A decision diAgrAm bAsed reliAbility evAluAtion method for multiple phAsed-mission systems metodA oceny niezAwodności systemów wielofAzowych w opArciu o diAgrAmy decyzyjne

The multiple phased-mission system (MPMS) exists widely in practical engineering, such as aviation, spaceflight and navigation fields. Its distinct characteristic is that the system usually performs multiple missions and each mission consists of different phases. In this paper, we mainly focus on the reliability analysis for MPMS when the components have to accomplish different missions successively. A new modeling method is proposed for MPMS analysis based on the binary decision diagram (BDD) and multi-state multi-valued decision diagram (MMDD). Through this method, different phases of missions are combined with in the whole system by certain merging rules according to the operating time of a common component. Then, the system reliability can be calculated by the common calculation methods of decision diagrams by generating the through. Finally, two case studies are implemented to demonstrate the generation of BDD/MMDD models and the evaluation of system reliability. The experiment results verified the efficiency and accuracy of the proposed modeling methods.

Article citation info:

A decision diAgrAm bAsed reliAbility evAluAtion method for multiple phAsed-mission systems metodA oceny niezAwodności systemów wielofAzowych w opArciu o diAgrAmy decyzyjne
The multiple phased-mission system (MPMS) exists widely in practical engineering, such as aviation, spaceflight and navigation fields.Its distinct characteristic is that the system usually performs multiple missions and each mission consists of different phases.
In this paper, we mainly focus on the reliability analysis for MPMS when the components have to accomplish different missions successively.A new modeling method is proposed for MPMS analysis based on the binary decision diagram (BDD) and multi-state multi-valued decision diagram (MMDD).Through this method, different phases of missions are combined with in the whole system by certain merging rules according to the operating time of a common component.Then, the system reliability can be calculated by the common calculation methods of decision diagrams by generating the through.Finally, two case studies are implemented to demonstrate the generation of BDD/MMDD models and the evaluation of system reliability.The experiment results verified the efficiency and accuracy of the proposed modeling methods.

Introduction
Phased-mission systems (PMS) are very common in practical engineering, where the mission of system usually consists of multiple, consecutive, and non-overlapping phases in operation [12,20,21].A simple example is that the phases of car-driving mission include start, acceleration, deceleration, and stop.During each phase, the system has to complete the specific task and may be subject to different stresses and environmental conditions as well as different reliability requirements [12].Moreover, the system's functioning principle of different phases may change, and hence it is necessary to establish distinct models for each phase.
Accurate reliability analysis of PMS must consider the statistical dependencies of components across different phases, as well as the dynamics of system configurations, success criteria, and component behavior.In the previous study, researchers mainly focused on binary reliability models for PMS. Park and Yoo [11] introduced an iterative Lagrange technique to maximize the mission reliability of PMS by apportioning subsystem reliabilities according to multiple resource constraints.Dugan [2] proposed an automated analysis method of PMS based on the discrete-state continuous-time Markov model.Kim and Park [4] put forward three cases, whose phase durations are deterministic, random variables exponential distribution, to compute the mission reliability based on Markov model.Somani and Trivedi [13] proposed a Boolean algebraic method to analyze PMS reliability, and the failure criterion in each phase can be expressed as a fault tree.Ma and Trivedi [8] described an efficient Boolean algebraic algorithm which combines the fault trees of all the phases into a single fault tree with repeated events.Zang et al [24] established a method based on binary decision diagram (BDD) to analyze the reliability of PMS.Jung et al [3] proposed a BDD algorithm for coherent fault tree, where the truncated if-then-else (ite) connectives and subsuming could be performed in the progress of the BDD structure construction.
Recently, more and more researchers have been concentrated on multi-state systems (MSS) and multi-state PMS.Tang and Dugan [18] built the dependence-BDD for reliability analysis of PMS with multimode failures by applying dependence algebra.Xing and Dai [22] proposed a new modeling approach called multi-state multi-valued decision diagrams (MMDD) for the analysis of multi-state systems.Shrestha and Xing [14][15][16] introduced reliability analysis of multistate PMS with unordered and ordered states, and used MMDD to analyze the importance of components.Levitin and Xing [5,6] introduced a recursive algorithm based on conditional probability and an efficient recursive formula based on the branch and bound method for reliability evaluation of non-repairable PMS.Xing and Amari [23] put forward an efficient method to evaluate the reliability of k-out-of-n systems with identical components subject to phased-mission requirements and imperfect fault coverage.Wang and Xing [19] established an algorithm for competing failure analysis in PMS with functional dependence in one of the phases.Zang and Bai [25] proposed a mathematical model for success probability analysis of PMS based on minimal path set and system state analysis methods.Mo and Xing [9,10] built a new analytical method based on multi-valued decision diagrams for reliability analysis of non-repairable PMS with multimode failures.Li and Tao [7] combined the Bayesian networks with event tree and fault tree analysis to analyze PMS based on conditional probability by giving expression of the phase-dependency.
Multiple phased-mission systems (MPMS) have been applied in a wide range of engineering fields, where a system consists of multiple missions.The state of the component at the end of a mission will be the beginning state of the same component in the next mission.In MPMS, each mission also consists of multiple, consecutive, and non-overlapping phases which are accomplished in sequence.For example, the operational process of landing gear involves two missions: take-off and landing.The take-off mission involves speed skating, lifting, and climbing phases.And the landing mission involves landing gear drop-down, level flight, drift down, and skating phases.The landing gear system needs to complete both two missions for success flight.Compared with PMS, the analysis of MPMS is more difficult because a component may work during two missions in sequence.In PMS, for the component working in different phases, all the phases can be merged as one by existing algorithms.But in the MPMS, it is usually assumed that a component have to work in two missions in sequence.System structure and the environmental conditions will make the state of components more complex.So we need generate some new phases for the common component which works on the two missions' time nodes, and then combine all the phases.
The remainder of this paper is organized as follows.Section 2 presents the basic concept and phase-dependent operation algorithm of BDD and MMDD respectively.Section 3 describes the reliability evaluation methods of MPMS based on BDD and MMDD.Two examples are illustrated in Section 4 to show the efficiency and accuracy of the proposed modeling methods.Section 5 gives conclusion and points out the future work.

Basic concept of BDD
BDD is a rooted, directed acyclic graph representation of a Boolean expression based on Shannon decomposition rule [1].It has two sink nodes (outputs), labeled as '1' and '0', which represent a binary-state system being either operational or failed.Let ( 1, 2,..., ) .In general, the (ite) format for expressing Boolean expressions F (representing the system state structure function) in variable r A x based on Shannon's decomposition is: In practical engineering, non-sink node usually corresponds to the component's state.By traversing the BDD's all paths with each path pointing to sink node '1', the probability of occurrence of the system can be calculated.
Each non-sink node in BDD usually has two outgoing edges, called 0-edge and 1-edge, respectively.Supposing there are two sub-BDD models G and H, then they could be encoded with the Boolean expression in the ite format, as:

H y H y H ite y H H ite y H H
Phased-mission systems (PMS) are systems in which multiple non-overlapping phases of tasks are accomplished in sequence for a successful mission.To combine different phases, the operation rules for combing two sub-BDD models G and H are as: where the symbol ◊ represents a logic operation (AND or OR) between two sub-BDD models, the index( ) is assigned to each variable to indicate its position in the propagation order of all BDD variables.For example, index If G and H are connected with "OR" operator, then the combination process of two sub-BDD models in Fig. 2 is shown in Fig. 3.
Generally, the combination process of sub-BDD models could be concluded as follows: Compare the two sub-BDD models, it is clear that (1) index index ( ) ( ) a a = . According to the rules of equation ( 1), we have .Accroding to the rules of equation ( 1), we can get Simplify the process is as follows: Because the results of (1) Because the 0-edge of node (2) a is point to the same sub-tree as the 0-edge of node b at right.One of the two same sub-tree can be reduced.

Basic concept of MMDD
The MMDD is a multi-state extended form of BDD [16].It is a multi-valued logic structure for the natural representation of the MSS and is widely used in MSS reliability analysis [17].The nodes MMDD are also divided into two types: sink nodes and non-sink nodes.MMDD only has two sink nodes, labeled '1' and '0', which indicate that the system is either in state '1' or in state '0'.Non-sink node in MMDD can have more than two edges where each edge represents a possible state of the components.
According to [17], Logical expression F in MMDD can be represented as follows: case( , , , ) Each non-sink node is associated with a multi valued state vari- x is in state m , the value of F is '1'; otherwise the value is '0', as shown in Fig. 4.

Phase-dependent operation of decision diagram
In 1999, Zang et al [24] published a paper about the application of BDD for phased-mission systems and derived a special phased-dependent operation (PDO) as in equations ( 3) and ( 4).Let component r A be used in both phase i and j , i j < .Using ite format, , i j F F express the Boolean expressions of F is in phase i and j , while ri A x denoted the state variable of component r A in phase i .Then we have: , , Because BDD modeling process depends on the order of the variables, there are two types of ordering methods: forward PDO and backward PDO.For the forward PDO, the variable order is the same as the phase order .In the backward PDO, the variable order is the reverse of the phase order .For combined operations, the same component belongs to two sub-BDDs but in different phases.
To deal with the MPMS problems, we derive a new MMDD operation for Phase Algebra in this paper based on the results of [24].Similarly, two types of ordering methods are considered: forward PDO and backward PDO.Let component A appear in both phase i and j , i j < , then we have: For the forward PDO, (i) If is failed in phase i and further it is irreparable, then it keeps failed in phase j , i.e, This derivation uses the equation: is not relevant to H.

For the backward PDO, (ii)
If r

A
x is operational in phase j , then it must be operational in phase i , i.e, rj A x n ) ) This derivation uses the equation:

Reliability evaluation methods based on decision diagram
The proposed BDD and MMDD methods for MPMS are established according to the following assumptions: (1) Each mission consists of multiple non-overlapping phases; (2) The component completes its missions in sequence.In binary-state MPMS, both a system and its components have two and only two states: functioning or failed, which are labeled as '1' and '0', respectively.In multi-state MPMS, a system has two and only two states, while the components may have more than two states.
In order to evaluate the reliability of MPMS by the BDD and MMDD methods, we need to build the system structure function for each phase.The logical expression (representing the failure of the system in the phase) can be obtained by complementing the system structure function.The graphical representation of the logic expressions in terms of logic AND/OR gates gives the fault tree model for each phase.Based on the generated fault tree models, the proposed BDDbased analysis and the MMDD-based analysis can be performed in the following four steps: Step 1: New Phase Generation.One component participates in multiple missions consecutively.We can see that a new system consists of two missions and there's a common component which works on two missions sequentially.The new system is divided into many phases by the common component which works on the two missions' time nodes.
Step 2: Single-Mission BDD/MMDD Generation.Traditional method can be used for the generation of BDD model for each phase.In particular, equation ( 1) is applied to generate BDD based on the fault tree.The MMDD model is generated according to equation (2).
Step 3: Multiple Missions of BDD/MMDD Merged for the Same Phase.Based on the result from the Step 1, we need to merge the two missions of BDD/MMDD in the same phase.Each phase of the BDD/ MMDD is generated by performing the logic OR operation of the single phase BDDs/MMDDs generated in Step 2. In a binary-system, equation ( 4) is applied when operation is performed on two variables of different elements.In a multi-state system, equation ( 6) is applied when operation is performed on two variables of different elements.
Step 4: Generation of BDD/MMDD for MPMS.In this step, the entire MPMS is generated by performing the logic OR operation on all the merged BDD/MMDDs generated in Step 3. In a binary-system, equation ( 4) is applied when operation is performed on two nodes which belong to the same component but in different phases.In a multi-state system, equation ( 6) is applied when operation is performed on two nodes that belong to the same component but in different phases.Step 1: In the system consisting of mission P and mission Q , the common component 1 A works in both missions sequentially.The new system is divided into three phases by the common component  1, which shows the computed conditional reliability for each component at each phase, and all the components fail exponentially with constant failure rate.
Step 2: A single-mission single-phase BDD is generated.In a binary-state system, both the system and its components have only two states: '1' and '0', which represents the binary-state system and its components' state: either operational or failed.
In mission P of the phase 1, there are two components: 1 A and 2 A .Each component has two states (0, 1).When 1 A and 2 A are in state '1', the system is normal.Fig. 6 shows the BDD model about A .Each component has two states (0, 1).When both 3 A and 4 A are in state '1', the system is normal.Fig. 7 shows the BDD model about 1 Q .In mission Q of the phase 2, three are three components: A in the state '1', the system is also normal.Fig. 8 shows the BDD model about 2 Q .In mission P of the phase 3, there are two components: 1 A and 2 A , and each component has two states (0, 1) .When 1 A is in state '1'; or 1 A in state '0', 2 A in state '1', the system is normal.A are in state '1', the system is normal.Fig. 10 shows BDD model about 3 Q .
Step 3: BDDs of two sub-missions merged for the same phase.By applying equation ( 4) with the order of , the new BDD for each phase is presented in Fig. 11.Step 4: Generation of BDD for MPMS.Perform logic OR operation to combine the merged BDDs of three phases by applying equation (4) as shown in Fig. 12.
Finally, according to the built BDD for entire MPMS, the overall system reliability is 0.829358.

MMDD example
We assume that a multi-state system is composed of two missions: mission P and mission Q .The step-by-step analysis of the multistate MPMS is given as follows.
Step A completes its task in mission Q and starts its task in mission P , meaning that phase 2 P consists of four components ( 7A , 8 A , 9 A , 4 A ).The new system will be divided into two phases: 1) Phase 1 includes 1 P and 1 Q ; 2) phase 2 includes 2 P .For the entire system, component 4 A involves in two missions during different time periods.Table 2 shows the computed conditional reliability for each element at each phase.
Step 2: Built MMDD for each phase.In phase 1 of mission P , the Component 1 A has three states (0, 1, and 2); component 2 A and A have two states (0, 1).When component 1 A is in state '1', 3 A is in state '1', the system is normal.When component 1 A is in state '2' and 2 A in state '1', the system is normal.When component 1 A is in component.Let w denote the total number of components in the system.The two states of component r A represented by a Boolean variable, denoted by r A x .Each Boolean variable r Ax can be represented using the if-then-else (ite) format as ( ,1, 0) r A ite x

Fig. 1 .
Fig. 1.Binary decision diagram that the position of the r A y is behind the position of the r A x in the order.To clearly explain the operation rules in equation (1), the detailed examples of two sub-BDD models G and H are shown in Fig.2[24].For sub-BDD models G in Fig.2 (a), we know that ⋅ = ⋅ , then we can get G a c a b c a c b c = ⋅ + ⋅ ⋅ = ⋅ + ⋅ .For sub-BDD models H in Fig.= + , then we can get ( ) H a b c a c a b a c a c a b c = ⋅ + + ⋅ = ⋅ + ⋅ + ⋅ = ⋅ + .
are the same, the node b 0-edge and 1-edge all point to the node c.So the node b at left can be removed.
able r A x , and r A x m = means that the component r A is in state m .The m F can take one of two values: "1" or "0", indicating that F is in or not in state m(m=0,1,2,…,n) respectively.The non-sink note r A x has ( 1)n + possible states and can be in a particular state at a specific time.So the logical expression of F has( 1) n + possible values.For example

−
to G.In the forward PDO, index( ri A x )<index( rj A x ) when phase i j < .The new MMDD node of the combined sub-MMDD is ri A x .The 0-edge of node ri A x is generated, and the operation is applied to 0 G and 0 H .In order to generate the m-edge of node ri A x sub-MMDD model H together.In the backward PDO, index( rj A x )<index( ri A x ) when phase i j < .The new MMDD node of the combined sub-MMDDs is rj A x .The n -edge of node rj Ax is generated, and the operation is applied to n G and n H .In order to generate the m-edge of node rj A x in a combined MMDD, the operation is applied to and the other sub-MMDD model G together.

1 A , 3 A
An example is presented to illustrate the application of modeling method based on BDD for a system with one component being engaged in two missions: mission P and mission Q .Mission P needs two components, 1A and 2 A .Mission Q needs three components, , and 4 A .During different time periods, component 1A participates in mission P and mission Q .

1 A
which works on the two missions' time nodes.The i P and i Q denoted the phases of the mission P and mission Q .The first phase consists of 1 P and 1 Q .The second phase consists of 2 Q .The third phase consists of 3 P and 3 Q .The input parameters of each component are shown in Table

1 P
, where ri A denoted the component r A in phase i .In mission Q of the phase 1, there are two components:3 A and 4

4 A 3 A and 4
Fig.9shows the BDD model about 3 P .In mission Q of the phase 3, there are two components: 3A and , and each component has two states (0, 1).When both component

Fig. 6 .
Fig. 6.The BDD model of the phase 1 of mission P

1 : 1 Q
New phase generation.Depending on the mission time node of component 4 A , mission Q has one phase: 1 Q .Mission P can be divided into two phases: 1 P and 2 P .In mission Q , phase consists of three components ( 4

Table 1 .
Input parameters about each component.