specifies distribution types. Based on these findings and Markow process, it then deduces computational model of operational availability. It also defines reliability, maintainability, supportability and other constraint condition and establishes optimizing equations to find feasible solution and optimal solution. At last, it tests correctness and feasibility of established model by case study. After knowing the requirement of operational availability and operational method provided here to define corresponding level of reliability, maintainability, supportability, it transforms operational-using requirement to technical design requirement in support for further research.


Models of operational availability
The warship equipment needs to guarantee various missions carried out away from shore base, and the resources taken on the ship is limited while the support resources is hard to replenish, in order to guarantee the equipment readiness, a deep study on the operational availability is what it calls for to establish the models to calculate operational availability, and offer theoretical support to improve designation and optimize allocation of resources.
Before the models are established, some explanations and assumptions are made as follows: (1) Two levels of spare parts supply system are considered, assume that all spare parts in local level can be fulfilled in the intermediate level warehouse.
(2) Assume that local level spare parts have 3 individuals, and when the third ran out, a request for the 4th is made to the base. If the equipment faces no absence of any failure before the spare parts' arrival, 3 parts will be supplied, and the extra ones will be sent back to the base; if the equipment faces any failure and caused delay, the failed one will be changed and 3 more supplied. The time from application to acquirement is t, which obeys index distribution with parameter.
(3) The longevity and maintenance time obeys index distribution with and parameter, and can be analyzed through Markov process.
(4) When the spare parts are used up and the requested part doesn't arrive but the equipment fails, thus is the logistice delay until the part arrives, mean logistic delay time is the average time of logistic delay.
(5) When any failure is noticed of the equipment, immediate maintenance should be made to its normal form. Only replacement of the equipment is made on the ship and the maintenance staff is enough at hand.
The working and local level storage conditions are expressed with mn E .The subscript m stands for the working condition of the equipment,0 for normal working condition,1 for logistic delay condition, 2 for maintenance condition. The subscript n stands for local warehouse condition, 0 for the spare parts ran out and request to intermediate level warehouse for supply is already made, 1 for 1 spare part, 2 for 2 spare parts, 3 for 3spare parts, therefore 9 conditions are defined, 03 E stands for normal working condition with 3 spare parts; 22 E stands for under maintenance with 2 spare parts; 02 E stands for normal working condition with 2 spare parts; 21 E stands for under maintenance with 1 spare part; 01 E stands for normal working condition with 1 spare part; 20 E stands for under maintenance with 0 spare part; 00 E stands for normal condition with 0 spare part; 10 E stands for under maintenance with 0 spare part; 23 E stands for under maintenance with 3 spare parts. We consider any condition that doesn't satisfy the assumptions doesn't exist.
The time of requesting for the spare part obeys index distribution with γ parameter, the final spare part's life span also obeys index distribution with λ ,parameter, therefore the supplied spare part can arrive before the last available element fails(event A) or after(event B).These two events should be discussed separately.
According to the assumption, when the last spare part is used, the application for more spare part is made, that is the last spare part come into working condition and the application is made coincide. If random variable X is used to express the elements' lifespan, Y to express the application time, then the possibility of event A happens is ( ) Because the time of applying for spare part obeys index distribution with γ parameter, the longevity of the element obeys also index distribution with λ parameter, plus X and Y are independent so the combined probability density function are as follows: The solution of ( ) Dxz means the range where X is less than or equal to Z. Assume that failure happens at t1, and the spare part arrives at t2, then: Due to the fact that the failures happens and the spare part arrives are two independent events, so the combined probability density function is : If the spare part arrives after the last element fails, it will cause delay for (t2-t1). Regardless of the delay time caused by other factors, the mean logistic delay time is: According to the two-dimensional random variable function and its distribution, we can conclude that: Among them Dt1t2represents a range 1 2 t t d . In the following, we will take consideration of event A. Complementary element arrived before the last available element had broken. At this moment, the condition aggregate EA is {1,2,3,4,5,6,7}, normal condition aggregate WA is {1,3,5,7}, and breakdown condition aggregate FA is {2,4,6}. The figure of condition transformation is reflected in Figure 1.

Fig. 1. Condition transformation of event A.
Condition transformation matrix QAcan be showed in the following from condition transformation figure: According to bibliography [16] : As for all i,j E, lim ( ) 1, 2, ,7 As for any j E, lim ( ) 0 j t P t of c We can choose term limit value from both sides according to equation (10), and then obtain: Solve it to: , we can make a conclusion: DOI: 10.1051/ , 020 (2017)  710002010 According to the definition of o A , we know: When we take consideration of event B, complementary element arrived after the last available element had broken, and maintenance delay condition occurred before re-complementary element's arrival. At this moment, the condition aggregate EB is {1,2,3,4,5,6,7,8,9}, normal condition aggregate WB is {1,3,5,7}, and breakdown condition aggregate FB is {2,4,6,8,9}. The figure of condition transformation is reflected in Figure 2.

Fig. 2. Condition transformation of event.
Condition transformation matrix QBcan be showed in the following from condition transformation figure: According to bibliography 16 : As for all i,j E, lim ( ) 1, 2, ,9 ij j t P t P j of ,9 As for any j E, lim ( ) 0. j t P t of c We can choose term limit value from both sides according to equation (15), and then obtain: , , , 0,0, ,0 P P P Q 9 B , 0 , 0 , , 0 Solve it to: A , MTBF,MTTR and MLDT can be calculated according to known parameter λ, μ and γ in model (8)and (19). But thisset of parameters may not be the optimal parameters. In the process of equipment designation, we concern more about limit value of o A and the feasible solution and optimal solution of design parameter limited by other term, thus we need to come up with an optimized method.

Ascertain limit term and establish optimal relational models
According to o A , the larger MTBF is, the smaller MTTR is, the smaller MLDT is and the higher o A is. But in fact, as the result of the limitation of technical level and other objective conditions, MTBF, MTTR and MLDT can only change in a range, so do parameter λ, μ and γ. Besides, maintenance delay time (event B) made by the drawn-out application of replacement part can be cut down by optimized allocation of maintenance system, so we can prescribe a limit that maintenance delay time takes up proportion in unusable time no more than limit value. Unusable time is made of average maintenance delay time and average repair time. Maintenance delay time taking up proportion in unusable time no more than limit value is equal to average repair time taking up proportion in unusable time more than limit value. Then regard available cost's limit value, range of λ, μand γ, the rate average repair time takes up in unusable time, average maintenance delay time as limit condition, and the maximal value of o A as target function, to establish optimized relational models:  There are three variables λ, μand γ in optimal relational model (20), which both has linear constraint and nonlinear constraint, belonging to multivariate nonlinear constraint optimized problem. According to optimal theory, we can obtain optimal solution which meets the limit condition and optimal value of corresponding parameters by solving optimal relational model through numerical value MATLAB 18. We know that under the limitation of current technical level and other usage conditions, some equipment's MTBF is more than 500h and less than 1000h, MTTR is larger than 4h and less than 25h, and average time from the application of replacement parts to the arrival of them is more than 72h and less than 120h, as well as oaccept A is 0.96, lower limit value of MLDT is g, and the rate MTTR takes up in unusable time should be more than k,so the optimal relational model (20) can be transformed into (23).  When k is 0.85 and g is 6h, we can obtain relation figure showed in Figure 3 according to the relation between expected time ( ) E t from replacement parts' application to their arrival and Ao. After calculating, when E(t)is 81.1h, Ao arrives at the maximal value. At this moment, the three dimensional relational model of MTTR, MTBF and Ao can be showed in Figure 4.
From Table 1, we know that when k are same, the smaller g is, the larger the maximal value of operational availability is, and MTTR, MLDTand E(t) will gradually decrease. When g are same, the smaller kis, the larger the maximal value of operational availability is, and the optimal value of other parameters won't change, but only the optimal value of MTTR will decrease. The optimal value of MTBFwill choose the maximal value of the range all the time. Combining the relationship figure of MTBF, MTTR and Ao, as well as relationship figure of E(t)and Ao, we will analyze the phenomenon above. According to formula (8), we know that MLDT is influenced by λ and γ, it is inversely proportional to MTBF, and it is in proportion to E(t). When lower limit k that average repair time takes up proportion in unusable time are same, the lower limit value gof MLDT decreases and the constraint that MLDTputs on MTBF and E(t) also decreases. At this moment, MTBF will be bigger in its co-domain and E(t) will be smaller in its co-domain, as reflected in Figure 4 that the maximal value of Ao then will increase.
According to formula(22), we know that k is in proportion to MTTR and inversely proportional to MLDT. When g are same, and k decreases, the constraint that the lower limit that average repair time takes up proportion in unusable time puts on MTTR and MLDTalso decreases. At this moment, MTTR will be smaller in its co-domain, as reflected in Figure 4 that the maximal value of Ao then will increase. Although the constraint that k puts on MLDT decreases, the constraint that g puts on MLDT doesn't change, so that the optimal value of other parameters won't change.
We can know from Figure(4) that when k is 0.85, g is 6h, and t is 81.1h, maxAo that has concerns constraint conditions is the maximal value 0.9912 as MTBF is 1000h. When MLDT is inversely proportional to MTBF, and it is in proportion to E(t), MLDT's contour line can be showed in Figure(5). Under current constraint conditions, when g is between 5h and 7h, the minimal value of MLDT constrains the range of MTBF's and E(t)'s feasible region to be pentagon area in the lower right corner above contour line gall the time. When Ao is the maximal value, MTBFcan always be the maximal value 1000h in its feasible region. If taking costs, time limit and other constraints into consideration, MTBF's value should be further restricted.

Conclusion
Operational availability is an integrity norm as same importance as other functional norms of equipment, and it is an insurance of equipment that it can finish tasks perfectly when needed. This paper analyzes the influencing factors of DOI: 10.1051/ , 020 (2017) 710002010 operational availability and has a research on the process of replacement parts' maintenance based on fully understanding of the importance of operational availability. And based on Markow process, it establishes relational models of operational availability and defines constraint condition and establishes optimizing equations to find feasible solution and optimal solution. At last, it tests correctness and feasibility of established model by case study. The arithmetic established here transforms operational-using requirement to technical design requirement and ascertains target of equipment's reliability, maintainability and supportability. It provides theoretical support and quantitative reference of parameter setting when testing equipment. Although the arithmetic provided here can solve some physical problem, the optimal solution may have some distinction to reality without consideration of costs, limit time and other constraints. So it is the main problem we should solve in the future.